Design and control of a space based two link

Design and control of a space based two link
Calhoun: The NPS Institutional Archive
Theses and Dissertations
Thesis Collection
1992-09
Design and control of a space based two link
manipulator with Lyapunov based control laws
Sorensen, Dennis R.
Monterey, California. Naval Postgraduate School
http://hdl.handle.net/10945/23618
NAVAL POSTGRADUATE SCHOOL
3NIA 93943-5002
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DESIGN AND CONTROL OF A SPACE BASED
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TWO LINK
MANIPULATOR WITH LYAPUNOV BASED CONTROL LAWS
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Dennis R. Sorensen
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September 1992
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Non-Linear Control System Design, Robotics,
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Satellite Attitude
Control
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Flexible Spacecraft Simulator (FSS) at the Naval Postgraduate School
Abstract (continue on reverse
9
The
flexible
appendage with a two
was modified by replacing the
was designed to simulate motion
main body, a two link manipulator, and
link robotic manipulator. This experimental setup
of a spacecraft about the pitch axis. The spacecraft consists of a
momentum wheel with actuator to control the pitch attitude of the spacecraft. Position information from the main
body and manipulators were obtained from Rotary Variable Displacement Transducers (RVDT's). The equations
of motion were developed assuming that the main body and manipulator were rigid bodies. The resulting
coupled, nonlinear, time variant equations of motion were used to analyze the dynamics and kinematics of the
body, manipulator as well as the interaction between the two. Three different control strategies were developed
using Lyapunov's Second or Direct Method. With the first controller, simple linear feedback of position and
velocity information with constant gains was used to position the manipulator and stabilize the main body. A fifth
order polynomial was used to generate a reference trajectory for the second controller. This trajectory was used in
conjunction with a tracking Lyapunov controller to position and stabilize the system. In the third controller, a
near-minimum-time technique was used to generate a reference trajectory. This reference trajectory was employed
to design a tracking controller similar to that used in the polynomial reference controller.
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Brij N. Agrawal
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(408) 646-3338
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DESIGN AND CONTROL OF A SPACE BASED TWO LINK MANIPULATOR
WITH LYAPUNOV BASED CONTROL LAWS
by
Dennis R. Sorensen
Lieutenant Commander, United States Navy
B.S., Iowa State University, 1981
M.S. Astronautical Engineering, Naval Postgraduate School, 1991
Submitted in partial fulfillment
of the requirements for the degree of
ENGINEER OF AERONAUTICS AND ASTRONAUTICS
from the
NAVAL POSTGRADUATE SCHOOL
September 1992
ABSTRACT
The Flexible Spacecraft Simulator (FSS)
was modified by replacing
the flexible
at the
Naval Postgraduate School
appendage with
a
two
link robotic
This experimental setup was designed to simulate motion of a
manipulator.
spacecraft about the pitch axis.
link manipulator, and
the spacecraft.
The spacecraft
momentum wheel
consists of a
main body, a two
actuator to control the pitch attitude of
Position information from the
main body and manipulators was
obtained from Rotary Variable Displacement Transducers
(RVDT). The equations
of motion were developed assuming that the main body and manipulator were
rigid bodies.
were used
The
resulting coupled, nonlinear, time invariant equations of
to analyze the
motion
dynamics and kinematics of the main body and
manipulator as well as the interaction between the main body and manipulator.
Three different control strategies were developed using Lyapunov's Second
or Direct Method.
With
the first controller, simple linear feedback of position and
velocity information with constant gains
and
stabilize the
main body.
A
fifth
was used
to position the
manipulator
order polynomial was used to generate a
reference trajectory for the second controller.
This trajectory was used in
conjunction with a tracking controller to position and stabilize the system. In the
third controller, a
trajectory.
near-minimum-time technique was used
to generate a reference
This reference trajectory was employed using a tracking controller
similar to that used in the polynomial reference controller.
111
u*
ACKNOWLEDGMENTS
Many
notable
people have contributed
among
laboratory
these people
manager
but the most
to the success of this project
was Mr. Rafford Bailey.
Mr. Bailey was the
for the Spacecraft Robotics Simulator and
was primarily
responsible for the design, manufacture, and assembly of the manipulator. Mr. Pat
Hickey and Mr. John Moulton of the Department Of Aeronautics were
instrumental in the manufacture of manipulator components in a timely and
Air Force Captain Gary Yale and Dr. Hyochoong Bang
expeditious manner.
provided invaluable assistance with
many
of the theoretical and analytical
aspects for the design and analysis of the control system.
Junkins, the George
visiting
J.
Eppright Chair Professor
at
Texas
A&
Professor John L.
M University and
Co-Chair of the Space Systems Academic Group
at
the
Naval
Postgraduate School presented a firm foundation for the non-linear control
theory and Lyapunov stability theory that formed the theoretical basis for the
control law design.
Finally, to Professor Brij N. Agrawal,
who
provided the
funding, spacecraft attitude, dynamics and control expertise, and was responsible
for the overall coordination
and success of
IV
this project.
93943. 5002
;
TABLE OF CONTENTS
I.
INTRODUCTION
H.
EXPERIMENTAL SETUP
4
A.
SPACE ROBOTICS SIMULATOR DESCRIPTION
4
B.
EXPERIMENT DESCRIPTION
4
C.
HI.
1
Main Body
1.
Spacecraft
2.
Two
3.
Granite Table
10
4.
AC -
10
5.
VAX 3100 Series Computer
Link Manipulator
100
SYSTEM INTEGRATION
5
7
11
11
ANALYTICAL MODEL
14
A
COORDINATE SYSTEMS
14
B.
EQUATIONS OF MOTION
17
C.
D.
1.
Lagrange's Equations
2.
Kinetic Energy
3.
Equation
4.
Applied Torques
Of Motion
Of Motion
LYAPUNOV STABLE CONTROLLER DESIGN
17
17
18
19
20
1.
Lyapunov Stable Controller With Linear Feedback
21
2.
Lyapunov Stable Tracking Controller
24
REFERENCE TRAJECTORY GENERATION
25
1.
Polynomial Reference Trajectory
25
2.
Minimum-Time-Maneuver
29
3.
E.
IV.
Near-Minimum-Time Rigid Body Maneuver
30
ANALYTICAL MODEL SUMMARY
34
RESULTS AND DISCUSSION
36
A.
REFERENCE MANEUVER
36
B.
LINEAR CONSTANT GAIN FEEDBACK CONTROLLER
37
C.
POLYNOMIAL REFERENCE TRACKING CONTROLLER
41
D.
NEAR-MINIMUM-TIME REFERENCE TRACKING
CONTROLLER
V.
45
CONCLUSIONS
A.
52
SUBJECTS FOR FUTURE RESEARCH
54
APPENDIX A
56
APPENDIX B
A.
B.
C.
,
SYSTEM KINETIC ENERGY
1.
Body
2.
Body 2
3.
Body
4.
Total Kinetic Energy
57
57
Kinetic Energy
57
Kinetic Energy
57
3 Kinetic Energy
57
1
57
EQUATIONS OF MOTION
1.
Mass Matrix
2.
Coriolis
And
57
57
Centrifugal Acceleration Terms:
POLYNOMIAL REFERENCE TRAJECTORY
58
59
1.
Vector Polynomial Describing Tip Position:
59
2.
Boundary Conditions And Polynomials
59
3.
Vector Position
59
4.
Vector Velocity
59
VI
5.
Vector Acceleration
60
D.
TWO LINK INVERSE KINEMATICS
60
E.
NEAR MINIMUM TIME RIGID BODY MANEUVER
61
1.
Near-Minimum-Time Maneuver
61
2.
Torque Shaping Function
61
3.
Boundary Conditions
62
4.
Reference Angular Displacements
5.
Near-Minimum-Time Relationships
And
Velocities
APPENDIX C
A.
B.
C.
1.
Main Program For Linear Feedback Simulation
2.
Linear Feedback Equation
Of Motion Function
POLYNOMIAL REFERENCE TRACKING CONTROLLER
1.
Polynomial Reference Maneuver Tracker Main Program
2.
Polynomial Reference Tracking Equations Of Motion
65
65
69
72
72
Function
75
Polynomial Reference Trajectory Function
77
NEAR-MINIMUM-TIME RERENCE TRACKING
CONTROLLER
D.
64
65
LINEAR FEEDBACK MATLAB SIMULATION PROGRAM
3.
62
80
1.
Main Program
80
2.
Near-Minimum-Time Equations Function
83
3.
Near-Minimum-Time Reference Function
86
MISCELLANEOUS FUNCITONS
91
1.
Equation Of Motion Coefficient Funciton
91
2.
Inverse Kinematics Function
94
vu
APPENDIX D
95
A.
LINEAR CONSTANT GAIN FEEDBACK CONTROLLER
95
B.
POLYNOMIAL REFERENCE TRACKING CONTROLLER
99
C.
NEAR-MINIMUM-TIME REFERENCE TRACKING
CONTROLLER
103
REFERENCES
Ill
INITIAL DISTRIBUTION LIST
113
Vlll
LIST OF TABLES
TABLE 2.1
Momentum Wheel
TABLE 2.2
Link Actuator Characteristics
TABLE 2.3 Kepco
Power Supply
TABLE 2.4 Mass And
TABLE A.l
Actuator Characteristics
Characteristics
Inertia Characteristics
Servo Motor Characteristics
IX
6
8
9
13
56
LIST OF FIGURES
Figure 2.1
Spacecraft Robotics Simulator
5
Figure 2.2
Two
7
Figure 2.3
Control System Integration
Figure 3.1
Inertial
And
Figure 3.2
Inertial
Angles
Figure 3.3
Polynomial Reference Maneuver
26
Figure 3.4
Bang-Bang Minimum Time References
29
Figure 3.5
Normalized Input Shaping Funciton With
a=
0.1
31
Figure 3.6
Normalized Input Shaping Funciton With
a=
0.25
31
Figure 4.1
Reference Maneuver
Figure 4.2
Linear Constant Gain Feedback Joint Position
Link Manipulator
12
Local Coordinate Systems
And
15
Vectors
16
37
Time History
WithGp=l AndGv=5
Figure 4.3
39
Linear Constant Gain Feedback Joint Velocity Time History
WithGp=l AndGv=5
Figure 4.4
39
Linear Constant Gain Feedback Torque
Time History With Gp=l
AndGv=5
Figure 4.5
Linear Constant Gain Feedback
History With
Figure 4.6
40
Gp=l And Gv=5
40
Polynomial Reference Tracking Controller Joint Position Time
History With Gp=l,
Figure 4.7
Momentum Wheel Speed Time
Gv=5 And Maneuver Time Of 2.5
Seconds....43
Polynomial Reference Tracking Controller Joint Velocity Time
History With Gp=l,
Gv=5 And Maneuver Time Of 2.5
Seconds.. ..43
Figure 4.8
Polynomial Reference Tracking Controller Torque Time History
With Gp=l, Gv=5 And Maneuver Time Of 2.5 Seconds
Figure 4.9
Polynomial Reference Tracking Controller
44
Momentum Wheel
Speed Time History With Gp=l, Gv=5 And Maneuver Time Of
2.5
Figure 4. 1
44
Seconds
Near-Minimum Time Tracking
Figure 4.11
History With
Figure 4. 1 2
47
Controller Joint Velocity
Time
Gp=l And Gv=5, Maneuver Time Of 2.5 Seconds
Near-Minimum Time Tracking
Controller Torque
2.5
Time History
Seconds
,
And a
= 0.25
48
Near-Minimum Time Tracking
History With
And a =
Figure 4.14
,
Wheel Speed Time
Gv=5, Maneuver Time Of 2.5 Seconds
,
48
Gp=l
,
Controller Joint Position
Time
Gv=5, Maneuver Time Of 2.5 Seconds
,
49
0.1
Near-Minimum Time Tracking
History With
And a =
Controller
0.25
History With
Figure 4. 1 5
Gp=l
Near-Minimum Time Tracking
And a =
,
47
0.25
With Gp=l And Gv=5, Maneuver Time Of
Figure 4. 1 3
,
0.25
Near-Minimum Time Tracking
And a =
Time
Gp=l And Gv=5, Maneuver Time Of 2.5 Seconds
History With
And a =
Controller Joint Position
Gp=l
,
Controller Joint Velocity
Time
Gv=5, Maneuver Time Of 2.5 Seconds
,
49
0.1
XI
Figure 4. 1 6
Near-Minimum Time Tracking
With Gp=l
,
Controller Torque
Time
History
Gv=5, Maneuver Time Of 2.5 Seconds And
,
a=
50
0.1
Figure 4.17
Near-Minimum Time Tracking
History With
And a =
Figure D.l
Gp=l
,
Controller
Wheel Speed Time
Gv=5, Maneuver Time Of 2.5 Seconds
,
50
0.1
Linear Constant Gain Feedback Joint Position
Time History
With Gp=0.1 And Gv=0.2
Figure D.2
95
Linear Constant Gain Feedback Joint Velocity
Time History
With Gp=0.1 And Gv=0.2
Figure D.3
96
Linear Constant Gain Feedback Torque
Time History With
Gp=0.1 AndGv=0.2
Figure D.4
Linear Constant Gain Feedback
History With Gp=0.1
Figure D.5
98
Gp=l
,
Gv=5 And Maneuver Time Of 5 Seconds
99
Gp=l
,
Gv=5 And Maneuver Time Of 5
Seconds... 100
Polynomial Reference Tracking Controller Torque Time History
With Gp=l
Figure D.8
And Gv=0.2
Polynomial Reference Tracking Controller Joint Velociy Time
History With
Figure D.7
Momentum Wheel Speed Time
Polynomial Reference Tracking Controller Joint Position Time
History With
Figure D.6
97
,
Gv=5 And Maneuver Time Of 5 Seconds
Polynomial Reference Tracking Controller
Speed Time History With Gp=l
5 Seconds
,
101
Momentum Wheel
Gv=5 And Maneuver Time Of
102
xu
Near-Minimum Time Tracking
Figure D.9
History With
a=
,
1
,
Controller Joint Velocity
Time
Gv=5, Maneuver Time Of 5 Seconds
,
And
104
0.25
And a =
,
Controller Torque
Time History
Gv=5, Maneuver Time Of 5 Seconds
,
0.25
1
Near-Minimum Time Tracking
History With
Figure D. 1 3
Gp=l
Near-Minimum Time Tracking
a=
And
103
With Gp=l
Figure D. 1 2
,
0.25
History With
a=
Time
Gv=5, Maneuver Time Of 5 Seconds
Near-Minimum Time Tracking
Figure D. 1
Figure D. 1
Gp=l
Controller Joint Position
Gp=l
,
Controller
Wheel Speed Time
Gv=5, Maneuver Time Of 5 Seconds And
,
106
0.25
Near-Minimum Time Tracking
History With
Gp=l
,
Controller Joint Position
Time
Gv=5, Maneuver Time Of 5 Seconds And
,
a = 0.1
Figure D. 14
107
Near-Minimum Time Tracking
History With
Gp=l
,
Controller Joint Velocity
Time
Gv=5, Maneuver Time Of 5 Seconds And
,
a = 0.1
Figure D. 15
Figure D. 1 6
108
Near-Minimum Time Tracking
With Gp=l
05
,
Time History
Gv=5, Maneuver Time Of 5 Seconds And
,
Near-Minimum Time Tracking
History With
Controller Torque
Gp=l
,
Controller
a = 0.
1
.
1
09
1
10
Wheel Speed Time
Gv=5, Maneuver Time Of 5 Seconds And
,
a = 0.1
xm
INTRODUCTION
I.
During the past few years there has been a significant increase
robotics.
in the
use of
Applications range from performing routine tasks in manufacturing to
The
deep sea and interplanetary space exploration.
extraterrestrial
and
environment has become the focus of research for future industrial
development and
scientific exploration.
and cost of manned space
upon robotics
interplanetary
With the hazards of
flight, researchers will
become
for assembly, service, and repair of
the exploration of space
itself.
Due
this
environment
increasingly dependent
equipment
in space as well as
to the requirements for terrestrial
and space
applications, there has been a significant increase in theoretical and experimental
research in the areas of robotic dynamics and control.
Space based robotic applications
stabilize the manipulator.
energy that
is
terrestrial applications in
A
space based manipulator,
when
forces on the spacecraft. In addition, there
added
to the system.
To
is
momentum wheel
is
is
one
provided to
repositioned, imparts
no
friction to dissipate
counteract these forces and moments, an
system normally consisting of thrusters
attitude control
in
combination with a
used to stabilize the spacecraft. Control system problems are
exasperated as the mass of the load that
relation to the
from
For the space based applications, no support
important area.
moments and
differ
mass of
the spacecraft.
is
being positioned becomes larger in
It is this
interaction
between spacecraft and
the motion of a manipulator that warrants further research.
The primary
objective for any control system
range of operating conditions while
performance.
It is
still
is
to
remain stable over a wide
providing adequate levels of
desired to meet this objective in the face of
hardware
characteristics,
changing loads as well as unmodeled disturbances and system
dynamics. These requirements and restrictions present the control system design
engineer with significant challenges.
Before the system can be analyzed, the equations of motions must be
determined. The equations of motion for a robotic system can easily be developed
through Lagrange's equations and are in the form of a set of second order
differential equations.
These equations are coupled and nonlinear with
trigonometric and higher order terms. Attempts to simplify these equations result
in equations of
motion
that are valid over a limited range of
boundary conditions.
The current
motion or for specific
trend in trajectory control requires highly
nonlinear maneuvers that are valid over a wide range of applications and
operating conditions.
These requirements dictate
that the full,
nonlinear
equations be used to describe the motion of the system. With the introduction of
the nonlinear equations of motion,
many
traditional tools in control theory used
to analyze linear, time-invariant systems are not available or are meaningless.
Recently, research by Junkins [Ref.
in using
1]
and Bang [Ref.
2] has revived interest
Lyapunov's second method for a flexible structure control system design.
This technique
is
very attractive because
it
can be applied to nonlinear, time
invariant, systems with guaranteed stability for a
important feature of Lyapunov's second method
Lyapunov function and
wide range of conditions.
is
the
freedom
the corresponding feedback control law.
An
to select the
The Lyapunov
function can be selected based on physical insight and the control law can be
selected to ensure that the system
positive definite and
is
is stable.
normally related
mechanical natural systems.
The
to the
The Lyapunov function must be
system energy for a large class of
control law can be selected such that the
Lyapunov function
or system energy will always decrease to zero or
some
equilibrium point.
The purpose of
Lyapunov
this thesis is to
apply a general methodology for finding
stable control laws for stabilizing the spacecraft
controlling a
two
main body while
link manipulator attached to that spacecraft.
description of the experimental setup
is
discussed in Chapter
the physical characteristics of the manipulator,
main body,
simulation, and data collection equipment. In Chapter
and the equations of motion are developed.
III,
II.
A
complete
Topics include
actuators, sensors, test,
the coordinate systems
Three different control strategies
were developed using Lyapunov's Second or Direct Method.
With
the first
controller, simple linear feedback of position and velocity information with
constant gains was used to position the manipulator while stabilizing the main
body.
A
fifth
order polynomial was used to generate a reference trajectory for the
second controller.
This trajectory was used in conjunction with a tracking
controller to position and stabilize the system.
minimum-time technique was used
In the third controller, a near-
to generate a reference trajectory.
This
reference trajectory was employed with a tracking controller similar to that used
in the
polynomial reference controller.
Chapter IV. Chapter
for future research.
V
includes a
Simulation results are presented in
summary
of the conclusions as well as topics
II.
A.
EXPERIMENTAL SETUP
SPACE ROBOTICS SIMULATOR DESCRIPTION
The Spacecraft Robotics Simulator (SRS)
is
a modification to the Flexible
Spacecraft Simulator (FSS) used for previous work by Hailey [Ref.
Watkins [Ref.
4].
and replacing
this
central
The FSS was modified by removing
is
and
the flexible appendage
appendage with a two link manipulator. The SRS consists of a
main body with a two
main body
3],
link robotic manipulator.
provided by a single
momentum wheel
Pitch axis control of the
driven by an electric servo-
motor. The central body was constrained to rotational motion only by an I-beam
mounted over
the over the granite table.
supported by
air
bearings that float upon a thin cushion of air on an optical
quality granite surface.
A
servo-motors.
The main body and manipulator were
Each of
the
two
links
were positioned via geared
Rotary Variable Displacement Transducer
(RVDT) was
obtain position information at each joint for position feedback.
designed to simulate a zero-gravity environment
in
EXPERIMENT DESCRIPTION
The SRS
is
composed of the following major components:
Spacecraft
Two
Main Body
Link Robotic Manipulator
RVDT Position Sensors
Granite Table
Electrical
Power Supplies
AC- 100
VAX 3100 Series Computer
used to
This setup was
two dimensions. The SRS
depicted in Figure 2.1.
B.
DC
is
Figure 2.1 Spacecraft Robotics Simulator
1.
Spacecraft
Main Body
The spacecraft main body
aluminum
diameter,
disk.
consists of a rigid, 7/8 inch thick, 30 inch
The main body
is
dimensional planar motion of a spacecraft about
supported by three
bearings
is
air
designed to simulate the two
its
pitch axis.
The main body
bearings spaced at 120 degree intervals.
capable of supporting a load of 60 pounds.
A
Each of
is
the
fourth air bearing
supported by an overhead I-beam constrains the spacecraft main body to
rotational
body on a
motion only. The
air
bearings are designed to float the spacecraft main
thin film of air supplied
R30D, was connected
by an external
air source.
to the rotor of the air bearing
A RVDT,
model
by a bellow-type device.
The
RVDT was manufactured by
Schaevitz Sensing Systems was used to measure
angular displacements of the spacecraft main body.
Attached to the spacecraft main body was a 10.7 kilogram
momentum wheel and
a torque to the
momentum
servo motor.
The momentum wheel was designed
to apply
main body by increasing or decreasing the angular velocity of
wheel.
The motion of
the spacecraft about
controlled by the torque generated by this
model JR16M4CH/F9T used
PMI
steel
to drive the
industries. Characteristics of this
momentum
its
wheel.
pitch axis
the
was
The servo motor,
momentum wheel was manufactured by
motor are presented
in Table 2.1.
TABLE 2.1 Momentum Wheel Actuator Characteristics
Units
Characteristic
PMI Motion
Manufacturer
Technologies
JR16M4CH-1
Model Number
Rated Speed
rev per minute
3000
Rated Power
horse power
1.4
Rated Torque
inch-pound
31.85
Rated Current
amps
7.79
Rated Voltage
volts
168
Outside Diameter
inches
7.4
Height
inches
4.5
Weight
pounds
17.5
An
integral analog tachometer,
Industries, Inc.
was mounted on
model ARS-C121-1A, manufactured by Watson
the servo-motor to
measure angular
velocity.
A
more
detailed description of the motor,
body can be found
2.
Two Link
Attached
the manipulator
Department
in [Ref. 3],
and [Ref.
wheel, and spacecraft main
4].
Manipulator
to the
main body was a two
were designed and
at the
momentum
built
link manipulator.
Components
for
by the Aeronautics and Astronautics
Naval Postgraduate School. All components for the arm were
manufactured from 7075 and 6061 series aluminum. All components were
connected with
SAE
grade 8
medium carbon chrome
alloy cap screws.
of the manipulator can be found in Figure 2.2.
Figure 2.2
Two Link Manipulator
A
picture
The
links of the manipulator
manufactured by PMI.
mounted on
pointing
A
were positioned by two geared, servo-disk motors
third motor, identical to the first
the tip of link
two
for the purpose of
some type of antenna.
At each
presented in Table 2.2.
two actuators and was
orienting a simulated tool or
Characteristics for the link actuators are
joint, a
RVDT, model R30D,
manufactured by
Schaevitz was used to measure relative angular displacement.
TABLE
2.2
Link Actuator Chars icteristics
Units
Characteristic
PMI Motion Technologies
Manufacturer
9FGHD
Model Number
Rated Speed
rev per minute
22
Rated Torque
inch-pound
80
Rated Current
amps
5.6
Rated Voltage
volts
12
Outside Diameter
inches
4.75
Height
inches
3
Weight
pounds
3.2
The power
supplies used to drive the actuators of the manipulators were
manufactured by Kepco
Inc. of Flushing,
Bipolar Power Supplies were designed to be
linear amplifiers.
The
BOP
power supply
New
fast,
is
an
York.
The Kepco
programmable,
all
series
BOP
fully dissipative,
solid-state design, featuring
integrated circuit operational amplifiers in the control circuit section and silicon
power
transistors
mounted on
special fan-cooled heat sinks in the
8
complementary
power
stage.
Characteristics for the
Kepco model
BOP
20-10 power supply are
presented in Table 2.3.
TABLE 2.3
Kepco Power Supply
Output Power
watts
200
Max. Input Current
amps
5.5
Max. Input Power
watts
540
DC. Output Range
volts
±20
amps
±10
volts/volt
2
amps/volt
1
Closed Loop Gain
Bandwidth
Rise
BOP
18 (voltage
kilohertz
Time
Recovery Step Load
The
Characteristics
kilohertz
6 (current mode)
microseconds
20 (voltage mode)
microseconds
60 (current mode)
microseconds
80 (voltage mode)
microseconds
20 (current mode)
can be operated in either the voltage or current
bipolar control channels. These
mode)
modes
mode through two
are manually selectable through the front
control panel or through remote signals. Each of the principal control channels
is
protected by bipolar limit circuits. All control and limit channels are connected to
the output stage via an "Exclusive-Or" gate so that only one channel
of the
full
BOP
output at any one time. The
output range by a
±10
BOP
output can be
volt signal applied to either
is in
control
programmed over
one of the inputs
its
to the
voltage or current channel.
by a
to
+10
The
limit control channels can be remotely controlled
volt signal applied to there respective inputs.
Granite Table
3.
The
entire mechanical
manipulator were supported by
air
assemblage, including the main body and
bearings that float on a thin cushion of air on a
granite table with dimensions of 8 feet by 6 feet by 10.5 inches thick.
of the table was highly polished to optical quality grade
valley).
The smooth
A
The
surface
(0.001 inch peak to
surface allows the air bearings to float freely over the surface
of the granite table to minimize the effects of friction on the motion of the main
body and manipulator.
The
gravity induced accelerations.
granite table
was carefully leveled
The mass of the
to eliminate
table provided an extremely stable
platform upon which to conduct the experiments.
4.
AC
-
100
The AC- 100
is
a microprocessor based, programmable, real time control
system manufactured by Integrated Systems International,
California.
The AC- 100 was designed
to
Inc. of
San Jose,
automate the development of real-time
systems by combining graphical modeling tools with a real-time controller. In
addition to modeling and controlling, the
collection and storage.
Intel
The AC-100
AC- 100 was
also capable of data
consists of the following major components:
80386 Application Processor,
Intel
80386 Multibus
II
Input
/
Output
Processor, Intel 80386 Communication Processor, Intel 80387 Coprocessor,
Weitek 3167 Coprocessor, Analog To Digital and Digital To Analog Digital Data
Translation
DT2402
Input/ Output Board,
Two - INX-04 Encoder
and Digital To
Analog Servo Boards, Ethernet Interface Module, and Cabinet Enclosure and
Power Supply
10
The software
a
AC- 100
tools used with the
include a Design Package and
Run-Time Package. The Design Package included Matrix x System
Build, and
,
Auto Code.
These tools are used for analysis, design, and code generation
The Run-Time Software Package provides
respectively.
Interface
(GUI) cross compiler, device
the Graphical User
drivers, data acquisition,
interface required to run software code generated
and Ethernet
by Auto Code on the AC- 100.
VAX 3100 Series Computer
5.
The
VAX
3100 Series Model 30 workstation was configured with
8
megabytes of main memory, a 19 inch (diagonal) color monitor, two 104 megabyte
The
Winchester hard disks and a mouse.
VAX
workstation
is
capable of 2.8
Million Instructions Per Second (MIPS).
C.
SYSTEM INTEGRATION
The AC- 100
interface.
is
integrated with a
In this system, the
the control system.
VAX
VAX
3100 Series computer
3100 computer was used
Auto Code was then used
linked to the
form of
in turn
C
-
AC- 100
were used
is still
the Ethernet interface.
2.3.
two
to
The
A
system
This code was then
control system, in the
conuol the servo-amplifiers which
provided through the
VAX
Manual
computer acting through
block diagram of the integration of the control system
link manipulator and spacecraft
Mass and
and design
to drive the actuators of the manipulator in real time.
control of the system
for the
via the Ethernet interface.
code software can then be used
to analyze
to convert the final control
design into fully compiled and linked "C" computer code.
down
via the Ethernet
inertia characteristics are
main body are presented
summarized
11
in
Table 2.4
in Figure
Reaction
Wheel
VAX-3100
Matrix
X
System Build
Autocode
RVDT
RVDT
/
Motor
RVDT
/
Motor
/'Motor
Granite
Table
Figure 2.3 Control System Integration
12
TABLE 2.4 Mass And
Body
Arm
Characteristic
Units
Value
Mass
kg
2.09
Inertia (1)
kg-m^
0.3102
Inertia (2)
kg-m^
0.032
cm
36.45
Mass
kg
2.47
Inertia (1)
kg-m^
0.3542
Inertia (2)
kg-m^
0.054
Of Mass(3)
cm
34.90
Mass
kg
52.73
Inertia (1)
kg^n^
0.3542
Mass
kg
10.67
Inertia (1)
kg^n^
0.0912
1
Center
Arm2
Center
Main Body
Momentum Wheel
Of Mass(3)
20
cm
Moment Of Inertia About Arm Axis Of Rotation.
Moment Of Inertia About Center Of Mass.
Center Of Mass Location With Respect To Axis Of Rotation.
Center
Notes: (1)
(2)
(3)
Inertia Characteristics
Of Mass
13
ANALYTICAL MODEL
III.
The Spacecraft Robotics Simulator (SRS)
which
is
consists of a central
main body
The motion of
main body
attached a two link robotic manipulator.
move
constrained to
in a plane resulting in a
the
to
is
system with three degrees of
freedom. This planar motion constraint greatly simplifies the problems associated
with the derivation of the equation of motion and control system design yet
retains the
modeled
most
critical analytical
as rigid structures.
elements.
still
The manipulator and main body were
Lagrange's equation was used to derive the
equations of motion. This section will
first
describe
how
the equations of motion
were derived and then how the control system was developed.
A.
COORDINATE SYSTEMS
Before the spacecraft attitude and manipulator can be controlled the
dynamics of the system must be carefully defined and understood. The
in this process, is to
determine the equations of motion for the system.
In this model, the
main body
different coordinate systems
this
problem was fixed
is
constrained to rotational motion only.
were used for
in inertial
this analysis.
xi, yi, z\,
is
fixed in the
xi axis pointing toward the attachment point for link
is
The Ni, N2, N3
Four
axis for
space coincident with the axis of rotation of the
main body. The coordinate system,
system
first step
1.
The
main body with
the
xi, yi, z\ coordinate
obtained by rotating about the N3 axis of the inertial coordinate, by an
angle of 61
.
In a similar
axis of rotation of link
1
way, the coordinate systems, X2, yi,
Z2, is fixed at the
and points toward the attachment point for link
2.
The
x 2> Y2> Z2 coordinate system can be obtain by rotating about the N3 axis of the
14
.
inertial
coordinate system by an angle of 02-
fixed in the
2.
body of
link
The coordinate system,
X3, y3, Z3,
is
2 with the X3 axis pointing toward the end point of link
This coordinate system
is
obtained by rotating by an angle,
3
about the N3
axis of the inertial system. Coordinate systems are presented in Figure 3.1
O
Servo Motor
Manipulator
Arm
Local Coordinate Axis
Figure 3.1 Inertial
And
Local Coordinate Systems
All angles, Q\, 62, and (%, as well as vectors r \
,
r*2,
and F3
are defined in terms
of inertial coordinates where these quantities are defined in Figure 3.1
15
A
Center
O
Of Mass
Servo Motor
fr* --"
^\
Manipulator
?
/A
Inertial
Arm
N2
|/\ e
rl/
Momentum
m -2^/Linkl
yf/^\
lx
Wheel
/.
y^\^2
As>
o
./^
\
*^
\
\
ei
Satellite
r2,
T
Y*
^P cm.
Main Body
and r3 describe
832
=
021
=02-01
63
-
02
i
Jr
Figure 3.2 Inertial Angles
,
X
S>Ni
_^^^^^k.
N3
>v
vectors ri
3
rv,
c-
The
#32
Axis
/
y'
Link 2
l
Cm
And
Vectors
the location of the center of
the manipulators in terms of inertial coordinates.
mass of each of
The angles 621, 631, and 632
represent relative displacements of the joints and can be derived from the inertial
coordinates 81, 62, and 83 in such a
@32
=
63
-
way
@2-
16
that 621
= 02
-
61, 631
=
63
-
9i,
and
B.
EQUATIONS OF MOTION
A momentum
The motion of
body.
the
wheel was used
momentum wheel
assumed
It is
1.
is
main body.
de-coupled from the motion of the main
momentum wheel
that the
control the motion of the
movement
to apply a control torque to the
is
only an actuator that
is
used to
main body and counteract torques generated by
the
of the manipulator.
Lagrange's Equations Of Motion
Lagrange's equation for n-dimensional dynamic systems are stated as
4j—
—=
-
dtldqi/
Qj where i =
,
1, 2, 3,...,
n
(3.1)
dqi
L = T-V
V
-
Potential
T
-
Kinetic Energy
qi
- i
qj
- 1
Generalized Coordinate
th
- i
Qi
For
this particular
to describe its
velocities are
2.
Energy
Generalized Velocity
Applied
Non
conservative Force
problem, the system requires three degrees of freedom
governing equations of motion.
chosen
to
be in terms of
T
q = {ei,e 2 ,e 3
}
q = {ei,e 2 ,e 3
)
T
inertial
The generalized coordinates and
coordinates as presented below.
= (qi,q2,q3)T
= {qi,q2,q3)
T
(3-2)
(3-3)
Kinetic Energy
The
total kinetic
first
step
is
to determine the total kinetic
energy of the system
is
given by
17
energy of the system. The
T=XTi
(3.4)
i=l
Ti =
2
ll
ei
i
:th
Ii - i
i
=
i
(3.5)
.?i)
i
Mass
1
X f
rir i=
3.
lm (?
Mass Moment Of Inertia About Center Of Mass
:th
mi -
A more detailed
+
1
=
1
(3.6)
1? l^Bi 0j (y 4 -yj)
1
algebraic procedure can be found in
Appendix B.
Equation Of Motion
Lagrange's equations can be applied to equation (3.4) and the terms can
be arranged in matrix form so that the
M(e) e + G(e,
e)
=
final
form can be written as
q
(3.9).
(3.9)
where
rriiimnmn
M(6) =
rri2i rti22 rri23
L11131
m32rri33
(3.10)
and
mn = II
+ L?
[mi
+ m2 +
m22 = Icm2 + L2 m2
1113]
'^1
U
+
rri4
L4
(3.11)
+m3
(3.12)
I
™33 =
Ic.m.3
mn = m2i =
mi 3
=m
3
i
+
Lc.m.3
m
(3.13)
3
Li L2 cos(G2i)
^ ksLi +m
= Li Lcm3 cos(9 3 i) m3
18
3
(3.14)
(3.15)
m2
3
= m32 = L2 Lcm
3
m
cos(6 32 )
(3.16)
3
G(l)
G(9,9) = G(2)
(3.17)
LG(3)
G(l)=[(m 2 L L c m )+(m 3 L
1
.
.
2
1
L2)]sin(e2i)e2 +
(3.18)
LiL c m
(m 3
.
.
)sin(e 3 i)e 3
3
G(2)=-[(m 2 L L c m )+(m 3 L L 2 )]sin(e 21 )ei
1
.
.
1
2
(3.19)
•2
(m 3 L 2 L c m )sin(e 31 )e 3
.
G(3) =
-
(m 3 L i L cm
.
3)
.
3
sin (6 2 i) 6!
-
(3.20)
(m 3 L 2 L cm )sin(e 32 )e 2
.
4.
3
Applied Torques
D' Alembert's principle for virtual work expression was used to determine
the expressions for
Q
for the equations of motion.
As
written,
Q
is
a vector
containing the torques applied by the actuators at each joint in terms of inertial
coordinates.
At
this point
local coordinates.
The
it
will be beneficial to rewrite the
virtual
work applied
to the
Q
system
vector in terms of
is
given by the
following equation.
5w = X,QiSe ^i;5w
1
i=l
The elements of
Q
i=l
are in terms of torques applied to inertial coordinates.
each actuator applies a local torque,
Q
in
(3.21)
j
it
was
beneficial to rearrange (3.21) and write
terms of the local torques applied by the individual actuators.
19
Since
The
virtual
work described
in
equation (3.21) can be described in terms of these local torques
as follows.
5W
=Ui6ei-u 2 56i
1
(3.22)
5W 2 =
112582-113862
(3.23)
5W 3 =
u 3 59 2
(3.24)
In matrix format
Q
can be written in terms of a control influence matrix
B
and the
local torque vector, u as written in (3.25).
Q
=
Bu=[Qi,Q 2 ,Q 3r
Qi =
111
Q2 = u2
(3.25)
-u 2
(3.26)
u3
(3.27)
-
Q3 = U 3
(3.28)
where
1-1
B=
1
Ul
u = u2
-1
LO
Lu 3
1.
C LYAPUNOV STABLE CONTROLLER DESIGN
This design for a stable non-linear controller
Stability Theory.
This theory
method. This theory
is
is
also
known
as
covered in greater detail
is
based upon Lyapunov's
Lyapunov's Second or Direct
in [Refs. 1, 2, 14,
and
15].
To
review the Lyapunov Stability Theory, a system without any external forces or
torques
this
is
assumed. This system
is
assumed
system, a positive definite function
is
20
to
have a single equilibrium
assumed
to be
state.
For
an exact integral of the
system under some idealization. This function
is
selected to satisfy the requirements that
positive everywhere else. This function
is
zero
it is
may
termed a Lyapunov function and
at the
desired equilibrium and
be represented by the system
total
energy or the Hamiltonian of the system for most of the time invariant mechanical
systems. If this system
some
increased to
Lyapunov
perturbed from
its
state, the
energy
state is
Depending upon the nature of
positive energy level.
system dynamics dictate that the
not increase with time, then the system
•
equilibrium
the
function, one of the following conclusions can be drawn.
If the
•
is
If the
initial
energy of the system does
is stable.
energy of the system monotonically decreases with time for
conditions, and eventually goes to zero, then the system
is
all initial
asymptotically
stable.
•
If the
energy increases for any
•
If the
energy measure neither increases, nor decreases as a function of time
initial
condition, then the system
is
unstable.
then no conclusion can be drawn.
Despite the power of
Lyapunov functions
description of the
found
1.
in [Refs.
1
this theory, there is
no unified process
that globally satisfy stability requirements.
Lyapunov
and
to find candidate
A
more complete
stable controller design for flexible structures can be
2].
Lyapunov Stable Controller With Linear Feedback
For
this application, a
simple non-linear controller with linear feedback of
position and velocity information
Lyapunov function
is
for this application
u
is
the primary objective.
of the form
= E + f(5e,,5e2,5e 3
21
The candidate
(3.29)
)
where E
is
defined as the "work energy" of the Lyapunov function and
positive function of
that renders the
80 1, 802, and
803. Note that "f
Lyapunov function
=
equilibrium point. In addition, 80j
'
is
"f" is a
a pseudo potential energy
positive definite with respect to the
-
0j
0j f
where
,
0j, is
new
a constant that describe
the final joint angle.
(ei,
2 , 03, ei,
2,
The Lyapunov function
U
=
The "work energy
e 3) f =
(0if,
2 f,
o, o, o)
E+ V -^80!
rate",
(ui
-
0.30)
can be differentiated to obtain
in (3.30),
(3.31)
E, of (3.31) can be directly obtained from (3.22) through
(3.24) and the "pseudo energy rate" of the system
E=
e 3 f,
u2 ) 80i +
(u 2
"Pseudo Energy Rate" =
defined in (3.33).
+ u 3 80 3
u 3 ) 80 2
-
is
(3.32)
-^- S0j
JT
i=l
^
(3.33)
l>
Equations (3.32) and (3.33) can be substituted into (3.31) to form equation (3.34)
which can be further simplified
U
=
(ui
- 112)
to equation (3.35).
+
80i
(112 -
3
= 80i Ul
-
U2 +
+
113
803
(3.34)
.
?f
£
U
U3) 802
5(50.)
-7
+ 80 2 U 2
r
"
U3 +
^663
^50!
(3.35)
+ 80 3 |u 3 +
3(563),
Based on
which
is
(3.35),
it is
evident that a function can be selected such that
the stability condition in the
Lyapunov
laws are chosen to satisfy the following relations
22
sense.
U
<
0,
Therefore, the control
"
Ul
U2
+
= - g '- 8ei
(336)
= - g2 8G2
(337)
3(lj
-
U2
U3
+
«
-
^Jr*-
803
(338)
where,
gu >
0,
g 2v
>
0,
g 3v
>
0, g,
>
0,
g 2p
>
0,
and g 3p >
for
f>0
In addition,
U
By
=
2
(
2
2
+ g 2v 60 2 + g 3v 56 3
gi ¥ 89!
(3.40)
)
selecting an appropriate function for "f" equation, the stabilizing control laws
(ui, U2, U3) satisfying the
U
<
requirement can be
built.
For
this case,
assume
that "f" is defined as follows
f=i(gipSe?+g2 P 8e 22 +g 3p 59?)
where
gi P
>
0,
g 2p
>
0,
g 3p
>
For a controller using pure linear feedback, (3.36-3.38) can be simplified
as
described below.
U3=-g*89 3 -g3»893
(3.42)
U2=u 3 -g*582-g2,8e 2
(3.43)
ui=U2-gi 5e -g 69
(3.44)
p
1
lv
1
23
5Gi
=
6i
-
9 if
50i
=
6i
-
6
,
fori
=1,2,3
(3.45)
(3.46)
if
where
6^
2.
=
fori
= 1,2,3
Lyapunov Stable Tracking Controller
In a similar
way, the Lyapunov stable tracking controller can also be
derived.
Lyapunov
[Ref.
In this application, a function generator
1].
stability is not
proven here but
trajectory for the controller to follow.
polynomial was used
With
this
is
discussed in
was used
detail in
to generate a reference
type of controller, a
to generate a desired tip trajectory with a
loop inverse kinematic solution in one case.
more
two
fifth
order
link closed
In another case a "near- minimum-
time" torque shaping scheme was utilized to generate a reference trajectory. Both
these trajectory generators will be discussed in
The
more
detail in following sections.
control torques required for this reference tracking controller are presented
next.
5u 2 = 5u 3
-
g 2p 5e 2
8ui = 8u 2 -gi p
-
g 2v 50 2
8ei-g l¥
89i
(3.48)
(3.49)
where
89i
=
6 iref
(3.50)
861
= 61-6^
(3.51)
8ui
= Ui-u^
(3.52)
6i
-
24
The
control laws presented above can be rigorously proven to satisfy the
Lyapunov
Stability condition if 50j
and 50j are very small.
REFERENCE TRAJECTORY GENERATION
D.
For the tracking controller developed previously while discussing the
Lyapunov
stable tracking controller requires
some reference
Here
trajectories.
trajectory refers to a time history of position, velocity, and acceleration for each
degree of freedom. The primary consideration for generating a trajectory
must
the fact that the trajectory
calculated.
In this thesis,
two
first
of
all
be smooth and second
it
lies in
must be
easily
different techniques will be used to generate the
required trajectory information to stabilize and control the system.
The
first trajectory
polynomial
that will be discussed involves using a fifth order
to describe the path of the tip of the link three of the manipulator.
the second trajectory, a near-minimum-time
In
maneuver using input torque shaping
will be used to generate the desired trajectory.
1.
Polynomial Reference Trajectory
For the tracking controller discussed
difficult if not impossible to obtain the
in the previous section,
it is
usually
open-loop solutions for the theoretical
reference system of differential equations. For practical considerations
it is
often
advantageous to design the control system that will follow an easy to calculate
path.
For robotic applications,
a
polynomial of order 3 or higher
is
often used to
specify the position of the end of the manipulator. For this application, imagine a
two
link manipulator attached to the spacecraft
3.3.
In this case, the
maneuver attempts
main body
to position link
as depicted in Figure
two from an
of 20 degrees to a final position of 40 degrees and link three
40
to
60 degrees. For
a
two
is
initial
angle
maneuvered from
link manipulator, there exists a closed loop solution
25
to the inverse kinematic
closed loop solution
is
problem
that
can be quickly and easily computed. The
described in more detail in appendix B.
For
this
manipulator, given the beginning and final coordinates of the manipulator, the
vector r
of time.
is
used to specify the position of the end point of link two as a function
This vector
is
presented in Figure 3.3.
Reference Maneuver
Polynomial
•-.._
N?
N,
Figure 3.3 Polynomial Reference Maneuver
26
1
.
?(t)=?(to) + f (t)[F(tf)-7(to)]
To
simplify the calculations,
given the
start
time
x
=
to
t
(3.53)
can be written as a function of normalized time,
and finish time
Note
tf.
that for to
<
t
<
tf,
then
<T<
^^
T,
1
(3.54)
t f-
A
fifth
order polynomial was used for this application. This allowed the
user to specify the beginning and final velocities and accelerations as well as the
beginning and final positions of the
the
tip.
The polynomial
for this controller
was of
form
f(x)
and was subject
= Ci +
C2 X
+
C3 X 2
to the following
f(0)
=
0,f(0)
=
+
C4 X 3
+
C 5 X4
+
C 6 X5
(3.55)
boundary conditions.
0,f(0)
=
f(l)=l,f(l) = 0,f(l) =
The
resulting expression for f(x)
is
obtained as
f(x)=10x 3 -15x4 + 6x 5
(3.56)
Equation (3.56) can be differentiated with respect to time.
These
expressions were utilized to calculate velocities and accelerations and are
presented in equations (3.57) and (3.58).
f(x)
= 30x 2 -60x 3 + 30x4
f(x)
= 60 x- 180
x2
+ 120
x3
(3.57)
(3.58)
Equation (3.53) can also be differential with respect to time to give the velocity
and acceleration of the
tip
of link three purely as a function of time.
Vector Position:
27
1
F(T)=-?(to)+f(T)[F(tf)-7(t
(3.59)
)]
Vector Velocity:
r(T)
=
r(tf)-r(to)~
f(T)
(3.60)
tfto
Vector Acceleration:
?(T)
=
(tf)-r(to)
f(T)
(3.61)
2
L
With
(tf - to)
the position of the tip
and 632 can be solved for directly
known
as a function of time, the angles 821
and 63 via the two link closed loop
as can 62
solution of the inverse kinematics problem.
assumed
At
this point 61, 61,
The Jacobian can be used
to be comparatively small.
and 61 are
all
to define the
relationship between the velocity and angular velocity as well as the tip
acceleration and angular acceleration as described in the following equations.
r
where
H
is
=Hje|
(3.62)
the Jacobian corresponding to the given configuration.
H=
- 1
2
- 13
sin (63)
(3.63)
12
r
sin (6 2 ),
cos (8 2 ),
I3
cos (e 3 )
=H(0) + H(e]
-
H=
2
cos (e 2 ),
(3.64)
- 13
cos (e 3 )
(3.65)
- 1
2
sin (0 2 ),
With the above equations,
- 13
sin (83)
the position (angular and cartesian), velocity
(angular and cartesian), and acceleration (angular and cartesian), can
determined as functions of time.
With
28
this information,
all
be
and the physical
B
'
characteristics of the system, the reference torques can also be determined using
equation presented below.
_1
=
u re f
[m
(e)
e + G(e,e)]
0.66)
Minimum-Time-Maneuver
2.
For an ideal case, of a single degree of freedom system, the
minimum
time
achieved by applying the
maximum
available torque for one-half of the time required to complete the
maneuver
required to perform a particular maneuver
maximum
followed by the remaining half with the
a controller where the torque
rise to a torque
is
is
always operated
maximum
at its
value and gives
shaping function, position, velocity, and accelerations profiles as
presented in Figure 3.4. This type of controller
bang
negative torque. This results in
is
sometimes referred
to as a bang-
controller.
1.2
_,.•
0.8
|
0.4
^
*
y
^
•*
*
.
.-'''
"S.
E
^-
<
..._
1
-0.4
-
—
ahape
-
-
-
:
1
unction
__
-
Referer ce Angle
£
Referer ce Velocity
-0.8
Rpfprpnrp Arrplpratinn
-1.2
:
1
-0.2
0.2
0.4
0.6
0.8
Normalized Time
Figure 3.4 Bang-Bang
Minimum Time
29
References
1.2
For an application with a single degree of freedom, the relationship
between the angular acceleration and the applied torque
(3.67) with the applied torques given
by equation
is
given by equation
(3.68).
I6 = u
(3.67)
<
Umax, for
<
t
£
2
/
U =
\
-U max for
,
0, for
Bang-Bang control theory
detail in [Refs.
3.
1
and
^<t<
<
tf
for
(3.68)
tf
t
minimum
time maneuvers
discussed in more
2].
Near-Minimum-Time Rigid Body Maneuver
A
more general case of
the bang-bang,
controller.
manipulator from the
initial to final
time controller
controller does
positions in the
primary drawback of the bang-bang controller
trajectory.
minimum
The bang-bang
near-minimum-time
which
is
is
the
maneuver
the
minimum amount
is
of time.
The
the rapid rise of the torque
This results in a rapid acceleration followed by a rapid deceleration
will require actuators with instantaneous switching capacity
which
is
practical and a robust structural design for the manipulators themselves.
not
By
introducing an input shape function, the instantaneous rise in the torque
trajectory can be reduced, resulting in slightly smaller accelerations
which
will in
turn require smaller actuators and reduced structural requirements.
comaparison, the input shaping function for
in Figures 3.5
The
a=
0.1
and
a=
For
0.25 are presented
and 3.6 respectively.
differential equations
above can be modified so
I 6ref
used to describe the
minimum
time maneuver
that they are of the form.
=
Uref
= U max
f(At,t f ,t)
30
(3.70)
—
\
"
i
,
1.2
---..
/
0.8
4
**.
.-•*"
|
1
0.4
o.
E
1
#
***.,
»»*
*
-""
,.-.*"
#
-
_ .
»
\
<
•
-
iN
5hape
-0.4
—
"
-0.8
:
I
—
"-lli.._
unction
Referer ice Angle
/
Kctcrcr ice Velocity
'
—
=====
-1.2
1
1
0.2
0.1
0.4
0.3
0.6
0.5
0.8
0.7
0.9
Normalized Time
Figure 3.5 Normalized Input Shaping Funciton With
1.2
a = 0.1
T
---...
._-
0.8
.''
N.
,''
3
0.4
\
*°
**.
*
-*"
%
.-"\
D.
E
...'*''
<
1
-*SL*S.
-0.4
o
-
Shape Function
-
Reference Angle
y
Z
Reference Velocity
-0.8
Referen LC /\CCCIC aiiun
-1.2
1
0.1
1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Normalized Time
Figure 3.6 Normalized Input Shaping Funciton With
31
a=
0.25
Where
the input shaping function
is
suggested in the form.
&
=1-
2
M
<
for
t
< At
Ati
=
f(At,tf,t)=
2
"
3
1
(M
for At <
-i+M
<
ti
3-2(^1,
\2At
=
t
\
-
1
2At
for t2 <
t
for
<
<
t
<
t2
t3
- 2
3-21^
3
At
ti
II
hH
for
At
t3
<
t
<
tf
(3.69)
where
tf
= Maneuver Time
ti
= ts - At
t2
=ts+At
=
t3
At
tf -
At = atf = Rise Time
ts
= P
To determine
maximum
tf
= Switching Time, where
(3
0.5
a relationship between the angular acceleration and the
available torque
M(e)e +
we
G(e, e)
begin with the non-linear equations of motion
=
Bu
@.70)
where
M(6)
-
Mass Matrix
G(0, 6)
u
=
-
-
Coriolis Acceleration
Local Torques
32
Terms
Equation (3.70)
order terms.
is
linearized by eliminating
These linearization process
all
non-linear cosine, sine, and higher
results in the following equation for an
th
1
link.
I 6ref
=
Uref
I -
= U max
f(At,t f ,t)
(3 71
)
Moment Of Inertia About Axis Of Rotation
Equation (3.71) can be reorganized into the form of the following equation.
Umax f(At
6ref =
'
tf ' t}
(3.72)
In this equation, Umax
is
a design parameter that
is
determined by the
characteristics of the actuator used to drive the manipulators. For this application,
all
the
the actuators are the
a variable.
I,
same geared motor.
the linearized
f(At,tf,t)
mass moment of
can be modified by varying
inertia,
is
determined by the mass
characteristics of the system and does not vary with time or changing geometry.
Given
a
maximum
torque Umax, and an input shaping function with suitable
boundary conditions, the reference position and velocities can be determined by
integrating the following equations piece wise of the time interval determined by
the
maximum torque.
OreKO = 6 +
^ax
f(At,t f ,T)
dx
(3.73)
./to
OreKt)
= 6 + Go
(t-to)
+
Sm
33
f(At,t f ,x 2 )
dx 2 dX]
(3.74)
where
t
t
are the
=
=
6(0)
= Go
e(t f)
tf
boundary conditions
= 6f
at the start
setting
t
=
tf
1-
i-
a+
2
L4
G(tf)
=
The
Appendix B.
complete the maneuver can be obtained by
to
below and obtained from Appendix B.
in the equation presented
ef =smax
=
and completion of the maneuver.
resulting reference equations are presented in
The near-minimum-time
6(0)
-J-
a2
t
2
f
(3.75)
10
I
By doing
and the
this,
maximum
^ =
a relationship between the time to complete the
torque available can be determined for an
,2/1
18
U
this application, the
l a+
a+
Xa
2
10
a76)
{X,,)
2)
)
time derived with the above equations represents the
near-minimum-time required
E.
link.
J Wtt-a
maXl
For
i
maneuver
to
perform the maneuver for only one joint.
ANALYTICAL MODEL SUMMARY
This research project analyzes
how
a
two
link manipulator can be controlled
from a spacecraft main body while minimizing the
upon the main body
the links and
diagrams.
effect of manipulator
In general one can calculate the interaction force
main body by using the Newtonian approach with
Based on
this analysis, the
more smoothly
performed, the smaller the interactive force will be.
were developed. In the
first controller, linear
34
Two
that the
motion
between
free
body
maneuver
is
basic type of controllers
feedback of position and velocity
information from the joints was used to control the endpoint position.
In the
second type of controller, position and velocity information are used
in
conjunction with a reference trajectory to control endpoint position.
For
this type of controller,
reference trajectory. In the
two
two
first,
different
schemes were used
to generate the
a fifth order polynomial in conjunction with the
link closed loop solution for the manipulator inverse kinematics and the
Jacobian were used to generate the reference trajectory.
minimum-time torque shaping technique was used
trajectory.
modeled
to
In the second, a near-
determine the reference
All components, including the manipulator and main body, are
as rigid bodies.
35
RESULTS AND DISCUSSION
IV.
Analytical results from the simulation are presented in this section.
different techniques
manipulator.
were used
to control
Three
and stabilize the position of the
Results from the linear feedback controller with constant gains,
polynomial reference tracking controller and near-minimum-time reference
tracking controllers are presented sequentially.
A.
REFERENCE MANEUVER
The
three
most import
criteria
with respect to this research are the end
position, joint torques, and the effect of manipulator
main body.
motion upon the main body
on the spacecraft or the
communications
is critical
effect
controllers, a standard reference
an
the servo-
effect of the manipulator
to the excitation of flexible structures
upon antenna pointing accuracy
compare
to
maintain a
the performance of the three different
maneuver was developed. For
desired rotation of the main body
move from
due
The
if
for
link.
able to effectively
inertial
is critical
Joint torques are important to verify
actuators can perform the desired maneuvers.
from an
attitude of the
Final position for the endpoint of the manipulator
performing assembly tasks.
To be
motion on the
tip final
was
zero.
Link
angle of 20 degrees to 40 degrees.
initial inertial
This reference maneuver
is
1
this
maneuver, the
was programmed
to
move
Link 2 was programmed
to
angle of 40 degrees to a final position of 60 degrees.
depicted in Figure 4.1
36
Reference
N2
Maneuver
A
Final
Position
k2
Polynomial
w
w
Jm
s-O
S^Y
S /
Initial
Position
$
W$f
Linkl
)egfees
e 3 = 40 Degrees
Momentum
Wheel
82 = 20 Degrees
8
1|
~
Degrees
N,
Main Body
N-
Figure 4.1 Reference Maneuver
B.
LINEAR CONSTANT GAIN FEEDBACK CONTROLLER
The theory behind
in
more
detail in
Chapter
velocity gains of 5
G v = 0.2
The
)
the linear constant gain feedback, controller
(
Gv
=
III
and [Ref.
1,15].
5
were used
for these plots.
)
plots are presented in
Appendix
linear feedback controller
the easiest to implement.
was
D
Position gains of
1
Small gain
(
(
is
described
Gp =
Gp =
1
0.1
)and
and
for comparison.
the simplest controller conceptually and
This controller provided stable control over a wide
37
range of gains and for a wide variety of maneuvers.
mass or
inertia characteristics
was required
and velocity gains, and beginning and
No knowledge
for this controller.
final
of the system
Only the position
arm positions were required
to
calculate the required control torques to perform the maneuver.
The
joint time history for this case
velocity time history
is
is
presented in Figure 4.2 and the joint
Performance of
presented in Figure 4.3.
this control
system was directly related to gain selection. The system was relatively sensitive
to small
the
changes
in the position
and velocity gains. This resulted
maneuver time and control system damping. The system was
gains and maneuvers evaluated and
it
to optimize
was no way
manipulator and the
stable for all
Even with system
stability not really
to systematically select position
and velocity gains
maneuver time or torques
The magnitude of
changes to
could be proven by Lyapunov stability
theory that the system was globally stable.
in question, there
in
to achieve desired
performance measures.
the gains directly impacted the performance of the
stability of the
main body.
Larger gains generated larger
control torques and reduced the time required to perform the maneuver.
damping of
this
the system could also be varied with the gain selection.
movement on
the
main body was small but
nearly 2 degrees. Although this
still
cause degradation
in
is
still
The
The
effect of
resulted in a displacement of
apparently a small displacement, 2 degrees
communication with the
satellite
due
may
to antenna
pointing errors.
Time
history plots for torques and
Figures 4.4 and 4.5 respectively.
was characterized by
momentum wheel
The response of
this
speed are presented
system
in
to the controller
a very rapid increase in torque immediately after initiation of
38
~
70
60
1
M
50
40
j.-'""
-
30
20
—
-—
s
Main Body Angular
Displacement
Joint
10
1
Angular Displacement
Joint 2 Angular Displacement
i
-10
1
1
10
8
Time
14
12
16
18
20
(Sec)
Figure 4.2 Linear Constant Gain Feedback Joint Position Time History With
G p =l and G v =5
Ai\
i
I
o
sv
|
•-
Main Body An gular
--
\
Velocity
'\
,
~ Link
l-\ -
>
-
-
*
1
Angular velocity
Link 2 Angular vciutiiy
...
<
\
c
^''
\/
"
^**'V
**ic_
—
.
10
15
Time
20
25
30
(Sec)
Figure 4.3 Linear Constant Gain Feedback Joint Velocity Time History With
G p =l and G v =5
39
1
—
0.7
,
i
0.6
——
0.5
—
-
Main Body Torque
—
B
0.4
«
9
0.3
l^lll*
1
1
UlljUk.
CT
i
1
£
1
0.2
1
-
0.1
-
",\r
-0.1
i
Time
30
25
20
15
10
(Sec)
Figure 4.4 Linear Constant Gain Feedback Torque Time History With
and
G p =l
G v =5
980
10
20
15
Time
25
30
35
40
(Sec)
Figure 4.5 Linear Constant Gain Feedback Momentum Wheel Speed Time
History With G p =l and G v =5
40
the
was
maneuver followed by
a rapid decrease after a
similar to an impulse input that
commanded motion
few seconds. This response
became more pronounced with increased
or with increased gains. Large gains are desirable to achieve
acceptable levels of performance but must be balanced against actuator
characteristics
,
magnitude of the maneuver, and
commanded
large and sudden increase in
A
structural considerations..
torque could saturate the actuator or
excite flexible appendages attached to the spacecraft.
The
large initial torques, depicted in Figures 4.4 and 4.5, correspond to
momentum
rapid changes in the speed of the
required during the
initial
portion of the
wheel.
maneuver
Large speed changes are
to generate large toques to
counteract the forces generated by the manipulator.
experience with the
actuator and
FSS and
power supply,
analysis of electrical
it
is
Based on previous
power requirements of
the
reasonable that the actuator will be able to
compensate for the torques generated by the movement of the manipulator.
C.
POLYNOMIAL REFERENCE TRACKING CONTROLLER
To
rectify the
problems encountered with the simple linear constant gain
feedback controller a reference tracking controller was implemented.
previous case, there was a large
to correct.
initial error for
In the
which the system was attempting
In this controller, a smooth, fifth order polynomial
was used
to
generate a reference trajectory so that the correction could be evenly partitioned
over a specified maneuver time. With
position, velocity, and acceleration
As
this information, reference torques,
were obtained
as a
angular
smooth function of
time.
a result the initial "jump" generated by the linear feedback controller
eliminated and the manipulator was able to track a smooth path from
41
was
initial to
final conditions.
A
Lyapunov
variation of the
stable control
law used
in the
previous controller was used with this reference trajectory controller.
With
this controller, the
specified but
was
subject to the characteristics of the actuators and the task being
Two maneuver
performed.
time required to perform the maneuver could be
times were analyzed.
perform the maneuver was fixed
at 2.5
determine joint torques, velocities, and
near the torque limit of the actuators.
In the first case, the time to
seconds.
This time was selected to
momentum wheel
speeds while operating
In the second case, a time of 5 seconds
used to determine the effect of maneuver time on the actuators. These plots for
the 5 second
maneuver time
are included in
Appendix
D to demonstrate the effect
of maneuver time upon joint torques and velocities.
Position and velocity time history of the manipulator for a
maneuver time of
2.5 seconds are presented in Figure 4.6. and 4.7 respectively.
momentum wheel
Torque and
speed plots for a maneuver time of 2.5 seconds are presented
in
Figures 4.8 and 4.9 respectively.
Movement
of the manipulator did not cause measurable disturbance to the
main body. Manipulator torques were smoothly
was able
to counter these torques effectively
The
motion by the main body.
motion
is
desirable since
it
applied, and the control system
and in a timely manner, negating
stability of the
main body during manipulator
will reduce pointing errors for unstablized
imaging
devices or high gain antenna communications systems.
The polynomial reference
controller demonstrated stable operation over a
wide variety of operating conditions and feedback gains. The magnitude of the
gains had
little
or no effect
upon the maneuver time since
specified in the reference maneuver.
Commanded
42
this characteristic
torques closely tracked the
was
70
60
50
40
E
—a.
30
5
- ""
*
J
MainB od>
..--'""
.2
a
-
_„.
Angular
Displac ernent
20
'
Joint
Angular Displacement
1
10
~~
Joint 2
Anj ular Displacement
-10
0.5
1.5
1
Time
2.5
(Sec)
Figure 4.6 Polynomial Reference Tracking Controller Joint Position Time
History With Gp=l, G v =5 And Maneuver Time Of 2.5 Seconds
18
—""*.
„ * *
"*-.
16
X
>/
14
*
»
\.
4
12
%
\
10
,
/
\
/
8
*
\.
',
6
L.
/
9
5 4
city
c
<
i
w
2
Link 2 Angular Velo city
,'
^r
\
•'''
-2
0.5
1.5
1
Time
2.5
(Sec)
Figure 4.7 Polynomial Reference Tracking Controller Joint Velocity Time
History With G p =l, G v =5 And Maneuver Time Of 2.5 Seconds
43
I
0.8
-
Main B ody Torque
0.6
/•''' .-
--..
?
0.2
\
***
'
0.4
!"*,
X*
-
Link
\
Link 2 Torque
""-
v
^^**
er
©
H
rorque
1
^ ^l
%
m
..
-0.2
\x
s>...
-0.4
-0.6
/
x
• *
**
*y
/
/
x
-0.8
-1
-
C
1
0.5
1.5
1
Time
2
2. 5
(Sec)
Figure 4.8 Polynomial Reference Tracking Controller Torque Time History
With G p =l, G v =5 And Maneuver Time Of 2.5 Seconds
10R0
i
icnn
^—
.
*
1
980-
1a.
930
1
C/3
J!
Rsn
i
Rin
1
.
i
7Rn
.
()
0.5
1.5
1
Time
:I
2.5
(Sec)
Figure 4.9 Polynomial Reference Tracking Controller Momentum Wheel
Speed Time History With Gp=l, G v =5 And Maneuver Time Of 2.5 Seconds
44
reference torques with no discernible difference between the reference and
commanded
The polynomial reference tracking
torques.
smooth position control commands
controller, there
With the tracking
for all maneuvers.
were no discontinuities or sudden changes
The performance was independent of
controller provided
the
in torque
commands.
magnitude of the commanded maneuver
Maneuver time could be
with no effect of gain selection being noted.
but was subject to actuator and structural characteristics.
specified
Feedback
errors
remained small throughout the maneuver as the controller smoothly tracked the
reference. This resulted in smoothly changing torque
commands
with smooth and
predictable motion of the manipulator.
D.
NEAR-MINIMUM-TIME REFERENCE TRACKING CONTROLLER
The theory behind
Chapter
case,
III
a was
the
and [Ref. 1,2].
near-minimum-time tracking controller
Two cases
set equal to 0.25.
for this controller
were
a
the
maneuver
time.
this reference are
a
approaches
wave with
a period of
of 0.1 was used.
zero, the input torque shape approached that of a square
As
This shape more closely approximated the
bang-bang controller.
A
maneuver times of
described in
selected. In the first
The shape function generated by
sinusoidal in shape. In the second case, an
is
2.5 seconds
minimum
was used so
time
that the
polynomial reference and the near-minimum-time control system performance
characteristics could be compared.
Similar plots with a maneuver time of 5
seconds can be found in Appendix D.
The general performance of
controller provided
some of
previous control systems.
the
near-minimum-time reference tracking
the advantages and disadvantages of the
two
Since this was a tracking controller and the feedback
45
errors
were generally small, the commanded torques were generally smooth with
only minor disturbances to the main body.
Near minimum-time reference position and velocity time history of the
manipulator for a maneuver time of 2.5 seconds with
Figure 4.10. and 4.11 respectively.
time of 2.5 seconds with
respectively.
a =
Momentum wheel
a =
0.25 are presented in
speed plots for a maneuver
0.25 are presented in Figures 4.12 and 4.13
Near minimum-time reference position and velocity time history of
the manipulator for a
maneuver time of 2.5 seconds with
Figure 4.14. and 4.15 respectively.
time of 2.5 seconds with
a =
Momentum wheel
0.1
a=
0.1 are presented in
speed plots for a maneuver
are presented in Figures 4.16 and 4.17
respectively.
With
a =
0.25 the input torque shaping closely resembles that of the
polynomial reference trajectory tracking controller.
reference trajectory, the manipulator links
final conditions.
Note
As with
moved smoothly from
the
polynomial
the initial to the
that for this controller, there is only a very small angular
displacement of the main body.
Joint velocities for the manipulator links follow a
ending with zero velocities.
controller, there is
links.
With
smooth
path, beginning and
this controller, note that unlike the
previous
no phase difference between the movement of the manipulator
Both links move with the same velocity
also displays a very small angular velocity.
46
at the
same
time.
The main body
70
T
,
60
^
-a
4)
50
a
1
c
E
V
u
A
'
40
i
30
Q.
Main Body Angular
„ ••••""
.2
Q
Displacement
20
""
1
Angular Displacement
Joint 2
Angular Displacement
Joint
10
,
i
-10
0.5
2.5
1.5
Time
(Sec)
Figure 4.10 Near-Minimum Time Tracking Controller Joint Position Time
History With Gp=l and G v =5, Maneuver Time Of 2.5 Seconds , And a = 0.25
18
~i
16
14
1
s
}»
6- io
»>»
|8
>
ir
Link
I
Velocity
Angular ve locity
-/
5
c
<
Link 2 Angular ve Iocity
—
^r^Z.
0.5
—
**
1.5
1
Time
2.5
(Sec)
Figure 4.11 Near-Minimum Time Tracking Controller Joint Velocity Time
History With G =l and G v =5, Maneuver Time Of 2.5 Seconds , And a = 0.25.
p
47
1.5
_
r
1
Main Bo dy Torque
/
^
-
,*•
-...
x
i
Link
T arque
1
0.5
E
Z
>i^*'
^a
—
-'""
*^»
Link 2 Torque
**V
..^--' ~^r
cr
E
H
-0.5
>v
*"»..
."
,.
/
-1
.,5c
>
0.5
1.5
1
•
Time
2
2.5
(Sec)
Figure 4.12 Near-Minimum Time Tracking Controller Torque Time History
With G p =l and G v =5, Maneuver Time Of 2.5 Seconds And a = 0.25
,
1100
1050
~
1000
E
950
22
9oo
850
800
750
0.5
1.5
1
Time
2.5
(Sec)
Figure 4.13 Near-Minimum Time Tracking Controller Wheel Speed Time
History With G =l and G v =5, Maneuver Time Of 2.5 Seconds , And a = 0.25
p
48
70
60
•
I
Q
50
a
40
„„.--*"
.J
-
a
Q
Main B ody Angular
...—--*"*""
.2
Displac ement
20
Angular Displacement
Ia
io 4
Joint 2 Angular
uispiacemem
10
0.5
2.5
1.5
1
Time
(Sec)
Figure 4.14 Near-Minimum Time Tracking Controller Joint Position Time
History With G =l and G v =5, Maneuver Time Of 2.5 Seconds And a = 0.1
p
18
,
-
!
16
•
14-
§«
B
1
"5
>
io
-
—
8
Main Body Angular Velocity
6
J
is
IB
<
Link
1
Angular Velocity
4
Link 2 Angular Velocity
2
-2
-
0.5
()
1.5
1
Time
2 5
2
(Sec)
Figure 4.15 Near-Minimum Time Tracking Controller Joint Velocity Time
History With G =l and G v =5, Maneuver Time Of 2.5 Seconds And a = 0.1
p
,
49
1
0.8
_
,
.
—
_
/
__
v
0.6
/
Main Body Torque
0.4
?
2
"v
/'/
>
t
Link
if
0.2
"
**«
*f *
1
Torqu e
Link 2 Torqu e
L_
V
S
3
V \
3-0.2
1
-0.4
s
-0.6
1
\
-0.8
0.5
2.5
1.5
1
Time
(Sec)
Figure 4.16 Near-Minimum Time Tracking Controller Torque Time History
With Gp=l and Gy=5, Maneuver Time Of 2.5 Seconds And a = 0.1
,
1
inn
insn
.
L^»~
i
—''"'^
(RPM)
1
1a
Wheel
950
8
RSfl
i
800
-
()
0.5
1.5
Time
:\
2.5
(Sec)
Figure 4.17 Near-Minimum Time Tracking Controller Wheel Speed Time
History With G p =l and G v =5, Maneuver Time Of 2.5 Seconds , And a = 0.1
50
As
more
a
square.
to track
goes toward zero, the shape of the input torque reference becomes
The square corners of
the torque input are difficult for the controller
and required larger gains. This characteristic
depicted plots with
required.
Even with
a =
0.1.
Note
large gains, the
that rapid
to the
optical
As
best noted in the previous
changes
main body was
The jerky motion of the manipulator disturbed
but detectable position changes.
is
the
in the
still
disturbed by motions.
main body and resulted
in the previous controllers,
main body can degrade communications
imaging devices.
51
wheel speed are
in small
any disturbance
links or cause pointing errors for
V.
CONCLUSIONS
Three different control systems were simulated and analyzed.
controllers
were
stable
effective manner.
out to be
how
and were able
The most
to position the
significant difference
effective the control system
was
All three
manipulator in a timely and
between the controllers turned
stabilizing the
main body while
positioning the manipulator.
The
linear feedback controller
the easiest to implement.
was
the simplest controller conceptually and
This controller provided stable control over a wide
range of gains and for a wide variety of maneuvers. Large gains were required to
achieve acceptable levels of performance.
Large gains corresponded
torques and large displacements of the main body.
had
the
to
achieve
all
the control system
was implemented,
it
was not possible
to
of these objectives for the given reference maneuver.
The polynomial reference
two
Position and velocity gains
be selected to balance control torques, system performance, and motion of
main body. As
classic
to large
trajectory for robotic applications represents a
approach for generating a reference trajectory to position and control a
link manipulator.
This approach also offers the advantage that the reference
trajectory can be pre-calculated off line prior to the
maneuver.
These pre-
calculated values for the reference trajectory can be used in conjunction with the
controller to
minimize
real time computational requirements.
The polynomial reference tracking
control
commands
for all maneuvers.
controller provided
With the tracking
discontinuities or sudden changes in torque
smooth position
controller, there
were no
commands. The performance was
independent of the magnitude of the commanded maneuver and only slightly
52
Maneuver time could be
affected by gain selection.
actuator and
structural
specified but
was subject
Feedback errors remained small
characteristics.
throughout the maneuver as the controller smoothly tracked the reference.
resulted in smoothly changing torque
to
commands
This
with smooth and predictable
motion of the manipulator. The polynomial reference tracking controller was the
most effective controller implemented with respect
and timely
to accurate
positioning of the manipulator, and stabilization of the main spacecraft body.
The general performance of
controller provided
some of
the near-minimum-time reference tracking
the advantages and
two previous control systems.
some of
the disadvantages of the
Since this was a tracking controller and the
feedback errors were generally small, the commanded torques were generally
smooth with only minor disturbances
The near-minimum-time
maximum
to the
main body.
for the reference
maneuver was determined by
torque capability of each actuator and the magnitude of the maneuver.
The maximum of
specified as the
the three times calculated for each of the three joints
maneuver
time. For each maneuver, one joint
the limiting case for the selected
would operate
toque capability in order to complete the maneuver
at less
at the
than the
same time
Two
of
maximum
as the other
completed the maneuver.
Large gains were required
to force the controller to closely track the
reference input as the input torque shape approached the
time.
was
would then become
maneuver based on these requirements.
the three remaining actuators therefore
links
the
Even with
larger gains, the controller
torques generated by the manipulator and
were observed.
53
was not able
some small
to
minimum maneuver
completely negate the
rotations of the
main body
Two
different control laws
linear feedback
was
utilized.
were evaluated.
In the first, constant gains with
This controller proved to be the easiest to implement
but did not effectively stabilize the main body during
manipulator.
The second
successful
controller
way
two
different references
to generate a reference trajectory.
upon mass and
controller represents a traditional and
of positioning the manipulator.
complex than the
the
control law entailed using a tracking controller with
The reference tracking
constant gains.
movements by
With the reference tracking
were used. In the
first
a polynomial
was used
This technique proved to be only slightly more
linear feedback controller with constant gains,
inertia characteristics of the
was dependent
manipulator and was the most
effective at positioning the manipulator while minimizing the motion of the
main
body. In the second, a near-minimum- time technique was employed to generate a
reference trajectory.
This reference was more complex, but provided the
capability to position the manipulator in near-minimum-time.
This technique
provided greater flexibility in shaping the input torque reference but was not as
successful as the polynomial reference tracking controller in stabilizing the main
body.
In the end, the polynomial reference tracking controller provided the best
performance of the three controllers simulated.
A.
SUBJECTS FOR FUTURE RESEARCH
Lyapunov theory represents only one methodology
system design.
Many
for nonlinear control
other methodologies including neural networks, adaptive
controllers, sliding controllers, and robust controllers
among
others provide
additional areas for research.
With the addition of
the vision server to the Spacecraft Robotics Simulation
Lab, the capability to track a target as well as the endpoint of the manipulator will
54
become
tasks.
available.
Once
This provides the capabilty to perform end point tracking
this is successful
with a stationary target, the same can be repeated
with a moving target.
The design of
the manipulators allows for the replacement of the rigid
manipulator arm design with a flexible arm design.
Control system design for
accurately and quickly positioning the flexible manipulator provides another area
for future research.
The minimum time
joint
complete the maneuver was the longest of the three
to
maneuver times and was based on
the
maximum
actuator and the magnitude of the maneuver.
torque capabilities of the
With the near-minimum-time
reference tracking controller, the switching time for the maneuver was specified
to
be one half of the maneuver time. This methodology resulted
the three actuators
working
minimum-time equations
at
maximum
in another
torque.
way,
it
may
By
This
may
only one of
parameterizing the near-
be possible to optimize the
solution in terms of a variable switching time and variable
of the actuators.
in
maneuver time
for each
allow optimization of the torques or time required to
perform the maneuver, maneuver time or torque requirements.
55
APPENDIX A
TABLE
Characteristic
A.l Servo Motor Characteristics
Units
Manipulator
Momentum
Actuator
Wheel
Actuator
9FGHD
Model Number
Gear Reduction Ratio
147.51
:
JR16M4CH-1
1
1:1.0
rev per minute
17
3000
Rated Torque
inch-pound
80
31.85
Rated Current
amps
5.6
7.79
Rated Voltage
volts
12
168
Peak Torque
inch-pound
119
341.43
Peak Current
amps
62
79.3
Peak Voltage
volts
13.2
7.67
Outside Diameter
inches
4.75
7.4
Height
inches
3.1
4.5
Weight
pounds
3.2
17.5
oz-in/amp
3.23
69.01
volts/krpm
2.39
51
Terminal Resistance
ohms
0.95
1.4
Average Friction Torque
oz-in
2.5
11
Viscous Damping Constant
oz-in/krpm
0.3
7.84
Moment Of Inertia
oz-in/sec A 2
0.0052
0.084
Rated Speed
Torque Constant
Back
EMF Constant
56
APPENDIX B
A.
SYSTEM KINETIC ENERGY
1.
Body
1
Kinetic Energy
ti =^ii
2.
2
e,
(B.i)
Body 2 Kinetic Energy
t2 = £
2
ic.m. 2
+
e2
m 2 L! L m
+ a- m 2 L?. m
+
c
m2
1-
.
,
.
2
e!
2
l? e 2
9 2 cos(e 2 i)
(B.2)
2
2
2
3.
Body 3 Kinetic Energy
t3 = 1
m
+ m
+
4.
2
ic.m.3
3
3
03
+ i
m
3
[Lf
2
>
er +
.
U ne
2
t 2
~
2
L 2 0i e2cos(02i)+m3L3Lc. m
L 2 L c m 6 2 9 3 cos (e 32
Lj
.
.3
.
+
.
3
um
-
t 2
.
3
2l
e 3 ~J
ei e 3 cos(e 3
(B.3)
i)
)
Total Kinetic Energy
3
T=X
(B.4)
Ti
i=l
B.
EQUATIONS OF MOTION
1.
Mass Matrix
The mass matrix
is
a function of the
mass and
system as well as the geometry of the system.
inertia characteristics of the
The mass matrix
is
defined in
equation (B.5) where the elements of the mass matrix are described as follows in
equations (B.6) through (B.I
1).
57
mnmi 2 mi3
m 22 m 23
m m 32 m 33
M(9) = m2i
(B.5)
31
where
mil =
Hl22
=
Ii
+ L?
Icm.2
+ m2 + rm] +
[mi
"Cjn.2
+ L2 m2
L2
m33 =
+
Icm.3
mn =m2i =
mn = m3
i
Lcm.3
Coriolis
And
L4
(B.6)
+ m3
(B.7)
>
"^
Li L2 cos(0 2 i)
(B.8)
m2
Lcjn. 2
j
j
+ m3
(B.9)
= Li Lc m3 cos(0 3 i) m3
(B.10)
m
(B.ll)
m23=m3 2= L2 Lcm.3 cos(6 32
2.
1114
)
3
Centrifugal Acceleration Terms:
G(l)
G(9,9) = G(2)
(B.12)
G(3)
G(l)=[(m 2 LiL c m )+(m 3 LiL 2 )]sin(G 2 i)e 2 +
.
.
2
(B.13)
(m 3 LiL c m 3)sin(e 3 i)e 3
.
.
G(2)=-[(m 2 LiL c m 2)+(m 3 LiL 2 )]sin(e 2 i)ei
.
.
+
(B.14)
(m 3 L 2 L c m )sin(e 3 i)e 3
.
G(3) = - (m 3 Li
Lc m
.
.
3)
.
3
sin (0 2 i) 6?
58
-
(m 3 L 2 L c m
.
.
3)
sin (e 32 )
e2
(B.15)
C.
POLYNOMIAL REFERENCE TRAJECTORY
1.
Vector Polynomial Describing Tip Position:
F=F(to) +
t
(B.16)
f(t)[F(tf)-?(to)]
= Ll1o
,o<t<tf,0<i<l
tf-to
f(T)
2.
= C] +C2T + C3T 2 +
C4 T 3
+ C5T 4 + C6T 5
Boundary Conditions And Polynomials
f(0)
= 0,ftO) = 0,f(0) =
f(l)=l,f(l)
= 0,f(l) =
f(T)=10x 3 -15x 4 + 6T 5
3.
f(T)
= 30T 2 -60x 3 + 30T 4
f(x)
= 60 x
- 1
80 T 2 +
1
(B.18)
(B.19)
20 T 3
(B.20)
Vector Position
r(T)
4.
(B.17)
=
r(t
)
+
r
f(T)[i (tf)-r(to)]
(B.21)
Vector Velocity
F (t)= f( x)
(lM^lM)
I
tf-to
(B .22)
I
59
1
Vector Acceleration
5.
r(T)
=
f(T)
(F(tf)-r(to)l
(B.23)
2
(t f - to)
D.
TWO LINK INVERSE KINEMATICS
Law Of Sines
& Cosines:
xl
+ yl=ii +
021
=82-81
i2
+ 2i
i2
1
cos(e 2 -ei)
(B.24)
(B.25)
(B.26)
sin (e 2 i
)
= ± Vl-cos(8 2 i)
2
(B.27)
821=^2(^21)^
\C0S18 2
1
(B.28)
);
Velocity Vector:
r
=
[H] 9
-
H=
(B.29)
2
sin (62),
-
b
sin (0 3 )
I3
cos
(B.30)
12
cos (9 2 ),
(63)
Acceleration Vector:
r=[H] e +[h] e
(B.31)
60
H=
E.
- 12
cos
- 1
2
sin (0 2 ),
b
-
(62),
cos
(83)
(B.32)
sin (0 3 )
- 13
NEAR MINIMUM TIME RIGID BODY MANEUVER
1.
Near-Minimum-Time Maneuver
I
6ref
= U re f = U max
tf -
2.
Maneuver Time
t]
= ts - At
t2
= ts + At
=
tf -
At
At =
a tf =
=
=
tg
(B.33)
Moment Of Inertia About Axis Of Rotation
I -
t3
f(At,tf,t)
(3 tf
Rise Time
=
Switching Time, where p
0.5
Torque Shaping Function
3-2-1
/
-te
=
f (At, t f,t)
= 1-2 (LlL
=
W 2At
l\
OreKO = Go +
^
I
t
<
3.2M1
t
< At
\
tj
for
t,
<
t
<
t2
l2At
I
-1+ i^a
\
for At <
1
\2
=
.=
<
for
\At/J
-
2
1
for
t2
<
t
3-2 t-t3
)
(B.34)
<
t3
for
t3
<
t
<
tf
At
At
f(At,tf,T)
dT
61
(B.35)
GreKt)
= 6 + Go
^
+
(t-to)
f(At,t f ,T 2 )
dx 2 dii
(B.36)
Boundary Conditions
3.
to
t
=
=
=e
e(o)
0;
e
tf ;
(tf)
= ef
e(o)
=o
e(tf)
=o
Solution for the piece wise integration of equations (B.35) and (B.36) for
the given time interval provides us with refeence angular displacements and
angular velocity.
Reference Angular Displacements
4.
<
t
Velocities
< At
lllflillf
2
e ref= Umax At
\AV
L4
3
Umax
t
<
t
\
LVAti
I
At <
And
_
U
(B.37)
10 VAt
^
4
t
(B.38)
2 \At
ti
6 ref = Mmax
J-t
2
e re f =
2
-ItAt +
2
^-(At)
v
;
5
(B.39)
20
^(t-iAt
(B.40)
62
1
tl
<
t
<
1
t2
a+l
23<x2.3
L20
4
=^x
e rPf
1
(2Atf
t-ti
.2
i
f
f
1
2At
1
.
2
/
+
t?
8
4
t-ti
f
1
+ 1 t-ti
2At
^
5
+
f
|
5
/
(ltf -|At)(t-
(B.41)
2At
I
tl )
I
e ref = ymax
I
t2
<t<
L _lAt +
2At
t-til
2Ati
2
I
+ (LLti
2 (t>iLf
2At
1
/
(B.42)
2At
\
t3
2
tf
Qref
=
20
- max
4
= %bsl
-t
1 (tf
3At)
-
(t -
2)
(B.43)
i(t-^-At
'i
ref
+
8
+
ti
+
-i-
t2 -
At
(B.44)
2
t3
<t<
tf
1
-J-a 2
U
e ref = 2m&
r
20
L
5fflas.(.
i-
a+
2
t^lAtft-tsHAtf
4J
(B.45)
.
e raf =
-
1
2
t-t 3
f
V
t-t 3
f+1
(
At
lAt +
4
'
'
At
4
_
j
'
J_ t-t 3
^
5
(
10
I
At
At
(B.46)
\
At
/
63
\
At
/
2
I
2At
I
o
;
At time
t
=
tf
u max
Qref =
J- -
L4
1a+
-i-
2
10
a2
t
(B.47)
.
e ref =
5.
2
tf
(B.48)
Near-Minimum-Time Relationships
tfi
"iHe
=
fi
-e
J"
Um -(4"2 a+
Umax;
—
(B.49)
a2
io
i,(e fl -eo,)
^4-2 a+
m
)
(B.50)
a:
64
APPENDIX C
A.
LINEAR FEEDBACK MATLAB SIMULATION PROGRAM
1.
Main Program For Linear Feedback Simulation
%
%
Constants
Lenght of Manipulators
L(l)= 15*2.54/100;
L(2) = 17*2.54/100;
L(3) = 17*2.54/100;
L(4) = 20/100;
%
Distance
From Axis Of Rotation To Center Of Mass
Lcm(l) = 0/100;
Lcm(2) = 36.45/100;
Lcm(3) = 34.9/100;
%
Mass Of Major Components
(kg)
m(l) = 54.69;
m(2) = 2.09364;
m(3) = 2.466;
m(4) = 10.667;
%
Inertias
Of Major Components (kg-m A 2)
1(1)
= 4.32132;
1(2)
= 3.20338e-2;
1(3)
= 5.38398e-2;
1(4)
= 0.0912;
65
%
%
%
t0
Integrate Equations
Matlab Runge-Kutta 4 Routine
Boundary Conditions
=
0;
tfinal
xO =
tol
Of Motion Using Commercial
=
=
40;
[0.0; 0.0; 0.0;
0*pi/180; 20*pi/180; 40*pi/180];
le-8;
trace=l;
Call "lfbrk" Function
[t,x,uu]
%
And
Integrate Equations
Of Motion
= rku2('lfbrk',t0,tfinal,x0,tol, trace);
Program To Calculate Motor Torque, Current, and Voltage
TF=1.41e-2;
%N-m
KT=2.28e-2;
% N-m/amp
% N-m/rad/sec
% Volt/rad/sec
% ohms
KD=2.00573e-5;
KE=2.28271e-2;
RT=0.95;
j=length(t)
fori=l:j;
amp22(i) = (uu(i,2)/148.51 +
volt22(i)
=
KE
* x(i,2)
+ RT
amp33(i) = (uu(i,3)/148.51 +
volt33(i)
=
KE
* x(i,3)
+ RT
TF + KD
* x(i,2)* 148.51
) /
KT;
) /
KT;
* amp22(i);
TF + KD
* x(i,3)* 148.51
* amp33(i);
end
66
%
%
Program To Calculate Motor Torque, Current, and Voltage
For
Momentum Wheel
TF=0.0777;
%N-m/amp
KD=5.29131e-4;
% N-m/rad/sec
KE=0.487 11;
% Volt/rad/sec
RT=1.4;
%ohms
Iw=0.0912;
% kg-mA 2
% rad/sec (=
thd0=104.7;
lOOOrpm)
fori=l:j;
tor(i)
=
uu(i,l);
thddw(i)
thdw(i)
=
tor(i)/Iw;
= thddw(i)
amp(i)
=
volt(i)
= KE
(tor(i)
* t(i)
+
+ TF + KD
* thdw(i)
thdO;
* thdw(i)
+ RT
* amp(i);
xl(i)=L(l)*cos(x(i,4));
yl(i)=L(l)*sin(x(i,4));
x2(i)=xl(i)+L(2)*cos(x(i,5));
y2(i)=yl(i)+L(2)*sin(x(i,5));
x3(i)=x2(i)+L(3)*cos(x(i,6));
y3(i)=y2(i)+L(3)*sin(x(i,6));
end
%
Store Data For Plotting Later
datal=[t,x*180/pi,uu];
,
,
,
,
data2=[t,xl ,yl ,x2 ,y2 ,x3
) /
,
,y3'];
data3=[t,thdw'*30/pi,thddw'*30/pi];
67
KT;
,
,
data4=[t,amp ,amp22',amp33',volt ,volt22',volt33'];
%
Save Data In Text Format
save lfb9.dat datal
/ascii
save lfbl0.dat data2
/ascii
save lfbl 1 .dat data3 /ascii
save lfbl2.dat data4 /ascii
68
2.
Linear Feedback Equation
%
Function Containing Linear Feedback Equations
function [xdot,Ul]
%
%
Of Motion Function
=
Of Motion
lfbrk(t,x)
Constants
Lenght of Manipulators
L(l)
=
15*2.54/100;
L(2) = 17*2.54/100;
L(3)= 17*2.54/100;
L(4) = 20/100;
%
Distance
From Axis Of Rotation To Center Of Mass
Lcm(l) = 0/100;
Lcm(2) = 36.45/100;
Lcm(3) = 34.9/100;
%
Mass Of Major Components
(kg)
m(l) = 54.69;
m(2) = 2.09364;
m(3) = 2.466;
m(4) = 10.667;
%
Inertias
Of Major Components (kg-m A 2)
1(1)
= 4.32132;
1(2)
= 3.20338e-2;
1(3)
= 5.38398e-2;
1(4)
= 0.0912;
69
%
Angles
thd(l)
=
x(l);
thd(2)
=
x(2);
thd(3)
=
x(3);
%
Convert Input Variables Into Variable Used In This Function
th(l) =x(4);
th(2) =x(5);
th(3) =x(6);
%
Final Conditions For Manipulator
thfl=0;
thf2=40*pi/180;
thf3=60*pi/180;
thdfl=0;
thdf2=0;
thdf3=0;
%
Linear Feedback Control
Law
gip=i;
g2p=.i;
g3p=.l;
glv=.2;
g2v=.2;
g3v=.2;
70
And Main Body
%
Control Torques
U(3)=-g3p*(th(3)-thf3)-g3v*(thd(3)-thdf3);
U(2)=U(3)-g2p*(th(2)-thf2)-g2v*(thd(2)-thdf2);
U(l)=U(2)-glp*(th(l)-thfl)-glv*(thd(l)-thdfl);
U1=U';
%
Calculate Coeffients For
The Equation Of Motion And
[MM,GM] = mgm(th,thd);
B = [l,-l,0;0,l,-l;0,0,l];
thdd=inv(MM)*(B*Ul-GM*);
xdot=[thdd;x(l);x(2);x(3)];
71
Integrate
B.
POLYNOMIAL REFERENCE TRACKING CONTROLLER
1.
Polynomial Reference Maneuver Tracker Main Program
% EOM3.m Program
% Constants
% Lenght of Manipulators
L(l)= 15*2.54/100;
L(2) = 17*2.54/100;
L(3) = 17*2.54/100;
L(4) = 20/100;
%
Distance
From Axis Of Rotation To Center Of Mass
Lcm(l) = 0/100;
Lcm(2) = 36.45/100;
Lcm(3) = 34.9/100;
%
Mass Of Major Components
(kg)
m(l) = 54.69;
m(2) = 2.09364;
m(3) = 2.466;
m(4) = 10.667;
%
Inertias
Of Major Components (kg-m A 2)
1(1)
= 4.32132;
1(2)
= 3.20338e-2;
1(3)
= 5.38398e-2;
1(4)
= 0.0912;
72
;
%
t0
Boundary Conditions And Integration Time
=
0;
tfinal
xO =
tol
=
=
10;
[0.0; 0.0; 0.0;
0*pi/180; 20*pi/180; 40*pi/180];
le-8;
trace=l
%
Integrate Equations
[t,x,uu]
Of Motion
= rku2('eom3rk',tO,tfinal,xO,tol,trace);
j=size(t);
% Calculate Electrical Power Requirement For Manipulator Actuators
% Motor Parameters
TF=1.8e-2
%N-m
KT=2.28e-2
% N-m/amp
% N-m/rad/sec
% Volt/rad/sec
KD=2.00573e-5
KE=2.2827 le-2
% ohms
RT=0.95
j=size(t)
for i=l:j;
amp22(i) = (uu(2,i)/148.51 +
volt22(i)
= KE
* x(i,2)
+ RT
amp33(i) = (uu(3,i)/148.51 +
volt33(i)
=
KE
* x(i,3)
+ RT
TF + KD
* x(i,2)
) /
KT;
) /
KT;
* amp22(i);
TF +
KD
* x(i,3)
* amp33(i);
end
%
%
Program To Calculate Motor Torque, Current, and Voltage
Momentum Wheel
73
%
Motor Parameters
TF=0.0777
KT=0.48732
KD=5.29131e-4
% N-m
% N-m/amp
% N-m/rad/sec
% Volt/rad/sec
% ohms
KE=0.48711
RT=1.4
% kg-m A 2
% rad/sec (= lOOOrpm)
Iw=0.0912
thd0=104.7
j=size(t)
for i=l:j;
tor(i)
= uu(l,i);
thddw(i)
thdw(i)
=
tor(i)/Iw;
= thddw(i)
amp(i)
=
(tor(i)
volt(i)
=
KE
* t(i)
+
+ TF + KD
* thdw(i)
thdO;
* thdw(i)
+ RT
) /
* amp(i);
end
74
KT;
2.
Polynomial Reference Tracking Equations Of Motion Function
function [xdot,Ul] =eom3rk(t,x)
%
Constants
L(l)= 15*2.54/100;
L(2)= 17*2.54/100
L(3) = 17*2.54/100
L(4) = 20/100;
Lcm(l) = 0/100;
Lcm(2) = 36.45/100;
Lcm(3) = 34.9/100;
%
Mass
m(l) = 54.69;
m(2) = 2.09364;
m(3) = 2.466;
m(4) = 10.667;
%
Inertia
1(1)
= 4.32132;
1(2)
= 3.20338e-2;
1(3)
= 5.38398e-2;
1(4)
= 0.0912;
%
Angles
thd(l)
=
x(l);
thd(2)
=
x(2);
thd(3)
=
x(3);
th(l) =x(4);
75
th(2) =x(5);
th(3) =x(6);
[MM,GM] = mgm(th.thd);
%
Gains
glp=1.0;
g2p=1.0;
g3p=1.0;
glv=5.0;
g2v=5.0;
g3v='5.0;
[Uref,thref,thdref,thddref]
= eom3ref(t,L,Lcm,m,I);
dU(3)=-g3p*(th(3)-thref(3))-g3v*(thd(3)-thdref(3));
dU(2)=dU(3)-g2p*(th(2)-thref(2))-g2v*(thd(2)-thdref(2));
dU(l)=dU(2)-glp*(th(l)-thref(l))-glv*(thd(l)-thdref(l));
Ul=Uref+dU';
U=U1';
B = [1,-1,0;0,1,-1;0,0,1];
,
thdd=inv(MM)*(B*U -GM');
xdot=[thdd;x(l);x(2);x(3)];
76
3.
Polynomial Reference Trajectory Function
= eom3ref(t,L,Lcm,m,I);
function [Uref,thref,thdref,thddref]
thref(l)=0;
%
t0
tf
Initial
=
=
%
And
Final
And
Final Vector Positons
Time For Maneuver
0;
10;
Initial
r3x0 = L(l)*cos(thref(l))+L(2)*cos(20*pi/180)+L(3)*cos(40*pi/180);
r3y0 = L(l)*sin(thref(l))+L(2)*sin(20*pi/180)+L(3)*sin(40*pi/180);
r3xf
= L(l)*cos(thref(l))+L(2)*cos(40*pi/180)+L(3)*cos(60*pi/180);
r3yf
= L(l)*sin(thref(l))+L(2)*sin(40*pi/180)+L(3)*sin(60*pi/180);
%
Calculate Reference
tau
= (t-t0)/(tf-t0);
Maneuver
r3x = r3x0
+
(
r3xf
-
r3x0
)
*
(
10
* tau A 3
-
15 * tau A4
+
6 * tau A 5
);
r3y = r3y0
+
(
r3yf
-
r3y0
)
*
(
10
* tau A 3
-
15 * tau A4
+
6 * tau A 5
);
r3xd =
(
r3xf
-
r3x0
) / (
tf -
r3yd =
(
r3yf
-
r3y0
) / (
tf
-
tO )*
(
tO )*(
30
30
* tau A 2
* tau A 2
-
-
60
60
* tau A 3
* tau A 3
+ 30
+ 30
* tau A 4);
* tau A 4);
r3xdd = (r3xf-r3x0)/((tf-t0) A 2)*(60*tau-180*tau A 2+120*tau A 3);
r3ydd = (r3yf-r3y0)/((tf-t0) A 2)*(60*tau-180*tau A 2+120*tau A 3);
77
if(t>tf);
r3x=0;
r3y=0;
r3xd=0;
r3yd=0;
r3xdd=0;
r3ydd=0;
end
%
Determine Inverse Kinematics
th23
=
kink(r3x-L(l)*cos(thref(l)),r3y-L(l)*sin(thref(l)),L(2),L(3));
thref(2)
=
th23(l);
thref(3)
=
th23(2);
%
Calculate Joint Velocites Using Jacobean
H=
[-L(2)*sin(thref(2)),-L(3)*sin(thref(3));....
L(2)*cos(thref(2)),L(3)*cos(thref(3))];
thd23 = inv(H) *
[
r3xd; r3yd
thdref(2)
=
thdref(3)
= thd23(2);
%
];
thd23(l);
Calculate Joint Acceleration Using Jacobean
Hdot=[-L(2)*thdref(2)*cos(thref(2)),-L(3)*thdref(3)*cos(thref(3));..
-L(2)*thdref(2)*sin(thref(2)),-L(3)*thdref(3)*sin(thref(3))];
78
thdd23=inv(H)*([r3xdd;r3ydd]-Hdot*[thdref(2);thdref(3)]);
thddref(2)
= thdd23(l);
thddref(3)
= thdd23(2);
% Calculate Reference Control Torques
[MM,GM] = mgm(thref,thdref);
B = [l,-l,0;0,l,-l;0,0,l];
Uref = inv(B)
*
(
MM * thddref + GM'
79
);
C
NEAR-MINIMUM-TIME RERENCE TRACKING CONTROLLER
1.
Main Program
%
%
nmtl program
Constants
L(l)
=
15*2.54/100;
L(2) = 17*2.54/100
L(3) = 17*2.54/100
L(4) = 20/100;
Lcm(l) = 0/100;
Lcm(2) = 36.45/100;
Lcm(3) = 34.9/100;
%
Mass
m(l) = 54.69;
m(2) = 2.09364;
m(3) = 2.466;
m(4) = 10.667;
%
Inertia
1(1)
= 4.32132;
1(2)
= 3.20338e-2;
1(3)
= 5.38398e-2;
1(4)
= 0.0912;
%
Maximum Torque
umax=0.300;
%
Shaping Function Coefficient
80
;
;
alfa=0.1;
beta=0.5;
tO=0;
MM(1,1) = I(l)+L(l) A 2*(m(2)+m(3))+L(4) A 2*m(4);
MM(2,2) =
1(2)
+ m(2)
*
Lcm(2) A 2 + m(3)
MM(3,3) =
1(3)
+ m(3)
*
Lcm(3) A 2;
%
Initial
And
* L(2) A 2;
Final Manipulator Position
thO=[0*pi/l 80;20*pi/l 80;40*pi/l 80]
thf=[0.01 *pi/l 80;40*pi/l 80;60*pi/l 80]
%
Determine
Minimum Time To Perform Maneuver
T(l)=sqrt(MM(l,l)*(thf(l)-thO(l))/...
(umax(l)*(l/4-l/2*alfa+l/10*alfa A 2)));
T(2)=sqrt(MM(2,2)*(thf(2)-thO(2))/...
(umax(l)*(l/4-l/2*alfa+l/10*alfa A 2)));
T(3)=sqrt(MM(3,3)*(thf(3)-thO(3))/...
(umax(l)*(l/4-l/2*alfa+l/10*alfa A 2)))
T_max=max(T)
tf=T_max
delt=alfa*tf
tf=T_max
dt=alfa*tf
ts=beta*tf
tl=ts-dt
81
;
t2=ts+dt
t3=tf-dt
echo
off;
t0=0;
trace=l;
tol=le-8;
xO=[0;0;0;0*pi/l 80;20*pi/l 80;40*pi/l 80]
%
Integrate Equations
[t,x,uu]
=
Of Motion
rku2('nmtrk',tO,tf,xO,tol,trace);
82
2.
Near-Minimum-Time Equations Function
function [xdot,U]
=
nmtrk(t,x);
umax(l)=0.300;
%
Constants
L(l)= 15*2.54/100;
L(2)
= 17*2.54/100;
L(3) = 17*2.54/100;
L(4) = 20/100;
Lcm(l) = 0/100;
Lcm(2) = 36.45/100;
Lcm(3) = 34.9/100;
%
Mass
m(l) = 54.69;
m(2) = 2.09364;
m(3) = 2.466;
m(4) = 10.667;
%
Inertia
83
1(1)
= 4.32132;
1(2)
= 3.20338e-2;
1(3)
= 5.38398e-2;
1(4)
= 0.0912;
%
Angles
thd(l)
=
x(l);
thd(2)
=
x(2);
thd(3)
=
x(3);
th(l)
=x(4)
th(2)
=
th(3)
=x(6)
x(5)
[MM,GM] = mgm(th,thd);
%
Gains
glp=.10;
g2p=.10;
g3p=.10;
glv=.50;
g2v=.50;
g3v=.50;
84
[MM,GM] =
mgm(th,thd);
[Uref,thref,thdref,thddref]
=
nmt_ref(t);
dU(3)=-g3p*(th(3)-thref(3))-g3v*(thd(3)-thdref(3));
dU(2)=dU(3)-g2p*(th(2)-thref(2))-g2v*(thd(2)-thdref(2));
dU(l)=dU(2)-glp*(th(l)-thref(l))-glv*(thd(l)-thdref(l));
Ul=Uref+dU*;
u=ur,
B = [l r l,0;0,l,-l;0,0,i];
xdot =
[
inv(MM)
*
(
B
* U'
-
GM'
85
);
x(l); x(2); x(3)
];
3.
Near-Minimum-Time Reference Function
%
Reference Generator Function
function [Uref,thref,thdref,thddref]
=
nmt_ref(t);
umax(l)=0.300;
%
Constants
L(l)= 15*2.54/100;
L(2) = 17*2.54/100
L(3) = 17*2.54/100
L(4) = 20/100;
Lcm(l) = 0/100;
Lcm(2) = 36.45/100;
Lcm(3) = 34.9/100;
%
Mass
m(l) = 54.69;
m(2) = 2.09364;
m(3) = 2.466;
m(4) = 10.667;
%
Inertia
1(1)
= 4.32132;
1(2)
= 3.20338e-2;
1(3)
= 5.38398e-2;
1(4)
= 0.0912;
MM(1,1) = I(l)+L(l) A 2*(m(2)+m(3))+L(4) A 2*m(4);
86
;
;
+ m(2)
*
Lcm(2) A 2 + m(3)
MM(3,3) = K3) + m(3)
*
Lcm(3) A 2;
MM(2,2) =
1(2)
*
L(2) A 2;
thO=[0*pi/l 80;20*pi/l 80;40*pi/l 80]
thf=[0.01 *pi/l 80;40*pi/l 80;60*pi/l 80]
alfa=0.1;
T(l)=sqrt(MM(l,l)*(thf(l)-thO(l))/(umax(l)*...
(l/4-l/2*alfa+l/10*alfa A 2)));
T(2)=sqrt(MM(2,2)*(thf(2)-thO(2))/(umax(l)*...
(l/4-l/2*alfa+l/10*alfa A 2)));
T(3)=sqrt(MM(3,3)*(thf(3)-th0(3))/(umax(l)*...
(l/4-l/2*alfa+l/10*alfa A 2)));
T_max=max(T);
tf=T_max;
dt=alfa*T_max;
tl=tf/2-dt;
t2=tf/2+dt;
t3=tf-dt;
umax(3)=MM(3,3)*(thf(3)-thO(3))/...
A 2*(l/4-l/2*alfa+l/10*alfa A
(tf
2));
umax(2)=MM(2,2)*(thf(2)-thO(2))/...
A 2*(l/4-l/2*alfa+l/10*alfa A
(tf
2));
umax(l)=MM(l,l)*(thf(l)-thO(l))/(tfA 2*(l/4-l/2*alfa+l/10*alfa A 2));
%
Near Minimum Time Reference Maneuver
87
if(t>=0&t<=dt);
f=(t/dt) A 2*(3-2*(t/dt));
ff=dt*((t/dt) A 3-(l/2)*(t/dt)M);
fff=(dtA 2)*((l/4)*(t/dt)M-(l/10)*(t/dt) A 5);
elseif (t>dt
& t<=tl
);
f=i;
ff=(t-(l/2)*dt);
fff=((l/2)*t A 2-(l/2)*t*dt+(3/20)*dt A 2);
elseif (t>tl
f
=
ff
1 -
=
& t<=t2);
2*(((t-tl)/(2*dt)) A 2*(3-2*((t-tl)/(2*dt))));
-(l/2)*dt+tl+2*dt*(((t-tl)/(2*dt))-2*((t-tl)/(2*dt)) A 3+.
((t-tl)/(2*dt)) A 4);
fff
=
((23/20)*alfa A 2-(3/4)*alfa+(l/8))*tfA 2+(2*dt) A 2*...
((l/2)*((t-tl)/(2*dt)) A 2-(l/2)*((t-tl)/(2*dt)) A4...
+(l/5)*((t-tl)/(2*dt)) A 5)+((l/2)*tf-(3/2)*dt)*(t-tl);
elseif (t>t2
& t<=t3);
f=-l;
ff=-t+tl+t2-(l/2)*dt+2*dt*(((t2-tl)/(2*dt))-...
2*((t2-tl )/(2*dt)) A 3+((t2-tl )/(2*dt)) A 4);
fff=(-(21/20)*alfaA 2+(l/4)*alfa+l/8)*tfA 2+...
(l/2)*(tf-3*dt)*(t-t2)-(l/2)*(t-(l/2)*tf-dt) A 2;
88
;
elseif (t>t3
f
& t<=tO;
-l+(((t-t3)/(dt)) A 2*(3-2*((t-t3)/dt)));
=
ff=((l/2)*dt+dt*(-(t-t3)/dt+((t-t3)/dt) A 3-(l/2)*((t-t3)/dt) A4));
fff=(-(l/20)*alfa A 2-(l/2)*alfa+l/4)*tfA 2+(l/2)*dt*(t-t3)+...
dt A 2*(-(l/2)*((t-t3)/dt) A 2+(l/4)*((t-t3)/dt) A 4-(l/10)*((t-t3)/dt) A 5);
elseif (t>tf);
f=0;
ff=0;
fff=0;
end
thref( 1 )=(umax( 1
)/MM( 1
,
1 )) *fff+thO( 1 );
thref(2)=(umax(2)/MM(2,2))*fff+thO(2);
thref(3)=(umax(3)/MM(3,3))*fff+thO(3);
thdref( 1 )=(umax( 1 )/MM( 1
,
1
))*ff
thdref(2)=(umax(2)/MM(2,2))*ff;
thdref(3)=(umax(3)/MM(3,3))*ff;
thddref(l)=umax(l)*f/MM(l,l);
thddref(2)=umax(2)*f/MM(2,2);
thddref(3)=umax(3)*f/MM(3,3);
[MM,GM] = mgm(thdref,thref);
89
B = [1,-1,0;0,1,-1;0,0,1];
Uref = inv(B)
*
(
MM * thddref + GM'
90
);
D.
MISCELLANEOUS FUNCITONS
1.
Equation Of Motion Coefficient Funciton
function
%
[MM.GM] =
mgm(th,thd);
Constants
L(l)= 15*2.54/100
L(2) = 17*2.54/100
L(3) = 17*2.54/100
L(4) = 20/100;
Lcm(l) = 0/100;
Lcm(2) = 36.45/100;
Lcm(3) = 34.9/100;
%
Mass
m(l) = 54.69;
m(2) = 2.09364;
m(3) = 2.466;
m(4) = 10.667;
%
Inertia
1(1)
= 4.32132;
1(2)
= 3.20338e-2;
1(3)
= 5.38398e-2;
1(4)
= 0.00912;
91
%
Subroutine
To
Calculate
Mass and "G" Matrix
th21=th(2)-th(l);
th31=th(3)-th(l);
th32 = th(3)
%
-
th(2);
'Mass' Matrix
MM(1,1) = I(l)+L(l) A 2*(m(2)+m(3))+L(4) A 2*m(4);
MM(1,2) = ((m(2)*L(l)*Lcm(2))+(m(3)*L(l)*L(2)))*cos(th21);
MM(l,3) = m(3)*L(l)*Lcm(3)*cos(th31);
MM(2,1) = MM(1,2);
MM(2,2) =
1(2)
+ m(2)
MM(2,3) = m(3)
*
*
Lcm(2) A 2 + m(3)
*
L(2) A 2;
L(2) * Lcm(3) * cos(th32);
MM(3,1) = MM(1,3);
MM(3,2) = MM(2,3);
MM(3,3) = K3) + m(3)
%
'G'
*
Lcm(3) A 2;
Matrix
cl22 = ((m(2)*L(l)*Lcm(2))+(m(3)*L(l)*L(2)))*sin(th21);
cl33 = m(3)*L(l)*Lcm(3)*sin(th31);
c211 =- ((m(2)*L(l)*Lcm(2))+(m(3)*L(l)*L(2)))*sin(th21);
c233 =
m(3)*L(2)*Lcm(3)*sin(th32);
92
c311 =-m(3)*L(l)*Lcm(3)*sin(th31);
c322 =
-
m(3)*L(2)*Lcm(3)*sin(th32);
cl=[0,0,0;...
0,cl22,0;...
0,0,cl33];
c2=[c21 1,0,0;...
0,0,0;...
0,0,c233];
c3=[c31 1,0,0;...
0,c322,0;...
0,0,0];
GM(l) = thd*cl
*thd';
GM(2) =
thd * c2 *
thd*;
GM(3) =
thd * c3 *
thd';
93
2.
Inverse Kinematics Function
function theta=kink(x,y,Ll,L2)
%
Subroutine
To Determine
Inverse Kinematic Solution
cl2=(x A 2+y A 2-Ll A 2-L2 A 2)/(2*Ll*L2);
sl2=sqrt(l-cl2 A 2);
thetal2=atan2(sl2,cl2);
minv=inv([Ll+L2*cl2,-L2*sl2;L2*sl2,Ll+L2*cl2]);
c=minv*[x;y];
theta(l )=atan2(c(2),c( 1 ));
theta(2)=theta( 1 )+theta 12;
94
APPENDIX D
LINEAR CONSTANT GAIN FEEDBACK CONTROLLER
A.
70
"*--.
--*
60
*
50
,.-**'
40
Main Bodv Ancular
30
—
Displacement
20
Joint
10
*
<
- - -
1
Joint 2
Angular Displacement
Angular Displacement
— ..
-10
10
20
15
Time
25
30
35
40
(Sec)
Figure D.l Linear Constant Gain Feedback Joint Position Time History With
G =0.1 And G v =0.2
p
95
c
i
.
/
1
/
1
Mail i Body Angular
Velc city
v
*~
1—
*
*
1
Link
1
Angular Velocity
Link
i.
/Miguiai veiuuuy
(Deg/Sec)
\
•
Velocity
\ \
!
/
-,
„-"
—
Angular
-
i.
t
%
^
m
m
**
**
*
y^T-^
*
* " "
^•*-^">--^
*
i
V
e=r
..''/
1
1
c
1
5
10
20
15
Time
25
30
35
40
(Sec)
Figure D.2 Linear Constant Gain Feedback Joint Velocity Time History
With G p =0.1 And G v =0.2
96
—
0.07
i
0.06
V
Main Body Torque
0.05
0.04
?
0.03
^3
0.02
f
0.01
z
H
i
i
L
nk
1
—
Torque
_v
........ iL nk 2 Torque
V
\\
1
Vi
\
*
\v
.
0.00
\*
-0.01
'T^
^
\\\
*^j i? -------- rrrr^^
"
mm
\--.
s
/
-0.02
-0.03
10
20
15
Time
25
30
35
40
(Sec)
Figure D.3 Linear Constant Gain Feedback Torque Time History With
G p=0.1 And G v =0.2
97
Figure D.4 Linear Constant Gain Feedback Momentum Wheel Speed Time
History With G =0.1 And G v =0.2
p
98
B.
POLYNOMIAL REFERENCE TRACKING CONTROLLER
70
I
60
i
50
40
*
j
30
*»
*
~
*
Main Body Angular
Displacement
20
ngular Displacement
u
c
10
<
Join) 2 ^ ngular Displacement
-10
1
1
0.5
1.5
2
2.5
Time
3.5
4.5
(Sec)
Figure D.5 Polynomial Reference Tracking Controller Joint Position Time
History With Gp=l and G v =5 And Maneuver Time Of 5 Seconds
99
1
.-'*'
'""***.,
—
.'
/
IV
/
Q
1
i
••
—
—
/
-/-—
-
\
>
Main Body Angular Velocity
!
/'
—
**--
/
Link
1
\
Angular Velocity
\
Link 1 Angular Velocity
,''
S
------- -
0.5
.
l
-
1.5
2.5
Time
3.5
4.5
(Sec)
Time
Figure D.6 Polynomial Reference Tracking Controller Joint Velociy
Seconds
Of
Time
5
History With G p =l and G v =5 And Maneuver
100
0.25
1
0.2
Main Body Torque
0.15
1
1
0.1
y
l
S
.»
..-•
-
"-.
'
--
I
E
Z
0.05
£'•
^
"
" *"*
a-
©
1
Torque
Link 2 Torque
"['"""-—
v
3
L.
—
Link
*
*"*^.
.-**""
-»..
•'/
-0.05
//
\%,
-0.1
\*
-0.15
i
%
V
i
-
—
- *
'
-0.2
1
-0.25
1
0.5
1
1.5
2
2.5
Time
3
3.5
4.5
(Sec)
Figure D.7 Polynomial Reference Tracking Controller Torque Time History
With Gp=l and G v =5 And Maneuver Time Of 5 Seconds
101
1(W0
-
1020
-
1000
f
980
-
\
z
Z
960
_
cr
j2
940
|\
920
900
I
i
i
RRO
0.5
1
1.5
2
2.5
Time
3
3.5
4
4.5
1
(Sec)
Figure D.8 Polynomial Reference Tracking Controller Momentum Wheel
Speed Time History With Gp=l and Gv=5 And Maneuver Time Of 5 Seconds
102
C.
NEAR-MINIMUM-TIME REFERENCE TRACKING CONTROLLER
70
60
I
50
| 40
E
u
u
« 30
D.
n
?
,
—
__"
.--"""
Main Body Angular
....—-
Displacement
20
J2
I
Joinl
I
1
A ngular
Displacement
10
5
Joint 2 Angular uisp lacement
-10
0.5
2.5
1.5
Time
4.5
3.5
(Sec)
Figure D.9 Near-Minimum Time Tracking Controller Joint Position
Time
History With G =l , G v =5, Maneuver Time Of 5 Seconds And a =
0.25
p
,
103
9
8
|9
§
Main Body Angular
1s
Veloc
—
0.5
~y~
Link
1
Angular Velocity
Link
2
Angular Velocity
2.5
1.5
Time
V
3.5
4.5
(Sec)
Figure D.10 Near-Minimum Time Tracking Controller Joint Velocity Time
History With G p =l , G v =5, Maneuver Time Of 5 Seconds , And a = 0.25
104
0.4
0.3
Main B ody Torque
0.2
'
- -
<*
/
e
tV
01
2"
o
s
_ Link
f
---
yf'\
3
Link
1
Torque
*
*\
1 ic
>v
e
>""*»..
H
**"-—..
-
_——-
]ljr
.
^-
*
-0.1
^.
-0.2
"-._-»*"
-0.3
0.5
1.5
2
Time
Figure D.ll
With
3
2.5
4.5
3.5
(Sec)
Near-Minimum Time Tracking
Controller Torque
Time History
G p =l G v =5, Maneuver Time Of 5 Seconds And a = 0.25
,
,
105
1040
-
1090
-
I
1000
w
1
.2
960
-
s
940
Q9fl
-
ono
880
-
0.5
2.5
1.5
Time
4.5
3.5
(Sec)
Figure D.12 Near-Minimum Time Tracking Controller Wheel Speed Time
History With G =l , G v =5, Maneuver Time Of 5 Seconds And a = 0.25
p
,
106
70
-
60
-
|
50
|
40
3
«
30
a
-
„•
Main Body Angular
- -••'"
_ _ „-.--
.83
?
20-
I
10
Displacement
Joint
<
1
Joint 2
-10
Angular Displacement
A ngular Displacement
-
()
0.5
1
1.5
2
2.5
Time
3
3.5
4
4.5
5
(Sec)
Figure D.13 Near-Minimum Time Tracking Controller Joint Position Time
History With G p =l , G v =5, Maneuver Time Of 5 Seconds , And a = 0.1
107
9
8
Tt
7
"8
Main Body Angular
Veloc ity
Link
c
Link
1
Angular Velocity
i.
Angular Velocity
'
-—^=—
0.5
2.5
1.5
Time
4.5
3.5
(Sec)
Figure D.14 Near-Minimum Time Tracking Controller Joint Velocity Time
History With G p =l , G v =5, Maneuver Time Of 5 Seconds And a = 0.1
,
108
0.25
-
«
-
Main Body Torque
0.15
/
/
Link
y
-
/'
f
Z
1
Torque
/
0.05
Link 2 Torque
A
J;/
3
4
\\
I" -0.05
H
t
'
-0.15
I
V
-0.25
0.5
1
1.5
2
2.5
Time
3
3.5
4
4.5
5
(Sec)
Figure D.15 Near-Minimum Time Tracking Controller Torque Time History
With G p =l , G v =5, Maneuver Time Of 5 Seconds , And a = 0.1
109
IVJJU
j£
-
IUUU
1a
J
__.
950
900
/
-
2
3
Time
(Sec)
Figure D.16 Near-Minimum Time Tracking Controller Wheel Speed Time
History With G p =l , G v =5, Maneuver Time Of 5 Seconds And a = 0.1
,
110
.
REFERENCES
1.
2.
Junkins, John L., Kim, Youdan, An Introduction To Dynamics And Control
Of Flexible Structures, to appear, American Institute Of Aeronautics and
Astronautics Education Series, pp. 73-97, 102-136, 1992.
Bang, Hyochoong, Maneuver And Vibration Control Of Flexible Space
Structures By Lyapunov Stability Theory, Phd Dissertation, Texas
University, College Station, Texas, September 1992.
A&M
3.
Hailey, Jeffrey A., Experimental Verification of Attitude Control
Techniques For Flexible Spacecraft Slew Maneuvers, Master's Thesis,
Naval Postgraduate School, Monterey, California, March 1992.
4.
Watkins Jr., R.J., The Attitude Control Of Flexible Spacecraft, Master's
Thesis, Naval Postgraduate School, Monterey, California, June 1991.
5.
Integrated Systems, Inc.,
6.
MATRIXX
Core, 7 tn ed., January 1990.
Kepco, Inc., Model BOP 20-10M Power Supply Instruction Manual,
Hushing, New York, 1987.
7.
Integrated Systems Inc.,
8.
Greenwood, Donald
Prentice-Hall,
AC-100 Users Guide, V2.4.05, February 1991
T., Principles
New Jersey,
Of Dynamics, 2nd
ed., pp.
239-291,
1988.
9.
Kalman, R.E., Bertram, J.E., "Control Systems Analysis And Design Via The
Second Method Of Lyapunov, Part 1, Continuous Time Systems", Journal
Of Basic Engineering, pp. 371-393, June 1960.
10.
Craig, John,
11.
Johnson, Eric R., Servomechanisms, pp. 209-230, Prentice-Hall,
1963.
12.
Lauer, Henri, Lesnick, Robert U., and Matson, Leslie E., Servomechanism
Fundamentals, 2nd ed., pp. 349-486, McGraw Hill, New York, 1960.
13.
Klafter, Richard, D., Chmielewski, Thomas, A., and Negin, Michael, Robotic
Engineering, An Integrated Approach, Prentice Hall, New Jersey, 1989.
Introduction To Robotics Mechanics
Addision-Wesley Publishing, New York, 1989.
J.,
Ill
And
Control, 2 n " ed.,
New
Jersey,
14.
Slotine, Jean Jacques E., and Li, Weiping, Applied
88-97, 169-228, 266-271, 392-421, Prentice Hall,
Nonlinear Control, pp.
New
Jersey, 1991.
15.
Asada, H., and Slotine, J.J.E., Robot Analysis And Control, pp. 133-184,
John Wiley and Sons, New York, 1986.
16.
Nye, Theodore W., LeBlanc, David J., and Cipra, Raymond J., "Design And
Modeling Of A Computer Controlled Planar Manipulator", The Kinematics
Of Robot Manipulators, pp. 191-201, MIT Press, Cambridge, MA, 1987.
112
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No. Copies
1.
Defense Technical Information Center
2
Cameron
Station
Alexandria, Virginia 22304-6145
2.
Library,
Code 52
2
Naval Postgraduate School
Monterey, California 93943-5002
3.
4.
5.
Chairman, Code AA
Department Of Aeronautics and Astronautics
Naval Postgraduate School
Monterey, California 93943-5000
Chairman, Code SP
Department Of Aeronautics and Astronautics
Naval Postgraduate School
Monterey, California 93943-5000
Professor Brij N. Agrawal,
Code AA/Ag
1
1
2
Department Of Aeronautics and Astronautics
Naval Postgraduate School
Monterey, California 93943-5000
6.
Hyochoong Bang, Code AA/Bn
Department Of Aeronautics and Astronautics
Dr.
1
Naval Postgraduate School
Monterey, California 93943-5000
7.
LCDR Dennis
Sorensen
Naval Air Warfare Center
Point
Mugu,
CA
1
-
Weapons Division
93042-5000
113
rxnuA Liunwn
i
1 POSTGRADUATE SCH
MONTEREY. CALIFORNIA 93943-5002
Thesis
S66596
c.l
Thesis
S66596
c.l
Sorensen
of
Design and control
link
two
a space based
manipulator with Lyapunov
based control laws.
Sorensen
Design and control of
a space based two link
manipulator with Lyapunov
basedcontrol laws.
J5WC4>
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