Modelling the control strategies for riding a motorcycle. PhD thesis

Modelling the control strategies for riding a motorcycle. PhD thesis
Rowell, Stuart (2007) Modelling the control strategies for
riding a motorcycle. PhD thesis, University of
Nottingham.
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MODELLING THE CONTROL STRATEGIES
FOR RIDING A MOTORCYCLE
GEORGE GREEN LIBRARY OF
SCIENCE AND ENGINEERING -r
Stuart Rowell, MEng
Thesis submitted to the University of Nottingham
for the degree of Doctor of Philosophy
July 2007
Abstract
Computer simulation models are increasingly necessary as a design tool for modern
vehicles, for which a subcategory relates to motorcycles.
employed for a variety of applications,
an important
Simulation models can be
area of which relates to the
motorcycle's dynamic responses. The response of a motorcycle is heavily dependent
on the rider's control actions, and consequently a means of replicating the rider's
actions provides an important extension to this area.
The application of mathematical
control techniques for replicating the motorcycle
rider's control actions is presented in this thesis, detailing specifically the techniques
of optimal control and model predictive control. The work begins with modelling the
dynamics of the motorcycle using standard procedures.
The application of optimal
control to a motorcycle rider is not new, but the available results have been extended
significantly over those previously available, allowing further insights into the behaviour and therefore applicability of this strategy to modelling a motorcycle rider.
Use of the model predictive control approach is new in the field of motorcycle rider
modelling, and a similarly extensive parametric
study is conducted to evaluate the
suitability of this approach, and to highlight the similarities and differences between
this and the optimal control approach.
Both controller models were simulated over a standard single lane-change manoeuvre.
Comparison of the relative performances of the two control approaches confirmed
strong similarities between the techniques, particularly
permitted
when the modelled rider is
an extensive knowledge of the approaching road path to follow. When
this knowledge is restricted, differences were apparent between the two, suggesting
the predictive control approach is capable of better performance here, and therefore
represents a more robust control strategy.
An option of the predictive control ap-
proach allows more elaborate target paths for the rider to follow to be set. However,
ii
defining the target path for the rider model to follow as the road centreline, and then
permitting the controller itself to select the most appropriate course to take, has also
been shown to be the more suitable option.
The predictive control technique for motorcycle rider modelling is shown to be a
theoretically
suitable application.
Further work is suggested to validate the results
presented here. If it can be confirmed that the model accurately captures a motorcycle
rider's actions, this will prove a very useful tool for the understanding
of a motorcycle
rider's control actions, with potential benefits towards rider safety and furthermore
as a design tool for the motorcycle industry.
Acknowledgements
My thanks are extended to my project supervisors, Dr. Atanas Popov and Dr. Jacob
Meijaard at the University of Nottingham.
been approachable
Throughout
the project they have always
and willing to provide assistance,
information and suggestions
whenever necessary. Their wealth of knowledge is something to aspire to, and has
many times been a source of guidance and insight.
Gratitude
must also be given to the Engineering and Physical Sciences Research
Council, who provided the funding necessary for this project.
Finally, the support of my fellow students should not go unmentioned;
John Coul-
tate, for the instigation of many much needed coffee breaks; Mark Robinson, regular
provider of both the afternoon crossword and general humour in the office; Paul
Houlston, who has always been willing to share his knowledge to help others on request; and to Chris Larmer, Paul Baalham and Robin Elliot for providing welcome
conversations away from the topic in hand.
iii
Contents
Abstract
i
Acknowledgements
iii
Table of Contents
iv
List of Figures
ix
List of Tables
xv
Nomenclature
xvi
1 Introduction
2
1
1.1
Motivation
3
1.2
Tables
5
1.3
Figure
Literature
...............
,
.
6
7
Review
2.1
Introduction
....
7
2.2
Motorcycle Stability Analysis
9
2.3
Tyre Modelling
13
.
iv
CONTENTS
2.4
2.5
v
Rider Control . . . . . . .
16
2.4.1
Visual Perception.
16
2.4.2
Rider Analysis and Modelling.
18
2.4.3
Optimised Rider Models .
24
Summary
2.5.1
. . . . . . . . . . . . .
28
Objectives and Thesis Outline
29
3 Motorcycle Modelling
31
3.1
Introduction....
31
3.2
Coordinate System
32
3.3
Tyre Model . . . .
33
3.3.1
Tyre Force Equations
34
3.3.2
Validation of Tyre Model
37
3.4
Motorcycle Model
39
.
3.4.1
Motorcycle Equations of Motion
3.4.2
Validation of the Equations of Motion
54
3.4.3
Validation of the Advanced Tyre Motorcycle Model
56
3.5
Motorcycle Model Conclusions
3.6
Figures
..............
40
57
.
59
69
4 Rider Preview
4.1
Introduction.
4.2
Road Preview Shift Register .
69
·
.
70
4.2.1
Global Coordinates Preview .
·
.
70
4.2.2
Local Coordinates Preview
·
.
73
.
vi
CONTENTS
4.3
Rider Preview Conclusions.
79
4.4
Figures
81
5 Optimal Control Rider Model
84
5.1
Introduction
.
84
5.2
Optimal Control Theory .
85
5.2.1
Algebraic Riccati Equation Solution: Numerical Method.
90
5.2.2
Algebraic Riccati Equation Solution: Analytical Method .
91
5.2.3
Application to the Riding Task
95
5.2.4
Optimal Gains
5.3
. . . . . . . . .
••
"
••••
10
97
••••••••••
98
Optimal Control Rider Model Results
5.3.1
Low Speed Optimal Control Model .
99
5.3.2
High Speed Baseline Parameter Set.
108
5.3.3
Local Coordinate Preview.
109
5.4
Optimal Control Conclusions
5.5
Tables.
5.6
Figures
111
...................................
116
••••••••••••••
117
t
••••••••••••••••••••
128
6 Model Predictive Control Rider Model
6.1
Introduction.
6.2
MPC Theory
................................
••••••••••••••••
10
"
"
•
128
"
•••••••••
"
•
130
6.2.1
Linear Prediction Model .
"
"
. 132
6.2.2
Non-Linear Prediction Model
........
"
. 139
6.2.3
Reference Path Definition
6.2.4
MPC Optimal Gains . . . . . . . . . . . . . . . . . . . . . . . . 150
.....................
143
vii
CONTENTS
6.3
6.2.5
Application to Motorcycle Rider Modelling
152
6.2.6
MPC Theory Conclusions
155
. . . . . . . .
156
Model Predictive Control Rider Model Results
6.3.1
Low Speed Baseline Prediction Model
158
6.3.2
High Speed Linear Prediction Model
165
6.3.3
Non-Linear Prediction Model
172
6.3.4
Reference Path Definition
. .
174
6.4
Model Predictive Control Conclusions
177
6.5
Tables.
179
6.6
Figures
180
203
7 Performance Comparisons of Control Techniques
7.1
Introduction
7.2
Comparison Results
203
.....
204
7.2.1
Comparison 1 - Baseline Parameters,
Low Speed
204
7.2.2
Comparison 2 - Baseline Parameters,
High Speed .
205
7.2.3
Comparison 3 - High Speed, Loose Control
7.2.4
Comparison 4 - Limited Preview, Loose Control
206
7.2.5
Comparison 5 - Yaw Error Minimisation.
207
7.2.6
Comparison 6 - Short Control Horizon . .
208
7.2.7
Comparison 7 - Very Low Speed
. . .
.................
..................
206
208
210
7.3
Performance Comparison Conclusions
7.4
Table ..
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
7.5
Figures
...................................
215
CONTENTS
viii
8 Conclusions
8.1
221
Model Analysis
222
8.1.1
State Gains
222
8.1.2
Preview Gains
223
8.1.3
Reference Path
225
8.2
Coordinate System . .
226
8.3
Non-Linear Prediction
226
8.4
Simulation Results
227
8.5
Final Conclusions and Further Work
..
229
Bibliography
232
List of Publications
241
A Motorcycle Data
242
A.1 Motorcycle Data
242
A.2 Geometric Details
243
A.3 Inertial Properties
243
A.4 Tyre Properties
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
245
B VRML Simulation Model
245
B.1 Introduction.
B.2 Coding ....
.
,.
. 245
B.2.1
Motorcycle Body
...............
246
B.2.2
Animation....
...............
247
List of Figures
1.1
Comparative use of motorcycles in the major European countries, [27]
3.1
SAE coordinate system, motorcycle image from [34]
6
59
59
3.2 ISO coordinate system, motorcycle image from [34] .
3.3 Normalised lateral tyre forces with slip ratio and wheel camber, simple
tyre model, parameter values as in Appendix A
60
3.4
Simplified 'bicycle' model of the motorcycle
60
3.5
Normalised lateral tyre forces with slip ratio and wheel camber, ad-
. .
vanced tyre model, parameter values as in Appendix A.
3.6
and a are approximately
Slat
and a are approximately
Difference in final orientation following translation,
tion (left), and translation,
3.9
Slat
equal . . . ..
61
Normalised lateral tyre forces comparison, 50° wheel camber, rear tyre,
where for small slip angles
3.8
61
Normalised lateral tyre forces comparison, 0° wheel camber, front tyre,
where for small slip angles
3.7
. . . . . . ..
rotation, translation
equal . . . ..
translation,
(right).
62
rota-
. . . . . . ..
62
Vectorial definition of front frame mass centre in displaced coordinates
63
3.10 Real parts of the system matrix eigenvalues, simple tyre motorcycle
model
"
63
3.11 Root locus plot of the system matrix eigenvalues, simple tyre motorcycle model . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . ..
ix
64
LIST OF FIGURES
x
3.12 Capsize mode, simple tyre. Eigenvalue = 0.079733
3.13 Wobble mode, simple tyre. Eigenvalue = -5.2281
3.14 Weave mode, simple tyre. Eigenvalue
= -2.8095
..
. . .
64
± 54.061i
65
± 18.067i. . . . . ..
65
3.15 Real parts of the system matrix eigenvalues, advanced tyre motorcycle
model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
66
3.16 Root locus plot of the system matrix eigenvalues, advanced tyre motorcycle model
.
66
3.17 Capsize mode, advanced tyre. Eigenvalue = 0.060997 .
67
3.18 Wobble mode, advanced tyre. Eigenvalue = -2.6314 ± 49.419i
.
67
. .
68
4.1
Road preview information in discrete steps for Np = 6 . . . . . .
81
4.2
Update of road preview information in discrete steps for Np = 6
81
4.3
Update of road preview in local coordinates
82
4.4
Update of motorcycle global position when moving one step ahead in
3.19 Weave mode, advanced tyre. Eigenvalue
= -1.0804
± 19.934i
. . . . . . . . . . . .
reference frame of step k using a local coordinates approach
4.5
. . . . ..
Conversion of new preview point information from global to local coordinates
83
5.1
Single lane change path, not to scale
5.2
Path following, v
5.3
State gains, v
5.4
Steer torque and roll angle, v = 10 mIs, Tp = 3.0
5.5
Principal individual state torque contributors,
= 10 ta]«, Tp = 3.0 s, ql
= 10 mIs,
Tp = 3.0
S, ql
= 5000 m-2
= 5000 m-2
S,
.
117
•
117
118
••
ql = 5000 m-2
•
Principal individual state torque contributors,
ql = 5000 m-2, local coordinates
118
v = 10 mIs, Tp = 3.0 s,
ql = 5000 m-2, global coordinates
5.6
82
119
v
= 10 mIs,
Tp
= 3.0 s,
119
LIST OF FIGURES
xi
= 10 mis,
Tp
= 3.0 S, ql = 5000 m-2
5.7
Preview gains, v
5.8
Path error correction capabilites of a motorcycle
5.9
Steer torque, v
= 10 mis,
= 10 mis,
5.10 Path errors, v
= 10 mis,
5.11 Steer torque, v
5.12 State gains, v
= 10 mis,
5.13 Preview Gains, v
5.14 Contrasting
121
•
= 3.0 s, ql = 1000, 5000 &
Tp
121
10000 m-2
= 3.0 S, ql = 1000, 5000 & 10000 m-2•
Tp
= 10 mis,
Tp
road information
= 10 mis,
5.16 Preview gains, v
5.17 Steer torque, v
Loose control aims for more distant target
5.18 Path errors, v
= 1.5 s, 3.0 s & 4.5 s, ql = 5000 m-2
Tp
= 10 mis,
= 10 mis,
= 10 mis,
Tp
Tp
Tp
= 40 mis,
5.21 Preview gains, v
= 10 ta]«
124
= 1.5 s, 3.0 s & 4.5 s, ql = 5000 m-2
124
= 1.5 s, 3.0 s & 4.5 s, ql
Tp = 1.5
S, ql =
5000 m-2
= 5000 m-2
125
= 1.5 s, 3.0 s & 4.5 s, ql = 5000 m-2
125
ql = 5000 m-2
126
Tp
S,
= 3.0 S, ql = 5000 m-2
& 40 mis, Tp
5.22 Path comparison, global vs.
123
•
5.19 Path following, v = 40 txi]», Tp = 3.0
5.20 Steer torque, v
123
of tight (left) and loose
(dashed line), resulting in corner cutting . . . . . . . . . . . . . . .
5.15 State gains, v
122
122
= 3.0 S, ql = 1000, 5000 & 10000 m-2
requirements
120
120
= 3.0 s, ql = 1000, 5000 & 10000 m-2
Tp
(right) control strategies.
...
= 3.0 S, ql = 5000 m-2
Tp
• • • • • • •
= 3.0 S,
local coordinate
ql
••••
• • • • •
126
= 5000 m-2
127
system, v = 10 mis,
•••••••••••••••••••••••••
127
6.1
Path definition in Model Predictive Control, [9] ...
180
6.2
Linearisation of a non-linear system at two points ..
180
6.3
Added complexity of non-linear prediction model (right) against linear
6.4
prediction model (left)
181
Typical reference path definitions in MPC systems
181
LIST OF FIGURES
xii
6.5
Definiton of a linear reference path, Np = 6 . . . . . . . . . . . . .
6.6
Definiton of a linear error reference path, Np
6.7
Exponential reference path definitions
6.8
Exponential error reduction reference path definitions
183
6.9
Single lane change path, not to scale . . . . . . . . . . .
184
6.10 Path following, v
6.11 State gains, v
= 10 mis,
6.12 Preview Gains, v
6.13 Steer Torque, v
Tp
Tp
= 10 mis,
= 10 mis,
6.14 Path following, v
6.15 State gains, v
= 10 mis,
= 10 mis,
Tp
Tp
Tp
= 3.0 S,
ql
ql
= 3.0 S, ql
183
•
184
•.......
185
= 5000 m-2•
= 5000 m-2
= 1000 & 10000 m-2•
6.20 State gains, v
186
187
= 3.0 S, ql = 1000, 5000 & 10000 m-2
Tp = 3.0
6.19 Path following, v = 10 mis, Tp
results overlapping
186
•••••••••
S,
ql
187
of tight (left) and loose
(right) control strategies, showing typical path aim (dashed line)
= 10 mis,
185
• • • • • •
= 3.0 S, ql = 1000, 5000 & 10000 m-2
road information requirements
6.18 Steer torque, v
182
. . . . . . . . .
= 3.0 S, qi = 5000 m-2
= 3.0 S,
6.16 Preview gains, v = 10 mis, Tp
6.17 Contrasting
. . . . . . . . . .
= 3.0 S, ql = 5000 m-2
Tp
= 10 mis,
=6
182
...
188
= 1000, 5000 & 10000 m-2
= 1.5 s, 3.0 s & 4.5 s, ql
•
= 5000 m-2,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
= 10 tul«;
Tp
= 1.5 s, 3.0 s & 4.5 s, ql = 5000 m-2
6.21 Preview Gains, v = 10 mis, Tp
= 1.5 s, 3.0 s & 4.5 S, ql
• • •
190
6.22 Steer Torque, v = 10 mis, Tp = 1.5 s, 3.0 s & 4.5 s, ql = 5000 m-2
ql = 1000 m-2
189
= 5000 m-2,
paths coincident
6.23 Path Errors, v
188
=
10 mis, Tp
=
• •
190
3.0 s, Tu = 3.0 s, 1.5 s & 0.5 s,
••••••••••••••••••••••••••••••
191
6.24 State Gains, v = 10 mis, Tp = 3.0 s, Tu = 3.0 s, 1.5 s & 0.5 s,
ql = 1000 m-2
.......•......................
191
LIST OF FIGURES
xiii
v=
6.25 Preview Gains,
ql
10
mis,
=
Tp
3.0 s, Tu
=
3.0 s, 1.5 s & 0.5 s,
= 1000 m-2
192
= 40 mis, Tp = 3.0 S, ql = 5000 m-2
6.26 Path following, v
6.27 State gains, v
=
10
mls
& 40 ta]«, Tp
= 3.0 S, ql = 5000 m-2
6.28 Total lateral preview error; effects of increased forward speed
6.29 Preview gains, v
=
10 ta]» & 40
192
.•...•.•.
. • • • •
. . . . . 193
mis, Tp = 3.0 s, ql = 5000 m-2
6.30 Path following performance with reduced wheel inertia, v = 40
Tp = 3.0
S,
ql = 5000 m-2
6.31 Steer torque, v
6.32 Steer angle, v
=
6.33 State gains, v
= 40 mis, Tp = 3.0 s, ql =
6.34 Preview gains, v
6.35 Steer torque, v
10
= 40 mis, Tp =
=
ql = 5000 m-2
= 5000 m-2
= 3.0 S, ql = 5000 m-2
40
mis,
Tp
=
40
mis,
Tp
=
• • •
•••••••••
ql = 5000 m-2
=
40
mis,
= 10 tal«,
196
= 5000 m-2
197
3.0 s, Tu
=
3.0 s, 1.5 s & 0.5 s,
198
3.0 s, Tu = 3.0 s, 1.5 s & 0.5 s,
198
Tp = 3.0 s, Tu = 3.0 s, 1.5 s & 0.5 s,
Tp
199
= 3.0 S, ql = 5000 m-2
= 10 mis, Tp = 3.0 S, ql = 5000 m-2
6.42 Path following, v
196
197
••••••••••••••••••••••••••••••
6.40 Path following, v
195
1000, 5000 & 10000 m-2
••••••••••••••••••••••••••••••
6.39 Preview gains, v
194
195
1000,5000 & 10000 m-2
1.5 s, 3.0 s & 4.5 s, ql
=
194
mis,
••••••••••••••••••••••••••••••
ql = 5000 m-2
6.41 Steer torque, v
Tp
ql
= 40 mis, Tp = 4.5 S, ql = 5000 m-2
6.37 Preview gains, v
6.38 State gains, v
mls & 40 mis,
= 40 mis, Tp = 3.0 S, ql =
6.36 Preview gains, v
• • •
• • • • • • • • • • • • • • • • • • • • •
= 10 mls & 40 mis, Tp = 3.0 S,
193
= 10 mis, Tp = 3.0 S, ql = 5000 m-2,
•••
••••
199
200
Linear reference
path
200
6.43 Linear reference path, showing alternative points of aim; limit of preview horizon (a), half way point of preview horizon (b)
201
xiv
LIST OF FIGURES
6.44 Path following, v
=
10 mis, Tp
= 3.0 S, ql = 5000 m-2,
Linear error
reduction reference path
6.45 Path following, v
=
201
10 mis, Tp
= 3.0 S, ql = 5000 m-2,
Exponential
error reduction reference path . . . . . . . . . . . . . . . . . . . . . . . 202
= 10 mis,
= 3.0 S, ql = 5000 m-2
7.1
Path following, v
7.2
Path errors, v
= 10 mis,
Tp
= 3.0 S, ql = 5000 m-2
215
7.3
State gains, v
= 10 mis,
Tp
= 3.0 S, ql = 5000 m-2
216
7.4
Preview gains, v
7.5
Path errors, v = 40 mis, Tp
= 3.0 S, ql
= 5000 m-2
.
" "
" "
7.6
Path errors, v = 40 mis, Tp
= 3.0 S, ql
= 1000 m-2
"
" "
"
" "
217
7.7
Path errors, v = 10 mis, Tp
= 1.5 S, ql
= 1000 m-2
..
"
..
218
7.8
State gains, v = 10 mis, Tp = 1.5
= 1000 m-2
"
."
" "
218
7.9
Preview gains, v = 10 mis, Tp = 1.5
= 10 mis,
Tp
Tp
215
•••••••••
= 3.0 S, ql = 5000 m-2
S, ql
S, ql
7.10 Path following, v = 10 mis, Tp = 3.0
S,
216
= 1000 m-2
q2 = 5000 rad-2
"
.
217
" "
219
• • • • • • • •
219
" "
" "
7.11 Path errors, v = 10 mis, Tp = 3.0 s, Tu =1.5 s & 0.5 s, ql = 5000 m-2 220
7.12 State gains, v = 4 mis, Tp = 3.0
S, ql
= 1000 m-2
220
A.1 Definitions of bicycle model dimensions
242
B.1 VRML Motorcycle Model
248
..
B.2 VRML Animation snapshots
249
List of Tables
1.1 Average C02 emissions by vehicle type, [22] ......
· ........
5
1.2 Accident statistics, percentages by road user type, [23]
· ........
5
5.1 Low speed controller parameter sets, optimal control .
· ...........
116
5.2 High speed controller parameter sets, optimal control.
..................
116
6.1 Low speed controller parameter sets, model predictive control .
179
6.2 High speed controller parameter sets, model predictive control.
179
7.1
214
Controller comparison parameter sets
xv
Nomenclature
Variables:
a
Distance, perpendicular
b
Distance, motorcycle origin to rear tyre contact patch
Distance, perpendicular
from steer axis to motorcycle origin
from steer axis to front wheel centre
Generic body damping coefficient
Steer damper coefficient
Cr,I, Cr,r
Radius, to centre of tyre torus; front, rear tyre
e
Distance, perpendicular
from steer axis to front frame mass
centre
Error, predicted controlled output to reference path,
ith
preview
point
e
Error vector, predicted controlled output to reference path,
complete preview
f
External force (singular) on a body
f(x)
Generic function in terms of generic variable x
f
Tyre forces vector
9
Gravitational
hi, hr
Height, frame mass centre; front, rear frame
constant
MPC prediction step number
Integer index 1, ... , n
Iteration number
Controller gain vector for a generic system
Generic body stiffness coefficient
Lateral position state gain
Exponential path relaxation length
xvi
xvii
NOMENCLATURE
m
Number of controlled outputs
Generic body mass
Front frame assembly mass (inc. front wheel)
Rear frame assembly mass (inc. rear wheel)
n
Number of motorcycle states
p
Number of controller inputs
p
Augmented vector state
q,q,q
Vector of generalised coordinates, velocities, accelerations
Lateral position cost function weighting parameter
Yaw angle cost function weighting parameter
Steer torque cost function weighting parameter
r
Toroidal radius of tyres
Generic tyre normal spin
Slat
Generic tyre non-dimensional lateral slip ratio
Slong
Generic tyre non-dimensional longitudinal slip ratio
Stat
Generic tyre non-dimensional total slip ratio
t
Sample time
u
Control input (single)
u
Control input (vector)
ii
Small step change in u
v
Linear forward speed
v
Linear velocity vector
x, x(t)
Longitudinal position
X
Longitudinal velocity
a
8 (t)
,
t
Generic system state vector
x
Small step change in x
y, y(t)
•
8y(t)
Lateral position
y, iJt
Lateral velocity
y
Generic system output vector
Y/
Lateral position of MPC reference path
Motorcycle lateral position in global coordinates
Yli
Road point lateral position in local coordinates,
point
Yn
New motorcycle lateral position in local coordinates
ith
preview
NOMENCLATURE
xviii
Yr
Road point lateral position in global coordinates
YJ
Vector of MPC reference path
Y9
Road preview vector, global coordinates
YI
Road preview vector, local coordinates
Yr
Road preview vector
Yrn
Road preview vector, new information into shift register
z
Vertical displacement
z
Combined motorcycle-preview
state vector
A
Combined motorcycle-preview
A-matrix
Af
Reference path state space shift-register A-matrix, linear path
Ag
Reference path state space shift-register A-matrix, global coordinates, states component
Al
Reference path state space shift-register A-matrix, local coordinates, states component
Reference path state space shift-register A-matrix, local coordinates, global position component
Preview state space shift-register A-matrix
Reference path state space shift-register A-matrix, linear error
reduction path, set path component
Reference path state space shift-register A-matrix, linear error
reduction path, states component
Combined motorcycle-preview
B-matrix
Reference path state space shift-register A-matrix, linear reference path
Preview state space shift-register B-matrix
Combined motorcycle-preview
C-matrix
Damping matrix, generic system
c; Ca,,, c..
Tyre stiffness, with sideslip; generic, front, rear tyre
c; C-y,/, a.,
Tyre stiffness, with camber; generic, front, rear tyre
NOMENCLATURE
Cl,
q,/' Cl,r
x~
Ratio of tyre stiffness to normal force, with cornering; generic,
front, rear tyre
D
Combined motorcycle-preview
Dsmatrix
EM
Euler Matrix
F
Prediction model state space system matrix relating to system
states for a generic system
Flat
Lateral force, generic tyre
Fiong
Longitudinal force, generic tyre
Fn
Normal force, generic tyre
Fy, FY,/, Fy,r
Lateral tyre force, steady state; generic, front, rear tyre
G
Prediction model state space system matrix relating to system
inputs for a generic system
Coordinate vector, frame mass centre; front, rear
Hamiltonian vector
Identity matrix, dimension i
Moment of Inertia, about single axis
Generic inertia matrix
II,xz
Moment of inertia, front frame xy-axes cross couple
II,x
Moment of inertia, front frame about x-axis
Ij,y
Moment of inertia, front frame about y-axis
Ij,z
Moment of inertia, front frame about a-axis
Ir,x
Moment of inertia, rear frame about x-axis
Ir,y
Moment of inertia, rear frame about y-axis
t.;
Moment of inertia, rear frame about z-axis
Ir,xz
Moment of inertia, rear frame xy-axes cross couple
I/w,x
Moment of inertia, front wheel about z-axis
I/w,y
Moment of inertia, front wheel about y-axis
I/w,z
Moment of inertia, front wheel about z-axis
Irw,x
Moment of inertia, rear wheel about x-axis
Irw,y
Moment of inertia, rear wheel about y-axis
Irw,z
Moment of inertia, rear wheel about a-axis
Cost function value
Jordan form of Euler matrix
Partitioned
components of
JEM
xx
NOMENCLATURE
JI
Jacobian matrix of tyre force equations with respect to the state
vector
Jacobian matrix of motorcycle equations of motion with respect
to the state vector
Jacobian matrix of motorcycle equations of motion with respect
to the input vector
K
Gain matrix, generic system
Kb
Stiffness matrix, generic system
Kp,K2
Preview gains vector
KII, Kl
State gains vector
Kn,J, Kn,r
Tyre Stiffness, normal to surface; front, rear tyre
M
Generic tyre moment about z-axis
M
Prediction model state space outputs matrix relating to system
states for a generic system
Mass matrix, generic system
Discrete number of preview points
Discrete number of control points
Prediction model state space outputs matrix relating to system
inputs for a generic system
N
Number of iterations of full simulation
P
Algebraic Riccati equation solution
Pij
Partitioned
Q
Cost function output weighting matrix
R
Cost function input weighting matrix
n; RI, n;
Rolling radius; generic, front, rear tyre
RI
Rotation matrix, yaw
R2
Rotation matrix, roll
R3
Rotation matrix, steer
Reta
Rotation matrix, front fork inclination
SQ
Square root of Q
SR
Square root of R
T
System kinetic energy
Tp
Preview horizon time
Tu
Control horizon time
components of P
xxi
NOMENCLATURE
Tw
Dead-time horizon time
T
System modal matrix
Tij
Partitioned
T*
Inverse of T
Ttj
Partitioned
U
Potential energy
W
Virtual power of external forcing
YJ, Y,.
Dynamic lateral tyre force; front, rear tyre
a
Tyre sideslip angle
"Y
Tyre camber angle
8, 8(t)
Motorcycle steer angle
u1-,
81JP
components of T
components of T*
Motorcycle steer rate
feq
Ratio of camber stiffness to cornering stiffness, advanced tyre
e
Error vector, reference path to predicted uncontrolled output,
complete preview
T/
Front fork inclination angle
()
Function of final states and inputs in controller cost function
Predictive controller gains matrix
Lagrangian multiplier
Friction coefficient; generic, front, rear tyre
PjI Pr
Toroidal tyre radius; front, rear tyre
Relaxation length; front, rear tyre
</>,
</>(t)
Motorcycle roll angle
/p,
8t}/)
Motorcycle roll rate
1/1, 1/1(t)
Motorcycle yaw angle
. ,(t)
1/1, t
Motorcycle yaw rate
1/1J
Reference path yaw angle
1/19
Motorcycle yaw angle, global coordinates
1/11,
Road point yaw angle in local coordinates,
1/1n
New motorcycle yaw angle, local coordinates
ith
preview point
xxii
NOMENCLATURE
1/Jr
Road point yaw angle in global coordinates
w
Angular velocity, single rotation
w
Angular velocity vector
X
Vector of predicted future state vectors
<p
Vector of predicted future output vectors
A
Small step change in a variable
e
Function of states and inputs in controller cost function
Y
Vector of predicted future input vectors
Abbreviations:
OC
Optimal Control
MPC
Model Predictive Control
ARE
Algebraic Riccati Equation
DOF
Degree(s) of Freedom
ISO
International
SAE
Society of Automotive Engineers
VRML
Virtual Reality Modelling Language
Standardizations
Organization
General Notations
•
A
above a scalar, vector or matrix indicates a predicted value
• subscript v after a scalar, vector or matrix indicates specific reference to the
motorcycle model
Chapter 1
Introduction
With increasing frustration due to road traffic congestion and growing concerns over
environmental issues, transport
remains an area in which alternatives are constantly
being sought and questions raised about the viability of large multi-seat motor vehicles, often with only a driver aboard, travelling on the roads everyday. Prototype
vehicles using advanced power sources, or employing expensive high-grade materials
have been proposed, yet amongst all this concern, there exists already a form of motorised transport
which seems much more suitable to single-occupant journeys and
could therefore help to alleviate some of the current concerns.
The popularity of motorcycles has seen a steady increase in recent years, with annual
sales rising from 93,289 in 1997 to 133,938 at the end of 2004 [55]. In recognition
of the increasing relevance of motorcycles as a viable transport
option, the Govern-
ment launched the UK's first National Motorcycle Strategy in 2005, covering topics
including environmental issues, infrastructure,
and safety. The European Agenda for
Motorcycle Safety [301had covered similar areas a year previously.
The motorcycle as a form of transport
has a much smaller impact, both on the
road due to dimensional size, and on the environment,
physical mass.
A recent DEFRA (Department
due to smaller engines and
for Environment,
Food and Rural
Affairs) report [22] provided some useful statistics on vehicle C02 emissions (Table
1.1), confirming the lower environmental impacts of motorcycles compared with other
forms of motorised transport.
Furthermore, the greater manoeuvrability
of motorcycles, particularly in traffic, means
1
CHAPTER
2
1: INTRODUCTION
less time spent static in queues with the engine still running, and thus a more efficient
journey. The vast majority of commuting journeys are made with minimal luggage,
and therefore the limited storage space associated with motorcycles should not, in
the main, be a hindrance. Despite some impracticalities,
the use of motorcycles and
scooters is more widespread in mainland Europe (Figure 1.1), quite possibly on account of more conducive weather, though the numbers are still relatively low.
A more important
issue that may influence the use of motorcycles concerns safety.
Relatively speaking, motorcycles are not a safe form of transport.
In the years 1994-
2005, they accounted for between 13% and 21% of deaths and serious injuries on UK
roads (Table 1.2), despite making up only around 2% of the total traffic by numbers.
This is in main due to the fact that when an accident occurs a motorcycle rider lacks
the protective structures
and safety features afforded to car drivers.
Furthermore,
the more complicated rider strategy required to control a motorcycle, the relatively
greater instabilities,
and the smaller visual impact of motorcycles relative to cars
mean that motorcycle riders, and in particular inexperienced riders, are, in general,
at greater risk of accidents in the first place. Collectively, these have prompted Government action [23], with the safety strategy document setting a target to reduce the
number of people killed or seriously injured by 40 % by 2010. A greater understanding
of the riding strategy and requirements of motorcycle riders could therefore provide
useful information both for rider training, road infrastructure,
and also potentially in
the development of electronic rider aids that could assist the rider's control.
A range of motivations for computer simulation methods were outlined in [87]. From
a more commercial perspective, motorcycle manufacturers,
as with any industry, con-
stantly strive to make their product more appealing to the customer, whose decision
will be influenced by a number of factors. Some of these will be objective, such as cost
or comfort of riding, a greater number may be more subjective, for instance visual
styling or riding qualities. In order to develop a better product, it is useful to have
some understanding
of the more subjective elements, and if possible develop means
of quantifying these in a more defined, mathematical
way.
Historically, these subjective qualities could only be assessed by the development of
physical prototypes which would be extensively assessed by test riders, who would
give feedback to the design departments.
The development of such prototypes and the
necessary testing is both financially and time intensive. Customers will always desire
CHAPTER 1: INTRODUCTION
3
more for less, and consequently manufacturers
must find ways to achieve this if they
are to remain in business. Thus, the product must be better, yet the customer does
not always expect to pay a premium for this. Furthermore, manufacturers are aiming
always to reduce their time-to-market,
such that they can respond more readily to
either the market needs or potential threats from products offered by their rivals.
The widening use of computer simulation methods has enabled these requirements to
be met; modern computer systems are now relatively cheap, can be run all day and
every day if needed, negate the need for expensive and time-consuming
prototype
development, and can generate vast quantities of data in a relatively short time.
There is therefore an increasing need for accurate simulation tools to aid in the motorcycle design process, and indeed these can and have been employed for modelling
various aspects of the motorcycle's design. Of the constituent elements of the whole
process, the most difficult element to model concerns the rider, and the simulation of
his or her control actions.
Compared with a twin-track vehicle such as a car, a motorcycle requires a greater
degree of involvement from the rider, requiring control both to guide the motorcycle
and to remain upright at low speeds while riding. Furthermore,
the control process
of the rider involves control via both the steering system and through the influence of
body weight movement on the machine. Consequently, this makes the modelling of
motorcycle riders a less straight-forward problem. Because the riding process is a more
demanding task, it therefore becomes more important
to consider the implication of
design changes on the ability of a rider to control the machine, and to assess how the
rider may need to adjust his control actions in light of the motorcycle's characteristics.
Advanced simulation tools will facilitate this process and therefore have also potential
commercial benefits.
1.1
Motivation
As a result of the statistics and information provided, the suitability of motorcycles
has and will for some time to come be debated by those either in favour or against
them, and the aim here is not to fuel the debate further. Rather, the objective here
is to consider how engineering research work can be applied to the greater good of
CHAPTER
4
1: INTRODUCTION
motorcycle knowledge, understanding
and design.
The motivation for this research work is to develop motorcycle rider simulation tools.
These will enable a simulated motorcycle to be manoeuvred along a set course, that
could therefore be employed as a design tool for future motorcycles, or to assist in the
modifications of existing machines. Furthermore,
development of rider models may
provide a useful insight into the control strategies employed by motorcycle riders,
which could find use in improving road safety for motorcycle riders.
The last two decades in particular have seen the development of a number of vehicle
handling control systems, including anti-lock braking systems, electronic stability
control and active suspensions. Almost universally, these systems apply to twin-track
vehicles. It is suggested that this, in the most part, is as a result of the relatively
simpler control strategy for twin-track vehicles and thus the relatively more simple
procedures for developing the necessary computer code to define the strategies of
these systems. It is therefore also proposed that the development of motorcycle rider
control models may find use in the subsequent development of electronic motorcycle
chassis enhancement systems.
CHAPTER
1.2
5
1: INTRODUCTION
Tables
I Vehicle
Type
I Size
Label
I gC02/km I
Small
159.2
Medium
188.0
Large
257.7
Hybrid
Medium
109.7
petrol-electric
Large
194.7
Small
72.9
Medium
93.9
Large
128.6
Petrol Car
Motorcycle
Table 1.1: Average C02 emissions by vehicle type, [22]
Killed
Killed and Seriously Injured
1994-98
2003
2004
2005
1994-98
2003
2004
2005
Car Occupants
49.25
50.43
51.88
52.33
48.80
46.46
47.00
45.46
Motorcyclists
13.05
19.75
18.16
17.78
13.59
20.56
19.35
20.24
5.20
3.25
4.16
4.62
7.83
6.48
6.72
7.34
28.17
22.06
20.83
20.96
24.49
21.32
21.77
22.17
Others
4.33
4.50
4.97
4.31
5.30
5.18
5.16
4.79
All Road Users
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
Cyclists
Pedestrians
Table 1.2: Accident statistics, percentages by road user type, [23]
CHAPTER
1.3
6
1: INTRODUCTION
Figure
Sweden
Finland
Portugal
Poland
Austria
Netherlands
Italy
France
Spain
Greece
Germany
Denmark
Czech Repeblic
Belgium
United Kingdom
0
1.2
1
0.8
0.6
0.4
0.2
% of passenger-km by powered two-wheeled vehicles
1.4
Figure 1.1: Comparative use of motorcycles in the major European countries, [27]
Chapter 2
Literature Review
2.1
Introduction
Chapter 1 discussed some of the motivations and needs for the development of simulation and analysis methods with reference to motorcycles. The ability to replicate
realistically a number of physical features, characteristics
and phenomena associated
with dynamic systems can have significant bearings upon their design, manufacture
and understanding
to ultimately improve the effectiveness of the modelled system.
With specific reference to a motorcycle and rider, in order to generate a realistic
simulation model, the process must be formed by the combination of a number of
systems. Broadly, these systems comprise the modelling of the dynamic response of
the motorcycle, the correct modelling of the tyre behaviour and the manner in which
the rider operates in order to control the motorcycle.
These systems themselves
consist of a number of smaller subsystems, and thus in total a combined motorcyclerider simulation model may have a vast array of variables that ultimately affect the
performance of the model.
The dynamic response is clearly heavily influenced by the structural
the motorcycle.
properties of
These structural
properties include both the geometric dimensions
of the motorcycle'S construction,
and the physical stiffness and damping properties
of the motorcycle'S components.
Additionally a number of other factors may also be
relevant, for example the addition of external mass in the form of luggage or a pillion
passenger.
7
CHAPTER 2: LITERATURE REVIEW
8
The forces acting on the motorcycle will have a significant influence upon the vehicle
behaviour, and come from a number of sources. These can include aerodynamic forces,
the forces exerted by the rider, but perhaps most fundamentally
the forces exerted
by the interaction of the tyres with the ground. These forces primarily influence the
direction of the motorcycle's motion, but also play a significant role in the stability
characteristics
of the motorcycle. Additionally, a number of other sources of forcing,
such as acceleration or braking forces, will have an impact upon the condition of the
vehicle.
Finally the influence that the rider has upon the motorcycle, assuming that limit
conditions for tyre forces or stability conditions are not exceeded, ultimately dictates
the response of the motorcycle. The rider has influences on the motorcycle due not
only to the direct forces that are applied to the motorcycle controls, but also indirectly
as a consequence of the structural influence his body mass and body movements have
upon the stability and response of the motorcycle.
This final influence is far more
prevalent for a motorcycle than with a car, for example, where the driver's body mass
is virtually static with regard to the vehicle.
The combination and interaction of all these factors determine the stability and handling characteristics
of the motorcycle and how the motorcycle may respond to a
rider's control actions.
As a consequence, a vast array of research work has been
conducted in the fields of motorcycle stability, tyre modelling, rider analysis and the
combination of all to form advanced motorcycle-rider
simulation models, which can
ultimately be used to improve the performance of a real motorcycle and rider through
better design, manufacture and understanding
of the physical characteristics.
This chapter will therefore outline the significant research work conducted in and
around these areas, exploring the opportunities
the field of motorcycle-rider
control.
that exist for novel research work in
Section 2.2 gives an outline of the historical
research into motorcycle behaviour, focusing primarily on the modes of motion and
the instabilities of motorcycles, considering both the detailed modelling of motorcycle
as a physical structure and how the identified modes of motion are influenced by design
features of the motorcycle. Tyre modelling will be covered in Section 2.3, as this has a
significant influence upon the motorcycle responses and modes that will be covered in
the first section. These two sections provide important background knowledge to the
main review in Section 2.4, which will focus on the rider's control of the motorcycle,
CHAPTER 2: LITERATURE REVIEW
9
and the work that has been conducted previously for both the understanding
of a
motorcycle rider's control process and subsequently in the modelling of these control
actions to form simulation models of motorcycle riders.
2.2
Motorcycle Stability Analysis
The analysis of a motorcycle's response characteristics
has been investigated by a
number of authors, with ever more complex multi-body dynamics models appearing
regularly. The earliest studies considered bicycle models, being simplified motorcycle
models without, for example, the inclusion of suspension characteristics.
Whipple was one of the earliest authors to produce stability analysis models of a
bicycle [106]. His work resulted in the equations of motion for a bicycle, including
rider control inputs, and also identified some of the classic instability modes of bicycles
and motorcycles, subsequently to be coined as the weave and capsize modes. Other
early authors on the subject, with similar objectives and results, included Bower [8]
and Dohring [24]
With more specific reference to motorcycles, being the subject of this thesis, it was
not until the early 1970's that the area begin to receive significant attention with the
emergence of greater research work. Of these, Sharp was one of the first, producing
work of a similar nature to the work of Whipple many years earlier, but focused more
specifically on motorcycle research.
Sharp [80] generated the linearised equations of motion for a bicycle model representation of a motorcycle, also exploring the fundamental
modes of motion for a
motorcycle, which were identified as wobble, weave and capsize modes, terms since
adopted almost universally. His inclusion of a tyre model, not a feature in [106], also
resulted in the identification of a new mode, termed weave.
Capsize was identified as a non-oscillatory instability related to the tendency of the
motorcycle to fall onto its side like an inverted pendulum.
Wobble was defined as
the high frequency oscillation of the front steering system in a shimmy motion, often
compared with that of a shopping trolley wheel. Finally, the weave mode was identified as a combined oscillation involving simultaneous roll and yaw of the motorcycle,
such that the motorcycle followed a high-frequency slalom-like trajectory.
10
CHAPTER 2: LITERATURE REVIEW
A parameter
study was conducted using the model as a means of identifying the
qualities required to produce stable motorcycles.
approach as a potential
Sensing the possibilities of the
design tool, several research works aimed to develop the
techniques used for more advanced motorcycle models, exploring further aspects of
motorcycle design and their respective influences upon the motorcycle's behaviour.
Related works by Kane [42] and Sharp [81] investigated the influence of frame flexibility on the stability characteristics,
with each author modelling the flexibility in
subtly different ways. Both research works found significant influence of frame flexibility upon the stability characteristics
the influence of mass distribution
of the motorcycle, while Kane also included
and front fork trail.
In a similar vein, the effects of front fork flexibilities upon motorcycle behaviour were
assessed by Roe [75], leading to the proposal of a new front fork design idea which,
following extensive testing, was found to provide superior stability characteristics.
Over subsequent years a number of researchers continued to investigate the influences
of various structural
structural
elements of the motorcycle,
models of a motorcycle.
leading to increasingly complex
Notable authors included Sharp [83, 84, 85],
Cossalter and his co-workers [16, 17,21] and Nishimi [66].
In parallel with research work to investigate further structural
aspects of the frame
stiffness, it was natural that the most compliant structural element of the motorcycle,
namely the suspension, should also be included in the models. Jennings [40] was one
of the first to explore this avenue, followed soon after with further work by Sharp
[82].
Other aspects influential upon the motorcycle's stability characteristics were also worthy of investigation.
The effects of aerodynamic forces on moving bodies were already
well appreciated as a result of aerospace research work and physical experiences. FUndamental effects on motorcycles were suspected, but not fully understood, and so at
around the same time that structural
influences on motorcycle behaviour were being
explored, the influence of aerodynamic
properties upon the stability modes of mo-
torcycles was investigated by Cooper [12], also later covered in Foale [31] and Hucho
[36].
Earlier works had hinted towards the importance of mass distribution
with regard to
CHAPTER
2: LITERATURE
11
REVIEW
the stability modes, and the inclusion of accessories such as luggage racks on the rear
of a motorcycle could clearly have a significant impact upon this. Research of this
area was addressed by Otto [67].
Cornering, acceleration and deceleration were known by experience to change a motorcycle's dynamic characteristics.
Koenen [45] completed research work into the
dynamic behaviour of motorcycles, for both straight running and cornering conditions.
While resonant vibrations of the motorcycle and, in particular
the steering
system, are undesirable, the impact of vibrational modes is somewhat less dangerous
in a straight running situation than in a cornering condition; the steer torques and
frequencies associated with such vibrational modes make it very difficult for a rider
to overcome them, and in a cornering situation this severely impacts upon a rider's
ability to apply any necessary steer control with potentially disastrous consequences.
Since mass distribution
was known to have an effect upon the stability of the mo-
torcycle, it seemed logical that acceleration and deceleration would cause a dynamic
weight shift and therefore have some impact upon the motorcycle's response. During
cornering the tyre contact patch moves significantly around the tyre's cross-section,
additional external forces are introduced, and the geometry of the motorcycle is affected due to relative movements of the front and rear frames by the steer angle.
Limebeer et al. [48] investigated the acceleration/deceleration
case, extending the
earlier stability models of Sharp to include these effects, finding theoretical results
consistent with riding experiences. The results were also presented in [28].
The braking case was also considered in detail by Meijaard and Popov in [56], and
later extended to include suspension deflection, aerodynamic
drag and stabilising
rider control in [59]. Extensive studies considered the implications on motorcycle
stability for varying proportions of front and rear brake force distribution,
and the
cases of a locked front or rear wheel.
It was already well known that the rider could have a significant influence on the
motorcycle by the movement of his body weight on the machine.
Early research
work had simplified the analysis by considering the rider as a rigidly attached body,
but with greater understanding
of a motorcycle'S principal stability characteristics,
it became apparent that this factor ought to be considered in some detail.
A considerably more advanced model was therefore developed by Nishimi et al. [66],
CHAPTER
2: LITERATURE
REVIEW
12
extending earlier models to include flexibilities of the motorcycle rider upon the motorcycle, specifically a lateral freedom of the rider's lower body to move his position
on the seat, and a rotation of the upper body relative to the lower body about a
longitudinal axis, representative
of the rider leaning on the motorcycle.
The influence of the rider's upper body mass upon the stability of the motorcycle was
considered further by Sharp and Limebeer [93]. The rider was allowed a roll freedom
relative to the rear frame and also a yaw freedom. Roll freedom had already been
considered by Katayama et al. [43]. However, it was postulated
that when a rider
senses excessive levels of steering oscillation, such as the onset of a wobble or weave
instability, there is a tendency for the rider's upper body to tense in an attempt to
control the oscillations, leading to a yaw motion of the riders upper body fed by
the steering oscillations.
The responses to initial steer disturbances
were modelled,
analysing the stability modes that resulted.
The ever-developing capabilities of computer simulations extended the range of possibilities for parameteric
investigations
further.
Consolidating
his earlier stability
analysis work, Sharp [85] revisited the problem, modelling the motorcycle dynamics
through the use of symbolic software programs. The results from these new investigations agreed with findings in earlier work, and reconfirmed that amongst the most
important parameters in motorcycle design were the mass centre location of the rear
frame and its distance to the front and rear wheels, and the tyre relaxation length.
An eleven degree-of-freedom model was developed by Cossalter and Lot [18], with
the model described through the use of direct mathematical
modelling as opposed to
the use of commercial dynamics analysis software. Reasons for this choice included
the ability to more realistically model the complexities unique to motorcycles such as
tyre forces at high wheel camber angles. The modelling incorporated a detailed tyre
model including a complex shape definition and the consideration of tyre deformation
under loads.
Earlier work in motorcycle analysis had been used to study the straight running
.stability of motorcycles,
and the classic instability
modes of capsize, wobble and
weave considered. Apart from [45], previous works had not included the investigation
of cornering modes, with particular
difficulties surrounding mainly the question of
correctly modelling the tyre characteristics
in a cornering condition. Cossalter et al.
CHAPTER 2: LITERATURE
13
REVIEW
[20] therefore also developed his earlier work and used modal analysis to model the
stability of the motorcycle for both straight running and cornering behaviour.
The
modal analysis consisted of calculation of the steady-state conditions, linearisation of
the equations, finally the solution of the eigenvalue problem.
While the vast majority of motorcycle simulation work was conducted with regard to
road-going machines, James [39] also presented the stability behaviour of an off-road
motorcycle through the use of experimentations
on an instrumented
motorcycle and
an analytical simulation model.
Finally, an extensive benchmarking of Whipple's [106] bicycle model was conducted
by Meijaard et al. [62], providing both an extensive review of the bicycle stability
literature, and the linearised equations of motion of Whipple'S model.
This literature review, covering here published work on motorcycle stability characteristics, is by no means exhaustive. The great wealth of work already covered should
be apparent, and the extent to which the level of motorcycle stability modelling has
now advanced.
This review has intended to cover only some of the more relevant
findings in this area in order to provide the necessary background information, and
to guide the reader toward sources of greater detail in this area.
2.3
Tyre Modelling
Since the tyre forces represent the primary external forcing on the motorcycle-rider
system, and therefore play such a fundamental role in the stability characteristics
the motorcycle, a realistic representation
of the tyre's characteristics
of
is an essential
component of the complete modelling.
Experimental studies of vehicle tyres have highlighted the non-linearity of the forces
generated as a function of slip and camber, although within the range of moderate slip
ratios, linear models can be used without significant loss of accuracy for the modelling
of motorcycle dynamic response. However, with increased complexity of motorcycle
models, and with increasing demands to understand
the absolute performance char-
acteristics, more realistic non-linear tyre models have been sought.
De Vries and Pacejka [100] considered the impact of tyre modelling upon the stability
CHAPTER 2: LITERATURE REVIEW
14
of a motorcycle, employing both a relatively simple first-order relaxation model based
on lateral velocity and tyre sideslip, and a more complex rigid ring tyre model. Both
models were compared against a dynamic tyre test on a rotating drum, and both
subsequently applied to a simple bicycle model. For moderate speeds and manoeuvres,
the simpler tyre model was seen to give acceptable performance, showing any weakness
only at high speeds by not accounting for the gyroscopic effects of the mass of the
tyre belt.
Cossalter and his collaborators have presented a number of papers on the detailed
modelling of motorcycle tyre geometries and the resulting forces [16, 17, 18]. These
led to more detailed definitions of tyre contact patch locations for a cambered tyre,
and the forces and moments that are generated as a result, allowing a move away
from disc-model tyres that had been used previously.
The significant influence of tyre characteristics
in respect of scooter and motorcycle
handling was the subject of further investigations by Cossalter et al. [19], comparing
the characteristics
for a range of different tyre types. The lateral force due to wheel
camber was seen to be a more significant lateral force provider than pure sideslip,
with the most interesting point to arise from the tests conducted being the observation of the importance
of, in particular,
of twisting (steer) torques resulting from the cambering
the front wheel, and of the effects that this steer torque can have
upon the handling characteristics
of the motorcycle. This was a factor that had not
been considered in previous studies, for example [104], in attempting
to assess the
requirements of a motorcycle with good handling qualities as perceived by the rider.
An area of particular difficulty yet of great importance concerns the consideration of
the tyre forces generated by the parallel processes of lean and side-slip. The "Magic
Formula" method [68] had been shown to produce very good analytical representations of the non-linear tyre forces, and so the inclusion of this method into a more
complete motorcycle analysis was desirable. However, the complexity of the calculations involved, and the task of determining the formula's parameter values through
experimentation,
made this task computationally
tella [91] therefore looked at a normalisation
very demanding.
Sharp and Bet-
of parameters approach as a means of
reducing the complexity of the task and hence making the integration of the "Magic
Formula" method into a complete model more feasible.
CHAPTER 2: LITERATURE REVIEW
15
The "Magic Formula" tyre model was implemented on a motorcycle simulation model
and compared against both a more simple tyre model and experimental
data by
Tezuka et al. [97]. The simple tyre model comprised a carpet plot model, generating the necessary tyre forces and moments from the interpolation
of a data set of
experimental tyre test results. The two tyre models were applied to a eight-bodied
motorcycle model, simulated in a steady turn. An instrumented
motorcycle and rider
was tasked with the same manoeuvre, and the results compared.
For straight run-
ning, the 'Magic Formula' method showed superior correlation to the experimental
results, while cornering produced similar results from both methods.
Sharp et al. [92] also combined tyre geometry models by Cossalter and his co-workers
[16, 17, 18] with the application of the "Magic Formula" tyre model [68]. The tyre
model that resulted was extensively validated against available test data for modern high performance front and rear motorcycle tyres.
Furthermore,
a monoshock
rear suspension arrangement and rider upper body lean freedom was included in the
motorcycle dynamics model, resulting in an advanced motorcycle dynamics model.
Meijaard and Popov produced a number of research papers developing advanced tyre
simulation models. [58] developed a string model of a motorcycle tyre, allowing for
deflections of the string elements both laterally and longitudinally to develop lateral
and longitudinal tyre forces with lateral and longitudinal slip and with camber of
the tyre.
Later tyre models by the same authors [61] defined the tyre contact as
a deflected point of the tyre carcass and developed the tyre forces via a simplified
representation
of the full "Magic Formula", seen to capture the response of the full
"Magic Formula" with some accuracy.
A number of research papers have considered the modelling of twin-track vehicle
tyres, a good deal fewer consider the greater complexity of a motorcycle tyre with
its far more extensive use of camber thrust as a lateral force generator.
general understanding
introduction.
For a more
of tyre force generation, [31] and [13] also provide an excellent
16
CHAPTER 2: LITERATURE REVIEW
2.4
Rider Control
The previous research areas considered were largely concerned with understanding the
dynamic response of the motorcycle alone. The aim of this thesis will be focused more
towards an understanding and representation of the rider as a controller. However, the
rider's control actions will clearly be influenced by the way in which the motorcycle
will respond to his control inputs, and so the preceding references all provide vital
background knowledge.
The task at hand, however, is the understanding
of the rider's control characteristics.
This will concern the rider's active control characteristics and the use made of sensory
information in selecting a control input, which is clearly a fundamental requirement
in the development of a realistic motorcycle-rider
2.4.1
model.
Visual Perception
The inputs that a rider applies to the steer controls of a motorcycle are based upon
his interpretation
of the approaching road that he is tasked with following, and his
knowledge and understanding
of the motorcycle's response.
Clearly the visual per-
ception part is a fundamental requirement in order to accomplish the task, and so a
relevant branch of research work concerns the understanding
interpretation
of a motorcycle rider's
of his visual horizon. Observations made by Miyamaru et al. [65] had
hinted at the effects that correct modelling of a rider's visual road perception could
have upon the ability of a rider control simulation. Identifying the link between visual
input and control decision is not an easy task to do; primarily, determining exactly
what the rider is looking at, and what proportion of the observed road information
is stored and how much is discarded, is not readily obtained.
Donges [25] attempted
to address this question, investigating the manner in which
the driver perceives the road ahead and uses this information in the control task. The
presented theoretical model consisted of a two-mode control strategy, with the future
direction of the vehicle influenced by the distant preview (guidance) information
and the more immediate preview information, termed stabilisation information, used
to stabilise the vehicle motions resulting from the guidance control.
strategies were considered to act in parallel.
Both control
17
CHAPTER 2: LITERATURE REVIEW
An experimental simulation was set up, in which drivers were tasked with following a
randomly curving track using a driving simulator.
A mathematical
control strategy
to replicate the driver's road perceptions and control actions was compared with the
experimental
results, comprising an anticipatory
open-loop control and a compen-
satory closed-loop control element. The former represented the guidance information
as a feedforward controller, while the latter represented the stabilisation information
as a multiple input, single output feedback controller.
Comparisons between the actual drivers and the mathematical control strategy showed
similar results, with the control strategy able to anticipate forthcoming changes in
roadway curvature and to begin to alter the steering input in advance of that curvature, as would a real driver. The pertinent
findings of this research were that a
driver's visually-influenced control actions are based on both distant and near preview
observations to control fully the heading and attitude of a vehicle.
In a similar study, Land and Horwood [47] also used a simulated road preview, with
defined road edges, for experimental
drivers to attempt
to follow. By varying the
extent of the approaching road visible by the test driver, the effect on the driver's
control task of the previewed road information suggested, in agreement with [25],
that the distant preview was used to determine the curvature of the road, while the
near preview information was primarily used for controlling the position-in-lane
of
the vehicle.
The accuracy of the modelling of a human vehicle controller clearly relies upon the
appropriate selection of information input to the driver, and so MacAdam [51] investigated the relative importance of sensory stimuli to the driver to understand
are the more important
for the task of driving.
The physical attributes
which
of drivers,
including the use of visual preview, adaptive control and the presence of an internal
vehicle model concept, were also discussed. A control model, combining a number of
sensory and physical driver control elements coupled with a vehicle dynamics model,
was presented.
This model was used to simulate a tyre blow-out while negotiating
a double lane-change manoeuvre as an example of the model's ability to represent a
human drivers' ability to conduct both planned manoeuvres and also to react to unexpected changes to the task. As a result of such modelling, a number of key features
necessary for a realistic representation
of a human driver were identified.
18
CHAPTER 2: LITERATURE REVIEW
2.4.2
Rider Analysis and Modelling
With an appreciation of the complex processes that occur between information input
and control output, a far greater quantity of research work has been involved with
trying to understand
the control procedures of a motorcycle rider and to replicate
them effectively. Broadly, the work in this area can be broken into the specifically
experimental work, where instrumented
equipment was used to gain an insight into
the control actions, and the theoretical side, in which some effort is made to replicate
the rider's control actions in some computational way. Naturally, a cross-over between
the two exists, in which theoretical models are compared with experimental results.
Unlike a twin track vehicle driver who relies primarily on control of the steering
system, a motorcycle rider can apply control action to influence the heading direction
of the motorcycle either by direct control to the steering system or to the motorcycle
itself via forces and torques applied by the body to the seat, footpegs, or by body
movement in order to influence the lean condition and hence heading direction of the
motorcycle. A number of research works have aimed to understand
the use made of
these options in the control process applied by a motorcycle rider.
Weir [102] was one of the first authors to study in some detail the control procedures
of a motorcycle, investigating the relationships between a number of the motorcycle's
states and the principal control inputs available to the rider. The analyses suggested,
for a single-loop controller, that roll control was the primary objective of the rider and
was best achieved through the steer torque control input. Other good input-output
relationships
included heading angular rate via rider lean and roll angle via rider
body lean. Control of the lateral position, as a means of stabilising the motorcycle,
was poor for all control options. The results of the single-loop control strategies were
extended to the multiple-loop case, where steer torque to roll angle and upper body
lean to heading angle and lateral position was considered to be the most representative
control strategy of a motorcycle rider. The influence of the task upon these findings
was assessed by parametric
speed.
By understanding
variation of the motorcycle model and of the forward
the link between system states and control inputs, and
the influence of the design parameters
upon this link, the approach had potential
application as a design tool to affect the perceived handling of a motorcycle.
In separate studies, Rice [74] and Weir and Zellner [103, 104] also produced useful
19
CHAPTER 2: LITERATURE REVIEW
research findings in the attempt to understand a rider's control process. Rice experimentally measured a number of motorcycle and rider parameters for four different
riders performing a lane change manoeuvre, each having identical equipment but varied levels of riding experience and physical stature.
Measured parameters included
yaw, roll and steer angles, steer torque and rider lean angle relative to the motorcycle.
Weir and Zellner [103] meanwhile modelled the control task theoretically, allowing in
a rider model both steer torque and upper body lean torque as control inputs, and
using three key control strategies, comparing their theoretical findings with instrumented machines performing similar lane change manoeuvres.
Like Rice, they also
sought to investigate the specific requirements of a good-handling motorcycle [104],
performing experimental results of a similar nature to those of Rice and again taking
measurements of motorcycle states and rider control inputs.
The experimental
findings of Rice suggested that rider upper body lean was a sig-
nificant element of the control process, though the author was not able to define to
what extent this featured. It was noted that different riders displayed different riding
styles; this style included the severity of countersteer to initiate a turn, and the relative amounts of body lean, body lateral movement and steer torque control inputs.
It was suggested that the relative use that the rider makes of these control options
could be based either on a personal preference or as a consequence of varying levels
of riding experience.
These results were mildly contradictory
to the findings of Weir and Zellner [104].
As with the work by Rice, the method employed an instrumented
motorcycle, here
performing a constant radius cornering test and a single lane change manoeuvre, but
with subtly different dimensions. Interesting observations made in these experiments
suggested that for some riders the upper body lean angles relative to the motorcycle
were near zero, implying that the rider remains upright relative to the motorcycle,
consequently needing to apply greater levels of steer torque.
This goes against the
findings of Rice, though it is possible that the length of the lane change manoeuvre
and the relative experiences of the rider's may have had some influence upon this.
It is reasonable to assume that to complete the manoeuvre over a shorter distance
a more severe turn is required, and by leaning the upper body such as to keep the
upper body vertical, as Rice found, there will be a force applied to the machine that
will tend to help to lean the motorcycle into the turn.
It is possible that for the
CHAPTER 2: LITERATURE REVIEW
20
more gentle manoeuvre such a control input is not required, or in the case of a less
experienced rider, may not be a control technique that the rider has developed fully.
Weir and Zellner [103] modelled the theory that a complete manoeuvre consists of
three distinct phases, identified previously in [102]. In order to initiate manoeuvres
the rider model operated in a precognitive control strategy, whereby the rider executed
a manoeuvre using previous experience, instinctively countersteering to initiate a turn
for example. A compensatory control was found to be the dominant method for steady
state riding, be it in a straight line or negotiating a curve, with the rider establishing
the manoeuvre and then reacting to any deviations to this path. Finally, a pursuit
control method was suggested, whereby knowledge of the system inputs allowed the
rider to apply a feedforward control to the benefit of the motorcycle's performance,
including use of either the throttle
or brakes for example.
At track-racing
level,
control of the throttle on the exit of turns can be used to increase the rear tyre slip
angle and thus help to yaw the motorcycle, in addition to providing accelerative drive
out of the corners.
A series of control systems to replicate these features was presented and assessed.
Using this model, variations in speed, tyre response and front wheel trail were made,
and it was concluded that the resulting equations of motion for the motorcycle-rider
system could allow a fuller understanding
of what are the criteria for a motorcycle
with good handling qualities, and how these may be achieved through parameter
variation. The method, it was suggested, may also find use in a rider training role.
The question of controllability of motorcycles and bicycles was considered by Seffen
et al. [79]. The concept of controllability is a common descriptor of dynamic systems
which determines the ability of the control inputs to exercise control over the system
states [38]. Seffen et al. obtained a numerical coefficient, based on the control problem
Grammian matrix, which gave an indication of the degree of controllability over the
system, and applied this to both a motorcycle model and a bicycle model, conducting
a parameter study on both to investigate the effects of design variables upon the ease
of control.
Yokomori et al. [108] investigated the ability of a rider to maintain the stability of a
motorcycle at low speed with no hands on the handlebars, and thus using only body
lean and forces on the footpegs to affect the stability of the motorcycle, comparing
21
CHAPTER 2: LITERATURE REVIEW
these results with some experimental tests.
Other works of interest to the field of rider control include the work of Miyamaru et a1.
[65], with the objective of constructing
an appropriate
real-world motorcycle-riding
simulator. The long-term aim was to develop a physical, rather than computer based,
simulator for a motorcycle, such that riders can practice the control and riding of a
motorcycle without needing to venture on to the highways, with a view to improving
rider training and hence safety. The results offer interesting observations into the
difficulties that exist in simulating the control tasks applied by a rider on a moving
motorcycle.
In studying the control actions required to ride a motorcycle,it
was observed that
at low speeds the rider actively controls the steer angle, being the primary method
of directional control.
turn, but thereafter
At higher speeds some steer effort is required to initiate a
the steer angle is essentially set depending on the roll angle,
as it is the balancing of centripetal
and weight forces and hence roll angle which
governs the turn radius. The distribution
of control through steering and roll control
then varies accordingly through the speed range from low to high speeds. Further
pertinent
observations included that in assessing the road ahead of the motorcycle
while turning, a rider will tend to look to the inside of that turn, rather than straight
ahead.
A parameter
study was conducted with regard to the rider's head position
compared to the motorcycle yaw angle during a slalom manoeuvre, finding that by
adding a coefficient of 0.3 multiplied by the yaw rate to the motorcycle yaw angle
gave a suitable head angle, leading to improved simulation results.
Detailed works were conducted by Prem and Good [70] and by Katayama et al. [43].
Both groups developed rider control strategies that were tasked with a particular
manoeuvre that was later compared with experimental observations. Prem and Good
[70] modelled the rider allowing for steer torque and upper body lean; the steer torque
inputs were taken to be developed directly as a result of the upper body lean, with
the resulting torques developed due to stiffness in the rider's arms. Katayama et al.
[43] allowed independent control of steer torque, lower body control torque and upper
body control torque.
Prem and Good [70] modelled their proposed system based on previous work by Weir
[102]. In this earlier modelling, the rider steer torque was used as the primary means
22
CHAPTER 2: LITERATURE REVIEW
for stabilising the roll angle of the motorcycle while the rider's upper body roll angle
and lower body lateral position were used as the primary means for affecting the yaw
angle and lateral position of the motorcycle. The proposal being made and tested by
Prem and Good, contradicting with [102], was that the rider's upper body lean angle
is the primary control method, and that steer torque is linked to the roll angle; as
the upper body is rolled, the stiffness of the arms will naturally apply a steer torque
without the rider needing to control the arms independently.
The simple rider control model from [102] was adapted such that the rider steer torque
was linked to the rider upper body lean angle. This model would allow roll stabilisation of the motorcycle but not control the path following or yaw of the motorcycle.
A more complicated model, based on Weir's multiple loop path control method [102],
fed back the lateral position, yaw angle and roll angle outputs in order to control the
rider upper body lean and linked steer torque.
Experimental tests were conducted with several riders of varying experience attempting an obstacle avoidance manoeuvre on an instrumented
motorcycle.
From the
tests, key observations made were that a skilled rider will use noticeably more severe
amounts of counter-steer to initiate a turn independently
However, once the initial counter-steer,
of upper body lean angle.
turn-in phase of the turn is completed, for
both riders the upper body lean angle and steer angle were seen to be closely coupled,
supporting the proposed theory of linked body roll and steer torque.
Doth of the theoretical control models were found to yield broadly similar responses,
with appropriate control gains applied. However the key conclusions drawn were that
the unskilled rider can be represented by the simpler single feedback loop controlling
the upper body lean, whereas to replicate the skilled rider required the multiple loop
feedback model to correctly replicate the rider's actions. This implies that a skilled
rider is able to control steer torque and upper body lean angle independently, whereas
the less skilled rider cannot achieve this so readily.
By contrast, Katayama et al. [43], considered the rider as a two mass system represented as an upper body and the lower body, treated as upside-down connected
pendula; the lower mass pivoted about the longitudinal
axis of the motorcycle at
ground level with the upper mass then pivoting about the longitudinal axis of the
lower body mass. The model itself was an extension of Sharp's model [80] with the
23
CHAPTER 2: LITERATURE REVIEW
addition of the two rider degrees of freedom.
Rider control covered three inputs;
steer torque, lower body control torque and upper body control torque, calculated by
means of proportionality
to motorcycle roll angle and a calculated heading error of
the motorcycle based on a simple preview method.
A lane-change manoeuvre at fixed speed was studied with the three control methods
considered separately. It was found that steer torque was the most important parameter, the lower body influence was found to be
control influence at
b of the
-la of the steer effect for the particular
steer with the upper-body
lane-change manoeuvre con-
sidered, suggesting that upper body movement is primarily to maintain rider comfort
and of limited influence upon the control of the motorcycle attitude.
The lower-body
movement is considered to assist the steer torque as a means of control.
The modelling method was compared with experiment by using 12 riders on instrumented machines.
Different riders were found to employ varying control strategies
(riding styles) with regard to magnitudes of initial steer torques for instance, and
as such the simulation required the control coefficients to be adjusted appropriately.
However, assuming this was done the simulation was capable of replicating the measured motorcycle-rider
generic motorcycle-rider
behaviour with good accuracy. This suggests, however, that a
simulation can have only limited accuracy due to the heavy
influence of rider style upon the motorcycles behaviour.
Previous research works, including [lO4], had sought to provide some means by which
to determine the qualities required for a good handling motorcycle.
With a similar
aim, Cossalter et al. [16] calculated the steer torques necessary to complete a given
manoeuvre using an appropriate mathematical
model. The model itself was a generic
four-body bicycle model, with the influence of suspension movement neglected, as
the consideration
of a steady turn should result in fixed wheel positions relative
to the motorcycle frames.
Changes to the physical attributes
of both tyres and
the motorcycle influence these torques, and so analysis of this type may enable a
motorcycle to be designed that exhibits particular characteristics with regard to steer
torque felt by the rider, and hence influence the rider's perception of its handling.
24
CHAPTER 2: LITERATURE REVIEW
2.4.3
Optimised Rider Models
A number of research works have involved analysing the rider's control actions and
seeking to replicate these via the use of a mathematical
control strategy.
Another
branch is concerned with the development of rider models that attempt to optimise
the mathematically
defined riding task and hence replicate the rider's control actions
accordingly.
Considering first the approach taken to optimise the vehicle performance based on
system dynamics, Cossalter et al. [14] developed a technique termed the Optimal
Manoeuvre Method.
This approach essentially employed an optimisation
strategy
with several performance criteria forming the cost function. These elements included
maximising the distance travelled over a manoeuvre, trajectory
constraints to keep
the motorcycle within the road width, costs to ensure the ratio of lateral front wheel
force to longitudinal thrust/braking
forces were not excessive, and other less sig-
nificant performance factors. Experimental
performance evaluation with respect to
handling and manoeuvre capabilities would always be heavily influenced by rider
ability and control strategies, and in limit cornering conditions rider style and handling preferences may adversely affect experimental results. The Optimal Manoeuvre
Method was therefore developed that would use an optimal preview approach in order
to simulate the manoeuvre of a motorcycle, having the advantage of applying, within
reason, whatever input controls were necessary, through steer torque, throttle and
braking actions, such that the motorcycle performed the required manoeuvre in the
most efficient manner specific to that motorcycle. This would therefore give a more
appropriate measure of ultimate machine performance, on the assumption that a rider
would be able to apply the necessary control strategy to achieve this, reasoning that
a highly skilled rider would be able to adapt himself to the particular motorcycle and
therefore maximise its performance.
Simulations were compared with telemetric data
for a motorcycle negotiating an S-bend on a race track, and found to agree well. The
findings were also presented by Cossalter in [15].
Cossalter's co-authors Biral and Da Lio applied the Optimal Manoeuvre Method to
the modelling of a vehicle driver [6]. The principles of the approach were the same
as for the motorcycle case, with similar simulations and results.
A review of driver control models was made by Guo and Guan [33] before presenting
25
CHAPTER 2: LITERATURE REVIEW
the development of their own optimal preview driver control models. In particular, the
broad areas of compensatory and preview tracking models were compared, suggesting
that the latter provides a far superior path tracking ability.
The optimal preview
control model of MacAdam [50] was reviewed, and the concept extended to produce
more intricate driver control models considering the lateral accelerations of previewed
road position and orientation
information.
Initially based on a global coordinate
system, the model was adapted to local coordinates, displaying good performance.
The control of forward speed was discussed, as was the concept of a reference path,
being a path that the driver may desire to follow, rather than necessarily the centreline
of some constrained road path.
Di Puccio et al. [73] compared three types of driver control model for path tracking
of a four-wheeled vehicle; a simple preview tracking model using single point preview,
the second a simple fuzzy-logic controller, the third a more detailed fuzzy controller
aimed at capturing the driver's behavioural characteristics.
The single point preview
model represented a very simple prediction model, determining the future position
of the vehicle based on the current vehicle states. The fuzzy control models however
proved unable to realistically reproduce human behaviour.
A very similar preview-tracking approach was taken by Sharp et al. [86]. Again with
application to a twin-track vehicle, Sharp and his co-authors successfully generated
a driver control model capable of steering a vehicle model along a road path.
The
road information was presented to the driver model as a series of discrete road points,
and the controller generated a steer control input using PIn theory to minimise the
lateral errors between the previewed road and a preview arm projected directly ahead
of the vehicle, this time using a multiple preview point approach.
The theory was
later applied to the case of a motorcycle [76], but the application was seen to be
inappropriate
to the task. The requirement to countersteer a motorcycle was not well
catered for by this approach, leading to ineffective path following.
The possibility of a control system to assist in motorcycle stabilisation
as an aid
to safety was investigated by Kamata and Nishimura [41]. This control system was
evaluated through implementation
of such a device on a computer simulation of a
motorcycle. Use of such a control device was shown to be beneficial in reducing the
roll angle of the motorcycle following a disturbance
over a range of vehicle speeds.
Though the paper covered only theoretical work, it was indicated that subsequent in-
26
CHAPTER 2: LITERATURE REVIEW
vestigations by the authors would look at adapting such a system to a real motorcycle
and evaluating the effects experimentally.
The use of computer-aided methods for motorbike handling and stability analysis was
further investigated by Styles [96]. Particular emphasis was placed on the steer torque,
and to identification of the important
contributors to this torque. Important
design
parameters for the motorcycle were varied and the effects upon the steer torques reexamined. The important contributors were found to be the moment arising from the
normal tyre force, the gyroscopic torque in roll and the aligning moment resulting
from the lateral force through mechanical trail.
A method of optimal preview was also investigated, with the aim of assisting future
motorcycle development. Manoeuvre simulations were again run and parameter variations made. The stability was analysed by observing the eigenvalues resulting from
the state space matrices used in the modelling.
A driver control strategy was also proposed by Antos and Ambr6sio [1] for a twintrack vehicle. The vehicle model, with some simplifications made, was combined with
a multivariable bilinear control methodology presented elsewhere. The controller used
an optimal control strategy, without preview, to force the vehicle to follow a predetermined 'ideal' path, with information on this path input by means of coordinates,
section lengths and section curvatures. Essentially, current vehicle position and states
relative to the ideal path were combined with an optimal controller that applied a
steering control to the vehicle front wheels and rotational torques to all four wheels.
The weightings for the controller cost function were user-defined and close path following was achieved, although the lack of any form of visual preview would seem to
be in contradiction
with the driving tasks employed by a human driver.
Modelling of the control strategies for riding a motorcycle was investigated by Huyge
et al. [37] with the use of a multibody motorcycle model coupled with a separate
model to replicate the rider's control actions, in this case an applied handlebar torque.
The biomechanical information for the rider was included in the multibody motorcycle
model, though the movement of the rider's body mass was not used as a direct control
input to the motorcycle as had been done by other authors previously. The strategy
adopted in regard of the rider's control used a target roll angle to be input to the
model, which corresponded to a particular turn radius for a specific motorcycle at a
27
CHAPTER 2: LITERATURE REVIEW
given speed. The controller of the motorcycle applied a steer torque to the handlebars
in order to generate the required roll angle and to correct if the target roll angle
was not being met. The presented example of an S-bend manoeuvre showed good
path following results.
However, it is known that a rider will countersteer prior to
commencing a turn, and so will necessarily need to employ some form of preview to
initiate the turn. While countersteer is shown in the model, it was not in the form of a
preview control to initiate the turn, but appears instead to be applied in a regulatory
manner to adopt the target roll angle only once the turn has begun.
Building on earlier works on driver preview control [98], Sharp [94] adapted the optimal control technique to the application of a motorcycle rider's control actions. As
with the driver model, the optimal control element related to a visual preview of the
road, for which the controller calculated a series of gain values in order to minimise a
combined cost of lateral path position error and steer torque input. The control input
parameters available to the rider were steer torque applied to the handlebars and an
upper body lean torque. The path following performance was seen to be very successful, and with the results suggesting, in agreement with earlier work by Katayama [43]
in particular,
that the primary control method was through steer torque and with
the influence of rider lean torque an order of magnitude less important than the steer
torque. By calculating a series of gain values against the previewed road path, this
effectively allowed the relative importance of the previewed road to the task of path
following to be assessed. In agreement with Land and Horwood [47], the near preview
information was seen to be more important
in regard of maintaining the position of
the motorcycle relative to the road.
Other interesting
areas of research into rider control that have recently appeared
include the use of predictive control. The mechanics of the method are not dissimilar
to an optimal control technique, which has been applied in a number of ways to assess
both motorcycle performance [4, 15] and driver/rider
performance [51,94]. However,
unlike optimal control, predictive control generates a set of anticipated future states,
and by the comparison of predicted future states with target future states, a control
input is generated that attempts to find a balance between minimising the state errors
and the control input required.
Prokop and Cole, the latter with his co-workers, separately developed predictive control driver models. Prokop [72] developed an extensive application of this technique
28
CHAPTER 2: LITERATURE REVIEW
in modelling the control actions of a twin-track vehicle driver, allowing control of
steer angle, throttle and braking inputs. Using the plant of the vehicle dynamics, the
output several seconds ahead of the vehicle could be predicted and coupled with a
PID controller in order to guide the system to the targets set by the nature of the
road. A number of simulated manoeuvres were compared with experimental manoeuvres, concluding that the method was conceptually capable of replicating a driver's
control actions.
The controller developed by Cole et al.
[11] was a more representative application of
the complete model predictive control technique. As with Prokop's model, a prediction of future output states was made, but in this case the control input determined by
minimisation of a quadratic cost function, in a similar manner to the optimal control
approaches.
The investigations showed encouraging similarity to the conceptually
accurate optimal control method, but limited path-tracking
results were available
against which to judge its applicability to the task of modelling vehicular control.
2.5
Summary
The literature review has covered the wide range of research areas with relevance to
motorcycles and motorcycle riders. These have been broadly defined as the areas of
motorcycle stability characteristics,
tyre modelling and rider control modelling.
The first of these, motorcycle stability analysis, has been extensively studied by a
number of researchers over many years. A great deal of the early simulation models
have been incorporated
into commercial software packages, such that results can be
achieved relatively easily via these programs. Extensive work has been conducted into
developing ever more complex models to represent further structural
characteristics
of a motorcycle, to the point where there is limited scope to establish any novel yet
constructive avenues of research.
Tyre modelling continues to be an important area for research. The tyre is the principal provider of external forcing to the motorcycle and hence primarily affects its
behaviour.
The "Magic Formula" is widely regarded as being capable of success-
fully generating realistic tyre forces and moments, and without access to tyre testing
apparatus,
the scope for research work into motorcycle tyre behaviour is restricted.
29
CHAPTER 2: LITERATURE REVIEW
Since high quality dynamic representations
of motorcycles are now widely understood
and available, and the rider plays such a significant part in this response, the modelling
of the dynamic behaviour of the combined motorcycle-rider
becomes the next goal.
In recent years a number of control strategies have been employed as a means of
replicating the control actions of the rider. While some of these proved successful in
being capable of applying suitable control to a dynamic model of a motorcycle, the
question should be asked whether the goal is to generate an appropriate control input
or to replicate the control strategy employed by a human rider. In the latter case,
this must include the consideration of how the rider interprets information available
to him, and how this information is used in performing his control task.
To date, there are few controller models that have been applied specifically to replicating the control actions of a motorcycle rider, and those that have been demonstrated appear worthy of more detailed investigations.
The objective of this thesis
will therefore be to replicate the contemporary rider control models and conduct more
extensive parameter studies to establish particular strengths or weaknesses of these
methods. In particular, the principles of model predictive control strategies, hitherto
not applied to the modelling of a motorcycle rider, appear to provide the necessary
elements for replicating the complex actions of a motorcycle rider between problem
interpretation
and control solution.
2.5.1 Objectives and Thesis Outline
In light of the findings outlined in the literature,
the goal of this thesis will be to
developed the concept of model predictive control for the specific application of a
producing a motorcycle rider control model. The motivations for this work are covered
in Chapter 1.
The starting point will be the generation of an appropriate basis upon which to assess
the control techniques. Chapters 3 and 4 will therefore detail the specifics of modelling
the motorcycle dynamics and the rider's preview respectively. Combined, these two
elements will form the motorcycle-rider
model that will form the basis of the more
detailed rider control modelling.
The rider modelling will consider the application
of two specific control theories.
Chapter 5 will cover the use of optimal control for replicating the rider's control
30
CHAPTER 2: LITERATURE REVIEW
actions. Both the theory and application will be covered, with an extensive parameter
study conducted to evaluate the performance. Subsequently, Chapter 6 will consider
the application of predictive control techniques for the same aim, again with detailed
theory and parameter studies conducted.
A cautionary
note to the reader is needed here.
Both optimal control and model
predictive control techniques can be considered 'optimal' approaches, since they both
aim to minimise a cost function to provide the theoretically
input.
best possible control
However, the distinction in this thesis is made between optimal control and
predictive control techniques, which will be more clearly defined by the theory in the
relevant chapters (Chapters 5, 6).
For both control techniques, extensive parameter studies, to understand to a suitable
extent the behaviour of the control system, will be conducted to assess the applicability of the approaches to the task.
The aim is that this will result in a suitable
control strategy for replicating the actions of the rider, with good potential for future
development and application.
A feature of predictive control that is considered to
be suitable for the application here is the ability to include constraints on the modelling, which can therefore be used to represent limits in the available road width, the
physical limits of steering lock, and any other limitations pertinent to the task.
The two techniques will then be compared and contrasted in Chapter 7, with final
conclusions drawn in Chapter 8.
Chapter 3
Motorcycle Modelling
3.1
Introduction
In order to generate a motorcycle-rider
simulation
tool, an appropriate
dynamic
model of the motorcycle is required, enabling the response of the motorcycle to internal and external forces to be determined.
The response of the dynamic model must
correctly replicate the response of the real motorcycle to these forces if the combined
controller model is to be assessed correctly. Without a correct dynamic model, any
conclusions drawn about the performance of the controller model may be meaningless.
A motorcycle is a complicated piece of machinery, consisting of many thousands of
parts, each with its own physical properties.
constrained
Many of these parts are not completely
within the motorcycle as a whole, having freedoms to rotate, translate
or a combination of both. The dynamics of the engine and driveline are a complex
system, and every structural
component on the motorcycle has associated with it
some mass, damping and stiffness properties.
In order to develop a simulation tool for the dynamics of a modern motorcycle, the
system must be simplified considerably in order to capture the fundamental physical
characteristics of the motorcycle with sufficient computational
efficiency. The purpose
of the research work presented in this thesis is to develop a controller representative
of a human rider, and so provided that the dynamic motorcycle model employed
replicates the fundamental responses of a motorcycle to forcing, then detailed dynamic
models to capture intricate dynamic characteristics
31
are not considered necessary, and
CHAPTER
3: MOTORCYCLE
32
MODELLING
the use of a simplified model can therefore be justified.
be employed to generate the appropriate
Computer software can
equations of motion, which provides the
dynamic response of a generic motorcycle model to the dominant internal and external
forces.
This chapter will therefore detail the processes and techniques used in order to generate the dynamic response of a simplified motorcycle model. The model itself was
generated through direct mathematical
in [80j. The mathematical
methods and based on the much-cited model
procedure will initially be presented, and the resulting
model subsequently validated against [80j. A more detailed tyre model than in [80j
will be presented, validated and implemented to the motorcycle model, resulting in
a model more capable of accurately replicating the motorcycle responses at high tyre
slip and camber angles. This model will be used as the motorcycle model to which the
rider control strategy will be applied, to enable a path following task to be simulated.
The combined motorcycle-rider
simulation model using a novel control strategy will
form the goal of this thesis, and so this dynamic response model will form the platform on which the control model will be evaluated. The novelty of the work is largely
restricted
to the control modelling of the rider; the modelling of the motorcycle's
dynamics is a tool on which the control model will be applied.
3.2
Coordinate System
Before performing any type of dynamic modelling, it is essential to define a clear
coordinate
system that is to be used, and ensure that all dimensions, forces and
velocities conform to those coordinates.
arbitrary,
The choice of coordinate system can be
provided that this coordinate system remains consistent throughout
the
modelling.
There are, however, conventions that are commonly used, and for vehicle modelling
the typical coordinate systems used are the SAE (Society of Automotive Engineers) or
ISO (International
Standardization
Organization) systems. Both coordinate systems
consist of a set of three orthogonal axes, where, with respect to a vehicle, the zaxis is aligned with the direction of travel.
The SAE coordinate system has the
y-axis aligned positive right and the z-axis positive down (Figure 3.1), while the ISO
coordinate system aligns the axes positive left and positive upwards (Figure 3.2).
CHAPTER
3: MOTORCYCLE
33
MODELLING
In both coordinate systems the roll, pitch and yaw rotation directions are right-handrule rotations about the z-, y- and z-axes respectively.
For a global coordinate system, the axes remain fixed at the origin of the simulation,
while for a moving coordinate system simulation the coordinate system remains fixed
to and moves with the vehicle.
translate
longitudinally
ground-based
In a general moving coordinate system the axes
and laterally and yaw, roll etc. with the motorcycle.
For
vehicle modelling, the coordinate system rotates only with the yaw
rotation of the vehicle such that the z- and y-axes remain constrained to the ground
plane, even when the vehicle itself adopts a roll angle.
For the motorcycle model employed here, the SAE coordinate system is used. The
motorcycle's origin is defined by the position of the rear frame centre of mass projected
vertically down to the ground plane with the motorcycle upright. The z-, y- and zaxes form an orthogonal set of axes, with the z- and y- axes in the ground plane
and the z-axis perpendicular
to the ground plane projecting vertically down. The
z-axis projects ahead of the motorcycle, and the y-axis projects to the right of the
motorcycle. A yaw rotation 1jJ constitutes
a rotation of the entire motorcycle about
the vertical z-axis to define the yaw frame. A pitch angle 6 defines a rotation of the
motorcycle about it's y-axis, i.e. in the yaw frame, to define the pitch frame. A roll
angle
,p
constitutes
a rotation of the motorcycle about the z-axis of the motorcycle
in the pitch frame to define the roll frame. Finally, a rotation 8 of the front frame
about the steer axis of the motorcycle in the roll frame constitutes
the steer, and
hence defines the steer frame. The motorcycle model used is described in detail in
Section 3.4.
Positive lateral tyre forces are in the positive lateral direction for the motorcycle
model used here.
Hence, positive slip is in the negative lateral direction.
Thus,
positive tyre camber results in positive lateral forces.
3.3
Tyre Model
Initial modelling work employed a simple linear tyre model,
(3.1)
CHAPTER
3: MOTORCYCLE
34
MODELLING
where lateral forces were the sum of the product of sideslip angle a with tyre sideslip
stiffness Co, and the product of camber angle 'Y with camber stiffness C'Y' The
detailed definitions of sideslip, camber angles and other useful definitions can be
found in Appendix 1 of [68]. This simple tyre model was used to establish the early
motorcycle dynamic models, before a more advanced tyre model was introduced.
The lateral tyre forces generated by this simple tyre model for a range of slip ratios
and wheel camber angles of a front tyre are shown in Figure 3.3, for which the tyre
parameter values can be found in Appendix A.
The advanced tyre model that has been employed here is based on work by Meijaard
and Popov [60], where a non-linear model was developed to capture the behaviour of
the lateral and longitudinal forces and moments about the vertical axis of a motorcycle
tyre. The work was conducted using a commercial multibody simulation program [3].
Here, the approach is modelled using symbolic mathematical
coding [53]. For small
angles of sideslip, the response of a tyre can be represented quite well by a linear
model, but to capture the response for larger slip angles typically experienced at
higher speeds, a more advanced model is required. The tyre model itself is obtained
by a simplification of the "Magic Formula" by Pacejka [68]. This section will outline
the fundamental
steps applied to obtain the tyre model, and the validation of this
model against the original work by Meijaard and Popov.
3.3.1
Tyre Force Equations
Full details of the tyre model used can be found in [60]. Here, an overview of the
technique will be presented.
Tyre forces are generated by the combination of deformation of the tyre in the contact
patch and slip of the tyre's contact patch relative to the ground surface. Both result
in slip velocities between the wheel and the ground surface, and therefore the initial
steps are concerned with calculating the slip velocities of the wheel and tyre as a rigid
body in the lateral and longitudinal directions.
This is achieved through vectorial
definitions of the wheel centre and contact patch location, and knowledge of the
wheel spin velocity and wheel forward speed.
The vector corresponding to the wheel spin axis is obtained by rotating a unit vector in
CHAPTER
3: MOTORCYCLE
35
MODELLING
the positive lateral direction by the yaw, roll and, for the front wheel, fork inclination
and steer angle. The component of this vector in the z-direction is equivalent to the
sine of the camber angle, and thus the camber angle of the wheel is obtained by taking
the arcsine of this component.
The axial spin rate of the wheel is obtained by the division of the forward speed
with the nominal undeflected tyre radius. The motorcycle will be assumed to run at
a constant forward speed, and thus the axial spin rate is not required as a variable
here. This spin rate, being about an axis perpendicular
to the wheel plane, is then
translated via the rotations of yaw, roll and, if appropriate,
fork inclination and steer
to define the spin velocity of the wheel in the motorcycle's reference frame.
The tyre contact patch is defined by a single point in the contact patch area, the
exact position being the combination of several vectors. The starting point for the
definition of a generic tyre can be an arbitrary point in the ground plane. For the
more specific application here, the origin will be taken as the origin of the motorcycle
coordinate system. This origin is located at the intersection of a projection vertically
down from the rear frame's centre of mass with the ground plane when the motorcycle
is upright (Figure 3.4). First, the position of the wheel centre in this reference frame
is defined. A vector in the plane of the wheel projected down towards the contact
patch then defines the centre of the toriodal profile of the tyre, and finally, a vector
projection vertically down, with length equal to the toroidal radius of the tyre, defines
the undeflected tyre contact point.
The intersection of a vertical line through this undeflected contact point with the road
surface is used to define the deflected, actual contact point of the tyre with the road
surface. The division of the tyre's vertical stiffness with the vertical displacement of
the deflected contact point relative to the undeflected point can provide the vertical
force acting on the tyre.
For the application here, the tyre forces were calculated
from statics.
Tyre forces result from deformation of the tyre carcass as the tyre moves into the
contact patch area, coupled with the tyre's stiffness properties.
This distortion of
the tyre effectively results in a higher axial spin rate for a given forward wheel speed
than would be experienced for a solid wheel with the same rolling radius. The wheel
in the contact patch area can therefore appear to move relative to the road surface,
CHAPTER
3: MOTORCYCLE
36
MODELLING
leading to the definition of a tyre slip velocity.
This slip velocity can be obtained numerically by considering an increased effective
wheel radius, with the wheel rotating at the same angular velocity as for a solid wheel
with the actual tyre rolling radius. A new effective radius is therefore defined, being
part way between the undeflected and deflected tyre contact points.
The angular velocity of the wheel is obtained from the forward speed of the wheel
and the actual rolling radius of the tyre. Thus, use of this angular velocity with the
increased effective rolling radius leads to an increased effective contact patch speed
relative to the wheel centre.
The summation of this contact patch speed with the
speed of the wheel centre results in the effective slip velocity in the contact patch.
This process is done independently for the longitudinal and lateral tyre contact patch
velocities to independently control the behaviour of the tyre's lateral and longitudinal
force properties with slip. Dimensionless slip quantities are then obtained by the
division of the tyre contact point slip velocities with the velocity of the wheel centre
in the plane parallel to the ground plane.
The lateral and longitudinal tyre forces and tyre moment about the vertical axis are
obtained by a simplified version of the "Magic Formula". Specifically, the expressions
are [60]
(3.2a)
(3.2b)
(3.2c)
where C, is the ratio of the cornering stiffness to normal force, J.Lw is the generic
tyre friction coefficient,
feq
is equivalent to the ratio of camber stiffness to cornering
stiffness, Fn is the normal force,
longitudinal slip quantities,
Stot
Slat
and
Slang
are the dimensionless lateral and
is the dimensionless uni-directional
is the normal wheel spin, and finally
slip quantity,
Sn
Rw is the nominal wheel radius.
In this application, the longitudinal tyre force is not required, since the motorcycle
modelling will be conducted at a constant forward speed.
CHAPTER
3.3.2
3: MOTORCYCLE
37
MODELLING
Validation of Tyre Model
The tyre model, described in the preceding section, was coded using Maple symbolic
software [53] and validated against the original tyre model in [60], itself validated
against the full Magic Tyre Formula in [68]. The original validation of the model
considered the lateral tyre forces generated for a representative range of tyre camber
angles and sideslip ratios. All parameters for the motorcycle tyre in this application
were as in [60], allowing a direct comparison of the results to ensure the correct
implementation
of the approach, and can be found in Appendix A.
First, however, a reminder about sign conventions is required. The referenced paper
considers positive lateral slip to be in the positive lateral direction. The lateral force
generated by slip is directed so as to oppose the slip direction, and thus positive
lateral tyre force is in a direction opposite to positive slip, and hence in a direction
opposite to positive lateral displacement.
This convention is adopted temporarily to
validate the tyre model against the original paper.
The lateral tyre forces were obtained for a range of tyre camber angles from 0° to 50°,
typical of the operating range of a road-going motorcycle tyre, with the lateral slip
ratio in the range -0.3 to +0.3, and for the tyre parameter
values in Appendix A.
The lateral tyre forces and moments are presented in Figure 3.5, where the non-linear
response, characteristic
of pneumatic rubber tyre, is clearly apparent.
Considering first the tyre operating at zero camber angle, i.e. in the upright position,
it is seen that in the slip ninge -0.05 to +0.05 the response of the tyre lateral force
is close to linear. Models that employed a linear tyre model would therefore show
reasonable accuracy within these operating ranges. Outside this range, the ratio of
lateral tyre force to lateral slip diminishes as tyre force saturation
begins, at which
point a linear model would give rise to incorrect results, and at slips approaching
+/ -
0.3 for zero tyre camber angle the tyre is close to fully saturated
maximum lateral force obtained.
with the
The tyre moments about the vertical axis show
similar responses, with a small linear operating range about zero slip, in the range
approximately
-0.05 to +0.05.
The tyre moments peak somewhat earlier than for
the lateral forces, at lateral slips of approximately
decaying away as tyre saturation
+/ -
0.08, with the moments then
sets in with increasing lateral slip.
CHAPTER
3: MOTORCYCLE
38
MODELLING
As a motorcycle tyre is cambered, a lateral force is generated in the direction of the
tyre's lean, a characteristic known as camber thrust. The camber thrust force acts in
the direction in which the tyre is cambered such that a positive camber angle, in which
the top of the tyre is deflected in the positive lateral direction relative to the contact
patch, results in a lateral tyre force in the positive lateral direction.
It was noted
previously that positive slip is considered to be in the positive lateral displacement
direction, and thus positive lateral tyre force is in the negative lateral displacement
direction. Thus, a positive tyre camber should result in a negative lateral tyre force.
The effect of camber on the lateral forces generated by the tyre are again apparent
in Figure 3.5.
Considering first the case for zero lateral sideslip, it is seen that
as the camber of the tyre is increased, the tyre lateral forces become increasingly
negative, as expected, i.e the effect of camber is to generate a lateral tyre force in the
direction of the lean of the tyre. As the tyre camber increases, the cumulative gains in
lateral force with each additionallO°
of camber diminish. Expected lateral tyre force
properties can be found from experimental studies of tyres, carried out by both tyre
manufacturers and by research institutions.
Results from the former are well-guarded.
However, detailed information on typical tyre behaviour can be found in [68], while
other references of a broader nature include [13, 31]. Further combinations of sideslip
and camber give results that would be expected of a motorcycle tyre, showing the
combined effects of lateral force generation through sideslip and camber thrust that
have been outlined previously.
As a final point of interest, and to give justification
for the use of this advanced
tyre model, the lateral force generated by the simple and linear tyre models are
compared, here for the 00 camber angle condition of the front tyre (Figure 3.6) and
the 500 camber condition of the rear tyre (Figure 3.7). The responses of the linear
tyre model are seen to be a good approximation
to more realistic tyre forces over
only a narrow range of sideslip ratios, and outside this narrow range the tyre forces
generated by the linear tyre would not be representative
of a real motorcycle tyre.
This advanced tyre model, coded using Maple symbolic software [53] has therefore
been validated against the original model in [60], itself originally validated against
[68], giving confidence that the advanced tyre model employed here gives a realistic
representation
of a motorcycle tyre across a broad range of operating conditions for
lateral slip and tyre camber angle. This advanced tyre model was therefore employed
CHAPTER
3: MOTORCYCLE
39
MODELLING
in modelling the dynamic response of the motorcycle to control inputs subsequently
used in the motorcycle-rider
3.4
simulations.
Motorcycle Model
The modelling of the motorcycle itself is considered here.
A motorcycle is a ma-
chine formed from many smaller subassemblies and parts, each of which has its own
characteristic dynamic behaviour, and when combined, the response of the whole motorcycle is influenced by the interaction of all subcomponents
with each other. For
ultimate accuracy of the dynamic behaviour of the motorcycle, it would be necessary to consider the individual contributions
of all such subassemblies.
To perform
such a calculation would, however, require extensive numerical calculation, and thus
a simplified motorcycle model was employed here.
The fundamental
characteristic
response of the motorcycle is dictated by the inter-
action of the principal bodies of the motorcycle and their movement relative to each
other, and so in the interests of computational
efficiency, the motorcycle has been
modelled as a simplified four-bodied bicycle. This implies that the motorcycle is represented to consist of front frame, rear frame, front wheel and rear wheel, without
the inclusion of suspension freedoms. The front frame refers to the front forks, brake
calipers, handlebars and any fixtures that move with the steering, such as lighting assemblies or mudguards. The rear frame is taken to represent the combined structural
and geometric properties of the rear chassis structure,
engine and drivetrain assem-
bly, seat structure, fuel tank and any other rigidly attached fittings to this combined
structure.
The front and rear wheel bodies consist of all rotating wheel parts, namely
the wheel itself, the tyre and all rotating brake components.
This simplified motorcycle model is depicted in Figure 3.4, with a coordinate system as
described in Section 3.2. The front frame attaches to the rear frame via a revolute joint
inclined through the steering inclination angle
'T},
with rotational velocity restrained
by the use of a steering damper. The wheels similarly attach via revolute joints along
their spin axes. Suspension deflections are not accounted for in the model used here,
and hence motions of pitch and heave are not included.
In principle, some pitch
motion will result from the geometry of the tyres and steer system whenever the
steer angle is non-zero, and also through deformation of the tyres. In practice, the
CHAPTER
3: MOTORCYCLE
40
MODELLING
effects of this on the pitch angle of the motorcycle are small, and considered to have
a negligible impact upon the dynamic behaviour of the motorcycle. The wheels are
axi-symmetric, and thus it is not necessary to include the rotation angles of the wheels
as specific system states. The actual rotational angles of the wheels are unimportant;
only the angular velocities are required which can be readily obtained with knowledge
of the forward speed and rolling radius of the tyres. The non-linear characteristics
of
the tyre's lateral forces are obtained using the simplified "Magic Formula" approach
presented in [60], and described in Section 3.3.
The geometric and physical properties of the motorcycle are taken from [80]. The
values used for the motorcycle were typical for a contemporary motorcycle at the time
of publication of the referenced paper. The fundamental designs of motorcycles have
not changed dramatically since the publication of the cited paper, and hence use of the
data is still considered valid in modelling the response of a typical motorcycle, and the
model used here is therefore essentially an extension of the original motorcycle model
presented in [80]. Many more advanced motorcycle models have been developed in
the literature
particular.
[81], consisting of additional flexures of the motorcycle's structure
in
However, the object of the work presented in this thesis is concerned with
the modelling of the rider, and so the exact details of the motorcycle model used for
the simulation process are not a primary consideration, provided that the fundamental
response and primary modes of the motorcycle are captured.
The motorcycle model
developed in [80] has been used extensively for both academic and non-academic
applications, and can therefore justifiably be used as a reference model on which to
base the development of a rider control strategy that is the goal of this thesis.
3.4.1 Motorcycle Equations of Motion
This section will outline the basic theory that was used to obtain the simplified
motorcycle model's equations of motion.
An array of commercial computer programs are available that are capable of modelling
the dynamic response of physical systems. Popular codes include generic dynamics
programs such as MD Solutions [63] or Virtual.Lab
[99], and more specific programs,
for example CarSim [10] and BikeSim [5]. Such programs typically require the definition of the principal structural
characteristics
of the bodies of the systems and
CHAPTER
3: MOTORCYCLE
MODELLING
41
information regarding the coupling between the bodies. The programs build up the
complete structure from knowledge of the sub-bodies and their interactions to obtain
the dynamic response of the entire combined assembly.
The approach taken here develops the dynamic response of the motorcycle from fundamental mathematical principles. The equations of motion are developed symbolically,
using a symbolic software code Maple [53], and derived using Lagrange's theory. Fundamentally, the positions, motions and physical properties of the system's bodies are
defined, and then by consideration of the energy of the system as a function of the
system's degrees of freedom, the equations of motion are obtained.
These equations of motion are initially obtained symbolically, and are then exported
to Matlab [54] where the numerical responses and motion simulation of the motorcycle
are performed.
The reasons why the equations are initially developed symbolically
will become clear in subsequent sections of this thesis.
The theory of Lagrange can be broken down into a few fundamental
steps.
The
first step involves the definition of the physical properties of the motorcycle's main
bodies and the definition of the principal degrees of freedom allowing the energy of
the system, both kinetic and potential,
to be obtained.
on the system are then defined, and the instantaneous
The external forces acting
virtual power of these forces
obtained. The combination of internal energies and the external energies as a result of
external forces are then used to obtain the dynamic response of the entire motorcycle
assembly.
Theory
The Lagrangian method for deriving the equations of motion can be found in many
dynamics textbooks [64, 101]. The method involves calculating the kinetic and potential energy of a system and using these functions to generate the equations of motion.
Explicitly, for an unforced system the Lagrange's equations are defined by:
(3.3a)
where T is the kinetic energy of the system, U is the potential energy, and q is the
vector containing the degrees of freedom. Extending this to include the influence of
CHAPTER
3: MOTORCYCLE
42
MODELLING
external forcing on the system, the Lagrange's equations become
(3.3b)
where W is the sum of the virtual powers of any external forcing on the system.
Specifically, these powers can include aerodynamic
forces and tyre contact patch
forces.
The kinetic and potential energies of the motorcycle system are broken down into
the individual contributions
made by the front and rear frames and the front and
rear wheels. The kinetic energy comes from two sources: a translational
component
resulting from linear velocities and a rotational component from angular velocities of
the bodies. The generic linear and angular kinetic energies of an object can be defined
by !mbv2 in a single direction, where mb is the body mass and v is the velocity of
the body, and
!Jw2
for a single rotation direction, where I is the object's moment
of inertia and w is the angular velocity of the body. For the more general case of
translation
and rotations in a six degrees of freedom system (3 linear, 3 angular),
these expressions extend to !vTmbv, where v is the vector of linear velocities of a
body, and the rotational kinetic energy is defined by !wTlw where w is the vector of
angular velocities of a body and I is the inertia matrix of the body in three dimensions.
Since the body is moving in three-dimensional
space, the energy terms should account
for motion in all three dimensions.
The derivatives of the energy terms with respect to the degrees of freedom of the
motorcycle are obtained from the Jacobian matrix of the energies of the motorcycle in
all six directions (3 orthogonal linear directions and 3 orthogonal angular directions)
with the degrees of freedom of the motorcycle system.
In a similar way, the first term in the Lagrange's equations (3.3b) is obtained from
the time derivative of the Jacobian of the kinetic energy terms with respect to the
time derivatives of the motorcycle degrees of freedom.
The virtual powers result from the product of the magnitudes
of the forces and
moments, taken in the 3 principal orthogonal directions and the 3 principal orthogonal
rotations, with the velocities of the corresponding points of application of forces and
moments, again in the 3 principal directions and rotations. In this application, these
CHAPTER
3: MOTORCYCLE
43
MODELLING
are the tyre contact forces, acting at the tyre contact patches.
The motorcycle model considered here has five degrees of freedom, detailed in Section
3.4; longitudinal and lateral displacement of the motorcycle, yaw and roll rotations
of the motorcycle and steer rotation of the front frame relative to the rear frame.
Consequently, the application of Lagrange's equations results in five second-order
equations of motion.
Implementation
The fundamental
in Maple
principle of the Lagrange's equation has been outlined.
Further
details, if required, can readily be obtained from the referenced texts [64, 101]. Here,
the application of the theory using Maple symbolic software [53] will be outlined. To
fully detail every line of code would be both tedious to the reader and an uneconomical
use of space within this thesis, and so only the key inputs and steps required to obtain
the desired equations of motion are presented. While automatic modelling codes are
readily available, the equations of motion were developed in Maple using Lagrange's
theory explicitly in order to retain more complete control of the modelling detail.
This does not therefore represent any significant novelty to the literature; the model
developed will provide a tool for the subsequent work on rider control.
Fundamentally,
the coding of Lagrange's theory can be broken down into several
broad processes:
1. Define preliminary vectors and matrices necessary for the calculations.
2. Define the body positions and rotations, and hence linear and angular
velocities.
3. Calculate the total kinetic and potential energies.
4. Calculate all applied forces and velocities of points of application to obtain
total virtual power.
5. Form the Lagrange's equation.
6. Manipulate into the state space format and export to Matlab.
CHAPTER
3: MOTORCYCLE
44
MODELLING
The preliminary vectors and matrices included the definition of the state vector and
the definition of rotation matrices corresponding to the yaw, roll, steer and front
frame inclination angles. The vector q of the system degrees of freedom, consisting
of global longitudinal x and lateral position y, global yaw angle 1jJ, roll angle <p and
steer angle 0, together with their first and second derivatives were input as
q
:- <x, y, psi, phi, delta>;
q_dot
:- <dxdt, dydt, dpsidt, dphidt, ddeltadt>;
q_dot_dot
:. <d2xdt2, d2ydt2, d2psidt2,
d2phidt2,
d2del tadt2>;
(3.4)
It should be noted in this instance that the degrees of freedom are not specifically
defined as functions of time, even though they are taken to be so. Similarly, the
derivatives are not expressed as functions of time as far as the computer code is
concerned.
During the coding, it was at times necessary to take derivatives with
t, and sometimes with respect to the states, which are functions of
respect to time
time themselves.
Maple cannot perform differentiation
with respect to functions,
only with respect to variables, and so it became necessary during coding to change
definitions of variables between explicit functions of time and independent variables,
for instance substituting
a~t)
with dxdt and vice versa, so that the derivative of an
expression with respect to the time derivative of x(t), i.e. dxdt, could be taken.
Definitions of the rotation matrices concerning the yaw angle 1jJ, roll angle <p, front
frame inclination angle TJ and steer angle 0 are required.
These matrices were used
.
to orientate the coordinate system as necessary to define the positions of the bodies
.
and positions of key features of the motorcycle in the arbitrary displaced state. Using
as an example the matrix RI, corresponding to a yaw rotation, the matrix and its
transpose are input as
Ri
< sin(psi(t»
< 0
trJRi
I
:- «cos(psi(t»
I
0
I
1 »;
:- transpose(Ri);
I
-sin(psi(t»
cos(psi(t»
I
I0
0 >,
>,
(3.5)
CHAPTER
3: MOTORCYCLE
45
MODELLING
leading to
COS(1P(t»
R1
=
sin(1P(t»
[
tr_R1
-
-sin(1P(t»
0
cOS(1P(t»
o
0
0
1
cOS(1P(t»
sin(1P(t»
0
-sin(1P(t»
cOS(1P(t»
0
[
1
1
(3.6)
001
With the degrees of freedom of the motorcycle and the corresponding rotation matrices defined, the physical dimensions and orientations of the motorcycle assembly can
be fully defined in space. To define the position of a body within a moving coordinate
system it is necessary to fix a rotation convention. By changing the order of rotations
and translations, the final position of a body is changed significantly, as demonstrated
in Figure 3.8, and so a single convention must be used consistently. The convention
used here, in common with the general practice of vehicle dynamics, is yaw rotation
first, followed by pitch rotation then roll rotation. The front frame is then defined by
a rotation for the steer axis inclination followed by the steer rotation itself. For the
application here, suspension deflection of the motorcycle is not permitted,
the small
influence of the steering geometry on the pitch of the motorcycle has been neglected,
and hence no pitching motion of the motorcycle is considered.
The following code example defines the location of the front frame mass centre. The
starting point is the origin of the global coordinate system. In the displaced coordinate
system the location of the front frame mass centre can be defined by [32]:
G1 :-
<x(t),
y(t),
+ R1.«O,
0>
rt*tan(phi(t»,
+ R2.«c1,
-rt>
(3.7)
0, -(h1-rt»
+ (Reta.R3.<e,
0, 0»»;
In this expression the first vector defines the position of the motorcycle's origin within
the global coordinate system (Point a, Figure 3.9). In the second expression, the coordinate system is rotated by the yaw angle using the vector R1, and in this intermediate
coordinate system the intermediate
origin is displaced to the centre of the toroidal
CHAPTER
3: MOTORCYCLE
46
MODELLING
tyre profile (Point b, Figure 3.9). The third expression rotates the coordinate system
by the roll angle (matrix R2), in which a translation
moves the coordinate system
to the front frame steering joint (Point c, Figure 3.9). Finally, the last expression
rotates by the front frame inclination angle (matrix Reta), then by the steer angle of
the front frame (matrix R3), followed finally by the location of the front frame mass
centre relative to the front frame steering joint (Point d, Figure 3.9).
In order to calculate the kinetic energy of the motorcycle, it is necessary to know
the velocities of the centres of mass. Having defined the locations in space of these
points, defined now as specific functions of time, it is a simple procedure to obtain the
velocities of the points. These are simply the time-derivatives of the position vectors,
and can be readily obtained using the derivative function diff
Gl_dot
:-
in Maple:
map(diff,Gl.t);
(3.8)
In a similar manner, the angular velocities of the bodies are calculated. The individual
angular velocities resulting from yaw, roll and steer were defined earlier, and so the
angular velocity of the bodies is simply the combined effect of all these rotations,
paying specific attention to the order in which the rotations are applied. Using again
the front frame as an example, the angular velocity may be defined as:
omega_l
:=
evalm( Steer...Rate
+ ( tr...R3. tr...Reta.Roll...Rate
(3.9)
+ ( tr...R3.tr-Reta.tr-R2.Yaw..Rate»)
Similar expressions for the rear frame and the wheels are derived. It should be noted
here that the angular velocities of the wheels will be the combination of rotations due
to yaw and roll, plus the contribution
of their spin velocities about their spin axes.
The energy expressions necessary in the derivation of the Lagrange's equation can now
be obtained. The kinetic energies of the bodies are comprised of the individual bodies
translational
and rotational energies, calculated by !vT mbv and !wTIw respectively.
CHAPTER
3: MOTORCYCLE
47
MODELLING
For the front frame, this is coded with
T1_Linear := evalm«1/2)*(m1*transpose(G1_dot).G1_dot»;
T1JRotational
:- evalm«1/2)*(transpose(omega_1).I1.omega_1»;
(3.10)
where m1 is the mass of the front frame. Similar terms express the kinetic energies of
the rear frame, and the rotational kinetic energy of the wheels.
With regard to the angular velocities of the front and rear wheels, as explained in [80],
the kinetic energy of the wheels needs to be included since they form a substantial
part of the overall vehicle mass. The masses of the front and rear frames includes the
wheels, and so their contribution to the energies of the system by their linear motion
and rotation with the frames is accounted for. However, the wheels have additional
energy due to rotations about their spin axes, and so this must also be accounted for.
Thus, the rotational kinetic energy for the wheels alone, assuming no spin relative to
the frames, is calculated.
The rotational kinetic energy with the spin included was
then calculated and subtraction of the former from the later leaves only the rotational
kinetic energy contribution
from the wheel spin speed, which is then added to the
rotational kinetic energy of the appropriate frame.
The potential energy of the motorcycle is calculated next, and is simply the summation of the front and rear energies, with the wheel masses included in the frame
masses. This is obtained from the mass multiplied by gravitational constant 9 and by
the height above ground, which is simply the z-axis component of the body position
vector in three dimensional space:
Potential
:- < m1*g*<O I 0 I -1>.G1 + m2*g*<O I 0 I -1>.G2 >;
(3.11)
To form the necessary components of the Lagrange's equations (3.3b), the energy
terms must be differentiated with respect to the state vector q. Again this can be
achieved by taking the Jacobian matrix which can be readily implemented in Maple.
At this point it is necessary to remove any specific definition of variables as functions
of time, such that the energy expressions can be differentiated with respect to the
CHAPTER
3: MOTORCYCLE
48
MODELLING
states. Applying the jacobian command, the necessary derivative expression of the
kinetic energy is obtained:
dT_dq :-jacobian(Kinetic.
(3.12)
q);
where q is the vector of the degrees of freedom, as seen earlier.
The third term
in (3.3b) is similarly obtained by taking the Jacobian matrix of the potential energy
expression with respect to the state vector.
The first term in (3.3b) requires subtle manipulation
of the expressions. As before,
the Jacobian matrix of the kinetic energy term, with specific references to functions of
time replaced, is taken with respect now to the time-derivative of the state vector. To
complete the first term, the time derivative of this new expression must be obtained.
However, at this point the terms in the expression are not specific functions of time,
and so the time derivative of this expression would result in a zero matrix. Thus, the
specific time-dependencies
of the terms must first be reinstated,
and once done the
time derivative of this new expression can then be sought.
There is also one additional term which is included at this point concerning the
Rayleigh dissipation energy of the steering damper.
~Co(a~t))2
This energy is calculated from
where Co is the steer damping coefficient, and again the jacobian func-
tion is used to evaluate the derivatives of this energy with respect to the system
degrees of freedom:
dR_dqdot :-
jacobian(Rayleigh.Dissipation.
q_dot);
(3.13)
The Rayleigh dissipation term is added to the left hand side of equation (3.3b) to
obtain the unforced system dynamic response. To complete the modelling, the effects
of external forcing on the motorcycle must be accounted for. The virtual power is
calculated by the multiplication of the forces by the virtual velocities of the points of
application of the force, both linearly and rotationally
as appropriate.
The positions
of the points of application are therefore calculated first, in a process very similar to
that used to establish the mass centres of the bodies of the motorcycle (3.7). The
variables within these expressions are given as functions of time, enabling the linear
velocities of the points to be once again obtained by simply taking the derivative of
CHAPTER
3: MOTORCYCLE
49
MODELLING
the expression with respect to time. The rotations of the points and hence the angular
velocities are obtained in a similar process, and combined with the linear velocities to
form 6 x 1 velocity vectors of the linear and angular velocities, where the order of the
elements is the
X-,
y- and a-lateral velocities, and
X-,
y- and z-axis angular velocities.
The last step in calculating the virtual power is to create a vector of the actual
forces and moments acting at the point, with the order of these forces and moments
corresponding to the order of the velocities in the 6 x 1 velocity vector. In the case
of the model used here, this refers only to the external tyre contact patch forces.
The virtual power is then simply the product of the force and moment vector with
the linear and angular velocity vector. By multiplying the velocity vector with the
transpose of the force and moment vector, each velocity element is multiplied by the
corresponding force/moment element to obtain the single total virtual power for that
point. If this point corresponded to the front tyre contact patch, for instance, the
same process would also be required for the velocities, forces, moments and resulting
virtual power of the rear contact patch.
Summating all the virtual power terms relating to the external forces then allows their
inclusion in (3.3b) and hence the formation of the equations of motion for the forced
dynamic response of the motorcycle. This equation will be in terms of all degrees of
freedom and the associated derivatives.
State Space formulation
Thus far, the equations of motion for the motorcycle are expressed as a 5 x 1 vector of second-order symbolic equations where each element corresponds to each of
the degrees of freedom, and also as a 2 x 1 vector consisting of two first-order equations relating to the dynamic forces of the front and rear tyres. For the subsequent
numerical operations to model the response of the motorcycle, these equations will
be formed into a combined state space representation.
The five second-order equa-
tions will therefore be formed into ten first-order equations, combined with the two
first-order tyre equations to result in twelve first-order equations of motion.
CHAPTER
3: MOTORCYCLE
Fundamentally,
50
MODELLING
the form of a second-order equation of motion for a generic single
degree of freedom system is
(3.14)
which, when extended to the multiple degrees of freedom case, becomes
(3.15)
where the matrices Mb, Cb and Kb are the mass, damping and stiffness matrices of
the dynamic system response, the vector q is the vector of generalised coordinates,
and matrix Ju is the matrix relating the system inputs vector u to the generalised
coordinates of the system.
The first-order state space model is obtained by defining a new state vector, consisting
of the states and their first derivatives:
x=
Making this substitution,
[:
1
(3.16)
the second order system (3.15) is converted to a first-order
system by forming the standard state space model:
(3.17)
For the problem here, rather than forming individually the matrices Mb, Cb and Kb,
the Jacobian matrix approach can be taken to obtain the equivalent of these more
readily. Thus the Jacobian matrix of the 4 x 1 vector of the equations of motion with
respect to the states vector [qT i{lT is taken, resulting in a Jacobian matrix, termed
Ja, equivalent in form to
[Kb Cbl. Similarly, the Jacobian matrix of the equations of
motion with respect to the input vector u can be taken, to give Ju•
To complete the formation of the state space matrix form, the matrix M, is required.
This is again readily obtained by taking the Jacobian matrix of the 5 x 1 equations
CHAPTER
3: MOTORCYCLE
51
MODELLING
of motion with respect to the second-order terms of the generalised coordinates,
q.
Multiplying the negative inverse of this matrix with the previous Jacobian results in
_M;lJ8,
a matrix form equivalent to [_M;lKb -M;lCb]. The complete state space
model can then be formed as
(3.18)
For the case here, the complete problem consists of the five second-order equations of
motion and two first-order equations relating to the tyre lateral force responses, and
these equations must be solved jointly.
The two additional states relate to the front and rear dynamic lateral tyre forces, and
are given as
(3.19)
f= [~]
Thus, when combined with the previous zero- and first-order terms of the generalised
coordinates, the extended state space vector of the motorcycle becomes
(3.20)
The state space model of the combined first- and second-order equations of motion
can be solved in the same manner as previously.
The Jacobian matrix JII of the
second-order equations of motion is taken with respect to the now extended state
vector
Xv,
and similarly the Jacobian matrix of the first-order tyre forces equations
of motion is taken with respect to
Xv
to give JI
CHAPTER
3: MOTORCYCLE
52
MODELLING
Thus, the complete state space representation
of the motorcycle model is achieved
with
05x7
[~]=
[
I5x5
s,
]
[ M;lJs
(3.21)
]
which is then equivalent to the continuous-time
dynamic response model of the mo-
torcycle,
(3.22)
where
Xv
is the vector of vehicle states,
Av
is the state space dynamic response matrix,
B, is the state space input response vector and u is the actual vehicle control input,
all in continuous-time.
By modelling the process in this symbolic way, the non-linearity
of motion is retained.
The linearisation
of the equations
only occurs once the numerical values of
the motorcycle's states are input, which, in subsequent motion simulation, will be
conducted at each discrete step of the simulation.
Thus, at each simulation step, a
new valid linear state space model will be obtained.
Obtaining the symbolic model using the steps outlined above was not entirely straightforward however. For the mass matrix Ms, the inverse is required, and due to the
complexity of the equations of motion, the expressions contains many thousands of
terms.
To find the symbolic inverse of the mass matrix is therefore an incredibly
demanding problem from a computational
perspective.
Consequently, it was found
necessary to apply some simplification procedures to assist with this problem.
Specifically, before forming the mass matrix Mb, a vector of the second order terms
from the equations of motion vector was extracted.
The expand operation in Maple
was used first, in order to expand any compound or embedded expressions.
combine operation was used next, with the trigonometric
The
extension trig used, to
combine together the trigonometric expressions and thus simplify the number of terms
within each element of the vector. The final operation consisted of a Taylor series
expansion, to the fourth-order,
to further reduce the size of the second-order ex-
CHAPTER
3: MOTORCYCLE
MODELLING
53
pressions, before the Jacobian operation was used on this new vector to obtain the
simplified mass matrix Mb. With the reduced size of the elements in the mass matrix,
obtaining the inverse of the matrix as required became a task that was achievable
within the computational
abilities of the software [53] and personal computer used.
Finally, the necessary expressions are exported to Matlab [54], where the numerical
operations were conducted to calculate the system's response and hence validate the
model generated.
Within Matlab, the numerical values for the states, which vary
during the simulation, were substituted.
As the values would be constantly changing,
then obtaining the state space equations symbolically meant that the complicated
operation of solving the Lagrange's equations was only required once. The numerical
forms could then readily be obtained by substitution
of the appropriate state values.
The differential equations resulting from the Lagrange's equations are of a continuous nature, having the form of (3.22).
certainly operate in a continuous-time
While a motorcycle rider would almost
manner, for the purpose of modelling what
could be a random road path a discrete-time approach brings useful simplification to
the modelling process. Thus, the symbolic continuous-time
exported to Matlab [54], the numeric values substituted,
state space matrices were
and the state space model
then converted to discrete-time using the function command c2dm built in to Matlab
to convert the state space model from continuous- to discrete-time, thus adopting the
form
(3.23)
where
x, (k) is the vector of vehicle states, A, (k) is the state space dynamic response
matrix, Bv(k)
is the state space input response vector and u(k) is the actual vehicle
control input, all at the
5,
kth
step.
The time-step set for this discretisation
was
to
therefore able to capture oscillations with frequencies up to 25 Hz. The objective
of the work was not to investigate the vibrational
characteristics
cle, but to develop a model on which to apply a control strategy.
of the motorcyThe omission of
higher-frequency modes brought about by this frequency threshold was not therefore
considered important
in the context of the work.
CHAPTER
3.4.2
3: MOTORCYCLE
54
MODELLING
Validation of the Equations of Motion
The model generated was based on a simplified motorcycle model using the application of well-proven principles of Lagrange's theory. However, the execution of these
principles, using symbolic software programming, leaves open the potential for errors
to creep in, both in the way that the computer code may deal with the mathematical procedures required, and from the possibility for programming
arisen during the model generation.
behaviour of these mathematical
errors to have
Thus it becomes necessary to ensure that the
models is consistent with expectations,
and that
any subsequent modelling work conducted will have an accurate base from which to
build.
The simplified motorcycle model used (Figure 3.4) was taken from [801. This work
identified the principal vibrational
modes of the motorcycle, showing both the fre-
quency and damping of these modes across the typical speed range of the motorcycle.
The starting point for the modelling was therefore to replicate the model in [80], employing initially the simple linear tyre model, as in the original work. To compare the
model here with the original model by Sharp, the stability modes of the motorcycle
were assessed by obtaining the damping and, where appropriate,
the frequencies of
the principal vibration modes, namely for the wobble, weave and capsize modes originally identified by Sharp. These damping ratios and frequencies are obtained from
the eigenvalues of Av in the state space equations of motion of the motorcycle.
The real parts of the eigenvalues, representing the damping ratios of the modes, are
presented in Figure 3.10. With the advances in computing power since the original
work of Sharp, the analysis of the modes over the full speed range with a high resolution can readily be obtained quickly using even a modest personal computer.
The
results in Figure 3.10 therefore cover the speed range from 2 m/s up to 50 m/s at a
resolution of 0.1 m/s, enabling the behaviour of the results to be examined in detail.
The response closely resembles the original results in [80], with the capsize, weave
and wobble modes readily identified. The root locus plot of the eigenvalues is shown
in Figure 3.11. This also replicates the results of Sharp's model, presented in [28].
The capsize mode appears mildly stable at low speeds; the real part of the eigenvalue
at 2 m/s is approximately
-3.5 and is non-oscillatory.
As the speed increases the
mode becomes marginally unstable at a speed of approximately
10 m/s and above,
CHAPTER
3: MOTORCYCLE
55
MODELLING
remaining of the order of 0.08 for speeds from 10 m/s and upwards, but is easily
stabilised by the rider's control.
The weave mode is mildly unstable at low speed, with real part 2.6 and imaginary
part of 1.3 rad/s,
The imaginary parts of the modes, indicative of the oscillation
frequencies, are shown in Figure 3.11. The weave mode becomes stable at approximately 6 mis, with increasing stability reaching a maximum at just over 15 m/s of
-6.6 with frequency of approximately
11.6 rad/s, before decaying gradually as the
speed increases, but remaining still stable.
The wobble mode is stable over the full speed range considered.
part of the eigenvalue corresponding
56.0 rad/s.
to this mode is -5.0
At 2 m/s the real
and with frequency of
As the speed increases, the real part reaches a minimum of -6.9 with
corresponding frequency of approximately
55.2 rad/s at 13 m/so The stability of the
mode decreases almost linearly as the speed increases further, with a stability margin
of -0.8 and frequency 57.9 rad/s at 50 m/so
Confirmation of these principal modes is obtained by components of the modal plots.
Figures 3.12, 3.13 and 3.14 show the magnitudes of the eigenvectors corresponding
to the three key modes of capsize, wobble and weave at a forward speed of 25 m/s
respectively.
At the forward speed of 25 mis, the capsize mode (Figure 3.12) is marginally stable
and non-oscillatory.
The mode results in small angles of roll and steer, and greater
levels of yaw angle as the motorcycle gradually veers off to one side from the straight
running as a result of this stability mode. Lateral tyre forces and lateral accelerations
are low.
The wobble mode (Figure 3.13) is associated with significant oscillation of the front
frame relative to the rear frame, while the rear frame remains relatively unaffected,
and the motorcycle continues along a straight path, resulting in significant front tyre
forces, steer angles and steer rates.
Any changes to the yaw and roll angles are
therefore small and with rear tyre forces low, features which are all apparent in the
modal plot.
The weave mode (Figure 3.14) consists of oscillations of both the front and rear
frames in anti-phase, leading to significant variation of both roll and yaw states as
CHAPTER
3: MOTORCYCLE
56
MODELLING
the motorcycle performs a slalom-like progress along a straight path, and significant
lateral forces from both the front and rear tyres.
The rear tyre force is naturally
larger due to the rear weight bias of the motorcycle. The variations in the motorcycle
states expected from this mode are apparent in the modal plot.
The results here all correspond very closely to the original results in [80], giving
confidence in the correct modelling of the motorcycle when employing the simple tyre
model.
Having confirmed the correct behaviour of the motorcycle model with the
simple tyre, the model was extended by the application of the advanced tyre model.
3.4.3
Validation of the Advanced Tyre Motorcycle Model
With confidence in the programming of the motorcycle model using the simple tyre
model and of the coding of the advanced tyre model, the two elements were combined
together.
Again, the stability of the model in the speed range 2 to 50 m/s was analysed by
considering the eigenvalues of the state space model of the motorcycle's equations of
motion, presented in Figure 3.15. The frequency response of the motorcycle model is
shown by the root locus plot (Figure 3.16). The identified modes are again confirmed
by modal analysis (Figures 3.17, 3.18, 3.19). It is apparent that the results of the
eigenvalue analysis differ from the results when the simple tyre model was employed,
however the basic traits of the modes are still identifiable and are not dissimilar.
The capsize mode is relatively unchanged with the introduction of the advanced tyre
model. The mode shows greater stability at lower speed, with an eigenvalue real part
of -4.4 at 2 mis, compared with -3.5 when employing the simple tyre model. Again,
the mode becomes mildly unstable at approximately
10 m/so
The wobble mode can again be identified, showing a similar trend to the simple tyre
model, but with notable differences. This mode was seen to be stable over the full
speed range when the simple tyre model was employed, but with the introduction
of the advanced tyre model it is unstable at low speeds, rapidly becoming stable as
the speed is increased, crossing into the stable range at approximately
becoming unstable again at approximately
6 mis, then
39 m/so Although the simple tyre model
wobble mode did not become unstable again over the speed range considered, it is
CHAPTER
3: MOTORCYCLE
MODELLING
57
apparent from the stability plot (Figure 3.10) that this seems likely to happen at
speeds not significantly greater than 50 m/so The frequency of this oscillatory mode
with the advanced tyre model ranges from 50.6 rad/s at 2 mis, to 49.9 rad/s at the
peak stability at 15 mis, to 54.0 rad/s at 50 m/so
The behaviour of the weave mode is not dissimilar when the advanced tyre model is
employed, though the changes to the stability of the mode are more dramatic over
the speed range than with the simple tyre model. The mode is again unstable at
low speed, with eigenvalue real part 2.6 and frequency 1.6 rad/s at 2 m/so
The
mode becomes stable at just over 5 mis, with peak stability of the mode at 12 mis,
with corresponding
frequency of 12.7 rad/s,
again to reach a stable value of approximately
of approximately
The stability of the mode decreases
-0.7 with a corresponding frequency
22.0 rad/s.
Comparing the capsize mode for the simple and advanced tyre models, it is seen that
the advanced tyre model shows slightly greater magnitude of the lateral state and
slightly lower roll angle state, suggesting slightly larger lateral tyre force generation
with roll.
Although the model now presented cannot directly be validated against previous literature, both the original motorcycle model with the simple tyre and the independent
advanced tyre model have been separately validated against expected behaviour. The
combination of the two would therefore also be expected to show correct representation of the real behaviour of a motorcycle.
3.5
Motorcycle Model Conclusions
This chapter has presented the mathematical
model of the motorcycle dynamics that
will subsequently be combined with the rider control model. The simplified motorcycle model and the theory used to generate the equations of motion has been presented.
The theory for both a simple and an advanced tyre model have also been shown, with
the motivation for the advanced model discussed.
Validation of the advanced tyre model was carried out against the original source of
the model [60) to confirm the correct implementation
of the procedure.
CHAPTER
3: MOTORCYCLE
MODELLING
58
The much-cited model from [80], employing the simple tyre model, was reproduced
and validated to ensure initially that the mathematical
procedure used was correct.
With confidence of the correct response, the validated advanced tyre model was incorporated
into the motorcycle model, and the response of the new hybrid model
analysed. The net effect of this analysis suggested that, while the dynamic response
of the motorcycle model was changed by the introduction of the advanced tyre model,
the response is fundamentally
changed only subtly from the simple tyre model ver-
sion. Thus, the motorcycle model generated here utilising the advanced tyre method
can justifiably be employed with confidence that the response of the model is suitably
representative
of a real motorcycle.
Subsequent chapters will present the theory of the rider model and the control strategies that will be employed for this task. The suitability of the control strategies will
be assessed by their ability to guide and control a motorcycle model, which will be
the model presented here. The results shown in this chapter give confidence of the
correct dynamic response of the model, and so the control actions to be generated
will therefore be based upon an accurate model of the motorcycle's behaviour, and
this control can be applied to the model in the knowledge that the subsequent response will be consistent with what may be expected for a real motorcycle and rider
combination.
CHAPTER
3.6
3: MOTORCYCLE
59
MODELLING
Figures
x
z
Figure 3.1: SAE coordinate system, motorcycle image from [34]
z
x
Figure 3.2: ISO coordinate system, motorcycle image from [34]
CHAPTER
3: MOTORCYCLE
60
MODELLING
-F,
-- •••
F, 0 Degrees
10 Degrees
F, 20 Degrees
F, 30 Degrees
F, 40 Degrees
F, 50 De rees
a
Figure 3.3: Normalised lateral tyre forces with slip ratio and wheel camber, simple
tyre model, parameter values as in Appendix A
Figure 3.4: Simplified 'bicycle' model of the motorcycle
CHAPTER
3: MOTORCYCLE
MODELLING
61
1.2
0.8
-0.2
-0.1
0.1
0.2
-- F, 0 Degrees
--F,
10 Degrees
F, 20 Degrees
F, 30 Degrees
0.3 - -F, 40 Degrees
... F, 50 Degrees
-M, ODe rees
-1.2
Figure 3.5: Normalised lateral tyre forces with slip ratio and wheel camber, advanced
tyre model, parameter values as in Appendix A
1.5
_- -----
----
_--
Figure 3.6: Normalised lateral tyre forces comparison, 0° wheel camber, front tyre,
where for small slip angles
Slat
and a are approximately equal
CHAPTER
3: MOTORCYCLE
~
'1 -0.3
...
-0.2
MODELLING
62
0.3
-0.1
Simple Tyre
Advanced
T re
L-_-'--___:'_';_:_'--'-~I
Figure 3.7: Normalised lateral tyre forces comparison, 500 wheel camber, rear tyre,
where for small slip angles
Slat
and a are approximately equal
Figure 3.8: Difference in final orientation followingtranslation, translation, rotation
(left), and translation, rotation, translation (right)
CHAPTER
3: MOTORCYCLE
63
MODELLING
z
Figure 3.9: Vectorial definition of front frame mass centre in displaced coordinates
"
2
0
-2
I
-4
8.
-6
'I
'il
G
0::
-8
\
\
\\
Capsize
\
-10
i
\\
-12
\
\
\
-14
-16
0
\
\
5
10
15
20
25
30
35
40
45
50
Forward Speed (m/a)
Figure 3.10: Real parts of the system matrix eigenvalues, simple tyre motorcycle
model
CHAPTER
3: MOTORCYCLE
64
MODELLING
60r--------r------~r_------._------_,--------._------_,
Wobble
c.../
40
20
~
.
1
~
0
-- •••-.-.-.-.-
••- ••- •••-.------------
Cl
'iii
I
.
-20
-40
_60L_
_L
-50
~L_
-30
-40
_L
~
~
~
~
o
-10
-20
10
Re{eigenvalues)
Figure 3.11: Root locus plot of the system matrix eigenvalues, simple tyre motorcycle
model
3.00E-03
...0...
Q)
>
e 2.00E-03
Q)
til
jjj
0
tI
e
Q)
c
0
Q.
1.00E-03
E
0
Q.
x
y
Yr
Y1
ax1()t
ayl()t
a\jJ/()t
a~/at.
Motorcycle States
Figure 3.12: Capsize mode, simple tyre. Eigenvalue = 0.079733
a61at.
CHAPTER
3: MOTORCYCLE
65
MODELLING
1.S0E-04
~
i
1.00E-04
c
CD
Cl
iii
OS
!l
c
CD
c
&.
S.OOE-OS
E
e
2.
O.OOE+OO +--,---,y
x
Yr
Yt
OX/fJI.
aylfJI.
aljJ/fJI.
a~/fJI.
a~/fJI.
Motorcycle States
Figure 3.13: Wobble mode, simple tyre. Eigenvalue
=
-5.2281 ± 54.061i
1.00E-04
~0
...
u
7.S0E-OS
~
c
CD
Cl
W
0
S.OOE-OS
!l
c
CD
c
0
Q.
E 2.S0E-OS
0
2.
c.....,
O.OOE+OO
x
Y
ljJ
~
~
Yr
Yt
aylfJI.
axlfJI.
aljJ/fJI.
a~/fJI.
~/fJI.
Motorcycle States
Figure 3.14: Weave mode, simple tyre. Eigenvalue
=
-2.8095 ± 18.067i
CHAPTER
3: MOTORCYCLE
66
MODELLING
4
2
0
-2
-4
I
18. -6
'II
~
0::
-8
-10
-12
-14
-16
0
5
10
15
35
25
30
Forward Speed (mJa)
20
40
50
45
Figure 3.15: Real parts of the system matrix eigenvalues, advanced tyre motorcycle
model
~~------r-------~------~-------r-------r-------,
Wobble
40
0-~
20
I
1 O~------------------~~-.---I
,
~
~
\
-zc
'-----
.__
-..,
~... ..-::110 ...., __
" ......
~50~------~--------3~0--------~2~O--------~10--------0L-------~10
Re(elgenvaluea)
Figure 3.16: Root locus plot of the system matrix eigenvalues, advanced tyre motorcycle model
CHAPTER
3: MOTORCYCLE
67
MODELLING
3.00E-03
..
1:"
0
u
~
c
2.00E-03
CD
Cl
m
'0
sc
CD
c
0
Q.
1.00E-03
E
0
Q.
O.OOE+OO +---~
x
Yr
y
Yfax/at.
aylat.
aljl/at.
~/at.
a~/at.
Motorcycle States
Figure 3.17: Capsize mode, advanced tyre. Eigenvalue = 0.060997
1.S0E-04
1:"
su
~
c
1.00E-04
CD
Cl
...
jjj
0
!l
e
CD
c
0
Q.
S.OOE-OS
E
0
Q.
O.OOE+OO +--,--r-
x
y
Motorcycle States
Figure 3.18: Wobble mode, advanced tyre. Eigenvalue = -2.6314 ± 49.419i
CHAPTER
3: MOTORCYCLE
68
MODELLING
1.00E-04
~
~GI
7.50E-05
>
C
GI
Cl)
...
iii
0
5.00E-05
II
c
GI
C
0
Q,
E
0
2-
2.50E-05
.__,
L..,-
O.OOE+OO
x
y
IjJ
<p
15
Yr
Yt
axJat
Oylat
aljJl at
a<plat
O15lat
Motorcycle States
Figure 3.19: Weave mode, advanced tyre. Eigenvalue = -1.0804 ± 19.934i
Chapter 4
Rider Preview
4.1
Introduction
A human rider operates by observing the road ahead and taking in information about
the motorcycle's condition.
Knowledge of the road information gives him a path to
follow, and knowledge of the motorcycle's condition then informs him of his position
relative to the intended path, the lean angle, steer angle, yaw angle and associated
rates.
This information is used to decide what control input to apply in order to
achieve the task of following the intended path and control the motorcycle.
It is therefore logical that a simulation controller should replicate this process and
thus take in information pertaining to both the road information and the motorcycle
states. Thus, the combined motorcycle-rider
simulation model is formed to combine
the dynamic response model of the motorcycle to internal and external forcing with
knowledge of the approaching road path, updated with progress along the road.
A fundamental consideration when undertaking the modelling of any dynamic system
of bodies is the selection of an appropriate coordinate system. In principle, the choice
of coordinate
system should have no effect on the outcome of a system response,
provided that dimensions, forces and motions of the bodies are modelled correctly
and consistently
within the selected coordinate
system.
With careful selection of
a coordinate system, however, the dynamic modelling can, in some cases, be made
simpler and more appropriate
to the system being represented.
69
CHAPTER 4: ruDER PREVIEW
70
For the case of modelling vehicle drivers, the choice of coordinate system is an interesting question. On the one hand, the task being undertaken by a simulated driver is
to follow a chosen road path. The road path is a fixed, global feature, and so it may
seem appropriate to model the motorcycle and road preview using global coordinates.
This will make the definition of the road path simple and straightforward,
and at any
time during a simulation the position of the vehicle relative to the globally fixed road
path, and thus the performance of the driver control model, can be determined easily.
On the other hand, the task is to simulate the driver's actions.
within the global coordinates,
The driver moves
without necessarily having knowledge of the layout
of the road path in global coordinates.
It may therefore seem more appropriate
to
consider the task from the driver's moving perspective and thus consider the motorcycle states and road preview path relative to the driver at all times. This becomes
a more complicated approach, since throughout
the simulation the road information
interpreted by the driver must be continuously redefined into the moving coordinate
system, and in addition the global position of the vehicle must be obtained in order
to assess the performance of the controller in achieving the overall global path following task. Although the latter approach appears the more complicated, it intuitively
seems the more correct method of modelling a vehicle driver.
This chapter will outline the steps necessary to generate a discrete-time state space
vehicle model with road preview operating in both a fixed, global coordinate system,
and a moving, vehicle-fixed local coordinate system.
It will be seen in subsequent
analysis of controller methods that the local approach can, in certain circumstances,
be necessary for correct performance of the controller.
4.2
4.2.1
Road Preview Shift Register
Global Coordinates Preview
Initially, the rider model operating in global coordinates is considered. This method
is the relatively more simple approach and will outline the fundamental
principles
of the shift-register process used to update the rider's preview information, initially
presented in [98] and also later employed in [94].
CHAPTER
4: RlDER PREVIEW
71
In driving a vehicle or riding a motorcycle, a driver will base his control actions on
knowledge of the vehicle conditions and the requirements
of the task, in this case
following the road path he can see. It is therefore logical that the system state vector
z(k) should comprise information pertaining to both the vehicle states and the road
states. The vehicle state vector xv(k) is therefore augmented by the addition of the
vector of road preview information,
yr(k).
[98] and [94] included only the lateral
deviation of the road path, calculating the yaw angles trigonometrically.
Here, the
road preview consists of both the lateral position and yaw angle of previewed discrete
road information points in global coordinates, such that
z(k) = [ xv(k)
Yr(k)
1
(4.1a)
with
(4.1b)
where Yr, (k) and 1/Jr,(k) are the lateral positions and yaw angles of the ith road preview
point at the kth simulation iteration step (Figure 4.1). The states in the vehicle state
vector
Xv
are here defined in global coordinates.
The rider's road information is therefore stored as lateral positions of the road and
the yaw angle of the road at discrete points along the path, and at the start of the
simulation the first Np road information points are loaded, where Np is the number
of preview points selected.
As the motorcycle progresses along the road in the simulated motion, the road information must be updated in light of the rider's new viewpoint relative to the road,
and the new information that has come into his limited preview horizon at the forward limit of this horizon. The spacing of the discrete road points in the initial road
preview vector Yr is such that following one iteration step the road information point
Yrl
is the road information point
previous 1/Jrl becomes the new
Yrl
at the previous iteration step, and likewise the
v«. as depicted
the discrete points of the road path.
in Figure 4.2. Here, the dots represent
Filled dots imply that these road points are
stored in the rider model's road preview information vector Yr' At each step, as the
road preview is updated the 'old' values of Yrl and 1/Jrl are discarded, such that at
any simulation iteration step k the rider model has only the preview information for
CHAPTER 4: ruDER PREVIEW
72
the Np road points ahead of him at that step.
A simple shift-register matrix can therefore be used that will perform this change in
the vector Yr, such that all the road information is moved up by one position in the
vector Yr with each successive discrete iteration step of the motorcycle simulation. At
each step, the previous road information Yrl and
v-. are discarded,
and the new road
information point YrNp and 1/JrNp must be introduced to the vector Yr' The process is
represented by a discrete-time state space expression, formed as
+ 1) = ApYr(k) + BpYrn(k)
Yr(k
(4.2a)
where
Ap=
0
1m
0
0
0
0
0
1m
0
0
0
0
0
1m
0
0
0
0
0
0
0 0 0 0
1m
0 0 0 0
0
(4.2b)
(NpmxN"m)
0
0
0
(4.2c)
Bp=
0
(Npmxm)
Yrn =
YrNP+l(k)"]
[
(4.2d)
1/JrNp+l (k)
and m is the number of parameters in each discrete road preview point, in this case 2.
The updating of the road preview points in a global coordinate system is a relatively
straight-forward
process, since the numerical values of the global coordinates do not
change with the motion of the motorcycle, only their location within the road preview
information vector will need changing.
CHAPTER 4: R1DER PREVIEW
73
Thus, both the motorcycle dynamics model (3.23) and the rider's road preview information (4.2) are now represented by simple discrete-time state space models. In
line with the aim to form a single combined motorcycle-preview
representation
of the
riding task, these can now be readily formed into a simple combined rider-preview
state space model having the following structure:
xv(k
[ yr(k
1
+ 1) =
+ 1)
[AV(k)
0
1
[XV(k) 1 + [ Bv(k) 1 u(k) + [
1
Yrn(k)
Ap
Yr(k)
0
Bp
0
0
(4.3)
The states of a combined motorcycle-rider
model should include both the information
the rider has of the vehicle conditions and also of the road conditions, and the combination therefore of both the vehicle state space model and the road preview state
space model is a logical and justified approach.
The control that the rider applies
will be based on both vehicle and road information,
and so a control strategy will
be developed in order to achieve this condition. This was the approach presented in
Sharp [94].
Controller gains will subsequently be calculated that are applied to the states of both
vehicle and road preview, thus developing a realistic representation
of the control
actions of a motorcycle rider.
4.2.2
Local Coordinates Preview
The modelling of the rider's preview in global coordinates,
taken from [94], is an
elegant and simple way of updating the rider's visual preview. However, it will be
seen that it can at times be advantageous to represent the rider's preview in a local
coordinate system, and thus the combined rider-preview model will now be converted
to operate in a vehicle-fixed coordinate system. It is suggested that this representation
of the combined motorcycle-rider
in which a rider would operate.
system is a more accurate assessment of the manner
A similar observation was made by Cole et al.
[11], and necessary modelling modifications to a driver preview model were made to
accomplish the task. The goal here is the same, though the details of the process are
subtly different. Fundamentally,
the processes required to achieve the rider-preview
model in vehicle-fixed coordinates are not dissimilar to the globally fixed coordinates,
74
CHAPTER 4: RIDER PREVIEW
though the details of the process differ.
It should be noted here that the following theory and subsequent simulations relate to
a path which is close to straight running, such that small angle theories are assumed
without significant loss of accuracy.
To extend to a more general model, the x-
coordinate of the motorcycle would also be required as a generalised coordinate in
the road preview information.
Previously, the motorcycle state vector was combined with the vector of previewed
road information to generate the combined motorcycle-preview
vector (4.1a). If the
motorcycle is set to start from the global origin, then the initial local road preview
information is the same as the initial global road preview information.
For subsequent
iterations, the values Yri(k) and 'lfJri(k) for i = 1 ... Np in the road preview vector and
the motorcycle's state vector xv(k) will be replaced by the equivalent information in
local coordinates, Le.
z(k) = [ xl(k)
Yl(k)
1
(4.4a)
with
(4.4b)
The order of the elements in the state vector
However, the states in
Xl
Xl
(k) is unchanged compared with xv(k).
are now defined relative to the moving coordinate system.
The state space model remains essentially the same, but now at each step of the
simulation the lateral position and yaw angle states are reset to zero when the new
local coordinate system is defined with each iteration step. The steer and roll angles
are unchanged.
In the global coordinates system, the road preview information was updated
simple shift register process (4.2).
structure,
by a
It is desirable if possible to retain this simple
but modify it in such a way that the state vector gives the road preview
information in local rather than global coordinates and updates it accordingly.
At the
kth
step, the rider has preview of Np points ahead of him, defined in his local
coordinate frame at step k. As the motorcycle advances one step to k
+ 1, the
road
preview information must be updated to account for his motion and defined in the
CHAPTER
75
4: RIDER PREVIEW
new local coordinate frame at k
From step k to step k
+ 1, the
+ 1.
local coordinates frame will move both laterally and as
a rotation due to yaw of the motorcycle. There is also longitudinal motion, accounted
for by the shifting of the road preview information in the preview vector. Thus, the
shift-register process must move the road preview information in the preview vector
as before, but also take account of the change in the preview information on account
of the lateral and yaw shifts of the coordinate system with each successive simulation
step.
Figure 4.3 shows a diagrammatic
being updated from step k to k
representation
+ 1.
of the road preview information
When the motorcycle is at the origin of the
local coordinates system for step k, both the yaw angle and lateral position are zero
in the local coordinates.
In the coordinate system of step k, the new lateral position
and yaw angle over the step k to k
motorcycle has moved to step k
+ 1 are
+ 1, the
defined as Yn(k) and 'l/Jn(k).
When the
motorcycle lateral position and yaw angle
will be reset to zero, being then at the origin of the local coordinate frame at k
Thus, the preview information of the ith preview point for the (k
+ 1)th
+ 1.
step (which
was the (i + 1)th point at the step k) can be calculated as
YI. (k
+ 1) = YlH1 (k) - Yn(k)
'l/Jl;(k + 1) = 'l/JIHl (k) -
- 'l/Jn(k)di
(4.5)
'l/Jn{k)
where di = (i - 1)vt, where v is the forward speed, assumed constant, t is the discrete
time step, and hence (vt) is the distance travelled in one iteration step.
The equivalent expressions for the global coordinate system were
Yr; (k
'l/Jr;{k+
+ 1) = Yr;+1 (k)
1)
(4.6)
= 'l/JrHl{k)
Thus, the aim is to modify the state space representation
preview model to perform this calculation
of the combined motorcycle-
for all the previewed road information
points and thus modify the shift-register to operate in local coordinates.
The vehicle state vector already includes both the lateral displacement and yaw angles
of the motorcycle, and so the obvious way in which to achieve this would be a simple
76
CHAPTER 4: RlDER PREVIEW
modification to the lower part of the state space matrix A to use these values and
hence modify the road preview. This could be achieved with
Xl
[
(k
YI(k
1)] [ s; (k) 0]
+
+ 1)
Al(k)
=
Ap
[Xl
(k) ]
YI(k)
(4.7a)
The states Yn(k) and tPn(k) are calculated by the discrete-time state space model over
the iteration step from k to k
at k
+ 1, but
+ 1. Strictly,
they should perhaps be defined as terms
defining them as terms at k avoids the confusion as to which reference
frame they are in. For confirmation, Yn(k) and tPn(k) are the lateral displacement
and yaw angle of the motorcycle achieved over the step k to k + 1, but in the reference
frame of step k, before the states are reset to local values (zero) in the new local frame
at k
+ 1.
The matrix Al (k) that will update the local road preview in accordance
with (4.5) and for j = 1, .., n, where n is the number of vehicle states, is
-Av(k)(1,j)
-Av(k)(2,j)
-Av(k)(I,j)
- vtAv(k)(2,j)
-Av(k)(2,j)
(4.7b)
AI(k) =
-Av(k)(I,j)
- (Np - 2)vtAv(k)(2,j)
-Av(k)(2,j)
-Av(k)(I,j)
- (Np - 1)vtAv(k)(2,j)
-Av(k)(2,j)
Thus Av(k)(I,j)
and Av(k)(2,j)
refer to the rows in the Av(k)
the lateral position (row 1) and yaw angle (row 2) respectively.
is unchanged.
matrix calculating
The matrix Ap
This then correctly updates the road preview information
in local
coordinates following one iterative vehicle step to achieve the structure of (4.5).
Following the iteration step, the vehicle states relating to the vehicle position must
then be reset in local coordinates,
and are therefore set to zero. This refers only
to the lateral position and yaw angle states of the motorcycle; all other states (roll
angle, steer angle, tyre forces and velocities) must not be reset. This does not imply
that the corresponding rows of the state space matrices are also zero: the elements
in these rows consists of terms relating to all the generalised coordinates, not all of
CHAPTER 4: RIDER PREVIEW
77
which are set to zero.
To complete the iteration step, the new road information point as a result of the
rider's forward motion must be included at the forward limit of his preview horizon.
In the same manner that the information
in the combined vehicle-state
vector is
modified, the new information point must also be converted to the new coordinate
system.
The preview points within the rider's preview are, with each successive iteration step,
converted from one set of local coordinates to the new coordinate system and need
only be adjusted to account for the changes in lateral position and yaw angle over
that iteration step. The new preview information fed into the rider's preview at the
limit of the horizon must however be converted from the global coordinates to the
new local coordinate system. This requires a similar calculation to that employed on
the rest of the preview information, but making use of the global lateral position and
global yaw angle of the motorcycle in the adjustment
made to the road information.
Assuming that the trajectory is close to straight running such that small angle theory
can be assumed without significant loss of accuracy, the new global lateral position
Yg and yaw angle 1/Jg of the motorcycle at the step k
values at k, i.e. Yg{k),
k
+ 1 which,
+ 1 can
be achieved from the
plus the change in the local values over the iteration step k to
as before, are termed Yl{k)
and 1/J1{k). Strictly, they are the state values
when the motorcycle is at the position k
+ 1, but
in the reference frame of k, before
they are reset to zero again in the new local coordinate frame at step k
+ 1 (Figure
4.4), such that
+ 1)
1/Jg{k + 1)
Yg{k
+ Yg{k) + 1/Jg{k)vt
= 1/J1{k) + 1/Jg{k)
= Yl{k)
(4.8)
Thus, the new preview information point fed in to the rider's preview at the limit of
+ 1 must be transferred from the global
step k + 1 (Figure 4.5), achieved with
the horizon at the step k
local coordinates at the
= [ YrNp (k
+ 1) -
Yg{k
+ 1) -
vTp1/Jg{k
1/JrNp (k + 1) -1/Jg{k + 1)
coordinates to the
+ 1)
1
(4.9a)
78
CHAPTER 4: RIDER PREVIEW
which, by substitution
1=
YINp(k+1)
[ 1/JZN(k
p
of (4.8), results in
1
[YrNP(k+1)-{Yh(k+1)+yg(k)}-Npvt{1/Jh(k+1)+1/Jg(k)}
+ 1)
1/JrNp(k
+ 1) -
{1/Jh(k
+ 1) + 1/Jg(k)}
(4.9b)
The global position and yaw angles are appended onto the end of the rider-preview
state vector, such that
xv(k)
yz(k)
[
]
= [Y(k)
1/Jh(k)
1/J(k) Yl1 (k)
...
YINp(k)
1/JINp(k)
Yg(k)
yik)
(4.9c)
and thus the complete discrete-time matrix structure for operating in local coordinates
is achieved with
o ]
An
1m
[xv (k)]
Yl(k)
+ [BV 0(k)
yy(k)
]
0
u(k)+
[
0 ]
Bn Yrn(k)
0
(4.10a)
where Av(k)
is the vehicle discrete-time dynamics matrix, Bv(k)
Matrices Al(k)
dynamics input vector.
and Ap(k)
is the discrete-time
are as defined previously, and,
with j = 1, .. , n as before,
0
0
0
0
0
0
0
0
(4.10b)
An=
0
0
0
0
-1
0
-vNpt
-1
(Npmxm)
CHAPTER 4: RlDER PREVIEW
79
+ NpvtAv(k)(2,j)
Ag(k) = [ Av(k)(l,j)
Av(k)(2,j)
4.3
]
(4.10c)
()mxn
o
0 0 0
000
o
0 0 0
o
1
(4.10d)
0 1 0
Rider Preview Conclusions
This chapter has covered the detail regarding the rider's road preview, and the manner
in which it is formulated into a discrete-time
state space model in line with the
motorcycle dynamics (Chapter 3). The theory of this shift-register process and the
combination with a vehicle dynamics model was detailed in [98] and considered the
global coordinates
system.
The inputs to the combined motorcycle-rider
system
considered here consist of the road information and the rider's steer torque control
input.
The steer torque control input will be determined by the use of a control strategy,
representing the control process of the motorcycle rider. Two control theories will be
analysed; optimal control, covered in Chapter 5, and predictive control, detailed in
Chapter 6. Thus, the combined motorcycle-preview
model presented in this chap-
ter will be the platform on which the two control strategies will be modelled and
evaluated.
The rider's road preview has also been presented in a local coordinates form. A similar
local-coordinates
approach was taken in [11], where the road preview information
was retained in a global coordinates system, but converted to local coordinates when
being used to calculate the control inputs. Thus, the approach used here, for which
the road information itself is in local coordinates,
represents a departure
from the
previous literature.
The majority of the modelling will be conducted in a global coordinates system, as
CHAPTER 4: RIDER PREVIEW
this represents the simpler of the two approaches.
80
In principle the choice of coordi-
nate system should have no effect on the performance of the system, provided that
any measurements or calculations made are consistent with the choice of coordinate
system.
However, it will be seen in Chapter 5 that the use of a local coordinates
road preview system can have significant benefits to the path following accuracy of a
limited-preview optimal control strategy due to assumptions made in the modelling
of the controller, and so the inclusion of a local coordinates model is included here
for subsequent use.
CHAPTER 4: RIDER PREVIEW
4.4
81
Figures
road path
i=1
y
motorcycle
Figure 4.1: Road preview information in discrete steps for Np = 6
step 1
road path
..
motorcycle
step 2
..
..
step 3
Figure 4.2: Update of road preview information in discrete steps for Np = 6
CHAPTER 4: RIDER PREVIEW
82
i+ 1 in frame (k),
i in frame (k+ 1)
~---------------------------------------~
Figure 4.3: Update of road preview in local coordinates
- _.
~----------~----------------------------------------~~
Figure 4.4: Update of motorcycle global position when moving one step ahead in
reference frame of step k using a local coordinates approach
83
CHAPTER 4: RIDER PREVIEW
~~:-~------.
~(k+l)
~----------------~--------------------------~~
Figure 4.5: Conversion of new preview point information from global to local coordinates
Chapter 5
Optimal Control Rider Model
5.1
Introduction
The application of optimal control techniques is widespread in the field of control
engineering and for good reason.
Optimal control is a control strategy capable of
balancing a number of performance requirements in order to generate a system input
that will strike the best balance between the often conflicting requirements
of the
controlled system.
Typically, an optimal control technique can be used to balance the requirements of system accuracy against control effort required. These two factors commonly act against
each other, with more accurate control requiring a greater control force. Weighting
parameters
are applied to the output and input variables to permit tuning of the
control system, enabling the controller to be biased towards high performance accuracy or conversely to minimise the control inputs.
performance parameters
Furthermore,
if a number of
are present, the relative importance of these can be tuned
using the weighting factors, and likewise for multiple control inputs.
These features make optimal control a suitable approach for the modelling of a vehicle
driver.
A vehicle driver has broadly two choices. On the one hand, he can follow
the road path very accurately, though this may be at a high cost with regard to
his steer control inputs.
Conversely, he may elect to cut corners of the road path,
simplifying his control inputs but at a cost to the accuracy of his path following. Thus,
the optimal control strategy for this application will aim to balance path following
84
CHAPTER
85
5: OPTIMAL CONTROL RIDER MODEL
accuracy against steer effort input.
The chapter will begin by introducing the theory of the optimal control approach,
including both the generation of the cost function and associated relevance to the control problem, and how the optimal controller gains are calculated in order to provide
the theoretically optimal control input. An extensive parameter study is conducted
to extend the work of Sharp [94] and to obtain a fuller insight into the behaviour
and suitability of the optimal control approach for a range of situations,
including
variations of the preview horizon length, the forward speed and the cost function
error weighting parameters.
The results that are presented will form a benchmark
against which the predictive control approach, to be covered in Chapter 6, can be
compared. At the time of writing the predictive control approach has not specifically
been applied to the modelling of a motorcycle rider, and will therefore form the main
area of interest in this thesis. The direct comparisons between the optimal control
approach presented in this chapter, and the predictive control approach, detailed in
Chapter 6, will be drawn in Chapter 1, and will ultimately aim to determine the more
effective control strategy for modelling a motorcycle rider.
5.2
Optimal Control Theory
An optimal controller is a mathematical
means of generating a controlling input to a
system that will balance the requirements of system accuracy against control effort
input. This balance is achieved by generating a cost function, consisting of accuracy
and input effort components, which is to be minimised by the controller. Weighting
factors applied to both the output
and input variables allow the contribution
in
the cost function of the individual elements to be varied, thus affecting the relative
contribution of output and input variables to the overall cost. Consequently, a greater
bias can be applied to inputs or outputs
with appropriate
selection of weighting
parameters.
This chapter begins by presenting the optimal control theory for an arbitrary dynamic
system.
The specific application to the motorcycle riding task will be detailed in
Section 5.2.3.
For a general discrete-time
system at the kth iteration
step with states x(k) and
CHAPTER
5: OPTIMAL CONTROL RIDER MODEL
86
control input u(k), the calculated system outputs y(k) and system states x(k
+ 1)
following one iteration step can be expressed as [95]
x(k
+ 1) = J[x(k),
u(k), k]
(5.1)
y(k) = ~[x(k), u(k), k]
The equations of motion can be represented by a linearised state space expression
which will capture the system dynamics and provide the required system outputs,
having the form
x(k
+ 1) =
y(k)
+ B(k)u(k)
A(k)x(k)
= C(k)x(k)
+ D(k)u(k)
(5.2a)
(5.2b)
where the matrices A(k), B(k), C(k) and D(k) represent the discrete-time state space
matrices at the
kth
step as shown in Chapter 3.
In general control terms, accurate system performance
is usually achieved with a
high control cost; accurate control of a robot arm, for instance, would require large
actuator forces if rapid but accurate movement is required. Similarly, in following a
curving road path significant steer inputs may be required if the path is to be followed
accurately. In the case of twin-track vehicles, the steering is achieved by control of the
steer angle, whereas for the case of a motorcycle the directional control is achieved via
the control of the steer torque applied to the handlebars. These torque inputs however
can be reduced if some element of corner-cutting
is made in order to reduce the com-
plexity of the path to be followed, and the optimal control technique therefore seeks
to calculate an appropriate system input u(k) that will provide a suitable compromise
between the conflicts of system accuracy and control effort, based on weighting values
to define the relative importance of system performance characteristics.
The standard approach used, taken from [95], is to define a cost function incorporating
the sum of the weighted squares of calculated system outputs and measured system
inputs, expressed as
(5.3)
CHAPTER
87
5: OPTIMAL CONTROL RIDER MODEL
where J(k) is the cost function, y(k) is the vector of measured system outputs, u(k)
is the vector of system inputs, and Q(k) and R(k) are the weighting matrices on y(k)
and u(k) respectively, all at the kth iteration step. In the general case, both Q(k)
and R(k) are symmetric positive semi-definite matrices. In the applications that will
follow, they are diagonal matrices with the elements on the diagonal corresponding
to the weightings on the states and control inputs, ql, q2 ... qm and
TI, T2 ...
Tp
respectively, with m the number of controlled outputs and p the number of control
inputs in the cost function.
With D(k)
= 0, since
the required output information is contained solely within the
system states, and making the substitution
y(k) = C(k)x(k),
the cost function can
be expressed as
The output matrix C(k) and weighting matrices Q(k) and R(k) remain constant in
this application, and so the identifier (k) can be omitted for clarity. Over a predetermined number of iterations, the optimum performance is sought that minimises the
cost over the sum of all the iterations, and so the cost function becomes
N-I
J =~
L (xT(k){CTQC}x(k)
+ uT(k)Ru(k))
(5.5)
k=O
which can be expressed as
N-I
J=
L 8[x(k),
(5.6)
u(k), k]
k=O
In the general case, a cost can be placed on the final system states after a predetermined number N of iterations, such that the cost function is extended to
N-I
J = O[x(N), u(N), N]
+ L 8[x(k),
u(k), k]
(5.7)
k=O
Due to constraints within a system, the minimum cost and therefore the theoretical
optimal solution may not be a feasible option for that system, and therefore some
CHAPTER
88
5: OPTIMAL CONTROL RIDER MODEL
constraint must be placed upon the calculation of the optimal gains to account for
these. The main physical constraints on a system are the equations of motion; the
response of a system cannot violate the equations of motion, and thus the optimal
control solution must therefore take account of these constraints.
These are included
by appending the equations of motion into the cost function, multiplied by an appropriate factor, the Lagrangian multiplier vector ..x(k) . The complete cost function,
combining costs on system states, system input and the dynamics of the system, is
thus given by the extension of (5.7):
N-l
J = O[x(N), u(N),N]+
L (8[x(k),
u(k), k]_..xT(k
+ I)[x(k + 1) -
f(x(k},
u(k), k}])
k=O
(5.8)
This then represents a cost function consisting of final system states cost, control
input cost and measured system output.
For the purpose of modelling a motorcycle
rider, the simulation is continuous and the final system states x(N)
are not achieved,
and so this element can be omitted from the cost function in this application, leading
to
N-l
J=
L (8[x(k),
u(k}, k] - ..xT(k
+ I)[x(k + 1) - f(x(k), u(k), k)])
(5.9)
k=O
At this stage, we introduce the concept of the Hamiltonian, which is defined by
H(k)
= 8[x(k),
u(k), k]
+ ..xT(k + I)f[x(k),
u(k), k]
(5.10)
This therefore simplifies (5.9) to
N-l
J=
L (H(k)-..xT(k+
k=O
I)x(k+ I))
(5.11)
CHAPTER
This expression still represents the constrained
cost function, and so the optimum
solution is obtained by minimisation of this function.
turbations
89
5: OPTIMAL CONTROL RIDER MODEL
Applying the method of per-
using the concept of small variations in the state and input vectors. The
theory presented here is taken from [95].
where
x(k) = x(k)
+ Do8(k)
(5.12a)
u(k) = ii(k)
+ A1](k)
(5.12b)
Do represents a small change of the variable. These expressions are substituted
into the cost function (5.11). Since the optimal solution is sought, then the aim is to
minimise the cost function, and, to find the minimum of the function, the following
two conditions must be satisfied:
[)J
lim -
a .....
o[)Do = 0
and
[)2 J
lim [) A2
a ....
o L.l. > 0
(5.13)
These requirements lead to the Euler equations, defined as
~~~} = CT(k)Q(k)C(k)x(k)
~~g]=
R(k)u(k)
+ AT(k)>'(k + 1) = >'(k)
+ BT(k)>'(k + 1) = 0
(5.14a)
(5.14b)
In order to obtain >.(k+ 1) and hence u(k) from (5.14b), the solution to (5.14a) must
be sought. >'(k) is unknown at this stage, and so an initial estimate of its solution is
made, setting
(5.15)
>'(k) = P(k)x(k)
Where P(k)
is an unknown matrix that could be viewed as systems of coupled
quadratic equations [107]. Making use of (5.15) and (5.2a), (5.14b) can be rearranged
to give
u(k) = -[BT(k)P(k
+ l)B(k) + R(k)t1BT(k)P(k
+ l)A(k)x(k)
(5.16)
CHAPTER
90
5: OPTIMAL CONTROL RIDER MODEL
To obtain the optimal control input u(k) therefore requires P(k+1)
to be calculated.
Assuming the weighting matrices Q(k) and R(k) to be invariant as before, the iteration step identifier k can be omitted for clarity. Combining (5.14a), (5.15), (5.16)
with (5.2a) leads to
P(k)x(k)
= CT(k)QC(k)x(k) + AT(k)P(k + 1)(A(k)x(k)
- B(k)[BT(k)P(k
+ l)B(k) + Rj-lBT(k)P(k + l)A(k)x(k))
(5.17)
The term x(k) in (5.17) can be cancelled to simplify the expression to the Algebraic
Riccati Equation:
P(k)=AT (k)P(k+1)A(k)
_AT (k)P(k+l)B(k) [BT(k)P(k+ l)B(k)+RJ-l
BT (k)P(k+1)A(k)+cT
(k)QC(k)
(5.18)
If P(k
+ 1) can
be obtained, then from (5.16) the optimal input u(k) can be found.
This can be re-expressed as u(k) = -Kx(k),
where the optimal control gain K is
given by
K = [BT(k)P(k
+ l)B(k) + Rj-1BT(k)P(k + l)A(k)
(5.19)
5.2.1 Algebraic Riccati Equation Solution: Numerical Method
Two techniques exist for the solution of the Algebraic Riccati Equation (ARE) for
P (k
+ 1),
being the numerical or analytical.
iterative method.
The simpler method is the numerical
For this approach, an initial value for P(k
using equation (5.18), a value for P(k) can be obtained.
the correct result, as an initial estimate for P(k
+ 1)
is selected and,
This, however, will not be
+ 1) was made.
Thus, a new value of
P(k+ 1), equal to the value of P(k) just obtained, is taken as the new initial estimate
and the process repeated.
With each successive iteration, the estimate of P(k
will become closer to the real value of P(k
+ 1).
When the estimated value P(k
+ 1)
+ 1)
and the value of P(k) that results from (5.18) become equal within a desired accuracy
level, then the matrix P(k
+ 1) Can be used to obtain the optimal controller gain and
CHAPTER
91
5: OPTIMAL CONTROL RIDER MODEL
hence input.
The choice of initial value for P(k+ 1) at the start of the iterative process is arbitrary;
the closer the initial estimate is to the final solution, then the sooner the iterative
process will obtain the result, but by the very nature of the process the final solution
will gradually and eventually be obtained. Typically however, the initial value P(k+1)
is chosen as the eigenvalues of the matrix A(k).
Convergence of the Riccati equation via numerical methods, and hence the solution,
is not always guaranteed however. More details of the theory can be found in [38,
46], where it is shown that convergence is guaranteed provided that the system is
controllable and observable, or is exponentially stable.
Although this iterative process is fundamentally
vantage is the potentially
straight-forward,
one notable disad-
high processing time required to solve a problem in this
way. This factor will depend to some extent on the accuracy of the initial estimate
to the final solution, and the accuracy tolerances that are applied to the iterative
process. Depending on the application, this potentially high processing time may be
considered important.
5.2.2
Algebraic Riccati Equation Solution: Analytical Method
The alternative
approach is the analytical method.
Mathematically
this is a more
complex process, but can have the advantage of being computationally
more efficient,
calculating the solution to P(k) in a single calculation.
Calculating u(k) from (5.14b) and substituting
x(k
and rearrangement
+ 1) = A(k)x(k)
into (5.2a) gives
- B(k)R-IB(k)T'x(k
+ 1)
(5.20)
of (5.14a) results in
>'(k
+ 1) = -A-T(k)Qx(k)
+ A -T(k)>'(k)
(5.21)
CHAPTER
92
5: OPTIMAL CONTROL RIDER MODEL
The combination of (5.20) and (5.21) therefore results in
x(k
+ 1) = (A(k) + B(k)R-IBT(k)A
-T(k)Q)x(k)
- B(k)R-IBT(k)A -T(k)'>"(k)
(5.22)
The matrices A(k)
and B(k)
will change with each iteration
step, but with the
iteration identifiers k omitted for clarity, the expressions (5.21) and (5.22) can be
written as a single discrete-step matrix expression:
x(k + 1)
[ '>"(k + I}
1
This relationship
T
= [ A + BR-1B
A-TQ
-A -TQ
T
-BR-1B A-T
A-T
1
[X(k)
'>"(k)
1
(5.23)
can be represented by introducing the augmented vector p(k)
=
[x(k),'>"(k)]T, and defining the Euler Matrix EM, such that
p(k
+ 1) = EMP(k)
(5.24)
The general solution to (5.24) can be represented by
(5.25)
where J EM is the Jordan matrix of the Euler matrix (5.23) and T the modal matrix
of the system, whose columns are the eigenvectors of the Euler matrix, arranged with
the matrices partitioned into the n stable and n unstable eigenvalues and eigenvectors
[95] such that
JEM=
Jll
[
o
0]
(5.26a)
J22
(5.26b)
(5.26c)
CHAPTER
5: OPTIMAL CONTROL RIDER MODEL
where the components Til, Th etc. represent the partitioned
93
components of the
inverse of the matrix T, and are not equal to TIl, TIl etc.
The first n columns are the eigenvectors of the stable roots, and the second n columns
refer to the eigenvectors of the unstable roots. Using this partitioned structure of the
matrices, (5.25) can be formed into a vector-matrix expression:
x(k) ] = [TU
Tl2]
[ 'x(k)
T2l T22
[J~l
0
(5.27)
which can be expanded to give
x(k) = (TuJ~l Til + Tl2J~2T:h)x(O) + (Tl1J~l Ti2 + Tl2J~2T22)>'(0)
(5.28a)
>'(k) = (T2lJ~lTh + T22J~2T2l)X(O)+ (T2lJ~lTi2 + T22J~2T22)>'(0) (5.28b)
Since a stable solution is required, we must therefore eliminate from (5.28) any instance of the unstable matrix J~2' Thus, it must be the case that for both (5.28a)
and (5.28b),
(5.29)
and therefore
(5.30a)
(5.30b)
By elimination of the unstable terms matrix J~2' the expressions (5.28a) and (5.28b)
simplify to
x(k) = TuJ~lTilx(O) +Tl1J~lTi2>'(O)
(5.3Ia)
>'(k) = T21J~lTilX(O)+ T21J~lTi2>'(O)
(5.3Ib)
CHAPTER
94
5: OPTIMAL CONTROL RIDER MODEL
The substitution
of (5.30b) into (5.31a) and (5.31b) results in
x(k) = TUJ~l[Til - Th(T22)-lT21]X(O)
(5.32a)
..\(k) = T21J~1[Th - Ti2(T22)-lT21]X(O)
(5.32b)
Making the temporary substitution
J~dTh - Ti2(T22)-lT21]x(O) = (3, expressions
(5.32a) and (5.32b) can be simplified to
x(k) = Tn{3
(5.33a)
..\(k) = T21{3
(5.33b)
Referring back to (5.15), we can therefore now express the Riccati equation solution
as
(5.34)
Recall that the columns of the matrix T consist of the eigenvectors of the Euler
matrix EM, arranged such that the leftmost columns contain the stable eigenvectors.
The upper half of these, i.e. the quadrant
Tu, corresponds to the stable roots of the
system, while the lower half, the element T21, consists of the eigenvectors associated
with the eigenvalues of the Lagrangian multiplier.
The eigenvectors can be sorted based on the magnitudes
of the eigenvalues [98],
such that the n stable eigenvalues, associated with magnitudes less than unity, are
selected, forming the elements [TE
TT2]T. Thus, T21 and TIl can be obtained, and
(5.34) solved to obtain the Riccati equation solution and hence generate the optimal
controller gains.
Therefore, via this approach the Riccati equation solution can be obtained simply
from the eigenvalues of the Euler matrix in a single step, assuming that the eigenvalues
of the Euler matrix are symmetrically distributed
with respect to the imaginary axis
in the complex z-plane, with half stable and half unstable, and with no eigenvalues
equal to zero.
CHAPTER
95
5: OPTIMAL CONTROL RIDER MODEL
The eigenvalue-eigenvector method was found to be a suitable approach for the modelling of a car driver using a similar optimal control strategy [98], and so was initially
adopted here due to the relatively lower computational
demand with respect to pro-
cessing time of the method compared with the numerical approach.
However, as is often the case with iterative processes, small errors can quickly develop
into much larger errors. While the eigenvalue-eigenvector method proved acceptable
for relatively gentle manoeuvres for which state values such as roll angle and state
accelerations were low, more severe manoeuvres led to numerical errors in the code
which ultimately caused the program to fail. While this problem was not explored
in great detail, it is suspected that this is a result of the numerical limitations
in
computing. The method requires the inverse of the matrix element Tu to be obtained,
and if the elements in this matrix become such that the problem is ill-conditioned,
then this would result in the computational
problems experienced.
If the solution were obtained by using the numerical iterative method to solve the
lliccati equation then this problem could be avoided, and so the code was therefore
adapted to use the iterative solution method instead where necessary.
5.2.3 Application to the Riding Task
The theory for obtaining optimal controller gains for a generic dynamic system has
been covered in the preceding sections.
The specific application to the motorcycle
rider modelling task is now outlined here.
Chapter 4 detailed the shift-register procedure that is used to provide the road preview
information element, leading to the combined motorcycle-preview
model, having the
form
xv(k + 1)
[ Yr(k+1)
1= [
Av(k)
0
o
Ap
1
[xv(k)
Yr(k)
1+ [
Bv(k)
0
1
u(k)
+[
0
Bp
1
Yrn(k)
(5.35)
In principle therefore, the solution method for the optimal control input follows the
theory outlined for a generic dynamic system outlined in preceding sections, replacing
the matrices for the generic system with the matrix forms given in (5.35).
CHAPTER
96
5: OPTIMAL CONTROL RIDER MODEL
Since the objective of the model is to follow the previewed road path, the output
element of the cost function (5.4) can be set to provide the errors between the lateral
path following and the path heading angle by appropriate
selection of the output
matrix C(k):
C(k) =
[0o 1 0 -1
0 1
0
o
... 01
-1
...0
(5.36)
mxn+Npm
where the road preview information consists of both lateral positions and heading
angles of the road path in global coordinates. The lateral motorcycle position and yaw
angle are the 2nd and 3rd elements in the motorcycle state vector, thus multiplying by
1, while the lateral position and heading angle of the target path at the first preview
point (corresponding to the motorcycle's target position on the path) are the elements
at n
+ 1 and
n
+ 2 in
the combined motorcycle-preview
vector z(k), multiplied by
-1. When only the lateral position information is known, the target heading angle
can still be deduced by simple trigonometry
[98].
The corresponding matrix Q(k) is
Q(k) = [ql
o
0
q2
1
(5.37)
where the elements ql and q2 respectively weight the lateral path following error and
heading angle error of the motorcycle relative to the target road path.
The vector of control inputs u(k) consists only of a steer torque input, and is therefore
represented as u(k}. The corresponding weighting matrix R(k} consists therefore only
of a single element, r, weighting the steer torque control input.
The number Np of preview points in the road information vector is determined by
the length (in time) of the preview horizon Tp and the discrete sampling time t, i.e
Np
= Tp/t
CHAPTER 5: OPTIMAL CONTROL RIDER MODEL
5.2.4
97
Optimal Gains
A necessary condition for the analytical method presented previously is for the system
state matrix A(k) to be non-singular, and for the case of the motorcycle dynamics
alone (Av(k», this condition is satisfied. However, when the road preview is appended
to the state space representation
of the motorcycle-rider
as in (5.35), the augmented
system state matrix then becomes singular. The problem is still solvable, finding the
solution to the Riccati equation and subsequently the optimal gains, but requires a
manipulation
process developed in [71] and also later for a car-driver
application in
[98].
Essentially, this requires that the state space representation of the combined motorcyclepreview model is partitioned into smaller sub-matrices that represent the motorcycle
dynamics and the road preview separately, i.e
(5.38)
where the subscript 1 relates to the motorcycle dynamics and the subscript 2 to the
road preview. Thus, Kl(k) would be the element of the gain vector K(k)
applied to
the motorcycle states, and K2(k), the elements corresponding to the preview element.
The matrix Av{k) in (4.3) is non-singular, and so for the case with no preview included
(i.e. K(k)
= Kl(k)
and P(k)
= PuCk»
the analytical method can be used to solve
the optimal control problem without difficulty. Manipulation
of the ARE, shown in
detail in [98], shows that by solving first the non-preview case to obtain KI(k) and
Pll(k)
the matrix P12(k) can be obtained, and that with these matrices it is then
possible to obtain K2{k),
previewed road path.
the element of the optimal preview gain relating to the
Forthwith therefore, the matrices KI{k) and K2(k)
will be
referred to as Ks, the state gains, and Kp, the preview gains, respectively.
Thus the controller gains for the combined rider-preview model were obtained which
then enabled the simulated model to perform the path following task. The simulation
itself was run as a discrete time iterative model, and thus at each step of the discrete
time simulation the control problem using the preceding theory was solved to generate
the required steer input and hence follow the target path.
CHAPTER
5.3
98
5: OPTIMAL CONTROL RIDER MODEL
Optimal Control Rider Model Results
The work covered here and the model developed aim to replicate and extend the
work conducted by Sharp [94]. There, the model employed the same optimal control
strategy and was applied to a broadly similar motorcycle model. However, the model
used here has what is believed to represent a more advanced and realistic tyre model,
and additionally with the non-linearity of the motorcycle dynamics accounted for by
continuously re-evaluated
linearisations
of the motorcycle state space equations of
motion during the simulated motion. The results presented in [94] appeared to represent well the actions of a motorcycle rider, and the technique will therefore be used
as a benchmark against which to assess the performance of a model predictive control
technique that was subsequently applied to the rider model. Direct comparisons and
conclusions of the two techniques are drawn in Chapter 7.
The optimally-controlled
motorcycle rider model is tasked with a simple path follow-
ing exercise, involving a simple single lane change consisting of a lateral shift of 3.5 m
over a forwards distance of 20 m (Figure 5.1). This is an ISO standard manoeuvre
commonly used in the assessment of vehicle manoeuvre performance.
The optimal control strategy has several variables that are set to define the operating
characteristics
of the controller, including weighting factors on the system outputs,
weighting factors on the system inputs, the preview horizon and more fundamental
parameters such as the sampling time of the discrete model.
The specific application of the optimal control theory and the relevant terms were
covered in Section 5.2.3. The weighting parameters in the cost function will provide
the primary means of varying the system's performance;
the weighting factor ql
applied to the lateral path error, q2 weighting the yaw angle error, and r on the
steer torque control input. In addition, the preview horizon time Tp will be varied to
change the distance to which the rider is able to see ahead. Division of the preview
horizon Tp by the discrete time interval
t will then give the number of discrete preview
points.
Details relating to the nature of the manoeuvre task itself can be varied. This can
include both the nature of the road path and the forward velocity v at which the
rider attempts the manoeuvre. In order to maintain some consistency and thus allow
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the characteristics
of the controller alone to be assessed, the road path will remain
fixed for the purpose of these investigations, remaining as a 3.5 m lateral shift over a
forward distance of 20 m. The speed v at which the rider attempts this manoeuvre
however will be changed, to account for low and high speed conditions.
The presented results will consider the implications on the control task of variation of
the principal variables outlined above, independently
of each other. This will permit
clear observations to be drawn on the effects of parameter variation on the control
task. Furthermore,
these observations will permit conclusions to be made regarding
the applicability of the control strategy to the modelling of a human motorcycle rider.
5.3.1 Low Speed Optimal Control Model
Initially, a baseline model will be evaluated that will provide a model against which
changes to the controller settings can be assessed. This baseline model will aim to
provide a system with a moderate performance and balance between accuracy and
effort, with sufficient but not excessive preview, and the speed will be relatively low.
Subsequently, parameters will be varied individually to assess the implications that
they have upon the performance characteristics
of the controller.
The full range of
parameter sets for low speed are presented in Table 5.1.
Baseline Parameter Set
The ability of the model to track the path is assessed first. Figure 5.2 presents the
path followed by the model using the initial baseline parameters
(Set 1), where it is
observed that the model is able to successfully negotiate the manoeuvre and tracks
the path well after the manoeuvre phase is complete. Additionally, the countersteer
associated with riding a motorcycle is evidenced by a slight deviation away from the
turn direction before the lane change begins.
The controller's gains are considered next, where the magnitudes of the gains may
be thought of as representative
of the importance placed the state elements by the
rider for the control task. Due to variation of the state space model on account of the
model's non-linearity, some variation of the gains occurs over the manoeuvre.
gain values that will be presented are for the motorcycle approximately
The
one quarter
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of the way through the lane-change, at the point where the roll angle on the initial
left turn is the greatest.
The controller state gains that give rise to the control are presented in Figure 5.3,
appearing to suggest that the highest gain value is associated with the yaw angle
state. The gain in the roll angle state is the next largest, with all other state gains
of significantly lower magnitude.
However, the significance of the magnitudes of the state gains needs to be clarified due
to the different units and magnitudes of the states. The gains are therefore compared
by the contributions
steer torque.
that they individually make to the complete control input, the
The total steer torque, overlaid with the motorcycle's
roll angle, is
presented in Figure 5.4, showing the expected pattern of a main (negative torque)
countersteer to initiate the manoeuvre, increasing to a positive peak to stabilise the
resulting roll and eventually reduce this roll, decaying as the roll angle then reduces
followed by a negative torque to stabilise the motorcycle once it is upright again.
The contribution made to this total torque by each state is the sum of the state gain
value and the state value itself, and therefore it is more representative
to consider the
controller in these terms. The use of global coordinates here complicates the analysis
somewhat, as the significant torque contribution
by the significant but opposite contribution
from the lateral position is offset
made by the road preview.
These are
therefore removed from the analysis, together with a number of additional states that
have a much more insignificant contribution to the overall steer torque, to leave only
the contributions
made by the yaw angle, roll angle and their respective velocities,
and the lateral velocity of the motorcycle. Figure 5.5 shows the contributions
to the
total steer torque of the states and state gains over the path following simulation.
From this plot, it is apparent that the contribution to the steer torque resulting from
the yaw angle gain is still the most significant, followed by that resulting from the
roll angle.
The same analysis was also made for the motorcycle operating in a local coordinate
manner.
In local coordinates,
the situation where the motorcycle lateral position
and road preview information contribute
contributions
equal and opposite non-zero steer torque
in straight running does not arise, and thus the steer torque contri-
butions made by the individual states are much more easily understood.
shows again the steer torque contributions
Figure 5.6
arising from the yaw angle, roll angle and
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their respective velocities, and the lateral velocity of the motorcycle. The pattern is
similar, however the roll angle and road preview information now provide the largest
contributions
to the overall steer torque, while the contribution
made by the yaw
angle component is virtually nil and is therefore not shown in the figure.
These findings suggest two things.
Firstly, the local coordinates
approach would
appear to suggest that roll angle control is the primary objective, followed by control
of the lateral position of the motorcycle. These statements
are in strong agreement
with Weir [102], who suggested that roll control was the primary objective of the
steer torque control, and lateral position and yaw angle control were slightly weaker
objectives met by movement of the rider's upper body. Secondly, the observation
that, in local coordinates,
the locally observed yaw angle of the motorcycle has a
much smaller significance on the future path of the motorcycle is in strong agreement
with the observations of Cole et al. [11]. In a local coordinate system, the yaw angle
will, in general, always be relatively small, and so this result may not be surprising.
Considering now the preview gains (Figure 5.7), these also show fundamental characteristics. The gains are initially zero, rising to a peak in the middle preview distance
before decaying gradually towards zero, implying minimal influence on the control
task of the road information at this distance ahead. Thus, it appears that the rider
model places no importance on the road observed directly in front of the motorcycle,
with the road in the middle preview being of most influence to the control task, and
with the distant previewed information being of reducing importance as the preview
distance increases much beyond 20 m ahead.
In attempting
to optimise the path following exercise, the controller steers the mo-
torcycle in order to stay on course.
For road errors perceived directly in front of
the motorcycle, there is nothing that can be done about these errors; the motorcycle
would need to stop and be physically moved in order to correct any errors here. A
short distance ahead of the motorcycle, the rider will have some opportunity to correct
any path errors by steering, but may not be completely successful due to the limited
forward distance available and limited steering capabilities (Figure 5.8). As the distance ahead of the motorcycle at which lateral deviations are observed increases, so
the rider begins to have sufficient opportunity
to correct any path following errors
that are detected in the visual preview, and this may explain the increase in the preview gains as the preview distance increases. At greater preview distances, the rider
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has an excess of time available to take any necessary control actions. The correction
of these potential path-tracking
errors are of lower importance than those for which
the available time is limited and, consequently, as the preview distance increases further, the preview gains begin to diminish away towards zero. By considering the
capabilities of a motorcycle rider in this manner, then the preview gains achieved by
the controller would appear to offer a good representation
of the process.
The steer torque that results from the controller gains and that leads to the path following performance seen is shown in Figure 5.9. The steer torque shows the expected
characteristics,
with a countersteer in advance of the turn to initiate the manoeuvre,
leading to a peak torque as the rider begins to bring the motorcycle back upright
to change the turn direction, reducing again and finishing with another countersteer
torque to arrest the roll movement as the motorcycle is brought back upright.
These initial observations on the performance of the optimal control strategy appear
encouraging; the path following task is successfully accomplished, achieved with realistic steer torques and with controller gains that would appear to fit well intuitively
with the expected control characteristics
of a motorcycle rider. These results were
also originally found by Sharp [94]. This therefore gives some confidence in extending
the parameter set to further investigate the performance of the control strategy.
Cost Function
Weighting
Influence
The weighting parameter associated with the lateral path error
initial value for
ql
was 5000 m-2, and additionally,
ql
is now varied. The
1000 m-2 and 10000 m-2 are
now used. The lower value should lead to a less accurate path following performance,
the higher value a closer following of the path due to their influence on the optimal
control cost function.
The path following errors for the three weighting parameters are presented in Figure
5.lD. In line with expectations,
the increase and decrease in the values for
direct influence on the path-tracking
ql
have a
accuracy achieved, with the controller displaying
less accurate path following and therefore greater path errors termed 'loose' control,
with the more accurate path following performance resulting from higher values of
referred to as 'tight' control [94].
ql
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The changes in the path following performance stem from the change in steer torques
generated by the controller model. It is seen (Figure 5.11) that the loose control is
associated with early initiation of the manoeuvre, leading ultimately to lower peak
torque values, and in this manner, the rider model is sacrificing path-tracking
accu-
racy in order to reduce the magnitude of the steer torques he must apply. This is to be
expected, since the optimal control cost function comprises elements relating to both
the output performance and the control input effort. An increased bias to either performance or effort optimisation,
achieved via increased weighting parameters,
would
naturally be expected to reduce the emphasis on the other cost function element, and
vice versa. So it is then that a decrease in the output performance bias appears to
result in an increase in the control effort bias. Similarly, tight control, indicative of
low path errors, leads to correspondingly
higher peak steer torques applied over a
much shorter duration.
These steer torques arise from the controller gains, with variation of the controller
settings naturally
affecting the controller gains.
Figures 5.12 and 5.13 show the
state and preview gains for all three error weighting values. Increases in ql are seen
to produce increases in all the state gains; the ratios between them however remain
constant. The effect on the preview gains is also to increase them, but more interesting
is the shift that is observed in the distribution
of the gain values. With increased
qI, indicative of tighter control, the bias in the preview gains moves closer to the
motorcycle.
In other words, the distance ahead of the rider corresponding to his
most important
road information point moves closer to him. This implies that the
rider is focusing on the road closer to him, which would seem like an expected result.
Loose control is associated with minimisation of control effort, and to achieve this a
rider may be expected to select the most efficient path from an initial point 'A' to a
final point 'B', travelling as directly as possible to minimise the control effort required
(Figure 5.14). In order to select the most efficient path, a complete knowledge of the
road path is desirable, and so the rider would be expected to be looking further down
the road and considering a distant target as being more important
in deciding his
control strategy. Consequently it might be expected that the preview gains would be
more evenly distributed,
and extending to a greater distance ahead.
Conversely, tight control is associated with accurate path tracking, where the rider
will attempt
to position his motorcycle on the target path at all times during the
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104
motion, irrespective of the path some distance ahead. The focus is on ensuring that
throughout the simulation, where the motorcycle is immediately about to move to still
accurately follows the path, regardless of the path some greater distance ahead. Thus
the rider would be expected to focus more attention on the near-preview information,
and less on the far-preview distance (Figure 5.14).
Thus, the expected actions of a human rider with a change in path following strategy
appear to be represented well by the optimal control technique. The changes in the
preview gains observed with the change in ql appear to be an acceptable representation of a human rider's control actions, adding credibility to the applicability of the
approach.
Preview Horizon Effects
The preview horizon time Tp available to the rider is now varied to assess the impact
that this parameter has upon the control performance of the optimal control strategy.
The preview horizon is set to 4.5 s and 1.5 s to represent excessive and limited preview
respectively (Sets 4 & 5, Table 5.1).
The rider model was initially allowed 3.0 s of visual road preview. At a forward speed
of 10 mis, this was seen to be sufficient to allow the preview gains to diminish to zero
(Figure 5.7), suggesting that the rider had enough preview information to make the
necessary control actions for complete control. It might therefore be expected that
further additional preview information would have minimal impact upon his control
actions.
The controller gains and resulting steer torques and path following error (Figures
5.15,5.16,5.17,5.18)
largely seem to support this view. There is seen to be minimal
difference between the plots as the allowable preview is increased from 3.0 s to 4.5 s,
suggesting minimal impact on the controller performance as a result of the increased
preview horizon.
A rider generates a control action based on the knowledge that he has of the road
and of the motorcycle. If the rider were presented with new road information which
had minimal impact upon his required control actions, it is reasonable that no attention would be paid to this new information,
and therefore the attention paid to
CHAPTER
5: OPTIMAL CONTROL RIDER MODEL
105
the original information would not change. Hence, if the rider's preview knowledge
is already sufficient to determine an appropriate
gain distribution
control strategy, then the preview
would be expected to show minimal difference.
However, if the rider were presented with new information that would influence his
control problem, it seems likely that the rider's control would be expected to adjust
in light of this; some of the rider's attention would additionally be focused on this
new information, and consequently the level of attention paid to the rest of the road
information reduced slightly, modifying the preview gains.
By a similar token, it
would seem equally likely that the reverse would be true, and that if relevant road
information regarding the control task were removed, then the attention paid to the
remaining road information would be changed.
The state gains (Figure 5.15) show identical gains for all three preview horizon lengths
considered.
The reduction in the available preview information available to the rider appears to
have minimal effect on the distribution of the preview gains (Figure 5.16), at least for
the 10
mls
forward speed case. Irrespective of the horizon length, the preview gains
are identical; the only change resulting from a reduction of the horizon is that the
preview gains are simply truncated,
and thus the preview gain curves overlap each
other when plotted together.
The steer torque control input that results from a change in the preview horizon is
shown in Figure 5.17. For preview horizons of 3.0 sand 4.5 s, the control inputs are
virtually identical.
For the reduced horizon of 1.5 s, the resulting torque is seen to
be modified slightly, which may appear to agree with the expected modification of
control resulting from the change in road information available.
However, the effect of this preview limitation
and modified steer torque input on
the path following performance of the motorcycle is seen to be detrimental.
broad characteristic
The
is unchanged, with initial countersteer leading the motorcycle to
follow the lane change before straightening
up after the turn to follow the straight-
running road section. Figure 5.18 shows the path errors that result from the change in
allowable preview horizon, with the cases for Tp = 3.0 s and 4.5 s being so similar that
the traces follow each other almost identically.
accurately
tracked the straight
Whereas previously the motorcycle
section, it is noted that the limited preview case
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now results in a path tracking error, essentially a steady-state
error in pure control
terms, following the manoeuvre. The controller gains, notably the preview gains, have
not been recalculated to account for the reduction in available preview information,
but have merely been truncated,
and evidently this can lead to poor path tracking
performance if the gains are reduced too far.
An initial thought, in order to in some way compensate for these lost gains terms, may
be to introduce some form of scale factor to compensate for the lost terms. However,
without knowledge of the full set of gains, it is not possible to determine what the
scaling factor should be in order to obtain equivalency of the sums of the gains, and
so any scaling factor chosen could not be accurately determined.
It is not unreasonable to expect a human motorcycle rider to still be able to track
a path, even with limited preview. The transient performance may well suffer, but
if given only, say, 1.0 m of visual preview, the rider would still have knowledge of
whether his final steady state road position were on the path that he should be
following or offset from it, and would therefore, given sufficient time, be able to track
back across the road to recover the target path.
This control characteristic
would appear to be a limitation of the optimal control
technique in modelling a human motorcycle rider operating in a global coordinate
system. The limitation is worthy of further investigation, and is therefore considered
mathematically.
The control input u(k), in this case the steer torque, that ultimately
trajectory
of the motorcycle is generated by the combination
from the multiplication
dictates the
of torques resulting
of states and state gains, and of road lateral preview and
preview gains:
(5.39a)
where K" is the vector of state gains, xv(k)
is the state vector, Kp the vector of
preview gains and Yr(k) is the vector of the previewed road path. At the exit of the
turn and return to straight running, the vehicle states xv(k) all tend to zero, apart
from the state representing the vehicle lateral position, y, and thus at the exit of a
turn,
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5: OPTIMAL CONTROL RIDER MODEL
(5.39b)
where ks,y is the state gain corresponding to the lateral position y(k) of the motorcycle
In the steady state following the manoeuvre, u(k) is zero, and thus
ks,yy(k) = -KpYr(k)
(5.39c)
Np
(5.39d)
::} ks,yy(k) = - LKp,Yr,(k)
i=l
With sufficient preview horizon Np
-+ 00,
this condition is achieved with y(k)
=
Yrl (k), and the motorcycle accurately tracks the path. Similar observations were in
[90] for the case where the preview horizon is sufficiently long such that the controllers
gains tend to zero, in which case the controller's performance is independent of the
coordinate system.
The case is now explored for the condition of limited preview,
such that the gains have not necessarily reached minimal values.
With limited preview, in steady state the steer torque still becomes zero, and the
above condition given in (5.39d) will still hold. Although the preview gain values
remain the same in magnitude, the gain vector itself becomes truncated
as a conse-
quence of the limited horizon, and hence
Np
LKpi
i=l
00
'" LKp,
(5.3ge)
i=l
The vector Yr(k) and state gain K, are unchanged, and thus it must be the case that
(5.39f)
and hence,
(5.39g)
Thus for any finite preview distance, there will be a steady-state
error in the path-
tracking performance when the rider is modelled in this way. An infinite preview
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5: OPTIMAL CONTROL RIDER MODEL
horizon is not a practical option, so the significance of the reduction must be analysed.
The significance of this error effect can be assessed by the value of (L:~1 KPi -
L:~1 Kpi)i the smaller this difference, the smaller will be the steady state error in
the controller.
Thus, provided that the controller gains omitted are insignificantly
small, the reduction in preview horizon should have an insignificant impact upon the
accuracy of the controller model. If the preview horizon is so short that the preview
gains have not reached close to zero values at the limit of the preview horizon, then the
value of (L:~1 KPi steady-state
L:~
KpJ will not be insignificantly small and thus significant
path following errors are likely to result.
Thus it would appear that reductions in the preview distance available to the rider
model can have detrimental
effects on the performance of the controller, and this
needs to be considered when such a control technique is applied to control tasks of
this nature.
Low Speed Modelling Conclusions
Optimal control appears to be a useful technique for the representation
of a human
motorcycle rider, with the ability to replicate realistic control actions in order to complete a path following task. The nature of the controller gains, which are indicative
of the use made of available information
in generating the control actions, appear
realistic of the human rider.
The effects of control variables, namely related to the path following accuracy variables, are capable of influencing the balance that a rider makes between accuracy and
control effort. However, the technique is not without limitations as seen when the
preview horizon is reduced.
5.3.2 High Speed Baseline Parameter Set
The speed of the motorcycle model is now increased to 40 m/s. This will enable
any changes in the control task as a result of forward speed to be analysed.
The
parameter sets are as for the lower speed case, but with v = 40 mls (Table 5.2).
The performance of the controller at higher speed is fundamentally
similar to that
seen at the lower speed, with the path following performance (Figure 5.19) displaying
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similar traits but with greater levels of corner cutting. The nature of the gains that
lead to these control inputs are similar to those seen at lower speed, but notably have
higher magnitudes.
Steer torques are also significantly increased with the increase in
forward speed (Figure 5.20).
Fundamentally,
it is believed that the increase in forward speed leads to increases
in the gyroscopic torques provided by the rotating wheels. The increased gyroscopic
torques mean that, for a given steer input, a greater steer torque is required to overcome these gyroscopics, and the result of this is seen in all aspects of the controller's
performance.
At higher speeds, the ratio of the steer torque to the steer angle increases. In order
to minimise the combined cost function of path following (dictated by steer angle)
and steer torque, then as speeds increase the emphasis shifts in favour of minimising
steer torque rather than steer angle. Consequently, the path chosen by the rider tends
towards a greater level of corner-cutting
as speeds increase.
It has been seen before that a looser control strategy, involving greater levels of corner
cutting, is associated with an increased use of the full preview information, where peak
preview gains may be reduced, but the distribution
preview distance.
characteristic
of gains is extended to a greater
The increase in speed shows this characteristic
(Figure 5.21), a
which is not unlike the switch from tight to loose control strategies
(Figure 5.13). In other words, at higher speeds but with identical controller parameter
settings, the corner cutting characteristics of the motorcycle rider's control actions are
seen to be greater than for the lower speed case. Figure 5.21 also shows a significant
oscillatory pattern in the preview gains. This arises due to the minimal damping of
the wobble mode at this forward speed (see also Figure 3.15).
5.3.3 Local Coordinate Preview
The initial motivation for modifying the controller to operate in local coordinates was
partly because this is arguably the way that a motorcycle rider would operate, but
more importantly in an attempt to eliminate the steady-state tracking errors that can
occur using global coordinates with a limited preview horizon. In principle, a change
of coordinates should not have an impact upon the results of a robust control strategy.
However, the path-following errors that result are a consequence of using theory that
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assumes an infinite horizon. When the horizon is long, terms that are lost from the
preview gains are zero and therefore not significant.
When the preview horizon is
much shorter, more significant terms are lost, leading to errors in the controllers'
performance.
It is now investigated whether these errors are overcome through the
use of a local coordinate system.
The controller strategy remains essentially unchanged, with the controller operating
on a preview road in his local coordinate frame. This, in principle, is no different
to the motorcycle beginning from the origin of a global coordinates frame. However,
with each iterative step the road preview is updated as local coordinates information
(4.9c). The final two points on the preview vector track the global position and yaw
one step ahead of the preview horizon and are not included in the control problem.
As before, the steady-state
is achieved when the steer torque is reduced to zero,
arising when
(5.40a)
which, when the all states bar the lateral position settle to zero, simplifies to
(5.40b)
In global coordinates the left hand side was not equal to zero following a manoeuvre,
and thus the right hand side would be required to be equal and opposite.
Kp meant that this was achieved when y( k) '"
steady state error arose in the straight-path
Yrl
Limited
(k) in the global case, and thus a
path-tracking
following the manoeuvre
part of the task.
By using local coordinates, the target
Yrl
(k) is zero when the rider regains the target
path. The right hand side will tend to zero, as will the left hand side and thus y(k).
Although the reduction in Kp will affect the transient performance of the controller,
the steady state local lateral position of the motorcycle should not be affected by the
limited vector Kp, thus forming a more robust optimal control rider model.
This is confirmed by comparison of two simulations run to identical parameters.
One
simulation is set to operate using the original global coordinate system, the other in
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local coordinates.
The model is intentionally run with limited preview horizon, such
that the effect of the modified coordinate system is more clearly seen.
The vehicle parameters
are: v = 10 mis, ql = 5000 m-2,
q2 = 0 rad-2,
r = 1,
Tp = 1.0 s. The path following results, shown in Figure 5.22, clearly demonstrates
the beneficial effect of modelling the road using local coordinates.
Although the transient response of the global coordinates model appears better, the
large steady state error in the path tracking ability of the model is both undesirable
and unrealistic of the actions of a human rider. If a rider had only limited forward
vision, due possibly to weather conditions, night-time riding or obstructions
from
other vehicles ahead of him, the transient response may justifiably be compromised,
but the rider would still be able to see sufficient road path to know whether he
is on target (position-in-lane
control), or displaced to one side or the other, and
consequently would eventually be able to track back onto the target path if needed.
This represents a notable and important weakness of the optimal control approach for
more general rider-control modelling. While it has been shown that these limitations
can be corrected through the use of a local coordinate system, the local approach,
covered in Section 4.2.2, is a more complex approach than global coordinates.
Re-
gardless of the coordinate system used, the rider still has a knowledge of both the
motorcycle's position and heading, and the position and heading of the road. He can
therefore deduce the relative difference between the two to determine whether he is
off the target path and therefore is required to exercise some control in order to regain
the target path.
Thus, a controller that is capable of accurate performance regardless of the coordinate
system used would provide a more robust and representative
control strategy, and
therefore be more suitable for the task at hand. For the optimal control approach,
this appears to be achieved only when a local coordinates approach is employed.
5.4
Optimal Control Conclusions
The application of optimal control theory has been made to the modelling of a motorcycle rider, initially developed by Sharp [94]. The work here has aimed to generate
a more extensive set of results to gain further insight into the characteristics
of the
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method, and to highlight any strengths or weaknesses that may transpire from the
more extensive analysis.
The findings of the work here agree well with those in [94], giving confidence that
the results here are correct. As before, the characteristics
provide a good representation
application.
of the controller appear to
of a motorcycle rider, showing strong suitability of the
State gain results were not presented in [94], and thus no conclusions
could be drawn on the relevance of the control of the motorcycle's states. The findings
presented here have shown that for all cases the pattern of the state gains remains
consistent, with the yaw angle and roll angle gains having the largest magnitudes.
The more relevant question of how these relate to the physical control applied to
the motorcycle show subtly different results. For the global coordinate system, the
contribution arising from the yaw gain is again the largest, followed by the roll angle
contribution.
However when the coordinate system is switched to local, the results are
notably changed, with the roll angle contribution being the most significant, followed
by the lateral position contribution,
and with the yaw angle providing a much smaller
overall contribution.
Since the local coordinate system is arguably more representative
of the way that
a human rider operates and interprets information, this would tend to suggest that
roll angle and lateral position control are the more important
motorcycle rider. Encouragingly,
considerations
for a
these findings are in strong agreement with Weir
[102], where the links between a number of information inputs and possible control
outputs found the roll angle to steer torque contribution
to be the most important,
followed by lateral position control, which was best controlled by movement of the
rider's body mass.
The analysis of the preview gains again shows agreement with the results of Sharp
[94], and also to the wider results provided by Donges [251 and by Land and Horwood
[47], where the important aspects of a car driver's preview were discussed. The results
had suggested the distinction between guidance control, biasing the distant preview,
and the position-in-lane control provided by the near preview information.
Changing
the riding strategy required through the cost function weighting parameters was seen
to change the influence of the near and far preview information in accordance with
either guidance or position-in-lane riding approaches, thus agreeing with the theories
of the referenced papers.
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The technique is not, however, without its limitations.
coordinate system, particularly
to result in a poor representation
Notably, the use of a global
when combined with limited road preview, appears
of the rider's actions. Although it was found that
this problem could be overcome if the simulations were conducted in a moving coordinate system, it remains a fact that for anything other than an infinite horizon there
will be some steady-state
permitted,
errors resulting, though provided that sufficient preview is
these errors can be reduced to insignificant values.
Useful analysis relating to the global versus local coordinates problem was made by
Cole et al. [11], who noted, in agreement with earlier results by Sharp and Valtetsiotis
[90] and the results here, that the question of global versus lateral coordinates has
no impact upon the path following abilities of the controller when sufficient preview
is allowed. The sufficiency of the preview is defined as the preview gains reaching
zero as before.
This is because the change in the previewed lateral displacements
brought about by a change of reference origin and multiplied by the preview gains
is compensated for by the change in the corresponding state multiplied by its state
gain, the two effectively being equal and opposite. As the theory assumes an infinite
horizon, then this change of reference frame will work provided that the preview gains
diminish to zero such that no significant terms are omitted.
In the case where the preview gains are reduced significantly, the theoretical calculation is still based on the assumption of an infinite horizon, and consequently there
appears no reason why the calculated gain values should differ, as is the case. However, with the loss of significant terms from the preview gains, the contributions from
the state gain multiplied by the state and the preview gains multiplied by the states
are no longer equal and opposite when the vehicle is on the target path, and consequently the steady state errors that were observed result when a non-local reference
frame is used. For a local coordinates
approach and accurate path following, both
the state contribution and the preview contribution should, strictly, still be equal and
opposite for correct path following. However, this occurs now with both contributions
equal to zero, since the local lateral and yaw displacements are zero for steady running and the local road preview for a steady path, when on that target path, is also
zero. The loss of zero terms in the preview resulting from a short preview horizon
therefore has no effect on the numerical result and hence the performance of the local
coordinates controller.
CHAPTER
114
5: OPTIMAL CONTROL RIDER MODEL
Cole et al.'s results suggested that, even with the road path in the state vector defined
by global coordinates, the truncation
of the preview horizon could be compensated
for by a change in the state gains that would correspond to the 'lost' contribution
from the shortened preview and therefore still result in accurate path following. Their
model employed a conversion matrix that would effectively recalculate the global road
information into a local picture as observed in the moving vehicle, and in this way the
controller was effectively set to operate in a local coordinates manner, even though
the road path in the state vector was still defined in a global reference frame. Thus,
the control problem was effectively being solved for a local coordinates problem, but
using global road information.
To retain accurate path following for a shortened preview horizon, the loss of significant road preview terms in the control problem can be overcome in two ways. Either
the controller gains are calculated in such a way as to compensate for these, as was
the case in [11], where the road preview was global but the controller modified to
operate in a local manner, or the road preview information converted directly into
the local coordinates and the controller structure kept the same, as was the case here.
Either way, both controllers effectively operate using a local picture of the road, and
should therefore generate comparable path-following performances.
Thus the statement is again made that for accurate path following using this approach,
the problem must be solved for a local coordinates approach for accurate results with
short horizons.
Whether this is achieved by conversion of the preview gains or by
conversion of the road information may be a question of preference, however both
should produce the same results.
The results presented here using optimal control methodology have extended the understanding of the technique for this application.
Chapter 6 will present the predictive
control technique for modelling the rider, an approach which has some significant similarities to the optimal approach presented in this chapter, but also some notable and
possibly significant differences. The predictive control rider model will be presented
with the same tasks as for the optimal approach here, and so the optimal control
results shown here will be used as a measure against which the performance of a
model predictive control rider model can be compared.
Some weaknesses of the op-
timal control approach have been highlighted here, and so the goal the predictive
control method will be to replicate the positive features while also aiming to correct
CHAPTER
5: OPTIMAL CONTROL RIDER MODEL
115
the limitations found. Finally, the direct comparisons of the optimal control results
and the predictive control results will then be drawn in Chapter 7.
CHAPTER
5.5
116
5: OPTIMAL CONTROL RIDER MODEL
Tables
Set 1
Set 2
Set 3
Set 4
Set 5
Parameter
Baseline
Loose Control
Tight Control
Long Preview
Short Preview
v [ms-l]
10
10
10
10
10
3
3
3
4.5
1.5
ql [m-2]
5000
1000
10000
5000
5000
q2 [rad-2j
0
0
0
0
0
r [(Nm)-2]
1
1
1
1
1
Tp
[SI
Table 5.1: Low speed controller parameter sets, optimal control
Set 6
Set 7
Set 8
Set 9
Set 10
Baseline
Loose Control
Tight Control
Long Preview
Short Preview
40
40
40
40
40
Tp [sI
3
3
3
4.5
1.5
ql [m-2]
5000
1000
10000
5000
5000
q2 [rad-2]
0
0
0
0
0
r [(Nm)-2]
1
1
1
1
1
Parameter
v
[ms="]
Table 5.2: High speed controller parameter sets, optimal control
CHAPTER
5: OPTIMAL CONTROL RIDER MODEL
117
5.6 Figures
3.5m
Road Path
Figure 5.1: Single lane change path, not to scale
4
3.5
3
!
2.5
'!
2
~
-Target
-Path
!I 1.5
oS
to
Go
0.5
0
-0.5
20
40
60
80
Path Longitudinal (m)
Figure 5.2: Path following, v
= 10 mis,
Tp
= 3.0 S, ql = 5000 m-2
CHAPTER
118
5: OPTIMAL CONTROL RIDER MODEL
800
600
::!
c-
400
·ea
e
Cl)
.5 200
I/)
0
Yr
x
Yfax/at.
ay/at. 8ljJ/at. 8~/at. 8o/at.
-200
State
Figure 5.3: State gains, v
= 10 mis,
Tp
= 3.0 S,
ql
= 5000 m-2
20
0.2
15
0.15
10
0.1
E
5
0.05
~ca
..
0
0
'Cl
~
Cb
::s
Cl'
0
I-
.:..
Cl)
c:
cC
80
-Torque
-Roll
-0.05 ~
-5
0::
-10
-0.1
-15
-0.15
-20
-0.2
Path Longitudinal
Figure 5.4: Steer torque and roll angle, v
(m)
=
10 mis, Tp
=
3.0
S,
ql
=
5000 m-2
119
CHAPTER 5: OPTIMAL CONTROL RIDER MODEL
100
50
E
0
~
······psi
70
10
CD
:s
e-
- -dydt
- • - dpsidt
-dphidt
-50
0
I-
8 --phi
~
CD
S -100
I/)
-150
...........
..
-200
Path Longitudinal (m)
Figure 5.5: Principal individual state torque contributors,
ql = 5000 m-2,
v
=
10 mis, Tp
= 3.0 s,
global coordinates
100
50
E
~
0
70
10
CD
:s
e-
-50
J
-100
~
~
I/)
·····psi
8 ---phi
- -dydt
- - - dpsidt
-d
hidt
-150
-200
Path Longitudinal (m)
Figure 5.6: Principal individual state torque contributors,
ql
= 5000 m -2, local
coordinates
v = 10 mis, Tp
=
3.0 s,
CHAPTER
120
5: OPTIMAL CONTROL RIDER MODEL
2
1.5
Q,
~
C
"i
C)
~
0.5
.!!!
>
!
Q.
0
3
-0.5
-1
Preview Distance (m)
Figure 5.7: Preview gains, v
= 10 mis,
Tp
= 3.0 S, ql = 5000 m-2
Achievable
Unachievable
I
Target Path
/
/
,/
/
Minimum turn radius
,/
Figure 5.8: Path error correction capabilites of a motorcycle
CHAPTER
5: OPTIMAL CONTROL RIDER MODEL
121
20
4
3.5
15
3
10
E
!.
5
2.5
:[
2
'!
1.5
-I
QI
::l
~
...
0
0
-Torque
.!
IV
-
-Path
.c
IV
D-
-5
0.5
-10
0
-15
-0.5
Path Longitudinal (m)
Figure 5.9: Steer torque, v
= 10 mis, Tp = 3.0 S, ql = 5000 m-2
0.2
0.1
E
i -0.1
t::
w
;; -0.2
:.
-0.3
-0.4
-0.5
Path Longitudinal
Figure 5.10: Path errors, v
= 10 mis,
Tp
(m)
= 3.0 S, ql = 1000, 5000 & 10000 m-2
122
CHAPTER 5: OPTIMAL CONTROL RIDER MODEL
25
20
....
,
15
-
E 10
~
~
5
I-
0
Q)
...0
tr
80
-5
100
-q1
= 1000 ml\(-2)
- -q1 = 5000 ml\(-2)
... q1 = 10000 ml\(-2)
-10
-15
Path Longitudinal
(m)
Figure 5.11: Steer torque, v = 10 mis, Tp = 3.0
S,
ql = 1000, 5000
2
& 10000 m-
1000
Oq1 = 1000 ml\(-2)
.q1 = 5000 ml\(-2)
.q1 = 10000 ml\(-2)
800
III
::.:::
600
C
.iij
o 400
....fI
Q)
en
200
0
x
y
4J
cp
Yr
YfaxJat
ay/at
o4'/at
otfi/at
ofJ/at
-200
State
Figure 5.12: State gains, v
= 10 mis,
Tp
= 3.0
S,
ql = 1000, 5000
2
& 10000 m-
CHAPTER
123
5: OPTIMAL CONTROL RIDER MODEL
3.0
......
,,
2.0
Cl.
:.::
C 1.0
'ji
C)
~
'>'"
e
Il.
0.0
30
-1.0
,,
,
" '
"
-2.0
Preview Distance (m)
Figure 5.13: Preview Gains, v
= 10 mis, Tp = 3.0 S, ql =
1000,5000 & 10000 m-2
,
,,
"",'
,,'
,,"
"
~
Figure 5.14: Contrasting
road information
(right) control strategies.
Loose control aims for more distant target (dashed line),
resulting in corner cutting
requirements
Not Important
of tight (left) and loose
CHAPTER 5: OPTIMAL CONTROL RIDER MODEL
124
800
600
~
r:,-
400
DTp=1.5s
'iij
C!)
.Tp
S
:I
I/)
= 3.0 s
.Tp=4.5s
200
0
Yr
-200
State
Figure 5.15: State gains, v
= 10 mis,
Tp
= 1.5 s, 3.0 s & 4.5
S,
ql
= 5000 m-2
2
Tp = 4.5
Tp 3.0 s
Tp = 1.5 s
=
40
Preview Distance (m)
Figur
5.16: Pr vi w gain, v
= 10 mis,
s
Tp
= 1.5 S, 3.0 S & 4.5 s,
ql
= 5000 m-2
CHAPTER
125
5: OPTIMAL CONTROL RIDER MODEL
20
15
10
E
~
5
•
-Tp=
--Tp
;,
e-0
O k....,...---,r--"'T'""I+r--,--.,..t.....,.-.,...-......---..
...
__
"F"'..,..-~--.
1.5s
= 3.0s
T = 4.5 s
60
-5
-10
-15
Path Longitudinal
Figure 5.17: Steer torque, v
= 10 mis,
Tp
(m)
= 1.5 s, 3.0 s & 4.5 S, ql = 5000 m-2
0.6
0.5
0.4
E
..~
....
w
0.3
-Tp-1.5s
---Tp .. 3.0 s
_•• Tp..4.5 s
0.2
.J:.
Cl..
0.1
0
60
70
-0.1
-0.2
Path Longitudinal
Figure 5.18: Path errors, v
= 10 mis,
Tp
(m)
= 1.5 s, 3.0 s & 4.5 S, ql = 5000 m-2
CHAPTER
126
5: OPTIMAL CONTROL RIDER MODEL
4.5
4
3.5
3
:[
2.5
i!
2
~
...
.s::.
1.5
ftI
Cl.
1
0.5
0
-0.5
40
80
120
160
200
240
280
320
360
Path Longitudinal (m)
Figure 5.19: Path following, v
= 40 ia]»,
Tp
= 3.0 S, ql = 5000 m-2
60
4.5
50
4
3.5
40
E
3
30
~
2.5
ID
:::I
20
2
~
0
I-
...
10
!
I/)
:[
i!
~
..I
-Torque
-Path
1.5 .c
1ii
Cl.
0
280 320 360
-10
0.5
-20
0
-30
-0.5
Path Longitudinal (m)
Figure 5.20: Steer torque, v
= 40 mis,
Tp
= 3.0 S, ql = 5000 m-2
CHAPTER
127
5: OPTIMAL CONTROL RIDER MODEL
2
1.5
a-
lii:
rE
~
Cl
-v=
10mls
--v=40m/s
0.5
~
'>
e
a.
0
-0.5
·1
Preview Time (s)
Figure 5.21: Preview gains, v
= 10 m/s
& 40 mIs, Tp
= 3.0 S, ql = 5000 m-2
5
4.5
4
-.._______.-
3.5
[
3
f
-Global
- -Local
2.5
3
=a.
2
II
-Tar et
1.5
0.5
0
-0.5
20
40
60
100
80
Path Longitudinal
120
140
(m)
Figure 5.22: Path comparison, global vs. local coordinate system, v = 10 mIs,
Tp = 1.5
S, ql
= 5000 m-2
Chapter 6
Model Predictive Control Rider
Model
6.1
Introduction
As the name suggests, Model Predictive Control (MPC) is a technique in which a
prediction of a system's behaviour is made in order to generate some form of optimised
control input. The prediction is based on knowledge of the system's dynamic response
characteristics
and the calculation of a set of future control inputs.
The predicted
future system states can be compared against a set of future target states, with the
set of future control inputs continuously re-evaluated, if necessary, in an attempt to
ensure that the predicted future states match the target future states [35].
Widespread use of predictive control has been applied to the chemical engineering
industry, where the need to predict the future behaviour of chemical reactors and other
similar delayed-response systems was answered through the use of predictive control
techniques. In recent years use has been made of MPC techniques in numerous other
areas, including the modelling of human controllers for the task of driving [11, 72].
The application of predictive control methods to the simulation of a vehicle driver
seems an entirely appropriate
viewed information
choice. In driving/riding
of the approaching
a vehicle, the pilot has pre-
road path to follow and knowledge of the
vehicle's condition. A motorcycle rider will also have knowledge of how the machine
is likely to respond to certain control inputs as a result of previous riding experience,
128
CHAPTER
6: MODEL PREDICTIVE
CONTROL RlDER MODEL
129
and can therefore subconsciously plan his future control actions in order to follow the
road path he sees ahead of him. This process of evaluating the road and vehicle and
forming the necessary future control strategy is evaluated in a continuous manner
throughout
the riding task.
With riding experience, for instance, a motorcycle rider will know that to change
his path heading to, say, the left, he will be required to countersteer to the right at
some distance prior to the point at which he wishes to start the left turn.
While
this may not be a conscious process, the rider has to some extent predicted the
future response of the motorcycle to his initial countersteer to the right based on his
knowledge of typical system response. Similarly, if he were wanting to stop, he would
know that a certain braking distance is required to achieve this, and therefore makes
a subconscious prediction for how long it will take to slow down in choosing the point
at which he will start to brake, and balances this knowledge against the severity of
braking that he is prepared to tolerate.
The riding task that the rider is presented with will change continuously
due to
changes in both the road path as he progresses along a path, and due to the changing
vehicle states during motion of the motorcycle. The predicted future control actions
that the rider has will therefore also be continually updated
in light of the ever-
changing task. The changes in the task may be gradual, such as the changing direction
of the approaching road path, or quite sudden, such as a sudden loss of road traction
as a result of water, debris or oil on the road, causing the vehicle to respond in an
unexpected manner.
Breaking the riding task into small discrete time-steps, although at an instant the
rider may have control predictions for the full picture of his riding task, he will only
ever make use of and apply his first control prediction, before re-evaluating the control
problem and at the next instant using the next first control prediction in a continuous
process.
This chapter covers the theory of MPC techniques and the application to the modelling of a motorcycle rider.
The chapter begins with the detailed theory of the
prediction model, for both a linear prediction model and the more realistic non-linear
prediction model that will account for changes to the motorcycle states over the
course of the prediction horizon.
The opportunities
available for the definition of
CHAPTER
6: MODEL PREDICTIVE
130
CONTROL RIDER MODEL
a reference path for the motorcycle to attempt
to follow are considered, presenting
the theory for the application of different reference path definitions. The theory for
predictive controller gains is outlined before the strategy is applied to the motorcycle
model attempting
a lane change manoeuvre.
A wide range of controller parame-
ters is considered, including preview and control horizons, forward speed and cost
function weighting parameters,
definitions.
together with the effects of different reference path
Hence the applicability of the technique in replicating the actions of a
motorcycle rider and the detailed characteristics
of the controller will be ascertained.
Subsequently, in Chapter 7, the comparisons between this approach and the previous
optimal control approach detailed in Chapter 5 will be made, in order to present the
potential advantages that may be found when using the predictive control approach.
6.2
MPC Theory
The MPC approach [52] consists of two fundamental parameters that differentiate it
from other similar techniques such as optimal control, namely the prediction model
and the reference path definition that the controller attempts to follow.
Model predictive control, as suggested by the name, forms a prediction model to
anticipate
the future response of the system using a known set of future control
inputs, and by making use of the known system response to controlling inputs.
A
motorcycle rider, for example, will have a reasonably accurate knowledge of how the
motorcycle will respond to his controlling inputs, and therefore it can be said that
he has knowledge of the system response to control inputs. In riding a motorcycle,
the rider will be looking ahead at the road and subconsciously will have anticipated
his future control inputs in response to the road path he sees ahead of him, and
consequently he may have anticipated
a set of predicted future control inputs. This
is the primary difference between an MPC approach and an optimal control approach
(Chapter 5), where a control input is generated only for the motorcycle's
current
position and based on the current motorcycle states only.
The second fundamental difference concerns the system output that the rider is aiming
to follow. The MPC technique is defined with two output paths, known as the set
path and the reference path. The set path is the absolute target of the system which
it ultimately aims to follow, in this case the road centreline.
It is unlikely that the
CHAPTER
6: MODEL PREDICTIVE
CONTROL RIDER MODEL
131
system will follow exactly the set path. In the case of the motorcycle, as seen with
the optimal control approach in Chapter 5, there is always some element of corner
cutting. The reference path is therefore a newly defined path that takes the system
from some position displaced from the set path, and that returns to the set path over
some time and distance. This condition is depicted in Figure 6.1.
The reference path itself can be defined in any manner chosen. It can be a simple
step, such that the reference path is the set path, it can be defined by a linear path
from the current system states to target system states over a finite time, or indeed
can be an exponential, quadratic or any other chosen path definition. The reference
path, as its name suggests, is simply a reference which the system aims to follow.
It is accepted that the motorcycle will not follow the road path exactly, with some '"""?
corner-cutting
instantaneously;
likely. A motorcycle is not capable of correcting a lateral path error
the rider must steer, and with forward motion the lateral position of
the motorcycle on the road can be changed, and thus any lateral path errors resolved.
A rider will therefore aim to follow a path that will take him from his current, possibly
displaced, position to his target path position at some point in his future.
Another feature of predictive control, which is of less use for the application here~
includes the capacity to deal with system response lag. Such control is widely used in! ~dl~
the chemical engineering industry, where the response to control is often not realised
I)..d;J'~'~
until some time in the future, and is in general known as the dead-time or dead-zone.
Thus, the prediction element of the control strategy can take account of the delay
that will exist between control input and output response and can therefore predict
the need to apply the necessary control in advance. The ability to replicate this lag ~
in the control strategy can, for some applications, be a vital component.
These features of MPC make the application to modelling a motorcycle rider potentially more suitable than the optimal control technique. For simplicity, the system is
initially defined with the reference path and set path equal, such that the effects of
the prediction model can be more closely compared with the results of the optimal
control technique.
Subsequently, some analysis is made into the effects of different
reference path definitions on the performance of a rider model simulated using MPC
techniques.
It is necessary, at this point, to clarify some of the terminology that will subsequently
CHAPTER 6: MODEL PREDICTIVE CONTROL ruDER MODEL
132
be used, primarily concerning the distinction between the simulation and the prediction.
The goal here is to create a model that replicates a motorcycle rider negotiating a
path manoeuvre.
The complete representation
of the motorcycle following the path
is the simulation model. Thus, the simulation is the actual path that the motorcycle
model follows as a result of the rider model's control actions. The simulation model,
using optimal control techniques, was also presented in Chapter 5.
In addition to thi~L~he M_:p_Q__!l.pproacll_~~_~J>,!~_9.ict_i_()I1JI19del.
The simulation model
used here is a discrete sample simulation. .At each discrete step of the simulation,
the control strategy forms a prediction model, itself formed as series of discrete step,
which will predict the future path of the motorcycle up to a finite horizon, from that
particular discrete simulation step .. The prediction model for that simulation step will
be used to calculate the required control input for that one simulation step. Once the
control has been applied, and the motorcycle simulation moves to the next discrete
step, a new prediction model will be formed up to the finite horizon from the new
discrete simulation step, the new control input calculated, and the simulation model
moved to the next simulation step.
The principles of the simulation and prediction models are, in fact, almost identical,
however the distinction between the simulation and the prediction model is important
to be aware of.
~.1
Linear Prediction Model
MPC techniques develop a prediction model that aims to anticipate the future response of the controlled system based on a set of future control inputs. Comparison
between the predicted future outputs and the required future outputs defined by the
,r
reference path result in a set of errors. ~he controller forms a cost function combining
these errors with the control input cost and, in a similar manner to the optimal control approach (Chapter 5), determines the best set of future control inputs to balance
output error minimisation against control input effort.
As with the optimal control approach, the starting point in the analysis of MPC
techniques is with the equations of motion for the dynamic system response, formed
)J~w.~
~ '1
CHAPTER
6: MODEL PREDICTIVE
CONTROL RIDER MODEL
133
into the convenient linearised discrete-time state space system model, as presented in
Section 3.4.1. For a generic system at the kth step, this takes the form
+ 1)
. x(k
+ B(k)u(k)
= A(k)x(k)
y(k) = C(k)x(k)
+ D(k)u(k)
(6.1a)
(6.1b)
where the vectors x(k), y(k) and u(k) represent respectively the system states, system
output and system input, and the matrices A(k), B(k), C(k) and D(k) are the
discrete-time state space matrices, all at the kth step.
The distinction between the prediction and the simulation is emphasised again here.
The expression given in (6.1), when used with the actual state values and control
inputs, generates the actual system output and the states of the system at the next
step. This therefore represents a discrete step in the simulation. Here, we use the
dynamic response model of the system (based on the predicted states x(k» given by
this expression to determine a predicted future set of states and a predicted future
system output using predicted future states and control input:
x(k
y(k
+ 1)
+ 1)
=
= A(k)x(k)
C(k
+ B(k)u(k)
+ l)x(k + 1) + D(k + l)u(k + 1)
(6.2a)
(6.2b)
Since the state space matrices A(k), B(k), C(k) and D(k) are dependent on the
system states x(k), they too will be predicted when x(k) is used, and hence are given
here as A(k), B(k), C(k) and D(k).
At step k, the state vector x(k
+ 1) is not
available directly in (6.2b), but can be
obtained by substitution of (6.2a):
x(k
y(k
+ 1)
= C(k
+ 1)
= A(k)x(k)
+ l)[A(k)x(k)
+ B(k)u(k)
+ B(k)u(k)] + D(k + l)u(k + 1)
(6.3a)
(6.3b)
CHAPTER
6: MODEL PREDICTIVE
CONTROL RIDER MODEL
134
The distinction here is drawn between the optimal control output, where the system
output relates to the system's current states, and predictive control where in effect the
output is the predicted future path. The error that the controller aims to minimise
is between the states of the system at the current kth step, and the target states at
the kth step. For the optimal control approach, the first preview point is the system
information relative to the system's current position i.e. the road information for
the motorcycle's current discrete simulation point. The rider preview element and
corresponding state space formualtion for this condition was given in Chapter 4.
For the predictive control approach, the output considered is the future predicted')
output y(k
+ 1),
011
and the subsequent minimisation of this predicted future outpu.:)
against a future target. The preview horizon is therefore subtly different. The preview
horizon for the predictive controller will begin at the first prediction point in the
rider's horizon, in other words one step ahead of the motorcycle's current position,
i.e. at k
+ 1.
" e/vj
~}~p
The theory for the state space modelling of this predictive control
rider preview is no different from the approach given in Chapter 4. Provided that
the initial information fed into the rider's road preview vector corresponds to the
information one step ahead, and the new information fed into the preview vecto
similarly accounts for the correct new information point, the shift-register mat .
form is no different.
~
discrete-time state space representation of the system dynamics given in (6.3)
can be used to predict the output of the. system to some predetermined number of
iteration steps in the future [11]. The prediction begins from the current simulation
point, for which the vector x(k), and the matrices A(k), B(k), C(k) and D(k) are
known, and are x(k), A(k), B(k), C(k) and D(k). The input vector u(k) is, at this
point, a predicted input still, but will subsequently be recalculated as u(k), which
will be used for the simulation step. Thus, the system information at step k is used
...----__.----.~ .. --"~-"'---'-'-' .
to predict the first output in the rider's preview horizon, y(k
"-
----.,.--~
..-
..
,
,,-,
~~-...-.-.,.
With knowledge or prediction of the second control input u(k
+ 1)
.
+ 1), a similar process
can be applied to determine the system output after the second iteration:
x(k
+ 2)
y(k
+ 2) =
= A(k
C(k
+ l)x(k + 1) + B(k + l)u(k + 1)
(6.4a)
+ 2)x(k + 2) + D(k + 2)u(k + 2)
(6.4b)
~~
CHAPTER
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135
CONTROL RIDER MODEL
Only the current states x(k) and state space matrices A(k), B(k), C(k) and D(k),
for the current step, are exactly known, and therefore the solution to (6.4) cannot be
solved directly. Howeverthe result from (6.3) can be employed here, and, substituted
appropriately in (6.4), will result in the predicted system response at k
x(k
y(k
+ 2)
+ 2)
= A(k
+ 2 as:
+ I)[A(k)x(k) + B(k)u(k)) + B(k + I)u(k + 1)
(6.5a)
+ 2){A(k + I)[A(k)x(k) + B(k)u(k)) + B(k + I)u(k + I)}
+D(k + 2)u(k + 2)
= C(k
(6.5b)
For a linear prediction model, it is assumed that the state space matrices A(k
B(k
+ ip),
C(k)(k
+ ip)
and D(k)(k
+ ip),
+ ip),
where ip is the number of the prediction
step, are invariant over the full prediction, defining them now simply as A, B, C and
D. That is to say that at the kth iteration step, the linear state space matrices are
obtained and it is assumed that these matrices are valid over the full prediction at the
kth
step. However,when the controller gains are calculated and the simulation moved
to the next step, the state space matrices are re-evaluated, and these new matrices
used for the full prediction horizon at that next simulation step.
Furthermore, the system output is assumed to be based solely on the system states,
and not the control input. Thus, the matrix D is set to zero and can therefore be
removed from the expressions.
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136
CONTROL RIDER MODEL
With the simplifications of the predicted state space matrices made, the expressions
(6.2) and (6.4) can be redefined as
x(k + I}
=
Ax(k} + BU(k}
y(k + 1)
=
=
=
Cx(k+
=
=
=
Ax(k + I} + Bu(k + I}
=
=
=
Cx(k +2}
x(k + 2}
y(k + 2}
~
1)
C[Ax(k) + BU(k}]
CAx(k) + CBu(k)
A[Ax{k}
A2x{k}
(6.6)
+ BU(k)] + Bu{k + 1)
+ ABu(k} + Bu(k + 1)
C[Ax(k + 1) + Bu(k + 1)]
CAx{k + 1) + CBu(k + I}
is seen from the above expressions that a pattern
is starting to emerge for the
prediction of the future states and outputs using the initial vehicle states x(k) and
the predicted future control inputs u(k), u(k
+ I}....
This pattern
can readily be
formed into a state space expression, calculating all future system states and system
outputs using the initial states x(k) and vector of predicted future control inputs
u(k), u(k
+ I}, etc.:
i(k+1}
i(k+2}
i(k+3}
A
B
o
o
liCk}
AB
B
o
u(k+l}
AB
o
li(k+2}
B
u(k+(Np-l})
(6.7a)
CHAPTER
6: MODEL PREDICTIVE
CONTROL RIDER MODEL
137
and similarly for the system output
y(k+1)
CA
y(k+2)
y(k+3)
CA3
x(k)
y(k+Np)
CANp
(6.7b)
CB
0
0
u(k)
CAB
CB
0
u(k+1)
CA2B
CAB
0
u(k+2)
CANp-1B
CANp-2B
CB
+
u(k+(Np-l»
which can be represented by
X(k) = F(k)x(k)
+ G(k)Y(k)
(6.8a)
~(k) = M(k)x(k)
+ N(k)Y(k)
(6.8b)
where X(k), Cf!(k) and Y{k) represent the full vectors of all predicted future states
vectors, the vectors of predicted future outputs and the vectors of predicted future
inputs at the
kth
step respectively.
The matrices F(k),
a particular
G(k),
M(k)
and N(k)
have been defined here specifically for
iteration step k, while the constitutive
elements given in (6.7b) do not
show such time-dependency.
A reminder is made to the elements of the matrices
F(k), G(k), M(k)
at the simulation step k for a linear prediction model,
and N(k):
the system state space matrices A(k),
B(k),
C(k) and D(k) are used to form all of
the elements in (6.7), assumed invariant over the prediction horizon, and, only for
clarity, are defined as A, B, C and D in (6.7).
To this point, the assumption has been made that the preview horizon and the control
horizon are equal, and that there are consequently an equal number Np of future road
CHAPTER
6: MODEL PREDICTIVE
138
CONTROL RIDER MODEL
preview points and future control inputs.
Details relating to the discrete preview
horizon can be found in Chapter 4.
This condition is not strictly necessary, and it is possible that the control horizon can
be set shorter than the preview horizon. Defining the number of control inputs in the
control horizon as Nu, where Nu is the control horizon time Tu divided by the discrete
time step interval t, the control inputs can be held constant from the control horizon
up to the prediction horizon, such that the future control input vector is defined as
Y(k) = [uT(k)
+ 1)
uT(k
...
uT(k
+ (Nu
-1))
...
uT(k
+ (Nu
-1))
r
(6.9)
In this case, columns Nu to Np of the matrices G(k) and N(k) are all multiplied by
the control input u(k+ (Nu -1)), and thus these columns can be summed such that
the expressions (6.7a) and (6.7b) become
i:(k+1)
A
x(k+2)
A'l
x(k+2)
A3
x(k+Np)
AN"
+
x(k)
(6.lOa)
B
0
0
u(k)
AB
B
0
u(k+1)
A2B
AB
0
u(k+2)
AN,,-lB
ANp-2B
EN,,-NuAiB
,=0
u(k+(Nu-l»
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139
CONTROL RIDER MODEL
and again for the system output
y(k+1)
CA
y(k+2)
CA2
y(k+3)
CA3
y(k+Np)
CANp
+
x(k)
CB
0
0
u(k)
CAB
CB
0
u(k+1)
CA2B
CAB
0
u(k+2)
u(k+(Nu-1))
(6.10b)
and the matrices F(k), G(k), M(k), and N(k) are now redefined using the modified
expressions (6.10a) and (6.10b).
~erefore
results in a state space representation
of the predicted future states
and future outputs of a modelled system allowing for different control and prediction
horizons. Replacing the generic state space matrices A, Band
C with the motorcy-
cle state space matrices Av(k), Bv(k) and Cv(k), the prediction model can be used
in defining a control strategy to control the motorcycle and accomplish the path
following task.
6.2.2
Non-Linear Prediction Model
The theory of MPC and the prediction model, used to anticipate the system output
/
I
at some point in the future, is the characteristic
feature of MPC that differentiates it
from optimal control techniques. The controller attempts to control the system such
that the anticipated future states match the target future states. It should therefore
be apparent
that a fundamental
requirement
of an MPC controller is to have an
accurate prediction model if useful controller performance is to be obtained.
For the case of time-invariant
systems, the theory has already been discussed through
I
I
CHAPTER
6: MODEL PREDICTIVE
140
CONTROL RIDER MODEL
the derivation of equations (6.1) through to (6.10). The time-invariant
nature of the
state space model presented eased the analysis considerably, as at each iteration step
k of the model simulation, the sub-matrix elements of the generic prediction model
matrices were multiples of only three fixed matrices, A(k}, B(k} and C(k}.
Non-linear state space models will have matrices that will vary linearly with changes
in the system state vector, and thus may be time-variant.
In such cases, it is necessary
to linearise the equations of motion for a specific set of states. The model can then
be used as a linear model within small variations only of the individual states, before
system accuracy degrades excessively. Typically, the model state space matrices will
be linearised for a set of system states at each iteration step, and with each subsequent
iteration step the model state space matrices will then be re-linearised for the new
system states (Figure 6.2). For a system such as a motorcycle, whose states change
significantly over time, it is necessary to re-linearise repeatedly the state space model
to retain some accuracy of the linearised state space model.
The prediction model described in Section 6.2.1 makes use of the system state space
matrices A(k}, B(k} and C(k) over the preview horizon, yet the predicted system
states may change significantly from the actual model states to the states at another
point in the predicted future horizon. Therefore use of a fixed state space representation of the system, applied over the whole prediction horizon, may not be accurate.
For technical accuracy, it would be necessary to evaluate and line arise the prediction
model at the first step of the prediction horizon, before moving forward to predict
the next system response in the predicted horizon, re-evaluating
and re-linearising
the model, and repeating up to the limit of the prediction horizon.
Specifically, the prediction of the vehicle states one step ahead of the motorcycle will,
as before, be given by
x(k
y(k
+ 1)
+ I}
= C(k
= A(k}x(k)
+ B(k)u(k}
+ l}x(k + I} + D(k + l}u(k + 1)
(6.11a)
(6.11b)
The predicted state space equations A(k), B(k}, C(k) and D(k), linearised at k, are
valid over the step k to k
also valid.
+ 1, and
therefore the prediction over this iteration step is
CHAPTER 6: MODEL PREDICTIVE CONTROL RlDER MODEL
141
At the prediction point k + 1, the system states will be different from at the kth point.
The system state space matrices are calculated based on the values of the states, and
therefore A(k) ::;.A(k
+ 1),
+ 1) etc.
B(k) ::;.B(k
Again, the output will be assumed
to have no dependency on the input, and therefore D(k) is set to zero. Over the
prediction iteration k
x(k
+ 1 to k + 2, the
+ 2)
= A(k
correct prediction is therefore given by
+ l)x(k + 1) + B(k + l)u(k + 1)
y(k + 2) = C(k + 2)x(k + 2)
Extending
(6.12a)
(6.12b)
this sequence up to the prediction horizon, the appropriate
prediction
matrices again take the form
X(k) = F(k)x(k)
+ G(k)Y(k)
(6.13)
cp(k) = M(k)x(k)
+ N(k)Y(k)
(6.14)
except that this time the appropriate matrices are given by
.A(k)
.A(k+l).A(k)
F(k)=
(6.15)
.A(k+2).A(k+l).A(k)
(Npxn)
B(k)
G(k)=
0
0
.A(k+l)B(k)
B(k+l)
0
.A(k+2).A(k+1)B(k)
.A(k+2)B(k+l)
0
Np-l
II A(k
i=l
Np-l
+ i):8(k)
II A(k
i=2
+ i)B(k + 1)
B(k+Np-l)
(NpnxNpn)
(6.16)
CHAPTER 6: MODEL PREDICTIVE CONTROL ruDER MODEL
142
C(k+l)A(k)
C(k+2)A(k+l)A(k)
M(k)=
(6.17)
C(k+3)A(k+2)A(k+l)A(k)
(Npmxn)
O~+~AW
C(k+2)A(k+l)
o
C(k+3)A(k+l)B(k+l)
o
O(k+2)A(k+l)A(k)
N(k)=
o
0
C(k+3)A(k+2)A(k+l)A(k+l)B(k)
(NpmxNpn)
(6.18)
This approach is technically more correct than the method outlined by equations
(6.1) to (6.10), although from a computational
perspective it is considerably more
demanding.
For the simplified approach, at each simulation iteration step, the state space matrices
are calculated, and from this the prediction model matrices F(k), G(k), M(k) and
N(k) are calculated once. These are then used to predict the future system output
and consequently generate the controller gains and input.
Using the more technically correct approach outlined here, at each iteration
of the simulation model, the prediction of the system states at (k
step ahead of the motorcycle, are calculated.
vector x(k
+ 1)
1), matrices A(k
prediction of x(k
+ 1),
step
i.e. one
To achieve this, the predicted state
+
is calculated using matrices A(k), B(k) and C(k).
Using x(k
+ 1), B(k + 1) and C(k + 1) are then calculated,
+ 2), subsequently A(k + 2), B(k + 2) and C(k + 2),
allowing the
x(k
+ 2),
etc.,
gradually populating the matrices F(k), G(k), M(k) and N(k), with the necessary
terms and thus eventually allowing the necessary controller gains and system input
to be obtained for one iteration step of the simulated model's motion (Figure 6.3).
The value of
u; can be large,
typically in excess of 100, and consequently, this makesl
the simulation process using a full non-linear prediction model a considerably m~
~Ir
V, I
CHAPTER
6: MODEL PREDICTIVE
computationally
demanding task.
143
CONTROL RIDER MODEL
The implications of this will be assessed in the
results, to determine how critical it is to the performance of the controller to model
the prediction in this way, or whether acceptable performance can be achieved with
the relatively more simple prediction model.
6.2.3
Reference Path Definition
A key feature of the MPC technique that differentiates it from optimal control is the
use of a reference path.
__
This is a path that the system output attempts
to follow
in order to take the system from a displaced state to the target
state in a defined
.
..
manner over a defined time step. In the case of a motorcyclemodel;
~hi!,!
..~oIl!lt_i_t1J.tes-t__
a path that a rider would follow to take him from a position off the1deal;1riferided
road path to regain the target path over a certain forward distance.
The definition of how the system is tasked to get from the displaced state to the target
state is open to interpretation
into the MPC system.
and is another control variable that can be introduced
Typically, the system might be tasked with regaining the
target path by a linear, exponential or sinusoidal return path, for example.
These
three conditions are depicted in Figure 6.4.
»>specific definitions
VTwo
are made here relating to the terminology that will be used.
For the path information, a subscript 'r' will relate to the actual road, the set path,
and the subscript'
f' will relate
to the reference path.
The applicability of the various options for defining the reference path is discussed in
the results sections. For now, the theory required to define a typical reference path
is outlined.
Linear Reference Path
A linear reference path is considered first, and can be achieved with relative ease.
With knowledge of the current system state and the final, target state, the intermediate points are simply interpolated
points (Figure 6.5).
linearly based on the difference between the two
CHAPTER
6: MODEL PREDICTIVE
CONTROL RIDER MODEL
144
Assuming the current system state to be at zero, the ith reference path point at the
kth iteration step YJ. (k), for i = 1, ... ,Np, is achieved with
(6.19)
Extending this to the general case of non-zero current motorcycle position Yv(k),
results in
(6.20)
which simplifies to
(6.21)
Writing the expression explicitly for all discrete points in the preview horizon, this
becomes
vt. (k)
~YrNp
Yh(k)
-#;YrNp
Yh(k)
=
YfNp-l (k)
~YrNp
+ (NN:1)Yv(k)
+ (NN:2)Yv(k)
+ (NN:3)Yv(k)
(6.22)
NN:1YrNp + ~Yv(k)
YfNp(k)
YrNp
The required form for the definition of the reference path is as a discrete-time state
space model such that it can be combined with the motorcycle model as defined
previously (4.2). In the discrete-time
state space representation
of the system, the
system states, which includes the path information, are required to be calculated for
the system at the (k
+ 1)th
step, not at the kth step as given by (6.22). Thus, in
calculating the new reference path for the (k
+ 1)th
motorcycle from the position of the motorcycle at (k
(k
+ 1).
step, this path should take the
+ 1) to
the target road point at
145
CHAPTER 6: MODEL PREDICTIVE CONTROL ruDER MODEL
The
ith
reference path point for the iteration step (k
+ 1) that
should be generated
by the state space model, from (6.21), is therefore given by
YJ.(k
i
+ 1) = Np YrN
(k
I'
where the position of the motorcycle Yv(k
yv(k
+ 1) =
+ 1) +
Np-i
N Yv(k
P
+ 1) is given
Av(k)(2,j):XV(k)
+ 1)
(6.23)
by
+ Bv(k)(2,j)u(k)
(6.24)
and where the selection of the second rows of the matrices Av(k)
and Bv(k)
corre-
spond to the location of the lateral position state in the motorcycle state vector and
j is the index 1, ... , n, indicating that the whole of the second row is selected.
Using the result from (6.23), the discrete-time state space representation
(6.22) be-
comes
+ 1)
+ 1)
YI1 (k
Yh(k
Y/3(k+l)
YJNp-l (k
YIN I' (k
+ 1) + (Nk-l)Yv(k + 1)
(k + 1) + (Nk-2)Yv(k + 1)
(k + 1) + (Nk-3)Yv(k + 1)
.J.-YrN
(k
Hp
I'
J-YrN
=
JVp
I'
J-YrN
Hp
I'
I'
I'
I'
+ 1)
+ 1)
(Npxl)
If the road preview vector Yr(k)
is extended to include (Np
points, then the first terms in the right-hand-side
from ApYr(k),
(6.25)
+ 1)
road information
of (6.25) can be readily obtained
where
1
0
0
N;
0
0
N;
0
0
N;
2
3
(6.26)
Ap=
0
0
NN~l
0
0
1
0
0
0
.
(Np+1xn)
CHAPTER
6: MODEL PREDICTIVE
The matrix Ap is time-invariant,
CONTROL RIDER MODEL
146
and therefore does not require the time-step identi-
fier (k).
The second element terms in the right-hand-side
(N~- i) yv(k
p
+ 1) =
of (6.25) are obtained from
(N~- i) (Av(k)(2,j)Xv(k)
+ Bv(k)(2,j)u(k))
(6.27)
p
Thus, the second terms in the right-hand-side
of (6.25) can be obtained with
(6.28)
where
NN:1 Av(k)(l,j)
NN:2 Av(k)(l,j)
NN:3 Av(k)(l,j)
(6.29)
kAv(k)(l,j}
p
o
o
and
NN:1Bv(k)(1,j)
NN:2Bv(k)(1,j)
NN:3Bv(k)(1,j)
(6.30)
kBv(k)(l,j)
p
o
o
The preceding theory has considered the reference path defined only by the lateral
position of the road path, but can be extended to include the yaw angle using a
similar approach.
CHAPTER
6: MODEL PREDICTIVE
147
CONTROL RIDER MODEL
Extending the previous definition of the discrete-time state space system model (6.1)
to include (Np
+ 1) preview
points for a linear reference path, the whole process can,
using the matrices defined above, be re-expressed again in the simple state space form
using the matrices defined above:
o ] [xv(k)
Ap
y,(k)
1+ [
Bv(k)
B,(k)
1 () + [0 1
uk
Yrn(k)
Bp
(6.31)
where Y rn (k) represents again the new preview point fed into the system, corresponding here to a point Np
is the corresponding
+2
steps ahead of the current motorcycle position, and Bp
matrix to input this new information into the rider's preview
as given in Chapter 4, (4.2). This matrix form therefore generates the future road
path that the rider attempts to follow as a linear path between the motorcycle's current position and the actual road path information at the limit of the rider's preview
horizon.
Linear Error
Reduction Path
The previous reference path definition was for a linear reference path, going directly
from the motorcycle's
position to some target position at the limit of the rider's
preview, taking no account of the path direction in between.
Here, the definition
is subtly different, with the linearity being in the reduction of the error between
the set path and the reference path.
Thus, if the set path curves, so too will the
reference path, but the error between the two will be a linear function that will see
the reference path gradually track back onto the set path by the limit of the preview
horizon (Figure 6.6).
To achieve this, the reference path is taken as the set path less some multiple of the
difference between the motorcycle and the relative set path position. This multiple
is a linear function from one to zero, such that the error between the set path and
reference path gradually and linearly reduces to zero also.
The preview horizon includes Np preview points, where the first preview point is one
step ahead of the motorcycle (Figure 6.6). Thus, there are Np discrete steps between
CHAPTER
6: MODEL PREDICTIVE
148
CONTROL RIDER MODEL
the motorcycle and the limit of the preview.
The function that therefore gives the linear function from zero at the motorcycle to
one at the limit of the preview horizon is, for the ith preview point, given by 1- NIt.
p
Numerically then, the ith point on the reference path is given by
(6.32)
which can be simplified to
(6.33)
It is seen from this definition that, unlike the direct linear reference path, in this case
the full set path information, i.e. Yr;(k) for i = 1, ... , Np, is required. This naturally
makes this approach more complicated, as the complete system state vector must
now therefore include the motorcycle states, the set path road information and the
reference path road information in the rider's preview horizon. Since the required
format for the modelling is the state space representation, (6.33) must also be written
in a discrete-time state space form, such that Y!; (k
+ 1) can be obtained.
The set path information is updated by the simple shift-register matrix. Thus, Yr, (k+
1) = Yri+l (k) as before.
The motorcycle lateral position Yv(k
+ 1) is also readily
obtained by using the ap-
propriate elements of the state space matrices for the motorcycle dynamics as seen
previously, i.e. Yv(k
+ 1) = Av(k)(2,j)xv(k)
+ Bv(k)(2,j)u(k).
Thus, the necessary
complete system state space model can be achieved with
where the matrices Av(k) and Bv(k) are as defined in 3.23, Ap and Bp are as in 4.2,
and additionally,
CHAPTER
6: MODEL PREDICTIVE
149
CONTROL RIDER MODEL
:l
ilv(k)(2,j)lf :2
ilv(k)(2,j)lf :3
ilv(k)(2,j)lfJV
JV
il8(k)
=
JV
(6.35)
ilv(k)(2,j)#,;
0
0
ilr=
(1-~)
0
(1-~)
(If,,xn)
0 0
0
0 0
0
0
0
0
0
0
0 0
0
0
0
0
0 0
(1-#,;)
0
0
0
0 0
0
(6.36)
(If,,xlf,,)
Bv(k)(2, j) lfJV:1
Bv(k)(2, j) lfJV:2
:3
Bv(k)(2,jlfJV
(6.37)
Bv(k)(2,j)#,;
o
(N"xp)
Exponential Reference Path
The definition of the reference path as an exponential curve from the current vehicle
state to the target vehicle state follows from the theory outlined for the linear error
reduction case. To define an exponential path using the linear reference path theory
would be incorrect, as highlighted by Figure 6.7. Rather than gradually recovering
the set path, the reference path could potentially overshoot the set path and return
to it non-tangentially.
Although the path is indeed an exponential path, no account is taken for the trajectory
of the path, only the final target point.
exponentially
The proposed form is therefore to reduce
the error between the motorcycle and the reference path, using the
CHAPTER
6: MODEL PREDICTIVE
150
CONTROL RlDER MODEL
method outlined for the linear error reduction approach above. This would lead to a
path that may typically look something like the situation presented in Figure 6.8.
Specifically,an exponential growth from zero to one can be defined by
f(x) = 1- exp(-xJ1r)
(6.38)
where lr is a value that defines the relaxation length of the exponential growth. Thus,
modifying (6.33) to account for an exponential reduction results in
(6.39)
Comparing (6.39) with (6.33), it can be seen that by using the same approach as for
the linear error reduction reference path, the state space matrices As(k), Ar, Bf(k)
and Bfn(k) for the exponential error reduction reference path will be the same as
those for the linear error reduction case, modified only such that the terms
replaced by exp(l-
NJrpi
are
i)J1r).
6.2.4 MPC Optimal Gains
¥Ptimal
control approach formulated a cost function consisting of elements relat-
ing to the system's output accuracy and control effort input. Weighting parameters
were included in the cost function, such that the relative contributions of each to the
overall cost could be varied, and as such the controller's bias towards accuracy or
control efficiencycould be adjusted.
The MPC approach is very similar; the notable difference between the two is that
for MPC, the output error penalised in the cost function is the error between the
predicted future controlled system output and the reference path. The theory for the
generation of the model predictive controller gains is as follows [U, 52].
Considering the lateral positions of the motorcycle and the path, the error function
of an individual preview point is defined as
(6.40)
CHAPTER
6: MODEL PREDICTIVE
where 11i(k) the predicted output of the
output at the
kth
151
CONTROL RlDER MODEL
ith
preview point and YJ;(k) is the reference
iteration step. Extending to the full system preview, the vector of
all errors in the preview horizon is therefore defined by
e(k) = cp(k) -YJ(k)
=
(M(k)x(k)+N(k)l'(k))
-YJ(k)
= N(k)l'(k)
-e(k)
(6.41)
As with the optimal control cost function, the MPC cost function is formed by the
weighted sum of squares of the path followingerrors and the control input costs, such
that
N-l
J =
L (eT(k)Qe(k)
+ l'T(k)Rl'(k))
(6.42)
k=O
where Q and R are again invariant weighting matrices on the system outputs and
control inputs. Here, they take the form of diagonal matrices, where the elements on
the diagonal correspond to the weightings on the outputs [ql, qz, ... ], repeating Np
times, and the weightings on the inputs
[Tb T2, ••• ]
repeating Nu times.
The weighting vectors on the output and inputs are decomposed as
Q = S~SQ
(6.43)
R=S~SR
in which case the cost function (6.42) can alternatively be expressed as
J =
[SQ(N(k)l'(k)
- e(k))]
2
(6.44)
SRl'(k)
where SQ and SR are the square symmetric roots of the matrices Q and R. In this case
they are symmetric, diagonal matrices and the square root can readily be obtained
as the squares of the individual terms on the diagonals. For more elaborate matrices,
other methods such as single value decomposition or Choleski decomposition may be
required.
Thus, the control input l'(k) that minimises the cost function J is obtained from the
least-squares solution [52]of the equation
152
CHAPTER 6: MODEL PREDICTIVE CONTROL RIDER MODEL
SQ(N(k)l'(k)
[
- e(k))
1
=0
(6.45)
SRY(k)
The optimal gains for the controller can be obtained by making the substitution
l'(k) = K(k)e(k)
(6.46)
and using QR decomposition, such that
,,(k) = [ SQ:(k) ] \ [ SoQ
1
(6.47)
This then gives a full matrix of dimension (Nup) x (Npm) where p and m are respec- (
tively the numbers of control inputs and controlled outputs. Each row of the matrix
I ~
K(k) corresponds to the optimum gain multipliers for each of the
!
Nu
future control
I
inputs. The optimum future inputs are obtained by multiplying each row of this \
matrix K(k) with the vector of future errors, s, i.e. using the previous substitution
Y(k) = K(k)e(k). At each iteration step of the program, it is only the first element
u(k) in the future control input vector l'(k) that is used, and hence it is only the first
row that is required to calculate the next control input. At the next iteration step,
the problem is re-evaluated, a new matrix K(k) calculated and hence the next new
control input u(k) calculated. Thus, the optimal MPC input applied to the system
can be defined as
u(k) = k(k)e(k) = k(k)[y,(k)
- M(k)x(k)]
(6.48)
where k(k) refers to the first column of the gains matrix K(k).
6.2.5
Application to Motorcycle Rider Modelling
The preceding section outlined the fundamental mathematical principles of the MPC
theory, using generic states vector x(k) and generic input vector u(k). The specific
application of the MPC technique to the motorcycle rider model is now explored.
~h r I
_6tA.
CHAPTER
6: MODEL PREDICTIVE
153
CONTROL RIDER MODEL
In defining the theory of MPC, the discrete-time
state space model of the system
dynamics was given as
x(k
+ 1)
+ B(k)u(k)
= A(k)x(k)
y(k) = C(k)x(k)
(6.49a)
+ D(k)u(k)
(6.49b)
Equation (4.3) showed how the motorcycle dynamic response model was combined
with a state space representation
motorcycle-preview
Xv(k + 1)
[ Yr(k+1)
1
of the rider's road preview to develop the combined
model given as
= [ Av(k)
0
o
Ap
1
[Xv(k)
Yr(k)
1+ [
Bv(k)
0
1
u(k)
+[
0
Bp
1
Yrn(k)
(6.50)
where all terms have the same meanings as before; xv(k) is the vector of the motorcycle states, Yr(k) the vector of the road preview information, Yrn(k) is the vector
of the new road preview point information, and Av(k) and Bv(k) are the state space
matrices of the motorcycle dynamics, all at the
are time-invariant
kth
step. The matrices Ap and Bp
state space matrices for the road preview shift-register algorithm.
u(k) is the system input vector. Here, only a single input, steer torque, is present,
and so u(k) can simplify to u(k).
The control input has no effect on the approaching road path, it is a defined feature,
and therefore to generate a prediction model for the road path would be unnecessary.
Omitting the (k) for simplicity of presentation,
the predicted future states that are
generated in the MPC technique are therefore related to the motorcycle dynamics
alone, and are given as
154
CHAPTER 6: MODEL PREDICTIVE CONTROL RIDER MODEL
Xv (k+l)
Av
x,,(k+2)
x,,(k+3)
A~
x,,(k+Np)
Ar:
x,,(k)
p
+
(6.51a)
B"
0
0
u(k)
A"B"
B"
0
u(k+l)
A~B"
A"Bv
0
u(k+2)
Ar:p-1B" Ar:p-2B"
Bv
u(k+Np-l)
Similarly, for the system output
y,,(k+l)
C"A"
y,,(k+2)
CvA~
Y,,(k+3)
C"A~
y,,(k+Np)
CvAr:P
+
x(k)
(6.51b)
C"B"
0
0
u(k)
C"A"B"
C"B"
0
u(k+l)
CvA~Bv
CvAvBv
0
u(k+2)
C"B"
u(k+Np-l)
C"Ar:p-1B" c"Ar:p-2B"
The prediction model for the motorcycle dynamics can be represented by
Xv{k
+ 1) = Fv{k)xv{k) + Gv{k)'l'{k)
(6.52)
(6.53)
If necessary, the rightmost (Np
-
Nu) columns of the matrix Gv{k) can be summed
as shown for the generic controlled system (6.10). The control inputs from the limit
of the control horizon (Nu) up to the limit of the preview horizon (Np) are considered
,!
155
CHAPTER 6: MODEL PREDICTIVE CONTROL RIDER MODEL
invariant as the value u(Nu) at the limit of the control horizon. Thus, all the rightmost
columns of the matrix Gv(k) multiply by the same input u(Nu), which can be achieved
by summating the rightmost Np - Nu columns and multiplying singularly by u(Nu).
The solution for the optimal steer torque input control followsthe theory defined by
equations (6.40) to (6.48).
If the reference path y,(k) is defined as the actual road path Yr(k) that is stored in
the combined motorcycle-preview state vector, then, from (6.48), the control input
can be defined as
(6.54)
As both vectors xv(k) and Yr(k) are combined in the motorcycle-preview state vector
z(k), then equation (6.54) can be neatly written in the vector form
u(k)
=-
[kv(k)Mv(k)
-kv(k)]
[xv(k)
]
Yr(k)
=-
[kv(k)Mv(k)
-kv(k)]
[ z(k) ]
(6.55)
Thus, by solving (6.47) as before for the motorcycle dynamics model to obtain kv(k),
the optimal control for the motorcycle followingthe road path can be readily obtained.
Noting the structure of (6.55), it can be seen that the controller gain elements applied to the motorcycle states and to the road states can be identified readily. The
gains element kv(k)Mv(k)
-kv(k)
multiplies by the motorcycle states, while the element
multiplies by the road preview information. Consequently, kv(k)Mv(k)
and
- kv (k) will henceforth be referred to as the state gains K, and the preview gains Kp
respectively.
6.2.6 MPC Theory Conclusions
The theory for the solution of a model predictive controller has been presented in
some detail, including the freedom to select different reference paths, and with both
linear and non-linear prediction models. Where appropriate, the theory has been
compared and contrasted to the optimal control approach.
CHAPTER
6: MODEL PREDICTIVE
156
CONTROL RlDER MODEL
Based on the theory presented in this chapter, it is believed that the model predictive control approach may be a more realistic representation
of a motorcycle rider's
actions. A rider would certainly have some anticipation of how the motorcycle would
respond to control actions, and would therefore have some subconscious planning of
control actions based on knowledge of the approaching road path and likely system
behaviour.
Furthermore,
the freedom to define a reference path that takes the rider
from the motorcycle'S instantaneous
itively seems representative
state to some target state in the future intu-
of the manner in which the motorcycle would move from
one position, quite possibly off the target path, in a direction that will aim to regain
the target path.
These observations on the theory of model predictive control and the requirements
of a suitable model to replicate the actions of a motorcycle rider would appear to
suggest that the former will adequately meet the requirements
of the latter.
The
results of the application of model predictive control to the rider model will aim to
answer this question more definitively.
6.3
Model Predictive Control Rider Model Results
The motorcycle-rider
model is tasked with a simple single lane-change path to follow,
comprising a lateral shift of the road path of 3.5 m over a forward distance of 20 m
(Figure 6.9). Analysis of the controller's performance in attempting
be used to assess the appropriateness
this task will
of using such a control strategy in modelling
a human motorcycle rider and to discuss any rider control requirements
that may
transpire from the analysis.
Several parameters
regarding the performance of both the motorcycle and the rider
model can be varied, and their effect on the control task assessed. Concerning the
motorcycle, the inherent physical properties
details or inertial properties,
can be altered.
of the motorcycle, such as geometric
Almost certainly these will influence
the control task and hence the path following performance of the motorcycle-rider
model. The scope for adjustment
of the physical parameters
is vast, ranging from
the major influences such as steering geometry, to the more minor such as individual
component inertias. The task here is to study and evaluate primarily the controller,
and as such the motorcycle model remains fixed throughout
all of the path following
CHAPTER
6: MODEL PREDICTIVE
simulations.
157
CONTROL RIDER MODEL
Details of the motorcycle model used can be found in Chapter 3 and
Appendix A.
Detailed analysis will therefore be limited to variation of the controller parameters
alone, yet even here there remains still plenty of scope for adjustment.
The trajectory-
following performance of an MPC controller is primarily affected by the cost function
(6.42) composed of accuracy and effort components with weightings applied to the
individual components of each part.
The controller then attempts
cost function, effectively a mathematical
to minimise this
compromise between contributions
made by
the two components.
Inthis application the cost function penalises the lateral path and the yaw angle errors
against the handlebar steer torque input. From a rider's perspective, the requirement
to apply large steer torques is undesirable, so penalising this parameter is intended to
keep steer torques to acceptable levels. Weighting factors are applied to all measured
(simulation) values in the cost function, and so there exist m weighting parameters
relating to the output accuracy and p weighting parameters
applied to the control
inputs.
The cost function weighting parameters therefore form the initial modelling variable,
and secondly the horizons available to the rider model can also be varied. In the case
of an MPC strategy, three horizons exist; the preview horizon Tp, the control horizon
Tu, and the dead-zone horizon Tw. The dead-zone horizon was mentioned briefly in
Section 6.2. The horizon represents a system lag between control input and system
response.
For systems with large response lags, particularly
chemical processes for
example, this can be a very useful feature. For the motorcycle, the effect of control
input on the system's response is assumed instantaneous.
Any response lag that may
be experienced is assumed to be accounted for by the relaxation length property of
the tyre's lateral force response (Chapter 3). The dead-zone horizon is not required
here and is therefore always zero. The remaining horizons Tp and Tu can be varied
independently
with the proviso that the control horizon cannot exceed the preview
horizon.
Finally, details of the simulation itself can be varied. The nature of the path to be
followed in terms of both the geometric path and of the road surface can be varied, and
the speed at which the motorcycle-rider
attempts the manoeuvre changed. Variation
CHAPTER
6: MODEL PREDICTIVE
CONTROL RlDER MODEL
158
of the road geometry to be followed will naturally influence the control applied by
the rider, however, the task here is to study the controller, and the fundamental
influence of the controller settings upon the path following performance. These would
be expected to remain broadly fixed regardless of the road geometry.
Changes to
the road surface, such as elevation and surface roughness, both factors in real-world
riding scenarios, would similarly be expected to have minimal influence on the broad
performance of the control strategy, and thus opening up the simulation parameters to
include factors relating to changes in the road geometry and surface would complicate
the analysis unnecessarily.
constant throughout
The parameter
The road path and surface will therefore be maintained
all parameter studies.
studies will therefore focus on four key variables; the lateral path
error cost weighting ql, the preview horizon Tp, the control horizon Tu and the forward
speed v. In addition, the effect of non-linear prediction models will be evaluated. The
results will be discussed in specific parts, with initially the controller performance
analysed for a single set of controller parameters,
and characteristics
such that the generic behaviour
of the MPC controller can be assessed and compared with the
performance of an optimal control strategy using the same controller parameters.
Following this, parameter
studies will be made by progressively varying the four
controller variables to draw further conclusions about the behaviour and suitability
of MPC techniques to the task of motorcycle rider modelling.
6.3.1
Low Speed Baseline Prediction Model
This part of the results section is concerned with results generated using a linear
prediction model, discussed in Section 6.2.1. The linear prediction model is a considerably more computationally
efficient process, and therefore allows for a large range
of results to be generated readily.
controller parameters
This model is used to study the effects of the
for a range of settings, enabling the effects of individual pa-
rameters alone to be studied.
The implications
of the using the linear prediction
model compared to the technically more accurate non-linear prediction model will be
investigated later in the results, to determine how critical this aspect is in the overall
performance of the model predictive control rider model.
In considering the performance of the controller with reference to the task, several
CHAPTER
6: MODEL PREDICTIVE
CONTROL RIDER MODEL
159
output variables will be analysed. Primarily, the path following performance of the
controller is observed in attempting to followthe path, since this is the fundamental
task for the controller. Additionally, the state and preview gains, and the steer
torque generated, will be examined, with the results discussed both qualitatively and
quantitatively.
Baseline Parameter Set
The initial simulation parameters, set to act as a baseline giving a simulation with
moderate accuracy and preview levels at a low speed, are presented in Table 6.1 Set 1.
This set of values will subsequently be extended to provide variation in all parameters,
with the influence upon the controller's characteristics as a result of these changes
analysed.
Figure 6.10 shows the path following performance of the controller with the initial
baseline parameters. It is seen that the path following performance is good, with
the model being able to track the target path with acceptable accuracy. There is
an initial deviation away from the path where the motorcycle rider model initiates
the manoeuvre with countersteer and there is a small overshoot at the end of the
manoeuvre phase, after which the controller recovers the path and followsthe straightrunning section of the path exactly.
The controller gains are considered next. The state gains are presented in Figure
6.11, and are seen to be not dissimilar from those achieved using optimal control
methods. As before, the largest state gain corresponds to the yaw angle, followedby
the roll angle gain, and with all other gains of considerably lower significance. The
yaw angle directly influences the errors between the predicted path heading for the
rider and the target path, and so it is perhaps unsurprising that, as with the optimal
control approach, this state is associated with the largest gains. With fixed rider,
the cornering radius of a motorcycle is directly influenced by the roll angle of the
motorcycle. The roll angle therefore has a direct bearing on the future path of the
motorcycle, at least over the immediate preview distance, and thus the errors between
predicted future path and target future path. It is therefore unsurprising that the
roll angle of the motorcycle should also figure significantly in the rider's control task.
The preview gains are presented in Figure 6.12. As with the state gains, these show
CHAPTER
6: MODEL PREDICTIVE
160
CONTROL RIDER MODEL
considerable similarity to gains achieved using optimal control strategies, with the
significant characteristics
of a peak gain in the middle preview distance, decaying
towards zero values at the limit of the available preview.
The steer torque that results using this controller is presented in Figure 6.13. It
shows smooth development of the steer torque as might be expected of a real rider,
and includes the necessary countersteer
torque needed to initiate the manoeuvre.
The path following of the motorcycle is also overlaid on this plot, allowing the path
following that results from the steer torque to be seen.
Consideration
of the gains and resulting control inputs suggests that the baseline
parameters can represent a controller that appears to have sufficient knowledge of the
approaching road path to make good judgement of control required to accomplish the
task. The preview gains tend to zero, indicating that the information at the limit of
the rider's preview has minimal significance, the steer inputs resulting are smooth,
progressive and representative
of the approach that may be expected of a real rider,
and ultimately the path following ability of the controller is judged to be acceptable.
The behaviour is not unlike the results achieved using optimal control methods, a
technique that has previously been judged to represent well the control strategies of
a motorcycle rider [94].
Cost Function
Weighting
The baseline parameter set is extended to consider first three different values of lateral
error cost function weighting, ql. The initial value of ql was set at 5000 m -2 which
gave rise to moderately
close path following performance.
Additionally
values of
1,000 m-2 and 10,000 m-2 are now considered, which will aim to represent looser
and tighter control respectively (Table 6.1, Sets 2 & 3).
The results of this are presented graphically. The path following is shown in Figure
6.14, and as may be expected, the higher weighting results in closer following of the
path, showing reduced corner-cutting.
tively similar characteristics,
Both controller weightings display qualita-
and for both controller settings the motorcycle follows
the path accurately after the manoeuvre has been completed (not explicitly shown).
The difference in the path following performance results from changes in the controller
CHAPTER 6: MODEL PREDICTIVE CONTROL RlDER MODEL
161
gains that are generated. Figures 6.15 and 6.16 show the state gains and the preview
gains respectively for all the path error weightings considered, with ql = 1000, 5000
and 10,000 m-2•
It is seen that the state gains are increased as a result of an increase in the patherror weighting. All states are influenced, though the ratio of the magnitudes of the
individual state gains remains constant for all values of ql. The rate of change of the
state gains is seen to reduce with the increase in the path error weighting.
In considering the preview gains, tight control implies that the rider is concerned with
accurately following the path. Thus, during the simulation, the rider may be expected
to place greater emphasis on the road immediately ahead of him, and ensuring that
his trajectory
left).
will keep him on this path regardless of the control cost (Figure 6.17,
The road some distance ahead of him is of less concern; he must ensure he
tracks the path accurately, irrespective of whether this path is the most direct route
between the start and finish points. Similar observations were made for the optimal
control approach.
Conversely, lower values of ql result in the control input cost becoming more dominant
in the overall cost function.
Thus, a loose controller is concerned with minimising
the control input effort, achieved by attempting
to make the complete path easier to
follow. The rider might therefore be expected to consider the whole road that he can
see to evaluate the easiest path from his current position to his ultimate target at
the limit of his visual preview. The rider is less concerned with the road immediately
ahead of him, but is aiming instead to follow the road that can be seen ahead of
him efficiently. As a result, a loose controller would be expected to cut corners more
readily, in order to track the path globally yet efficiently.
If a controller is to model a human rider realistically, the controller must also reflect
these characteristics.
The preview gains achieved for the tight and loose control
strategies are therefore considered.
It was expected, based on the literature, that the emphasis of a tight control strategy
would be placed on road information in the rider's near preview, with less emphasis
given to the road seen in the distance.
Figure 6.16 showed the preview gains for all
three path error weightings considered. The changes in the gains reflect the expected
effects; higher error weighting ql resulting in tighter control stems from a shift of
CHAPTER 6: MODEL PREDICTIVE CONTROL ruDER MODEL
162
emphasis towards the near preview distance. Conversely, loose control considers the
observed path more broadly, with lower peak gain value and more evenly distributed
gains over the entire preview distance.
Confirmation of the correct performance of the controller model is made by considering the steer torque that the rider applies to the motorcycle'S handlebars. A loose
controller would be expected to begin to steer well in advance of the turn, in order to make the turn progressive and minimise the control effort he is required to
input, ultimately making the manoeuvre more efficient with regard to his control
input. Conversely, tight control might be characterised by later steer control inputs,
beginning closer to the turn, and correspondingly having to be larger in magnitude.
The steer torques for all three path error weightings are presented in Figure 6.18. The
loose control displays the required early turn initiation, lower ultimate steer torques
and a more gentle turn. Notable again is the countersteer applied at the beginning
of the manoeuvre.
Thus far, the task of assessing the controller's performance has considered several
aspects. These aspects have all suggested that the use of an MPC control strategy
replicates well the behaviour that may be expected of a real rider. 3.0 s of visual road
preview appears sufficient at this speed, showing good path following performance
and with the preview gains diminishing to zero for all path error weighting values
considered, with variations in the lateral path followingerror resulting from changes
to the controller's behaviour in a manner that appear consistent with the behaviour
expected of a human rider.
Preview Horizon
The influence of the controller cost function weighting parameters on the performance
of the controller as a whole has been investigated. The fundamental input to the
controller comes from the road information that the rider sees, and so logically the
preview and control horizons would be expected to have a significant impact upon the
controller's performance. The effects of both an increase and decrease in the available
preview information afforded to the rider model therefore require investigation.
Considered first is the case for an increased preview horizon (Table 6.1, Set 4). As
CHAPTER
6: MODEL PREDICTIVE
CONTROL RlDER MODEL
163
the rider initially appeared to have sufficient preview information available to him at
the forward speed of 10 mis, it may be expected that additional preview information
would have minimal significant impact on the controller's performance. Tp is increased
from 3.0 s to 4.5 s, representing what is believed to provide excessive visual preview.
The limited preview horizon case is also considered (Table 6.1, Set 5), for a reduction
in the preview horizon from 3.0 s to 1.5 s. For both controller sets, the results will be
analysed in the same manner as before, looking at path following performance, gain
distributions
and resulting control inputs.
For all three horizon lengths considered, the path following accuracy shows little difference (Figure 6.19); only when examining the data in detail for the loosest controller
(ql = 1000) is there a discernable difference. In analysing the influence of the path
error weighting function, it was seen that loose control was associated with fuller use
of the available preview information, such that the extra preview information enabled
the rider to improve the input-effort efficiency of his manoeuvre, consequently leading to the rider cutting the corner to a greater degree than previously. Conversely,
tight control placed the emphasis on the road information close to the motorcycle'S
position. It may therefore be expected that for tighter control additional preview information in the far preview region would not significantly affect the control strategy
applied during the manoeuvre.
The state gains for both Tp = 1.5 s and 4.5 s are compared with the baseline parameter
set (Figure 6.20). Minimal difference is observed for the two horizon values considered
compared to the baseline set.
For the 4.5 s case, it appears that the increase in
the amount of road information available to the rider model has, as predicted, had
minimal influence upon the state gains.
Likewise, the reduction in preview shows
barely perceptible differences.
The preview gains show similar findings, with minimal differences between the gain
values achieved with Tp
= 3.0 s compared
with Tp
=
4.5
8
(Figure 6.21). Again,
this suggests that for the current level of path following accuracy a preview time of
3.0 S is sufficient, and any additional preview information would be of little use to
the rider. The reduction in preview to 1.5 S again brings only subtle changes to the
pattern of the gains, suggesting that the reduction has had only a minor effect on the
rider's control.
CHAPTER 6: MODEL PREDICTNE CONTROL RIDER MODEL
164
As neither the state gains nor the preview gains are significantly affected by the
increase in the preview distance available to the rider, the consequential effect on the
steer torque is minimal (Figure 6.22).
It appears therefore that for the speed and path error weighting functions considered
here, additional
significantly.
preview information
has not affected the rider's control strategy
For riding with these path following parameters,
greater than 3.0 s
of visual preview appears to have minimal benefit to the rider.
Similarly, with a
reduction in the preview horizon to 1.5 s, the effects on the controller and the path
following performance appear to be of minor significance.
Control Horizon
The predictive control strategy also permits the control horizon to be set independently of the preview horizon.
For restricted control horizons, the control input is
assumed to be invariant from the limit of the control horizon up to the limit of the
preview horizon, such that a control input for the full preview horizon is still available (Section 6.2.1). This is a standard practice when employing predictive control
techniques [9, 35] A loose control strategy has been shown to favour a longer preview
horizon, and so this limited control horizon case is tested against a loose control strategy in order to provide a more challenging test, comparing three cases; the preview
horizon is fixed at 3.0 s, with initially a 3.0 s control horizon, then restricted to 1.5 s,
and the last case with the control horizon reduced further to 0.5 s (Table 6.1, Set 6).
As may be expected, some change in the controller's performance is observed. Figure
6.23 shows the path errors between the three controller settings. The shortest control
horizon case (0.5 s) displays the largest errors. Over approximately
the first 40 m of
the manoeuvre, the 1.5 s control horizon case displays the best performance.
In the
middle phase the 1.5 s and 3.0 s cases are very similar, while over the last phase of
the manoeuvre, from approximately
55 m to 65 m, the 3.0 s case proves superior to
the 1.5 s control horizon case.
The corresponding state and preview gains are given in Figures 6.24 and 6.25.
CHAPTER 6: MODEL PREDICTNE CONTROL RIDER MODEL
165
Low Speed Modelling Conclusions
Having conducting a parameter study for the motorcycle running at low speed, some
conclusions can be made regarding the performance and characteristics
of the control
strategy for this application.
In line with what may intuitively be expected, changes to the required path following accuracy affect the way in which the modelled rider uses the road information
available to him. A loose control strategy implies that the concern is biased towards
minimisation
of control effort, with accurate path following being less important.
Consequently, the rider makes more use of the full picture of the road ahead, thereby
allowing the most efficient route to be taken over the road that he can see. Conversely, tight control requires accurate path following, so regardless of where the road
may go some distance in the future, the prime consideration
is the immediate path
that the rider is about to encounter, and ensuring that his motion tracks this road
accurately. These characteristics
for tight and loose control are thus reflected in the
preview gains that represent the use that the rider makes of the observed road path.
If sufficient visual road preview is available in order to make the suitable compromise
between path accuracy and control effort, then any additional road preview at the
limit of the rider's preview appears to have no effect on the controller gains and hence
the control actions of the rider model. Again, this characteristic
representative
would appear to be
of the manner in which a human rider would operate.
Initial results for the application of the model predictive control strategy to motorcycle
rider modelling therefore appear encouraging, and the analysis is extended for the
motorcycle running at increased forward speeds.
6.3.2 High Speed Linear Prediction Model
Results presented thus far have been for one forward speed alone, v = 10 mis, where
it was observed that the controller produced good results for preview times Tp in
excess of 1.5 s. Variation of the control parameters
produced acceptable changes in
the controller's gains and consequently control actions.
The forward speed was therefore increased in order to assess the implications that
CHAPTER
6: MODEL PREDICTIVE
CONTROL RIDER MODEL
forward speed has upon the controller's
166
ability to guide the motorcycle along the
target path.
Baseline Parameter Set
As for the model running at v = 10 mis, the model has been run with a set of
baseline parameters,
but now at a forward speed of v = 40 m/s (Table 6.2, Set 7).
The modelling parameters were varied as at the lower speed, with the results analysed
and compared with the lower speed condition.
Examining first the path following capabilities of the controller (Figure 6.26), the
motorcycle continues to follow the intended path, negotiating the manoeuvre section
successfully, returning to follow the path accurately along the subsequent straightrunning section. Compared with the lower speed case (Figure 6.10), the motorcycle
was seen to overshoot at the turn exit to a greater degree than for the lower speed
case, cut the corner much more significantly, but ultimately achieved the manoeuvre.
The controller gains essentially represent a mathematical
associated with the output
balance between the cost
(path following) accuracy and the control effort (steer
torque) input. Higher speeds lead to an increase in gyroscopic forces associated with
the rotating wheels, and hence require a greater steer torque input effort to achieve
similar steer angles and thus path following performance.
lead to an increase of the contribution
Like for like, higher speeds
to the cost function made by the control
input effort. The path following error becomes relatively less of a priority at higher
speeds, and consequently the path following accuracy suffers. In this manner, the
behaviour with increased speeds is not unlike setting the controller to operate in a
looser manner, where the relative contribution
to the cost function is reduced, and
indeed this is reflected in the path following performance of the controller.
The state gains provide some interesting observations (Figure 6.27). The bulk of the
states gains are not changed significantly with the increase in forward speed, except
for the gain corresponding to the yaw angle, which sees an increase of nearly 250%.
A yaw angle error implies that the motorcycle is heading away from the intended
path. As the forward speed is increased, for a given iteration time step it is seen that
doubling the forward speed will double the lateral path error that is generated over
one iteration step (Figure 6.28). Perhaps then it is unsurprising that a 300% increase
CHAPTER
6: MODEL PREDICTIVE
CONTROL RIDER MODEL
167
in the forward speed and therefore the preview distance should result in a similar
increase to the gain applied to the yaw angle of the motorcycle.
Comparing the preview gains (Figure 6.29), it appears that with Tp = 3.0 s the gains
again come close to diminishing to zero, though perhaps not quite as completely as
seen at the lower speed. Also of note is the oscillatory response of the gains for the
higher speed. This condition is due to the minimal damping of the wobble mode at
this forward speed (Figure 3.15), seen also for the optimal control approach (Figure
5.21). Although the gains at the higher speed are clearly tending to zero it appears
that they have not yet reached a steady value, implying that a small increase in the
preview horizon may be required to achieve this. At the increased speed, the peak
gains obtained are smaller than had been seen for v = 10 mis, with peak gains of less
than 1.5 comparing with over 2.0 for the lower speed. This is an interesting result,
since the higher speeds are seen to increase the magnitudes of the state gains, while
seemingly reducing the magnitude of the preview gains. Considering again Figure
6.29, the trend seen here is again not dissimilar to that shown when the controller
running at v = 10 m/s was set to operate in a loose manner; the peak gains are lower
but the rider apparently is required to look further down the road to apply complete
control.
The increased gyroscopic forces are in part suspected to account for this; at the raised
speeds, the steer torques required to overcome the gyroscopics will increase, leading
to an increase in the relative contribution to the cost function relating to the control
effort. Consequently, since the ratio between the cost function's components is fixed,
the cost contribution
associated with the system output also increases.
Thus, the
errors associated with the path following also increase, and so the path following
performance appears to deteriorate.
To confirm that the deterioration
in path following accuracy is largely as a result
of increased gyroscopics and hence increased cost function contribution
due to the
system inputs, the inertias of the wheels were reduced, thus decreasing the gyroscopic
forces from the wheels that form a significant contribution
noeuvrability
of the motorcycle.
to the stability and ma-
If the hypothesis is correct, then the reduction in
wheel gyroscopics should lead to an improvement in the path following performance
achieved. The wheel moments of inertias were therefore halved, and the path following as a result is presented in Figure 6.30. Here, it is clearly seen that this reduction in
CHAPTER 6: MODEL PREDICTIVE
168
CONTROL RIDER MODEL
wheel inertia has had a significant impact on the path following accuracy. The path
following errors are not halved, since the gyroscopic forces are not the only forces
contributing to the steer torque required to steer the motorcycle, but it is apparent
that the contribution is nonetheless a notable element.
As anticipated, at higher speeds the steer torques applied by the controller are seen to
be significantly greater than had been seen at the lower speeds (Figure 6.31). Given,
in part, the relationship between forward speed and wheel gyroscopic forces this result
is perhaps not unexpected.
The magnitudes of the steer torques generated in this
simulation are likely to be greater than a rider would be capable of applying.
results here are a mathematical
The
solution, and this result adds weight to the need for
some form of limits or constraints that formed one of the motivations for using model
predictive control techniques. In spite of the increased steer torques, the steer angles
obtained are reduced compared with those achieved at lower speeds (Figure 6.32).
The comparisons drawn between the behaviour of the controller operating with the
higher speed baseline set indicate that, for all other parameters
fixed, the control
behaviour resulting appears to exhibit a more loose manner than at the the lower
speed. This agrees with the observation that increased speeds lead to higher control
contribution
to the cost function, and hence the controller must compromise path
accuracy at high speeds in an attempt to contain the control input effort required.
Cost Function Weighting
The influence of the output error cost function weighting parameter is again analysed
for the higher speed case (Table 6.2, Sets 8 & 9). It was observed that the influence upon the state gains, preview gains, steer torques and steer angles all showed
comparable behaviour to the lower speed case.
Reducing the cost weighting results in lower state gains, while increased cost weighting
results in higher state gains (Figure 6.33). For the looser control, the preview gains
are lower in magnitude but biased further forwards and more evenly distributed
the visual preview horizon, comparable to a looser control strategy.
over
However, it is
noted that for the preview gains to diminish to zero, as had been achieved at the
lower speed with Tp = 3.0 s, it becomes necessary to increase the preview horizon
to achieve the same results.
Figure 6.34 plots the preview gains against preview
CHAPTER 6: MODEL PREDICTIVE CONTROL ruDER MODEL
169
distance allowing 4.5 s of visual preview, where the preview gain pattern with regard
to the complete preview horizon is then comparable with Figure 6.16. Tighter control
was seen to result in higher peak preview gain values, biased more towards the near
preview.
The effects on the steer torques and hence steer angles are qualitatively
similar to
the effects seen at lower speed (Figure 6.35), and are consistent with the observations
made regarding rider control for the lower speed case.
Preview Horizon
The influence of the preview horizon on the controller performance is again analysed,
now at the higher speed (Table 6.2, Sets 10 & 11). As noted previously, to achieve
comparable preview gain distribution
at the higher speed it is seen to be necessary
to increase the length of the preview horizon, with the increased preview horizon
taken as 4.5 s. At the higher speed, comparison of the preview gains for the three
preview horizon times (Figure 6.36) shows minimal difference between Tp = 3.0 sand
Tp
=
4.5 s. However, the lower preview horizon, Tp
=
1.5 s, shows more notable
difference to the other two settings. In contrast, at the lower speed (Figure 6.21), all
three horizons gave very similar gain patterns.
These results appear to show that as the speed is increased an increased preview
time is ideally necessary to maintain the control performance of the motorcycle, and
as before, it is suggested that this is again the result of the increase in gyroscopic
forces experienced as the speed increases (Figure 6.30). Consequently, and in order to
maintain control torque effort, the rider must begin manoeuvres further in advance of
the turn as the speed is increased, which therefore requires a greater preview horizon
in order to facilitate this.
In spite of the apparent deterioration
of the controller gains for higher speeds with
short preview horizon, the model is still capable of negotiating the path competently.
What this result would appear to suggest is that the rider model's control may be
compromised by the limited preview, but this does not necessarily prevent successful
riding, as may be expected of a real rider.
CHAPTER 6: MODEL PREDICTIVE CONTROL ruDER MODEL
170
Control Horizon
Thus far, the control problem has dealt with the situation for which the rider's visual
preview horizon and his control horizon are the same. In other words, the rider plans
a control strategy for the full road picture that he has available. The MPC strategy
enables the two horizons to be set independently, thus allowing the control horizon
to be set shorter than the preview horizon. Relating this to a rider, this would imply
that the rider has knowledge of the road a long way ahead, but plans his control for
only a shorter time ahead of him.
It has been seen previously that for the low speed baseline parameter set (Table 6.1),
the controller's performance was good, and a reduction in the preview and control
horizon to 1.5 8 did not adversely affect the overall ability of the controller to complete
the task.
Ai;
the speed was increased, it was noted that the reduction in preview horizon from
3.0 s to 1.5 s did appear to have an effect on the preview gains. While the ultimate
performance of the controller was not significantly degraded, the effect of the reduced
preview horizon was apparent.
An appropriate point to begin to investigate the effect of a reduced control horizon
is therefore with the higher speed baseline parameter set. The simulation is therefore
run with a preview horizon Tp = 3.0 s, but with also a limited control horizons of
Tu
= 1.58 and 0.5 s (Set 12). Selecting these values will allow the performance to be
compared with earlier results with equal preview and control horizons.
The path followingperformance of all three control horizon settings was seen to result
in very similar results which were difficult to discern from one another. The path
errors, being the differencebetween the target path and the actual path achieved, are
therefore plotted in Figure 6.37, where it is seen that overall the performances of the
three controllers are similar. Only in the initial countersteer phase and at turn exit
are there any notable differences.
The maximum error during the initial phase is seen for the shortest control horizon,
as may be expected, but, interestingly, the best performance in this part is for the
1.5 8 control horizon. The 3.0 s control horizon falls midway between the two. At
turn exit, the roles are reversed; the shortest control horizon Tu = 0.5 s shows the
CHAPTER 6: MODEL PREDICTIVE
CONTROL RIDER MODEL
171
best performance, the 1.5 s case the worst, with again the 3.0 s case in between.
Both the state gains (Figure 6.38) and preview gains (Figure 6.39) show this similar
interesting pattern, whereby the values of the 3.0 s control horizon case fall midway
between the values of the 1.5 s and 0.5 s values.
High Speed Linear
Prediction
Model
Conclusions
The analysis for the two different forward speeds considered has generated
interesting conclusions, indicating the requirement
some
for greater preview information
for complete control as the speed is increased.
A human rider is constrained by physical limits of strength which ultimately
limit
the magnitudes of steer torque that can be applied to the motorcycle. Higher speeds
require higher steer torques due to the increase in gyroscopic forces (Figure 6.30), and
so as the rider's physical limits become more heavily used, he must instead find ways
to reduce this torque requirement.
This is most readily achieved by commencing the
turn earlier and more progressively, and essentially the rider may be forced to operate
with a looser control strategy by limitations of physical strength.
In a real riding situation, there are additional steps that a rider can take in order
to reduce the steer torque requirement
in order to overcome the gyroscopic forces.
Typical techniques include more emphasised countersteer
to force the bike into the
lean and movement of body weight, in order either to force the bike to lean into the
turn or to reduce the angle to which the motorcycle itself must lean, by moving his
own bodyweight to achieve the necessary balance between overturning moments and
moments due to centripetal forces.
In performing a manoeuvre, a car is not required to lean into a turn in the manner
that a motorcycle must.
As such, the increased gyroscopic forces that result from
higher speeds are inconsequential
to a car driver.
It may therefore be suspected
that as speeds are increased a car driver has less requirement to increase his preview
horizon for complete control in the manner that a motorcycle rider, it appears, would
need to. It is widely regarded that motorcycle riders exercise much greater visual
preview, examining the road ahead to a greater distance than a car driver might, and
indeed the findings here would appear to support those beliefs and offer some feasible
CHAPTER 6: MODEL PREDICTIVE CONTROL RlDER MODEL
172
explanation of why this needs to be so.
6.3.3
Non-Linear Prediction Model
The analysis conducted so far used a prediction model that employed a fixed state
space model of the motorcycle for the entire prediction horizon, as discussed in Section
6.2.1. If a technically more correct prediction model were to be used, the technique
to obtain the prediction element is essentially the same, albeit computationally
demanding.
more
The theory of this approach was presented in Section 6.2.2. In regard
of the additional computational
load of the non-linear model, a typical simulation
conducted here required several hours for the non-linear prediction as opposed to
several minutes for the linear prediction. For off-line model simulation, this does not
technically pose a problem.
However for the processing of a significant number of
modelling conditions, as considered within this thesis, the additional computational
time becomes a more relevant consideration.
In order to assess how critical the choice of a technically more correct non-linear
prediction model is to the modelling of the motorcycle rider, the full non-linear prediction model has been run for the set of low-speed baseline parameters presented in
Section 6.3.1.
For the linear prediction model running at moderately low speeds, the path following
performance was seen to be quite acceptable, with the motorcycle tracking the path
well and with smooth progress during the manoeuvre (Figure 6.10) The control gains
generated (Figures 6.11, 6.12) seemed consistent with what might logically be expected and the steer torque (Figure 6.13), while not completely smooth, also seemed
consistent with the actions that may be expected of a human rider.
The full non-linear model was run using the same modelling parameters
the same path.
and over
The path following performance of the model was examined first,
and compared with the linear prediction
model results obtained previously.
The
differences between the linear and non-linear prediction models are hard to discern;
both models follow broadly the same path, and it is only when the performance is
examined in detail that the differences become apparent.
Figure 6.40 presents the
final phase of the manoeuvre, as the motorcycle is exiting the turn phase and returning
to straight running. By examination of this phase, it is apparent that the non-linear
CHAPTER 6: MODEL PREDICTIVE
CONTROL RIDER MODEL
173
model performs better, tracking the target path more closely, both during the turn
and as the motorcycle begins to straighten its path.
The steer torques generated by the controller are also examined, and once again compared with the linear prediction model results obtained previously. For the linear
prediction model, the steer torques appeared acceptable (Figure 6.13), albeit not entirely progressive over the course of the manoeuvre. Figure 6.41 presents the steer
torques for both the linear and non-linear prediction models, and from this the superior performance of the non-linear prediction model is apparent, the steer torques
generated using the full non-linear prediction model showing much smoother torque
inputs without the transient oscillatory behaviour apparent with the linear prediction
model.
The results of this are significant for two reasons. Primarily, it is apparent that the
performance of the controller is superior when a more realistic prediction model is
used. This result is not surprising, since the controller bases the control decision on
the predicted output, and if the predicted output and the actual system output to
a given control input differ, then clearly the system behaviour will suffer from less
accurate performance.
Secondly, although the full non-linear prediction model is seen to provide superior
performance, the difference in the resulting outputs are not so different as to call
into question the validity of the results and observations made in assessing the MPC
controller using the linear prediction model.
It is likely that for the higher speed case, the difference between the linear and nonlinear models will be more distinct. At higher speeds, the lean angles required to
negotiate turns are greater, and as such the change in lean angle over the manoeuvre
in the prediction horizon could be expected to increase. Consequently, the dynamics
of the motorcycle will change to a greater extent over the preview horizon, making the
difference between the linear prediction model and the non-linear prediction model
more marked. Additionally, at the higher speed the preview distance increases, meaning that a greater distance of road is seen in the rider's preview horizon, and therefore
the manoeuvre planned for a greater length of the road path. The greater the length
of the road path, the greater the potential changes in the road path that the rider
will see. Correspondingly, the potential changes in the motorcycle's predicted future
CHAPTER 6: MODEL PREDICTIVE
CONTROL RIDER MODEL
174
trajectory may be greater, adding further to the differences between the linear and
non-linear prediction models at the higher speed.
6.3.4 Reference Path Definition
The application of a reference path, often used in predictive control methods, concerns
the definition of a target path to follow, different to the set path, that the system
output will attempt to track. With this method, the intention is to take the system
output from a position away from the ideal and guide it back onto track in some
defined way.
The results presented so far have been for the case where the set path and reference
path were equal. If the motorcycle were displaced from the target path, this would
effectively result in a step path that the rider would attempt to follow. However, it
was postulated that this may not be entirely realistic of a human rider, and that once
displaced from the ideal path of the road, a rider may not attempt immediately to
recover the road centreline, but will aim instead to gradually guide himself back onto
the road path.
The manner in which the set path regains the target path can be defined in different
ways. Here, three methods were considered: linear path, linear error reduction path,
and exponential error reduction path. The performance of the controller using these
modified path definitions will be compared with the original definitions using only
the set path, to determine whether improved controller performance can be achieved.
Linear Set Path
The theory of the linear set path was covered in Section 6.2.3. Essentially, the linear
set path defines a path that takes the system output state from the current position to
the target position at the limit of the preview horizon. Making use of the motorcycle's
current position and a target position given as the set path at the limit of the rider's
preview horizon, the intermediate linear reference path is calculated. The theory of
a linear prediction model is applied to the motorcycle model, and the controller then
attempts to followthe single lane change manoeuvre as before.
The path followingability of the controller is assessed first. From the trace of the path
CHAPTER 6: MODEL PREDICTIVE
175
CONTROL RIDER MODEL
output (Figure 6.42), it is immediately apparent that this is not an entirely suitable
manner in which to model the controller's actions. Essentially, by modelling the set
path as a linear path from current position to final position, at each point during
the simulation the rider is ignoring the road path in his preview horizon, focusing
only on his distant target.
For very gentle manoeuvres, this may be an acceptable
approach, but realistically it is not suitable for the application here. Throughout
manoeuvre the rider is always attempting
the
to cut the corner by the maximum amount,
and hence the resulting performance is poor and unrealistic of the control actions of
a motorcycle rider.
If a target point other than the final preview horizon point were chosen, say a point
half way along the rider's preview horizon, then the path following performance of
the rider would be expected to improve. However, the rider would still be aiming to
follow a path that heavily cuts corners (Figure 6.43), and would therefore give inferior
performance to the use of the set path. No further analysis was therefore conducted
for this definition of reference path.
Linear Error Reduction Path
The definition of the reference path here is subtly different to the full linear reference path.
There, the path was taken as a straight path between current position
and a target position some way in the distance, with no consideration
of the path
information in between in the rider's preview horizon.
The reference path defined here is such that the error between the set path and the
reference path is reduced linearly over the full preview horizon, from the current
lateral error to zero at the horizon limit. This does not imply that the set path itself
is linear, as was the case for the linear set path defined previously; if the road curves,
so too will the set path, but it is the error between the two paths that will reduce
linearly over the forward distance.
The theory of this model was discussed in Section 6.2.3. The performance of the
controller using this strategy
is again analysed by considering the path following
performance and the controller's input actions. Displayed in Figure 6.44, the result
is notably different to the linear reference path definition assessed previously. In this
case, the rider model does not begin to cut the corner well in advance, and begins
CHAPTER 6: MODEL PREDICTIVE CONTROL RIDER MODEL
176
the manoeuvre phase as the actual manoeuvre of the road set path begins.
However, beyond this point, the path following is again inferior to the performance
when the set path and reference path were equal.
Essentially, by defining the set
path as a linear error reduction between current position and a target in the future,
a weaker path following target is being set for the rider model to attempt to follow,
and consequently the actual path following accuracy of the motorcycle, relative to
the actual road defined by the set path, is reduced.
Exponential Error Reduction Path
The reference path is now defined in such a manner that the path error diminishes
exponentially to the target path. The basic theory of this approach is similar to the
linear error reduction path, except that now the reduction of the path error over the
full horizon is an exponential decay.
The theory of this approach (Section 6.2.3) was not dissimilar to the theory for the
linear error reduction case. Therefore, it is not surprising that the resulting performance shows similar characteristics
to those of the linear error reduction method.
ABwith that method, the motorcycle does not begin the manoeuvre until the actual
road path begins to deviate. However, once the path does begin to deviate, the path
followed by the motorcycle does not track this path particularly
closely, as before
taking a more direct route towards the road path at the limit of the preview (Figure
6.45).
The magnitude of the path error is not as great as for the linear error reduction case,
though this magnitude is dependent on the value of lr, the exponential path relaxation
length, set here at values of 20 m and 50 m. The smaller this value, the closer the
exponential path comes to a step function, essentially the case when the set path
and reference path are set equal. The shorter the value, the more the reference path
tends towards the linear error reduction case. Here, the value of lr is such that the
exponential rise time is quite short, and consequently the path following appears to
be quite promising. However, if a longer rise-time were used, then the results would
tend more towards those seen for the linear error reduction case.
The conclusion for the use of an exponential error path reduction therefore must be
CHAPTER 6: MODEL PREDICTIVE
CONTROL RIDER MODEL
177
that, in line with the previous two approaches considered, an inferior path following
results is produced compared with the use of the set path as the target path to follow.
6.4
Model Predictive Control Conclusions
This chapter has considered the modelling of the control actions of a motorcycle rider
using the theory of model predictive control. Through the generation of a suitable cost
function, a control input is generated that aims to theoretically minimise the combined
costs associated with system output accuracy and system input effort required.
The method has been shown to be highly effective in replicating the control actions
of a rider. The controller was tasked with a lane-change manoeuvre, for a variety of
controller and task-dependent
conditions. Variations in forward speed, rider preview
allowances and cost function weighting parameters were all considered. Additionally,
the possibility to define a reference path that the system may attempt to track instead
of the actual road (set path) was explored.
On the whole, variations to the modelling conditions gave encouraging results.
In-
creased weightings on path errors were seen to reduce the errors in the resulting path
following of the motorcycle, sufficient rider preview was seen to result in the best
path following performance,
and increases to the forward speed of the motorcycle
produced the sort of results that may be expected of a real motorcycle and rider.
Not all investigations were entirely successful. A distinct feature of predictive control concerns the ability to define a reference path, distinct from the set path.
For
some controlled systems, this feature may be able to provide an improvement in the
required system response.
However, when applied to the road path that the con-
trolled motorcycle attempts
to follow, the resulting performance was inferior. Such
an approach would be worthy of further consideration only if the manner in which a
rider may recover a target path can proved experimentally
such as presented here.
to be of a specific nature
It is. likely that riders would have subconscious strategies
by which path errors would be corrected, but a firm answer as to what this strategy
may be would not be easy to obtain. It may be the case that the choice of recovery
strategy employed by the rider is dependent on a number of factors that may include
the forward speed of the motorcycle, the nature of the road path and severity of his
CHAPTER 6: MODEL PREDICTIVE
CONTROL RIDER MODEL
178
departure from the target path, all of which would be very difficult to model. Further
work may be able to address this area, though for now the availability of the reference
path feature is considered superfluous.
Additionally, a full non-linear prediction model has been compared with the more
computationally efficient linear prediction model. It was expected that this would
provide superior path followingperformance, and the results confirmed this. Ideally
then, modelling work of this nature should employ a full non-linear prediction model,
though the comparison of the two approaches showed that for moderately severe
manoeuvres the differences between the two were not so great as to consider the
simpler, linear prediction model wrong.
The apparent suitability of the predictive control strategy with regard to replicating
the control process of a motorcycle rider and the prediction element that his riding
task may include was cited as a strong reason for exploring this control strategy. The
aim was therefore to replicate the positive features of the optimal control approach
and the apparent suitability to the application, while also correcting some of the
limitations found. Both these goals appear to have been achieved, with the generic
characteristics of the optimal control approach retained. Due to the strong similarity
of the two methods, in order to provide a more detailed comparison of the two approaches Chapter 7 directly compares the two approaches resulting in more defined
conclusions about the relative performances of the two approaches.
CHAPTER 6: MODEL PREDICTIVE
6.5
179
CONTROL RIDER MODEL
Tables
Parameter
Set 1
Set 2
Set 3
Set 4
Set 5
Set 6
Baseline
Loose
Tight
Long
Short
Short
Control
Control
Preview
Preview
Control
v [ms-I]
10
10
10
10
10
10
T" [sI
3
3
3
4.5
1.5
3.0
Tu [sI
3
3
3
4.5
1.5
3.0,1.5,0.5
Tw [sI
0
0
0
0
0
0
ql [m-2]
5000
1000
10000
5000
5000
1000
q2 [rad-2]
0
0
0
0
0
0
r [(Nm)-2]
1
1
1
1
1
1
Table 6.1: Low speed controller parameter sets, model predictive control
Set 7
Set 8
Set 9
Set 10
Set 11
Set 12
Baseline
Loose
Tight
Long
Short
Short
Control
Control
Preview
Preview
Control
40
40
40
40
40
40
T" lsI
3
3
3
4.5
1.5
3.0
Tu [sI
3
3
3
4.5
1.5
3.0,1.5,0.5
Tw [sI
0
0
0
0
0
0
ql [m-2)
5000
1000
10000
5000
5000
5000
[rad-2]
0
0
0
0
0
0
r [(Nm)-2)
1
1
1
1
1
1
Parameter
v
Q2
[m/s]
Table 6.2: High speed controller parameter sets, model predictive control
CHAPTER 6: MODEL PREDICTIVE
6.6
180
CONTROL RIDER MODEL
Figures
t
Figure 6.1: Path definition in Model Predictive Control, [9]
x
x(t)
~----~------------------------------~~t
Figure 6.2: Linearisation of a non-linear system at two points
181
CHAPTER 6: MODEL PREDICTIVE CONTROL RlDER MODEL
Simulation Step
k+i
Simulation Step
k+i
··.................................... ..
,'
··
~
Predict Full
Horizon
·
''
:
iNptimes
:,
~
Generate
Control Input
.................................
~
Simulation Step
k+I+ 1
··
··
·
,
,,
Generate
Control Input
Simulation Step
k + i+ 1
Figure 6.3: Added complexity of non-linear prediction model (right) against linear
prediction model (left)
t
Figure 6.4: Typical reference path definitions in MPC systems
CHAPTER 6: MODEL PREDICTIVE
182
CONTROL RIDER MODEL
Road Path
Y, - Yy
Np
y
.;
1=1
1=2
1=3
1=5
1=4
Figure 6.5: Definiton of a linear reference path, Np = 6
i=1
i=2
i=3
i=4
i=5
i=Np
y
Lx
Figure 6.6: Definiton of a linear error reference path, Np = 6
CHAPTER 6: MODEL PREDICTIVE
CONTROL RIDER MODEL
--_.
Reference Path
Set Path
Path Longitudinal
Figure 6.7: Exponential reference path definitions
--_.
Reference Path
Set Path
Path Longitudinal
Figure 6.8: Exponential error reduction reference path definitions
183
CHAPTER 6: MODEL PREDICTIVE
184
CONTROL RlDER MODEL
3.5m
Road Path
20m
Figure 6.9: Single lane change path, not to scale
4
3.5
3
I2.5
i!
3
i
2
-Target
-Path
1.5
Q.
0.5
O+-------~ __=_~--r_------_.------~
20
40
80
60
-0.5
Path Longitudinal (m)
Figure 6.10: Path following, v = 10 mis, Tp = 3.0
S, ql
= 5000 m-2
CHAPTER 6:}'I
DEL PREDICTIVE
185
CONTROL RIDER MODEL
-«
YI
Mat
·200
State
Figur
6.11:
tat gains, v
=
10 mis, Tp
= 3.0 S,
ql
= 5000 m-2
2.51
2
1 51
0.
~
c
Ii• .:1
>
Q;
0
20
10
3
.,
·15
Preview Distance (m)
Fi
W"
6.12: Pr vi w Gain,
v = 10 mis, Tp = 3.0
S, ql
= 5000 m-2
CHAPTER
6: MODEL PREDICTIVE
186
CONTROL RIDER MODEL
15
4
3.5
10
3
2.5
5
e
2
!.
0
!
!
100
80
1.5
E
'!
.!
!l
-
-Torque
-Path
.J:
as
1
-5
CL
0.5
-10
0
-0.5
-15
Path Longitudinal (m)
Figure 6.13: Steer Torque, v
= 10 mIs, Tp = 3.0 S, ql = 5000 m-2
4
3.5
-i!
..... ;.;.,'
. ,,,
.',
.,..,
.'
.'
3
E 2.5
S
2
.!l 1.5
-5IQ
Cl.
.,.',
,,/
V
1
0.5
0
-0.5
50
40
60
Path Longitudinal (m)
Figure 6.14: Path following, v = 10 mIs, Tp
= 3.0 S, ql = 1000 & 10000 m-2
187
1200
1
1000
1
800
~
~
B
Gl
1
600
1
oq1 = 1000 m"(-2)
.q1 = 5000 m"(-2)
.q1 = 10000 m"(-2)
1
400
Cl)
200
o -t----."""'-"""'"
-200
J
x
Y
41
Yr
<P
YfaxJat
f7y/Cl c3IjJ/CI o<p/Cl o6/C1
State
Figur
6.15:
ta
gains v = 10 mis, Tp = 3.0
S,
ql
= 1000, 5000 & 10000 m-2
......
-q1
= 1000 m"(-2)
- -q1 = 5000 m"(-2)
- - - q1 = 10000 m"(-2)
"
,
/",--
.
:
.
•I
./
0.5
O~i
-05
-1
......
,
I
I
',
/
..
-.;
"-
-/:~
'\
'..
...'.
,
~": .... ":"" ......
,~:::
10
.....
20
:::r
30
".1:
.'
..
-1.5 ]
·2
Preview Distance (m)
igur
.1
r vi
gain', v = 10 ta] , Tp = 3.0 S, ql = 1000, 5000 & 10000 m-2
CHAPTER
6: MODEL PREDICTIVE
CONTROL RIDER MODEL
188
Figure 6.17: Contrasting road information requirements of tight (left) and loose
(right) control strategies, showing typical path aim (dashed line)
20
15
10
E
z
-•
~
~
0
~
5
0
-5
-10
-15
Path Longitudinal (m)
Figure 6.18: Steer torque, v = 10 mis, Tp = 3.0 S, ql = 1000, 5000 & 10000 m-2
189
':1
§:
~
2.5
-Tp=1.5s
Tp = 3.0s
_. - Tp = 4.5 s
2
S
ftI
-
..J
15
+
;S
IQ
Cl.
05
0
60
40
20
-0.5
Path Longitudinal (m)
Figur 6.1 : Path foll wing v
r
ult
OV
=
10 mis, Tp
=
1.5 5, 3.0
5
& 4.5
5,
ql
= 5000 m-2,
rl pping
800 ,
600
~
C
400
DTp= 1.55
.Tp = 3.05
= 4.5s
'ji
Cl
.T
S
J! 200
tn
0
x
III
Yr
-200
State
igor
'.2.
at gain, v
= 10 mis,
Tp
= 1.5 5, 3.0 5 & 4.5 5,
ql
= 5000 m-2
CHAPTER 6: MODEL PREDICTIVE
CONTROL RIDER MODEL
190
2.5
2
1.5
Q.
~
r!
-Tp·1.Ss
- -Tp-3.0 s
- - - T = 4.S s
~ 0.5
•
.!
>
0
!
D.
10
20
40
30
-0.5
-1
-1.5
Preview Distance (m)
Figure 6.21: Preview Gains, v = 10 mis, Tp = 1.5 s, 3.0 s & 4.5 s, ql = 5000 m-2,
paths coincident
15
10
5
e
!.
! 0 .f-,,~__,.--rl--.-__,.-t--.-__,.-""T7"_"'~-"""'__"'~
0
SO
...
60
-Tp·1.5s
- - Tp • 3.0 s
• - - T • 4.S s
-5
-10
-15
Path Longitudinal (m)
Figure 6.22: Steer Torque, v = 10 mis, Tp = 1.5 s, 3.0 s & 4.5
S, ql
= 5000 m-2
CHAPTER
6:
1 DEL PREDICTIVE
191
CONTROL RIDER MODEL
0.5 ,
-0.3
Path Longitudinal
Figur
ql
6.23: P th Error
=1
(m)
v = 10 mis, Tp = 3.0 s, Tu
3.0 s, 1.5 s & 0.5 s,
m-2
1
300
400
200
OTp= 3.05, Tu = 3.0 5
.Tp = 3.0 5, Tu = 1.55
.T
3.0 5, Tu 0.55
1
~
$
B
=
=
100
Cl)
·100
State
Figur
ql
=1
6,2:
m-2
ain
v
=
10 mis, Tp
3.0 s, Tu = 3.0 s, 1.5 s & 0.5 s,
CHAPTER
6: MODEL PREDICTIVE
CONTROL RIDER MODEL
192
0.7
0.6
0.5
0.4
:c 0.3
;;
C)
\
0.2
1
> 0.1
"
"'"
\
-,
" .....
\
-,
"
a
\
\
10
-0.1
I
I
I
-,
\
!
A-
-Tp
- 3.0 S, Tu - 3.0 S
---Tp· 3.0 S, Tu -1.5 S
; ...... T - 3.0 S, Tu - 0.5 S
'.,
c-.-.
-.
I
20
-0.2
-0.3
Preview Distance (m)
Figure 6.25: Preview Gains,
ql
'V
=
10 mis, Tp
=
3.0 s, Tu
200
240
=
3.0 s, 1.5 s & 0.5 s,
= 1000 m-2
4
3.5
3
-!
2.5
i
2
~
oS
1.5
!
..
A-
0.5
0
40
80
120
160
280
-0.5
Path Longitudinal
Figure 6.26: Path following,
'V
(m)
= 40 mis, Tp = 3.0
S, ql
= 5000 m-2
CHAPTER 6: 10DEL PREDICTIVE CONTROL RIDER MODEL
193
4000
..
3000
~
C
~
(!)
S
!!!
II)
1000
.v ==
Dv
1
10 m/s
40 m/s
0
y
'II
'II
0
Yr
YI
Mat
~/f!t
liIjIIat
~lJI.
iJO/at
-1000
State
Figur
6.27:
tat gains, v = 10 mls & 40 mis, Tp = 3.0
S,
ql
=
5000 m-2
reference path
Low Speed
reference path
predicted path
HIgh Speed
Figur
.2
tal lat ral preview error; effects of increased forward speed
CHAPTER 6: MODEL PREDICTIVE
CONTROL RIDER MODEL
194
2
1.5
,.
~
~
1
.'
, "
"'"":''''.~l\ ~
C
V v\
~
C)
1
>
-v=
••• v
\/-.l\
v,
0.5
'"
.,.
10mls
= 40 mls
,.'
'o; ~
I!
0
Q.
-0.5
-1
Preview Time (s)
Figure 6.29: Preview gains, v
= 10 m/s
& 40 mis, Tp
"
,,
3.S
,
-
..--- .. ...._ .._
I
I
3
I
I
I
2.S
E
= 3.0 S, ql = 5000 m-2
,
I
'!
2
1
S
!I
-Low
I
1.5
i
Gyro
- -Std. Gyro
-Tar
et
G.
O.S
0
-o.s
SO
100
150
200
250
300
PathLongitudinal(m)
Figure 6.30: Path following performance
Tp = 3.0
S, ql
= 5000 m-2
with reduced wheel inertia, v = 40 mis,
CHAPTER 6: MODEL PREDICTIVE CONTROL RIDER MODEL
60
195
..
o~
50
.0
o
o
...
····
··
•
0
40
30
0
e-~
20
;:,
10
o
•
e0
....
0
-10
-20
-30
•
'
0
··
·
0
0
0
..
.
0
I
;;oN hio\.
.....-._
\50
I.-V=
0
0
·· .
··......:.
0
10mls
• •• v = 40 mls
.....-.- ....._.
0
.~50· 300
200
"
\ .._..... .....
350
400
,0'
'
-40
Path Longitudinal (m)
Figure 6.31: Steer torque, v = 10 mls & 40 mis, Tp = 3.0
S, ql
= 5000 m-2
0.04
0.03
A
0.02
1
0.01
~Cl
.........~J ......
0
i
c
50
100
1"'0.
1f~""'''''''''2~O'
250
l-V.10mlS
300 --.v.40mls
.(1.01
US .(1.02
.(1.03
V
.(1.04
.(1.05
Path longitudinal (m)
Figure 6.32: Steer angle, v
= 10 mls
& 40 mis, Tp
= 3.0 S, ql
= 5000 m-2
196
CHAPTER 6: ~rODEL PREDICTIVE CONTROL RIDER MODEL
6000
5000
4000
III
~
.E 3000
Oq1
.q1
.q1
IV
e
...~
2000
aI
= 1000
= 5000
=
m"(-2)
m"(-2)
10000 m"(-2)
(/)
1000
0
-1000
State
Figur
6.33:
tat
gains, v
= 40 mis, Tp = 3.0
S,
ql
= 1000, 5000 & 10000 m-2
2
1.5
e,
:lC
1
j
j
C
.;;;
e
J
.f
..
0.5
>
GI
11-
0
-0.51
-1
J
Preview
igur 6.34: Pr vi w gain , v
Distance
= 40 mis,
(m)
Tp
= 4.5 S, ql = 5000 m-2
CHAPTER
6: MODEL PREDICTIVE
..
·····, ...
:,
80
60
-"2
.,
:,
40
E
20
S
:,
,
l
197
CONTROL RIDER MODEL
-q1
- -q1
- - - q1
.....
\~
\,
\0
= 1000 mll(-2)
= 5000 mll(-2)
= 10000 mA -2)
\
\~
I
I
~
0
;;
360
"'
a. ·20
-40
-60
Steer Torque (Nm)
Figure 6.35: Steer torque, v = 40 mis, Tp
= 3.0 S, ql = 1000,5000
& 10000 m-2
3.5
3
2.5
~
2
lit:
C
-Tp=1.5s
--Tp= 3.0s
1.5
'i
Cl
1
:.
e
0.
- -. T
= 4.5 s
0.5
0
-o.s
-1
Preview Distance (m)
Figure 6.36: Preview gains, v
= 40 mis,
Tp
= 1.5 s, 3.0 s & 4.5 S, ql = 5000 m-2
198
CHAPTER 6: 10DEL PREDICTIVE CONTROL RIDER MODEL
15
~
0.5
E.
e..
o
3
-Tp
= 3.0 S, Tu = 3.0 S
.. ···Tp
= 3.0 S, Tu = 0.5 S
+-_..o:::::::~h-+--+,--"""","""==i~------Tp = 3.0 5, Tu = 1.55
300
5
-0.51
t:
-1
-1.5
Path Longitudial (m)
Figur
ql
6.37: PI' vi w gain,
=5
v
=
40 mis, Tp
=
3.0 s, Tu = 3.0 s, 1.5 s & 0.5
S,
m-2
5000
4000
3000
:2
c
';;;
Cl
=
OTp 3.0 S, Tu = 3.0 S
.Tp = 3.0 5, Tu = 1.5 S
.Tp
3.0 s. Tu = 0.5 S
2000
=
s
.s
II')
1000
-1000
State
Figur
ql
=5
.3:
m-2
t t
gain,
v
=
40 mis, Tp
=
3.0 s, Tu
3.0 s, 1.5 s & 0.5 s,
CHAPTER
6: MODEL PREDICTIVE
199
CONTROL RIDER MODEL
1.5
: 0.5
Ii
.5
I
.tt
Cl
A.
40
" -Tp.
3.0 s, Tu • 3.0 S
---Tp • 3.0 s, Tu • 1.5 s
12 ······T • 3.0 s, Tu· 0.5 s
'\ ' \
0 ~-l41---,---------,----+~,~,~
80
\
-0.5
-1
-1.5
Preview Distance (m)
Figure 6.39: Preview gains, v
=
40 mis, Tp
=
3.0 s, Tu = 3.0 B, 1.5 B & 0.5 B,
ql = 5000 m-2
3.7
----------
3.5
e= 3.3
- -Non Linear
-Linear
!
!J
i
-Ta
et
3.1
A.
2.9
2.7
45
50
55
60
65
Path longitudinal (m)
Figure 6.40: Path following, v
= 10 m/B, Tp = 3.0 B, ql = 5000 m-2
CHAPTER 6: MODEL PREDICTIVE
200
CONTROL RIDER MODEL
15
10
5
E
~
•
..
0
:::I
CT
10
{!.
-5
-10
-15
Path Longitudinal (m)
Figure 6.41: Steer torque, v = 10
mIs, Tp = 3.0
S, ql
= 5000 m-2
<4
3.5
3
E
2.5
i!
2
!I
1.5
-Linear Path
-Target
S
..
5
II.
1
0.5
0
20
40
60
80
100
120
-0.5
Path Longitudinal (m)
Figure 6.42: Path following, v = 10 mIs, Tp = 3.0
path
S, ql
= 5000
m-2, Linear reference
CHAPTER
6: MODEL PREDICTIVE
201
CONTROL RIDER MODEL
Path
Preview Horzlon ~
I
Figure 6.43: Linear reference path, showing alternative points of aim; limit of preview
horizon (a), half way point of preview horizon (b)
4
3.5
3
I
2.5
'!
2
~
1.5
i
II.
0.5
0
20
40
60
80
100
120
-0.5
Path Longitudinal (m)
Figure 6.44: Path following, v
reduction reference path
=
10 mis, Tp
= 3.0 S,
qi
= 5000 m-2,
Linear error
CHAPTER 6: MODEL PREDICTIVE CONTROL RIDER MODEL
202
4
_ ........ ------
3.5
....
..' ....
3
I
'!
S
!I
..
oS
,..'"
,
,,
2.5
I
2
-lr-20m
- -jr= sum
I
I
I
,
1.5
-Ta
,
I
et
I
,
A-
I
,
I
0.5
I
0
10
20
30
40
60
50
70
80
-0.5
Path Longitudinal
Figure 6.45: Path following, v = 10
error reduction reference path
(m)
mis, Tp =
3.0
S, ql
= 5000 m-2,
Exponential
Chapter 7
Performance Comparisons of
Control Techniques
7.1
Introduction
Two principal control strategies, optimal control and model predictive control, have
been assessed for modelling the control actions of a human motorcycle rider. Both
techniques have their roots in the solution of a quadratic
cost function, but the
formulation of the elements of the cost function, notably of the cost associated with
the system output, varies.
Essentially, the mathematical
difference between the two methods centres on the way
in which the rider model determines the errors from the path that he is attempting
to follow and the definition of the optimisation horizon. Additionally, predictive control offers further controller tuning options compared with optimal control, allowing
the preview and control horizons to be set independently
and with the option of a
reference path that the system attempts to follow.
The results obtained in the preceding two chapters (Chapters
5 & 6) have shown
that, on the whole, both techniques are capable of applying appropriate
control to
a motorcycle model when attempting
Controller
a single lane change manoeuvre.
gains, indicative of the importance placed on the available system information by the
controller, were seen to be similar for the two methods considered.
The differences
between the methods therefore lie in the details, and so this chapter will aim to
203
CHAPTER 7: PERFORMANCE COMPARISONS OF CONTROL TECHNIQUES
identify and explore the fundamental
differences between the performances
204
of the
two approaches and the potential advantages that may be gained through the use of
predictive control techniques.
The characteristics
of the optimal control approach [94] were generally regarded to
provide a good representation
of the control actions of a motorcycle rider. For the
model predictive control approach to be considered a more suitable alternative,
characteristics
the
of the controller must therefore be similar, while also providing some
useful advantage over the optimal control approach and correcting any identified
weaknesses.
This chapter will therefore draw together the results of the optimal control (Chapter
5) and the predictive control (Chapter 6) approaches.
Specific controller conditions,
for example the short preview horizon case, will be examined and the direct comparisons of the two approaches compared.
The results of these direct comparisons
will enable the final conclusions of this research work, regarding the suitability of the
model predictive control approach to the motorcycle rider task, to be made. These
final conclusions are subsequently given in Chapter 8.
7.2
Comparison Results
Both control techniques have been applied to the motorcycle model, detailed in Chapter 3, combined with the road preview information, as shown in Chapter 4, for a range
of controller parameters.
For each control approach a number of elements of the con-
troller's behaviour were analysed.
A select range of these parameter
sets will be
directly compared, presented in Table 7.1.
7.2.1 Comparison 1 - Baseline Parameters, Low Speed
Both control techniques were applied to the motorcycle model using a baseline parameter set that permitted a moderate balance between path following accuracy and
control cost. The low speed baseline parameter set is therefore used initially to compare the performances of the two techniques.
The paths of the two control approaches when using this baseline set are considered
CHAPTER 7: PERFORMANCE COMPARISONS OF CONTROL TECHNIQUES
205
first (Figure 7.1). The paths are very similar, with both approaches following the
target path well. No obvious differences are initially apparent, and so to consider the
performances in greater detail, the path errors for the two approaches are analysed,
defined as the difference between the target (set) path and the actual path achieved
by the motorcycle model.
The results are presented in Figure 7.2 and again the
results for the two approaches are not too dissimilar.
Both show similar errors at
similar points along the path, and in this example the optimal control model actually
produces marginally lower total errors over the lane change phase. However, for the
model predictive control case, the path error returns to zero after the manoeuvre,
whereas for the optimal controller this is not the case, with a small steady error of
-0.002 m resulting after the lane change.
The broad controller characteristics
for the predictive control approach should be
largely similar to those of the optimal control approach. The state gains and preview
gains are compared in Figures 7.3 and 7.4, being almost identical and therefore suggesting that, if the controller gains for the optimal control approach were considered
representative of a human rider's actions, then so too can the gains produced by the
predictive controller.
1.2.2
Comparison 2 - Baseline Parameters, High Speed
The same comparison is drawn for the higher speed case, again using the baseline
parameters,
and again analysing the path differences that result for the optimal con-
troller and the model predictive controller (Figure 7.5). At the higher speed, the
relative performances of the two methods are again very similar to those seen at the
lower speed. However, at the increased forward speed the steady state error of the
optimal control model after the manoeuvre phase is more apparent.
The optimal controller has peak path following errors of 1.067 m and -1.509 m, and a
steady state final error of -0.166 m. By contrast, the predictive controller has peaks
of 1.357 m and -1.152 m, and a final error of zero.
CHAPTER 7: PERFORMANCE COMPARISONS OF CONTROL TECHNIQUES
206
7.2.3 Comparison 3 - High Speed, Loose Control
The previous analysis in Chapters 5 and 6 indicated that the both high speed and loose
control conditions require greater levels of preview for good control. The combined
case for high speed running with loose control is therefore considered. In line with the
results of the previous two comparisons, the optimal controller again shows significant
steady state errors (Figure 7.6). The point of note now is that the steady state path
error of the optimal controller is positive, indicating that the lane change has fallen
short of the 3.5 m lateral shift. For ql = 5000 (Figure 7.5), the error was negative,
indicating that the motorcycle had overshot the lane change, and had settled to a
steady state at a lateral path value greater than the target 3.5 m. This suggests the
possibility for some value of ql which, although not resulting in preview gains that
diminish to zero, may result in zero steady state errors and erroneously suggest a
sufficiently long preview horizon that the gains to diminish to zero.
7.2.4
Comparison 4 - Limited Preview, Loose Control
The case for limited preview is a situation that should be considered important when
assessing the applicability of the approach to the modelling of a human rider. It seems
logical that a rider can still exercise accurate path following even given limited visual
preview. If a rider were, for instance, following a large vehicle such as a lorry, due to
restricted knowledge of the road some way in advance, it may be expected that his
transient control behaviour would be compromised.
However, for a steady, straight
section of road, despite limited forward vision, the rider would still be expected to
be able to adopt the correct position on the road. In essence, limited preview should
not restrict the rider from eventually achieving the correct position-in-lane,
It has been seen in the earlier analysis of the control methods (Chapters 5 and 6)
that a loose control strategy is in general associated with a greater emphasis on the
distant road preview information.
From a control perspective then, the worst case
scenario would be a loose control strategy coupled with limited preview horizon.
The results would be expected to be similar to the high speed, loose control approach,
since increased speed in general requires greater preview. Increasing the speed without
increasing the preview horizon is therefore similar to keeping the speed the same but
CHAPTER
7: PERFORMANCE
COMPARISONS
OF CONTROL TECHNIQUES
207
reducing the preview horizon.
The two control methods are therefore analysed for the limited preview, loose control
situation, with a preview horizon Tp of 1.5 s and a path error weighting ql of 1000 m-2•
The path error comparison is drawn in Figure 7.7, highlighting, in common with the
previous analysis, the errors that can occur when the optimal control approach is
employed with less than ideal levels of preview allowed. During the manoeuvre phase,
the path errors for the predictive control approach peak at 0.25 m, decaying to zero
c.
after the manoeuvre section of the path.
By contrast,
the errors for the optimal
control approach peak at 1.41 m, before settling to 1.17 m after the manoeuvre.
In such a situation,
the knowledge that the rider has of the motorcycle does not
change. A rider would still have full knowledge of the yaw angle, roll angle, steer
angle and so on, and therefore it may be expected that the state gains would, qualitatively, not change significantly with a reduction in the preview horizon (Figure 7.8).
However, the rider's knowledge of the approaching road, which is also used by the
rider to determine his control inputs, does change significantly with a reduction in the
preview horizon, and so it may be expected that the rider would need to re-evaluate
the use that is made of this limited information.
Figure 7.9 compares the preview
gains for the limited horizon, loose control situation, and here a significant difference
is seen between the two control approaches. The superior path following performance
of the predictive controller has already been seen (lower path errors, Figure 7.7), and
so the change in the preview gain is suggested to be representative
of the change in
emphasis that the rider places on the limited road path information available.
7.2.5
Comparison 5 - Yaw Error Minimisation
The vast majority of results presented in this thesis have concerned the rider model
operating to minimise the lateral path error of his position relative to the target path.
The capability to minimise the yaw angle of the motorcycle relative to the path also
exists, and the comparison of the two control strategies operating in this manner are
drawn.
In a similar way to the lateral path error minimisation analysis, the optimal control
strategy results in a steady state error, this time between the heading of the target
path and the heading of the motorcycle.
Consequently, with forward motion the
CHAPTER 7: PERFORMANCE COMPARISONS OF CONTROL TECHNIQUES
208
motorcycle's lateral error progressively increases (Figure 7.10). Although yaw angle
error minimisation is not considered as the primary way in which a motorcycle rider
would assess his path following performance, it is nonetheless a strategy which could
be adopted in conjunction with lateral error minimisation, and is therefore worthy of
some consideration.
7.2.6
Comparison 6 - Short Control Horizon
One of the main features of model predictive control that distinguishes it from optimal control concerns the ability to set the control horizon shorter than the preview
horizon. In the case of a limited control horizon, the model predictive control strategy assumes the control to be invariant from the control horizon up to the preview
horizon, therefore providing the controller with a full control input up to the preview horizon. This strategy of control is therefore compared with the optimal control
model, for which the preview and control horizons are intrinsically equal.
Figure 7.11 presents the path errors resulting from the path following task using the
low speed baseline parameters,
with the control horizon reduced to both 1.5 s and
0.5 s for the predictive controller.
Even with limited control horizon, the predictive
controller is still capable of generating appropriate
controller gains, and hence the
performance comparison between the limited control horizon predictive controller and
the optimal controller is very close to the comparison with equal preview and control
horizons (Comparison 1). The predictive controller with limited control horizon is
still capable of completing the manoeuvre and returning to a zero steady state lateral
path error condition despite the restricted control horizon. Although in some respects
the reduced control horizon appears to deteriorate
the controller's performance, the
differences are not severe.
7.2.7
Comparison 7 - Very Low Speed
Above a. forward speed of 10
mis, the
stability characteristics
of the motorcycle are
relatively invariant (Chapter 3, Figures 3.10, 3.15), but below this speed is notably
different: the weave mode is an unstable low frequency oscillatory mode at low speeds,
while the capsize mode is more stable at low speeds. As a consequence, the control
required to stabilise and guide the motorcycle may change, and so the performance
CHAPTER 7: PERFORMANCE COMPARISONS OF CONTROL TECHNIQUES
209
of the two control approaches under such conditions is compared.
Miyamaru et al. [65] suggested that at moderate speeds and above, the rider's directional control is achieved by control of the roll angle of the motorcycle, while at
low speed the primary control technique to influence the motorcycle's trajectory
is
through control of the steer angle of the motorcycle. Doth the optimal and predictive
control strategies were therefore tasked with the same path following task, but this
time at a forward speed of only 4 mis, with both strategies reflecting the change in
the control technique suggested by Miyamaru et al. The state gains distribution
is
seen to change such that now the peak controller state gains are for the steer angle
state (Figure 7.12).
Also of some significance in this figure is the state gains relating to the tyre lateral
forces. For the optimal controller, these are minimal, while for the predictive controller they are not, with the front tyre gain being the more significant of the two by
some margin.
These observations are worthy of some thought, and are believed to be due to the
change in stabilities of the capsize and weave modes at low speeds. At low speed,
as the motorcycle begins to capsize, the geometry of the steering system is such that
a significant steer angle is generated, correcting the lean of the motorcycle [29]; it
is this characteristic
which gives the low speed capsize stability. However, this may
then result in a lean to the opposite side, where the same effect results, consequently
resulting in a low speed, low frequency weave behaviour.
A number of movie clips
are available in [26] and [78] that more clearly demonstrate
this combined low speed
capsize and weave. At low speeds, significant steer angles will therefore be expected,
and consequently relatively large tyre forces. furthermore,
the weakness of the gyro-
scopic forces at low speeds also increases the relative contribution that the tyre lateral
forces must provide in order to stabilise the capsize mode of the motorcycle.
For a low speed weave, the change in lateral position may be small, but the change in
heading angle is relatively much larger, and will therefore have a significant impact
on the predicted future path.
The optimal control strategy, for lateral path error control, bases the control decision
on the motorcycle's
current position relative to the target path alone.
suggested, will not change significantly.
This, it is
CHAPTER 7: PERFORMANCE COMPARISONS OF CONTROL TECHNIQUES
210
Predictive control bases the control decision on the predicted future path. Although
the lateral deviation may be small, the heading angle change will not be, leading to
a significant change in the motorcycle's trajectory and hence predicted future path.
The tyre forces influence the stability and the heading of the motorcycle. As the predictive controller would be expected to place more emphasis on reducing the change
in heading angle (and consequently future predicted path) during a low speed weave,
and the primary means of doing this is through the tyre lateral forces, then an increase in controller gains relating to the tyre forces may not be unsurprising.
As the
changes in the tyre lateral forces are dictated by the steer angle, itself dictated by
the steer rate, it is also perhaps unsurprising to see that the gains on the steer rate
for the predictive controller is larger in magnitude at these low speeds compared with
the optimal controller.
7.3
Performance Comparison Conclusions
This chapter has brought together a detailed comparison of the two control strategies
of optimal and predictive control.
The results have largely confirmed the strong
similarities of the two approaches for the majority of conditions, but importantly
has
drawn out the significant differences.
The inability of the optimal control approach to result in truly zero steady state errors following a manoeuvre was highlighted in Chapter 5, Section 5.3.1. Indeed, the
steady state errors will only truly reach zero when the preview horizon distance is
infinite. Cole et al. [l1J, with specific reference to a car steering task, investigated
and presented the fundamental reasons behind these observations, and the potential
benefits that predictive control could give for such a preview-limited case. Fundamentally, the optimal control theory employed here is based on the theory of an infinite
horizon, whereas the predictive controller calculates gains based on a horizon only up
to the preview horizon of the controller. Thus, provided that the optimal controller's
gains have reached zero, then the numerical loss of information
that results from
this infinite horizon assumption is insignificant to the numerical result, and the two
controllers essentially give the same results. When the preview horizon is shortened
significantly, the predictive controller calculates a new set of gains based on this new
limited horizon, while the optimal approach employs still an infinite horizon assump-
CHAPTER 7: PERFORMANCE COMPARISONS OF CONTROL TECHNIQUES
tion, with correspondingly
211
inferior results in such a case. The results presented in
this chapter have deliberately aimed to highlight this feature in a significant way, and
the result that a predictive controller will, by contrast and regardless of the horizon
length, result in a zero steady state error. Analysis in Chapter 5 showed how this
path following error with limited horizon could be overcome with the use of a local
coordinates approach. Similar observations were made by Cole et al. [11], though the
two approaches differed subtly.
These results have been emphasised by the use of higher speed, tighter control weightings and limited preview horizons. With less than a finite preview horizon, the optimal
control approach is not, in fact, able to seemingly apply an optimum control, as some
steady state error will always result. However, it was shown in Chapter 5 that while
a local coordinates approach does not, in theory, make a difference to the problem,
in practice the steady state path errors are seen to be reduced to zero.
For the optimal control approach, variation of the speed and the cost function error
weightings dictate the accuracy of the path following that results.
Although the
steady state values will never truly reach zero, sufficient preview length can result
in steady state errors which are minimal, and so the speed, error weightings and
preview horizon length are intrinsically linked in determining the magnitude of the
final steady state errors.
The predictive controller's path following performance is also affected by the speed,
error weightings and horizon lengths in a similar way to the optimal control method.
However, the steady state error has no dependency on these, and thus the error will,
given sufficient distance to reach the steady value, always be zero.
Important differences are also seen for the very low speed running condition of the two
controllers.
At low speed, the distribution
differences between the two approaches.
of the state gains show some significant
For the low speed case, a human rider
will sense the capsize of the motorcycle and apply a steering action to correct it.
During a capsize, the heading angle of the motorcycle is also changed significantly,
leading to a deviation from the target path.
With low forward speed the lateral
deviation that results from the heading angle change is small, and therefore does not
significantly affect the optimal controller, which will aim mainly to stabilise the roll
of the motorcycle.
However, the predictive controller anticipates that, although the
CHAPTER 7: PERFORMANCE COMPARISONS OF CONTROL TECHNIQUES
212
current lateral deviation may be small, the resulting heading angle change has more
significant consequences for the predicted future path, resulting in notably different
state gains for the predictive controller.
CHAPTER 7: PERFORMANCE COMPARISONS OF CONTROL TECHNIQUES
7.4
Table
213
CHAPTER 7: PERFORMANCE COMPARISONS OF CONTROL TECHNIQUES
t"Cl
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CHAPTER 7: PERFORMANCE COMPARISONS OF CONTROL TECHNIQUES
1.5
215
Figures
4
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_
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.5.
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~
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0.5
10
20
30
40
50
70
60
80
-0.5
Path Longitudinal
(m)
Figure 7.1: Path following, v = 10 mis, Tp = 3.0
S, ql
= 5000 m-2
0.25
0.2
0.15
!
0.1
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Path Longitudinal
(m)
Figure 7.2: Path errors, v = 10 mis, Tp = 3.0
S, ql
= 5000 m-2
C
CHAPTER 7: PERFORMANCE COMPARISONS OF CONTROL TECHNIQUES
216
800
700
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!
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Figur 7.3: State gains, v
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2
[DOcl
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20
Preview Distance (m)
Figur 7.4: Preview gains, v
= 10 ta]«,
30
Tp
= 3.0 S, ql = 5000 m-2
CHAPTER 7: PERFORMANCE COMPARISONS OF CONTROL TECHNIQUES
217
2
1.5
FOol
350-
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-1.5
-2
Path Longitudinal
1m)
Figure 7.5: Path errors, v = 40 mis, Tp = 3.0
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2
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Figure 7.6: Path errors, v
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1m)
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C
CHAPTER 7: PERFORMANCE COMPARISONS OF CONTROL TECHNIQUES
218
1.6
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Figure 7.7: Path errors, v = 10 mis, Tp = 1.5 S, ql = 1000 m-2
400
300
::!
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Figur 7.
Stat
gains, v = 10 mis, Tp
= 1.5 S, ql =
1000 m-2
c
CHAPTER 7: PERFORMANCE COMPARISONS OF CONTROL TECHNIQUES
219
1.5
15
Preview Distance (m)
Figure 7.9: Preview gains, v = 10 mis, Tp = 1.5
S, ql
2
= 1000 m-
4
3.5
3
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Path Longitudinal
Figur
(m)
7.10: Path following, v = 10 ia]«; Tp = 3.0
S,
2
oz = 5000 rad-
CHAPTER 7: PERFORMANCE COMPARISONS OF CONTROL TECHNIQUES
220
0.25
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Figur
7.11: Path errors, v = 10 txx]«, Tp = 3.0 s, Tu =1.5 s & 0.5
S,
2
ql = 5000 m-
400
300
200
100
~
c'
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-400
·500
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Figur
7.12: State gains, v = 4 mis, Tp = 3.0
S,
2
ql = 1000 m-
C
Chapter 8
Conclusions
The applicability
of both model predictive control and optimal control techniques
to the task of modelling a motorcycle rider have been demonstrated.
The latter is
not a new concept, having originally been presented by Sharp [94]. However, further
parameter studies have been conducted to obtain more insight into the characteristics
of the controller to assess some of the strengths and weaknesses of the approach.
Building on previous related work by MacAdam [51], Sharp [94], and Cole et al.
[11] amongst others, the application
of model predictive control to the modelling
of a motorcycle rider presented in this thesis is a novel application.
demonstrated
The work has
strong similarities to the optimal control approach, but importantly
has shown some significant and advantageous differences. For the broadest task of
rider modelling, the findings therefore suggested that the model predictive control
approach is the more suitable approach.
Specific features that differentiate the predictive control approach from the optimal
control approach have been assessed to determine the advantages that they mayor
may not offer. The availability of a reference path definition to the control problem
has been investigated and the mathematical
shown, concluding that the opportunities
manner in which it can be implemented
that this may offer are not considered ad-
vantageous to the application at this time.
Additionally, the implementation
of a non-linear prediction model as opposed to the
more common linearised prediction has been made. As may be expected the extra
computational
burden was shown to result in performance
221
advantages
as may be
'1
CHAPTER
8: CONCLUSIONS
222
expected for a non-linear plant model.
8.1 Model Analysis
The effectiveness of the model control strategy in replicating the control actions of
a motorcycle rider can be assessed by consideration of a number of factors relevant
to the controller, notably the controller gains, the performance of the controller in
completing the task and the way in which the controller's behaviour changes with
changes to the modelled riding task.
8.1.1 State Gains
The analysis of controllers of this nature usually focus primarily on the controller's
gains, since these essentially determine the way in which control is applied to the
system. The gains achieved here for both the optimal controller and the model predictive controller show good agreement to both experimental and anecdotal evidence
of the motorcycle riding task achieved by previous authors.
A number of research works, some of which are covered in the literature review
(Chapter 2), have sought to understand the important criteria that a vehicle driver
or rider attempts to control. For motorcycle riding, Weir [102]identified the control
link between roll angle and steer torque as the primary stabilising control loop, with
a slightly weaker reliance on the heading angle and lateral position of the motorcycle.
Miyamaru et al. [65]also concluded that for anything other than very low speed riding
the control task was concerned with roll angle stabilisation, and that this would then
dictate the trajectory, rather than the steer angle. This forms a distinct difference
between single and twin vehicle track controllers.
A correct rider control model should therefore reflect this trend, placing the greater
importance on controlling the roll angle of the motorcycle (which ultimately dictates
the trajectory), followed by heading angle and lateral position. The results of both
the optimal control and the model predictive control approaches have been shown to
display these trends.
Although the state gains associated with the roll angle were not numerically the
t~ll
CHAPTER
8: CONCLUSIONS
223
largest, when the contribution of the roll angle gain to the total steer torque was
assessed, it was shown that the roll angle provided the largest contribution, therefore
appearing to have the most significant influence upon the rider's control as required.
The yaw angle contribution formed the second largest contribution, again agreeing
with Weir [102].The contribution from the lateral position was low. The contribution
made by the steer angle of the motorcycle was found to be significantly smaller, which
is seen to be in agreement with the findings of Miyamaru et al. [65]. However, when
both controller models were run at low speeds, the steer angle gains were seen to be
notably greater, again in agreement with [65].
For the broad case of generous preview horizons and moderate path following task,
both the optimal controller and the model predictive controller produced near identical state gains. In this sense, both control approaches appear equally capable of
reflecting the control actions of the rider with reference to the motorcycle's states.
Only when the preview horizon is notably restricted do any differences emerge, with
the state gains of the predictive controller reduced compared with the optimal controller. We consider now why this may be and how it may reflect a motorcycle rider's
control process.
Manoeuvrability and stability are generally regarded as conflicting goals. When operating in a limited preview condition, the rider will have only limited time to react
to the road information presented to him that he is aiming to follow, and therefore
it seems reasonable that in such a situation the rider may favour a manoeuvrable
motorcycle. The motorcycle states are generally regarded as the meters by which the
rider stabilises the motorcycle [102],while the preview is used for guidance control.
A reduction in the gains applied to these states may suggest that the rider is operating in a manner in which the stabilisation of the motorcycle is of less concern, as
may be the case for restricted preview when manoeuvrability is the more pressing
requirement. In this sense, the predictive controller appears to be able to represent
this shift in the rider's priorities, while the optimal controller does not.
8.1.2
Preview Gains
A number of publications have sought to determine the control actions of a road user
with regard to the visual perception of the road available, with results obtained both
CHAPTER
8: CONCLUSIONS
experimentally
224
and theoretically
[25, 47]. Both control strategies considered here
were modelled for a wide range of controller parameters,
with the results obtained
here again giving good agreement with the previous literature.
The model predic-
tive control approach gives good agreement with the optimal control approach with
regard to the preview gains when sufficient preview is permitted.
With restricted pre-
view, the gains pattern changes notably for the model predictive controller, leading
subsequently to better controller performance.
In assessing a driver's use of preview information,
Donges [25] observed the paral-
lel features of guidance control based on the distant preview and position-in-lane
control via the near preview, observations later agreed with by Land and Horwood
[47]. These characteristics
are reflected in the preview gains produced by both the
optimal control and model predictive control strategies. The controllers were run for
both tight and loose control strategies, the former defining accurate path following
and hence accurate position-in-lane
control, the latter defining efficient following of
a more distant target, hence guidance control. The tight control resulted in a bias of
the preview gains to the near preview distance, therefore operating with the required
bias towards position-in-lane control, while loose control placed greater emphasis towards the more distant preview, hence biasing towards a guidance control strategy.
MacAdam [51] also observed that a vehicle driver would have a natural tendency to
utilise shorter preview times under manoeuvre-demanding
conditions. Tight control
can arguably be classed as a demanding manoeuvre when compared with a looser
strategy.
The agreement between the experimentally
observed patterns
the use of road preview information covered by previous literature,
concerning
and the results
obtained when using both the optimal control and model predictive controllers, highlights the suitability of both approaches for this task.
In the main, both control approaches show similar preview gains. Crucially, as the
preview horizon is reduced the gains for the predictive controller are modified, whereas
for the optimal controller they are not. When the preview information available is
reduced, it seems intuitive that the limited information remaining is more highly
regarded; a driver or rider would realistically be expected to concentrate harder on
the road far ahead when driving in thick fog compared with clear air, due largely to
the restricted reaction time available. Thus it is expected that the preview gain values
relating to the limited road information available would increase in recognition of the
CHAPTER
8: CONCLUSIONS
225
higher attention placed on this limited road preview information.
This characteristic
is observed with the predictive controller, but not the optimal controller.
8.1.3 ReferencePath
One of the features that differentiates the predictive controller from the optimal controller concerns the capability to identify a reference path, distinct from the set path.
Previous authors have suggested the separate definition of a distinct reference path,
including Guo and Guan [33], as potentially offering modelling advantages. The definition of a reference path distinct from the set path has therefore been investigated,
specifically with the definitions of linear, linear error reduction and exponential error reduction reference paths for the model predictive control approach.
Universally, the results demonstrated
was detrimental
fundamental
that defining the target road path in this way
to the overall performance of the rider control model.
While the
behaviour of the controller was unaffected, the performance certainly
was. The control model forms a measured balance between corner cutting, relative
to the set path, and the control input effort required. By defining a reference path, a
weaker trajectory, already accounting for a corner-cutting
allowance, is presented to
the controller. The controller then makes the same balance between path accuracy,
this time against the weaker reference path, and the control inputs. This results in
an actual trajectory which attempts to follow, to a tighter or looser extent depending
on the controller settings, an already weaker target path.
Therefore, unless a more elaborate reference path definition can clearly be demonstrated to produce superior controller performance compared with common reference
and set paths, then the recommendation
is made that the concept of a separately de-
fined reference path is not applied for the modelling of riding or driving tasks, instead
leaving the controller strategy to make its own judgement with regard to the level of
path simplification that it is prepared to tolerate.
Ideally, future research work will
investigate the manner in which a rider aims to regain a target path, such that the
concept of the reference path can be employed more usefully.
a
CHAPTER
8.2
8: CONCLUSIONS
226
Coordinate System
The modelling work conducted in this thesis has mostly been done using a global
coordinate system. This approach employs a simple shift-register algorithm for the
road preview [98], which is simple to employ, easy to understand and computationally
relatively simple.
A local coordinates approach has also been presented here. Arguably, the rider operates in a local coordinates manner, and so if the task is to model the rider's control
then it seems appropriate
to consider the problem from the rider's perspective.
For the optimal control approach, the use of a local coordinate system showed clear
advantages,
especially when limited visual preview horizons were available.
optimal control approach is to be used, the recommendation
If an
is therefore to employ a
local coordinates approach.
Chapter 5 showed how the limitations of the infinite horizon optimal controller when
using a short preview horizon were overcome if the problem were modelled using
a local coordinates approach.
The method shown here converted the shift-register
algorithm to update the preview information of the road path in local coordinates,
which corrected the errors seen when using a global coordinate system.
Cole et al.
[11] performed the similar process, but in their case modifying the controller's gains
multiplied by the global road picture to achieve the same results.
For the model predictive control case, limited preview horizons did not result in
any steady state errors of the kind seen for optimal control, and so the question of
local or global coordinates is not as important.
For simplicity, the global coordinates
approach is preferable, though the use of local coordinates more intuitively captures
the process from the rider's perspective.
The selection of coordinate system may
therefore be based on the relative importance of these merits as appropriate
to the
application.
8.3
Non-Linear Prediction
The majority of the results presented in this thesis have been obtained using a linear
prediction model. The dynamics model of the motorcycle vaxy non-linearly with each
CHAPTER 8: CONCLUSIONS
227
step of the motion simulation, but over the prediction horizon at each simulation step
were assumed invariant. Use of this approach simplifies the procedure markedly and
reduces computational
requirements.
The theory for a non-linear prediction model
was covered (Chapter 6, Section 6.2.2).
The results showed that, when using a full non-linear prediction, the resulting control
led to superior performance, suggesting that, ideally, this non-linear approach should
be adopted for the best controller performance.
However, the performance
when
using the linear prediction model also showed that the controller was still capable of
applying a suitable control input to achieve the required lane change task.
The question of the importance of the linear or non-linear prediction model therefore
depends upon the degree to which the model dynamic behaviour changes over the
length of the prediction horizon. For relatively gentle manoeuvres of the motorcycle,
the changes to the dynamics are not severe, and consequently the differences resulting
from the use of a linear prediction model compared with a non-linear prediction are
also not severe.
The suitability of the linear prediction model will therefore depend on the anticipated
change in system behaviour of the prediction horizon, the accuracy of results required,
and the computational
8.4
processing capacity available.
Simulation Results
The performance of the controller, with regard to the controller's gains, suggests that
J
both optimal control and model predictive control operate in a manner consistent
with what may be expected of a real rider. The actual performance of the motorcycle
that results from the riders control actions further confirms this.
The path following ability of both control models appear to accomplish the task,
displaying the qualities of a human rider such as a countersteer manoeuvre in anticipation of the turn. Donges [25] identified the anticipatory
control strategy.
element of a car driver's
The path tracking can be replicated by monitoring the lateral po-
sition of the motorcycle with regard to the intended path or by the yaw angle of
the motorcycle relative to the path. The original optimal control work by Sharp [94]
considered only lateral path error weightings, as the yaw angle error was considered
tl
~'r·
CHAPTER
228
8: CONCLUSIONS
secondary to the control problem.
When the preview horizon is sufficient such that the preview gains settle towards zero,
the performance of both approaches produce similar results.
However, for limited
preview horizons the model predictive control approach was shown to demonstrate
clear advantages.
In the case of the optimal controller, mathematical
analysis of a reduction of the
preview horizon to any value below infinite was suggested to result in a steady state
error between the target path and the actual path achieved; the magnitude of this error however only became significant as the preview horizon was significantly reduced,
below the point at which the preview gains become close to zero.
For the case of the 10 m/s baseline modelling parameters considered in the analysis,
following the manoeuvre the optimal control model tracked the path with a small
steady lateral position error of 0.002 m. At the higher speed of 40 mis, the lateral
steady path tracking error increased to 0.165 m. As the preview distances were halved,
these errors increased further, to 0.383 m and 1.813 m respectively.
By contrast, the preview horizon of the model predictive controller could safely be
reduced without the introduction
haviour was deteriorated
of steady state errors. Although the transient be-
as a result of the limited preview, the steady state path of
the motorcycle would still eventually return to the target path.
For all modelling
conditions considered for the model predictive controller, all lateral position errors
would return to a steady zero.
The behaviour of the model predictive controller
would appear much more suitable and representative
of a motorcycle rider in such a
situation.
This feature of steady state errors is also notable when the controller is set to operate
by minimisation of yaw angle errors. This aspect of control was considered in [94],
but results were not presented as it was not considered representative of a motorcycle
rider's actions. This case is, however, considered here for completeness. In the same
way that steady state errors were observed for lateral position control of the optimal
control approach, the same result is found for yaw angle control, except that in this
case a steady state error is between the final yaw angle of the motorcycle and the path.
Consequently the motorcycle follows a straight path but heading on a different angle
to the intended path. For the 10 m/s baseline parameters,
this resulted in a heading
CHAPTER
8: CONCLUSIONS
229
error of 0.028 rad, and for the higher speed case a heading error of 0.074 rad. As with
the lateral position control case, the model predictive controller results in zero steady
state errors, and thus the actual and intended paths are parallel. Although the path
does not exactly follow the intended path, the performance of the model predictive
controller in this situation is clearly superior to that of the optimal controller.
8.5
Final Conclusions and Further Work
Thus far, the conclusions have been made that both optimal control and model predictive control appear to provide a control strategy that represents the rider's control
actions well. Both approaches aim to minimise a cost function that includes the road
path, and therefore both aim to provide the best, or optimal, control input to achieve
this. 'While both approaches can therefore be considered as some form of 'optimal'
control, the distinction
is made between the mathematical
approaches of optim
control and model predictive control, as given in Chapters 5 and 6.
While both approaches appear capable of generating appropriate
follow a target path, in specific cases, notably the case ofrestricted
results that have been shown here demonstrate
has clear advantages. Mathematically,
control inputs to
visual preview, the
that the predictive control approach
the characteristic that a steady state error for
the optimal controller is in fact always present has been shown. However, provided
that sufficient preview is allowed such that the preview gains reach close to zero, this
error becomes insignificant.
Previous work [90] indicated the requirements
for zero
steady state errors by allowing sufficient preview, the analysis has been extended to
show how those steady state errors arise, and how they can be overcome with a local
coordinates controller definition.
A local coordinates
approach was presented which was shown to be theoretically
capable of correcting the steady state path following errors of a short preview horizon
optimal controller. Similar findings had been made by Cole et al. [11]. Although the
overall controller's strategies were comparable, the methods differed mathematically.
In the method presented here, the road preview element of the state vector contains
the road information explicitly in the rider's local preview, while the method of Cole et
al. retains a global definition and essentially converts the preview to local coordinates
implicitly.
CHAPTER
8: CONCLUSIONS
230
Model predictive control also offers other advantages that have yet to be explored.
One of the motivations for the use of model predictive control regarded the relative
ease with which hard constraints
can be included into the modeL
The reference
texts provide greater detail on the possibilities available and the processes required
[9,52]. The modelling work revealed that, without constraints, the theory produces
mathematical
answers which may not represent physically achievable values, with the
steer torques generated for the high speed running cases being a good example of such
a case. In addition to limits on the steer torques, these hard constraints
could be
used to account for physical restrictions such as road boundaries, steer angle limits
and acceleration and braking constraints if the modelling were extended to include
forward speed control.
The controller gains for the optimal and predictive control approaches were seen to be
close to identical whenlong preview horizons were permitted, but became significantly
different as the horizon was reduced. The gains for the predictive controller were seen
to change as the preview horizon was reduced, allowing the motorcycle to still follow
the path correctly. This suggests that a motorcycle rider must modify his perception
of the available road information in such limited preview conditions.
While the results presented here have included extensive parametric
remain only theoretical.
studies, they
The rider's control strategy with regard to the use of road
preview has been compared with experimental studies for twin-track vehicle driving
[25,47], and the experimental results for rider input control [74, 104]. Experimental
evaluation of a motorcycle rider's use of road preview would provide a useful addition
to enable validation of the results presented here, and to broaden the knowledge in
the wider field of motorcycle rider control.
More specifically, the theoretical results that have been presented in this thesis have
suggested that a non-linear prediction model is capable of superior path tracking
abilities, and certainly strictly the more correct form of prediction.
However, the
ability of the rider to account for these non-linearities in the motorcycle's response is
another unanswered question. It may be that a highly skilled and experienced rider
is able to account for the non-linearities
of the motorcycle's response in his output
prediction, and can therefore optimise the performance of the motorcycle to a greater
extent than a less skilled rider. This may somehow reflect the differences between,
say, professional motorcycle racers and a more average rider.
While this is only a
CHAPTER
231
8: CONCLUSIONS
conjecture, it would be an interesting,
although challenging, task to quantify more
firmly to what extent a rider may be able to predict the non-linear behaviour of the
motorcycle.
Overall, the applicability of model predictive control has been demonstrated
using a
somewhat simplified problem. The model was considered to run at a constant forward
speed, and the motorcycle model itself was a relatively simple model.
implementation
The initial
of a non-linear tyre model was hoped to provide the ability to model
more complicated manoeuvre strategies to further investigate the characteristics
advantages of using predictive control for motorcycle rider modelling.
that the results found here will provide encouragement
and
It is hoped
to develop more elaborate
rider control models employing these techniques, ultimately leading to both a broader
understanding
of the rider's control strategies and also potentially as an advantageous
tool for motorcycle design and analysis.
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List of Publications
• Rowell S., Popov A. A., Meijaard J. P., Modelling the control tasks for riding a
motorcycle, 19th Symposium of the International Association for Vehicle System
Dynamics (IAVSD), Milan, Italy, 29 August - 2 September 2005, Poster Paper
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• Rowell S., Popov A. A., Meijaard J. P., Modelling the control strategies for
riding a motorcycle, ISMA2006 Conference, Leuven, Belgium, 18 - 20 September
2006.
• Rowell S., Popov A. A., Meijaard J. P., Application of predictive control strategies to the motorcycle riding task, 20th Symposium of the International
As-
sociation for Vehicle System Dynamics (IAVSD), Berkeley, California, U.S.A.,
13 - 17 August 2007.
• Rowell S., Popov A. A., Meijaard J. P., Modelling the control tasks for riding a
motorcycle, 5th IFAC symposium on advances in automotive control, Monterey,
California, U.S.A., 20 - 22 August 2007.
241
Appendix A
Motorcycle Data
This Appendix section details the specific values for all necessary parameters of the
motorcycle used to generate the results in this thesis. The motorcycle model is based
on the simplified motorcycle model first presented by Sharp in [Al].
parameter
values, specifically for the more advanced tyre model introduced,
drawn from work by Meijaard and Popov [A2].
A.1
Additional
Motorcycle Data
b
Figure A.l: Definitions of bicycle model dimensions
242
were
APPENDIX A: MOTORCYCLE DATA
A.2
243
Geometric Details
a = 0.9347 m
e = 0.0244 m
b = 0.4798 m
hr
c = 0.0226 m
hI = 0.4561 m
A.3
= 0.4714 rad
= 0.6157 m
Inertial Properties
= 1.2338 m4
II,y = 0 m4
II,z = 0.4420 m4
II,zz = 0 m4
Ir,z = 31.1838 m4
Ir,y = 0 m4
= 21.0694 m4
Ir,xz = -1.7354 m4
Ilw,x = 0.7 m4
Ilw,1J = 0.7186 m4
Ilw,z = 0 m4
Irw,z = 0 m4
Irw,1J
C'Y,r = 1326.6232 N/rad
R,. = 0.3048 m
Ir,z
II,z
A.4
'fJ
Irw,z
= 1.0508 m4
= 0 m4
= 30.6472 kg
m2 = 217.492 kg
ml
Tyre Properties
Simple tyre model:
Ca,1
= 11174.38
N/rad
C'Y,I = 938.6124 N/rad
= 15831.8556
Tt
=0m
(JI = 0.24 m
R, = 0.3048 m
a; = 0.24384 m
Cr,1 = 0.2448 m
Kn,r = 142627 N/m
(JI = 0.24 m
Cr,r = 0.2448 m
ILl = 1.0
(Jr = 0.24384 m
C',I = 11.096
J.Lr = 1.0
ROI = 0.3048 m
cc- = 11.096
PI
x;
Pr = 0.07 m
Ca,r
N/rad
Advanced tyre model:
= 100672.07 N/m
= 0.07 m
ROr = 0.3048 m
Tt
=0m
APPENDIX A: MOTORCYCLE DATA
244
References
[AI]
Sharp R. S., The stability and control of motorcycles, Journal of Mechanical
Engineering Science, 13(5), 1971, pp. 316 - 329.
[A2] Meijaard J. P. & Popov A. A., Numerical continuation
of solutions and bi-
furcation analysis in multibody systems applied to motorcycle dynamics, Nonlinear
Dynamics, 43(1), 2006, pp. 97 - 116.
Appendix B
VRML Simulation Model
B.l
Introduction
As an aid to the simulation work, and to obtain a clearer understanding
of how the
motorcycle was behaving during the simulated manoeuvre, a Virtual Reality Modelling Language (VRML) animation was generated, with the aim to produce a simple
representation
of the motorcycle such that the system output could be conveyed eas-
ily. VRML is a low-level code that allows simple shapes and forms to be drawn in
three dimensions and, with suitable input data, freedoms and constraints placed upon
the objects, motion to be simulated.
The intention of this Appendix is not to form a detailed guide for the programming
of an animation using VRMLj suitable texts on the subject can readily be found
[Bl],
The aim here is to provide a brief outline of the animation model generated to aid
the understanding
of the motorcycle and controller performance during this research
work.
B.2
Coding
The code to draw and animate the motorcycle was written as a Matlab m-file, using
a conversion program to convert the Matlab code to VRML code. This program was
obtained from Schwab [B2].
245
APPENDIX
B.2.l
246
B: VRML SIMULATION MODEL
Motorcycle Body
The preliminary task was to draw a simple motorcycle model in VRML. This did
, not require a intricately detailed model; in fact, the simpler the model the easier
the behaviour would be to analyse.
representation
The model was therefore intended only as a
of the simplified motorcycle model that was used to determine the
motorcycle's dynamic response. This VRML model is shown in Figure Bd.
The VRML code allows simple three-dimensional
objects to be drawn with relative
ease. The frame was drawn as a series of cylinders, for which the two end points, the
cylinder diameter and colour are defined. The rider's frame was drawn via a similar
approach, with a simple sphere to represent the rider's head, defined by position,
diameter and colour. Spheres were also used at the ends of the frame cylinders to
provide a more aesthetically pleasing appearance.
Specifically, spheres and cylinders
are defined by entries of the following nature:
sphere (B.Dia.FrameColour)
(A-I)
cylinder(B.E.Dia.FrameColour)
where, here, B represents the centre of the sphere and is a 3 x I vector giving x-, yand z-coordinates of the position, Dia is the diameter of the sphere and FrameColour
is again a 3 x 1 vector giving the RGB ratios of the colour required. Additionally for
the cylinder, a second x-y-z 3 x 1 vector E is defined; the two position vectors then
define the centres of the ends of the cylinder.
The wheels were drawn as two-dimensional circles, extruded through a 3600 sweep
about the centre of the tyre cross-section to obtain the toroidal shape of the front
and rear wheels.
Shape {
geometry Extrusion {
crossSection [define cross - section shape
spine [define extrusion
}
}
spine
]
]
(A-2)
247
APPENDIX. B: VRML SIMULATION MODEL
The objects in the motorcycle model were constructed in a parent-child tree structure,
such that, for instance, the front frame is a child of the rear frame, since any motion
of the rear frame will influence the front frame, but a steer rotation of the front frame
will not influence directly the rear frame.
A series of frames were defined, in which objects can be drawn. Any movement of
these defined frames will move any objects drawn within and therefore attached to
these frames. The structure of the frames was defined in such a way as to reflect the
hierarchy of movement as defined by the coordinate system used. Thus, the first frame
drawn was the global reference frame, in which the motorcycle frame was drawn, in
which was drawn the yaw frame, then the roll frame, then the steer frame (Chapter
3).
Thus, the MOTORCYCLE
frame is defined, which has the YAWFRAME as a child,
which subsequently has the ROLLFRAME
as a child, and so on:
DEF MOTORCYCLE Transform
{
children [
DEF YAWFRAME Transform
{
(A-3)
children [
J}
l}
Thus, the order was: MOTORCYCLE
-+
YAWFRAME
-+
ROLLFRAME
-+
STEER-
FRAME.
The rear frame and the rider were drawn in the ROLLFRAME
and the front frame
structure drawn in the STEERFRAME.
B.2.2
Animation
The results from the Matlab lane change simulations were exported to the VRML
code, defining individually the lateral positions of the motorcycle frame, and the
rotations of the yaw, roll and steer angles.
Each rotation was defined as a four-column vector, where each row entry corresponds
APPENDIX
248
B: VRML SIMULATION MODEL
to one step of the iterative motorcycle simulation. The first three columns define the
X-,
y- and z-coordinates of the end of a vector from the origin. This vector forms
the axis about which the rotation will occur. The fourth column defines the angle of
the rotation. Thus, a rotation of 0.1 rad about the z-axis (roll) would be defined by
[1 0 0 0.1]' for example.
To fully simulate the motion, the individual frames defined when drawing the motorcycle (YAWFRAME, ROLLFRAME, STEERFRAME) are rotated by using the
appropriate four-column rotation matrices.
As the structure of the motorcycle is
drawn in the ROLLFRAME and STEERFRAME, then as these frames are rotated
the objects drawn within them also rotate, and hence the animation of the motion is
obtained.
Further options exist for defining, for example, the cycle time of the simulation and
the positions and orientations of camera angles.
For the reader wishing to gain
further insight into the possibilities offered by Virtual Reality Modelling Language,
many suitable texts can be found than will provide a more detailed and specific
introduction to the topic [Bl].
Figures
Figure B.1: VRML Motorcycle Model
APPENDIX
249
B: VRML SIMULATION MODEL
Figure B.2: VRML Animation snapshots
References
[Bl] Ames A. L., Nadeau D. R. & Moreland J. L., VRML 2.0 Sourcebook, John
Wiley & Sons, New York, U.S.A., 1997. ISBN 0-471-16507-7.
[B2] Schwab A. L., Website of Professor A. L. Schwab,
http://audiophile.tam.comell.edu/ als93j.
rv
Accessed 23rd July 2007.
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