# THE STEERING RELATIONSHIP BETWEEN OFF-ROAD MILITARY VEHICLE

```THE STEERING RELATIONSHIP BETWEEN
THE FIRST AND SECOND AXLES OF A 6X6
CARL-JOHANN VAN EEDEN
Submitted in partial fulfilment of the requirements for
the degree
M Eng (Mechanical Engineering)
in the
Faculty of Engineering, Built Environment and
Information Technology
University of Pretoria
2007
THE STEERING RELATIONSHIP BETWEEN THE
FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD
MILITARY VEHICLE
By
Carl-Johann van Eeden
Department of Mechanical and Aeronautical Engineering
University of Pretoria
Summary
The steering arrangement of a 6x6 off-road military vehicle was investigated, with the aim to determine if a
variable steering ratio between the first and second steering axle of the vehicle will make an improvement in
the steady and transient state handling of the vehicle.
Low speed manoeuvring was evaluated, comparing the vehicle steering geometry with Ackerman geometry.
For steady state handling, a bicycle model was developed, and constant radius simulations at various track
radii, vehicle speeds and steering ratios (ratio between the first and second steering axle) was performed.
For transient dynamic simulations, a mathematical model was developed that included a simple driver model
to steer the vehicle through a single lane change, again at various speeds and steering ratios.
The vehicle was instrumented, and actual constant radii tests, as well as single lane change tests were
performed. The measurements enabled the comparison of simulated and measured results. Although basic
mathematical models were used, acceptable correlation was obtained for both steady state and transient
dynamic behaviour.
The results indicated that for this specific vehicle geometry, where the centre of mass is above the second
axle, no marked improvement would be obtained by implementing a variable ratio steering system.
The mathematical model was changed to simulate a vehicle with longer wheelbase and different centre of
mass. With the new geometry, theoretical slip angles (and therefore tire wear) reductions were more
noticeable.
It was concluded that a variable ratio system between the front and second axle would not be an
economically viable improvement for this vehicle, since the improvement achieved will not warrant the
Keywords: Variable steering ratio, steady state handling, transient handling, multi axle vehicle, off-road,
bicycle model, three axle model, four axle model, GPS measurements, slip angle.
DIE STUURVERHOUDING TUSSEN DIE EERSTE
EN TWEEDE AS VAN `N 6X6 MILITÊRE VELD-RY
VOERTUIG
Deur
Carl-Johann van Eeden
Studie leier: Mnr P.S. Els
Departement Meganiese en Lugvaartkundige Ingenieurswese
Universiteit van Pretoria
Opsomming
Die stuurverhouding van ‘n 6x6 militêre veld-ry voertuig is ondersoek, met die doel om te bepaal of ‘n
variëerbare stuurverhouding tussen die eerste en tweede stuur as van die voertuig ‘n verbetering in die
Lae spoed manuveerbaarheid is bepaal en vergelyk met die Ackerman stuur geometrie. Vir gestadigde
hantering, is ‘n fietsmodel ontwikkel, en ‘n konstante radius simulasie vir verskeie radiuse, voertuigspoede en
stuurverhoudings (verhouding tussen die eerste en tweede stuur as) is gedoen. Vir ongestadigde hantering
simulasies, is ‘n wiskundige model saamgestel wat ‘n eenvoudige bestuurdermodel implementeer, om die
voertuig deur ‘n enkelbaan-verandering te stuur, weer by verskeie voertuigspoede en stuurverhoudings.
Die voertuig is geïnstrumenteer, en konstante radius toetse, sowel as enkelbaan verandering toetse is
uitgevoer. Die metings is vergelyk met simulasie resultate. Alhoewel basiese wiskundige modelle gebruik is,
Die resultate toon aan dat vir die spesifieke voertuig geometrie, waar die massamiddelpunt van die voertuig
bo-op die tweede as geplaas is, geen merkbare verbetering verkry word met ‘n variëerbare stuurverhouding
nie.
Die model is verander om ‘n voertuig te simuleer met ‘n langer wielbasis en ‘n nuwe massa middelpunt
posisie. Met hierdie geometrie kon die teoretiese gliphoeke (en daarom bandslytasie) verminder word.
Daar is gevind dat ’n variëerbare stuurverhouding tussen die eerste en tweede as van die spesifieke voertuig
nie ekonomies lewensvatbaar sal wees nie, aangesien daar nie ’n merkbare verbetering verkry word wat die
addisionele koste en kompleksiteit sal regverdig nie.
Sleutel woorde: Varieerbare stuur verhoudig, gestadigde toestand hantering, oorgangs hantering, multi as
voertuig, veld-ry, fiets model, drie-as model, vier-as model, GPS metings, glip hoek.
Acknowledgements
Dr Schalk Els
Dr Stefan Nell
Renette van Eeden
Karla van Eeden
Paragraph
Description
Page number
1
Background
15
2
Problem statement
17
3
Aim of the study
17
4
Literature study
18
4.1
Multi-wheel steering
18
5
Test vehicle information
23
6
Mechanical integration of steering ratio changes
25
7
26
7.1
Introduction
26
7.2
Low speed turning
26
7.3
High speed cornering
28
7.4
Introduction to the tire model
28
7.5
Tire cornering forces
30
7.6
Cornering equations
31
7.7
33
8
Tire wear
35
9
Test vehicle low speed turning requirements
37
10
Constant radius: expanding the cornering equations to a four axle
configuration
38
11
44
11.1
Developing the three axle bicycle model
44
12
Constant radius: simulation results, simplified bicycle model
47
13
Transient state handling
57
13.1
Equations of motion, two degree of freedom model, two axles
57
13.2
Equations of motion, two degree of freedom model, four axles
60
14
Test track dimensions
64
15
Driver/steering model
65
STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
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16
67
17
69
18
Vehicle measurements
75
18.1
Introduction
75
18.2
GPS measurements
75
18.3
Test equipment
75
18.4
Test vehicle and measurement positions
76
18.5
Test procedure
80
18.6
Steering calibration measurements
81
18.7
GPS measurements
85
18.8
86
19
Test results, single lane change
89
20
Simulated vs. measured results
91
20.1
Introduction
91
20.2
91
20.3
Single lane change
92
21
Vehicle specific characteristics
96
21.1
Vehicle geometry
96
21.2
Tire and driver model
99
21.3
Speed ranges
99
21.4
Tire wear
99
22
Conclusion
100
23
Recommendations
101
24
References
102
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 6 of 138 pages
Appendix A:
104
Matlab routine to convert linear measured displacement to angular
displacements
Appendix B:
107
Measured CCW plot data manipulation
Appendix C:
110
Constant radius test – Matlab M file
Appendix D:
122
Lane change – Matlab M file
Appendix E:
126
Lane change data manipulation – Matlab M file
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
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Alphabetical Symbols
a
Distance between the first and second axle
A11
Matrix coefficient row 1 column 1
A12
Matrix coefficient row 1 column 2
A21
Matrix coefficient row 2 column 1
A22
Matrix coefficient row 2 column 2
aa
Distance between the first axle and centre of mass
Ar
Ackerman ratio
ay
Lateral acceleration
b
Distance between the second axle and centre of mass
B1
Matrix coefficient row 1 column 1
B2
Matrix coefficient row 2 column 1
bb
Distance between the centre of mass and the second axle
c
Distance between the centre of mass and third axle
cc
Distance between the second axle and third axle
CCα
Cornering coefficient
CF
Force centre
CFα
Cornering force stiffness
CG
Centre of gravity
CT
Turn centre
Cyf1
Cornering stiffness of the first axle tire
Cyf2
Cornering stiffness of the second axle tire
Cyr
Cornering stiffness of the rear tire
Cαf
Cornering stiffness of the front tire
Cαr
Cornering stiffness of the rear tire
d
Distance between the third axle and fourth axle
di
Inner wheel steering angle
do
Outer wheel steering angle
e
Distance between first axle and third axle
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
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f
Distance between second and third axle
Fy
Lateral force
Fyf
Lateral (cornering) force at the front axle
Fyf1
Lateral (cornering) force at the first axle
Fyf2
Lateral (cornering) force at the second axle
Fyr
Lateral (cornering) force at the rear axle
Fyr1
Lateral (cornering) force at the third axle
Fyr2
Lateral (cornering) force at the fourth axle
g
Gravitational acceleration constant
G
Centre of mass
I
Second moment of inertia
K
L
Wheelbase
M
Mass of the vehicle
Mc
Moment due to combined rear axle forces
R
Sr
Steering ratio
t
Track width
u
Velocity component y direction
V
Forward velocity
Vay
Velocity component y direction
Vax
Velocity component x direction
Vchar
Characteristic speed
Vcrit
Critical speed
Wf
Wr
Xt
X position at time t
Yt
Y position at time t
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
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Greek Symbols
δ0
Outside wheel angle
δi
Inside wheel angle
δ
Ackerman angle
α re
Equivalent slip angle rear axle
α f1
Slip angle first axle
αf2
Slip angle second axle
α r1
Slip angle third axle
αr2
Slip angle fourth axle
∆
Off-tracking distance
δ1
Average steering angle first axle
δ2
Average steering angle second axle
αr
Slip angle third axle
α
General slip angle
ψ
ν
Path angle from the X – axis
τ
Time constant [s]
β
Attitude angle
ψ
ψt
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
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Abbreviations
Automatic Dynamic Analysis of Mechanical Systems
AFV
Armoured Fighting Vehicle
AMV
Armoured Modular Vehicle
ARMSCOR
Armourments Corporation of South Africa
BPW
Bergische Patentachsenfabrik GmbH
CVED
Combat Vehicle Electric Drive
Dynamic Analysis and Design System
EMAS
Electronic Multi Axle Steering
ETD
Electrical Technology Demonstrator
GPS
Global Positioning System
GVM
Gross Vehicle Mass
LWB
Long Wheel Base
OEM
Original Equipment Manufacturer
PCF
Positive Centre Feel
RMS
Root Mean Square
SAE
Society of Automotive Engineers
SANDF
South African National Defence Force
SUV
Sports Utility Vehicle
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
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List of Figures
Figure 1 : Rooikat AFV
15
Figure 2 : Rooikat steering system
15
Figure 3 : Rooikat hydraulic steering system
16
Figure 4 : 1915 Jeffery quad four-wheel-steer truck [1]
18
Figure 5 : 1940 Nash quad all-wheel-drive and steer [1]
18
Figure 6 : EMAS automatic hydraulic steering [2]
19
Figure 7 : Addax multi-wheel steer
19
Figure 8 : Patria AMV
20
Figure 9 : Multi-steer trailer and truck
20
Figure 10 : Steerable trailer axles from BPW [6]
21
Figure 11 : Multidrive truck-trailer combination [7]
21
Figure 12 : EMAS automatic hydraulic steering [2]
22
Figure 13 : Bison dimensions
23
Figure 14 : Test vehicle: Bison
24
25
Figure 16 : Ackerman steering angle
26
Figure 17 : Ackerman steering, eight wheel vehicle steering with first and second axles
27
Figure 18 : Measuring tire data
29
Figure 19 : Measured tire data
30
Figure 20 : Tire cornering force properties [9]
30
Figure 21 : Cornering of a bicycle model [9]
31
Figure 22 : Change of steer angle with speed [9]
34
Figure 23 : Tire life for different tire pressures [23]
35
Figure 24 : Ackerman steering Bison
37
Figure 25 : Slip angle diagram
38
Figure 26 : Detail slip angle diagram
39
Figure 27 : Scale drawing of an 8x8 geometry
42
Figure 28 : Bison three axle bicycle model
44
Figure 29 : RMS slip angles, constant radius test
48
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
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Figure 30 : RMS steering angles
49
Figure 31 : RMS slip angles and steering angles
50
Figure 32 : Slip and steer angles - 20 meter radius
51
Figure 33 : Slip and steer angles - 40 meter radius
52
Figure 34 : Slip and steer angles - 60 meter radius
52
Figure 35 : Slip angle vs. vehicle speed
53
Figure 36 : Steering angle vs. vehicle speed – 20 meter radius
54
Figure 37 : Steering angle vs. vehicle speed – 40 meter radius
54
Figure 38 : Steering angle vs. vehicle speed – 60 meter radius
55
Figure 39 : Steering angle vs. lateral acceleration – 20 meter radius
55
Figure 40 : Steering angle vs. lateral acceleration – 40 meter radius
56
Figure 41 : Steering angle vs. lateral acceleration – 60 meter radius
56
Figure 42 : Angles and velocity components [10]
57
Figure 43 : Velocity components at path points [10]
58
Figure 44 : Free body diagram [10]
59
Figure 45 : Single lane change track dimensions
64
Figure 46 : Driver model
65
Figure 47 : Simulink driver model
66
67
Figure 49 : Time to complete the maneuver
69
Figure 50 : Maximum slip angle
70
Figure 51 : Slip angles – 30 km/h
71
Figure 52 : Slip angles – 60 km/h
71
Figure 53 : RMS slip angles
72
Figure 54 : Simulated trajectory
73
Figure 55 : Simulated yaw angle - 30 km/h
74
Figure 56 : Instrumented vehicle on the test track
76
Figure 57 : Rear GPS
77
Figure 58 : Rear GPS antenna position
77
Figure 59 : First axle displacement meter position
78
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
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Figure 60 : Second axle displacement meter position
78
Figure 61 : Vehicle speed measurement
79
Figure 62 : Front GPS antenna position
79
Figure 63 : Lane change track dimensions
80
Figure 64 : Measuring steering angles. The position of the plumb line are shown
81
Figure 65 : Steering angle calibration, measured data
82
Figure 66 : Measured steering angles for steering wheel positions
82
Figure 67 : Calibrated steering angle
83
Figure 68 : First and second axle calibrated steering angles
84
Figure 69 : Steering ratio, 0.7
84
Figure 70 : Front and rear GPS data
85
Figure 71 : GPS lane change data, front and rear GPS, 4 vehicle speeds
86
Figure 72 : Lateral acceleration, constant radius test
87
Figure 73 : Yaw rate, constant radius test
87
Figure 74 : Roll, constant radius test
87
Figure 75 : Steering angle comparison, constant radius test
88
Figure 76 : Steering single and speed comparison, constant radius test
88
Figure 77 : Measured first axle wheel speed and lateral acceleration - 28 km/h
89
Figure 78 : Steering angles on first and second axles measured data - 28 km/h
90
Figure 79 : Roll and yaw measured data - 28 km/h
90
Figure 80 : (Result of Figure 34 and Figure 76)
91
Figure 81 : (Result of Figure 34 and Figure 76)
92
Figure 82 : Single lane change measured vs. simulated trajectories
93
Figure 83 : Measured steering angle single lane change - 43 km/h
94
Figure 84 : Simulated steering angle single lane change - 50 km/h
95
Figure 85 : LWB vehicle
96
Figure 86 : LWB slip angles
97
Figure 87 : Bison slip angles
98
Figure 88 : LWB vehicle, maximum slip angles
98
Figure 89 : 8x8 MAN
99
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
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1.
Background
Continuous improvement in driveline technology, led the SANDF and ARMSCOR to
investigate the integration of a Diesel-Electric transmission system in a combat vehicle.
Project CVED (Combat Vehicle Electric Drive) was initiated. The aim of the project was to
increase internal volume, reduce vehicle mass, and increase overall vehicle performance.
As part of the project, an investigation was conducted on the feasibility of changing the
existing Rooikat steering system to a lighter compact unit, since the mechanical steering
linkages in the current Rooikat weighs in excess of 1.5 tonnes. The investigation found that
that the Rooikat can be converted to use a complete hydraulic steering system that will
increase the available internal volume. The Rooikat is shown in Figure 1.
Figure 1 : Rooikat AFV
The Rooikat Armour Fighting Vehicle (AFV) original steering system is shown in Figure 2.
Figure 2 : Rooikat steering system
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
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Figure 3 : Rooikat hydraulic steering system
Because of the large steering forces generated and required, the system chosen and shown in
Figure 3 is a hydraulic system. Hydraulic cylinders, with built-in linear transducers to
determine the exact position of individual wheels is shown, mounted onto the existing trailing
arm steering components. The need for a large mechanical steering reduction box is
eliminated using the above concept.
The legal implications of the above conversion were excluded from this investigation. The
cost of the conversion was prohibitive, and it was decided that the existing system would be
retained.
The Rooikat steering investigation suggested that the steering wheel angles can be controlled
individually, and the question was raised if the individual control of steering wheel angles
would improve manoeuvrability and improve vehicle handling. It was decided to conduct an
investigation on individual steering angles on a multi-steer off-road vehicle.
As a first attempt, the possibility to vary the steering ratio between the first and second axles
was investigated. The simplification to look at axles instead of individual wheels makes it
possible to use a bicycle model in the theoretical analysis. The mathematical complexity to
vary the ratios between axles only is reduced compared to individual wheel control. If the
modification proved worthwhile, the concept may be expanded to full individual steering.
As a first choice, an 8x8 vehicle as test vehicle would have been ideal, since the 8x8
configuration is the norm in armoured fighting vehicles in the 20 tonne to 30 tonne class, due
to mobility and wheel load advantages. However, as will be discussed in following
paragraphs, only a 6x6 vehicle was available for tests, and was therefore used. Where
applicable, mathematical relationships for an 8x8 vehicle was derived, and then simplified to
make it applicable to a 6x6 twin steer vehicle.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
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2.
Problem statement
Investigate the control of individual steering axle ratio of a 6x6 off-road vehicle, to determine
the steering ratios that will lead to improved steady and transient vehicle handling at low and
high speed.
3.
Aim of the study
The aim of this study is to continue with the initial steering investigation, as done on the
CVED programme, by analysing and designing a new steering system, where the steering
ratio between the first and second axles can be controlled to improve manoeuvrability and
handling at low and high vehicle speeds.
The study will investigate and identify optimum steering ratio’s for a given vehicle speed, in
order to improve vehicle handling and reduce tire wear. The study will start with low speed
manoeuvrability, where the Ackerman principle is required. At increased vehicle speeds, the
appropriate steering wheel angles for steady state manoeuvres will be investigated. The
constant radius test was chosen for the steady state evaluation. For transient state conditions, a
steering wheel step input will be investigated, as being generated during a single lane change
manoeuvre.
The result of the study will be a recommendation regarding a steering axle relationship,
dependant on vehicle speed, enabling a hydraulic or other steering system, with appropriate
control system, to adjust steering axle ratios to improve vehicle handling. As a first attempt,
the steering ratio between the first and second axle is investigated. Should the study indicate a
marked improvement in handling or other benefit, the concept may be further expanded to
look at individual wheel angles as well.
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4.
Literature study
4.1. Multi-wheel steering
The concept of multi-wheel steering is not new. Some of the more notable four-wheel-steer
4x4’s are shown in Figure 4 and Figure 5. Examples include the 1900-1902 Cotta
Cottamobile, the 1904-1907 Four-wheel-drive truck, the 1906-1912 American trucks, the
1913-1928 Jefferey (Nash from 1916-1928) Quad 3-ton truck, the 1914 Golden West truck
and the 1915-1917 Beech Creek truck [1].
Figure 4 : 1915 Jeffery quad four-wheel-steer truck [1]
Figure 5 : 1940 Nash quad all-wheel-drive and steer [1]
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Four-wheel steering has found itself into commercial and production vehicles. An example of
the EMAS [2] Automatic Hydraulic Steering fitted to a Ford Pickup and SUV is shown in
Figure 6. The improved low speed manoeuvrability can clearly be seen.
Figure 6 : EMAS automatic hydraulic steering [2]
Various multi-axle steering vehicles have been developed to investigate and demonstrate the
advantages of multi-wheel steering on light and heavy vehicles.
In South Africa, the University of Pretoria developed a four-wheel steer vehicle based on
Volkswagen Golf driveline components to investigate four-wheel steering algorithms [3]. On
the heavy vehicle side, the best example of a 6x6 multi-wheel steer vehicle is the Addax [4],
developed by ARMSCOR and Ermetek. The vehicle comprised of three axles that can be
steered individually. The control of individual steering angles enables the evaluation of
different steering control philosophies. The vehicle is shown in Figure 7 in crab steer mode
(all wheels turned in the same direction) on the left hand side and opposite steer of the first
and last axle (with the central axle locked in the dead ahead direction) on the right hand side.
Figure 7 : Addax multi-wheel steer [4]
The study indicted that rear axle steering improved handling, reduced tire wear and improved
slow speed manoeuvring.
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Figure 8 : Patria AMV
As a second example, the Patria AMV [5], shown in Figure 8, steers mechanically with the
first two axles. When low speed manoeuvring is required at maximum steering lock, the last
axle could be counter-steered electrically to reduce the turning radius. The practical
implementation of all wheel steering on this type of vehicle is relatively simple, since the
wheel-stations on the first, second, third and fourth axles are identical. The last axle is locked
mechanically to eliminate steering in conventional steering mode. The Patria vehicle also has
the ability to skid steer, using the brake system to clamp either the left or right hand side
wheels. The skid steer system enables manoeuvring on surfaces where loose gravel is present.
(On high friction surfaces, skid steering is not recommended due to the high forces
transmitted to the driveline and suspension).
Multi-wheel steering is also used in truck-trailer combinations. Truck-tractors steer the second
axles of the truck, as well as the trailer axles to reduce turning radii and tire wear.
A trailer with a steerable second axle is shown in Figure 9.
Figure 9 : Multi-steer trailer and truck
Commercial trailer manufactures and axle system suppliers such as BPW [6], offers a range of
steerable trailer axles, as shown in Figure 10. A mechanical link, a hydraulic system or the
fifth wheel may be used to actuate the rear axle set.
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Figure 10 : Steerable trailer axles from BPW [6]
The multi-wheel drive/steering concept has been taken further by Multidrive [7]. The
Multidrive truck-trailer combination drives with all four axles. The trailer axles are linked to
the truck by a telescopic drive shaft. The trailer axles are also steered, by linking the trailer
axles to the truck chassis using an a-frame. Superior off-road manoeuvrability and mobility is
claimed. The concept is shown in Figure 11.
Figure 11 : Multidrive truck-trailer combination [7]
However, it should be noted that in all the examples above, the aim of a second steering axle
was to achieve low speed manoeuvrability, and reduced tire scrub, and not necessarily
improved handling at higher vehicle speeds.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
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Commercial off the shelf systems to control the individual axle angles have been developed,
and an example is shown in Figure 12.
Figure 12 : EMAS automatic hydraulic steering [2]
These systems are primarily used to reduce tire wear and improve slow speed handling, where
a reduced turning circle is of benefit.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
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5.
Test vehicle information
As mentioned in paragraph 1, the ideal choice for a test vehicle would have been a twin axle
steered 8x8 vehicle. However, due to availability, a 6x6 vehicle was used for the study.
The test vehicle used for the simulations and testing is the Bison weapon platform
(BKW460M) [8]. Bison was developed for the SANDF as a concept demonstrator, and was
never released for serial production. ARMSCOR uses the vehicle for tire testing and other
technology development work. The vehicle is fitted with an ADE447T engine and six speed
ZF WG200 automatic transmission.
The vehicle was evaluated in the unladen condition. Figure 13 indicates the main dimensions
and position of the centre of mass of the test vehicle. Figure 14 shows the test vehicle on the
test track. Table 1 indicates the measured wheel and axle mass, and Table 2 indicates the axle
distances of the test vehicle. Table 3 indicates vehicle dimensions.
Figure 13 : Bison dimensions
Table 1 : Measured axle mass
Axle number
Left wheel [kg]
Right wheel [kg]
Axle total [kg]
1
2370
2490
5220
2
2760
2310
5070
3
2120
1800
3920
Total
14210 kg
Table 2 : Axle distances
Axle number
Distance [mm]
1 to 2
2016
1 to 3
4656
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
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Table 3 : Bison parameters
Description
Parameter
Vehicle track width
2080 mm
Height of centre of gravity
1515 mm
Width of vehicle
2325 mm
Height of cab
3030 mm
Tires
16.00 x 20 Michelin XL radial
625 mm
Tire pressure
450 kPa
Figure 14 : Test vehicle: Bison
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
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6.
Mechanical integration of steering ratio changes
During the study planning phase, a high level investigation was done to determine the
feasibility and practical arrangements on Bison, should the steering ratio be changed.
As a first step, for testing purposes only, the existing vehicle’s steering system could be
modified to accept stationary ratio changes, should it be required. Incidentally, the Bison
vehicle is well suited to implement ratio changes. The steering system of the vehicle is
mounted on the outside of the vehicle, and the first and second axle is connected with a series
of mechanical links, as shown in Figure 15.
In Figure 15, A indicates the steering link between the first and second axle (yellow line), B
indicates the hydraulic cylinder providing power assistance, C indicates the steering link to
the front axle, and D (blue line), indicates the steering link to the second axle.
Ratio changes may be achieved by replacing link A with different lengths, or making an
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 25 of 138 pages
7.
7.1. Introduction
The cornering behaviour of a motor vehicle is an important performance mode often equated
with handling. "Handling" is a loosely used term meant to imply the responsiveness of a
vehicle to driver input, or the ease of control. As such, handling is an overall measure of the
vehicle-driver combination. The driver and vehicle is a "closed-loop" system - meaning that
the driver observes the vehicle direction or position, and corrects his/her input to achieve the
desired motion. For purposes of characterizing only the vehicle, "open-loop" behaviour is
used. Open-loop refers to vehicle response to specific steering inputs, and is more precisely
defined as "directional response" behaviour [9,10].
The most commonly used measure of open-loop response is the understeer gradient.
Understeer gradient is a measure of performance under steady-state conditions, although the
measure can be used to infer performance properties under conditions that are not quite
Open-loop cornering, or directional response behaviour, will be examined in this section. The
approach is to first analyse turning behaviour at low speed, and then consider the differences
that arise under high-speed conditions.
7.2. Low speed turning
At low speed, the tires of a vehicle does not need to develop lateral forces. The tires roll with
no slip angle and the vehicle negotiates a turn, as described in Figure 16.
Figure 16 : Ackerman steering angle
If the rear wheels have no slip angle, the centre of the turn is situated on the projection line of
the rear axle. Also, a perpendicular line from each of the front wheels should pass through the
point of turn. If the line does not pass through the same point, the front tires will fight each
other in the turn, with sideslip producing tire scrub. The ideal turning angles of the front
wheels are determined by the geometry, as seen in Figure 16.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 26 of 138 pages
For proper geometry in the turn, assuming small steering angles, the steering angles (small
angles) are given by:
Outside wheel angle δ 0 ≅
Inside wheel angle δ i ≅
L
(R + t / 2 )
(7.1)
L
(R − t / 2 )
(7.2)
The average angle of the front wheels (assuming small steering angles) is defined as the
Ackerman angle:
Ackerman angle δ = L / R
(7.3)
The term "Ackerman steering" or "Ackerman geometry" are often used to denote the exact
geometry of the front wheels shown in Figure 16. The correct angles are dependent on the
wheelbase of the vehicle and the angle of turn. Errors, or deviations, from the Ackerman
geometry in the left-right steer angles can have a significant influence on front tire wear.
Errors do not have a significant influence on directional response; however, they do affect the
centreing torques in the steering system. With correct Ackerman geometry, the steering
torques tend to increase consistently with steer angle, thus providing the driver with a natural
feel in the feedback through the steering wheel. With the other extreme of parallel steer, the
steering torques grows with angle initially, but may diminish beyond a certain point, and even
become negative (tending to steer more deeply into the turn). This type of behaviour in the
steering system is undesirable.
A four-axle vehicle, steering with the first and second axles, is shown in Figure 17.
Figure 17 : Ackerman steering, eight wheel vehicle steering with first and second axles
The individual steering angles, for a specific turning radius, can be determined by halving the
distance between the third and fourth axles, and connecting it perpendicular with the
individual steering wheels, as shown in Figure 17, as per [11].
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 27 of 138 pages
The other significant aspect of low-speed turning is the off-tracking that occurs at the rear
wheels. The off-tracking distance, ∆ , may be calculated from simple geometry relationships
as:
∆ = R[1 − cos(L / R )]
(7.4)
Using the expression for a series expansion of the cosine, namely:
cos z = 1 −
z2 z4 z6
+
− .....
2! 4! 6!
(7.5)
Then:
L2
∆≅
2R
(7.6)
For obvious reasons, off-tracking is primarily of concern with long-wheelbase vehicles such
as trucks and buses. For articulated trucks, the geometric equations become more complicated
and are known as "tractrix" equations.
7.3. High speed cornering
At high speed, the turning equations differ because lateral acceleration will be present. To
counteract the lateral acceleration the tires must develop lateral forces, and slip angles will be
present at each wheel.
7.4. Introduction to the tire model
The tire contact patch is the only contact between the wheeled vehicle and the road. It is
therefore important to review tire models that may be used in the theoretical analysis. A brief
background on tire models is given in the following paragraphs. Since actual measured tire
data was available, measured data was used during the theoretical analysis.
7.4.1. Tire model literature review
Most notable among tire researchers and tire model developers is Pacejka [12-18], who
developed the “Magic Formula Tire Model”. This model developed by Pacejka is a semiempirical one.
A variety of theoretical tire models are used in vehicle simulations. The models are often
adapted for specific needs and use. Levels of accuracy and complexity may be and have been
introduced in the various categories of utilization.
Many researchers in vehicle dynamics and four-wheeled-steer vehicles [19-21] have made use
of the following tire model:
Fy = α .C Fα
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
(7.7)
Page 28 of 138 pages
Where the side force Fy is equal to the product of the cornering stiffness coefficient CFα and
the slip angle α. Recent studies on four-wheel-steer have made use of the simple model, even
when more accurate models are available. The reason for this is that at low to moderate levels
of lateral acceleration this model proves satisfactory. Secondly and more importantly, the
model is easy to implement. The dynamic implementation of CFα is further discussed in
paragraph 7.5.
For the present study, measured tire data was available, as shown in paragraph 7.4.2.
Throughout the theoretical calculation, a second order polynomial has been fitted thought the
applicable measured data (tire load and tire pressure). Where required, the slope of the side
force slip angle was determined, and used as CFα.
In this study, the linear tire model was deemed sufficient as a first order calculation; in order
to determine first order tendencies for steady and transient state handling. If it was found that
large benefit might be obtained from implementing individual steering angles, the models
may be refined to include the non-linear side force slip angle relationship.
The handling manoeuvres that were performed, were not on the limit of the vehicle and the
tires, since it was envisaged that any benefit that may be obtained by including variable
steering axle ratios, should be implemented in normal driving situations, and not only in the
instance where on-the-limit driving requires marginal improvement.
7.4.2. Measured tire data
Measured side force vs. slip angle data was available, and was used to calculate the cornering
stiffness coefficient CFα. A typical tire data-measuring set-up is shown in Figure 18. A towing
vehicle pulls a trailer equipped with an axle of which the wheel slip angles may be adjusted.
The drawbar pull, as well as side force is measured, enabling the calculation of the tire side
force generated. The actual heading angle of the trailer is also measured with a fifth wheel.
Figure 18 : Measuring tire data
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 29 of 138 pages
The measured data is shown in Figure 19. As indicated, data for two axle loads (2370 kg and
4020 kg) and two tire pressures (400 kPa and 600 kPa) were available.
Side Force vs. Slip Angle Michelin XZL 16R20
30000
25000
Side Force [N]
20000
15000
10000
5000
0
-5000
-10000
0
2
4
6
8
10
Slip Angle [°]
400 kPa @ 2370 kg
400 kPa @ 4020 kg
600 kPa @ 2370 kg
600 kPa @ 4020 kg
Figure 19 : Measured tire data
From Figure 19 it may be seen that the measured values have force offsets at a zero slip angle.
Throughout the calculations in the investigation, the measured data was normalised by
adjusting the curves with a positive offset, to obtain a zero force at zero slip angle.
Unless stated otherwise, the value for CFα:
N
°
)
(7.8)
7.5. Tire cornering forces
Under cornering conditions, in which the tire must develop a lateral force, the tire will also
experience lateral slip as it rolls. The angle between its direction of heading and its direction
of travel is known as slip angle, α. These are illustrated in Figure 20 [9].
Figure 20 : Tire cornering force properties [9]
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 30 of 138 pages
As discussed in paragraph 7.4.1, the lateral force, denoted by Fy, is called the "cornering
force" when the camber angle is zero. At a given tire load, the cornering force grows with slip
angle. At low slip angles (5 degrees or less) the relationship is linear.
A positive slip angle produces a negative force (to the left) on the tire, implying that CFα must
be negative; however, SAE defines cornering stiffness as the negative of the slope, such that
CFα takes on a positive value.
The cornering stiffness is dependent on many variables. Tire size and type (radial- vs. bias-ply
construction), number of plies, cord angles, wheel width, and tread are significant variables.
For a given tire, the load and inflation pressure are the main variables. Speed does not
strongly influence the cornering forces produced by a tire.
Cornering coefficients are usually largest at light loads, diminishing continuously as the load
reaches its rated value (Tire & Rim Association rated load [22]). At 100 % load, the cornering
coefficient is typically in the range of 0.2 (kg cornering force per kg load per degree of slip
angle).
7.6. Cornering equations
The steady-state cornering equations are derived from the application of Newton's Second
Law along with the equation describing the geometry in turns (modified by the slip angle
conditions necessary on the tires). For purposes of analysis, it is convenient to represent the
vehicle by the bicycle model shown in Figure 21 [9].
Figure 21 : Cornering of a bicycle model [9]
At high speeds the radius of turn is much larger than the wheelbase of the vehicle. Then small
angles can be assumed, and the difference between steer angles on the outside and inside front
wheels is negligible. Thus, for convenience, the two front wheels can be represented by one
wheel at a steer angle, δ , with a cornering force equivalent to both wheels. The same
assumption is made for the rear wheels.
For a vehicle travelling forward with a speed of V, the sum of the forces in the lateral
direction from the tires must equal the mass times the centripetal acceleration [9].
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 31 of 138 pages
∑F
y
= Fyf + Fyr =
MV 2
R
(7.9)
where:
Fyf
=
Lateral (cornering) force at the front axle
Fyr
=
Lateral (cornering) force at the rear axle
M
=
Mass of the vehicle
V
=
Forward velocity
R
=
Also, for the vehicle to be in a moment equilibrium about the centre of gravity, the sum of the
moments from the front and rear lateral forces must be zero [9]:
Fyfb – Fyrc = 0
(7.10)
Thus:
Fyf = Fyrc/b
(7.11)
Substituting yields:
M V2/R = Fyr (c/b+1) = Fyr (b+c)/b = FyrL/b
(7.12)
Fyr = M b/L (V2/R)
(7.13)
But M b/L is simply the portion of the vehicle mass carried on the rear axle (i.e., Wr/g); thus
the lateral force developed at the rear axle must be Wr/g times the lateral acceleration at that
point. Solving for Fyf in the same fashion will indicate that the lateral force at the front axle
must be Wf/g times the lateral acceleration.
With the required lateral forces known, the slip angles at the front and rear wheels are also
established [9]. That is:
αf = WfV2/(Cαfg R)
and
(7.14)
αr = WrV2/(Cαrg R)
(7.15)
Using the geometry of the vehicle in the turn, from Figure 21, it can be seen that [9]:
δ =L/R + αf - αr
(7.16)
Now substituting for αf and αr gives:
2
L WfV
WrV 2
δ= +
−
R Cαf gR Cαr gR
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
(7.17)
Page 32 of 138 pages
δ =
L  W f Wr  V 2
+
−
R  Cαf Cαr  gR
(7.18)
where [9]:
δ
=
Steer angle at the front wheels (rad)
L
=
Wheelbase
R
=
V
=
Forward speed
g
=
Gravitational acceleration constant
Wf
=
Wr
=
Cαf
=
Cornering stiffness of the front tires (n/rad)
Cαr
=
Cornering stiffness of the rear tires (n/rad)
The equation is often written in a shorthand form as follows [9]:
δ =L/R + Kay
(7.19)
where:
K
=
ay
=
Lateral acceleration (g)
The above equation is very important to the turning response properties of a vehicle. It
describes how the steer angle of the vehicle must be changed with the radius of turn, R, or the
lateral acceleration, V2/(gR). The term (Wf/Cαf - Wr/Cαr) determines the magnitude and
direction of the steering inputs required. It consists of two terms, each of which is the ratio of
the load on the axle (front or rear) to the cornering stiffness of the tires on the axle. It is called
the "Understeer gradient", and will be denoted by the symbol, K, which has the units of
degrees/g. Three possibilities exist [9]:
1)
Neutral steer:
Wf / Cαf = Wr / Cαr → K = 0 → αf = αr
(7.20)
On a constant-radius turn, no change in steer angle will be required as the speed is varied.
Specifically, the steer angle required to make the turn will be equivalent to the Ackerman
angle, L/R. Physically, the neutral steer case corresponds to a balance on the vehicle such that
the "force" of the lateral acceleration at the CG causes an identical increase in slip angle at
both the front and rear wheels [9]:
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 33 of 138 pages
2)
Understeer:
Wf / Cαf > Wr / Cαr → K > 0 → αf > αr
(7.21)
On a constant-radius turn, the steer angle will have to increase with speed in proportion to K
(deg/g) times the lateral acceleration in g's. Thus it increases linearly with the lateral
acceleration and with the square of the speed. In the understeer case, the lateral acceleration at
the CG causes the front wheels to slip sideways to a greater extent than at the rear wheels.
Thus to develop the lateral force at the front wheels necessary to maintain the radius of turn,
the front wheels must be steered to a greater angle [9].
3)
Oversteer:
Wf / Cαf < Wr / Cαr → K < 0 → αf < αr
(7.22)
On a constant-radius turn, the steer angle will have to decrease as the speed (and lateral
acceleration) is increased. In this case, the lateral acceleration at the CG causes the slip angle
on the rear wheels to increase more than at the front. The outward drift at the rear of the
vehicle turns the front wheels inward, thus diminishing the radius of turn. The increase in
lateral acceleration that follows causes the rear to drift out even further and the process
continues unless the steer angle is reduced to maintain the radius of turn.
The way in which steer angle changes with speed on a constant-radius turn for each of these
cases is illustrated in Figure 22. With a neutral steer vehicle, the steer angle to follow the
curve at any speed is simply the Ackerman angle.
Figure 22 : Change of steer angle with speed [9]
With understeer the angle increases with the square of the speed, reaching twice the initial
angle at the characteristic speed. In the oversteer case, the steer angle decreases with the
square of the speed and becomes zero at the critical speed value.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 34 of 138 pages
8.
Tire wear
In the present study, tire wear may be one of the greatest advantages of variable ratio steering
systems. Information from tire manufactures on the reasons for tire wear is therefore given in
the following paragraph.
According to Michelin [23] the greatest impact on tire life is tire pressure, as illustrated in the
figure published on their web site, and shown in Figure 23. The figure indicates that if the tire
pressure drops below 60% or is increased above 120% of the recommended pressure per axle
load, the incorrect pressure will result in loss of service of more than 50%.
Figure 23 : Tire life for different tire pressures [23]
According to tire manufacturers and tire specialists (Supaquick [24], Yokohama [25], Dunlop
[26] and Continental [27]), tire wear is also influenced by:
•
Seasonal and climatic conditions
•
The road or track it is used on, including the surface characteristics, the path that is
followed (straight line compared to curves)
•
Incorrect inflation pressures
•
Incorrect tire fitment
•
Poor maintenance
•
Driving style
•
The load on the tire and the load rating of the tire
•
The rotational speed of the tire
•
The number of stop/start cycles, and the severity of the manoeuvres
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 35 of 138 pages
•
Correct tire fitment, balancing and rim size
A study on tire wear on a 6x6 all wheel steer vehicle was performed, as described in [4]. The
rear axle of the vehicle was used to evaluate tire wear, by driving through an obstacle course,
using different steering control strategies. An improvement of between 11% and 52% in tire
wear was achieved, where the steering geometry follows the Ackerman steering principle
closely, i.e. roll without slip.
As described in paragraph 4.1, trailer system manufactures have developed various steering
methods to reduce tire wear.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 36 of 138 pages
9.
Test vehicle low speed turning requirements
The Ackerman angle is defined as the steering angle that is required to enable the wheels on
the steering axle to roll without slip. For a three-axle vehicle, steering with the first and
second axle, the tuning point is defined as the intersection of the line parallel to the rear axle,
intersecting with lines perpendicular with the steering wheels.
Figure 24 : Ackerman steering Bison
The average Ackerman angle for the first axle is therefore the wheelbase (distance between
first and last axle) divided by the radius of turn, R.
The average Ackerman angle for the second axle is the distance between second and third
axle divided by the radius of turn. The Ackerman ratio between the second and first axle is
therefore:
e
=
distance between first and last axle
f
=
distance between second and last axle
Ar
=
Ackerman ratio
Ar =
f / R f 2640
= =
= 0.57
e / R e 4656
or (steering angle of 2nd axle) = 0.57 x (steering angle 1st axle) (9.1)
The above definition is therefore independent of the track width and the turning circle.
In the following paragraphs, two different steering ratios are used and mentioned. One is the
Ackerman steering ratio and the other the actual steering ratio. The Ackerman ratio is the
theoretical steering ratio based on the vehicle geometry, in this case approximately 0.57. An
actual steering ratio of the vehicle is also measured and referred to. This ratio was determined
by physical measurement, and is described in 18.6. (It was found that the measured ratio, in
the region of zero steering angle, is closer to 0.7).
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 37 of 138 pages
10. Constant radius: expanding the cornering equations to a four axle
configuration
As mentioned in the paragraphs above, it was originally intended to use an 8x8 vehicle. The
bicycle model can be expanded to a four-axle vehicle. The model is then simplified to a threeaxle vehicle, as described in the following paragraphs.
A simple bicycle model can be constructed to calculate the steering and slip angles of a fouraxle vehicle, steering with the first and second axles. Because of the constant angular speed,
the slip angles must give no moment about the centre of mass G. Rolling resistance and
aerodynamic forces are neglected. The vehicle is rotating around CT, the turn centre. The tire
forces are perpendicular to the wheels, and intersect at the force centre CF. The line of action
of the resultant force on the vehicle is through the CG towards the point CF. This force has the
desired centripetal component towards CT.
The steer angles for the vehicle can be defined as the angles between the perpendicular tire
forces from the front and rear axles.
Figure 25 shows a scaled layout of a vehicle negotiating a 20 meter radius turn. A positive
under steer angle (i.e. αf >αr ) is assumed. An enlarged view is shown in Figure 26.
CT
CF
Figure 25 : Slip angle diagram
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 38 of 138 pages
Figure 26 : Detail slip angle diagram
The steer angles for the vehicle can be defined as the angles between the perpendicular tire
forces from the front and rear axles.
The geometry of the vehicle is defined as:
R
=
a
=
Distance between the first and second axles
b
=
Distance between the second axle and centre of mass
c
=
Distance between the centre of mass and third axle
d
=
Distance between the third and fourth axle
t
=
Track width
The Ackerman angle (all angles in radians, unless noted otherwise and small angles assumed)
for the first axle is therefore:
First axle Ackerman : (a + b + c + d / 2) / (R)
Second axle Ackerman : (b + c + d / 2) / (R)
The steer angle of the third and fourth axles, relative to a datum line, can be shown to be:
Wheels _ angle _ to _ datum _ rear = α r1 −
(c + d )
c
- (α r 2 )
R
R
(10.1)
The first axle angle relative to the datum can be shown to be:
First _ axle _ angle _ to _ datum = α f 1 +
(a +b)
R
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
(10.2)
Page 39 of 138 pages
The second axle angle relative to the datum can be shown to be:
Second _ axle _ angle _ to _ datum = α f 2 +
(b)
R
(10.3)
The steering angle of the first axle is therefore the difference of the first axle and the reference
axle (third and fourth axle):
(a +b)
c
(c + d )
− [ α r1 − − ( α r 2 −
)]
R
R
R
(a +b) c (c + d )
+ −
δ 1 = α f 1 − α r1 + α r 2 +
R
R
R
(a+b−d )
δ 1 = α f 1 − α r1 + α r 2 +
R
Steering _ angle _ first _ axle : δ 1 = α f 1 +
(10.4)
The steering angle of the second axle is therefore the difference of the second axle and the
reference axle (third and fourth axle):
Steering _ angle _ 2nd _ axle : δ 2 = α f 2 +
(b)
c
(c + d )
− [ α r1 − − ( α r 2 −
)]
R
R
R
(b )
c
(c + d )
− α r1 + + α r 2 −
R
R
R
(b − d )
− α r1 + α r 2 +
R
δ2 = α f 2 +
δ2 = α f 2
(10.5)
For a vehicle travelling forward with a speed of V, the sum of the forces in the lateral
direction from the tires must equal the mass times the centripetal acceleration:
∑ Fy = Fyf1 + Fyf2 + Fyr1 + Fyr2 = M V2/R
(10.6)
where:
Fyf1
=
Lateral (cornering) force at the first axle
Fyf2
=
Lateral (cornering) force at the second axle
Fyr1
=
Lateral (cornering) force at the third axle
Fyr2
=
Lateral (cornering) force at the fourth axle
M
=
Mass of the vehicle
V
=
Forward velocity
R
=
Also, for the vehicle to be in a moment equilibrium about the centre of gravity, the sum of the
moments from the front and rear lateral forces must be zero.
Fyf1(a + b) + Fyf2(b) - Fyr1(c) - Fyr2(c + d) = 0
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
(10.7)
Page 40 of 138 pages
The lateral tire forces can be expressed as a function of the cornering stiffness and the slip
angle required to develop the cornering force:
αf1Cyf1(a + b) + αf2Cyf2 (b) - αr1Cyr1 (c) - αr2Cyr2 (c + d) = 0
(10.8)
αf1Cyf1 + αf2Cyf2 + αr1Cyr1 + αr2Cyr2 = M V2/R
(10.9)
A steering relationship between the first and second axle is chosen to follow the Ackerman
principle. Define the steering ratio between the first and second axle as Sr. The relationship is
a mechanical relationship between the first and second axle that is controlled by the steering
geometry.
δ2 = Sr x δ1
(10.10)
The equations can therefore be summarised below:
δ 1 = α f 1 − α r1 + α r 2 +
(a + b − d )
R
(10.11)
δ 2 = α f 2 − α r1 + α r 2 +
(b − d )
R
(10.12)
δ2 = Sr x δ1
(10.13)
αf1Cyf1(a+b) + αf2Cyf2 (b) - αr1Cyr1 (c) - αr2Cyr2 (c + d) = 0
(10.14)
αf1Cyf1 +αf2Cyf2 + αr1Cyr1 + αr2Cyr2 = M V2/R
(10.15)
The above relations are shown on a scale drawing of a typical 8x8 vehicle geometry in Figure
27.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 41 of 138 pages
CG
CF
CT
Figure 27 : Scale drawing of an 8x8 geometry
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 42 of 138 pages
The following six parameters are unknown:
αf1, αf2, αr1, αr2, δ1, δ2, Sr
The equations can be rearranged in matrix format:
1
0
-1
0
1
-1
δ1
(a+b-d)/R
0
1
0
-1
1
-1
δ2
(b-d)/R
-Sr
1
0
0
0
0
0
0
(a+b)Cyf1
(b)Cyf2
0
0
Cyf1
Cyf2
Cyr1
1
0
0
0
0
.
αf1
=
0
αf2
0
Cyr2
αr1
Mv2/R
0
αr1
iterate
-(c)Cyr1 -(c+d)Cyr2
To solve for the six unknown parameters using five equations, the steering angle for the first
axle may be chosen, and the slip angles calculated, by varying the steering angle, all the
possible solutions can be compared, and the steering angle suited for a specific radius and
vehicle speed or other requirement.
The unknowns may be solved using Gauss elimination, by writing the matrix in the form
[A]{x}={b}, ensuring that x is non-singular and calculating {x} = [A]/{b}.
The lack of a test vehicle necessitates that the above equations be simplified and tailored for a
6x6 vehicle. The process is described in paragraph 11. The method of solving the equations
using Matlab is also discussed.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 43 of 138 pages
11.1. Developing the three axle bicycle model
Figure 28 assumes a positive under steer angle, i.e. αf > αr
The steer angles of the first axle and the second axle are determined first.
The steer angles for the vehicle can be defined as the angles between the perpendicular tire
forces from the front and rear axles.
The model may be simplified to suit the 6x6 geometry, as shown in Figure 28.
Figure 28 : Bison three axle bicycle model
The rear axle angle relative to the datum can be shown to be:
Rear _ axle _ angle _ to _ datum _ rear = ( αr -
(b + c )
)
R
(11.1)
The first axle angle relative to the datum can be shown to be:
First _ axle _ angle _ to _ datum = α f 1 +
(a)
R
(11.2)
The second axle angle relative to the datum can be shown to be:
Second _ axle _ angle _ to _ datum = α f 2 −
(b)
R
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
(11.3)
Page 44 of 138 pages
The steering angle of the first axle is therefore the difference of the first axle and the reference
axle (third axle):
Steering _ angle _ first _ axle : δ1 = α f 1 - αr +
(a+b+c)
R
(11.4)
The steering angle of the second axle is therefore the difference of the second axle and the
reference axle (third axle):
Steering_angle_second_axle : δ2 = α f 2 - αr +
(c)
R
(11.5)
For a vehicle travelling forward with a speed of V, the sum of the forces in the lateral
direction from the tires must equal the mass times the centripetal acceleration.
∑Fy = Fyf1 + Fyf2 + Fyr = M V2/R
(11.6)
Therefore where:
Fyf1
=
Lateral (cornering) force at the first axle
Fyf2
=
Lateral (cornering) force at the second axle
Fyr
=
Lateral (cornering) force at the third axle
M
=
Mass of the vehicle
V
=
Forward velocity
R
=
Also, for the vehicle to be in a moment equilibrium about the centre of gravity, the sum of the
moments from the front and rear lateral forces must be zero:
Fyf1(a) - Fyf2(b) - Fyr(b + c) = 0
(11.7)
The lateral tire forces can be expressed as a function of the cornering stiffness and the slip
angle required to develop the cornering force:
αf1Cyf1(a) - αf2Cyf2 (b) - αrCyr (b + c) = 0
(11.8)
and substituting into equation 11.6:
αf1Cyf1 + αf2Cyf2 + αrCyr = M V2/R
(11.9)
A steering relationship (Sr) between the first and second axle is chosen to follow the
Ackerman principle. The relationship is a mechanical relationship between the first and
second axle that is controlled by the steering geometry:
δ2 = Sr x δ1
(11.10)
The equations can therefore be summarised below:
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 45 of 138 pages
δ1 = α f 1 − α r +
( a + b + c)
R
(11.11)
δ2 = α f 2 −αr +
(c )
R
(11.12)
δ2 = Sr x δ1
(11.13)
αf1Cyf1(a) - αf2Cyf2 (b) - αrCyr (b + c) = 0
(11.14)
αf1Cyf1 + αf2Cyf2 + αrCyr = M V2/R
(11.15)
The following five parameters are unknown:
αf1, αf2, αr, δ1, δ2, Sr
To solve for the five unknown parameters a matrix is compiled:
1
0
-1
0
+1
δ1
(a+b+c)/R
0
1
0
-1
+1
δ2
(c)/R
0
0
Cyf1
Cyf2
Cyr
0
0
(a)Cyf1
-(b)Cyf2
-(b+c)Cyr1
αf2
0
-Sr
1
0
0
0
αr
0
.
αf1
=
Mv2/R
A Matlab routine to solve the above equations is shown in Appendix C: Constant radius test –
Matlab M file.
The unknowns may be solved using Gauss elimination, by writing the matrix in the form
[A]{x}={b}, ensuring that x is non-singular and calculating {x} = [A]/{b}.
The following philosophy was followed:
i.
Define a set of steering ratios from 0.3 to 1.2
ii.
Define a set of vehicle speeds, from 20 km/h to 80 km/h
iii.
Define a vehicle mass and geometry
iv.
Solve for the steering angles and slip angles
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 46 of 138 pages
12. Constant radius: simulation results, simplified bicycle model
From paragraph 11, a matrix can be solved to determine the required steering angles for the
vehicle to perform a constant radius test. The following parameters are used as input to
calculate the steering angles and wheel slip angles:
i.
Vehicle speed : V
ii.
iii.
Steering ratio between first and second axle : Sr (Indicated as “Ratio” in the graphs)
iv.
Tire characteristics:
Measured tire data was used as input for the calculation
v.
Vehicle mass:
Results were calculated for the vehicle in the unladen configuration
vi.
Vehicle centre of mass:
Were calculated using the measured axle mass in the empty configuration
vii.
Vehicle geometry
In an attempt to determine the effect of the steering ratio (ratio between the average angle of
the first and second steering axle angle) on steering angles and slip angles, the model
parameters were varied to determine the effect of vehicle speed, radius of the track and
steering ratio (between the first and second axles).
The following assumptions were made, in order to simplify the calculations:
i.
No load transfer from left to right (and vice versa) was included
ii.
A linear tire model was used
iii.
Camber changes were neglected
iv.
Constant steering ratio between first and second axles was assumed
v.
Negative steering ratios were evaluated as well, even though it may not be a practical
solution
The results are shown and discussed in the following paragraphs.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 47 of 138 pages
6
4
2
6
4
2
8
6
4
2
0
0
0.5
1
0
0.5
1
0
0.5
1
Steering Ratio
Steering Ratio
Steering Ratio
Bison Const.R Test: Radius 40 meters, Speed 30 km/h Bison Const.R Test: Radius 40 meters, Speed 40 km/h Bison Const.R Test: Radius 40 meters, Speed 60 km/h
6
6
6
4
2
4
2
4
2
0
0
0.5
1
0
0.5
1
0
0.5
1
Steering Ratio
Steering Ratio
Steering Ratio
Bison Const.R Test: Radius 60 meters, Speed 30 km/h Bison Const.R Test: Radius 60 meters, Speed 40 km/h Bison Const.R Test: Radius 60 meters, Speed 60 km/h
6
6
6
0
4
3
2
1
0
0
0.5
Steering Ratio
1
5
RMS Slip Angle [°]
5
RMS Slip Angle [°]
RMS Slip Angle [°]
0
RMS Slip Angle [°]
0
RMS Slip Angle [°]
RMS Slip Angle [°]
0
8
RMS Slip Angle [°]
8
RMS Slip Angle [°]
RMS Slip Angle [°]
Bison Const.R Test: Radius 20 meters, Speed 30 km/h Bison Const.R Test: Radius 20 meters, Speed 40 km/h Bison Const.R Test: Radius 20 meters, Speed 60 km/h
10
10
10
4
3
2
1
0
0
0.5
Steering Ratio
1
5
4
3
2
1
0
0
0.5
Steering Ratio
1
Figure 29 : RMS slip angles, constant radius test
Figure 29 shows a matrix of radius and speed combinations. In the matrix columns from left
to right, speed values of 30 km/h, 40 km/h and 60 km/h are indicated. In the rows, results of
radii 20 meters, 40 meters and 60 meters are shown. The RMS values of the slip angles of the
three axles are calculated.
From the above, it is possible to determine the steering ratio where the slip angles are a
minimum. The steering ratio required to minimize the slip angles for low and high speed,
small and large radii are 0.5. As will be discussed, this may be used if for example it is
necessary to minimize tire wear for a specific application.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 48 of 138 pages
10
8
6
4
2
10
8
6
4
2
10
8
6
4
2
0
RMS Steering Angle [°]
0
10
8
6
4
2
10
8
6
4
2
10
8
6
4
2
0
RMS Steering Angle [°]
0
10
8
6
4
2
0
0
0.5
Steering Ratio
1
RMS Steering Angle [°]
0
0.5
1
0
0.5
1
0
0.5
1
Steering Ratio
Steering Ratio
Steering Ratio
Bison Const.R Test: Radius 60 meters, Speed 30 km/h Bison Const.R Test: Radius 60 meters, Speed 40 km/h Bison Const.R Test: Radius 60 meters, Speed 60 km/h
12
12
12
RMS Steering Angle [°]
0
RMS Steering Angle [°]
0
0.5
1
0
0.5
1
0
0.5
1
Steering Ratio
Steering Ratio
Steering Ratio
Bison Const.R Test: Radius 40 meters, Speed 30 km/h Bison Const.R Test: Radius 40 meters, Speed 40 km/h Bison Const.R Test: Radius 40 meters, Speed 60 km/h
12
12
12
RMS Steering Angle [°]
0
RMS Steering Angle [°]
RMS Steering Angle [°]
RMS Steering Angle [°]
Bison Const.R Test: Radius 20 meters, Speed 30 km/h Bison Const.R Test: Radius 20 meters, Speed 40 km/h Bison Const.R Test: Radius 20 meters, Speed 60 km/h
12
12
12
10
8
6
4
2
0
0
0.5
Steering Ratio
1
10
8
6
4
2
0
0
0.5
Steering Ratio
1
Figure 30 : RMS steering angles
Figure 30 indicates a similar matrix structure, but with the RMS angle of the steering axles
(first two axles) plotted. If an attempt is made to minimize the steering angle required during
steady state handling, regardless of tire wear and handling, Figure 30 indicates that for low
speed the steering ratio should be approximately 0.4 to 0.5. For higher speeds, results indicate
that the steering angle should be in the region of 0.3 for the given parameters.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 49 of 138 pages
4
2
6
4
10
8
2
6
0.5
1
1.5
-0.5
0
0.5
1
1.5
-0.5
0
0.5
1
1.5
Steering Ratio
Steering Ratio
Steering Ratio
Bison Const.R Test: Radius 40 meters, Speed 30 km/h Bison Const.R Test: Radius 40 meters, Speed 40 km/h Bison Const.R Test: Radius 40 meters, Speed 60 km/h
4
5
7
Slip Angle
Slip Angle
Slip Angle
Steering Angle
Steering Angle
Steering Angle
3
4
6
2
1
3
2
5
4
1
3
0.5
1
1.5
-0.5
0
0.5
1
1.5
-0.5
0
0.5
1
1.5
Steering Ratio
Steering Ratio
Steering Ratio
Bison Const.R Test: Radius 60 meters, Speed 30 km/h Bison Const.R Test: Radius 60 meters, Speed 40 km/h Bison Const.R Test: Radius 60 meters, Speed 60 km/h
3
3
4.5
Slip Angle
Slip Angle
Slip Angle
Steering Angle
Steering Angle
Steering Angle
2.5
2.5
4
2
1.5
1
0.5
-0.5
RMS Angle [°]
0
RMS Angle [°]
RMS Angle [°]
0
-0.5
RMS Angle [°]
0
RMS Angle [°]
RMS Angle [°]
0
-0.5
RMS Angle [°]
RMS Angle [°]
RMS Angle [°]
Bison Const.R Test: Radius 20 meters, Speed 30 km/h Bison Const.R Test: Radius 20 meters, Speed 40 km/h Bison Const.R Test: Radius 20 meters, Speed 60 km/h
8
10
14
Slip Angle
Slip Angle
Slip Angle
Steering Angle
Steering Angle
Steering Angle
6
8
12
2
1.5
1
0
0.5
Steering Ratio
1
1.5
0.5
-0.5
3.5
3
2.5
0
0.5
Steering Ratio
1
1.5
2
-0.5
0
0.5
Steering Ratio
1
1.5
Figure 31 : RMS slip angles and steering angles
Figure 31 summarises the above by again showing the matrix structure, with both the RMS
slip and steering angles plotted on the same axes. The results indicate that at slow speeds a
minimum steering angle and slip angle occurs at roughly the same steering ratio. At high
speed a different steering ratio is required to minimize either the slip angle or steering angle.
It was shown in equation 9.1 that the steering ratio of the Bison vehicle is approximately 0.57,
which, according to the simulation results, is therefore appropriate for minimising the slip
angles at low speed.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 50 of 138 pages
20
Alfa 1st Axle
Alfa 2nd Axle
Alfa 3rd Axle
Steer 1st Axle
15
Steer 2nd Axle
Angle [°]
Angle [°]
20
Alfa 1st Axle
Alfa 2nd Axle
Alfa 3rd Axle
Steer 1st Axle
15
Steer 2nd Axle
10
5
0
10
5
0.4
0.6
0.8
0
1
0.4
0.6
Ratio
1
20
Alfa 1st Axle
Alfa 2nd Axle
Alfa 3rd Axle
Steer 1st Axle
15
Steer 2nd Axle
Angle [°]
Angle [°]
20
Alfa 1st Axle
Alfa 2nd Axle
Alfa 3rd Axle
Steer 1st Axle
15
Steer 2nd Axle
10
5
0
0.8
Ratio
10
5
0.4
0.6
0.8
Ratio
1
0
0.4
0.6
0.8
1
Ratio
Figure 32 : Slip and steer angles - 20 meter radius
Figure 32 indicates a typical simulation result for a radius of 20 meters, and a speed increase
from 30 km/h to 60 km/h. The slip angles of the three axles and steering angles of the first and
second axles are plotted for the different steering ratios considered in Figure 32. At low speed,
an intersection of the lines of at least two axles may be found at a steering ratio of 0.5, and at
high speed (60 km/h) a ratio of 0.6 may be used to minimize the slip angles and therefore tire
wear.
The results of the 40 meter and 60 meter radius are shown in Figure 33 and Figure 34.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 51 of 138 pages
6
10
Alfa 1st Axle
Alfa 2nd Axle
Alfa 3rd Axle
8
Steer 1st Axle
Steer 2nd Axle
Angle [°]
Angle [°]
10
Alfa 1st Axle
Alfa 2nd Axle
Alfa 3rd Axle
8
Steer 1st Axle
Steer 2nd Axle
4
2
0
6
4
2
0.4
0.6
0.8
0
1
0.4
0.6
Ratio
1
6
10
Alfa 1st Axle
Alfa 2nd Axle
Alfa 3rd Axle
8
Steer 1st Axle
Steer 2nd Axle
Angle [°]
Angle [°]
10
Alfa 1st Axle
Alfa 2nd Axle
Alfa 3rd Axle
8
Steer 1st Axle
Steer 2nd Axle
4
2
0
0.8
Ratio
6
4
2
0.4
0.6
0.8
0
1
0.4
0.6
Ratio
0.8
1
Ratio
Figure 33 : Slip and steer angles - 40 meter radius
Figure 33 indicates that the slip angle of the second and third axles are equal at a steering ratio
of 0.5 for a radius of 40 meters and 30 km/h, and at 60 km/h the ratio moves to 0.4 to equalise
the slip angles on the axes. A similar result for the 60 meter radius is shown in Figure 34.
4
7
Alfa 1st Axle
Alfa 2nd Axle
6
Alfa 3rd Axle
Steer 1st Axle
Steer 2nd Axle
5
Angle [°]
Angle [°]
7
Alfa 1st Axle
Alfa 2nd Axle
6
Alfa 3rd Axle
Steer 1st Axle
Steer 2nd Axle
5
3
4
3
2
2
1
1
0
0.4
0.6
0.8
0
1
0.4
0.6
Ratio
4
3
4
3
2
2
1
1
0.4
0.6
0.8
Ratio
1
7
Alfa 1st Axle
Alfa 2nd Axle
6
Alfa 3rd Axle
Steer 1st Axle
Steer 2nd Axle
5
Angle [°]
Angle [°]
7
Alfa 1st Axle
Alfa 2nd Axle
6
Alfa 3rd Axle
Steer 1st Axle
Steer 2nd Axle
5
0
0.8
Ratio
1
0
0.4
0.6
0.8
1
Ratio
Figure 34 : Slip and steer angles - 60 meter radius
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 52 of 138 pages
Slip Angle [°]
15
First Axle Steering Ratios
Second Axle Steering Ratios
Third Axle Steering Ratios
10
5
0
30
35
40
45
50
Vehicle Speed [km/h]
55
60
Slip Angle [°]
15
First Axle Steering Ratios
Second Axle Steering Ratios
Third Axle Steering Ratios
10
5
0
30
35
40
45
50
Vehicle Speed [km/h]
55
60
Slip Angle [°]
15
First Axle Steering Ratios
Second Axle Steering Ratios
Third Axle Steering Ratios
10
5
0
30
35
40
45
Vehicle Speed [km/h]
50
55
60
Figure 35 : Slip angle vs. vehicle speed
In Figure 35 the slip angles are plotted against the vehicle speed, for three different track radii
and a range of different steering ratios (ratio’s of 0.3 to 1.1). It summarises the effect of
changing the steering ratio between the first and second axles, therefore changing the slip
angles of the tires. The effect is more dramatic for small radii, since the centrifugal reaction
forces are larger for the small radii.
The above results may be plotted as steering angle against vehicle speed, to determine over or
understeer behaviour. Selected results are shown in the following figures. For clarity, steering
ratios of 0.5, 0.7 and 1 were selected. The results for a radius of 20 meters, 40 meters and 60
meters are shown in Figure 36 to Figure 41. An increased steering angle for an increase in
speed is required, characteristic of an understeer vehicle. The minimum steering angle is
achieved with a steering ratio of 1, since it ensures the largest side-force to be generated by
the tire contact patch.
As a further check, consider the wheelbase of Bison to be 4.656 meters, and a radius of 20
meters is attempted. The ratio L/R or Ackerman steering angle for the first axle is therefore
0.2328 rad or 13.3 degrees. This value corresponds to the 14 degrees indicated at 30 km/h in
Figure 36.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 53 of 138 pages
24
R0.7, 1st Steering Axle
R0.7, 2nd Steering Axle
R0.5, 1st Steering Axle
R0.5, 2nd Steering Axle
R1, 1st Steering Axle
R1, 2nd Steering Axle
22
20
Steering Angle [°]
18
16
14
12
10
8
6
30
40
50
60
70
80
90
100
Speed [km/h]
Figure 36 : Steering angle vs. vehicle speed – 20 meter radius
12
R0.7, 1st Steering Axle
R0.7, 2nd Steering Axle
R0.5, 1st Steering Axle
R0.5, 2nd Steering Axle
R1, 1st Steering Axle
R1, 2nd Steering Axle
11
10
Steering Angle [°]
9
8
7
6
5
4
3
30
40
50
60
70
80
90
100
Speed [km/h]
Figure 37 : Steering angle vs. vehicle speed – 40 meter radius
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 54 of 138 pages
8
R0.7, 1st Steering Axle
R0.7, 2nd Steering Axle
R0.5, 1st Steering Axle
R0.5, 2nd Steering Axle
R1, 1st Steering Axle
R1, 2nd Steering Axle
7
Steering Angle [°]
6
5
4
3
2
30
40
50
60
70
80
90
100
Speed [km/h]
Figure 38 : Steering angle vs. vehicle speed – 60 meter radius
24
R0.7, 1st Steering Axle
R0.7, 2nd Steering Axle
R0.5, 1st Steering Axle
R0.5, 2nd Steering Axle
R1, 1st Steering Axle
R1, 2nd Steering Axle
22
20
Steering Angle [°]
18
16
14
12
10
8
6
2
3
4
5
6
7
Lateral Acceleration [m/s 2]
8
-3
x 10
Figure 39 : Steering angle vs. lateral acceleration – 20 meter radius
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 55 of 138 pages
12
R0.7, 1st Steering Axle
R0.7, 2nd Steering Axle
R0.5, 1st Steering Axle
R0.5, 2nd Steering Axle
R1, 1st Steering Axle
R1, 2nd Steering Axle
11
10
Steering Angle [°]
9
8
7
6
5
4
3
2
3
4
5
6
7
Lateral Acceleration [m/s 2]
8
-3
x 10
Figure 40 : Steering angle vs. lateral acceleration – 40 meter radius
8
R0.7, 1st Steering Axle
R0.7, 2nd Steering Axle
R0.5, 1st Steering Axle
R0.5, 2nd Steering Axle
R1, 1st Steering Axle
R1, 2nd Steering Axle
7
Steering Angle [°]
6
5
4
3
2
2
3
4
5
6
7
Lateral Acceleration [m/s 2]
8
-3
x 10
Figure 41 : Steering angle vs. lateral acceleration – 60 meter radius
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 56 of 138 pages
13. Transient state handling
The bicycle model of a two-axle vehicle has been expanded to include four axles, and then
reduced to three axles. The equations of motion are derived below. The equations have been
solved using Matlab and Simulink routines, and the results of the simulation are discussed. It
is also necessary to investigate transient response, as this situation is more likely to be
encountered in actual vehicle use.
13.1. Equations of motion, two degree of freedom model, two axles
The lateral acceleration can be expressed in terms of the absolute motion. The vehicle is
shown in Figure 42 in the XY plane, moving in the primary X-direction [10].
Figure 42 : Angles and velocity components [10]
The path angle from the X-axis is ν, the heading angle is ψ, and the attitude angle is β. The
total speed V tangent to the path may be resolved into Vax and Vay [10].
Vax = Vcosβ ≅ V
(13.1)
Vay = ν =Vsinβ ≅ Vβ
(13.2)
Vy ≡ u ≅ ν V
(13.3)
The vehicle has yaw angular velocity r = dψ/dt.
Vax is the component, in the x direction, of the absolute velocity.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 57 of 138 pages
Figure 43 : Velocity components at path points [10]
Figure 43 shows the vehicle position and orientation at time t and t + dt. The relative rotation
is r dt.
The absolute acceleration in the y direction is [10]
Aay=dt=dν cos(r dt) + V sin (r dt)
≅ dν + V r dt
(13.4)
.
Aay
=
ν
+
V
A free body diagram of the bicycle model, with the roll and load transfer neglected, are shown
in Figure 44. The vehicle is considered in the accelerating vehicle-fixed axis, and the
appropriate compensation force and moments are included. The vehicle has been subjected to
a small disturbance, now having a lateral velocity component ν. The attitude angle is [10]:
β = ν / Vax ≅ ν/V
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
(13.5)
Page 58 of 138 pages
Figure 44 : Free body diagram [10]
The slip angles, allowing for the attitude angle and the yaw rotation speed r, is as follows
[10]:
αf = ν/V + ar/V
(13.6)
αr = ν/V - br/V
(13.7)
Using a linear approximation for the tire forces, the forces from the front and rear tires are:
Fyf = 2 αf Cαf
(13.8)
Fyr = 2 αr Cαr
(13.9)
The acceleration in the vehicle fixed axis is zero, and the equations of motion can be
described as [10]:
∑ Fy = Fyf + Fyr – M Aay = 0
(13.10)
∑ M = a.Fyf – b.Fyr – I dr/dt = 0
(13.11)
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 59 of 138 pages
Substituting Fyf and Fyr from above, the following equations can be derived:
.
ν
=
A11 A12
.
ν
(13.12)
.
r
A21 A22
r
where:
 −2 
A11 = 
.(Cαf + Cαr )
 MV 
(13.13)


 2 
A12 = −V + 
.(a.Cαf − bCαr )
 MV 


(13.14)
 2 
A21 =  .(b.Cαr − a.Cαf )
 IV 
(13.15)
(
 2 
A22 = − . a 2 .Cαf + b 2 Cαr
 IV 
)
(13.16)
The above can be expanded further to include the steering angle δ, and the expanded
equations can be shown to be:
.
ν
=
A11 A12
.
ν
+
2Caf/M
.
(13.17)
δ
.
r
A21 A22
r
2Caf/I
The position of the vehicle on the road can be determined by calculating the X, Y and the
X t = ∫ Vx (τ )dτ = ∫ (V cos(ψ (τ ) ) − ν sin (ψ (τ ) )dτ
t
t
to
to
Yt = ∫ (V sin(ψ (τ ) ) − V (τ ) cos(ψ (τ ) )dτ
t
(13.18)
to
t
ψ t = ∫ r (τ )dτ
to
13.2. Equations of motion, two degree of freedom model, four axles
The above equations can be expanded to a four-axle vehicle, steering with the first and second
axles. Define the following:
Fyf1 =
Lateral (cornering) force at the first axle
Fyf2 =
Lateral (cornering) force at the second axle
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 60 of 138 pages
Fyr1 =
Lateral (cornering) force at the third axle
Fyr2 =
Lateral (cornering) force at the fourth axle
αf1
=
Slip angle first axle
αf2
=
Slip angle second axle
αr1
=
Slip third first axle
αr2
=
Slip fourth first axle
δ1
=
Steering angle first axle
δ2
=
Steering angle second axle
M
=
Mass of the vehicle
V
=
Forward velocity
R
=
a
=
Distance between the first and second axle
b
=
Distance between the second axle and centre of mass
c
=
Distance between the centre of mass and third axle
d
=
Distance between the third and fourth axle
The slip angles of the wheel combinations can be described by the equations below:
α f1 =
αf2 =
α r1 =
αr2 =
ν
V
ν
V
ν
V
ν
V
+
( a + b) r
− δ1
V
(13.18)
+
b.r
−δ2
V
(13.20)
−
c.r
V
(13.21)
−
(c + d ).r
V
(13.22)
δ 2 = Sr .δ 1 , where Sr is the steering ratio between the first and second axles.
∑ FY = May expanding to:

•
− 2Cαf 1α f 1 − 2Cαf 2α f 2 − 2Cαr1α r1 − 2Cαr1α r1 = M ν + Vr 


THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
(13.23)
Page 61 of 138 pages
and
∑ MG = IG .dr/dt expanding to:
•
− Fyf 1 (a + b) − Fyf 2 (b) + Fyr1 (c) + Fyr1 (c + d ) = I G r
(13.24)
By substituting and regrouping, the equations can be written in matrix format as follows:
.
ν
=
A11 A12
.
ν
B1
.
+
.
r
(13.25)
δ
B2
A21 A22
r
where:
 −2 
A11 = 
.(Cαf 1 + Cαf 2 + Cαr1 + Cαr 2 )
 MV 
A12 = V +
(13.26)
(
2
. ( a + b ).Cαf 1 + b.Cαf 2 - c.Cαr 1 - ( c + d ).Cαr 2
MV
)
(13.27)
 2 
A21 =  .(c.Cαr1 + (c + d ).Cαr 2 − (a + b).Cαf 1 − b.Cαf 2 )
 IV 
(
 2 
A22 = − . (a + b) 2 .Cαf 1 + b 2Cαf 2 + c 2Cαr1 + (c + d ) 2 Cαr 2
 IV 
(13.28)
)
(13.29)
 1 
B1 =  .(2.Cαf 1 + 2.Sr .Cαf 2 )
M 
(13.30)
1
B1 =  .(2.( a + b )Cαf 1 + 2.Sr .b.Cαf 2 )
I
(13.31)
The above system may be solved using Matlab Simulink. In this instance, Matlab
5.3.0.10183R11 was used, and solver were set to use ODE45 (Dormand Prince) variable step
solver settings.
13.3. Equations of motion, two degree of freedom model, three axles
Since the 6x6 configuration was used to validate the model, using the above logic, it can be
shown that the matrix values for a 6x6 configuration changes to:
A11 =
(
-2
. C αf 1 + C αf 2 + C αr 1
MV
A12 = - V -
(
)
2
. ( aa ).Cαf 1 - bb.Cαf 2 - (bb + cc ).Cαr 1
MV
(13.32)
)
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
(13.33)
Page 62 of 138 pages
(
)
A21 =
-2
. - ( cc + bb ).Cαr 1 + ( aa ).Cαf 1 - bb.Cαf 2
IV
A22 =
-2
. ( aa )2 .Cαf 1 + bb 2Cαf 2 + ( cc + bb )2 Cαr 1
IV
(
(13.34)
)
 1 
B1 =  .(2.Cαf 1 + 2.Sr .Cαf 2 )
M 
(
1
. 2.( aa )Cαf 1 - 2.Sr .bb.Cαf 2
I
B1 =
(13.29)
(13.30)
)
(13.31)
where:
Fyf1 =
Lateral (cornering) force at the first axle
Fyf2 =
Lateral (cornering) force at the second axle
Fyr1 =
Lateral (cornering) force at the third axle
αf1
=
Slip angle first axle
αf2
=
Slip angle second axle
αr1
=
Slip third first axle
δ1
=
Steering angle first axle
δ2
=
Steering angle second axle
M
=
Mass of the vehicle
V
=
Forward velocity
R
=
aa
=
Distance between the first and centre of mass
bb
=
Distance between the centre of mass and the second axle
cc
=
Distance between the second axle and third axle
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 63 of 138 pages
14. Test track dimensions
The dimensions of the track (single lane change) and the track centreline used for the
simulation is shown in Figure 45.
Figure 45 : Single lane change track dimensions
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 64 of 138 pages
15. Driver/steering model
Various studies have been done on driver models and track followers for dynamic simulation
[19, 20, 21] & [29]. The detail falls beyond the scope of this investigation.
A simple steering/driver model has been implemented to steer the vehicle through the single
lane change track. The steering model receives as input the following:
•
The vehicle speed (A constant speed is assumed).
•
The centreline of the track that needs to be followed.
•
The distance the driver is looking ahead of the vehicle (because speed is constant,
distance is related to time).
•
The distance to the side the driver is looking at (because speed is constant, distance is
related to time).
•
The calculated yaw angle of the vehicle in its current position.
A schematic layout of the driver model is shown in Figure 46.
Figure 46 : Driver model
The implementation of the driver model in Matlab Simulink is shown in Figure 47. As can be
seen, the predicted track position, and the current track position are added, after it is
multiplied by a constant factor, in this case 0.5. Multiple runs of the simulation were done to
determine the factor, changing the constant until acceptable results were obtained.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 65 of 138 pages
Figure 47 : Simulink driver model
It should be noted that the aim of the simulation is not to determine the absolute maximum
speed the vehicle may attain during the single lane change, but to compare different steering
geometry relationships, using the same control strategy to compare the effect on ratio
changes.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 66 of 138 pages
The Simulink model used to solve the path of the vehicle and the slip angles of the individual
axles are shown in Figure 48.
Inputs to the model are listed below:
•
Tire characteristics of the individual axles, Cαf
•
Geometry of the vehicle, axle distances
•
Position of the centre of mass
•
Yaw moment of inertia of the vehicle
•
Mass of the vehicle
•
Track geometry and centreline path the driver is looking at
•
Initial steering angle input
•
Distance the driver is looking ahead
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 67 of 138 pages
•
Distance the driver is looking to the side
•
Ratio between the first axle and second axle steering angle
•
Vehicle speed through the obstacle
As output, the following parameters are obtained:
•
X and Y and heading angle ψ are displayed as a function of time
•
Slip angles of the individual axles as a function of time
•
Maximum slip angle per axle during the manoeuvre
•
The length of the path the vehicle has followed, after a specific X distance
•
The steering angle as a function of time for the first and second axles
A Matlab programme calculating the path as a function of time, as well as the tire
characteristics is shown in Appendix B.
A Matlab programme plotting the data, and calculating the track length, and journey time is
shown in Appendix C.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 68 of 138 pages
It should be noted that the aim of the simulation is not to find the absolute maximum speed
that may be achieved through the lane change, but rather to use the results to investigate
steering ratio tendencies. It may be possible to change the driver model to increase or
decrease the speed of the vehicle through the obstacles, or improve the trajectory through the
lane change. The driver models were set to ensure that the simulated results follow the
measured results (as described in paragraph 18) as close as possible.
As a first attempt to compare the effect of the steering ratio on transient state handling, the
steering ratio was varied from a ratio of 0.3 to a ratio of 1 between the first and second axles.
The time required completing the lane change, for different vehicle speeds were calculated,
and the times compared. In all instances, the exact same driver model for a specific speed was
used. The results are shown in Figure 49.
Bison Single Lane Change
10
30 km/h
40 km/h
50 km/h
60 km/h
70 km/h
9
8
time to complete the track [s]
7
6
5
4
3
2
1
0
0.2
0.3
0.4
0.5
0.6
0.7
Steering Ratio
0.8
0.9
1
1.1
1.2
Figure 49 : Time to complete the maneuver
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 69 of 138 pages
From the figure it is clear that no measurable time difference through the obstacle is achieved
by changing the steering ratio, if the driver model is kept constant. In other words, a driver
with the same “driving ability” will not achieve a quicker time through the lane change with a
change in steering geometry.
A further factor that may be investigated is the slip angles that are induced between the tire
and the road during the manoeuvre, for a specific steering ratio and speed. The maximum slip
angle during the lane change manoeuvre is shown in Figure 50.
Bison Single Lane Change 30 km/h
Bison Single Lane Change 40 km/h
First Axle
Second Axle
Third Axle
Max slip angle
8
6
4
10
2
0.6
0.8
1
Steering Ratio
Bison Single Lane Change 50 km/h
First Axle
Second Axle
Third Axle
Max slip angle
8
6
4
2
0.4
0.6
0.8
1
Steering Ratio
Bison Single Lane Change 60 km/h
1.2
10
First Axle
Second Axle
Third Axle
8
6
4
2
0.4
0.6
0.8
1
Steering Ratio
Bison Single Lane Change70 km/h
0
0.2
1.2
10
0.4
0.6
0.8
Steering Ratio
1
1.2
First Axle
Second Axle
Third Axle
8
Max slip angle
4
0
0.2
0.4
10
0
0.2
6
2
Max slip angle
0
0.2
First Axle
Second Axle
Third Axle
8
Max slip angle
10
6
4
2
0
0.2
0.4
0.6
0.8
Steering Ratio
1
1.2
Figure 50 : Maximum slip angle
From the above, it can be seen that the maximum slip angle may be minimized when a
steering ratio of 0.6 is used for low speed (30 km/h). For speeds above 50 km/h, the steering
ratio will not minimize the maximum slip angle. The maximum slip angle is smaller for large
steering ratios (less than 0.5), understandable since one axle is turned through an angle more
than double the steering angle of the second axle.
When the simulated slip angles are plotted against time, it is clear that an “aggressive” driver
model has been used. The model “reacts” to the step input at the beginning of the lane change
(first peak), and has to compensate (again second peak) in order to complete the track. A more
sophisticated model may have resulted in a smoother transition through the track. Since the
aim of the study is not to determine the absolute maximum speed of the vehicle through the
course, the results are deemed sufficient to investigate the problem on concept level.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 70 of 138 pages
Bison Single Lane Change 30 km/h. First Steering Axle
15
Slip Angle [°]
10
5
0
-5
-10
-15
0
1
0
1
2
3
4
time [s]
Bison Single Lane Change 30 km/h. Second Steering Axle
5
6
5
6
15
Slip Angle [°]
10
5
0
-5
-10
-15
2
3
time [s]
Bison Single Lane Change 30 km/h. Rear Axle
4
15
ratio 0.3
ratio 0.4
ratio 0.5
ratio 0.6
ratio 0.7
ratio 0.8
ratio 0.9
ratio 1
ratio 1.1
Slip Angle [°]
10
5
0
-5
-10
-15
0
1
2
3
time [s]
4
5
6
5
6
5
6
Figure 51 : Slip angles – 30 km/h
Bison Single Lane Change 60 km/h. First Steering Axle
15
Slip Angle [°]
10
5
0
-5
-10
-15
0
1
0
1
2
3
4
time [s]
Bison Single Lane Change 60 km/h. Second Steering Axle
15
Slip Angle [°]
10
5
0
-5
-10
-15
2
3
time [s]
Bison Single Lane Change 60 km/h. Rear Axle
4
15
ratio 0.3
ratio 0.4
ratio 0.5
ratio 0.6
ratio 0.7
ratio 0.8
ratio 0.9
ratio 1
ratio 1.1
Slip Angle [°]
10
5
0
-5
-10
-15
0
1
2
3
time [s]
4
5
6
Figure 52 : Slip angles – 60 km/h
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 71 of 138 pages
The above conclusions is also illustrated when the RMS of the slip angles are calculated, as
shown in Figure 53.
Bison Single Lane Change 30 km/h
Bison Single Lane Change 40 km/h
4
2
0
0.2
0.4
0.6
0.8
1
Steering Ratio
Bison Single Lane Change 50 km/h
First Axle
Second Axle
Third Axle
RMS. slip angle
4
2
0
0.2
0.4
0.6
0.8
1
Steering Ratio
Bison Single Lane Change 70 km/h
0.4
0.6
0.8
1
Steering Ratio
Bison Single Lane Change 60 km/h
1.2
First Axle
Second Axle
Third Axle
6
4
2
0.4
0.6
0.8
Steering Ratio
1
1.2
First Axle
Second Axle
Third Axle
6
RMS. slip angle
2
0
0.2
1.2
7
4
0
0.2
1.2
6
First Axle
Second Axle
Third Axle
6
RMS. slip angle
RMS. slip angle
6
RMS. slip angle
First Axle
Second Axle
Third Axle
5
4
3
2
1
0
0.2
0.4
0.6
0.8
Steering Ratio
1
1.2
Figure 53 : RMS slip angles
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 72 of 138 pages
The simulated trajectory through the lane change manoeuvre, for different steering ratios, and
vehicle speeds are shown in Figure 54. As discussed above, the driver model ensures that the
vehicle completes the track, but does not simulate the behaviour of a human driver, as will be
shown in the following paragraphs when measured results are discussed.
Bison Single Lane Change 40 km/h
8
6
6
Distance [m]
Distance [m]
Bison Single Lane Change: 30 km/h
8
4
2
0
2
0
-2
-2
0
20
40
60
80
100
Distance [m]
Bison Single Lane Change: 50 km/h
120
8
8
6
6
Distance [m]
Distance [m]
4
4
2
0
0
20
0
20
40
60
80
100
Distance [m]
Bison Single Lane Change: 60 km/h
120
4
2
0
-2
-2
0
20
0
20
40
60
80
100
Distance [m]
Bison Single Lane Change: 70 km/h
120
40
60
80
Distance [m]
100
120
8
Distance [m]
6
4
2
0
-2
40
60
80
Distance [m]
100
120
Figure 54 : Simulated trajectory
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 73 of 138 pages
As a final example of typical output values that was achieved, the yaw angle of the simulated
vehicle at 30 km/h is plotted in Figure 55. The effect of the aggressive driver model may be
seen in the change in yaw rate to enable direction changes.
Bison Single Lane Change 30 km/h. Yaw Angle
30
ratio 0.3
ratio 0.4
ratio 0.5
ratio 0.6
ratio 0.7
ratio 0.8
ratio 0.9
ratio 1
ratio 1.1
25
20
Yaw Angle [°]
15
10
5
0
-5
-10
-15
0
5
10
15
time [s]
Figure 55 : Simulated yaw angle - 30 km/h
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 74 of 138 pages
18. Vehicle measurements
18.1. Introduction
As a second phase of the investigation, track measurements were taken to verify and compare
simulated results. Measured results would indicate the validity of the mathematical models,
and the assumptions that was made.
The Bison vehicle was instrumented, and constant radius tests as well as single lane changes
were performed as described below.
18.2. GPS measurements
During the planning phase of the vehicle tests, it was decided to investigate the use of dual
GPS systems. With a GPS system mounted in the front of the vehicle, and a second GPS
mounted on the rear of the vehicle, it should be possible, in theory, to determine the yaw
angle of the vehicle at any time during the steady state and dynamic handling tests.
The yaw angle data and steering angle data may then be combined to evaluate the slip angles
of any of the wheels. This information was previously difficult to calculate, since only the
yaw rate is normally measured using a gyro. To obtain the yaw angle, the yaw rate needs to
be numerically integrated to obtain the yaw angle. Numerical integration of measured data
may result in drift, which needs to be corrected for.
18.3. Test equipment
The test equipment used is summarised in Table 4:
Table 4 : Measuring equipment
No
Description
Make
Model
Serial number
1.
Pitch gyro
Humphrey rate transducer
RT01-0101-1
H64
2.
Roll gyro
Humphrey rate transducer
RT01-0101-1
H66
3.
Yaw gyro
Humphrey rate transducer
RT01-0101-1
H65
4.
Optical pick-up (speed)
Banner
QS18VP6LV
0135F
5.
Frequency to voltage converter
Turck
MS25-Ui
MS25-Ui1
6.
Frequency to voltage converter
Turck
MS25-Ui
MS25-Ui2
7.
Position transducers
CELESCO
PT510-0040-111-1110
G1303685A
8.
Accelerometers
VTI Hamlin
5g
VOI 52
9.
GPS
RaceLogic
VBOX Pro -001
n/a
DM5406 with 100Hz low
pass filters LMT010
n/a
LMT DAQ001
10.
Data acquisitioning system
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 75 of 138 pages
18.4. Test vehicle and measurement positions
Figure 56 shows the instrumented vehicle performing a constant radius test on the Gerotek
test track.
Figure 56 : Instrumented vehicle on the test track
The following was measured:
•
Using wire displacement meters, the steering angle of the first and second axles were
measured.
•
The lateral acceleration of the vehicle body was measured.
•
The vehicle speed was measured using a rotational pulse counter.
•
The absolute position of the front of the vehicle was measured using a GPS.
•
The absolute position of the rear of the vehicle was measured using a second GPS.
•
The driver used a handheld GPS that was used to determine vehicle speed while
performing the test.
•
The roll and yaw velocities of the body was measured using gyros.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 76 of 138 pages
Figure 57 : Rear GPS
Figure 58 : Rear GPS antenna position
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 77 of 138 pages
Figure 59 : First axle displacement meter position
Figure 60 : Second axle displacement meter position
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 78 of 138 pages
Figure 61 : Vehicle speed measurement
Figure 62 : Front GPS antenna position
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 79 of 138 pages
18.5. Test procedure
The vehicle tires were inflated to 450 kPa all round, where after the wheel masses were
measured, with the results indicated in Table 5:
Table 5 : Bison test wheel mass
Axle number
Left mass
Right mass
Axle total
1
2670 kg
2470 kg
5140 kg
2
2660 kg
2680 kg
5340 kg
3
2270 kg
2120 kg
4390 kg
Total
14870 kg
The vehicle was driven to the test track, where the steering calibration was performed.
As a basis ISO 4138, steady state circular test procedure [30], and ISO 3888 for severe lane
change manoeuvres [31] were used.
The vehicle was positioned at the skidpan track; the data capturing equipment was activated
while the vehicle was stationary. The constant radius test was initiated, by first driving at a
constant speed (against the engine governor to ensure a constant speed) in the gear closest to
the required speed, around the track. After completion of a lap, the speed was increased. The
procedure was completed clock-wise and anti-clockwise and the data capturing was ended
after the vehicle came to a standstill.
A hand held GPS was used to record the actual vehicle speed as a reference.
A single lane change track was laid out on the Gerotek test facility long straight track. The
track had the dimensions shown in Figure 63:
Figure 63 : Lane change track dimensions
Lane changes were performed at speeds of 28 km/h, 43 km/h, 51 km/h and 60 km/h.
The data capturing equipment was initiated, and the lane changes performed at a successive
increase in speed.
After the vehicle came to a standstill, the measuring equipment was again switched off.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 80 of 138 pages
18.6. Steering calibration measurements
The steering angle calibration was done in two phases.
•
The aim of the first phase was to determine the maximum measured steering angles of
the first and second axles relative to the vehicle centreline, when the steering wheel is
turned from lock to lock.
•
The second phase was to determine the relationship between the linear displacement
measured, and the absolute steering angles of the wheels, with the aim to determine the
steering ratio between the first and second axles.
18.6.1. Steering calibration results: maximum steering angles
The vehicle was parked on a level surface, with the wheels at zero degrees. The steering
wheel was rotated half a turn to the left, and the change in the angle between the wheel plane
and the vehicle body was measured. The steering wheel was rotated another half a turn and
the steering angle measured. The process was repeated until the end of travel stop was
reached. The process was repeated turning the steering wheel to the left hand side, and the
right hand side, for the left and right hand side wheels of the first and second axles.
angle
Figure 64 : Measuring steering angles. The position of the plumb line are shown
Figure 64 shows the measurement of steering angles. A plumb line, perpendicular to the
wheel plane was used to scribe the position of the tire angle relative to the direction of travel,
for the steering wheel increments. A long distance between the wheel and measuring point
was used to reduce measuring errors. The process was repeated for the 4 wheels of the first
and second axles.
18.6.2. Steering calibration results linear to angles
The second phase was the measurement of the steering angles using linear displacement
meters on the first and second axles. From the neutral position, the steering was first turned to
left lock, and then to right lock.
The wheel angle relative to the vehicle chassis measurement could be derived once the linear
measurements were transformed to angular values.
The linear displacements measured, are shown in Figure 65.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 81 of 138 pages
Figure 65 : Steering angle calibration, measured data
Wheel Angle [°]
In order to transform the linear displacements to angular values (as required by the
simulations), the actual wheel angles was measured, as described above.
25
20
15
10
5
0
-5
-10
-15
-20
-25
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
Number of Steering Wheel Turns
LH FRONT
RH FRONT
LH REAR
RH REAR
Figure 66 : Measured steering angles for steering wheel positions
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 82 of 138 pages
From Figure 66 it is clear that the steering system is not symmetrical. As the system will be
simplified to a bicycle model, the average of the left and right wheels are determined.
The angles used are shown in Table 6.
Table 6 : Measured wheel angles
Counter
clockwise
Clockwise
Maximum angle: 1st axle left
-16°
18°
Maximum angle: 1st axle right
-20°
12°
Maximum angle: 2nd axle left
-12°
9°
Maximum angle: 2nd axle right
-7°
10°
Average 1st axle
-18°
15°
Average 2nd axle
-9.5°
9.5°
A Matlab subroutine (Appendix A) converts the measured linear displacements to angular
displacements. The results are shown in Figure 67 and Figure 68.
Figure 67 : Calibrated steering angle
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 83 of 138 pages
Figure 68 : First and second axle calibrated steering angles
Lines may be fitted through the first and second axle measurements, taken from time 30
seconds to time 45 seconds, representing full lock left to full lock right turn. The steering ratio
between the first and second axle may be calculated, and is shown in Figure 69.
Figure 69 : Steering ratio, 0.7
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 84 of 138 pages
The data in Figure 69 shows that the steering ratio between the first and second axle may be
approximated as 0.7, for large steering angles. Small steering angles close to zero, results in
an infinite ratio since the ratio calculation divides by zero.
18.7. GPS measurements
As mentioned, two GPS systems were used to plot the position of the vehicle at all times. It
was found that the data was not accurate enough to determine the vehicle heading to such an
extent that slip angles could be determined.
As an example, when the data of the constant radius test from the first and second GPS is
overlaid, a result shown in Figure 70 is obtained.
Figure 70 : Front and rear GPS data
It was expected that the approach and departure lines to and from the circle would have been
plotted over one another, since the vehicle travelled in a straight line. The arrows in Figure 70
indicate the problem areas. From the above, it is clear that inaccurate slip angle data will be
calculated, if the GPS data was used. A similar result for the lane change data is shown in
Figure 71.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 85 of 138 pages
Figure 71 : GPS lane change data, front and rear GPS, 4 vehicle speeds
The inaccurate results may be contributed, amongst other things, to the GPS systems
switching between satellites during the test. A further problem may have been the surrounding
walls of the test track, deflecting satellite signals away from the equipment.
The GPS data was used to determine the trajectory through the lane change, but could not be
used to calculate slip angles, as can be seen in Figure 71.
18.8. Test results, constant radius test
The skidpan track at the Gerotek test facility [32] was used to perform constant radius tests.
The constant radius manoeuvre was first completed clockwise, at measured speeds of 10
km/h, 17.9 km/h, 27 km/h, 37 km/h and 40 km/h. The tests were repeated counter clockwise,
at the same speeds. The radius of the test was typically 55 meters.
The results of the constant radius tests are shown below. The lateral acceleration, yaw rate and
roll rate are shown as a function of time. During the test, the speed of the vehicle was
increased, while the data was captured. Figure 72 to Figure 76 indicates the filtered measured
values.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 86 of 138 pages
Figure 72 : Lateral acceleration, constant radius test
Figure 73 : Yaw rate, constant radius test
Figure 74 : Roll, constant radius test
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 87 of 138 pages
Figure 75 : Steering angle comparison, constant radius test
60
4
40
2
20
0
0
-2
0
50
100
150
200
time [t]
250
300
350
Vehicle Speed [km/h]
First Axle Steering Angle [°]
Constant Radius Test Counter Clock Wise
6
-20
400
100
2
50
0
0
-2
0
50
100
150
200
time [t]
250
300
350
Vehicle Speed [km/h]
Second Axle Steering Angle [°]
Constant Radius Test Counter Clock Wise
4
-50
400
Figure 76 : Steering single and speed comparison, constant radius test
As expected, an increase in steering angle is required to keep the vehicle on the same radius at
an increased speed. A detailed comparison of the measured and simulated results is given in
paragraph 20.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 88 of 138 pages
19. Test results, single lane change
An example of measurements obtained from the single lane change manoeuvre is shown in
Figure 77 to Figure 79. The lateral acceleration, yaw and roll velocities and wheel steering
angles are shown.
Figure 77 : Measured first axle wheel speed and lateral acceleration - 28 km/h
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 89 of 138 pages
Figure 78 : Steering angles on first and second axles measured data - 28 km/h
Figure 79 : Roll and yaw measured data - 28 km/h
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 90 of 138 pages
20. Simulated vs. measured results
20.1. Introduction
The aim of the measurements described in the paragraphs above, was to determine the
correlation between simulated and measured results. A correlation will confirm the
mathematical model, and will provide confidence in performance improvements, effected by
changing the steering system geometry of the vehicle.
Correlation is obtained when the measured and simulated results are compared. Results of
previous figures are combined and shown as Figure 80.
7
Alfa 1st Axle
Alfa 2nd Axle
Alfa 3rd Axle
Steer 1st Axle
Steer 2nd Axle
6
5
Angle [°]
4
3
2
1
0
-0.2
0
0.2
0.4
Ratio
0.6
0.8
1
60
4
40
2
20
0
0
-2
0
50
100
150
200
time [t]
250
300
350
Vehicle Speed [km/h]
First Axle Steering Angle [°]
Constant Radius Test Counter Clock Wise
6
-20
400
Figure 80 : (Result of Figure 34 and Figure 76)
Figure 34 (repeated as Figure 80) indicates the simulated first axle steering angle for a 60
meter radius test at 30 km/h (for a steering ratio of 0.7) equates to 4.8 degrees. Figure 75 and
Figure 76 indicates a measured angle of 5.3 degrees clockwise and 4.2 degrees anticlockwise. An average of the clockwise and anti-clockwise data is 4.75 degrees, similar to the
calculated vale of 4.8. The simulated angle for the second axle is 3.5 degrees, and the
measured angle is 3 degrees at 30 km/h.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 91 of 138 pages
Angle [°]
7
Alfa 1st Axle
Alfa 2nd Axle
6
Alfa 3rd Axle
Steer 1st Axle
Steer 2nd Axle
5
4
3
2
1
0
0.4
0.6
0.8
1
Ratio
60
4
40
2
20
0
0
-2
0
50
100
150
200
time [t]
250
300
350
Vehicle Speed [km/h]
First Axle Steering Angle [°]
Constant Radius Test Counter Clock Wise
6
-20
400
Figure 81 : (Result of Figure 34 and Figure 76)
Figure 34 indicates the simulated steering angle for a 60 meter radius test at 50 km/h (for a
steering ratio of 0.7) equates to 5.3 degrees. Figure 75 and Figure 76 indicates a measured
angle of 5.6 degrees clockwise and 5.3 degrees anti-clockwise. Once again the average of
5.45 degrees corresponds with the calculated 5.3 degrees. The simulated angle for the second
axle is 4 degrees, and the measured angle is 3.5 and 3.2 degrees for clockwise and anticlockwise tests, an average of 3.35 compared to 4 degrees at 50 km/h.
20.3. Single lane change
The simulated path that the vehicle followed through the single lane change obstacle course,
at different vehicle speeds, are compared with the measured path of the vehicle through the
single lane change (front and rear GPS data) in Figure 82. Also included in Figure 82 is the
simulated trajectories predicted for the vehicle when the steering ratio between the front and
rear axle is varied from 0.5 to 1.1.
Once again the problem with the GPS changing satellite reference may be seen in the results.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 92 of 138 pages
6
6
4
2
4
2
0
0
-2
-2
0
20
40
60
80
100
Distance [m]
Bison Single Lane Change: 50 km/h
120
8
8
6
6
Distance [m]
Distance [m]
Bison Single Lane Change 40 km/h
8
Distance [m]
Distance [m]
Bison Single Lane Change: 30 km/h
8
4
2
0
-2
40
60
80
100
Distance [m]
Bison Single Lane Change: 70 km/h
120
0
20
40
60
80
100
Distance [m]
Bison Single Lane Change: 60 km/h
120
2
0
20
20
4
-2
0
0
40
60
80
Distance [m]
100
120
8
SIMULATED
Distance [m]
6
4
2
MEASURED
0
-2
0
20
40
60
80
Distance [m]
100
120
Figure 82 : Single lane change measured vs. simulated trajectories
The following conclusions may be drawn from the above:
•
The simulated results match the measured results accurately during the approach, lane
change and exit of the course.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 93 of 138 pages
•
The simulated driver model is more aggressive, resulting in increased slip angles
because of larger steer movements.
•
For the same driver model, a change in the steering ratio at any practical speed will not
have a noticeable effect on the path that the vehicle will follow through the lane
change.
As a second comparison, the measured and simulated steering angles for the 43 km/h lane
change are shown in Figure 83 and Figure 84. As mentioned in the preceding paragraphs, the
simulation driver model is more aggressive than the test driver. The rate of change of steering
angle for the test driver is 2 degrees per second, and the simulation results show a steering
rate change of 15 degrees per second in some instances. However, the vehicle trajectory, and
journey time correlates with measured results.
Single Lane Change 43 km/h
4
Front Steering Angle
3
2
1
0
-1
-2
0
5
10
15
Time [sec]
20
25
30
0
5
10
15
Time [sec]
20
25
30
4
Front Steering Angle
3
2
1
0
-1
-2
Figure 83 : Measured steering angle single lane change - 43 km/h
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 94 of 138 pages
Bison Single Lane Change 50 km/h. First Steering Axle
30
ratio 0.3
ratio 0.4
ratio 0.5
ratio 0.6
ratio 0.7
ratio 0.8
ratio 0.9
ratio 1
ratio 1.1
25
20
Steering Angle [°]
15
10
5
0
-5
-10
-15
0
5
10
15
time [s]
Figure 84 : Simulated steering angle single lane change - 50 km/h
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 95 of 138 pages
21. Vehicle specific characteristics
Reasons for the small improvement in handling, that have been measured and simulated in the
preceding paragraphs, may be explained summarised as follows:
21.1. Vehicle geometry
The Bison vehicle is a 6x6 vehicle, steering with the first and second axles. The third axle is
not steerable, but acts as a reaction to the steering forces generated by the first and second
axles. The tires on the three axles are the same, and the centre of mass of the vehicle is close
to the second axle. (The vehicle was tested in the unladen condition.) As measured before the
tests commenced, the axle load on the first and second axles are approximately 5 tonnes for
first and second, and only 3.9 tonnes on the rear axle.
Based on these assumptions, the following may be postulated:
•
The side force of the front axle and rear axle is balanced around the centre off mass,
with the second axle (with a short lever arm close to the centre of mass) having little
effect on the moment equilibrium of the vehicle.
•
Therefore, by changing the steering ratio within practical ranges (to ensure
manoeuvrability at low speed), the effect on the vehicle system, at normal operating
speeds have little effect.
•
Following on the above, it may be concluded that the effect of changing the steering
ratio may be more pronounced in vehicles with long wheelbases, and vehicles with a
centre of mass further away from the second steering axle, as was the case with Bison.
To illustrate the above, the single lane change simulation was repeated for a fictitious 6x6
vehicle with the dimensions shown in Figure 85.
Figure 85 : LWB vehicle
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 96 of 138 pages
When the slip angles for the same driver model of Bison is compared to a vehicle with the
same mass, but different axle spacing, it can be seen that for a Long Wheel Base (LWB)
vehicle, a steering ratio may be chosen to minimize the slip angles for a specific speed. The
results are compared in the following figures. Figure 86 shows the simulated slip angles on
three axles for a single lane change at 40 km/h for a LWB vehicle, and Figure 87 shows the
slip angles for Bison.
The slip angles for a LWB vehicle may be minimized, because of the better force and lever
arm relations that exist. For the instance calculated, a 0.7 steering ratio the slip angles of the
first and second axles are minimized for 30 km/h. Figure 88 illustrates the above concept
when slip angles may be reduced by a factor of more than two for all vehicle speeds
simulated. When the results of Bison is evaluated, it may be seen that steering ratio changes
have little effect on the slip angles that are generated by the first axle, and to a lesser extend
the second axle.
LWB Single Lane Change 40 km/h. First Steering Axle
Slip Angle [°]
4
2
0
-2
-4
0.5
1
1.5
2
2.5
3
time [s]
LWB Single Lane Change 40 km/h. Second Steering Axle
3.5
4
4.5
Slip Angle [°]
0
-5
-10
0.5
1
1.5
2
2.5
3
time [s]
LWB Single Lane Change 40 km/h. Rear Axle
3.5
4
4.5
4
Slip Angle [°]
2
ratio 0.3
ratio 0.4
ratio 0.5
ratio 0.6
ratio 0.7
ratio 0.8
ratio 0.9
ratio 1
ratio 1.1
Ratio 1.1
0
-2
Ratio 0.3
-4
0.5
1
1.5
2
2.5
time [s]
3
3.5
4
4.5
5
Figure 86 : LWB slip angles
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 97 of 138 pages
Bison Single Lane Change 40 km/h. First Steering Axle
4
Slip Angle [°]
2
0
-2
-4
-6
-8
0.5
1
1.5
0.5
1
1.5
2
2.5
3
3.5
time [s]
Bison Single Lane Change 40 km/h. Second Steering Axle
4
4.5
Slip Angle [°]
0
-5
-10
2
2.5
3
time [s]
Bison Single Lane Change 40 km/h. Rear Axle
3.5
4
4.5
4
ratio 0.3
ratio 0.4
ratio 0.5
ratio 0.6
ratio 0.7
ratio 0.8
ratio 0.9
ratio 1
ratio 1.1
Slip Angle [°]
2
0
-2
-4
0.5
1
1.5
2
2.5
time [s]
3
3.5
4
4.5
Figure 87 : Bison slip angles
LWB Single Lane Change 30 km/h
LWB Single Lane Change 40 km/h
First Axle
Second Axle
Third Axle
Max slip angle
8
6
4
10
2
0.6
0.8
1
Steering Ratio
LWB Single Lane Change 50 km/h
First Axle
Second Axle
Third Axle
Max slip angle
8
6
4
2
0.4
0.6
0.8
1
Steering Ratio
LWB Single Lane Change 60 km/h
1.2
10
First Axle
Second Axle
Third Axle
8
6
4
2
0.4
0.6
0.8
1
Steering Ratio
LWB Single Lane Change70 km/h
0
0.2
1.2
10
0.4
0.6
0.8
Steering Ratio
1
1.2
First Axle
Second Axle
Third Axle
8
Max slip angle
4
0
0.2
0.4
10
0
0.2
6
2
Max slip angle
0
0.2
First Axle
Second Axle
Third Axle
8
Max slip angle
10
6
4
2
0
0.2
0.4
0.6
0.8
Steering Ratio
1
1.2
Figure 88 : LWB vehicle, maximum slip angles
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 98 of 138 pages
The above conclusion may be expanded to 8x8 vehicles as shown in Figure 89. Once again
the vehicle has a long wheelbase, with the centre off mass close to the middle of the vehicle.
Due the dual rear axles, spaced close together far from the centre of mass, the change in
steering ratio for this vehicle may have a smaller effect, than on a similar 6x6 configuration,
steering with two axles.
Figure 89 : 8x8 MAN
21.2. Tire and driver model
A further characteristic of the Bison vehicle is a stiff spring pack and limited wheel travel,
compared to similar vehicles in its class. At low speed, these factors reduces body roll and
may increase load transfer. Load transfer, combined with a linear tire model may be expanded
in future work to improve results.
21.3. Speed ranges
The aim of the study was to investigate the improvement of vehicle handling during normal
road use, to see if a marked improvement can be found. In other words, a marked
improvement in handling of a variable ratio system, during normal use, may convince a
potential customer to select a variable ratio steering system above a normal system, only if
improvement can be shown.
The test speeds that was used, was selected to reduce the risk of vehicle and accident damage,
and represent realistic road manoeuvres that may be performed by normal road users. The aim
was not to improve the maximum lane change speed of the vehicle or on the edge handling,
even though upper limit handling improvement may be a life saving feature.
21.4. Tire wear
The theoretical study indicated that selecting a speed specific steering ratio might reduce the
slip angles. Improved tire wear should be traded off against the user profile of the vehicle. If,
for example, the vehicle will be used predominantly off-road and in sandy conditions, reduced
tire scrub may not on its own justify the added cost of a variable ratio steering system. On the
other hand, if tight manoeuvring on tar roads is done, or long distances covered, tire wear and
tire cost may justify the increased cost of a variable ratio steering system.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 99 of 138 pages
22. Conclusion
The following is concluded:
i.
For the specific vehicle and tire characteristics, simplified mathematical models for
steady and transient handling was developed and produced acceptable results and
correlation for the purpose of this investigation.
ii.
The simulation results enabled the evaluation of the feasibility of changing the
steering system of the test vehicle to a variable ratio steering system.
iii.
The Matlab models that were developed, gave valuable insight into the handling of the
test vehicle during steady state constant radius tests, and transient state single lane
change manoeuvres.
iv.
A further characteristic of the test vehicle is a stiff spring pack and limited wheel
travel, compared to similar vehicles in its class. At low speed, these factors reduces
body roll and may increase load transfer. Load transfer, combined with a non-linear
tire model may be expanded in future work to improve results.
v.
It is not recommended to implement a variable ratio steering system in the test vehicle,
since it was determined that no measurable improvement in the steady-state or
transient-state handling will be achieved.
vi.
Based on the results of a similar study done on a 6x6 all-wheel steer vehicle [4], for
the same level of mechanical complexity, a far greater improvement in vehicle
handling, manoeuvrability and tire wear may be achieved by changing the vehicle to
steer with the first and last axle, compared to the effect of changing the steering
system to a variable ratio type steering system for the first and second axles.
vii.
Tire wear is a function of vehicle speed and slip angle amongst other parameters. It is
possible (at least in theory) to select a steering ratio to reduce tire wear at a specific
vehicle speed. For low speed manoeuvres, the Ackerman ratio may be selected to
reduce tire wear.
viii.
The Ackerman ratio selected for low speeds may be adjusted at higher vehicle speeds,
by selecting a steering ratio between the first and second axles that is closer to one.
This steering ratio may be selected to increase the side force that can be generated by
the second axle, and improve the vehicle response to steering inputs. A downside may
be increased tire wear.
ix.
Steering ratio changes, and the dynamic adjustment of steering ratios may however
have a measurable effect on vehicles with specific geometry. It was shown that the
wheelbase, distance to centre of mass, and axle loading plays a role in the response of
a vehicle to steering ratio changes.
x.
It would be possible to integrate a simple mechanical adjustment on the test vehicle, to
perform further testing if required.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 100 of 138 pages
23. Recommendations
The following can be recommended:
i.
That an 8x8 vehicle be investigated further, to determine the effect on a long
wheelbase vehicle with a centre of mass in the geometric centre of the vehicle.
ii.
That the use of GPS systems to measure slip angles be investigated further, as this
technology will eliminate inaccuracies in the conventional measuring chain.
iii.
The Simulink driver model should be developed further is order to obtain a smoother
“human” approach tendency through the single lane change.
iv.
The effect of load transfer during both the lane change and constant radius
manoeuvres should be investigated and included in future simulation models.
v.
The available measured tire data should be used with the aim to include non-linear tire
characteristics, especially when load transfer is investigated.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 101 of 138 pages
24. References
1.
Allen, J., Four-Wheel-Steer Jeeps, 4x4x4, JP Magazine, Primedia Magazines Inc,
www.jpmagazine.com, accessed 2 August 2004.
2.
Anon, EMAS Automatic Hydraulic Steering, www.v-s-e.nl/rechts.htm, accessed 28
January 2005.
3.
Kleine, S., Active control of a four-wheel-steered vehicle, University of Pretoria,
August 1996.
4.
Nell, S., The influence of all-wheel steer (6x6) on vehicle handling and tire wear,
Ermetek, 7th European ISTVS Conference, Ferrara, Italy, October 1997.
5.
Anon, Patria Vehicles, www.patria.fi, accessed Desember 2004.
6.
Anon, BPW Axles, www.BPW.de, accessed 1 February 2005.
7.
Anon, Multidrive Limited, North Yorkshire Y07 3BX, UK, www.multidrive.co.uk,
accesed 12 January 2005.
8.
Giliomee, C.L., RSA-MIL-STD for Mobility: Software Validation (Milestone 10),
Reumech Ermetek, R0015195-0626, March 1999.
9.
Gillespie, T.D., Fundamentals of Vehicle Dynamics, Society of Automotive
Engineers, Warrenville, 1994
10.
Dixon, Tires, Suspension and Handling, Second Eedition, Society for Automotive
Engineers, 1996.
11.
Wetall, T.T., Fighting vehicles, Brassey’s New Battlefield Weapons Systems and
Technology Series, Volume 1, 1991
12.
Pacejka, H.B., Modelling of the Pneumatic Tire and its impact on vehicle dynamic
behaviour, CCG-Lehrgang V2.03, Carl-Cranz-Gesellschaft e. V., Oberpfaffenhofen,
1986.
13.
Bakker, E., The Magic Formula Tire Model, 1st International Colloquium on Tire
models for Vehicle Dynamic Analysis, Netherlands, 21-22October, 1991.
14.
Dugoff, H., Fancher, P.S. and Segel, L, An analysis of tire traction properties and
their influence on vehicle dynamic performance, 1970 International Automobile
Safety Compendium, FISITA/SAE, Brussels, June 1970, SAE no 700377.
15.
Pacejka, H.B., and Fancher, P.S, Hybrid simulation of shear force development of a
tire experiencing longitudinal and lateral slip, Proceedings XIVth FISITA Congress,
London, June 1972, p 3/78.
16.
Savkoor, A.R., The lateral flexibility of pneumatic tires and its application to the
lateral rolling contact problem, SAE paper 700378, FISITA/SAE Congress, 1970
(Brussels).
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 102 of 138 pages
17.
Pacejka, H.B., Some recent investigations into dynamics and frictional behaviour of
pneumatic tires, the Physics of Tire traction, eds D.F. Hays and A.L. Browne,
Procedures of symposium held at General Motors Research Lab. (Mich. 1973), Plenum
Press, New York 1974.
18.
Nordeen, D.L., and Cortese, A.D., Force and Moment Characteristics of Rolling
Tires, SAE paper no 640028 (713A), 1963, 13p.
19.
Hirano, Y. et al, Development of Integrated System of 4WS and 4WD by H∞
Control, SAE Paper 930267, 1993.
20.
Metz, L.D. et al, Transient and Steady state performance characteristics of a two
wheel and four-wheel steer vehicle model, SAE Paper 911926, 1991.
21.
Yu, S. et al, A Global approach to vehicle control: Coordination of four-wheel
steering and wheel torque’s, Journal of Dynamic Systems, Measurement, and Control,
vol 116, Dec 1994, pp 659-667.
22.
Copley, OH, 1991 Yearbook, The tire and Rim Association Inc., 1991.
23.
Anon, Michelin Tire, www.michelintruck.com, accessed 2 February 2005.
24.
Anon, SupaQuick, www.supaquick.co.za, accessed 2 February 2005.
25.
Anon, Yokohama Tires, www.yokohama.co.za, accessed 2 February 2005.
26.
Anon, Dunlop Tire, www.dunlopng.com/care.htm, accessed 2 February 2005.
27.
Anon, Continental Tire, www.continental.co.za, accessed 2 February 2005.
28.
Nell, S., Die implemetering van `n aandryfmodel en baanvolger stuurmodel in
DADS (The Implementation of a drive and track follower model in DADS),
Ermetek, 0119305-0679-001, 1991.
29.
Sledge, N.H. Jr and Marshek, K.M, Comparison of Ideal Vehicle Lane Change
Trajectories , SAE, 971062.
30.
International Organization for Standardization, 1982.
31.
Anon, Road vehicles, Test track for a severe lane-change manoeuvre – part 2, ISO
3888, Second Edition, International Organization for Standardization, 2002.
32.
Anon, Gerotek, www.gerotek.co.za, accessed 2 May 2004.
33.
Strauss, L., Inleindende Fisika, Universiteit van Pretoria, 1st ed, 1980.
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 103 of 138 pages
Appendix A
Matlab routine to convert linear measured displacement to angular displacements
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 104 of 138 pages
% Subroutine converts measured linear wheel displacements
to Angular Displacements
grid on
zoom on
clear all
close all
%gem hoek van die linker en regter wiel word gebruik om die
as se hoek te gee
fname = 'strcal';
%1e as: lk 18° -16° rk 12° -20° avg -18° 15°
%2e as: lk 9° -12° rk 10° -7° avg -9.5° 9.5°
time=0:0.003333:(length(a)-1)*0.003333;
q=polyfit([-0.355 0.5],[15 -18],1);
figure(1)
s=polyfit([0.204 -0.206],[9.5 -9.5],1);
subplot(2,1,1);
%meter=-0.355
plot(time,(a(:,1)-mean(a(1:100,1)))*1,'b');% check cal values
%degree=q(1,1)*meter+q(1,2)
ylabel('First Axle Displacement Meter [m]');
%meter=0.5
title('Bison Steering Angle Calibration');
%degree=q(1,1)*meter+q(1,2)
xlabel('Time [sec]');
caldata(:,1)=q(1,2)+q(1,1)*((a(:,1)-mean(a(1:100,1))))*1;
grid on
caldata(:,2)=s(1,2)+s(1,1)*((a(:,2)-mean(a(1:100,2))))*1;
zoom on
caldata(:,3)=(a(:,3)-mean(a(1:100,3)))*1;
subplot(2,1,2);
caldata(:,4)=(a(:,4)-mean(a(1:100,4)))*1;
plot(time,(a(:,2)-mean(a(1:100,2)))*1,'b');
figure;
ylabel('Second Axle Displacement Meter [m]');
subplot(2,1,1);
xlabel('Time [sec]');
plot(time,caldata(:,1));
grid on
ylabel('First Axle Steering Angle [°]');
zoom on
title('Steering Calibration')
figure (2)
grid on
subplot(2,1,1);
zoom on
plot(time,(a(:,3)-mean(a(1:100,3)))*1,'b');
subplot(2,1,2);
ylabel('Wheel Speed');
plot(time,caldata(:,2));
title('Carl Test')
ylabel('Second Axle Steering Angle [°]');
grid on
title('Steering Calibration')
zoom on
grid on
figure (2)
zoom on
subplot(2,1,2);
plot(time,(a(:,4)-mean(a(1:100,4)))*1,'b');
ylabel('Lat Acc');
figure;
xlabel('Time [sec]');
subplot(2,1,1);
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 105 of 138 pages
plot(time,clgfilt(time,caldata(:,1),0.5,'blue'));
ylabel('First Axle Steering Angle [°]');
title('Steering Calibration')
grid on
zoom on
subplot(2,1,2);
plot(time,clgfilt(time,caldata(:,2),0.5,'red'));
ylabel('Second Axle Steering Angle [°]');
title('Steering Calibration')
grid on
zoom on
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 106 of 138 pages
Appendix B
Measured CCW plot data manipulation
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 107 of 138 pages
Measured CCW Plot data manipulation
plot(time,(a(:,2)-mean(a(1:100,2)))*1,'b');
clear all
ylabel('Second Str Ang');
close all
xlabel('Time [sec]');
fname = 'ccw';
grid on
zoom on
time=0:0.003333:(length(a)-1)*0.003333;
figure (2)
q=polyfit([-0.355 0.5],[15 -18],1);
subplot(2,1,1);
s=polyfit([0.204 -0.206],[9.5 -9.5],1);
plot(time,60*(a(:,3)-mean(a(1:100,3))),'b');
caldata(:,1)=q(1,2)+q(1,1)*((a(:,1)-mean(a(1:100,1))))*1;
ylabel('Wheel Speed');
caldata(:,2)=s(1,2)+s(1,1)*((a(:,2)-mean(a(1:100,2))))*1;
title('Carl Test')
caldata(:,3)=(a(:,3)-mean(a(1:100,3)))*1;
grid on
caldata(:,4)=(a(:,4)-mean(a(1:100,4)))*1;
zoom on
figure;
figure (2)
subplot(2,1,1);
subplot(2,1,2);
plot(time,caldata(:,1));
plot(time,(a(:,4)-mean(a(1:100,4)))*1,'b');
ylabel('First Axle Steering Angle [°]');
ylabel('Lat Acc');
title('Steering Calibration')
xlabel('Time [sec]');
grid on
grid on
zoom on
zoom on
subplot(2,1,2);
figure(3)
plot(time,caldata(:,2));
subplot(2,1,1);
ylabel('Second Axle Steering Angle [°]');
plot(time,(a(:,5)-mean(a(1:100,5)))*1,'b');% check cal values
title('Steering Calibration')
ylabel('Roll');
grid on
title('Carl Test')
zoom on
grid on
subplot(2,1,1);
zoom on
plot(time,(a(:,1)-mean(a(1:100,1)))*1,'b');% check cal values
subplot(2,1,2);
ylabel('Front Str Ang');
plot(time,(a(:,6)-mean(a(1:100,6)))*1,'b');
title('Carl Test')
ylabel('Yaw');
grid on
xlabel('Time [sec]');
zoom on
grid on
subplot(2,1,2);
zoom on
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 108 of 138 pages
figure;
caldata(:,1)=caldata1(:,1)-mean(caldata1(1:100,1))*1;
yaw=(a(:,6)-mean(a(1:100,6)))*1;
caldata(:,2)=caldata1(:,2)-mean(caldata1(1:100,2))*1;
latacc=(a(:,4)-mean(a(1:100,4)))*1;
caldata(:,3)=(a(:,3)-mean(a(1:100,3)))*14.54;
roll=(a(:,5)-mean(a(1:100,5)))*1;
caldata(:,4)=(a(:,4)-mean(a(1:100,4)))*1;
plot(time,clgfilt(time,yaw,0.07,'b'),'b');
%Filter data
title('Constant Radius Test Counter Clock Wise')
caldata(:,1)=clgfilt(time,caldata(:,1),0.07,'b');
caldata(:,2)=clgfilt(time,caldata(:,2),0.07,'b');
xlabel('Time [s]');
figure;
grid on
subplot(2,1,1);
zoom on
[AX,H1,H2] = plotyy(time,caldata(:,1),time,caldata(:,3),'plot');
figure
set(get(AX(1),'Ylabel'),'String','First Axle Steering Angle [°]')
plot(time,clgfilt(time,latacc,0.07,'b'),'b');
set(get(AX(2),'Ylabel'),'String','Vehicle Speed [km/h]')
title('Constant Radius Test Counter Clock Wise')
title('Constant Radius Test Counter Clock Wise')
ylabel('Lateral Acceleration');
grid on
xlabel('Time [sec]');
zoom on
grid on
subplot(2,1,2);
zoom on
[AX,H1,H2] = plotyy(time,caldata(:,2),time,caldata(:,3),'plot');
figure
set(get(AX(1),'Ylabel'),'String','Second Axle Steering Angle [°]')
plot(time,clgfilt(time,roll,0.07,'b'),'b');% check cal values
set(get(AX(2),'Ylabel'),'String','Vehicle Speed [km/h]')
title('Constant Radius Test Counter Clock Wise')
xlabel('Time [s]');
grid on
title('Constant Radius Test Counter Clock Wise')
zoom on
grid on
pack;
zoom on
figure
figure;
plot(time,caldata(:,1),time,caldata(:,2))
%Kalibreer Data
title('Constant Radius Test Counter Clock Wise');
%gem hoek van die linker en regter wiel word gebruik om die
as se hoek te gee
ylabel('Steering Angle [°]');
legend('First Axle','Second Axle');
%1e as: lk 18° -16° rk 12° -20° avg -18° 15°
grid on
%2e as: lk 9° -12° rk 10° -7° avg -9.5° 9.5°
zoom on
q=polyfit([-0.355 0.5],[15 -18],1);
pack;
s=polyfit([0.204 -0.206],[9.5 -9.5],1);
caldata1(:,1)=q(1,2)+q(1,1)*(a(:,1));
caldata1(:,2)=s(1,2)+s(1,1)*(a(:,2));
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 109 of 138 pages
Appendix C
Constant radius test – Matlab M file
clear;
close all;
%*****************************************************
%18 June 2003
%Updated 21 June 2004 - Diameter 84 meters
%Update 28 June 2004
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 110 of 138 pages
%**************VEHICLE
PARAMETERS***************************************
g=9.81;
ratio=0.8;
M=14210;%kg
w=2.325;%width
CyR1=2*Cy1(1,1);
a=1.937;%DISTANCE FIRST AXLE TO CENTRE OF MASS
b=0.169;%DISTANCE CENTRE OF MASS TO SECOND
AXLE
c=2.550;%DISTANCE SECOND AXLE TO THIRD AXLE
t=2.080;
%********************************TIRE
DATA****************************
x2370_600=[0 2 4 6 8 10]/57.3;
y2370_600=[-2000 6000 12000 15000 17000 18000];
x2370_400=[0 2 4 6 8 10]/57.3;
y2370_400=[-2000 7000 13500 16500 17500 19000];
x4020_600=[0 2 4 6 8 10]/57.3;
y4020_600=[-4000 5050 15000 21000 24000 25500];
x4020_400=[0 2 4 6 8 10]/57.3;
y4020_400=[-4000 5000 16000 22500 25000 27000];
%********************************ZERO TIRE
DATA****************************
offset1=y2370_600(1,1);
offset2=y2370_400(1,1);
offset3=y4020_600(1,1);
offset4=y4020_400(1,1);
for i = 1:6,
y2370_600(i)=y2370_600(i)-offset1;
y2370_400(i)=y2370_400(i)-offset2;
y4020_600(i)=y4020_600(i)-offset3;
y4020_400(i)=y4020_400(i)-offset4;
end;
%********************************FIT
PLOT****************************
a2370_600=polyfit(x2370_600,y2370_600,2);
a2370_400=polyfit(x2370_400,y2370_400,2);
a4020_600=polyfit(x4020_600,y4020_600,2);
a4020_400=polyfit(x4020_400,y4020_400,2);
R=15;
v=5/3.6;
A(1,1)=0;
A(1,2)=0;
A(1,3)=CyF1;
A(1,4)=CyF2;
A(1,5)=CyR1;
A(2,1)=0;
A(2,2)=0;
A(2,3)=a*CyF1;
A(2,4)=-b*CyF2;
A(2,5)=-1*(b+c)*CyR1;
A(3,1)=1;
A(3,2)=0;
A(3,3)=-1;
A(3,4)=0;
A(3,5)=1;
A(4,1)=0;
A(4,2)=1;
A(4,3)=0;
A(4,4)=-1;
A(4,5)=1;
A(5,2)=-1;
A(5,3)=0;
A(5,4)=0;
A(5,5)=0;
B(2,1)=0;
B(5,1)=0;
%===============================generate matrix for
plotting===============================
for i = 1:6,
y1(i,1)=(a2370_600(1,1)*(x2370_600(1,i))^2+a2370_600(1,2)*
x2370_600(1,i))+a2370_600(1,3);
y2(i,1)=(a2370_400(1,1)*(x2370_400(1,i))^2+a2370_400(1,2)*
x2370_400(1,i))+a2370_400(1,3);
y3(i,1)=(a4020_600(1,1)*(x4020_600(1,i))^2+a4020_600(1,2)*
x4020_600(1,i))+a4020_600(1,3);
y4(i,1)=(a4020_400(1,1)*(x4020_400(1,i))^2+a4020_400(1,2)*
x4020_400(1,i))+a4020_400(1,3);
%*********CORNERING
STIFFNESS*************************************************
Cy1(i,1)=(2*a2370_600(1,1)*(x2370_600(1,i))+a2370_600(1,2)
);
Cy2(i,1)=(2*a2370_400(1,1)*(x2370_400(1,i))+a2370_400(1,2)
);
Cy3(i,1)=(2*a4020_600(1,1)*(x4020_600(1,i))+a4020_600(1,2)
);
Cy3(i,1)=(2*a4020_400(1,1)*(x4020_400(1,i))+a4020_400(1,2)
);
end;
%*********use 2370 kg @ 4 bar for CORNERING
STIFFNESS***************************
CyF1=2*Cy1(1,1);
CyF2=2*Cy1(1,1);
maxR=10;
R=10;
v=0.5;%m/s
ry=0;
ratio=0.2;
%Parameters to vary R,v,ratio
for countRatio = 1:10,
ratio=ratio+0.1;
R=10;
for countR = 1:10,
R=R+5;%meter
v=(30/3.6)-2.7778;
for countv = 1:9,
ry=ry+1;
v=v+2.7778;%m/s
A(5,1)=ratio;
B(1,1)=M*(v)^2/(R)+0;
B(4,1)=c/R;
B(3,1)=(a+b+c)/R;
C=inv(A);
Slip=C*B*57.3;
Antw(ry,1)=fix(v*3.6);%speed
Antw(ry,3)=ratio;
Antw(ry,4)=Slip(3);%Alfa1st
Antw(ry,5)=Slip(4);%Alfa2nd
Antw(ry,6)=Slip(5);%Alfa3rd
Antw(ry,7)=Slip(1);%Delta1
Antw(ry,8)=Slip(2);%Delta2
end;
end,
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 111 of 138 pages
end,
groot=size(Antw);
spoed=20;
tel=0;
for j = 1 : 8,
spoed=spoed+10;
tel=0;
for i = 1:groot(1,1),
if Antw(i,1)==spoed
tel=tel+1;
ntw(i,3);'])%ratio
ntw(i,4);'])%Alfa1st
ntw(i,5);'])%Alfa2nd
ntw(i,6);'])%Alfa3rd
ntw(i,7);'])%Delta1
ntw(i,8);'])%Delta2
ntw(i,1);'])%speed
qrt((1/3)*(Antw(i,4)^2+Antw(i,5)^2+Antw(i,6)^2));'])%gem slip
angle
sqrt((1/2)*(Antw(i,5)^2+Antw(i,6)^2));'])%gem stuur hoek
end,
end,
end;
end;
spoed=20;
tel=0;
for j = 1 : 8,
spoed=spoed+10;
tel=0;
for i = 1:groot(1,1),
if Antw(i,1)==spoed
tel=tel+1;
ntw(i,3);'])
ntw(i,4);'])
ntw(i,5);'])
ntw(i,8);'])
ntw(i,1);'])
ntw(i,2);'])
qrt((1/3)*(Antw(i,4)^2+Antw(i,5)^2+Antw(i,6)^2));'])%gem slip
sqrt((1/2)*(Antw(i,5)^2+Antw(i,6)^2));'])%gem stuur hoek
end,
end,
end;
end;
spoed=20;
tel=0;
for j = 1 : 8,
spoed=spoed+10;
tel=0;
for i = 1:groot(1,1),
if Antw(i,1)==spoed
tel=tel+1;
ntw(i,3);'])
ntw(i,4);'])
ntw(i,5);'])
ntw(i,6);'])
ntw(i,7);'])
ntw(i,8);'])
ntw(i,1);'])
ntw(i,2);'])
qrt((1/3)*(Antw(i,4)^2+Antw(i,5)^2+Antw(i,6)^2));'])%gem
sqrt((1/2)*(Antw(i,5)^2+Antw(i,6)^2));'])%gem stuur hoek
end,
end,
end;
end;
%\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$\$\$\$\$\$\$\$
ntw(i,6);'])
ntw(i,7);'])
%\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$\$\$\$\$\$\$\$
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 112 of 138 pages
%************ plot radius 20 rms slip
angles*******************************************
figure;
subplot(3,3,1);
spoed=30;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Slip Angle [°]');
axis([0.3 1.1 0 10]);
grid;
subplot(3,3,2);
spoed=40;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Slip Angle [°]')
axis([0.3 1.1 0 10]);;
grid;
subplot(3,3,3);
spoed=60;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Slip Angle [°]')
axis([0.3 1.1 0 10]);
grid;
subplot(3,3,4);
spoed=30;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Slip Angle [°]')
axis([0.3 1.1 0 6]);
grid;
subplot(3,3,5);
spoed=40;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Slip Angle [°]')
axis([0.3 1.1 0 6]);
grid;
subplot(3,3,6);
spoed=60;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Slip Angle [°]')
axis([0.3 1.1 0 6]);
grid;
subplot(3,3,7);
spoed=30;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Slip Angle [°]')
axis([0.3 1.1 0 6]);
grid;
subplot(3,3,8);
spoed=40;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Slip Angle [°]')
axis([0.3 1.1 0 6]);
grid;
subplot(3,3,9);
spoed=60;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Slip Angle [°]')
axis([0.3 1.1 0 6]);
grid;
%************ plot radius 20 rms steering
angles*******************************************
figure;
subplot(3,3,1);
spoed=30;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Steering Angle [°]');
axis([0.3 1.1 0 12]);
grid;
subplot(3,3,2);
spoed=40;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Steering Angle [°]')
axis([0.3 1.1 0 12]);
grid;
subplot(3,3,3);
spoed=60;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Steering Angle [°]')
axis([0.3 1.1 0 12]);
grid;
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 113 of 138 pages
subplot(3,3,4);
spoed=30;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Steering Angle [°]')
axis([0.3 1.1 0 12]);
grid;
subplot(3,3,5);
spoed=40;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Steering Angle [°]')
axis([0.3 1.1 0 12]);
grid;
subplot(3,3,6);
spoed=60;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Steering Angle [°]')
axis([0.3 1.1 0 12]);
grid;
subplot(3,3,7);
spoed=30;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Steering Angle [°]')
axis([0.3 1.1 0 12]);
grid;
subplot(3,3,8);
spoed=40;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Steering Angle [°]')
axis([0.3 1.1 0 12]);
grid;
subplot(3,3,9);
spoed=60;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Steering Angle [°]')
axis([0.3 1.1 0 12]);
grid;
hold off;
%%************ plot radius 20 rms slip angle and
steering*******************************************
figure;
subplot(3,3,1);
spoed=30;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Angle [°]');
legend('Slip Angle','Steering Angle')
%axis([0.3 1.1 0 200]);
grid;
subplot(3,3,2);
spoed=40;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Angle [°]')
legend('Slip Angle','Steering Angle')
%axis([0.3 1.1 0 200]);
grid;
subplot(3,3,3);
spoed=60;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Angle [°]')
legend('Slip Angle','Steering Angle')
%axis([0.3 1.1 0 200]);
grid;
subplot(3,3,4);
spoed=30;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Angle [°]')
legend('Slip Angle','Steering Angle')
%axis([0.3 1.1 0 200]);
grid;
subplot(3,3,5);
spoed=40;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Angle [°]')
legend('Slip Angle','Steering Angle')
%axis([0.3 1.1 0 200]);
grid;
subplot(3,3,6);
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 114 of 138 pages
spoed=60;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Angle [°]')
legend('Slip Angle','Steering Angle')
%axis([0.3 1.1 0 200]);
grid;
subplot(3,3,7);
spoed=30;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Angle [°]');
legend('Slip Angle','Steering Angle')
%axis([0.3 1.1 0 200]);
grid;
subplot(3,3,8);
spoed=40;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
xlabel('Steering Ratio');
ylabel('RMS Angle [°]')
legend('Slip Angle','Steering Angle')
%axis([0.3 1.1 0 200]);
grid;
subplot(3,3,9);
spoed=60;
hold on;
meters, Speed ',num2str(spoed),' km/h'')'])
legend('Slip Angle','Steering Angle')
xlabel('Steering Ratio');
ylabel('RMS Angle [°]')
%axis([0.3 1.1 0 200]);
grid;
%*********************************************************************
**
%\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$\$\$\$\$\$\$\$
%*********************************************************************
**
*******************************************
figure;
subplot(2,2,1);
spoed=30;
hold on;
axis([0.3 1.1 0 20]);
xlabel('Ratio');
ylabel('Angle [°]');
grid;
legend('Alfa 1st Axle','Alfa 2nd Axle','Alfa 3rd Axle','Steer 1st
Axle','Steer 2nd Axle',-1);
hold off;
%*********************************************************************
**
subplot(2,2,2);
spoed=40;
hold on;
axis([0.3 1.1 0 20]);
xlabel('Ratio');
ylabel('Angle [°]');
grid;
legend('Alfa 1st Axle','Alfa 2nd Axle','Alfa 3rd Axle','Steer 1st
Axle','Steer 2nd Axle',-1);
hold off;
%*********************************************************************
**
subplot(2,2,3);
spoed=50;
hold on;
axis([0.3 1.1 0 20]);
xlabel('Ratio');
ylabel('Angle [°]');
grid;
legend('Alfa 1st Axle','Alfa 2nd Axle','Alfa 3rd Axle','Steer 1st
Axle','Steer 2nd Axle',-1);
hold off;
%*********************************************************************
**
subplot(2,2,4);
spoed=60;
hold on;
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 115 of 138 pages
axis([0.3 1.1 0 20]);
xlabel('Ratio');
ylabel('Angle [°]');
grid;
legend('Alfa 1st Axle','Alfa 2nd Axle','Alfa 3rd Axle','Steer 1st
Axle','Steer 2nd Axle',-1);
hold off;
%*********************************************************************
**
*******************************************
figure;
subplot(2,2,1);
spoed=30;
hold on;
axis([0.3 1.1 0 10]);
xlabel('Ratio');
ylabel('Angle [°]');
grid;
legend('Alfa 1st Axle','Alfa 2nd Axle','Alfa 3rd Axle','Steer 1st
Axle','Steer 2nd Axle',-1);
hold off;
%*********************************************************************
**
subplot(2,2,2);
spoed=40;
hold on;
axis([0.3 1.1 0 10]);
xlabel('Ratio');
ylabel('Angle [°]');
grid;
legend('Alfa 1st Axle','Alfa 2nd Axle','Alfa 3rd Axle','Steer 1st
Axle','Steer 2nd Axle',-1);
hold off;
%*********************************************************************
**
subplot(2,2,3);
spoed=50;
hold on;
axis([0.3 1.1 0 10]);
xlabel('Ratio');
ylabel('Angle [°]');
grid;
legend('Alfa 1st Axle','Alfa 2nd Axle','Alfa 3rd Axle','Steer 1st
Axle','Steer 2nd Axle',-1);
hold off;
%*********************************************************************
**
subplot(2,2,4);
spoed=60;
hold on;
axis([0.3 1.1 0 10]);
xlabel('Ratio');
ylabel('Angle [°]');
grid;
legend('Alfa 1st Axle','Alfa 2nd Axle','Alfa 3rd Axle','Steer 1st
Axle','Steer 2nd Axle',-1);
hold off;
%*********************************************************************
**
*******************************************
figure;
subplot(2,2,1);
spoed=30;
hold on;
axis([0.3 1.1 0 7]);
xlabel('Ratio');
ylabel('Angle [°]');
grid;
legend('Alfa 1st Axle','Alfa 2nd Axle','Alfa 3rd Axle','Steer 1st
Axle','Steer 2nd Axle',-1);
hold off;
%*********************************************************************
**
subplot(2,2,2);
spoed=40;
hold on;
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 116 of 138 pages
axis([0.3 1.1 0 7]);
xlabel('Ratio');
ylabel('Angle [°]');
grid;
legend('Alfa 1st Axle','Alfa 2nd Axle','Alfa 3rd Axle','Steer 1st
Axle','Steer 2nd Axle',-1);
hold off;
%*********************************************************************
**
subplot(2,2,3);
spoed=50;
hold on;
axis([0.3 1.1 0 7]);
xlabel('Ratio');
ylabel('Angle [°]');
grid;
legend('Alfa 1st Axle','Alfa 2nd Axle','Alfa 3rd Axle','Steer 1st
Axle','Steer 2nd Axle',-1);
hold off;
%*********************************************************************
**
subplot(2,2,4);
spoed=60;
hold on;
axis([0.3 1.1 0 7]);
xlabel('Ratio');
ylabel('Angle [°]');
grid;
legend('Alfa 1st Axle','Alfa 2nd Axle','Alfa 3rd Axle','Steer 1st
Axle','Steer 2nd Axle',-1);
hold off;
%################################################
##################################################
#########################################
%plot ratio vs speed for a specifc slip angle and turning radius
%################################################
##################################################
#########################################
20##########################
s=20;
a=0;
b=0;
c=0;
d=0;
e=0;
f=0;
g=0;
h=0;
j=0;
for w = 1:8
s=s+10;
for i=1:10
eval(['temp=data_R_20_v_',num2str(s),'(i,1);']);
if temp <= 0.32
a=a+1;
7) data_R_20_v_',num2str(s),'(i,2)
data_R_20_v_',num2str(s),'(i,3)
data_R_20_v_',num2str(s),'(i,4)
data_R_20_v_',num2str(s),'(i,5)
data_R_20_v_',num2str(s),'(i,6)
data_R_20_v_',num2str(s),'(i,1)];']);
end;
if temp <= 0.42
if temp >= 0.32
b=b+1;
7) data_R_20_v_',num2str(s),'(i,2)
data_R_20_v_',num2str(s),'(i,3)
data_R_20_v_',num2str(s),'(i,4)
data_R_20_v_',num2str(s),'(i,5)
data_R_20_v_',num2str(s),'(i,6)
data_R_20_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 0.52
if temp >= 0.42
c=c+1;
7) data_R_20_v_',num2str(s),'(i,2)
data_R_20_v_',num2str(s),'(i,3)
data_R_20_v_',num2str(s),'(i,4)
data_R_20_v_',num2str(s),'(i,5)
data_R_20_v_',num2str(s),'(i,6)
data_R_20_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 0.62
if temp >= 0.52
d=d+1;
7) data_R_20_v_',num2str(s),'(i,2)
data_R_20_v_',num2str(s),'(i,3)
data_R_20_v_',num2str(s),'(i,4)
data_R_20_v_',num2str(s),'(i,5)
data_R_20_v_',num2str(s),'(i,6)
data_R_20_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 0.72
if temp >= 0.62
e=e+1;
7) data_R_20_v_',num2str(s),'(i,2)
data_R_20_v_',num2str(s),'(i,3)
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 117 of 138 pages
data_R_20_v_',num2str(s),'(i,4)
data_R_20_v_',num2str(s),'(i,5)
data_R_20_v_',num2str(s),'(i,6)
data_R_20_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 0.82
if temp >= 0.72
f=f+1;
7) data_R_20_v_',num2str(s),'(i,2)
data_R_20_v_',num2str(s),'(i,3)
data_R_20_v_',num2str(s),'(i,4)
data_R_20_v_',num2str(s),'(i,5)
data_R_20_v_',num2str(s),'(i,6)
data_R_20_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 0.92
if temp >= 0.82
g=g+1;
7) data_R_20_v_',num2str(s),'(i,2)
data_R_20_v_',num2str(s),'(i,3)
data_R_20_v_',num2str(s),'(i,4)
data_R_20_v_',num2str(s),'(i,5)
data_R_20_v_',num2str(s),'(i,6)
data_R_20_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 1.02
if temp >= 0.92
h=h+1;
7) data_R_20_v_',num2str(s),'(i,2)
data_R_20_v_',num2str(s),'(i,3)
data_R_20_v_',num2str(s),'(i,4)
data_R_20_v_',num2str(s),'(i,5)
data_R_20_v_',num2str(s),'(i,6)
data_R_20_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 1.12
if temp >= 1.02
j=j+1;
7) data_R_20_v_',num2str(s),'(i,2)
data_R_20_v_',num2str(s),'(i,3)
data_R_20_v_',num2str(s),'(i,4)
data_R_20_v_',num2str(s),'(i,5)
data_R_20_v_',num2str(s),'(i,6)
data_R_20_v_',num2str(s),'(i,1)];']);
end;
end;
end;
end;
40##########################
s=20;
a=0;
b=0;
c=0;
d=0;
e=0;
f=0;
g=0;
h=0;
j=0;
for w = 1:8
s=s+10;
for i=1:10
eval(['temp=data_R_40_v_',num2str(s),'(i,1);']);
if temp <= 0.32
a=a+1;
7) data_R_40_v_',num2str(s),'(i,2)
data_R_40_v_',num2str(s),'(i,3)
data_R_40_v_',num2str(s),'(i,4)
data_R_40_v_',num2str(s),'(i,5)
data_R_40_v_',num2str(s),'(i,6)
data_R_40_v_',num2str(s),'(i,1)];']);
end;
if temp <= 0.42
if temp >= 0.32
b=b+1;
7) data_R_40_v_',num2str(s),'(i,2)
data_R_40_v_',num2str(s),'(i,3)
data_R_40_v_',num2str(s),'(i,4)
data_R_40_v_',num2str(s),'(i,5)
data_R_40_v_',num2str(s),'(i,6)
data_R_40_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 0.52
if temp >= 0.42
c=c+1;
7) data_R_40_v_',num2str(s),'(i,2)
data_R_40_v_',num2str(s),'(i,3)
data_R_40_v_',num2str(s),'(i,4)
data_R_40_v_',num2str(s),'(i,5)
data_R_40_v_',num2str(s),'(i,6)
data_R_40_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 0.62
if temp >= 0.52
d=d+1;
7) data_R_40_v_',num2str(s),'(i,2)
data_R_40_v_',num2str(s),'(i,3)
data_R_40_v_',num2str(s),'(i,4)
data_R_40_v_',num2str(s),'(i,5)
data_R_20_v_',num2str(s),'(i,6)
data_R_40_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 0.72
if temp >= 0.62
e=e+1;
7) data_R_40_v_',num2str(s),'(i,2)
data_R_40_v_',num2str(s),'(i,3)
data_R_40_v_',num2str(s),'(i,4)
data_R_40_v_',num2str(s),'(i,5)
data_R_40_v_',num2str(s),'(i,6)
data_R_40_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 0.82
if temp >= 0.72
f=f+1;
7) data_R_40_v_',num2str(s),'(i,2)
data_R_40_v_',num2str(s),'(i,3)
data_R_40_v_',num2str(s),'(i,4)
data_R_40_v_',num2str(s),'(i,5)
data_R_40_v_',num2str(s),'(i,6)
data_R_40_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 0.92
if temp >= 0.82
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 118 of 138 pages
g=g+1;
7) data_R_40_v_',num2str(s),'(i,2)
data_R_40_v_',num2str(s),'(i,3)
data_R_40_v_',num2str(s),'(i,4)
data_R_40_v_',num2str(s),'(i,5)
data_R_40_v_',num2str(s),'(i,6)
data_R_40_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 1.02
if temp >= 0.92
h=h+1;
7) data_R_40_v_',num2str(s),'(i,2)
data_R_40_v_',num2str(s),'(i,3)
data_R_40_v_',num2str(s),'(i,4)
data_R_40_v_',num2str(s),'(i,5)
data_R_40_v_',num2str(s),'(i,6)
data_R_40_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 1.12
if temp >= 1.02
j=j+1;
7) data_R_40_v_',num2str(s),'(i,2)
data_R_40_v_',num2str(s),'(i,3)
data_R_40_v_',num2str(s),'(i,4)
data_R_40_v_',num2str(s),'(i,5)
data_R_40_v_',num2str(s),'(i,6)
data_R_40_v_',num2str(s),'(i,1)];']);
end;
end;
end;
end;
60##########################
s=20;
a=0;
b=0;
c=0;
d=0;
e=0;
f=0;
g=0;
h=0;
j=0;
for w = 1:8
s=s+10;
for i=1:10
eval(['temp=data_R_60_v_',num2str(s),'(i,1);']);
if temp <= 0.32
a=a+1;
7) data_R_60_v_',num2str(s),'(i,2)
data_R_60_v_',num2str(s),'(i,3)
data_R_60_v_',num2str(s),'(i,4)
data_R_60_v_',num2str(s),'(i,5)
data_R_60_v_',num2str(s),'(i,6)
data_R_60_v_',num2str(s),'(i,1)];']);
end;
if temp <= 0.42
if temp >= 0.32
b=b+1;
7) data_R_60_v_',num2str(s),'(i,2)
data_R_60_v_',num2str(s),'(i,3)
data_R_60_v_',num2str(s),'(i,4)
data_R_60_v_',num2str(s),'(i,5)
data_R_60_v_',num2str(s),'(i,6)
data_R_60_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 0.52
if temp >= 0.42
c=c+1;
7) data_R_60_v_',num2str(s),'(i,2)
data_R_60_v_',num2str(s),'(i,3)
data_R_60_v_',num2str(s),'(i,4)
data_R_60_v_',num2str(s),'(i,5)
data_R_60_v_',num2str(s),'(i,6)
data_R_60_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 0.62
if temp >= 0.52
d=d+1;
7) data_R_60_v_',num2str(s),'(i,2)
data_R_60_v_',num2str(s),'(i,3)
data_R_60_v_',num2str(s),'(i,4)
data_R_60_v_',num2str(s),'(i,5)
data_R_60_v_',num2str(s),'(i,6)
data_R_60_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 0.72
if temp >= 0.62
e=e+1;
7) data_R_60_v_',num2str(s),'(i,2)
data_R_60_v_',num2str(s),'(i,3)
data_R_60_v_',num2str(s),'(i,4)
data_R_60_v_',num2str(s),'(i,5)
data_R_60_v_',num2str(s),'(i,6)
data_R_60_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 0.82
if temp >= 0.72
f=f+1;
7) data_R_60_v_',num2str(s),'(i,2)
data_R_60_v_',num2str(s),'(i,3)
data_R_60_v_',num2str(s),'(i,4)
data_R_60_v_',num2str(s),'(i,5)
data_R_60_v_',num2str(s),'(i,6)
data_R_60_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 0.92
if temp >= 0.82
g=g+1;
7) data_R_60_v_',num2str(s),'(i,2)
data_R_60_v_',num2str(s),'(i,3)
data_R_60_v_',num2str(s),'(i,4)
data_R_60_v_',num2str(s),'(i,5)
data_R_60_v_',num2str(s),'(i,6)
data_R_60_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 1.02
if temp >= 0.92
h=h+1;
7) data_R_60_v_',num2str(s),'(i,2)
data_R_60_v_',num2str(s),'(i,3)
data_R_60_v_',num2str(s),'(i,4)
data_R_60_v_',num2str(s),'(i,5)
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 119 of 138 pages
data_R_60_v_',num2str(s),'(i,6)
data_R_60_v_',num2str(s),'(i,1)];']);
end;
end;
if temp <= 1.12
if temp >= 1.02
j=j+1;
7) data_R_60_v_',num2str(s),'(i,2)
data_R_60_v_',num2str(s),'(i,3)
data_R_60_v_',num2str(s),'(i,4)
data_R_60_v_',num2str(s),'(i,5)
data_R_60_v_',num2str(s),'(i,6)
data_R_60_v_',num2str(s),'(i,1)];']);
end;
end;
end;
end;
figure
subplot(3,1,1);
hold on;
legend('First Axle','Second Axle','Third Axle',-1);
xlabel('Vehicle Speed [km/h]');
ylabel('Slip Angle [°]');
grid;
axis([30 60 0 15]);
subplot(3,1,2);
hold on;
legend('First Axle','Second Axle','Third Axle',-1);
xlabel('Vehicle Speed [km/h]');
grid;
axis([30 60 0 15]);
ylabel('Slip Angle [°]');subplot(3,1,3); hold on;
legend('First Axle','Second Axle','Third Axle',-1);
xlabel('Vehicle Speed [km/h]');
ylabel('Slip Angle [°]');
grid;
axis([30 60 0 15]);
figure;
hold;
xlabel('Speed [km/h]');
ylabel('Steering Angle [°]');
legend('R0.7, 1st Steering Axle','R0.7, 2nd Steering
Axle','R0.5, 1st Steering Axle','R0.5, 2nd Steering Axle','R1,
1st Steering Axle','R1, 2nd Steering Axle',-1);
%axis([30 60 0 18]);
grid;
figure;
hold;
xlabel('Speed [km/h]');
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 120 of 138 pages
ylabel('Steering Angle [°]');
legend('R0.7, 1st Steering Axle','R0.7, 2nd Steering
Axle','R0.5, 1st Steering Axle','R0.5, 2nd Steering Axle','R1,
1st Steering Axle','R1, 2nd Steering Axle',-1);
%axis([30 60 0 18]);
grid;
figure;
hold;
xlabel('Speed [km/h]');
ylabel('Steering Angle [°]');
legend('R0.7, 1st Steering Axle','R0.7, 2nd Steering
Axle','R0.5, 1st Steering Axle','R0.5, 2nd Steering Axle','R1,
1st Steering Axle','R1, 2nd Steering Axle',-1);
%axis([30 60 0 18]);
grid;
%################################# Plot Steering vs
Lateral Acceleration
10(q,5),'b+-');
end;
xlabel('Lateral Acceleration [g]');
ylabel('1st Axle Steering Angle [°]');
legend('Ratio 0.7','Ratio 0.5','Ratio 1',-1);
grid;
figure;
subplot(3,1,1);
hold;
v=30/3.6;
for q = 1:8
07(q,5),'r+-');
05(q,5),'g+-');
10(q,5),'b+-');
v=v+(10/3.6);
end;
xlabel('Lateral Acceleration [g]');
ylabel('1st Axle Steering Angle [°]');
legend('Ratio 0.7','Ratio 0.5','Ratio 1',-1);
grid;
subplot(3,1,2);
hold;
for q = 1:8
07(q,5),'r+-');
05(q,5),'g+-');
10(q,5),'b+-');
end;
xlabel('Lateral Acceleration [g]');
ylabel('1st Axle Steering Angle [°]');
legend('Ratio 0.7','Ratio 0.5','Ratio 1',-1);
grid;
subplot(3,1,3);
hold;
for q = 1:8
07(q,5),'r+-');
05(q,5),'g+-');
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 121 of 138 pages
Appendix D
Lane change – Matlab M file
%*********************************
%*** modified 29 Sept 2003*********
%****29 Sept 2004*******************
%non-linear tire model
% 90 km/h
%voorkyk=26;%meter
%tydregs=20/V;
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 122 of 138 pages
a4020_400=polyfit(x4020_400,y4020_400,2);
% 80 km/h
%voorkyk=24;%meter
%tydregs=20/V;
for i = 1:6,
y1(i,1)=(a2370_600(1,1)*(x2370_600(1,i))^2+a2370_600(1,2)*
x2370_600(1,i))+a2370_600(1,3);
y2(i,1)=(a2370_400(1,1)*(x2370_400(1,i))^2+a2370_400(1,2)*
x2370_400(1,i))+a2370_400(1,3);
% 70 km/h
%voorkyk=22;%meter
%tydregs=20/V;
% 60 km/h
%voorkyk=20;%meter
%tydregs=20/V;
y3(i,1)=(a4020_600(1,1)*(x4020_600(1,i))^2+a4020_600(1,2)*
x4020_600(1,i))+a4020_600(1,3);
y4(i,1)=(a4020_400(1,1)*(x4020_400(1,i))^2+a4020_400(1,2)*
x4020_400(1,i))+a4020_400(1,3);
%*********CORNERING
STIFFNESS*************************************************
Cy1(i,1)=(2*a2370_600(1,1)*(x2370_600(1,i))+a2370_600(1,2)
);
% 50 km/h
%voorkyk=17;%meter
%tydregs=20/V;
Cy2(i,1)=(2*a2370_400(1,1)*(x2370_400(1,i))+a2370_400(1,2)
);
% 40 km/h
%voorkyk=15;%meter
%tydregs=20/V;
Cy3(i,1)=(2*a4020_400(1,1)*(x4020_400(1,i))+a4020_400(1,2)
);
% 30 km/h
%voorkyk=10;%meter
%tydregs=20/V;
%*********************************
%cd c:\m\m_leers\lane_change;
close all;
clear;
delete dataleer.mat;
stopp=1;
%********************************TIRE
DATA****************************
x2370_600=[0 2 4 6 8 10]/57.3;
y2370_600=[-2000 6000 12000 15000 17000 18000];
x2370_400=[0 2 4 6 8 10]/57.3;
y2370_400=[-2000 7000 13500 16500 17500 19000];
x4020_600=[0 2 4 6 8 10]/57.3;
y4020_600=[-4000 5050 15000 21000 24000 25500];
x4020_400=[0 2 4 6 8 10]/57.3;
y4020_400=[-4000 5000 16000 22500 25000 27000];
%********************************ZERO TIRE
DATA****************************
offset1=y2370_600(1,1);
offset2=y2370_400(1,1);
offset3=y4020_600(1,1);
offset4=y4020_400(1,1);
for i = 1:6,
y2370_600(i)=y2370_600(i)-offset1;
y2370_400(i)=y2370_400(i)-offset2;
y4020_600(i)=y4020_600(i)-offset3;
y4020_400(i)=y4020_400(i)-offset4;
end;
%********************************FIT
PLOT****************************
a2370_600=polyfit(x2370_600,y2370_600,2);
a2370_400=polyfit(x2370_400,y2370_400,2);
a4020_600=polyfit(x4020_600,y4020_600,2);
Cy3(i,1)=(2*a4020_600(1,1)*(x4020_600(1,i))+a4020_600(1,2)
);
end;
%*********use 2370 kg @ 6 bar for CORNERING
STIFFNESS***************************
CyF1=2*Cy1(1,1);
CyF2=2*Cy1(1,1);
CyR1=2*Cy1(1,1);
%################################################
##################################################
###########
%********************************BISON VEHICLE
DATA****************************
a=1.707;%DISTANCE FIRST AXLE TO CENTRE OF MASS
b=2.106-a;%DISTANCE CENTRE OF MASS TO SECOND
AXLE
c=2.550;%DISTANCE SECOND AXLE TO THIRD AXLE
g=9.81;
t=2.080;
m=14870;%N
w=2.325;%width
I=(1/12)*(m/9.81)*(w^2+4.566^2)*(0.75);%kgm^2
%Enkel baan
baanlengte=75;%meter
ratio=0.2;
for runnr = 1:9
ratio=ratio+0.1;
Vkm_h=20;%km/h;
for sppoed = 1:7
Vkm_h=Vkm_h+10;%km/h
V=Vkm_h/3.6;%m/s
fintyd=75/V;
%&&&&&&&&&&&&&&&&&&&&&&&&&&&&
if Vkm_h == 90
voorkyk=17;%meter
Constant2 = 0.47;
tydregs=19/V;
trapinset=5.43;
begintrap=0;
end;
if Vkm_h == 80
voorkyk=15;%meter
Constant2 = 0.5;
tydregs=23/V;
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 123 of 138 pages
trapinset=5.43+2;
begintrap=0;
end;
if Vkm_h == 70
voorkyk=16;%meter
Constant2 = 0.5;
tydregs=35/V;
trapinset=5.43;
begintrap=0;
end;
if Vkm_h == 60
voorkyk=16;%meter
Constant2 = 0.5;
tydregs=30/V;
trapinset=5.43+0.5;
begintrap=0;
end;
if Vkm_h == 50
voorkyk=13;%meter
Constant2 = 0.5;
tydregs=23/V;
trapinset=5.43;
begintrap=0;
end;
if Vkm_h == 40
voorkyk=14;%meter
Constant2 = 0.5;
tydregs=23/V;
trapinset=5.43;
begintrap=0;
end;
if Vkm_h == 30
voorkyk=13;%meter
Constant2 = 0.5;
tydregs=25/V;
trapinset=5.43;
begintrap=0;
end;
%&&&&&&&&&&&&&&&&&&&&&&&&&&&&
n=round(fintyd/0.1);% aantal_stappe
x=ratio;
A11=-2*(CyF1+CyF2+CyR1)/(m*V);
A12=-(V+2*(CyF1*a-CyF2*b-CyR1*(b+c))/(m*V));
A21=-2*(CyF1*(a)-CyF2*b-CyR1*(b+c))/(I*V);
A22=-2*(((a)^2)*CyF1+(b^2)*CyF2+((c+b)^2)*CyR1)/(I*V);
A=[A11
A12
A21 A22];
B=[2*(CyF1+CyF2*x)/m
2*CyF1*(a)/I-2*CyF2*b*x/I];
voortyd=voorkyk/V;
voorkyk_tel=round(voortyd/0.1);
tbaan(1)=0;
for i = 2:n,
tbaan(i)=tbaan(i-1)+0.1;
end;
tydbaan=tbaan';
BAAN1=[tydbaan(1:n)];
stop=0;
for i = 1:n,
if tydbaan(i)>tydregs
BAAN2(i,1)=trapinset;
if stop==0
tydregs_tel=i;
stop=1;
end;
else
BAAN2(i,1)=0;
end;
end;
BAAN4=[(0*tydbaan(1:(tydregs_tel-voorkyk_tel))begintrap)
(0*tydbaan(((tydregs_telvoorkyk_tel+1)):n)+trapinset)];
BAANX=[BAAN1 BAAN2];
BAANX2=[BAAN1 BAAN4];
ywaarde=1/(V*0.5*voortyd);
sim('bison_fiets_sim');
%==============================================
=========bison_fiets_plot==========================
================
%INSETTE t, X, Y, RATIO, V
posisie=80;%meter waar die tyd vergelyk sal word.
groot=size(t);
plek=0;
stop=0;
for i=1:(groot-1),
if X(i) > posisie
if stop == 0
plek=i;
stop=1;
end;
end;
end;
%*********************bepaal
baanlengte**********************
lengte=0;
AlfaF1_tot=0;
AlfaF2_tot=0;
AlfaR1_tot=0;
AlfaF1_rms_tot=0;
AlfaF2_rms_tot=0;
AlfaR1_rms_tot=0;
for i=1:(plek-1),
lengte=lengte+sqrt((X(i+1)-X(i))^2+(Y(i+1)-Y(i))^2);
AlfaF1_tot=AlfaF1_tot+AlfaF1(i);
AlfaF2_tot=AlfaF2_tot+AlfaF2(i);
AlfaR1_tot=AlfaR1_tot+AlfaR1(i);
AlfaF1_rms_tot=AlfaF1_rms_tot+(AlfaF1(i))^2;
AlfaF2_rms_tot=AlfaF2_rms_tot+(AlfaF2(i))^2;
AlfaR1_rms_tot=AlfaR1_rms_tot+(AlfaR1(i))^2;
end;
AlfaF1_AVG=AlfaF1_tot/(plek-1);
AlfaF2_AVG=AlfaF2_tot/(plek-1);
AlfaR1_AVG=AlfaR1_tot/(plek-1);
AlfaF1_rms=sqrt(AlfaF1_rms_tot/(plek-1));
AlfaF2_rms=sqrt(AlfaF2_rms_tot/(plek-1));
AlfaR1_rms=sqrt(AlfaR1_rms_tot/(plek-1));
eval(['leer',num2str(runnr),'_',num2str(Vkm_h),'=[t(plek),X(plek)
,Y(plek),V,ratio,lengte,max(AlfaF1)*57.3,max(AlfaF2)*57.3,ma
x(AlfaR1)*57.3,AlfaF1_AVG*57.3,AlfaF2_AVG*57.3,AlfaR1_A
VG*57.3,AlfaF1_rms*57.3,AlfaF2_rms*57.3,AlfaR1_rms*57.3];'
]);
temp=[ratio,V*3.6,X'
ratio,V*3.6,Y'];
eval(['leerbaan',num2str(runnr),'_',num2str(Vkm_h),'=temp;']);
temp2=[ratio,V*3.6,t'
ratio,V*3.6,57.3*AlfaF1'
ratio,V*3.6,57.3*AlfaF2'
ratio,V*3.6,57.3*AlfaR1'
ratio,V*3.6,57.3*phi'
ratio,V*3.6,57.3*r'
ratio,V*3.6,57.3*stuur'];
eval(['gliphoek',num2str(runnr),'_',num2str(Vkm_h),'=temp2;']);
runnr;
if stopp == 1
stopp=0;
eval(['save dataleer
gliphoek',num2str(runnr),'_',num2str(Vkm_h),'
leer',num2str(runnr),'_',num2str(Vkm_h),'
leerbaan',num2str(runnr),'_',num2str(Vkm_h)]);
else
eval(['save dataleer
gliphoek',num2str(runnr),'_',num2str(Vkm_h),'
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 124 of 138 pages
leer',num2str(runnr),'_',num2str(Vkm_h),'
leerbaan',num2str(runnr),'_',num2str(Vkm_h),' -append']);
end;%if
clear X;
clear Y;
clear BAANX;
clear BAANX2;clear BAAN1;clear BAAN2;clear
BAAN3;clear BAAN4;
end;%for
end %for
bison_matriks_saam_opsomming
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 125 of 138 pages
Appendix E
Lane change data manipulation – Matlab M file
% Program plot data vir BISON SINGLE LANE CHANGE
% Run eers Bison_fiets, bison_fiets_sim.mdl, Bison_fiets_plot
% Bison_fiets_plot roep automaties Bison_fiets_data op.
% Inset tot die program is n dataleer wat die matrikse bevat
uitgeskryf deur bison_fiets_plot
%19 Aug 2003
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 126 of 138 pages
close all;
%clear;
%bison_fiets;
datamatriks_30=[leer1_30
leer2_30
leer3_30
leer4_30
leer5_30
leer6_30
leer7_30
leer8_30];
datamatriks_40=[leer1_40
leer2_40
leer3_40
leer4_40
leer5_40
leer6_40
leer7_40
leer8_40];
datamatriks_50=[leer1_50
leer2_50
leer3_50
leer4_50
leer5_50
leer6_50
leer7_50
leer8_50];
datamatriks_60=[leer1_60
leer2_60
leer3_60
leer4_60
leer5_60
leer6_60
leer7_60
leer8_60];
datamatriks_70=[leer1_70
leer2_70
leer3_70
leer4_70
leer5_70
leer6_70
leer7_70
leer8_70];
%======================================== TIME
TO COMPLETE TE TRACK
===============================================
===============================
%t(plek),X(plek),Y(plek),V,ratio,lengte,max(AlfaF1)*57.3,max(
AlfaF2)*57.3,max(AlfaR1)*57.3,AlfaF1_AVG*57.3,AlfaF2_AVG
*57.3,AlfaR1_AVG*57.3];']);
figure;
%plot(datamatriks_30(:,5),datamatriks_30(:,1),datamatriks_40(
:,5),datamatriks_40(:,1),datamatriks_50(:,5),datamatriks_50(:,1
),datamatriks_60(:,5),datamatriks_60(:,1),datamatriks_70(:,5),d
atamatriks_70(:,1),datamatriks_80(:,5),datamatriks_80(:,1),dat
amatriks_90(:,5),datamatriks_90(:,1));
bar(datamatriks_30(:,5),datamatriks_30(:,1),'stacked');hold on;
bar(datamatriks_40(:,5),datamatriks_40(:,1),'stacked','w');
bar(datamatriks_50(:,5),datamatriks_50(:,1),'stacked','g');
bar(datamatriks_60(:,5),datamatriks_60(:,1),'stacked','b');
bar(datamatriks_70(:,5),datamatriks_70(:,1),'stacked','y');
title('Bison Single Lane Change');
xlabel('Steering Ratio');
ylabel('time to complete the track [s]');
legend('30 km/h','40 km/h','50 km/h','60 km/h','70 km/h',-1);
grid;
hold off;
%======================================== TIME
TO COMPLETE TE TRACK
===============================================
===============================
%t(plek),X(plek),Y(plek),V,ratio,lengte,max(AlfaF1)*57.3,max(
AlfaF2)*57.3,max(AlfaR1)*57.3,AlfaF1_AVG*57.3,AlfaF2_AVG
*57.3,AlfaR1_AVG*57.3];']);
figure;
%plot(datamatriks_30(:,5),datamatriks_30(:,1),datamatriks_40(
:,5),datamatriks_40(:,1),datamatriks_50(:,5),datamatriks_50(:,1
),datamatriks_60(:,5),datamatriks_60(:,1),datamatriks_70(:,5),d
atamatriks_70(:,1),datamatriks_80(:,5),datamatriks_80(:,1),dat
amatriks_90(:,5),datamatriks_90(:,1));
plot(datamatriks_30(:,5),datamatriks_30(:,1));hold on;
plot(datamatriks_40(:,5),datamatriks_40(:,1));
plot(datamatriks_50(:,5),datamatriks_50(:,1));
plot(datamatriks_60(:,5),datamatriks_60(:,1));
plot(datamatriks_70(:,5),datamatriks_70(:,1));
title('Bison Single Lane Change');
xlabel('Steering Ratio');
ylabel('time to complete the track [s]');
legend('30 km/h','40 km/h','50 km/h','60 km/h','70 km/h',-1);
hold off;
%======================================== MAX
SLIP ANGLE
===============================================
==========================================
figure;
%plot(datamatriks_30(:,5),datamatriks_30(:,7),datamatriks_30(
:,5),datamatriks_30(:,8),datamatriks_30(:,5),datamatriks_30(:,9
),datamatriks_40(:,5),datamatriks_40(:,7),datamatriks_40(:,5),d
atamatriks_40(:,8),datamatriks_40(:,5),datamatriks_40(:,9),dat
amatriks_50(:,5),datamatriks_50(:,7),datamatriks_50(:,5),data
matriks_50(:,8),datamatriks_50(:,5),datamatriks_50(:,9),datam
atriks_60(:,5),datamatriks_60(:,7),datamatriks_60(:,5),datamatr
iks_60(:,8),datamatriks_60(:,5),datamatriks_60(:,9),datamatrik
s_70(:,5),datamatriks_70(:,7),datamatriks_70(:,5),datamatriks_
70(:,8),datamatriks_70(:,5),datamatriks_70(:,9),datamatriks_80
(:,5),datamatriks_80(:,7),datamatriks_80(:,5),datamatriks_80(:,
8),datamatriks_80(:,5),datamatriks_80(:,9),datamatriks_90(:,5),
datamatriks_90(:,7),datamatriks_90(:,5),datamatriks_90(:,8),da
tamatriks_90(:,5),datamatriks_90(:,9));
subplot(3,2,1);
bar(datamatriks_30(:,5),datamatriks_30(:,7:9),'stacked');
axis([0.2 1.1 0 10]);grid;
title('Bison Single Lane Change 30 km/h');
legend('First Axle','Second Axle','Third Axle',-1);
xlabel('Steering Ratio');
ylabel('Max slip angle');
subplot(3,2,2);
bar(datamatriks_40(:,5),datamatriks_40(:,7:9),'stacked');
axis([0.2 1.2 0 10]);grid;
title('Bison Single Lane Change 40 km/h');
legend('First Axle','Second Axle','Third Axle',-1);
xlabel('Steering Ratio');
ylabel('Max slip angle');
subplot(3,2,3);
bar(datamatriks_50(:,5),datamatriks_50(:,7:9),'stacked');
axis([0.2 1.2 0 10]);grid;
title('Bison Single Lane Change 50 km/h');
legend('First Axle','Second Axle','Third Axle',-1);
xlabel('Steering Ratio');
ylabel('Max slip angle');
subplot(3,2,4);
bar(datamatriks_60(:,5),datamatriks_60(:,7:9),'stacked');
axis([0.2 1.2 0 10]);grid;
title('Bison Single Lane Change 60 km/h');
legend('First Axle','Second Axle','Third Axle',-1)
xlabel('Steering Ratio');
ylabel('Max slip angle');
subplot(3,2,5);
bar(datamatriks_70(:,5),datamatriks_70(:,7:9),'stacked');
axis([0.2 1.2 0 10]);grid;
title('Bison Single Lane Change70 km/h');
legend('First Axle','Second Axle','Third Axle',-1);
xlabel('Steering Ratio');
ylabel('Max slip angle');
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 127 of 138 pages
%======================================== AVG
SLIP ANGLE
===============================================
===============================
figure;
subplot(3,2,1);
bar(datamatriks_30(:,5),abs(datamatriks_30(:,10:12)),'stacked'
);
axis([0.2 1.2 0 0.3]);grid;
title('Bison Single Lane Change 30 km/h');
legend('First Axle','Second Axle','Third Axle',-1);
xlabel('Steering Ratio');
ylabel('Avg. slip angle');
subplot(3,2,2);
bar(datamatriks_40(:,5),abs(datamatriks_40(:,10:12)),'stacked'
);
axis([0.2 1.2 0 0.3]);grid;
title('Bison Single Lane Change 40 km/h');
legend('First Axle','Second Axle','Third Axle',-1);
xlabel('Steering Ratio');
ylabel('Avg. slip angle');
subplot(3,2,3);
bar(datamatriks_50(:,5),abs(datamatriks_50(:,10:12)),'stacked'
);
axis([0.2 1.2 0 0.3]);grid;
title('Bison Single Lane Change 50 km/h');
legend('First Axle','Second Axle','Third Axle',-1);
xlabel('Steering Ratio');
ylabel('Avg. slip angle');
subplot(3,2,4);
bar(datamatriks_60(:,5),abs(datamatriks_60(:,10:12)),'stacked'
);
axis([0.2 1.2 0 0.3]);grid;
title('Bison Single Lane Change 60 km/h');
legend('First Axle','Second Axle','Third Axle',-1)
xlabel('Steering Ratio');
ylabel('Avg. slip angle');
subplot(3,2,5);
bar(datamatriks_70(:,5),abs(datamatriks_70(:,10:12)),'stacked'
);
axis([0.2 1.2 0 0.3]);grid;
title('Bison Single Lane Change 70 km/h');
legend('First Axle','Second Axle','Third Axle',-1);
xlabel('Steering Ratio');
ylabel('Avg. slip angle');
%======================================== rms
SLIP ANGLE
===============================================
===============================
figure;
subplot(3,2,1);
bar(datamatriks_30(:,5),abs(datamatriks_30(:,13:15)),'stacked'
);
axis([0.2 1.2 0 7]);grid;
title('Bison Single Lane Change 30 km/h');
legend('First Axle','Second Axle','Third Axle',-1);
xlabel('Steering Ratio');
ylabel('RMS. slip angle');
subplot(3,2,2);
bar(datamatriks_40(:,5),abs(datamatriks_40(:,13:15)),'stacked'
);
axis([0.2 1.2 0 7]);grid;
title('Bison Single Lane Change 40 km/h');
legend('First Axle','Second Axle','Third Axle',-1);
xlabel('Steering Ratio');
ylabel('RMS. slip angle');
subplot(3,2,3);
bar(datamatriks_50(:,5),abs(datamatriks_50(:,13:15)),'stacked'
);
axis([0.2 1.2 0 7]);grid;
title('Bison Single Lane Change 50 km/h');
legend('First Axle','Second Axle','Third Axle',-1);
xlabel('Steering Ratio');
ylabel('RMS. slip angle');
subplot(3,2,4);
bar(datamatriks_60(:,5),abs(datamatriks_60(:,13:15)),'stacked'
);
axis([0.2 1.2 0 7]);grid;
title('Bison Single Lane Change 60 km/h');
legend('First Axle','Second Axle','Third Axle',-1)
xlabel('Steering Ratio');
ylabel('RMS. slip angle');
subplot(3,2,5);
bar(datamatriks_70(:,5),abs(datamatriks_70(:,13:15)),'stacked'
);
axis([0.2 1.2 0 7]);grid;
title('Bison Single Lane Change 70 km/h');
legend('First Axle','Second Axle','Third Axle',-1);
xlabel('Steering Ratio');
ylabel('RMS. slip angle');
%=====================total RMS slip
angle==============================
figure;
subplot(3,2,1);
bar(datamatriks_30(:,5),abs(datamatriks_30(:,13)+datamatriks
_30(:,14)+datamatriks_30(:,15)),'stacked');
axis([0.2 1.2 0 25]);grid;
title('Bison Single Lane Change 30 km/h');
xlabel('Steering Ratio');
ylabel('Total RMS. slip angle');
subplot(3,2,2);
bar(datamatriks_40(:,5),abs(datamatriks_40(:,13)+datamatriks
_40(:,14)+datamatriks_40(:,15)),'stacked');
axis([0.2 1.2 0 25]);grid;
title('Bison Single Lane Change 40 km/h');
xlabel('Steering Ratio');
ylabel('Total RMS. slip angle');
subplot(3,2,3);
bar(datamatriks_50(:,5),abs(datamatriks_50(:,13)+datamatriks
_50(:,14)+datamatriks_50(:,15)),'stacked');
axis([0.2 1.2 0 25]);grid;
title('Bison Single Lane Change 50 km/h');
xlabel('Steering Ratio');
ylabel('Total RMS. slip angle');
subplot(3,2,4);
bar(datamatriks_60(:,5),abs(datamatriks_60(:,13)+datamatriks
_60(:,14)+datamatriks_60(:,15)),'stacked');
axis([0.2 1.2 0 25]);grid;
title('Bison Single Lane Change 60 km/h');
xlabel('Steering Ratio');
ylabel('Total RMS. slip angle');
subplot(3,2,5);
bar(datamatriks_70(:,5),abs(datamatriks_70(:,13)+datamatriks
_70(:,14)+datamatriks_70(:,15)),'stacked');
axis([0.2 1.2 0 25]);grid;
title('Bison Single Lane Change 70 km/h');
xlabel('Steering Ratio');
ylabel('Total RMS. slip angle');
%[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[rms vs
time]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]
spoed=20;
figure;
for spd=1:5
spoed=spoed+10;
subplot(3,2,spd);
eval(['plotyy(datamatriks_',num2str(spoed),'(:,5),abs(datamatri
ks_',num2str(spoed),'(:,13)+datamatriks_',num2str(spoed),'(:,1
4)+datamatriks_',num2str(spoed),'(:,15)),datamatriks_',num2st
r(spoed),'(:,5),abs(datamatriks_',num2str(spoed),'(:,1)))']);
%axis([0.3 1.2 0 15]);
grid;
eval(['title(''Bison SLC Total RMS ',num2str(spoed),'
km/h'');']);
ylabel('time [s]')
end;
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 128 of 138 pages
%======================================== LANE
CHANGE XY PLOT
===============================================
=========
hold off;figure;
hold on;
%===========================================PL
OT RATIO GRAPHS
===============================================
===========
V=[30
40 50 60 70];
for j=1:5
for i=1:8
eval(['q=size(leerbaan',num2str(i),'_',num2str(V(j)),')']);
if i == 1
kleur='c';
subplot(4,2,i);
title('Bison Single Lane Change. Steering Ratio 0.4 ');
xlabel('Distance [m]');
ylabel('Distance [m]');
lyn1=[15 1.8
15 7.230];
lyn2=[45 -1.8
45 3.630];
lyn3=[0.1 1.8
15 1.8];
lyn4=[45 3.63
100
3.63];
lyn5=[15 7.23
100
7.23];
lyn6=[0.1 -1.8
45 -1.8];
lyn7=[0.01
0.01
30 0.01];
lyn8=[30 0.01
30 5.430];
lyn9=[30 5.43
100
5.43];
lyn10=[100 3.63
100 7.23];
plot(lyn9(:,1),lyn9(:,2),lyn8(:,1),lyn8(:,2),lyn7(:,1),lyn7(:,2),lyn6(:
,1),lyn6(:,2),lyn5(:,1),lyn5(:,2),lyn4(:,1),lyn4(:,2),lyn3(:,1),lyn3(:,
2),lyn2(:,1),lyn2(:,2),lyn1(:,1),lyn1(:,2),lyn10(:,1),lyn10(:,2));
axis([0 100 -2.5 8]);hold on;grid;
eval(['plot(leerbaan',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),lee
rbaan',num2str(i),'_',num2str(V(j)),'(2,3:q(1,2)),kleur)']);hold on;
end;
if i == 2
kleur='y';
subplot(4,2,i);
title('Bison Single Lane Change. Steering Ratio 0.5 ');
xlabel('Distance [m]');
ylabel('Distance [m]');
lyn1=[15 1.8
15 7.230];
lyn2=[45 -1.8
45 3.630];
lyn3=[0.1 1.8
15 1.8];
lyn4=[45 3.63
100
3.63];
lyn5=[15 7.23
100
7.23];
lyn6=[0.1 -1.8
45 -1.8];
lyn7=[0.01
0.01
30 0.01];
lyn8=[30 0.01
30 5.430];
lyn9=[30 5.43
100
5.43];
lyn10=[100 3.63
100 7.23];
plot(lyn9(:,1),lyn9(:,2),lyn8(:,1),lyn8(:,2),lyn7(:,1),lyn7(:,2),lyn6(:
,1),lyn6(:,2),lyn5(:,1),lyn5(:,2),lyn4(:,1),lyn4(:,2),lyn3(:,1),lyn3(:,
2),lyn2(:,1),lyn2(:,2),lyn1(:,1),lyn1(:,2),lyn10(:,1),lyn10(:,2));
axis([0 100 -2.5 8]);hold on;grid;
eval(['plot(leerbaan',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),lee
rbaan',num2str(i),'_',num2str(V(j)),'(2,3:q(1,2)),kleur)']);hold on;
end;
if i == 3
kleur='m';
subplot(4,2,i);
title('Bison Single Lane Change. Steering Ratio 0.6 ');
xlabel('Distance [m]');
ylabel('Distance [m]');
lyn1=[15 1.8
15 7.230];
lyn2=[45 -1.8
45 3.630];
lyn3=[0.1 1.8
15 1.8];
lyn4=[45 3.63
100
3.63];
lyn5=[15 7.23
100
7.23];
lyn6=[0.1 -1.8
45 -1.8];
lyn7=[0.01
0.01
30 0.01];
lyn8=[30 0.01
30 5.430];
lyn9=[30 5.43
100
5.43];
lyn10=[100 3.63
100 7.23];
plot(lyn9(:,1),lyn9(:,2),lyn8(:,1),lyn8(:,2),lyn7(:,1),lyn7(:,2),lyn6(:
,1),lyn6(:,2),lyn5(:,1),lyn5(:,2),lyn4(:,1),lyn4(:,2),lyn3(:,1),lyn3(:,
2),lyn2(:,1),lyn2(:,2),lyn1(:,1),lyn1(:,2),lyn10(:,1),lyn10(:,2));
axis([0 100 -2.5 8]);hold on;grid;
eval(['plot(leerbaan',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),lee
rbaan',num2str(i),'_',num2str(V(j)),'(2,3:q(1,2)),kleur)']);hold on;
end;
if i == 4
kleur='r';
subplot(4,2,i);
title('Bison Single Lane Change. Steering Ratio 0.7 ');
xlabel('Distance [m]');
ylabel('Distance [m]');
lyn1=[15 1.8
15 7.230];
lyn2=[45 -1.8
45 3.630];
lyn3=[0.1 1.8
15 1.8];
lyn4=[45 3.63
100
3.63];
lyn5=[15 7.23
100
7.23];
lyn6=[0.1 -1.8
45 -1.8];
lyn7=[0.01
0.01
30 0.01];
lyn8=[30 0.01
30 5.430];
lyn9=[30 5.43
100
5.43];
lyn10=[100 3.63
100 7.23];
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 129 of 138 pages
plot(lyn9(:,1),lyn9(:,2),lyn8(:,1),lyn8(:,2),lyn7(:,1),lyn7(:,2),lyn6(:
,1),lyn6(:,2),lyn5(:,1),lyn5(:,2),lyn4(:,1),lyn4(:,2),lyn3(:,1),lyn3(:,
2),lyn2(:,1),lyn2(:,2),lyn1(:,1),lyn1(:,2),lyn10(:,1),lyn10(:,2));
axis([0 100 -2.5 8]);hold on;grid;
eval(['plot(leerbaan',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),lee
rbaan',num2str(i),'_',num2str(V(j)),'(2,3:q(1,2)),kleur)']);hold on;
end;
if i == 5
subplot(4,2,i);
kleur='g';
title('Bison Single Lane Change. Steering Ratio 0.8 ');
xlabel('Distance [m]');
ylabel('Distance [m]');
lyn1=[15 1.8
15 7.230];
lyn2=[45 -1.8
45 3.630];
lyn3=[0.1 1.8
15 1.8];
lyn4=[45 3.63
100
3.63];
lyn5=[15 7.23
100
7.23];
lyn6=[0.1 -1.8
45 -1.8];
lyn7=[0.01
0.01
30 0.01];
lyn8=[30 0.01
30 5.430];
lyn9=[30 5.43
100
5.43];
lyn10=[100 3.63
100 7.23];
plot(lyn9(:,1),lyn9(:,2),lyn8(:,1),lyn8(:,2),lyn7(:,1),lyn7(:,2),lyn6(:
,1),lyn6(:,2),lyn5(:,1),lyn5(:,2),lyn4(:,1),lyn4(:,2),lyn3(:,1),lyn3(:,
2),lyn2(:,1),lyn2(:,2),lyn1(:,1),lyn1(:,2),lyn10(:,1),lyn10(:,2));
axis([0 100 -2.5 8]);hold on;grid;
eval(['plot(leerbaan',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),lee
rbaan',num2str(i),'_',num2str(V(j)),'(2,3:q(1,2)),kleur)']);hold on;
end;
if i == 6
kleur='b';
subplot(4,2,i);
title('Bison Single Lane Change. Steering Ratio 0.9 ');
xlabel('Distance [m]');
ylabel('Distance [m]');
lyn1=[15 1.8
15 7.230];
lyn2=[45 -1.8
45 3.630];
lyn3=[0.1 1.8
15 1.8];
lyn4=[45 3.63
100
3.63];
lyn5=[15 7.23
100
7.23];
lyn6=[0.1 -1.8
45 -1.8];
lyn7=[0.01
0.01
30 0.01];
lyn8=[30 0.01
30 5.430];
lyn9=[30 5.43
100
5.43];
lyn10=[100 3.63
100 7.23];
plot(lyn9(:,1),lyn9(:,2),lyn8(:,1),lyn8(:,2),lyn7(:,1),lyn7(:,2),lyn6(:
,1),lyn6(:,2),lyn5(:,1),lyn5(:,2),lyn4(:,1),lyn4(:,2),lyn3(:,1),lyn3(:,
2),lyn2(:,1),lyn2(:,2),lyn1(:,1),lyn1(:,2),lyn10(:,1),lyn10(:,2));
axis([0 100 -2.5 8]);hold on;grid;
eval(['plot(leerbaan',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),lee
rbaan',num2str(i),'_',num2str(V(j)),'(2,3:q(1,2)),kleur)']);hold on;
end;
if i == 7
kleur='c';
subplot(4,2,i);
title('Bison Single Lane Change. Steering Ratio 1 ');
xlabel('Distance [m]');
ylabel('Distance [m]');
lyn1=[15 1.8
15 7.230];
lyn2=[45 -1.8
45 3.630];
lyn3=[0.1 1.8
15 1.8];
lyn4=[45 3.63
100
3.63];
lyn5=[15 7.23
100
7.23];
lyn6=[0.1 -1.8
45 -1.8];
lyn7=[0.01
0.01
30 0.01];
lyn8=[30 0.01
30 5.430];
lyn9=[30 5.43
100
5.43];
lyn10=[100 3.63
100 7.23];
plot(lyn9(:,1),lyn9(:,2),lyn8(:,1),lyn8(:,2),lyn7(:,1),lyn7(:,2),lyn6(:
,1),lyn6(:,2),lyn5(:,1),lyn5(:,2),lyn4(:,1),lyn4(:,2),lyn3(:,1),lyn3(:,
2),lyn2(:,1),lyn2(:,2),lyn1(:,1),lyn1(:,2),lyn10(:,1),lyn10(:,2));
axis([0 100 -2.5 8]);hold on;grid;
eval(['plot(leerbaan',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),lee
rbaan',num2str(i),'_',num2str(V(j)),'(2,3:q(1,2)),kleur)']);hold on;
end;
if i == 8
kleur='m';
subplot(4,2,i);
title('Bison Single Lane Change. Steering Ratio 1.1 ');
xlabel('Distance [m]');
ylabel('Distance [m]');
lyn1=[15 1.8
15 7.230];
lyn2=[45 -1.8
45 3.630];
lyn3=[0.1 1.8
15 1.8];
lyn4=[45 3.63
100
3.63];
lyn5=[15 7.23
100
7.23];
lyn6=[0.1 -1.8
45 -1.8];
lyn7=[0.01
0.01
30 0.01];
lyn8=[30 0.01
30 5.430];
lyn9=[30 5.43
100
5.43];
lyn10=[100 3.63
100 7.23];
plot(lyn9(:,1),lyn9(:,2),lyn8(:,1),lyn8(:,2),lyn7(:,1),lyn7(:,2),lyn6(:
,1),lyn6(:,2),lyn5(:,1),lyn5(:,2),lyn4(:,1),lyn4(:,2),lyn3(:,1),lyn3(:,
2),lyn2(:,1),lyn2(:,2),lyn1(:,1),lyn1(:,2),lyn10(:,1),lyn10(:,2));
axis([0 100 -2.5 8]);hold on;grid;
eval(['plot(leerbaan',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),lee
rbaan',num2str(i),'_',num2str(V(j)),'(2,3:q(1,2)),kleur)']);hold on;
end;
end;%for
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 130 of 138 pages
end;%for
hold off;
hold on;
%==============================================
===============================================
=====================================
%==============================================
===================PLOT XY == 2
===============================================
====
figure;
off30=910;
off40=910;
off50=910;
off60=905;
off70=910;
off80=910;
V=[30
40 50 60 70];
for j=1:5
for i=1:8
eval(['q=size(leerbaan',num2str(i),'_',num2str(V(j)),');']);
if j == 1
kleur='c';
subplot(3,2,j);
title('Bison Single Lane Change: 30 km/h ');
xlabel('Distance [m]');
ylabel('Distance [m]');
lyn1=[15 1.8
15 7.230];
lyn2=[45 -1.8
45 3.630];
lyn3=[0.1 1.8
15 1.8];
lyn4=[45 3.63
130
3.63];
lyn5=[15 7.23
130
7.23];
lyn6=[0.1 -1.8
45 -1.8];
lyn7=[0.01
0.01
30 0.01];
lyn8=[30 0.01
30 5.430];
lyn9=[30 5.43
130
5.43];
lyn10=[130 3.63
130 7.23];
plot(uitset_28_2(:,1)-off30,-1*uitset_28_2(:,2)22.5,uitset_28_1(:,1)-off30,-1*uitset_28_1(:,2)22.5,lyn9(:,1),lyn9(:,2),lyn8(:,1),lyn8(:,2),lyn7(:,1),lyn7(:,2),lyn6(
:,1),lyn6(:,2),lyn5(:,1),lyn5(:,2),lyn4(:,1),lyn4(:,2),lyn3(:,1),lyn3(:,
2),lyn2(:,1),lyn2(:,2),lyn1(:,1),lyn1(:,2),lyn10(:,1),lyn10(:,2));
axis([0 130 -2.5 8]);hold on;grid on;
eval(['plot(leerbaan',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),lee
rbaan',num2str(i),'_',num2str(V(j)),'(2,3:q(1,2)),kleur)']);hold on;
end;
if j == 2
kleur='y';
subplot(3,2,j);
plot(uitset_43_2(:,1)-off40,-1*uitset_43_2(:,2)22.5,uitset_43_1(:,1)-off40,-1*uitset_43_1(:,2)22.5,lyn9(:,1),lyn9(:,2),lyn8(:,1),lyn8(:,2),lyn7(:,1),lyn7(:,2),lyn6(
:,1),lyn6(:,2),lyn5(:,1),lyn5(:,2),lyn4(:,1),lyn4(:,2),lyn3(:,1),lyn3(:,
2),lyn2(:,1),lyn2(:,2),lyn1(:,1),lyn1(:,2),lyn10(:,1),lyn10(:,2));
axis([0 130 -2.5 8]);hold on;grid on;
eval(['plot(leerbaan',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),lee
rbaan',num2str(i),'_',num2str(V(j)),'(2,3:q(1,2)),kleur)']);hold on;
title('Bison Single Lane Change 40 km/h ');
xlabel('Distance [m]');
ylabel('Distance [m]');
lyn1=[15 1.8
15 7.230];
lyn2=[45 -1.8
45 3.630];
lyn3=[0.1 1.8
15 1.8];
lyn4=[45 3.63
130
3.63];
lyn5=[15 7.23
130
7.23];
lyn6=[0.1 -1.8
45 -1.8];
lyn7=[0.01
0.01
30 0.01];
lyn8=[30 0.01
30 5.430];
lyn9=[30 5.43
130
5.43];
lyn10=[130 3.63
130 7.23];
end;
if j == 3
kleur='m';
subplot(3,2,j);
title('Bison Single Lane Change: 50 km/h ');
xlabel('Distance [m]');
ylabel('Distance [m]');
lyn1=[15 1.8
15 7.230];
lyn2=[45 -1.8
45 3.630];
lyn3=[0.1 1.8
15 1.8];
lyn4=[45 3.63
130
3.63];
lyn5=[15 7.23
130
7.23];
lyn6=[0.1 -1.8
45 -1.8];
lyn7=[0.01
0.01
30 0.01];
lyn8=[30 0.01
30 5.430];
lyn9=[30 5.43
130
5.43];
lyn10=[130 3.63
130 7.23];
plot(uitset_52_1(:,1)-off50,-1*uitset_52_1(:,2)22.5,uitset_52_2(:,1)-off50,-1*uitset_52_2(:,2)22.5,lyn9(:,1),lyn9(:,2),lyn8(:,1),lyn8(:,2),lyn7(:,1),lyn7(:,2),lyn6(
:,1),lyn6(:,2),lyn5(:,1),lyn5(:,2),lyn4(:,1),lyn4(:,2),lyn3(:,1),lyn3(:,
2),lyn2(:,1),lyn2(:,2),lyn1(:,1),lyn1(:,2),lyn10(:,1),lyn10(:,2));
axis([0 130 -2.5 8]);
hold on;grid on;
eval(['plot(leerbaan',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),lee
rbaan',num2str(i),'_',num2str(V(j)),'(2,3:q(1,2)),kleur)']);hold on;
end;
if j == 4
kleur='r';
subplot(3,2,j);
title('Bison Single Lane Change: 60 km/h ');
xlabel('Distance [m]');
ylabel('Distance [m]');
lyn1=[15 1.8
15 7.230];
lyn2=[45 -1.8
45 3.630];
lyn3=[0.1 1.8
15 1.8];
lyn4=[45 3.63
130
3.63];
lyn5=[15 7.23
130
7.23];
lyn6=[0.1 -1.8
45 -1.8];
lyn7=[0.01
0.01
30 0.01];
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 131 of 138 pages
lyn8=[30 0.01
30 5.430];
lyn9=[30 5.43
130
5.43];
lyn10=[130 3.63
130 7.23];
plot(uitset_60_1(:,1)-off60,-1*uitset_60_1(:,2)21.5,uitset_60_2(:,1)-off60,-1*uitset_60_2(:,2)21.5,lyn9(:,1),lyn9(:,2),lyn8(:,1),lyn8(:,2),lyn7(:,1),lyn7(:,2),lyn6(
:,1),lyn6(:,2),lyn5(:,1),lyn5(:,2),lyn4(:,1),lyn4(:,2),lyn3(:,1),lyn3(:,
2),lyn2(:,1),lyn2(:,2),lyn1(:,1),lyn1(:,2),lyn10(:,1),lyn10(:,2));
axis([0 130 -2.5 8]);hold on;grid on;
eval(['plot(leerbaan',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),lee
rbaan',num2str(i),'_',num2str(V(j)),'(2,3:q(1,2)),kleur)']);hold on;
end;
if j == 5
subplot(3,2,j);
kleur='g';
title('Bison Single Lane Change: 70 km/h');
xlabel('Distance [m]');
ylabel('Distance [m]');
lyn1=[15 1.8
15 7.230];
lyn2=[45 -1.8
45 3.630];
lyn3=[0.1 1.8
15 1.8];
lyn4=[45 3.63
130
3.63];
lyn5=[15 7.23
130
7.23];
lyn6=[0.1 -1.8
45 -1.8];
lyn7=[0.01
0.01
30 0.01];
lyn8=[30 0.01
30 5.430];
lyn9=[30 5.43
130
5.43];
lyn10=[130 3.63
130 7.23];
plot(lyn9(:,1),lyn9(:,2),lyn8(:,1),lyn8(:,2),lyn7(:,1),lyn7(:,2),lyn6(:
,1),lyn6(:,2),lyn5(:,1),lyn5(:,2),lyn4(:,1),lyn4(:,2),lyn3(:,1),lyn3(:,
2),lyn2(:,1),lyn2(:,2),lyn1(:,1),lyn1(:,2),lyn10(:,1),lyn10(:,2));
axis([0 130 -2.5 8]);hold on;grid on;
eval(['plot(leerbaan',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),lee
rbaan',num2str(i),'_',num2str(V(j)),'(2,3:q(1,2)),kleur)']);hold on;
end;
end;
end;
%&&&&&&&&&&&&&&&&&&&&& ou cut
hold off;
%=================================== PLOT
GLIPHOEKE
===============================================
================================
spoed=20;
for s = 1:7
spoed=spoed+10;
figure;
for i=1:8
if i == 1
subplot(3,1,1);%F1
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
%kleur='c-<';
kleur='c';
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
eval(['title(''Bison Single Lane Change ',num2str(spoed),'
km/h. First Steering Axle'')']);
xlabel('time [s]');
ylabel('Slip Angle [°]');
axis([0 15 -15 15 ]);grid;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
eval(['title(''Bison Single Lane Change ',num2str(spoed),'
km/h. Second Steering Axle '')']);
xlabel('time [s]');
ylabel('Slip Angle [°]');
axis([0 15 -15 15]);grid;
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
eval(['title(''Bison Single Lane Change ',num2str(spoed),'
km/h. Rear Axle '')']);
xlabel('time [s]');
ylabel('Slip Angle [°]');
axis([0 15 -15 15]);grid;
end;
if i == 2
kleur='y';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
subplot(3,1,1);%F1
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 3
kleur='m';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
subplot(3,1,1);%F1
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 4
kleur='r';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
subplot(3,1,1);%F1
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 132 of 138 pages
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 5
kleur='g';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
subplot(3,1,1);%F1
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 6
kleur='b';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
subplot(3,1,1);%F1
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 9
kleur='b';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
subplot(3,1,1);%F1
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
end;
end;
legend('ratio 0.3','ratio 0.4','ratio 0.5','ratio 0.6','ratio 0.7','ratio
0.8','ratio 0.9','ratio 1','ratio 1.1');
end;
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 7
kleur='c';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
subplot(3,1,1);%F1
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 8
kleur='k';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
subplot(3,1,1);%F1
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,3);%F3
%=================================== PLOT
stuurhoeke
===============================================
================================
spoed=20;
for s = 1:7
spoed=spoed+10;
figure;
for i=1:8
if i == 1
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
%kleur='c-<';
kleur='c';
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(7,3:q(1,2)),kleur)']);hol
d on;
eval(['title(''Bison Single Lane Change ',num2str(spoed),'
km/h. First Steering Axle'')']);
xlabel('time [s]');
ylabel('Steering Angle [°]');
axis([0 15 -15 30 ]);grid;
end;
if i == 2
kleur='y';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(7,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 3
kleur='m';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(7,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 4
kleur='r';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(7,3:q(1,2)),kleur)']);hol
d on;
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 133 of 138 pages
end;
if i == 5
kleur='g';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(7,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 6
kleur='b';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(7,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 7
kleur='c';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(7,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 8
kleur='k';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(7,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 9
kleur='b';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(7,3:q(1,2)),kleur)']);hol
d on;
end;
end;
legend('ratio 0.3','ratio 0.4','ratio 0.5','ratio 0.6','ratio 0.7','ratio
0.8','ratio 0.9','ratio 1','ratio 1.1');
end;
%=================================== PLOT yaw
===============================================
================================
spoed=20;
for s = 1:7
spoed=spoed+10;
figure;
for i=1:8
if i == 1
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
%kleur='c-<';
kleur='c';
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
eval(['title(''Bison Single Lane Change ',num2str(spoed),'
km/h. Yaw Angle'')']);
xlabel('time [s]');
ylabel('Yaw Angle [°]');
axis([0 15 -15 30 ]);grid;
end;
if i == 2
kleur='y';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 3
kleur='m';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 4
kleur='r';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 5
kleur='g';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 6
kleur='b';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 7
kleur='c';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 8
kleur='k';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 9
kleur='b';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(7,3:q(1,2)),kleur)']);hol
d on;
end;
end;
legend('ratio 0.3','ratio 0.4','ratio 0.5','ratio 0.6','ratio 0.7','ratio
0.8','ratio 0.9','ratio 1','ratio 1.1');
end;
%==================================== DETAIL
SLIP ANGLES
===============================================
===
spoed=20;
for s = 1:7
spoed=spoed+10;
figure;
for i=1:9
if i == 1
subplot(3,1,1);%F1
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
%kleur='c-<';
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 134 of 138 pages
kleur='c';
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
eval(['title(''Bison Single Lane Change ',num2str(spoed),'
km/h. First Steering Axle'')']);
xlabel('time [s]');
ylabel('Slip Angle [°]');
axis([0 3 -15 10 ]);grid;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
eval(['title(''Bison Single Lane Change ',num2str(spoed),'
km/h. Second Steering Axle '')']);
xlabel('time [s]');
ylabel('Slip Angle [°]');
axis([0 3 -15 10]);grid;
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
eval(['title(''Bison Single Lane Change ',num2str(spoed),'
km/h. Rear Axle '')']);
xlabel('time [s]');
ylabel('Slip Angle [°]');
axis([0 3 -15 10]);grid;
end;
if i == 2
kleur='y';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
subplot(3,1,1);%F1
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 3
kleur='m';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
subplot(3,1,1);%F1
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 4
kleur='r';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
subplot(3,1,1);%F1
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 5
kleur='g';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
subplot(3,1,1);%F1
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 6
kleur='b';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
subplot(3,1,1);%F1
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 7
kleur='c';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
subplot(3,1,1);%F1
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 8
kleur='k';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
subplot(3,1,1);%F1
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,2);%F2
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 135 of 138 pages
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 9
kleur='b';
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
subplot(3,1,1);%F1
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(2,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,2);%F2
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(3,3:q(1,2)),kleur)']);hol
d on;
subplot(3,1,3);%F3
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(4,3:q(1,2)),kleur)']);hol
d on;
end;
end;
legend('ratio 0.3','ratio 0.4','ratio 0.5','ratio 0.6','ratio 0.7','ratio
0.8','ratio 0.9','ratio 1','ratio 1.1');
end;
%==================================PLOT
ANGLE=========================================
============
spoed=20;
for s = 1:7
spoed=spoed+10;
figure;
for i=1:8
eval(['q=size(gliphoek',num2str(i),'_',num2str(spoed),');']);
if i == 1
kleur='c';
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 2
kleur='y';
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 3
kleur='m';
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 4
kleur='r';
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 5
kleur='g';
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 6
kleur='b';
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 7
kleur='c';
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 8
kleur='k';
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
end;
if i == 9
kleur='m';
eval(['plot(gliphoek',num2str(i),'_',num2str(spoed),'(1,3:q(1,2)),
gliphoek',num2str(i),'_',num2str(spoed),'(5,3:q(1,2)),kleur)']);hol
d on;
end;
eval(['title(''Bison Single Lane Change ',num2str(spoed),'
km/h'')']);grid on;
xlabel('Time [t]');
ylabel('Yaw Angle [°]');
legend('ratio 0.3','ratio 0.4','ratio 0.5','ratio 0.6','ratio
0.7','ratio 0.8','ratio 0.9','ratio 1','ratio 1.1');
end;
end;
%==============================================
======== PLOT SUMMARY YAW ANGLE
===============================================
============
figure;
V=[30
40 50 60 70 80 90];
for j=1:7
for i=1:8
eval(['q=size(leerbaan',num2str(i),'_',num2str(V(j)),')']);
if j == 1
kleur='c';
subplot(4,2,1);hold on;
eval(['plot(gliphoek',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),glip
hoek',num2str(i),'_',num2str(V(j)),'(5,3:q(1,2)),kleur)']);hold on;
title('Bison Single Lane: Change: 30 km/h');
xlabel('Time [t]');
ylabel('Yaw Angle [°]');
grid on;
axis([-2 15 -20 30]);
end;
if j == 2
subplot(4,2,2);hold on;
kleur='y';
eval(['plot(gliphoek',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),glip
hoek',num2str(i),'_',num2str(V(j)),'(5,3:q(1,2)),kleur)']);hold on;
title('Bison Single Lane Change: 40 km/h');
xlabel('Time [t]');
ylabel('Yaw Angle [°]');
grid on;
axis([-2 15 -20 30]);
end;
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 136 of 138 pages
if j == 3
subplot(4,2,3);hold on;
kleur='m';
eval(['plot(gliphoek',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),glip
hoek',num2str(i),'_',num2str(V(j)),'(5,3:q(1,2)),kleur)']);hold on;
title('Bison Single Lane Change: 50 km/h');
xlabel('Time [t]');
ylabel('Yaw Angle [°]');
grid on;
axis([-2 15 -20 30]);
end;
if j == 4
subplot(4,2,4);hold on;
kleur='r';
eval(['plot(gliphoek',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),glip
hoek',num2str(i),'_',num2str(V(j)),'(5,3:q(1,2)),kleur)']);hold on;
title('Bison Single Lane Change: 60 km/h');
xlabel('Time [t]');
ylabel('Yaw Angle [°]');
grid on;
axis([-2 15 -20 30]);
end;
if j == 5
subplot(4,2,5);hold on;
kleur='g';
eval(['plot(gliphoek',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),glip
hoek',num2str(i),'_',num2str(V(j)),'(5,3:q(1,2)),kleur)']);hold on;
title('Bison Single Lane Change: 70 km/h');
xlabel('Time [t]');
ylabel('Yaw Angle [°]');
grid on;
axis([-2 15 -20 30]);
end;
if j == 6
subplot(4,2,6);hold on;
kleur='b';
eval(['plot(gliphoek',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),glip
hoek',num2str(i),'_',num2str(V(j)),'(5,3:q(1,2)),kleur)']);hold on;
title('Bison Single Lane Change: 80 km/h');
xlabel('Time [t]');
ylabel('Yaw Angle [°]');
grid on;
axis([-2 15 -20 30]);
end;
if j == 7
subplot(4,2,7);hold on;
kleur='c';
eval(['plot(gliphoek',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),glip
hoek',num2str(i),'_',num2str(V(j)),'(5,3:q(1,2)),kleur)']);hold on;
title('Bison Single Lane Change: 90 km/h');
xlabel('Time [t]');
ylabel('Yaw Angle [°]');
grid on;
axis([-2 15 -20 30]);
end;
end;%for
end;%for
%==================================PLOT YAW
ANGLE
VELOCITY=======================================
==============
figure;
V=[30
40 50 60 70 80 90];
for j=1:7
for i=1:8
eval(['q=size(gliphoek',num2str(i),'_',num2str(V(j)),')']);
if j == 1
kleur='c';
subplot(4,2,1);hold on;
eval(['plot(gliphoek',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),glip
hoek',num2str(i),'_',num2str(V(j)),'(6,3:q(1,2)),kleur)']);hold on;
title('Bison Single Lane: Change: 30 km/h');
xlabel('Time [t]');
ylabel('Yaw Angle Velocity [°/s]');
grid on;
axis([-2 15 -20 30]);
end;
if j == 2
subplot(4,2,2);hold on;
kleur='y';
eval(['plot(gliphoek',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),glip
hoek',num2str(i),'_',num2str(V(j)),'(6,3:q(1,2)),kleur)']);hold on;
title('Bison Single Lane Change: 40 km/h');
xlabel('Time [t]');
ylabel('Yaw Angle Velocity [°/s]');
grid on;
axis([-2 15 -20 30]);
end;
if j == 3
subplot(4,2,3);hold on;
kleur='m';
eval(['plot(gliphoek',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),glip
hoek',num2str(i),'_',num2str(V(j)),'(6,3:q(1,2)),kleur)']);hold on;
title('Bison Single Lane Change: 50 km/h');
xlabel('Time [t]');
ylabel('Yaw Angle Velocity [°/s]');
grid on;
axis([-2 15 -20 30]);
end;
if j == 4
subplot(4,2,4);hold on;
kleur='r';
eval(['plot(gliphoek',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),glip
hoek',num2str(i),'_',num2str(V(j)),'(6,3:q(1,2)),kleur)']);hold on;
title('Bison Single Lane Change: 60 km/h');
xlabel('Time [t]');
ylabel('Yaw Angle Velocity [°/s]');
grid on;
axis([-2 15 -20 30]);
end;
if j == 5
subplot(4,2,5);hold on;
kleur='g';
eval(['plot(gliphoek',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),glip
hoek',num2str(i),'_',num2str(V(j)),'(6,3:q(1,2)),kleur)']);hold on;
title('Bison Single Lane Change: 70 km/h');
xlabel('Time [t]');
ylabel('Yaw Angle Velocity [°/s]');
grid on;
axis([-2 15 -20 30]);
end;
if j == 6
subplot(4,2,6);hold on;
kleur='b';
eval(['plot(gliphoek',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),glip
hoek',num2str(i),'_',num2str(V(j)),'(6,3:q(1,2)),kleur)']);hold on;
title('Bison Single Lane Change: 80 km/h');
xlabel('Time [t]');
ylabel('Yaw Angle Velocity [°/s]');
grid on;
axis([-2 15 -20 30]);
end;
if j == 7
subplot(4,2,7);hold on;
kleur='c';
eval(['plot(gliphoek',num2str(i),'_',num2str(V(j)),'(1,3:q(1,2)),glip
hoek',num2str(i),'_',num2str(V(j)),'(6,3:q(1,2)),kleur)']);hold on;
title('Bison Single Lane Change: 90 km/h');
xlabel('Time [t]');
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 137 of 138 pages
ylabel('Yaw Angle Velocity [°/s]');
grid on;
axis([-2 15 -20 30]);
end;
end;%for
end;%for
THE STEERING RELATIONSHIP BETWEEN THE FIRST AND SECOND AXLES OF A 6X6 OFF-ROAD MILITARY VEHICLE
Page 138 of 138 pages
```