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On the synthesis of controls for general nonlinear constrained mechanical systems
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On the synthesis of controls for general nonlinear constrained mechanical systems
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Content
ON THE SYNTHESIS OF CONTROLS FOR GENERAL NONLINEAR
CONSTRAINED MECHANICAL SYSTEMS
by
Thanapat Wanichanon
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(AEROSPACE ENGINEERING)
May 2012
Copyright 2012 Thanapat Wanichanon
ii
Acknowledgements
I would like to convey my deep gratitude and appreciation to my research advisor
Professor Firdaus E. Udwadia, who has treated me like a family member and has taken
good care of me during my years at the University of Southern California. This
dissertation was possible through his guidance and constant encouragement. He has the
unique ability to understand the field of mechanics in a simple way, and his knowledge
has motivated me to pursue my goals in the theoretical areas of my interest. I have always
highly valued his advice, and I can never thank him enough.
I am also deeply grateful to my dissertation committee members Professor Henryk
Flashner, Professor Geoffrey Shiflett, and Professor Michael G. Safonov who have
provided a great deal of encouragement over the years. I am also deeply indebted to all
my early-day teachers. I would also like to thank all my friends who have always
supported me and trusted in my ability.
Finally, most importantly, I want to thank my parents for seeing the potential inside me
and helping me to bring that out. None of this would have been possible without your
unconditional love and support; I really love you “Mom and Dad”. You are my
inspiration. Your love and support have guided me along through the most important
parts of my life.
iii
The voyage to the completion of this dissertation would have been much harder without
this group of people in my life. So I dedicate this dissertation to them and their life-long
hard work.
iv
Table of Contents
Acknowledgements……………………………………………………….………..
List of Figures………………………………………………………………..……..
Abstract……………………………………………………………….….…………
Chapter 1 GENERAL INTRODUCTION…………………………………………
1.1 Motivation…………………………………………………...…………..
1.2 Background……………………………………………………….……...
1.3 Research Objectives……………………………………………….…….
1.4 The Study’s Contributions………………………………………….……
1.5 Organization……………………………………………………………..
Chapter 2 FUNDAMENTAL EQUATION FOR NONLINEAR
CONSTRAINED SYSTEMS……………..………………………….....
2.1 Introduction………………………………………………………………
2.2 Explicit Equations of Motion of Nonlinear Constrained Systems……….
2.2.1 Description of the unconstrained system……………………….……
2.2.2 Description of the constraints……………………………….…….…
2.2.3 Description of the constrained system………………………….……
2.3 Example of Nonlinear Constraint-Following Problems…………….……
2.3.1 A triple pendulum in the XY-plane…………………………………..
2.3.2 Numerical results and simulations of the constraint-following
problem…………………………………...…………………...……..
2.4 Summary………………………………………………………………….
Chapter 3 HAMEL’S PARADOX AND THE FOUNDATIONS OF
ANALYTICAL DYNAMICS………………………………………….
3.1 Introduction………………………………………………………………
3.2 On the Dynamics of Constrained Systems……………………………….
3.3 Four Illustrative Examples………………………………………………..
3.3.1 Example 1……………………………………………………………
3.3.2 Example 2……………………………………………………………
3.3.3 Example 3……………………………………………………………
3.3.4 Example 4……………………………………………………………
3.4 Summary………………………………………………………………….
Chapter 4 ON GENERAL NONLINEAR CONSTRAINED SYSTEMS…………
4.1 Introduction………………………………………………………………
4.2 System Description of General Constrained Mechanical Systems………
ii
vii
xvi
1
1
2
6
6
9
14
14
15
15
16
17
20
20
23
27
28
28
33
37
37
44
52
57
65
69
69
72
v
4.3 Explicit Equations of Motion for General Nonlinear Constrained
Mechanical Systems………………………………………….…………..
4.3.1 Positive Definiteness of the Augmented Mass Matrices…………….
4.3.2 Explicit Equation for Constrained Acceleration……………………..
4.3.3 Explicit Equation for Constraint Force………………………………
Table 4.1: The three-step conceptualization of constrained motion by
utilizing the auxiliary system ……………..........…….…………
4.4 Illustrative Examples……………………………………………………..
4.4.1 Example 1……………………………………………………………
4.4.2 Example 2……………………………………………………………
4.4.3 Example 3……………………………………………………………
4.4.4 Example 4……………………………………………………………
4.4.5 Example 5……………………………………………………………
4.5 Summary………………………………………………………………….
Chapter 5 METHODOLOGY FOR TRACKING CONTROL OF NONLINEAR
UNCERTAIN SYSTEMS………………….………...…………………
5.1 Introduction………………………………………………………………
5.2 The Description of Control Approach……………………………………
5.2.1 System description of the actual systems……………………………
5.2.2 The effect of uncertainties in mechanical systems…………......……
5.2.3 System description of the controlled actual systems…..…………….
5.3 Uncertainties in the Dynamics of Mechanical Systems…………..……...
5.4 Summary………………………………………………………………….
Chapter 6 GENERALIZED TRACKING CONTROLLERS FOR NONLINEAR
UNCERTAIN SYSTEMS ……...............................................................
6.1 Introduction………………………………………………………………
6.2 Closed-Form Controlled Actual System………………………………….
6.3 Motivation in Developing Compensating Controllers……………………
6.4 Summary………………………………………………………………….
Chapter 7 TRACKING CONTROLLERS BASED ON THE CONCEPT OF THE
GENERALIZED SLIDING SURFACE CONTROL …….....................
7.1 Introduction………………………………………………………………
7.2 Generalized Sliding Surface Controllers (
SS
G ).……………………..…...
7.3 Numerical Results and Simulations of the Generalized Sliding Surface
Control……………………………………………………………………
7.4 Summary………………………………………………………………….
Chapter 8 TRACKING CONTROLLERS BASED ON THE CONCEPT OF THE
GENERALIZED DAMPING CONTROL ………………………….....
8.1 Introduction………………………………………………….……….......
77
78
80
89
100
101
101
106
109
113
116
122
129
129
131
132
139
145
146
150
151
151
151
153
155
157
157
158
167
209
214
214
vi
8.2 Generalized Damping Controllers (
D
G ).………………………………...
8.2.1 Numerical Results and Simulations of the Generalized Damping
Controller (
D
G )………………...……………………...….....……
8.3 Generalized Damping Control (
D
G )………………………….……...…..
8.3.1 Numerical Results and Simulations of the Generalized Damping
Controller (
D
G )................................................................................
8.4 Summary………………………………………………………………….
Chapter 9 GENERAL CONCLUSIONS…………………………………………..
Chapter 10 FUTURE WORK……………………………………………………...
Bibliography…….………………………………………………………………….
Appendix A SIMULTANEOUS UNCERTAINTIES IN MASSES AND GIVEN
FORCES ON THE SYSTEM…………………………………..….…
Appendix B THE PROOF OF
T
xy x y
∞∞
≥ …...……………………….............
215
226
249
256
278
283
292
294
299
300
vii
List of Figures
Figure 2.1:
Figure 2.2:
Figure 2.3:
Figure 2.4:
Figure 2.5:
Figure 3.1:
Figure 4.1:
Figure 4.2:
Figure 4.3:
Figure 5.1:
Figure 5.2:
Figure 5.3:
Figure 6.1:
Triple pendulum with the datum at the origin O……………………
Trajectory of the mass
3
m in the XY-plane (in meters) of the triple
pendulum shown for a duration of 10 secs. The trajectory starts at
the circle and ends at the square…………...............…......................
Angular responses of the masses
1
() , am
2
() , bm
3
and ( ) cm (no.
of revolutions ( 360
))…….................................................................
Energies in the system (a)
1
() Et , (b)
2
() Et , (c)
3
() Et , and (d)
23
() () () Et E t E t = + (
2
/ sec kg m − )…………………………………
Control forces applied to the nominal system to satisfy
23
() () () Et E t E t = + (in Newtons)…………………………………...
Taken from Hamel [21], this figure shows the blade inclined at an
angle ϑ to the x-axis of an inertial frame of reference……………..
A wheel rolling down an inclined plane under gravity……………...
A two degree-of-freedom multi-body system……………………….
Decomposition of the multi-body system shown in Figure 4.2 using
more than two coordinates…………………...……………………...
The difference in trajectory responses of the mass
3
m in the XY-
plane over a period of 10 secs. (a) the nominal system and (b) the
actual system when the uncertainties in masses are prescribed as
1
0.1 kg m δ = ,
2
0.2 kg m δ = − , and
3
0.3 kg m δ = …………………..
Angular responses (no. of revolutions (360
)) of the masses
2
( ) bm
3
and ( ) cm of the actual system move away from those of the
nominal system as time increases…………………………………...
The block diagram of the controlled uncertain system….…………..
The control functions……………………………….….…………....
21
25
25
26
26
29
102
106
110
141
142
144
154
viii
Figure 7.1:
Figure 7.2:
Figure 7.3:
Figure 7.4:
Figure 7.5:
Figure 7.6:
Figure 7.7:
Figure 7.8:
Figure 7.9:
Location of the actual masses , 1,2,3
ii
m mi δ±= of the 1014
Monte Carlo simulation……………………………………..………
Probability density function of Γ occurring from the 1014 Monte
Carlo simulation of 10 ± percent uncertainties in masses of the
triple pendulum when using the estimate of q δ from equation
(5.3.5)..………………………………………………………………
Location of the actual masses , 1,2,3
ii
m mi δ±= of the 1014
Monte Carlo simulation……………………………………..………
Probability density function of Γ occurring from the 1014 Monte
Carlo simulation of 10 ± percent uncertainties in masses of the
triple pendulum when using the estimate of q δ from equation
(5.3.10)…...………………………………………………………….
Location of the actual masses , 1,2,3
ii
m mi δ±= of the 1014
Monte Carlo simulation……………………………………..……....
Probability density function of Γ occurring from the 1014 Monte
Carlo simulation of 50 ± percent uncertainties in masses of the
triple pendulum when using the estimate of q δ from equation
(5.3.5)..………………………………………………………………
Location of the actual masses , 1,2,3
ii
m mi δ±= of the 1014
Monte Carlo simulation……………………………………..………
Probability density function of Γ occurring from the 1014 Monte
Carlo simulation of 50 ± percent uncertainties in masses of the
triple pendulum when using the estimate of q δ from equation
(5.3.10)………………………………………………………………
SS
G - Trajectory responses (in meters) of the mass
3
m
over a period
of 10 secs. of (a) the nominal system and (b) the controlled system
are approximately the same while (c) the actual system yields a
totally different trajectory when the uncertainties in masses are
prescribed as
1
0.1 kg m δ = ,
2
0.2 kg m δ = − , and
3
0.3 kg m δ =
( 10% ± of nominal masses) and the uncertainty’s bound in equation
(7.3.3) is chosen to be 100 ′ Γ = ………………………..……………
171
171
172
173
174
174
175
175
178
ix
Figure 7.10:
Figure 7.11:
Figure 7.12:
Figure 7.13:
Figure 7.14:
Figure 7.15:
Figure 7.16:
Figure 7.17:
SS
G - Angular responses (no. of revolutions (360
)) of the masses
23
( ) and ( ) bm c m of the actual system move away from those of
the nominal system as time increases while those of the controlled
system track the nominal system very well…………………………
SS
G - Tracking errors (
c
θ θ − ) in displacement of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled
systems (in radians).….......................................................................
SS
G - Tracking errors (
c
θ θ −
) in velocity of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled systems
(in rad/s).…..………………………………………………………...
SS
G - (b) Additional force to compensate for uncertainties in
masses,
u
Q , is small when compared to (a) the control force on the
nominal system to satisfy the energy control,
c
Q , where (c) shows
the sum of
cu
QQ + (in Newtons)…………………………………...
SS
G - Trajectory responses (in meters) of the mass
3
m
over a period
of 10 secs. of (a) the nominal system and (b) the controlled system
are approximately the same while (c) the actual system yields a
totally different trajectory when the uncertainties in masses are
prescribed as
1
0.1 kg m δ = − ,
2
0.05 kg m δ = , and
3
0.2 kg m δ = −
and the uncertainty’s bound in equation (7.3.6) is chosen to be
100 ′ Γ = …………….……………………………………………….
SS
G - Angular responses (no. of revolutions (360
)) of the masses
23
( ) and ( ) bm c m of the actual system move away from those of
the nominal system as time increases while those of the controlled
system track the nominal system very well…………………………
SS
G - Tracking errors (
c
θ θ − ) in displacement of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled
systems (in radians).....……………………………………………...
SS
G - Tracking errors (
c
θ θ −
) in velocity of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled systems
(in rad/s)...…………………………………………………………...
179
180
181
183
186
187
188
189
x
Figure 7.18:
Figure 7.19:
Figure 7.20:
Figure 7.21:
Figure 7.22:
Figure 7.23:
Figure 7.24:
Figure 7.25:
SS
G - (b) Additional force to compensate for uncertainties in
masses,
u
Q , is small when compared to (a) the control force on the
nominal system to satisfy the energy control,
c
Q , where (c) shows
the sum of
cu
QQ + (in Newtons)……………..…………………….
SS
G - Tracking errors (
c
θ θ − ) in displacement of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled
systems when using 500 ′ Γ = (in radians)…………………………..
SS
G - Tracking errors (
c
θ θ −
) in velocity of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled systems
when using 500 ′ Γ = (in rad/s)..………….…………………………
SS
G - (b) Additional force to compensate for uncertainties in
masses,
u
Q , is small when compared to (a) the control force on the
nominal system to satisfy the energy control,
c
Q , where (c) shows
the sum of
cu
QQ + when using 500 ′ Γ = (in Newtons)……………
SS
G - Trajectory responses (in meters) of the mass
3
m
over a period
of 10 secs. of (a) the nominal system and (b) the controlled system
are approximately the same while (c) the actual system yields a
totally different trajectory when the uncertainties in masses are
prescribed as
1
0.5 kg m δ = ,
2
0.8 kg m δ = − , and
3
1.5 kg m δ = and
the uncertainty’s bound in equation (7.3.6) is chosen to be 800. ′ Γ =
SS
G - Angular responses (no. of revolutions (360
)) of the masses
23
( ) and ( ) bm c m of the actual system move away from those of
the nominal system as time increases while those of the controlled
system track the nominal system very well…………………………
SS
G - Tracking errors (
c
θ θ − ) in displacement of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled
systems (in radians)............................................................................
SS
G - Tracking errors (
c
θ θ −
) in velocity of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled systems
(in rad/s)……………………………………………………………..
191
193
194
195
197
198
199
200
xi
Figure 7.26:
Figure 7.27:
Figure 7.28:
Figure 7.29:
Figure 7.30:
Figure 7.31:
Figure 8.1:
SS
G - (b) Additional force to compensate for uncertainties in
masses,
u
Q , is small when compared to (a) the control force on the
nominal system to satisfy the energy control,
c
Q , where (c) shows
the sum of
cu
QQ + (in Newtons)…………………….......................
UC - Trajectory responses (in meters) of the mass
3
m over a period
of 10 secs. (a) the nominal system and (b) the controlled system are
approximately the same while (c) the actual system yields a totally
different trajectory when the uncertainties in masses are prescribed
as
1
0.1 kg m δ = ,
2
0.2 kg m δ = − , and
3
0.3 kg m δ = and the
uncertainty’s bound in equation (7.3.7) is chosen to be 100. ′ Γ = …..
UC - Angular responses (in degrees) of three masses
1
() , am
2
() , bm
3
and ( ) cm of the actual system are different from those of
the nominal system as time increases while those of the controlled
system track the nominal system very well…………………………
UC - Tracking errors (
c
θ θ − ) in displacement of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled
systems (in radians)…………………………………………………
UC - Tracking errors (
c
θ θ −
) in velocity of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled systems
(in rad/s)……………………………………………………………..
UC - Additional control forces to compensate for uncertainties in
masses,
u
Q (in Newtons)….........…………………………………...
D
G - Trajectory responses (in meters) of the mass
3
m
over a period
of 10 secs. of (a) the nominal system and (b) the controlled system
are approximately the same while (c) the actual system yields a
totally different trajectory when the uncertainties in masses are
prescribed as
1
0.1 kg m δ = ,
2
0.2 kg m δ = − , and
3
0.3 kg m δ =
( 10% ± of nominal masses) and the uncertainty’s bound in equation
(8.2.45) is chosen to be 100 Γ= ..………………………………..…
202
205
206
207
208
209
229
xii
Figure 8.2:
Figure 8.3:
Figure 8.4:
Figure 8.5:
Figure 8.6:
Figure 8.7:
Figure 8.8:
D
G - Angular responses (no. of revolutions (360
)) of the masses
23
( ) and ( ) bm c m of the actual system move away from those of
the nominal system as time increases while those of the controlled
system track the nominal system very well…………………………
D
G - Tracking errors (
c
θ θ − ) in displacement of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled
systems (in radians)…………………………………………………
D
G - Tracking errors (
c
θ θ −
) in velocity of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled systems
(in rad/s)...…………………………………………………………...
D
G - (b) Additional force to compensate for uncertainties in
masses,
u
Q , is small when compared to (a) the control force on the
nominal system to satisfy the energy control,
c
Q , where (c) shows
the sum of
cu
QQ + (in Newtons)…………………..……………….
D
G - Trajectory responses (in meters) of the mass
3
m
over a period
of 10 secs. of (a) the nominal system and (b) the controlled system
are approximately the same while (c) the actual system yields a
totally different trajectory when the uncertainties in masses are
prescribed as
1
0.1 kg m δ = − ,
2
0.05 kg m δ = , and
3
0.2 kg m δ = −
and the uncertainty’s bound in equation (8.2.48) is chosen to be
100 Γ= ..………..…………………………………………………...
D
G - Angular responses (no. of revolutions (360
)) of the masses
23
( ) and ( ) bm c m of the actual system move away from those of
the nominal system as time increases while those of the controlled
system track the nominal system very well…………………………
D
G - Tracking errors (
c
θ θ − ) in displacement of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled
systems (in radians)…........................................................................
230
231
232
234
237
238
239
xiii
Figure 8.9:
Figure 8.10:
Figure 8.11:
Figure 8.12:
Figure 8.13:
Figure 8.14:
Figure 8.15:
D
G - Tracking errors (
c
θ θ −
) in velocity of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled systems
(in rad/s)…..........................................................................................
D
G - (b) Additional force to compensate for uncertainties in
masses,
u
Q , is small when compared to (a) the control force on the
nominal system to satisfy the energy control,
c
Q , where (c) shows
the sum of
cu
QQ + (in Newtons)………………..………………….
D
G - Trajectory responses (in meters) of the mass
3
m
over a period
of 10 secs. of (a) the nominal system and (b) the controlled system
are approximately the same while (c) the actual system yields a
totally different trajectory when the uncertainties in masses are
prescribed as
1
0.5 kg m δ = ,
2
0.8 kg m δ = − , and
3
1.5 kg m δ = and
the uncertainty’s bound in equation (8.2.48) is chosen to be
800 Γ= ..………..…………………………………………………...
D
G - Angular responses (no. of revolutions (360
)) of the masses
23
( ) and ( ) bm c m of the actual system move away from those of
the nominal system as time increases while those of the controlled
system track the nominal system very well…………………………
D
G - Tracking errors (
c
θ θ − ) in displacement of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled
systems (in radians)…........................................................................
D
G - Tracking errors (
c
θ θ −
) in velocity of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled systems
(in rad/s)…...………………………………………………………...
D
G - (b) Additional force to compensate for uncertainties in
masses,
u
Q , is small when compared to (a) the control force on the
nominal system to satisfy the energy control,
c
Q , where (c) shows
the sum of
cu
QQ + (in Newtons)………………..………………….
240
241
243
244
245
246
247
xiv
Figure 8.16:
Figure 8.17:
Figure 8.18:
Figure 8.19:
Figure 8.20:
Figure 8.21:
Figure 8.22:
D
G - Trajectory responses (in meters) of the mass
3
m
over a period
of 10 secs. of (a) the nominal system and (b) the controlled system
are approximately the same while (c) the actual system yields a
totally different trajectory when the uncertainties in masses are
prescribed as
1
0.1 kg m δ = ,
2
0.2 kg m δ = − , and
3
0.3 kg m δ =
( 10% ± of nominal masses)………………………………………….
D
G - Angular responses (no. of revolutions ( 360
)) of the masses
23
( ) and ( ) bm c m of the actual system move away from those of
the nominal system as time increases while those of the controlled
system track the nominal system very well…………………………
D
G - Tracking errors (
c
θ θ − ) in displacement of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled
systems (in radians)….……………………………………………...
D
G - Tracking errors (
c
θ θ −
) in velocity of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled systems
(in rad/s)...…………………………………………………………...
D
G - (b) Additional force to compensate for uncertainties in
masses,
u
Q , is small when compared to (a) the control force on the
nominal system to satisfy the energy control,
c
Q , where (c) shows
the sum of
cu
QQ + (in Newtons)………………...…………………
D
G - Trajectory responses (in meters) of the mass
3
m
over a period
of 10 secs. of (a) the nominal system and (b) the controlled system
are approximately the same while (c) the actual system yields a
totally different trajectory when the uncertainties in masses are
prescribed as
1
0.1 kg m δ = − ,
2
0.05 kg m δ = , and
3
0.2 kg. m δ = − ….
D
G - Angular responses (no. of revolutions ( 360
)) of the masses
23
( ) and ( ) bm c m of the actual system move away from those of
the nominal system as time increases while those of the controlled
system track the nominal system very well…………………………
259
260
261
262
263
266
267
xv
Figure 8.23:
Figure 8.24:
Figure 8.25:
Figure 8.26:
Figure 8.27:
Figure 8.28:
Figure 8.29:
Figure 8.30:
D
G - Tracking errors (
c
θ θ − ) in displacement of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled
systems (in radians)….……………………………………………...
D
G - Tracking errors (
c
θ θ −
) in velocity of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled systems
(in rad/s).…………………………………………………………….
D
G - (b) Additional force to compensate for uncertainties in
masses,
u
Q , is small when compared to (a) the control force on the
nominal system to satisfy the energy control,
c
Q , where (c) shows
the sum of
cu
QQ + (in Newtons)………………..………………….
D
G - Trajectory responses (in meters) of the mass
3
m
over a period
of 10 secs. of (a) the nominal system and (b) the controlled system
are approximately the same while (c) the actual system yields a
totally different trajectory when the uncertainties in masses are
prescribed as
1
0.5 kg m δ = ,
2
0.8 kg m δ = − , and
3
1.5 kg m δ = ……
D
G - Angular responses (no. of revolutions ( 360
)) of the masses
23
( ) and ( ) bm c m of the actual system move away from those of
the nominal system as time increases while those of the controlled
system track the nominal system very well…………………………
D
G - Tracking errors (
c
θ θ − ) in displacement of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled
systems (in radians).………………………………………………...
D
G - Tracking errors (
c
θ θ −
) in velocity of the masses
1
() , am
2
() , bm
3
and ( ) cm between the nominal and the controlled systems
(in rad/s).…………………………………………………………….
D
G - (b) Additional force to compensate for uncertainties in
masses,
u
Q , is small when compared to (a) the control force on the
nominal system to satisfy the energy control,
c
Q , where (c) shows
the sum of
cu
QQ + (in Newtons)…………………..……………….
268
269
270
273
274
275
276
277
xvi
Abstract
This dissertation develops in a unified manner a new and simple approach for the
modeling and controlling of general nonlinear constrained mechanical systems. Since, in
general, the description of constrained mechanical systems is highly nonlinear, the
determination of the equations of motion of such highly nonlinear constrained
mechanical systems is considered one of the central problems in analytical dynamics.
Even though this problem has been the focus of numerous studies, several questions
remain unanswered at the present time. Of particular importance among these questions
is an appropriate approach to obtaining the equations of motion for constrained
mechanical systems when the mass matrix of the unconstrained mechanical system is
singular, or if there are uncertainties in the description of the systems, what should be
done to cancel the effects of this uncertainty while the constrained equations of motion
are being derived. Moreover, misinterpretations persist concerning the fundamental
conceptualizations for deriving these constrained equations of motion.
The theoretical framework developed in this dissertation provides a concrete explanation
that deals with those fundamental conceptualizations for deriving the equations of motion
for general nonlinear constrained mechanical systems. Utilizing these fundamental
conceptualizations, an explicit constrained equation of motion that works with systems
containing either positive semi-definite or positive definite mass matrices is also further
developed. And when considering uncertainties, the explicit equation of motion for
xvii
nonlinear constrained mechanical systems has been modified by augmenting additional
additive controllers that are able to guarantee tracking of reference trajectories of the
system with no uncertainty assumed. The new, simple, general, and closed-form
equations of motion for nonlinear constrained uncertain mechanical systems are thus
developed. The results herein provide deeper insights into the behavior of constrained
motion and open up new approaches to modeling complex, uncertain, constrained
mechanical systems, such as those encountered in real-life multi-body dynamics.
1
Chapter 1
GENERAL INTRODUCTION
1.1 Motivation
One of the central problems in analytical dynamics is the determination of equations of
motion for constrained mechanical systems. Having been formulated more than 200 years
ago, many engineers, mathematicians and scientists have since been arduously working
on this problem. In their pursuits, multiple techniques have been developed with the
objective of finding the best methods in the aspects of computational accuracy and
efficiency. Although, many techniques have been provided, the perfect method,
describing exactly physical motions of constrained systems has yet to be reached and
there also still exists a multitude of misinterpretations concerning the fundamental
conceptualizations on deriving constrained equations of motion. Nonetheless, these
fundamental conceptualizations are all valuable in their own right in the understanding of
constrained equations of motion. The following section will discuss in further detail of a
few of the techniques of current methodologies.
2
1.2 Background
The description of equations of motion for constrained mechanical systems is first studied
in great detail by Lagrange (1787) [25]. He invents the Lagrange multiplier method
specifically to handle constrained motion. In general, this method requires extensive
algebraic manipulations, especially when dealing with nonlinear system equations.
Subsequently, Gauss (1829) [14] introduces a general, new principle of mechanics for
handling constrained motion, which today is commonly referred to as Gauss’s Principle.
Afterwards, Gibbs (1879) [15] and Appell (1899) [2] independently discover the so-
called Gibbs-Appell equations of motion using the concept of quasi-coordinates. Later
on, Maggi’s equation (1901) [27], [28] and null space formulation (1981) [13], [20] are
introduced to eliminate Lagrange multipliers in the Lagrange multiplier method.
However, both methods require the independency of constraints and the invertability of
the mass matrix. The latter puts more difficulty in deriving the equations of motion for
constrained mechanical systems. Following Gauss’s Principle, Udwadia and Kalaba
(1992) [43] obtaine a three-set of equations of motion which utilize the concept of the
generalized inverse of a matrix and provides the explicit equations of motion, a so-called
fundamental equation, for more general constrained systems. However, Udwadia and
Kalaba’s method cannot handle a situation in the instance that the mass matrix does not
have an inverse (positive semi-definite mass matrices). Udwadia and Phohomsiri (2006)
3
[50] derive an explicit equation of motion for such systems with positive semi-definite
(singular) mass matrices. They find that under certain restrictions on the structure of the
mass matrix and the structure of the constraints, the equation of motion of the constrained
system becomes unique. However, Udwadia and Phohomsiri’s equation differs in
structure and form from the so-called fundamental equation (Udwadia and Kalaba) and it
is not easy to see the physical interpretation of their equation. More recently, Udwadia
and Schutte (2010) [51] develop a simpler set of equations that have the same form as the
fundamental equation and these equations are valid for systems whose mass matrices can
be singular (positive semi-definite) and/or positive definite. This is done through the
replacement of the unconstrained mechanical system that has the singular mass matrix
with an unconstrained auxiliary system which is derived from augmenting the singular
mass matrix of the unconstrained mechanical system by appropriately making it positive
definite. However, the structure form of this augmented mass matrix requires
considerable computational effort especially when complex multi-body systems are
involved.
To the conceptualization of all the above-mentioned work, the equations of motion for the
constrained mechanical systems have been handled by assuming that there are no
uncertainties in defining their physical motions. However, real life multi-body systems
4
are usually uncertain. The most important feature in determination of equations of motion
for constrained mechanical systems is thus how to connect these equations of motion to
real world practices. The perfect mathematical model, describing those exactly physical
motions have yet to be reached, considering all real-life physical systems have their own
uncertainties at various levels. Some of these uncertainties are mainly derived from
modeling errors, while others from outer disturbances. In regards to the modeling error,
the structures and designed parameters of the mathematical model which describes a
dynamical system cannot be known perfectly. For the latter, many disturbances from
external environments are unpredictable and random. The error, which has the greatest
impact in describing mechanical systems, and that poses itself as most problematic and
critical to arriving at the most accurate result, arises from the modeling error. This error
which we might refer to the uncertainty in the system stems from two general sources:
uncertainties in the knowledge of the physical system and uncertainties in the ‘given’
forces applied to the system. Consequently, in order to arrive at constrained equations of
motion that can best describe the actual physical practices, one must take into
consideration those uncertainties in physical systems and in forces applied to the systems.
In the past few years, several results in motion controls based on the uncertainties in the
knowledge of the physical system have been considered [6], [29], [34], [35], [37], [54],
5
and [55]. However, the uncertainties in the knowledge of the forces acting on the system
have not been conceived. The following will discuss a few of the techniques that have
been used in motion controls of nonlinear uncertain systems. Model-based controllers
proposed in [29], [35] guarantee the asymptotic convergence of motion errors. But
computations of the schemes are time-consuming due to dependence on the regressor
matrix. In [37] and [6], a mixed
2
/ HH
∞
adaptive tracking control approach and an
adaptive fuzzy approach are presented respectively. But the asymptotic convergence
property of both approaches depends on a strong assumption that the approximation
errors belong to
2
LL
∞
∩ . Moreover, [6], [29], [35], and [37] address the control of
configuration variables only related to nonholonomic constraints and ignore the variables
subject to the holonomic constraints. Refs. [54] and [55] consider a tracking control
method which can deal with holonomic constraints. However, the method proposed in
[55] bases on a reduced-dimensional dynamic description which transforms the systems
into the so-called chained form but not all the systems can be converted to chained
systems. Ref. [54] applies robust adaptive techniques to systems with uncertainties and
gets satisfactory performances. But the proposed control law is not unique for the cases
of holonomic and nonholonomic constraints. Later on, a unified tracking controller based
on the sliding mode controls for general class of nonholonomic systems with
uncertainties is proposed in [34]. However, the proposed sliding mode controller in [34]
6
considers only the case of nonholonomic constraints and uses only a specific signum
function as a switch function which, in general, causes chattering when the tracking
trajectories reach the sliding manifold.
1.3 Research Objectives
Summarize above work, the two major challenges for our research are (i) the
determination of a unified framework in deriving the equations of motion for general
constrained mechanical systems which may contain either positive semi definite or
positive definite mass matrices that also can be explained by physical explanations and
also has a simpler explicit form of the equations of motion, and (ii) the determination of a
general approach for obtaining the equations of motion for nonlinear uncertain
constrained mechanical systems for general class of both holonomic and nonholonomic
systems in the presence of uncertainties in the actual real-life systems.
1.4 The Study’s Contributions
In pursuing the above-mentioned goals, we expand on previous research. The following
work is divided into three main sections. The first section posits an explanation for the
paradox posed by Hamel in his 1949 book on Theoretical Mechanics. As mentioned
previously, although many techniques have been provided, there still exists a multitude of
misinterpretations concerning the fundamental conceptualizations on deriving
7
constrained equations of motions. For this reason, Hamel’s paradox is explored. The
explanation deals with the foundations of mechanics and points to new insights
concerning analytical dynamics. Thereafter, in order to provide a simple unified
framework in deriving general constrained mechanical systems that may contain singular
(positive semi-definite) mass matrices, a new, simple, general, explicit form of the
equations of motion for general constrained mechanical systems that may contain either
positive semi-definite or positive definite mass matrices is developed. This is done
through the replacement of the actual unconstrained mechanical system, which may have
a positive semi-definite mass matrix, with an unconstrained auxiliary system that is
subjected to the same holonomic and/or nonholonomic constraints as the actual
constrained mechanical system. The mass matrix of the auxiliary system is appropriately
augmented to make it positive definite in a much simpler, more general, and
computationally more efficient manner than versions found in previous works in order for
the so-called fundamental equation (Udwadia and Kalaba) to be directly and simply
applied to get in closed-form the acceleration of the actual constrained mechanical system
when the mass matrix of the actual unconstrained mechanical system is positive semi-
definite or positive definite. Furthermore, it is shown that by appropriately augmenting
the ‘given’ force acting on the actual unconstrained mechanical system, the auxiliary
system directly provides the constraint force required to satisfy the constraints imposed
8
on the actual mechanical system. Thus, irrespective of whether the mass matrix of the
unconstrained mechanical system is positive definite or positive semi-definite, a simple,
unified fundamental equation results in closed-form representation of both the
acceleration of the constrained mechanical system and the constraint force acting on it.
The results herein provide deeper insights into the behavior of constrained motion and
open up new ways of modeling complex, constrained mechanical systems, such as those
encountered in multi-body dynamics. Lastly, considering uncertainties in the two general
sources of mechanical systems—in system’s descriptions and in ‘given’ forces, the
fundamental equation from the conceptualization in the first section is modified to deal
with these uncertainties. The aim in this section is to develop a general control
methodology, which when applied to a real-life uncertain system, causes this system to
track a desired reference trajectory that is pre-specified from the fundamental equation.
We do this by augmenting an additional additive controller—based on a generalization of
the concept of general control schemes: the sliding surface control and the damping
control—to the fundamental equation, providing a general approach for obtaining the
equations of motion for a general nonlinear uncertain constrained mechanical system.
This leads to a set of closed-form nonlinear controllers that can guarantee tracking
reference trajectories under uncertainties. The additional additive controllers developed
herein can be used for a wide class of control laws that can be adapted to the specific
9
real-life practical limitations of a given, particular controller being used. Thus, in this
section, methodologies are developed to obtain sets of closed-form controllers for
nonlinear uncertain multi-body systems that can track a desired reference trajectory. The
sets of controllers show good transient behavior and are robust with respect to the
uncertainties involved in the description of the real-life system.
1.5 Organization
This study is organized into the following ten chapters:
Chapter 2: Fundamental Equation for Nonlinear Constrained Systems
In chapter 2, the conceptualization on deriving equations of motion for nonlinear
constrained mechanical systems is discussed. Using Gauss’s principle, three-set equations
of motion: the unconstrained system, the constraint equation, and the constrained system,
are all reviewed herein. With the help of the concept of a generalized inverse of a matrix,
the explicit equations of motion for general constrained systems are explained. The
formulation of the explicit equation of motion provides a nonlinear optimal controller for
nonlinear constrained systems so that the constraint trajectory is exactly tracked. No
approximations/linearizations are done related to the nonlinear nature of the system. An
example of a simple, though non-trivial, mechanical system is provided to demonstrate
10
the efficacy of the controller. Numerical results illustrate the effectiveness of the
approach.
Chapter 3: Hamel’s Paradox and the Foundations of Analytical Dynamics
In this chapter, the paradox posed by Hamel in his 1949 book on Theoretical Mechanics
is dealt with. The three-step conceptualization on deriving equations of motion of
constrained mechanical systems discussed within chapter 2 is utilized. Illustrative
examples are shown for both correct and incorrect approaches to demonstrate that
conflating the unconstrained system description with the constrained system description
results in the wrong equation of motion. The objective of this chapter is to provide a
unique methodology for deriving the equations of motion for constrained mechanical
systems, which minimizes the chances for miscalculation and misinterpretation.
Chapter 4: On General Nonlinear Constrained Systems
Chapter 4 deals with a new, simple, general, explicit form of equations of motion for
general nonlinear constrained mechanical systems that may contain either positive semi-
definite or positive definite mass matrices.
11
Chapter 5: Methodology for Tracking Control of Nonlinear Uncertain Systems
Since descriptions of real-life complex multi-body systems are usually uncertain, the
uncertainties which arise from the two general sources—uncertainties in the knowledge
of the physical system and uncertainties in the ‘given’ forces applied to the system—are
both introduced within this chapter. Considering these uncertainties, a closed-form
equation of motion that can deal with the uncertainties—the so-called controlled actual
system—is explained. The development of this equation is done by augmenting an
additional additive controller to the fundamental equation discussed in chapter 2. This
new, simple, and general closed-form equation of the controlled actual system is a
general formulation that is applied by several tracking control laws which is discussed in
Chapter 7 and Chapter 8.
Chapter 6: Generalized Tracking Controllers for Nonlinear Uncertain systems
Chapter 6 discussess in detail the advantages of the controlled actual system as well as
provides the background idea in developing a compensating controller to be used with the
proposed controlled actual system.
12
Chapter 7: Tracking Controllers based on the Concept of the Generalized Sliding
Surface Control
A controller based on a generalization of the notion of the sliding surface control is
developed in this chapter. When using this controller in the closed-form controlled actual
system described in chapter 5, tracking of reference trajectories is guaranteed within
desired error bounds.
Chapter 8: Tracking Controllers based on the Concept of the Generalized Damping
Control
In this chapter, two controllers based on the concept of the generalization of the damping
control are introduced. The first controller can relax the limitation on the forcing function
of a tracking controller being used in a general discontinuous control, while it is still able
to guarantee tracking of reference trajectories within specified error bounds.
The second controller developed in this chapter is also able to guarantee tracking of
reference trajectories within estimated error bounds. However, the formulation of the
controller does not need knowledge of the bound on the uncertainty as required by the
previous two controllers.
13
Chapter 9: General Conclusions
Chapter 9 outlines the fundamental conceptualizations in analytical dynamics from
chapters 2 and 3 as well as summarizes the procedures in modeling constrained
mechanical systems utilizing new methodologies from chapters 4, 5 and 6. Some
comparisons among the three controllers in chapters 7 and 8 are also discussed and
recapitulated.
Chapter 10: Future Directions
Lastly, chapter 10 proposes plans for future work and suggests follow-up studies that
could be based on this dissertation.
14
Chapter 2
FUNDAMENTAL EQUATION FOR NONLINEAR
CONSTRAINED SYSTEMS
2.1 Introduction
In this chapter, a general formulation to describe the dynamics of a general nonlinear
constrained mechanical system by following the conceptualization developed by Gauss
[14] is introduced. The formulation is done under d’Alembert’s principle (assumption)
which states that the total work done by all the forces of constraint under virtual
displacements sum to zero (Udwadia and Kalaba [47]). No approximations/linearizations
are done related to the nonlinear nature of the system. The conceptualization being used
provides a clear picture of the manner in which constrained mechanical systems need to
be conceptualized, so that the equations of motion that are then obtained properly reflect
physical observational evidence. An example of a simple, though non-trivial, mechanical
system is provided to demonstrate the efficacy of the proposed formulation. In the
following, a systematically modeling approach of a general nonlinear constrained
mechanical system is illustrated.
15
2.2 Explicit Equations of Motion of Nonlinear Constrained Systems
It is useful to conceptualize the description of nonlinear constrained systems in a three-
step procedure. We do this in the following way:
2.2.1 Description of the unconstrained system
The unconstrained system is conceptually the mechanical system in which the virtual
displacements are assumed to be all independent of one another. One obtains the
equations of motion that govern this unconstrained system by: (1) starting with the
assumption that no constraints exist between the coordinates that describe the
configuration of the system, (2) writing down the kinetic energy of the system under this
assumption, and then (3) using Lagrange’s equation, taking into account all the impressed
forces (both conservative and non-conservative) that are acting on the system. We note
that the inclusion of these impressed forces in Lagrange’s equations arises through a
determination of the work that they do under virtual displacements; and these
displacements are assumed to be independent when describing the unconstrained system.
Using Lagrange’s equations we can write the equations of motion of this unconstrained
system as
( , ) ( , , ) M qt q Q q qt =
(2.2.1)
16
with the initial conditions
00
( 0) , ( 0) , qt q qt q = = = =
(2.2.2)
where q is the n-vector of coordinates specifying the configuration of the
unconstrained system, the n by n matrix 0 > M is the mass matrix which is a function of
q, and t; and Q is the ‘impressed’ or ‘given’ force n-vector that acts on the system which
is a known function of , , and qq t . From equation (2.2.1), we find that the acceleration
of the unconstrained system is given by
1
: ( ,) ( , ,). a M qt Q q qt
−
= (2.2.3)
2.2.2 Description of the constraints
One now conceptualizes that the necessary m equality constraints are then imposed on
this unconstrained system that is described in our previous step. We suppose that the
unconstrained mechanical system is now subjected to the m constraints given by
( , , ) 0, 1, 2,..., ,
i
q qt i m ϕ = =
(2.2.4)
where rm ≤
equations in the equation set (2.2.4) are functionally independent. The set of
constraints described by (2.2.4) include all the usual varieties of holonomic and/or
nonholonomic constraints, and then some. We shall assume that the initial conditions
(2.2.2) satisfy these m constraints. Therefore, the components of the n-vectors
0
q and
0
q
cannot all be independently assigned. We further assume that the set of constraints (2.2.4)
17
are smooth enough so that we can differentiate them with respect to time t to obtain the
relation
( , ,) ( , ,), A qq t q bqq t =
(2.2.5)
where A is an m by n matrix whose rank is r, and b is an m-vector. We note that each row
of A arises by appropriately differentiating one of the m constraint equations in the set
given in relation (2.2.4).
2.2.3 Description of the constrained system
Using the information in the previous two steps, in the last step we bring together the
description of motion of the constrained mechanical system as
( , ) ( , , ) ( , , ),
c
M qt q Q q qt Q q qt = + (2.2.6)
where
c
Q is the constraint force n-vector that arises to ensure that the constraints (2.2.5)
are satisfied at each instant in time. Thus, equation (2.2.6) describes the motion of the
nonlinear constrained mechanical system.
Gauss in his landmark chapter goes on to show that the motion of the constrained system
is obtained by ensuring that the deviation of its acceleration from that of the
unconstrained system (weighted by the matrix M) is a global minimum at each instant of
time (Gauss [14]). It can be shown that this condition gives the explicit, so-called
18
fundamental equation of motion of the constrained system (Udwadia and Kalaba [45]) in
the form
1
( , , ) ( ) ( ),
TT
Mq Q q q t A AM A b Aa
−+
=+− (2.2.7)
wherein the various quantities have been defined in the previous two steps, and the
superscript ‘+’ denotes the Moore-Penrose inverse of the matrix. In the above equation,
and in what follows, we shall suppress the arguments of the various quantities unless
required for clarity.
We note that relation (2.2.7) is valid:
(a) whether or not the equality constraints are holonomic or nonholonomic,
(b) whether or not they are nonlinear functions of the generalized velocities and
coordinates, and
(c) whether or not they are functionally dependent.
We also note that from equation (2.2.7) the constraint (control) force that the
unconstrained system is subjected to because of the presence of the constraints (2.2.5)
can be explicitly expressed as
1
( , , ) ( , , )( ( , ) ) ( ( , , ) ( , , )).
cT T
Q q qt A q qt AM qt A b q qt Aa q qt
−+
= − (2.2.8)
19
Pre-multiplying both sides of equation (2.2.7) with
1
M
−
, a constrained acceleration of
nonlinear systems can be expressed as
11
( ) ( ),
TT
q a M A AM A b Aa
− −+
= + − (2.2.9)
where the second member in equation (2.2.9) represents the constrained acceleration that
guarantees constraint-following of the mechanical system.
When the unconstrained description (given by (2.2.1)) of the constrained system is such
that the n by n matrix M is singular, then equation (2.2.7) cannot be used since
1 −
M does
not exist. In that case equation (2.2.7) needs to be replaced by the equation
()
Q I A AM
q
b A
+
+
−
=
(2.2.10)
under the proviso that the rank of the matrix [ | ]
T
MA is n. This rank condition is a
necessary and sufficient condition for the constrained system to have a unique
acceleration—a consequence of physical observation of the motion of bodies. (Udwadia
and Phohomsiri [50])
While this step can be carried out through the use of Lagrange multipliers, equations
(2.2.9) and (2.2.10) are more convenient, and work in situations (such as, when the
constraints are not functionally independent) in which the Lagrange multiplier method
20
breaks down. They obviate the difficulties in finding the Lagrange multipliers, especially
when the constraints are nonlinear functions of their arguments.
2.3 Example of Nonlinear Constraint-Following Problems
To clarify our above-mentioned description we introduce an example to demonstrate the
applicability of the constraint-following control methodology suggested in the previous
section. It is straightforward to extend this example to more general situations.
2.3.1 A triple pendulum in the XY-plane
Consider a planar pendulum consisting of three masses
1
m ,
2
m , and
3
m suspended from
massless rods of lengths
1
L ,
2
L , and
3
L moving in the XY-plane (see Figure 2.1). The
inertial frame of reference is fixed at the point of suspension, O, of the triple pendulum
and the X-axis is taken as the datum for describing the potential energy of each mass. The
masses are constrained to move so that the total energy, () Et , of the system equals the
sum of the energies (kinetic and potential) of only the two masses m
2
and m
3
, i.e.,
2 3
() () () Et E t E t = +
where we have denoted ()
i
E t as the total energy of mass
i
m .
21
Figure 2.1: Triple pendulum with the datum at the origin O
We begin by writing the equation of the unconstrained system (corresponding to equation
(2.2.1)) using the generalized coordinate 3-vector
12 3
[, , ]
T
q θθ θ = whose components, in
the absence of any constraints, are independent of one another. Lagrange’s equations then
yield the relation
12 3 12 3
(; , , ) (; , , ) M q mm m q Q q mm m =
(2.3.1)
where the elements of the 3 by 3 symmetric matrix M are given by
2
11 1 2 3 1 12 2 3 1 2 12 13 3 1 3 13
22
22 2 3 2 23 3 2 3 23 33 3 3
( ); ( ) cos(); cos()
( ); cos(); ,
M m m m L M m m LL M m LL
M m m L M mL L M mL
θθ
θ
= ++ = + =
=+= =
(2.3.2)
and the elements of the 3-vector Q are given by
22
22
1 2 3 1 2 2 12 3 1 3 3 13 1 2 3 1 1
22
2 23 1 2 1 12 23 1 2 1 2 12 3 2 3 3 23 23 2 2
22
3 3 1 3 1 1 3 13 323 2
( ) sin( ) sin( ) ( ) sin
( ) sin( ) 2( ) sin( ) sin( ) ( ) sin
(2 )sin( ) (2
Q m m LL m LL m m m gL
Q mm L L mm L L m L L mm gL
Q mL L mL L
θθ θθ θ
θθ θθ θ θθ θ
θθθ θ θθ
= − + − − ++
= + − + − − +
= − + −
2 3 23 3 3 3
)sin( ) sin . m gL θθ θ −
(2.3.3)
In the above, we have denoted
ij i j
θ θθ = − .
Using the X-axis as the datum (see Figure 2.1), we next describe the energy constraint
23
EE E = +
which is equivalent to the relation
1
0, E =
(2.3.4)
where the energy
1
E of mass
1
m is give by
22
1 1 11 1 1 1
1
cos .
2
E m L m gL θθ = −
(2.3.5)
Since the system may not initially (at time t = 0) satisfy this constraint we modify the
constraint (2.3.4) using the trajectory stabilization relation [39],
11
0, EE α +=
(2.3.6)
where
α
is any positive real number. By (2.3.5) and (2.3.6) we then get the constraint
relation
2 22
11 1 1 1 11 1 1
1
: 0 0 = sin ( cos ) : .
2
Aq L q gL L gL b θ θθ α θ θ = −− −=
(2.3.7)
23
We note that the masses
i
m do not enter this constraint equation. The next step to obtain
the equations of motion of the constrained system using the information from equations
(2.3.1)-(2.3.3) and (2.3.7), is to bring together the description of motion of the
constrained mechanical system as (see equation (2.2.6))
,
c
Mq Q Q = + (2.3.8)
where
c
Q represents the forces of constraint that will be exerted on the bobs, causing
them to move so that at every instant of time the constraint (2.3.7) is satisfied. Since
0 M > , the determination of these constraint forces can be made through the use of the
fundamental equation as in [47] (see equation (2.2.8))
1
( , , ) ( ) ( ).
c TT
Q q q t A AM A b Aa
−+
= − (2.3.9)
Using equation (2.3.9) in equation (2.3.8) and pre-multiplying both sides of the equation
by
1
M
−
, we obtain the constrained acceleration of the system as (see equation (2.2.9)),
11
( ) ( ).
TT
q a M A AM A b Aa
− −+
= + − (2.3.10)
2.3.2 Numerical results and simulations of the constraint-following problem
The values of the three masses are taken to be
1
1 m = kg,
2
2 m = kg, and
3
3 m = kg, and
the lengths of the massless rods to be
1
1 L = m,
2
1.5 L = m, and
3
2 L = m. At t = 0, the
masses are located with the angles of
1
(0) 1 rad, θ =
2
(0) 0 rad, θ = and
3
(0) 0 rad θ = with
respect to the vertical Y-axis. The initial velocities of the three bobs are taken to be
24
1
(0) 0.001 rad/s, θ =
2
(0) 0 rad/s, θ =
and
3
(0) 0 rad/s θ =
. We note that these initial
conditions do not satisfy the constraint,
1
0 E = . The acceleration due to gravity is
downwards, and of magnitude
2
9.81 / g ms = , and the parameter α in (2.3.6) is chosen
to be
4
2
0.02 A where
2
A is the
2
L norm of the matrix A in (2.3.7). The numerical
integration throughout this section is done in the Matlab environment, using a variable
time step integrator with a relative error tolerance of
8
10
and an absolute error tolerance
of
12
10
.
Figure 2.2 plots the trajectory of mass
3
m of the triple pendulum in the XY-plane for a
period of 10 seconds. The trajectory starts at the circle and stops at the square as shown in
the figure. Figure 2.3 shows the angular responses, in numbers of revolutions of 360
, of
each of the masses as a function of time. The energies of the components of the system
are shown in Figure 2.4. We see that the total energy (E) is the sum of the energies from
the mass
2
m (
2
E ) and mass
3
m (
3
E ), i.e.
23
EE E = + . Figure 2.4(a) also shows the extent
of error in the satisfaction of this constraint
1
0 E = . The magnitude of this error is seen to
be commensurate with the relative error tolerance used in the numerical integration. In
Figure 2.5, we show the numerically obtained control force
c
Q on the system in order to
follow the desired control requirement
23
() () () Et E t E t = + .
25
Figure 2.2: Trajectory of the mass
3
m in the XY-plane (in meters) of the triple pendulum shown for a duration of 10
secs. The trajectory starts at the circle and ends at the square
Figure 2.3: Angular responses of the masses
12 3
( ) , ( ) , and ( ) a m bm c m (no. of revolutions ( 360
))
26
Figure 2.4: Energies in the system (a)
1
() E t , (b)
2
() E t , (c)
3
() E t , and (d)
23
() () () Et E t E t = +
( )
2
/ sec kg m −
Figure 2.5: Control forces applied to the nominal system to satisfy
23
() () () Et E t E t = +
(in Newtons)
27
2.4 Summary
A general formulation for describing the exact equations of motion of nonlinear
constrained systems has been shown. By using this formulation, we obtain the exact
closed-form solution to the constraint-following problem of energy control. Through use
of the fundamental equation, we capture all the nonlinearities of the nonlinear, non-
autonomous system of differential equations. The constraint (control) force that must be
applied to the system in the presence of the energy constraint is easily obtained. Also,
with the incorrect initial states, the error in satisfaction of the energy requirement
converges to zero exponentially.
In the next chapter, we provide a more thorough, detailed exploration of the
conceptualization of deriving the equations of motion for nonlinear constrained systems
presented in the previous sub-section of this chapter. We show that conflating the
unconstrained system description with the constrained system description results in the
wrong equations of motion of constrained systems.
28
Chapter 3
HAMEL’S PARADOX AND THE FOUNDATIONS OF
ANALYTICAL DYNAMICS
3.1 Introduction
We begin by stating the problem considered by Hamel and the subsequent paradox posed
by him in his book on theoretical mechanics (Hamel [19]). Hamel considers in one of his
examples the planar motion of the blade of an ice skate of mass m modeled as an edge
resting on a horizontal ice surface (see Figure 3.1). The center of mass of the blade is at
the point M, and it makes contact with the ice surface at the point B which is at a distance
s from M. The blade makes an angle ) (t ϑ with the x-axis of an inertial coordinate frame
at the instant of time t, and the coordinates of the point B at that instant are denoted by (x,
y). The blade’s motion is restricted so that its velocity is always along the line along
which it is inclined in the x–y plane.
29
Figure 3.1: Taken from Hamel [19], this figure shows the blade inclined at an angle ϑ to the x-axis of an inertial
frame of reference
Following Hamel, the kinetic energy of the system is given by
22 2
11
( ) [ cos sin ] ,
22
B
T m x y ms y x I ϑϑ ϑ ϑ = ++ − +
(3.1.1)
where
B
I is the moment of inertia of the blade about the point B. The non-holonomic
constraint equation that requires that the velocity of the blade be along the direction of its
length is given by
cos sin 0. yx ϑ ϑ −=
(3.1.2)
Hamel obtains the generalized (momenta) impulses as
sin ,
x
T
p mx ms
x
ϑϑ
∂
= = −
∂
(3.1.3)
B
M
y
x
s
ϑ
y
x
30
cos ,
y
T
p my ms
y
ϑϑ
∂
= = +
∂
(3.1.4)
and,
( cos sin ) ,
B
T
p ms y x I
ϑ
ϑ ϑϑ
ϑ
∂
== −+
∂
(3.1.5)
and gets
( sin ),
x
d
W mx ms
dt
ϑϑ = −
(3.1.6)
( cos ),
y
d
W my ms
dt
ϑϑ = +
(3.1.7)
and,
( ( cos sin )) ( sin cos ).
B
d
W ms y x I ms y x
dt
ϑ
ϑ ϑ ϑϑ ϑ ϑ = − ++ +
(3.1.8)
Using Lagrange’s equations and the Lagrange multiplier, λ, Hamel then writes the
equations of motion of the constrained system as
( sin ) sin
d
mx ms X
dt
ϑϑ λ ϑ − = −+
(3.1.9)
( cos ) cos
d
my ms Y
dt
ϑϑ λ ϑ +=+
(3.1.10)
31
( ( cos sin )) ( sin cos ) ,
B
d
ms y x I ms y x M
dt
ϑ ϑ ϑϑ ϑ ϑ − ++ + =
(3.1.11)
where X and Y are the components in the x and y directions of the impressed (or ‘given’)
force on the blade at B, and M is the impressed moment. In order to simplify our
discussion, we shall assume that these impressed forces and moments are zero, and that
the mass m of the blade is unity.
Using the constraint (3.1.2) the Lagrange multiplier, λ , can be eliminated from equations
(3.1.9)-(3.1.11), as Hamel does in his book, and the equations of motion for the system
can be written as
2
2
cos tan
sin .
( sin cos )
B
x s x
ys x
I sy x
ϑ ϑ ϑ ϑ
ϑ ϑϑ
ϑ ϑ ϑ ϑ
−
= +
−+
(3.1.12)
Hamel in his book goes on to state (and we quote):
“Having come up with the equations of motion, we can obviously make use of the
equation of constraint and therefore simplify the third equation [i.e., (3.1.11)] by
eliminating the first term; we would not have been permitted to work with the kinetic
energy
32
22 2
11
()
22
B
T m x y I ϑ
+
= ++
(3.1.13)
which is obtained by using the equation of constraint [in (3.1.1)]. Very obviously this
would have given the wrong equations.” (The square brackets above are placed by the
author for clarity.)
This quote points out that were we to substitute the constraint (3.1.2) in the expression for
the kinetic energy given in (3.1.1) and thereby eliminate the second term on the right-
hand side in this relation, we would obtain the expression
+
T given in equation (3.1.13).
The use of this amended kinetic energy,
+
T , along with the constraint (3.1.2) would then
have led us—in a manner similar to what we did before in using Lagrange’s equations
along with the multiplier λ —to the following equation of motion
tan
,
0
xx
yx
ϑϑ
ϑ
ϑ
−
=
(3.1.14)
which, as stated above by Hamel, would indeed be incorrect!
While Hamel is correct in what he writes, he leaves the reader unclear as to why such a
seemingly harmless substitution of the constraint equation (3.1.2) in the kinetic energy
33
expression T leads to the wrong equations of motion; for, after all, the system must
satisfy this constraint at each and every instant of time.
More importantly, from his statements that we have quoted above, he leaves the reader
unclear whether this situation is unique to this specific problem, or perhaps that it occurs
in only certain special problems. And if the latter were to be true, he leaves open the
question of identifying those special circumstances in which this seemingly harmless
substitution of the constraint equation(s)—which in any case must always be satisfied by
the system—in the expression for the kinetic energy of the system would lead to
erroneous results upon the application of Lagrange’s equations using the usual multiplier
method.
It is this paradox that Hamel brings up in his book that has led us to investigate it in
greater detail, and as shown below, its understanding will lead us to the very foundations
of analytical dynamics.
3.2 On the Dynamics of Constrained Systems
As a prelude to explaining what might be going on here it is useful to move from the
Lagrangian view of mechanics to the one developed by Gauss [14]. Both views
ultimately rest on d’Alembert’s principle (assumption) that states that the total work done
34
by all the forces of constraint under virtual displacements sum to zero (Udwadia and
Kalaba [47]). However, Gauss provides a clearer picture of the manner in which
constrained mechanical systems need to be conceptualized, so that the equations of
motion that are then obtained properly reflect physical observational evidence. His
conceptualization is done in three distinct steps as shown in Section 2.2.
1. Description of the unconstrained system: The unconstrained system is
conceptually the mechanical system in which the virtual displacements are
assumed to be all independent of one another.
In Hamel’s example,
T
y x q ] , , [ ϑ =
is the 3-vector of coordinates specifying
the configuration of the system (skate). Equations (3.1.9)-(3.1.11) with λ
set to zero are the unconstrained equations of motion of the system; they
correspond to equation (2.2.1). This is because setting λ to zero signifies
that the constraint (3.1.2) is assumed to be non-existent, and hence that the
coordinates (and the corresponding virtual displacements) are independent.
2. Description of the constraints: We now impose a set of constraints on this
unconstrained system system that is described in our previous step.
35
In Hamel’s example, the constraint that needs to be imposed on the
unconstrained system described earlier is given by equation (3.1.2). Upon
differentiating this equation once with respect to time we can express it in
the form of equation (2.2.5) where
0] cos sin [ ϑ ϑ − = A
(3.2.1)
and
sin cos . by x ϑ ϑϑ ϑ = +
(3.2.2)
3. Description of the constrained system: Using the information in the previous
two steps, we bring together in the last step the description of motion of the
constrained mechanical system as (see equation (2.2.9))
11
( ) ( ),
TT
q a M A AM A b Aa
− −+
= + − (3.2.3)
wherein
1
: a MQ
−
= and the various quantities have been defined in the
previous two steps, and the superscript ‘+’ denotes the Moore-Penrose
inverse of the matrix.
Of crucial importance in this three-step conceptualization of constrained systems is the
need to construe the unconstrained system in Step 1 above as one on which no constraints
36
are assumed to be imposed. Conflating the unconstrained system with the constrained
system—by imposing the constraints while obtaining the equations of motion for the
supposedly unconstrained system—would naturally then, in general, lead to the wrong
final set of equations for the constrained system.
In Hamel’s example, substitution of equation (3.1.2) into the kinetic energy of the system
given by equation (3.1.1), and use of the amended kinetic energy
+
T so obtained as in
relation (3.1.13), conflates the unconstrained system with the constrained system.
Lagrange’s equations obtained by using
+
T as the kinetic energy of the system therefore
cannot, in general, give us a correct description of the unconstrained equations of motion
of the system which we are supposedly looking for in Step 1 above, and also on which we
want to further impose our constraint given by (3.1.2), in order to finally get the
equations of motion of the constrained system.
We see then that it is this fundamental misunderstanding on how a general, constrained
mechanical system is to be conceptualized in analytical dynamics that leads us to this
seeming paradox, and eventually to the wrong results when using
+
T to obtain the correct
equations of motion.
37
If things are as stated above, then this should always be so. Let us explore if that is indeed
the case. In the spirit of Hamel’s example we do this through the use of four examples.
3.3 Four Illustrative Examples
Our aim is to keep the examples as simple as possible while still attempting to cull out as
much of the fundamental analytical dynamics from them as possible. This will help to
elucidate the somewhat interesting way in which one needs to conceptualize constrained
motion in mechanical systems so as to obtain the correct equations of motion that
describe them.
3.3.1 Example 1
Consider a particle of unit mass moving in 3-dimensional space. Using an inertial right-
handed coordinate system OXYZ let us assume that this particle’s position at any time t is
described by its coordinates (x, y, z) and that it is subjected to an impressed force whose
components along the X-, Y- and Z-directions are respectively ( , ,, , ,, )
x
Q x yz x yz t ,
( , ,, , ,, )
y
Q x yz x yz t , and ( , ,, , ,, )
z
Q x yz x yz t . The particle is subjected to the
nonholonomic constraint
2 22
z xy = + .
38
Our aim is to find the equation of motion of this constrained system. This happens to be,
in fact, a significant, problem in the history of analytical dynamics, first proposed by
Appell [1] in 1911.
In what follows we shall first provide the three-step, correct approach to conceptualizing
the constrained motion of this simple system and so obtain its proper equation of motion.
Then we shall see what happens when we conflate the aforementioned distinction
between the description of the unconstrained system and the constrained system.
3.3.1.1 The Correct Approach
Let us denote : [ , , ]
T
x y z
Q QQ Q = . We first conceptualize the unconstrained system in
which all the coordinates that describe the configuration of the particle are independent of
one another. The kinetic energy of the particle is
2 22
11 1
,
22 2
Tx y z = + + (3.3.1)
and Lagrange’s equation for this unconstrained system in which all the virtual
displacements are assumed independent of one another is trivially obtained as
( , ,, , ,, )
( , ,, , ,, ) .
( , ,, , ,, )
x
y
z
Q x yz x yz t x
y Q x yz x yz t
z
Q x yz x yz t
=
(3.3.2)
39
Note that the right-hand side of equation (3.3.2) arises while applying Lagrange’s
equation through a consideration of the virtual work done by the force Q under virtual
displacements, the components of which are assumed independent of one another. From
this relation we find that the acceleration of the unconstrained system [, , ]
T
x y z
a QQ Q = .
Our next conceptual step is to impose the constraint
2 22
z xy = + (3.3.3)
on this unconstrained system that is described by relation (3.3.2). Differentiating equation
(3.3.3) with respect to time once, we obtain
, z z xx y y = +
(3.3.4)
so that
[] A xy z = − (3.3.5)
and
0. b = (3.3.6)
The equation of motion of the constrained system—our last conceptual step—is obtained
by simply using relation (3.2.3), which trivially gives (
3
MI = )
1
1
The Moore-Penrose inverse of a row vector
1
12
is simply : [ , , . . ., ] ( )
TT
n
a a a a aa a
−
= .
40
22
2
22
2 22 2
22
2
( )
2
( )
2
()
2
x yz
x
x yz x y z
y
z
xy z
y z Q xy Q xz Q
z
xQ x
x Q yQ z Q x yQ x z Q yz Q
yQ y
x yz z
zQ z
x zQ y zQ x y Q
z
+− +
− − + − ++ +
= +=
++
−
+ ++
,
(3.3.7)
assuming that 0 ≠ z .
3.3.1.2 The Incorrect Approach
Were we to conflate the unconstrained system with the constrained system, and substitute
for
2
z from equation (3.3.3) in the expression for the kinetic energy T given in (3.3.1),
we would then obtain
2 2 22 22
11 1
() .
22 2
T x y xy xy
+
= + + += + (3.3.8)
We note that this substitution has eliminated z from the expression for the kinetic energy
+
T ! Were we to subsume that the system is constrained (since we have used the equation
describing the constraint in getting
+
T ), we would need to then say that the virtual
displacements are no longer independent and that the relation
z z xx y y δ δδ = +
(3.3.9)
41
must be satisfied by them so that the work done by the force Q under these virtual
displacements becomes
( ) ( ).
xz y z
xy z
xy
Q Q xQ Q y
zz
W Q xQ y Q z
δ δ
δδ δ
= + ++
= + +
(3.3.10)
Assuming, as before, that 0 ≠ z . Using equations (3.3.8) and (3.3.10) in Lagrange’s
equation, then equations of motion of the constrained system become
2
xz
z Q xQ
x
z
+
=
(3.3.11)
and
.
2
yz
z Q yQ
y
z
+
=
(3.3.12)
Using these expressions in (3.3.4) we then obtain
22
2
()
.
2
xy z
x zQ y zQ x y Q
z
z
+ ++
=
(3.3.13)
Equations (3.3.11) and (3.3.12) are not the same as the first two in (3.3.7). In fact, we
have obtained the wrong set of equations of motion! We note that the root cause of this
error is obfuscation of the difference between the unconstrained and the constrained
descriptions of the mechanical system.
42
The reader may think that we have performed a slight sleight-of-hand here in using
equation (3.3.9) to eliminate z δ from the virtual work expression in (3.3.10). While we
might justify our actions by saying that our elimination of z δ was prompted by the fact
that now
+
T concerns only 2 coordinates as shown in (3.3.8), and that we have already
assumed that the constraint is active since we have used it to get
+
T , we need to be more
careful and stick more closely to our three-step conceptualization of constrained motion
stated in Section 2.2 to see what exactly went wrong in the procedure we followed. So
let’s do that.
Having got the kinetic energy
+
T in (3.3.8) by conflating the unconstrained and
constrained systems, what would the Lagrange equations of motion of the ‘presumably
unconstrained’ system be? Recall that by ‘unconstrained’ we always mean that the virtual
displacements are independent. Using
+
T , the Lagrange equations we would get are
2 0 0
0 2 0 .
0 0 0
x
y
z
Q x
yQ
z
Q
=
(3.3.14)
The last equation must strike some alarm in the reader’s mind, but we must recall that
these are not the equations of motion of the actual system yet! They are the equations of
motion of our presumed unconstrained system. Now we need to go to our next step in the
43
conceptualization process and differentiate our constraint as we did in (3.3.4). Our final
step is to obtain the equations of motion of the constrained system by imposing this
constraint on our presumed unconstrained system, which is described by (3.3.14). This
last step as we saw in Section 2.2 is done usually through the use of relation (2.2.9),
which gives the description of the constrained system.
But the mass matrix M describing this unconstrained system (equation (3.3.14)) is now
singular! And so the unconstrained acceleration a is undefined! Luckily, we do have a
way of getting the unique equations of motion in this predicament, provided the matrix
[M | ]
T
A has full rank (Udwadia and Phohomsiri [50]); and this it does, when 0 ≠ z . In
fact the equations of motion of the constrained system are given now by (2.2.10) and they
yield exactly the same equations as (3.3.11), (3.3.12), and (3.3.13), which are, of course,
wrong!
Thus we see that confusing the constrained and the unconstrained system by imposing the
constraint in the kinetic energy (more, generally in the Lagrangian, as we shall see later
on) of the unconstrained system before applying the constraints as described in Step 2 of
our discussion in Section 2.2 causes us to get the wrong equations of motion.
44
The root cause, as before, is that we conceptualized the problem of constrained motion
incorrectly.
3.3.2 Example 2
Consider a particle of unit mass moving in 3-dimensional space subjected to no
externally impressed forces. The position of the particle at time t is given by its
coordinates (x, y, z) measured in an inertial frame of reference. The particle is subjected
to the nonholonomic constraint y cz x = , where c is a fixed constant.
We shall find the equation of motion of this particle and go a step further than we did in
our previous example by illustrating what might result if we progressively ‘mix up’ the
concepts of unconstrained and constrained dynamical systems, as we shall shortly
explain. We begin with the correct approach for conceptualizing the constrained system.
3.3.2.1 The Correct Approach
The kinetic energy of the unconstrained system is again given by
2 22
11 1
22 2
Tx y z = + + (3.3.15)
45
and since the impressed forces are all zero, Lagrange’s equation, under the assumption
that the coordinates are all independent and no constraints exist between them, is given
trivially by the equation
0
0.
0
x
y
z
=
(3.3.16)
This completes our first step in the conceptualization process; defining the unconstrained
system.
Next, we impose the constraint
, y cz x =
(3.3.17)
which when differentiated with respect to time gives
, y cz x cxz = +
(3.3.18)
so that (see (2.2.5))
[ 1 0] A cz = − and b cxz = . (3.3.19)
The last step of the conceptualization process is formed by using the fundamental
equation (2.2.9), which then gives
46
2
22
22
1
1
0
c xz z
cz
x
c xz
y
cz
z
−
+
=
+
(3.3.20)
as the correct description of the motion of our constrained system.
3.3.2.2 The Incorrect Approach
Let us again rewrite the kinetic energy T of the unconstrained particle, where we assume
that all the coordinates are independent, in the form
2 2 22
11 1 1
(1 ) ,
22 2 2
Tx y y z αα = + +− + (3.3.21)
where 01 α ≤< . Note that we have split the contribution of the term
2
) 2 / 1 ( y in T given
by (3.3.15) in two terms now. We now ‘mix up’ the concepts of the unconstrained and
constrained descriptions of the system by using the constraint to replace only part of the
kinetic energy. Thus we have
2 2 22
11 1 1
( ) ( ) (1 )
22 2 2
T x cz x y z αα α
+
= + +− + (3.3.22)
where we notice that the term
2
1
2
y α on the right-hand side of (3.3.21) has been replaced
through the use of the constraint equation (3.3.17). Note that when 0 α = , . T T =
+
47
Under the assumption that all the coordinates are independent, the Lagrange equation of
motion for the presumed unconstrained system, for a fixed value of α , becomes
22 2
22
1 0 0 2
0 1 0 0 .
0 0 1
c z x c xzz
y
z
c xz
αα
α
α
+−
−=
(3.3.23)
We use the word ‘presumed’ because we have, conceptually speaking, ‘partially’
confused the unconstrained and constrained descriptions of the system to the extent of
replacing the term
2
1
2
y α in T by using the constraint equation (3.3.17). Hence, these are
indeed not the correct equations of motion of the unconstrained system!
In fact, one can think of the parameter α as a measure of the extent to which we have
conceptually conflated the unconstrained description with the constrained description of
the mechanical system. For, in a manner of speaking, one can say that if we chose 0 α =
then TT
+
= and we have not conceptually engaged in conflating the two descriptions of
the system. On the other hand, when 1 α = , we have ‘completely’ confused the
unconstrained and the constrained descriptions, having completely replaced the entire
contribution made by the
2
y term to the kinetic energy, T, by using the constraint
equation (3.3.17). For 1 0 < < α , we could consider that we have done a ‘partial’
conflation, its extent increasing as α moves closer to unity.
48
From equation (3.3.23) we find that the acceleration of the presumed unconstrained
system is given by
2
22
22
2
1
0.
c xzz
cz
a
c xz
α
α
α
−
+
=
(3.3.24)
Noting that the constraint equation (3.3.17) upon differentiation can again be expressed
by relations (3.3.18) and (3.3.19), we now need to impose this constraint on our
‘presumably’ unconstrained system, which is described by (3.3.23). This is done through
the use of the fundamental equation (2.2.9), which then gives the relations (note, 1 α ≠ )
2
22
22
22
22
( 1)
1
(1 ) .
1
c xz z
x cz
c c z xz y
cz
z
c xz
α
α
α
+
−
+
− =
+
(3.3.25)
Not surprisingly, the equation of motion depends on the parameter α that describes the
extent to which we conflated the unconstrained and the constrained descriptions of our
system. When 0 α = and T T =
+
, equations (3.3.20) and (3.3.25) are identical, and we
get the correct equations of motion for the constrained system. But for other values of α
49
they are not the same. The substitution of the constraint in the kinetic energy leads to
incorrect equations of motion since the descriptions of the unconstrained and the
constrained system (described in Section 2.2) are then ‘partially’ confused.
Lastly, we point out another erroneous way of conceptualizing constrained systems. For
0 α ≠ , since we have already used the constraint equation in describing part of the
kinetic energy of the system so as to obtain
+
T , one might think that the virtual
displacements are no longer independent. This reasoning is, in fact, again not true. For
from the Lagrangian mechanics point of view, instead of the relations (3.3.23), we would
then get not a set of equations, but the sum of three terms adding up to zero. We would
obtain
{ } { } { }
22 2 2 2
(1 ) 2 (1 ) 0. c z x c x z z x y y z c x z z α α δ αδ α δ + + + − +− =
(3.3.26)
The virtual displacements, being no longer independent, must now satisfy the relation
. y cz x δδ =
(3.3.27)
Substituting (3.3.27) in equation (3.3.26), we get
{ } { }
22 2 2 2
(1 ) 2 (1 ) 0. c z x c x z z c z y x z c x z z α α α δ αδ + + + − +− =
(3.3.28)
50
Since by (3.3.27) we see that the constraint is only between y δ and x δ , the virtual
displacements x δ and z δ are independent of one another so that (3.3.28) yields the two
relations
22 2
(1 ) 2 (1 ) 0 c z x c x z z c z y αα α + + +− = (3.3.29)
and
22
0. z c xz α −= (3.3.30)
Also from the constraint equation (3.3.18), we obtain
, y cz x cxz = +
(3.3.31)
and using equations (3.3.29) on the right-hand side of (3.3.31) we obtain the differential
equation for y , so that the constrained system’s equation becomes
2
22
22
22
22
( 1)
1
(1 ) .
1
c xz z
x cz
c c z xz y
cz
z
c xz
α
α
α
+
−
+
− =
+
(3.3.32)
Equations (3.3.25) and (3.3.32) are the same, and both of them confirm that both these
ways of conceptualizing constrained motion are incorrect. Thus, once we conflate the
unconstrained and the constrained systems, the resulting system can no longer be thought
51
of, in general, as being either unconstrained or constrained. In other words, the virtual
displacements cannot be taken to be either dependent (as dictated by the constraint
equation) or independent (as when there are no constraints). Use of either of these
conceptualizations eventually leads to the wrong equations of motion for the system.
The reader might have noticed that when 1 α = the mass matrix in equation (3.3.23)—
our presumed unconstrained description of the system—becomes singular. Strictly
speaking then, since the matrix ] | [
T
A M has full rank, one would have to use equation
(2.2.10) (instead of (2.2.9)) to arrive at the constrained equation of motion as we did
before in the previous example. Were we to do that, we would again obtain equation
(3.3.32) with the right-hand side evaluated at 1 α = ; that is to say, the wrong equation of
motion.
In the next example we go a step deeper into understanding the problem, and show that
while the seemingly paradoxical situation exhibited by Hamel in Section 3.1 deals with
conflating the descriptions of the unconstrained and the constrained system (as described
in Section 3.2) by using the constraint equation to amend only the kinetic energy of the
system from T to
+
T , we could have come to such a situation, in general, by conflating
the two descriptions in other ways.
52
3.3.3 Example 3
Consider a particle of unit mass moving in 3-dimensional space subjected to no
externally impressed forces. The position of the particle at time t is given by its
coordinates (x, y, z) measured in an inertial frame of reference. Let the potential energy of
the particle be
2
) 2 / 1 ( x , and let the particle be subjected to the nonholonomic constraint
x c yz = , where 0 ≠ c is a fixed constant.
3.3.3.1 The Correct Approach
Since the kinetic energy of the unconstrained system is given by (3.3.15) and the
potential energy is
2
) 2 / 1 ( x , the Lagrangian of the unconstrained system is
,
2
1
2
1
2
1
2
1
2 2 2 2
x z y x L − + + = (3.3.33)
and the equations of motion for the unconstrained system are accordingly
0.
0
xx
y
z
−
=
(3.3.34)
Differentiating the constraint equation
, x c yz =
(3.3.35)
we get
53
x z y c y z c = +
(3.3.36)
so that
[ ] 0 A cz cy = and x b = . Using (2.2.9) the correct equation of motion for the
constrained system is then
2 22 22
22
0
0 .
() ()
0
()
x
xx
x xz
y cz
c yz c yz
z cy
xy
cy z
−
−
=+=
++
+
(3.3.37)
Again, we shall see what happens when we conflate the aforementioned distinction
between the description of the unconstrained system and the constrained system.
3.3.3.2 The Incorrect Approach
Were we to substitute for x from the constraint equation (3.3.35) into the Lagrangian L in
(3.3.33), we would get
, ) (
2
1
2
1
2
1
2
1
2 2 2 2
z y c z y x L − + + =
+
(3.3.38)
and the equations of motion of our unconstrained system would become
54
22 2
2 22
1 0 0
0
0 (1 ) 2 0 .
0
0 2 (1 )
x
c z c yz y
z
c y z c y
−− =
−−
(3.3.39)
Imposition of the constraint (3.3.35) on this presumably unconstrained system, (assuming
that
22
1 cz − and
22 2 2
1 (3 ) cx y z − ++ are both non-zero), yields on using equation (2.2.9),
22
2 22
22
2 22
0
1
.
2
1
2
x
xz c y
y
c x yz
z
xy c z
c x yz
+
=
++
+
++
(3.3.40)
From (3.3.40) and (3.3.37) we see that we have clearly obtained the wrong equations of
motion because we have conceptually mixed up our unconstrained and constrained
descriptions of our mechanical system.
As in the previous example, since we have already used the constraint equation in
describing the Lagrangian of the system so as to obtain
+
L , we might think that the
virtual displacements should no longer be considered independent. Working out the
55
equations from the Lagrangian mechanics point of view, we would, as before, get the
sum of three terms adding up to zero, and would obtain
{ } { } { }
22 2 2 2 2
(1 ) 2 2 (1 ) 0. x x c z y cy z z y cy z y cy z z δδ δ + − − +− + − =
(3.3.41)
The virtual displacements being no longer independent (since we have now
conceptualized the system with the Lagrangian
+
L as being constrained), they must
satisfy the relation
. z y yz δδ = −
(3.3.42)
Substituting (3.3.42) in equation (3.3.41), we get
{ }
2 2 22 2 22
(1 ) 2 2 (1 ) 0.
y
x x c z y cy z cy z y cy z z
z
δδ
+− − + − + − =
(3.3.43)
The constraint (3.3.42) causes the virtual displacements y δ and z δ to be related; but
x δ and z δ are independent of one another so that (3.3.43) yields the two relations
0 x = (3.3.44)
and
2 2 22 2 22
(1 ) 2 2 (1 ) 0.
y
c z y cy z cy z y cy z
z
−− + − + − =
(3.3.45)
56
Also from the constraint equation (3.3.36), we obtain
,
x cz y
z
cy
−
=
(3.3.46)
and substituting equations (3.3.46) in the left hand side of (3.3.45) we obtain the
differential equation for y . The constrained system’s equation then becomes
22
2 22
22
2 22
0
1
.
2
1
2
x
xz c y
y
c x yz
z
xy c z
c x yz
+
=
++
+
++
(3.3.47)
Equations (3.3.40) and (3.3.47) are the same, and both of them confirm that both these
ways of conceptualizing constrained motion are incorrect. Thus, once we conflate the
unconstrained and the constrained descriptions, the resulting system, in general, can no
longer be thought of as being either unconstrained or constrained!
So far we have been careful to choose constraints that have all been nonholonomic. This
leaves open the following question: If we had only holonomic constraints acting on a
mechanical system and we were to use the constraint equation(s) to amend the
Lagrangian L to
+
L as before, and then use the usual approach for obtaining the
57
constrained equations of motion, would we obtain the correct equations? In the next
section we show that in this special case we indeed will, and we explain the reason for
that.
3.3.4 Example 4
A particle of unit mass is moving in 3-dimensional Cartesian space. The impressed force
on the particle is [ , , ]
T
x yz
Q QQ Q = . The position of the particle in a rectangular inertial
coordinate frame is described by its coordinates (x, y, z). The particle is subjected to the
holonomic constraint
2
1
2
yz = .
3.3.4.1 The Correct Approach
Using the three steps outlined in Section 2.2 we have the following results for each step.
1. The equation describing the motion of the unconstrained system, assuming that all
the coordinates (and the corresponding virtual displacements) are independent is
.
x
y
z
xQ
yQ
zQ
=
(3.3.48)
2. To impose the constraint
2
1
2
yz = (3.3.49)
58
on this unconstrained description of the system, we differentiate equation (3.3.49)
twice with respect to time in order to put the constraint equation in the form of
equation (2.2.5), and get
2
. y zz z = + (3.3.50)
Thus, [0 1 ] Az = − and
2
b z = .
3. Using the fundamental equation (2.2.9), we then obtain
2 22
22
2
2
0
1
11
1
x
x
yz yz
y
z
yz
Q
xQ
z Q zQ z z Q zQ
yQ
zz
zQ z
z z zQ Q
z
−+ + +
=+=
++
−
−+ +
+
(3.3.51)
as the equation of motion of our constrained system.
Let us contemplate a suitable change of coordinates to convert the constrained holonomic
system to an unconstrained system (Pars [31]). This can be done because the constraint
has usable information about both the generalized coordinates and their derivatives. Let
us employ a new coordinate
2
1
2
qy z = − instead of the coordinate ‘ y’ so that we
describe the configuration of the system by the coordinates (, , ) xq z instead of (, , ) x yz .
59
The mapping from (, , ) x yz to (, , ) xq z must be 1-1, and since
1 00
(, , )
det det 0 1 0 1 0,
(, , )
01
xq z
x yz
z
∂
= = ≠
∂
−
(3.3.52)
so it is. Note that since from the constraint equation (3.3.49), we must have
2
1
() 0
2
qt y z =−= for all time, which in turn implies that () 0 qt ≡ and () 0 qt ≡ for all
time. In addition, we can differentiate
2
1
2
qy z = − , with respect to time, to obtain
. y q zz = +
(3.3.53)
Then, inserting equation (3.3.53) into the expression for the kinetic energy (we are
conflating the unconstrained and constrained descriptions here),
2 22
11 1
22 2
Tx y z = ++ (3.3.54)
of the system, yields
2 22
11 1
( ) .
22 2
T x q zz z
+
= + + + (3.3.55)
But since () 0 qt ≡ , we get
2 2 2 2 22
11 1 11
( ) (1 ) .
22 2 22
T x zz z x z z
+
= + + = ++ (3.3.56)
60
Note that this is tantamount to simply using the constraint (3.3.53) to replace the
2
(1/ 2)y
term in the kinetic energy—a key observation. From (3.3.53), after setting () 0 qt ≡ , we
get
. y zz δδ =
(3.3.57)
The generalized forces in the new coordinates (x, q, z) are obtained by equating the
virtual work done by the forces, given by
ˆˆ ˆ
,
x y z x q z
W Q xQ z z Q z Q xQ q Q z δ δ δδ δ δ δ = + += + + (3.3.58)
which gives
ˆˆ ˆ
() ,
x yz x q z
Qx zQ Q z Qx Q q Q z δ δ δ δδ + + = + + (3.3.59)
in view of (3.3.57). Noting now that () 0 qt ≡ so that () 0 qt δ ≡ , we get
ˆˆ
and ( ).
x x z yz
Q Q Q zQ Q = = + (3.3.60)
Since by (3.3.57) the constraint is only between y δ and z δ , the virtual displacements
x δ
and z δ are independent of one another. Applying Lagrange’s equations to the
coordinates (x, z), we obtain
x
x Q =
(3.3.61)
61
22
(1) .
yz
z z z z zQ Q + = −+ +
(3.3.62)
The time evolution of the last coordinate q of the triple (x, q, z) that we are using to
describe the configuration of the system is simply given by () 0 qt ≡ , as we saw before.
We see that the dynamical equation describing the evolution of q is trivial (i.e., () 0 qt ≡ )
and it is uncoupled from (3.3.61) and (3.3.62). We can interpret, from a physical
viewpoint, what has been done in arriving at equations (3.3.61) and (3.3.62) as an
elimination of one of the coordinates of our triplet (x, y, z) to obtain the dynamical
equations of the system in only the coordinates x and z. These coordinates may be
considered to be independent, since the corresponding virtual displacements x δ and z δ
in them can be specified independently. Thus, (3.3.61) and (3.3.62) represent, technically
speaking, the equations of motion of an unconstrained system (see Section 2.2). That the
equation describing the evolution of the coordinate () qt is uncoupled from these two
equations is an important observation, which we shall make use of later on.
To get back to our original configuration space coordinates (, , ) x yz , we use (3.3.50)
along with (3.3.61) and (3.3.62) to get
62
22
2
.
1
yz
z z Q zQ
y
z
++
=
+
(3.3.63)
Notice that in order to transform the coordinates back from (, , ) xq z to (, , ) x yz , we have
to use the equation (3.3.50) which causes the virtual displacements y δ and z δ to be
related, and so the equations (3.3.61), (3.3.62), and (3.3.63) now correspond to the
constrained system. They are indeed identical to (3.3.51), as they should.
We then see that the substitution of the holonomic constraint in the kinetic energy (or
Lagrangian) of the unconstrained system, namely,
2
1
(, , , , , ) (, , , , , ),
2
T x y zx y z T x z zx zzz → (3.3.64)
is actually just an appropriate change in coordinates from the original coordinates (x, y, z)
to (x, q, z), namely,
2
1
(, , , , , ) (, , , , , )
2
T x y zx y z T x q z zx q zzz →+ + (3.3.65)
in which the coordinate q is so chosen that () () 0 qt qt = ≡ , thus making the right-hand
sides of the expressions in (3.3.64) and (3.3.65) identical.
Were we to consider the system described by
2
1
(, , , , , )
2
T T x z zx zzz
+
= and the impressed
force Q as constituting an unconstrained system (since x and z are independent
63
coordinates now), and use Lagrange’s equations we would get the singular mass system
described by
2
2
1 0 0
0 0 0 .
0 0 (1 )
x
y
z
xQ
yQ
z zz Q
z
=
−+
+
(3.3.66)
According to Section 2.2, we would then need to impose the constraint (3.3.49) on this
unconstrained system, and obtain the description of the constrained system by further
using equation (2.2.10). Since the matrix [ | ]
T
MA has full rank, we indeed can use
equation (2.2.10) and if we did so, we would get the same (correct) set of equations of
motion (3.3.61), (3.3.62), and (3.3.63) as we did before.
Lastly, we consider what might happen if we were to partially conflate the kinetic energy
T of the system by expressing the term
22 2
11 1
as (1 )
22 2
yy y αα +− , and replacing
2
1
2
y α
by
2
1
()
2
q zz α + , while retaining the term
2
1
(1 )
2
y α − in our amended kinetic energy T
+
,
analogous to what we did in arriving at (3.3.22). With this, the kinetic energy
2 2 22 2 2 22
11 1 1 11 1 1
( ) (1 ) ( ) (1 ) .
22 2 2 22 2 2
T x q zz y z x zz y z α α αα
+
= + + +− + = + + − + (3.3.67)
64
Notice that we have now made a transformation (, , ) (, , , ) x yz x yz q → in going from T to
T
+
; but since () 0 qt ≡ , the equation of motion () 0 qt = is uncoupled from that for x, y,
and z. This amounts to a transformation from (, , ) (, , , 0) x yz x yz q →≡ , or simply from
(, , ) (, , ) x yz x yz → . One can then use the right-most expression for T
+
in (3.3.67)
(instead of T) as the kinetic energy of the system along with the constraint (3.3.49) to get
the correct equations of motion of the system. This happens because the use of the
constraint equation (3.3.53) in T to get T
+
as we did in (3.3.67), is tantamount to the
coordinate transformation from (, , ) (, , ) x yz x yz → , i.e. the same coordinate system, so
that T
+
is also the correct kinetic energy of the unconstrained system.
Thus, only in the special case when holonomic constraints are present, we can use T
+
and
the impressed forces to comprise a constrained system whose virtual displacements are
no longer independent as we did in (3.3.56)-(3.3.63), or we can consider it as an
unconstrained system on which the constraints need to be further applied as we did in
using (3.3.66) and later (2.2.10). Either of these conceptualizations will yield the correct
equations of motion. The reason for this is that such substitutions of the constraints in the
kinetic energy (Lagrangian) are tantamount to an appropriate special change of
coordinates. But this will not work for more general systems with nonholonomic (or a
65
combination of holonomic and nonholonomic) constraints since in that case, one cannot
demonstrate such a coordinate transformation.
3.4 Summary
The main contributions of this chapter are as follows:
(i) Hamel’s paradox is not limited to the specific skate problem considered by
him in his text. It brings out deeper issues in analytical dynamics—
specifically, the way in which one needs to conceptualize a constrained
mechanical system so that the equations of motion so obtained are consistent
with physical observation.
In doing this we are led to the view first proposed by Gauss which provides a
uniform three-step method of conceptualizing such systems. This three-step
conceptualization of constrained motion involves:
1. description of the unconstrained system in which the coordinates are all
independent of each other.
2. description of the constraints, and
3. description of the constrained system using the previous two descriptions.
66
It provides a systematic way to obtain the correct equations of motion—those
that are consistent with physical observation—of a constrained mechanical
system.
(ii) In general, substitution of the constraint equation(s) in the kinetic energy (or
in the Lagrangian) of a general constrained mechanical system conflates the
unconstrained and the constrained descriptions of such a system. The system
with such an amended kinetic energy, T
+
(or Lagrangian, L
+
), cannot, in
general, be considered either as (i) an appropriate description of the given
constrained system, or (ii) an appropriate description of the given
unconstrained system on which the constraints then need to be further
imposed.
(iii) In the special situation in which we have only holonomic constraints such
substitutions represent simply an appropriate change in coordinates, and the
use of the amended kinetic energy (Lagrangian) will then lead to the correct
equations of motion.
(iv) We note that to have a rigorous explanation of Hamel’s paradox, we are led
deeper to the foundations of analytical dynamics and to the use of newly
developed concepts that deal with singular mass matrices.
67
(v) Though Lagrange’s equations are over 200 years old, and the problem of
constrained motion has been worked on near-continuously by numerous
researchers since the time it was first conceived by Lagrange, the study of
analytical dynamics still has many interesting aspects that seem to need
considerable care, and still call for improved understanding.
As seen in this chapter, a significant problem in deriving the equation of motion for
constrained mechanical systems arises when the mass matrix of the unconstrained
mechanical system is singular. Although, Udwadia and Phohomsiri [50] provide a useful
result that works with the system with singular mass matrix (see (2.2.10)), their equation
differs in structure and form from the so-called fundamental equation (Udwadia and
Kalaba [43]), and it is not easy to see the physical interpretation of their equation. Thus,
in the next chapter we now go further to develop a simple unified explicit form of the
equations of motion for general constrained mechanical systems whose mass matrices
may or may not be singular by utilizing the three-step conceptualization of constrained
motion presented in this chapter. This is done through the replacement of the actual
unconstrained mechanical system, which may have a positive semi-definite mass matrix,
with an unconstrained auxiliary system that is then subjected to the same holonomic
and/or nonholonomic constraints as those applied to the actual unconstrained mechanical
68
system. The unconstrained auxiliary system is subjected to the same ‘given’ force as the
actual mechanical system, and its mass matrix is appropriately augmented to make it
positive definite so that the so-called fundamental equation (2.2.9) can then be directly
and simply applied to obtain the closed-form acceleration of the actual constrained
mechanical system. Furthermore, it is shown that by appropriately augmenting the
‘given’ force that acts on the actual unconstrained mechanical system, the auxiliary
system directly provides the constraint force that needs to be imposed on the actual
unconstrained mechanical system so that it satisfies the given holonomic and/or
nonholonomic constraints. Thus, irrespective of whether the mass matrix of the actual
unconstrained mechanical system is positive definite or positive semi-definite, a simple,
unified fundamental equation results that yields a closed-form representation of both the
acceleration of the constrained mechanical system and the constraint force. The results
herein provide deeper insights into the behavior of constrained motion and open up new
approaches to modeling complex, constrained mechanical systems, such as those
encountered in multi-body dynamics.
69
Chapter 4
ON GENERAL NONLINEAR CONSTRAINED SYSTEMS
4.1 Introduction
As mentioned before, the description of motion of nonlinear constrained mechanical
systems is an important problem in analytical dynamics that has been worked on by
numerous researchers. References 1-2, 7, 10, 14-16, 19, 25, and 36 give a brief sampling
of some of the researchers who have made substantial contributions; nevertheless, several
questions remain unanswered at the present time. A significant problem in deriving the
equation of motion for constrained mechanical systems arises when the mass matrix of
the unconstrained mechanical system is singular. Since the mass matrix then does not
have an inverse, standard methods for obtaining the constrained equations of motion,
which usually rely on the invertability of the mass matrix, cannot be used. For example,
the so-called fundamental equation developed by Udwadia and Kalaba [43] cannot be
directly applied. Observing this, Udwadia and Phohomsiri [50] derived an explicit
equation of motion for such systems with singular mass matrices. However, the structure
of their explicit equation differs significantly from their so-called fundamental equation
[43]. Recently, by using the concept of an unconstrained auxiliary system, Udwadia and
Schutte [51] developed a simpler explicit equation of motion that has the same form as
70
the so-called fundamental equation, and is valid for systems whose mass matrices may or
may not be singular. They do this by augmenting the singular mass matrix of the
unconstrained mechanical system by appropriately making it positive definite. However,
the structural form of this augmented mass matrix requires considerable computational
effort especially when complex multi-body systems are involved. In this chapter we
present a new alternative equation of motion for systems with positive definite and/or
positive semi-definite mass matrices that is in many respects superior to that proposed in
Ref. [51].
We consider an unconstrained auxiliary system that has a positive definite mass matrix
instead of the actual unconstrained mechanical system whose mass matrix may be
positive semi-definite. This unconstrained auxiliary system is subjected to the same
‘given’ force as that acting on the actual mechanical system, and when subjected to the
same constraints as the actual unconstrained mechanical system, provides in closed-form,
at each instant of time, the acceleration of the actual constrained mechanical system.
Similarly, by suitably augmenting the ‘given’ force that is acting on the actual
unconstrained mechanical system, we obtain from the auxiliary system the proper
constraint force acting on the actual unconstrained mechanical system in closed form. In
short, the auxiliary system obtained herein gives the equation of constrained motion of
71
the actual mechanical system in closed form whether or not the mass matrix is singular in
a much simpler, more straightforward, and more computationally efficient, manner than
in Ref. [51]. The proofs of our results are also much simpler, and they lead to deeper
insights into the nature of constrained motion of mechanical systems.
We briefly point out the importance of being able to formulate correctly the constrained
equations of motion for mechanical systems whose mass matrices are positive semi-
definite. When a minimum number of coordinates is employed to describe the
(unconstrained) motion of mechanical systems, the corresponding set of Lagrange
equations usually yields mass matrices that are non-singular [31]. One might thus
consider that systems with singular mass matrices are not common in classical dynamics.
However, in modeling complex multi-body mechanical systems, it is often helpful to
describe such systems with more than the minimum number of required generalized
coordinates. And in such situations, the coordinates are then not independent of one
another, often yielding systems with positive semi-definite mass matrices. Thus, in
general, singular mass matrices can and do arise when one wants more flexibility in
modeling complex mechanical systems. The reason that more than the minimum number
of generalized coordinates are usually not used in the modeling of complex multi-body
systems, though this could often make the modeler’s task much simpler, is that they result
72
in singular mass matrices, and to date systems with such matrices have been difficult to
handle within the Lagrangian framework. Several examples are provided in this chapter
showing how singular mass matrices can appear in the modeling of constrained
mechanical systems.
This is the reason it is useful to obtain in closed-form the general, explicit equations of
motion for nonlinear constrained mechanical systems whose mass matrices may or may
not be singular. Since such systems normally arise when modeling large-scale, complex
mechanical systems in which the modeler seeks to substantially facilitate his/her work by
using more than the minimum number of coordinates to describe the system, it is also
important to keep an eye on the computational efficiency of the equations so obtained.
4.2 System Description of General Constrained Mechanical Systems
It is useful to follow the three-step procedure of the conceptualization for the description
of a constrained mechanical system, S, as we stated before in Section 2.2. However, in
this section the so-called fundamental equation (2.2.9) and equation (2.2.10) are modified
in order to work with a non-ideal constraint. We do this in the following way:
First, we describe the so-called unconstrained mechanical system in which the
coordinates are all independent of each other. We do that by considering an
73
unconstrained mechanical system whose motion at any time t can be described, using
Lagrange’s equation, by (see (2.2.1))
( , ) ( , , ), M qt q Q q qt =
(4.2.1)
with the initial conditions (see (2.2.2))
00
( 0) , ( 0) , qt q qt q = = = = (4.2.2)
where q is the generalized coordinate n-vector; M is an n by n matrix that can be either
positive semi-definite ( 0) M ≥ or positive definite ( 0) M > at each instant of time; and Q
is an n-vector, called the ‘given’ force, which is a known function of q, q , and t. We
shall often refer to the system described by equation (4.2.1) as the unconstrained
mechanical system S.
Second, we impose a set of constraints on this unconstrained description of the system.
We suppose that the unconstrained mechanical system is now subjected to the m
constraints given by (see (2.2.4))
( , , ) 0, 1, 2,..., ,
i
q qt i m ϕ = =
(4.2.3)
where rm ≤
equations in the equation set (4.2.3) are functionally independent. The
presence of the constraints does not permit all the components of the n-vectors
0
q and
0
q
to be independently assigned. We shall assume that the initial conditions (4.2.2) satisfy
74
these m constraints. The constraints (4.2.3), which may be holonomic or nonholonomic,
if sufficiently smooth, can be differentiated appropriately with respect to time t to yield
the equation
( , ,) ( , ,), A qq t q bqq t =
(4.2.4)
where A is an m by n matrix whose rank is r, and b is an m-vector. We again note that
each row of A arises by appropriately differentiating one of the m constraint equations.
Using the information in the previous two steps, in the last step we bring together the
description of motion of the constrained mechanical system as (see (2.2.6))
( , ) ( , , ) ( , , ),
c
M qt q Q q qt Q q qt = + (4.2.5)
where
c
Q is the constraint force n-vector that arises to ensure that the constraints (4.2.4)
are satisfied at each instant in time. Thus, equation (4.2.5) describes the motion of the
actual constrained mechanical system, S. In what follows we shall suppress the arguments
of the various quantities unless required for clarity.
Equation (4.2.3) provides the kinematical conditions related to the constraints. We now
look at the dynamical conditions. The work done by the forces of constraints under
virtual displacements at any instant of time t can be expressed as [46]
75
() ( , , ) () ( , , ),
Tc T
v t Q q qt v t C q qt = (4.2.6)
where ( , , ) C q qt is an n-vector describing the nature of the non-ideal constraints which is
determined by physical observation and/or experimentation, and the virtual displacement
vector, v(t), is any non-zero n-vector that satisfies [45]
( , , ) 0. Aq q t v =
(4.2.7)
When the mass matrix M in equation (4.2.1) is positive definite, the explicit equation of
motion of the constrained mechanical system S is given by the so-called fundamental
equation [47]
1/2 1/2 1/2
( ) ( ) , q a M B b Aa M I B B M C
−+ − + −
=+ − + − (4.2.8)
where
1 1/2
, a M Q B AM
−−
= = , and the superscript “+” denotes the Moore-Penrose (MP)
inverse of a matrix [17], [32], [44]. We note that equation (4.2.8) is valid (i) whether or
not the equality constraints (4.2.3) are holonomic and/or nonholonomic, (ii) whether or
not they are nonlinear functions of their arguments, (iii) whether or not they are
functionally dependent, (iv) and whether or not the constraint force is non-ideal. We note
that the constrained mechanical system S is completely described through the knowledge
of the matrices M and A, and the column vectors Q, b, and C. The latter four are functions
of q, q , and t, while the elements of the matrix M are, in general, functions of q and t. In
76
what follows, we shall also denote the acceleration of the constrained system given in
equation (4.2.8) by ()
S
qq = .
However, when the unconstrained mechanical system given by (4.2.1) is such that the
matrix M is singular ( 0) M ≥ , the above equation cannot always be applied since the
matrix
1/2
M
−
may not exist. In that case, equation (4.2.8) needs to be replaced by
equation [50]
()
: ,
QC QC I A AM
qM
bb A
+
+
+
++ −
= =
(4.2.9)
under the proviso that the rank of the matrix
ˆ
|
TT
M MA =
is n. This rank condition is
a necessary and sufficient condition for the constrained mechanical system to have a
unique acceleration—a consequence of physical observation of the motion of systems in
classical mechanics.
However, the form of equation (4.2.9) when M is positive definite is noticeably different
from the form of the so-called fundamental equation (4.2.8). A unified equation of
motion that is applicable to both these situations is presented in Udwadia and Schutte
[51]. They considered an auxiliary system that has a positive definite mass matrix, which
is subjected to the same constraint conditions as the actual mechanical system that has a
77
singular mass matrix. This positive definite mass matrix of the auxiliary system is
expressed as
2
M AA α
+
+ , where α
is any non-zero real number and again the
superscript “+” denotes the Moore-Penrose (MP) inverse of the matrix [51]. However, the
use of the Moore-Penrose (MP) inverse of the matrix A is expensive to compute,
especially when the row and column dimensions of A are large.
In this chapter we uncover a new general equation of motion for constrained mechanical
systems by instead using the augmented mass matrix,
2
T
T
AG
M M A GA α = + , which is
simpler, more general and directly uses the so-called fundamental equation (4.2.8). The
function () t α is an arbitrary, nowhere-zero, sufficiently smooth (
2
C ) real function of
time, and ( ,) : ( ,) ( ,)
T
G qt N qt N qt = is any arbitrary m by m positive definite matrix whose
elements are sufficiently smooth functions (
2
C ) of the arguments. Thus greater
generality, simpler results, and greater computational efficiency are herein achieved.
Furthermore, the proofs of the various results are much simpler than in Ref. [51].
4.3 Explicit Equations of Motion for General Nonlinear Constrained
Mechanical Systems
From physical observation, the acceleration of a system in classical dynamics under a
given set of forces and under a given set of initial conditions is known to be uniquely
determinable. As shown in Ref. [50] a necessary and sufficient condition for this to occur
78
is that the rank of the matrix
ˆ
|
TT
M MA =
is n. We shall therefore assume throughout
this chapter that for the constrained systems we consider herein, the matrices M and A are
such that this condition is always satisfied. Thus we assume that the actual constrained
mechanical system under consideration is appropriately mathematically modeled and the
resulting acceleration of the system can be uniquely found.
4.3.1 Positive Definiteness of the Augmented Mass Matrices
Lemma 4.1: Let 0 M ≥ , let () t α be an arbitrary, nowhere-zero, sufficiently smooth (
2
C )
real function of time, and let ( ,) ( ,) ( ,)
T
G qt N qt N qt =
be any m by m positive definite
matrix [8] whose elements are sufficiently smooth functions (
2
C ) of the arguments. The
n by n augmented mass matrix
2
:
T
T
AG
M M A GA α = + is positive definite at each instant of
time if and only if the n by nm + matrix
ˆ
|
TT
M MA =
has rank n at each instant of
time.
Proof: (a) Consider any fixed instant of time. Assume that
ˆ
M has rank n; we shall prove
that the augmented mass matrix
2
T
T
AG
M M A GA α = + is positive definite at that instant.
We first observe that the matrix
T
AG
M is symmetric since M is symmetric as is
T
A GA .
Since the column space of the matrix A is identical to the column space of A α
79
( )
ˆ
() | .
T
MM
n rank M rank rank rank M A
AA
α
α
= = = =
(4.3.1)
We shall denote by () Col X the column space of the matrix X. Since
1/2
() ( ) Col M Col M = , and () ( )
T TT
Col A Col A N = because N is nonsingular, we get
( ) ( )
1/2
| | : ( ),
T TT
n rank M A rank M A N rank M αα = = =
(4.3.2)
where we have denoted
1/2
: |
TT
M M AN α =
.
Next, we consider the augmented mass matrix
T
AG
M . It can be expressed as
22
1/2
1/2
|
0.
T
T TT
AG
TT
T
M MA GA MA N NA
M
M AN
NA
MM
αα
α
α
=+=+
=
= ≥
(4.3.3)
Thus, the n by n matrix
T
AG
M must at least be positive semi-definite. But from (4.3.2),
() rank M n =
, hence
T
AG
M is positive definite.
(b) Consider any fixed instant of time. Assume that
2
T
T
AG
M M A GA α = + is
positive definite; we shall prove that
ˆ
M has full rank n at that instant.
80
From (4.3.3) and the assumption that 0
T
AG
M > , we have 0
T
T
AG
M MM = >
, so that
T
MM
has rank n, and hence () rank M n =
.
Since elementary row operations do not change the rank of a matrix, we find that
( )
1/2 1/2
1/2
( ) | .
T TT
MM
n rank M rank rank rank M A N
NA NA α
= = = =
(4.3.4)
And also since
1/2
() ( ) Col M Col M = and () ( )
T TT
Col A Col A N = , we have
( ) ( )
( )
1/2
ˆ
| | | ( ).
T
T TTT
rank M A N rank M A rank M A rank M = = =
(4.3.5)
Hence,
ˆ
() rank M n = , and the proof is therefore complete.
4.3.2 Explicit Equation for Constrained Acceleration
Having the auxiliary mass matrix
T
AG
M , which has been proved to be always a positive
definite matrix, we are now ready to begin implementing the explicit equations of motion
for the constrained acceleration of the system S that may have a positive semi-definite
mass matrix, 0 M ≥ . We begin by proving a useful result that will be used many times
from here on.
81
Lemma 4.2: Let A
+
denote the Moore-Penrose (MP) inverse of the m by n matrix A, then
( ) 0.
T
I A AA
+
−= (4.3.6)
Proof:
( ) ()
( ) 0.
TT TT T T
T TT T
I AA A A AAA A AA A
A AA A
+ ++
+
− =−=−
=−=
(4.3.7)
In the second equality above, we have used the fourth Moore-Penrose (MP) condition
(see [17], [32], [44]) and in the last equality, we have used the first MP condition. This
yields the stated result.
Recalling that our actual mechanical system has a mass matrix M that may be positive-
semi definite, and since
1/2
M
−
does not exist, we encounter difficulty in finding the
acceleration of the constrained mechanical system when using the fundamental equation
(see equation (4.2.8)). However, we note from Lemma 4.1 that the matrix
T
AG
M is
always positive definite when
ˆ
M has rank n. Moreover, this rank condition is a check
that our mathematical model appropriately describes a given physical system, since in all
physical systems in classical mechanics the acceleration must be uniquely determinable.
Were we then to use this matrix
T
AG
M (instead of M) as the mass matrix of an
‘appropriate’ unconstrained auxiliary system, subjected to the same constraints as the
82
actual unconstrained mechanical system, we would encounter no difficulty in using the
fundamental equation to obtain the acceleration of this constrained auxiliary system (see
equation (4.2.8)), since the mass matrix of this auxiliary system is positive definite! Our
aim then is to define this unconstrained auxiliary system in the ‘appropriate’ manner so
that the resultant constrained acceleration it yields upon application of the fundamental
equation always coincides with the acceleration of our actual constrained mechanical
system. We now proceed to show that this indeed can be done, and we demonstrate how
to accomplish this.
Consider any unconstrained mechanical system S,
(i) whose equation of motion is described by equation (4.2.1) where the n by n
mass matrix M may be positive semi-definite or positive definite, and whose
initial conditions are given in relations (4.2.2),
(ii) which is subjected to the m constraints given by equation (4.2.4) (or
equivalently by equation (4.2.3)) that are satisfied by the initial conditions
0
q
and
0
q as described by equation (4.2.2), and
(iii) which is subjected to the non-ideal constraint that is prescribed by the n-vector
( , , ) C q qt as in equation (4.2.6).
83
Recalling our assumption that in the actual mechanical system S,
ˆ
|
TT
M MA =
has
rank n at each instant of time.
Consider further an unconstrained auxiliary system
T
AG
S that has
(1) an augmented mass matrix given by
2
( , ) ( ) ( , , ) ( , ) ( , , ) 0,
T
T
AG
M M qt t A q qt G qt A q qt α=+ >
(4.3.8)
where () t α
is any sufficiently smooth function (
2
C would be sufficient) of time
that is nowhere zero, and ( ,) ( ,) ( ,)
T
G qt N qt N qt =
is any m by m positive definite
matrix with its elements sufficiently smooth functions (
2
C would be sufficient) of
its arguments, and
(2) an augmented ‘given’ force defined by
,
( , ,) ( , ,) ( , ,) ( ,) ( , ,)
T
T
A Gz
Q q qt Q q qt A q qt G qt z q qt = +
(4.3.9)
where ( , ,) z q qt is any arbitrary, sufficiently smooth m-vector,
(3) so that the equation of motion of this unconstrained auxiliary system is given by
,
( ,) ( , ,) ( , ,) ( ,) ( , ,) : ( , ,).
TT
T
AG AG z
M qt q Q q qt A q qt G qt z q qt Q q qt =+=
(4.3.10)
Similar to the conceptualization stated in section 4.2, the system described by
equation (4.3.10) is referred to as the unconstrained auxiliary system
T
AG
S .
84
(4) We shall subject this unconstrained auxiliary system
T
AG
S to (a) the same initial
conditions, and (b) the same constraints, which the unconstrained mechanical
system S is subjected to, as described in items (ii) and (iii) above.
We note from equation (4.3.10) that the unconstrained auxiliary system has at each
instant of time an augmented mass matrix
T
AG
M obtained by using the mass matrix,
0 M ≥ , of the unconstrained mechanical system S and augmenting it by
2 T
A GA α ; also,
the ‘given’ force,
,
( , ,)
T
A Gz
Q q qt , which the unconstrained auxiliary system is subjected to,
at each instant of time, is obtained by using the ‘given’ force ( , ,) Q q qt that the
unconstrained mechanical system S is subjected to, and augmenting it by
T
A Gz , where
the vector z is arbitrary (note that z could be taken to be the zero vector). Thus the
unconstrained auxiliary system
T
AG
S differs from the unconstrained mechanical system S
in that at each instant of time (a) it has an augmented mass matrix
T
AG
M , and (b) it is
subjected to an augmented ‘given’ force
,
T
A Gz
Q .
The two unconstrained systems (S and
T
AG
S ) when subjected to the same set of
constraints (both ideal and/or non-ideal at each instant of time) and the same set of initial
conditions yield, correspondingly, what we shall call the constrained mechanical system
and the constrained auxiliary system.
85
Having both the unconstrained mechanical system S and the unconstrained auxiliary
system
T
AG
S we shall now show the following result:
Result 4.1: The acceleration of the constrained mechanical system S obtained by
considering the unconstrained mechanical system and its constraints as described by (i)-
(iii), is identical with, and directly obtained from, the explicit acceleration of the
constrained auxiliary system
T
AG
S obtained by considering the unconstrained auxiliary
system and its constraints as described by (1)-(4).
Proof: As shown in Ref. [50], the acceleration,
S
q , of the constrained mechanical system
S is described by the equation (see equation (4.2.9))
()
: ,
S
QC QC I A AM
qM
bb A
+
+
+
++ −
= =
(4.3.11)
while the acceleration,
T
A G
S
q , of the constrained auxiliary system
T
AG
S
is given by (see
also equation (4.2.9))
,
()
:.
T
T
T
A G
T
A Gz
AG
SA
QC
I A A M Q A Gz C
qM
Ab b
+
+
+
+ − + +
= =
(4.3.12)
86
Let us consider first the term () I A AM
+
− of equation (4.3.11). Post-multiplication of
both sides of (4.3.6) by
2
GA α , yields
2
( ) 0,
T
I A A A GA α
+
−= (4.3.13)
so that
2
2
() () ()
( )( )
( ).
T
T
T
AG
I AA M I AA M I AA A GA
I A A M A GA
I A AM
α
α
++ +
+
+
− =− +−
=−+
= −
(4.3.14)
Using equation (4.3.14) in equation (4.3.11) thus yields
()
:.
T
AG
SA
QC QC I A AM
qM
bb A
+
+
+
+ + −
= =
(4.3.15)
We note that the acceleration of the constrained system is still the same even though the
mass matrix 0 M ≥ is replaced with the augmented mass matrix 0
T
AG
M > (see equations
(4.3.11) and (4.3.15)). Since the augmented mass matrix
T
AG
M is always positive
definite, the matrix
1/2
T
AG
M
−
now does exist. The drawback in using the fundamental
equation (4.2.8) is now resolved. Thus in order to obtain only the acceleration of a
constrained system, we can just simply replace the mass matrix 0 M ≥
with the
augmented mass matrix 0
T
AG
M > and use the fundamental equation (4.2.8).
87
Again, pre-multiplying and post-multiplying both sides of (4.3.6) by
T
AG
M and Gz
respectively, we have
( ) 0.
T
T
AG
M I A A A Gz
+
−=
(4.3.16)
Noting that for any matrix X, ()
TT
X XX X
++
= [32], from equation (4.3.15), we have
( ) |
( )( )
()( ) ()
T
T
TT
TT
S A AA
AG
TT
AA
AG
T TT
AA
AG AG
QC QC
q M M M M I AA A
bb
M M M I A A Q C A b
M M M I A A Q C M I A A A Gz A b
M
+
++
+
+
+
++
++
= = −
= − + +
= − + + − +
=
( )( )
( ) |
.
T
T
T
A G
T TT
AA
AG
T
TT
AA
AG
T
AS
M M I A A Q A Gz C A b
Q A Gz C
M M M I AA A
b
Q A Gz C
Mq
b
+
+
+
+
+
− + + +
+ +
= −
+ +
= =
(4.3.17)
The third equality above follows from equation (4.3.16) and the last from equation
(4.3.12). This proves the claim.
Since we know that at each instant of time the acceleration of the constrained mechanical
system S is the same as that of the constrained auxiliary system
T
AG
S (see equation
(4.3.17)), and also that the augmented mass matrix of the system
T
AG
S is positive
definite, we can directly apply the so-called fundamental equation (4.2.8) to the
88
unconstrained auxiliary system described by (4.3.10) to get
T
A G
S
q and therefore
S
q
explicitly as [47]
1/2 1/2 1/2
, ,
( ) ( ),
T T T T T TT T
S
AG z AG AG AG z AG AG AG AG
q a M B b Aa M I B B M C
−+ − + −
= + − + −
(4.3.18)
where
2
0,
T
T
AG
M M A GA α =+>
(4.3.19)
,
,
T
T
A Gz
Q Q A Gz = +
(4.3.20)
1 11
,,
,
TT TT T
T
AG z AG AG z AG AG
a MQ MQ M A Gz
− − −
= = +
(4.3.21)
and
1/2
.
TT
AG AG
B AM
−
=
(4.3.22)
Remark 4.1: We know that when
ˆ
M has rank n, the acceleration,
S
q , of the constrained
mechanical system S is unique and is explicitly given by equation (4.3.11). And since we
have shown that
T
A G
SS
qq = at each instant of time, the acceleration of the constrained
auxiliary system must be independent of the arbitrary (nowhere-zero) scalar function
() t α , the arbitrary m-vector z(t), and the arbitrary (positive definite) matrix ( ,) G qt ,
provided each of these three entities is a sufficiently smooth (
2
C ) function of their
arguments.
89
Remark 4.2: Since α , z, and G are arbitrary as just stated, we can further particularize
equation (4.3.18) by setting 1 α ≡ , 0 z ≡ , and
m
GI ≡ in describing our unconstrained
auxiliary system. Thus this unconstrained auxiliary system now has only an augmented
mass matrix
T
M AA + , and it is subjected to the same ‘given’ force as the unconstrained
mechanical system S. This unconstrained auxiliary system, when subjected to the same
constraints (kinematical and dynamical) as those placed on S, yields the acceleration of
the constrained mechanical system S, given by
1/2 1/2 1/2
( ) ( ),
T T T T T TT T
S
A A A A A AA A
q a M B b Aa M I B B M C
−+ − + −
=+ − + −
(4.3.23)
where
0,
T
T
A
M M AA =+>
(4.3.24)
1
,
T T
A A
a MQ
−
=
(4.3.25)
and
1/2
.
TT
A A
B AM
−
=
(4.3.26)
4.3.3 Explicit Equation for Constraint Force
So far, we have developed an unconstrained auxiliary system
T
AG
S which always has a
positive definite mass matrix, and we have used it in the so-called fundamental equation
(4.2.8) to directly yield the acceleration of the constrained mechanical system S. We now
90
further explore whether the constraint force
c
Q acting on the unconstrained mechanical
system S (that is brought into play by the presence of the constraints (ii) and (iii)
described earlier in Section 4.3.2) can be directly adduced from the equation of motion of
the constrained auxiliary system
T
AG
S . To show this, we begin by putting forward a
useful result.
Lemma 4.3:
1/2 1
,
TT T
TT
AG AG AG
M B AM A A
+−
=
(4.3.27)
where
T
AG
M is defined in equation (4.3.8) and
T
AG
B is defined in equation (4.3.22).
Proof:
1/2 1 1/2 1/2 1/2 1/2
1/2 1/2
1/2
( ) ()
TT T TT T T TTTT
T TT T TT T T
TT
T TT
AG AG AG AG AG AG A G AG AG AG A G
TT T T T
AG A G A G A G AG AG AG A G
T
AG AG
MB AM A MB AM M A MB B B
M BB B M B B B
MB
+− +− − +
++
= =
= =
= = .
T
A
(4.3.28)
In the third equality above, we have used the fourth MP condition and in the fifth we
have used the first MP condition.
From equation (4.2.5) we know that once we obtain the constrained acceleration ( )
S
qq =
from equation (4.3.18) of the mechanical system S, we can determine the constraint force
91
c
Q acting on the unconstrained mechanical system S (described by equation (4.2.1)) at
each instant of time from the relation
.
c
S
Q Mq Q Mq Q = −= − (4.3.29)
Alternatively, consider the equation of motion of the constrained auxiliary system
T
AG
S ,
which can be obtained by pre-multiplying both sides of the equation (4.3.18) by
T
AG
M .
We have
1/2 1/2 1/2
,
,,
( )( )
:,
T TT T T TT T
TT
T
S
AG AG AG AG z AG AG AG AG
c
A Gz A Gz
M q Q A Gz M B b Aa M I B B M C
QQ
+ + −
=+ + − + −
= +
(4.3.30)
where
1/2 1/2 1/2
,,
( ) ( ).
T TT T T TT T
c
AG z AG AG A G z AG AG AG AG
Q M B b Aa M I B B M C
+ + −
= − +−
(4.3.31)
We notice from equation (4.3.30) that under the same set of constraints (both ideal and
non-ideal) as those acting on the unconstrained mechanical system S, the constraint force
acting on the unconstrained auxiliary system
T
AG
S
(described by equation (4.3.10)) is
,
T
c
A Gz
Q ; the explicit expression for this force is given by equation (4.3.31).
92
We now explore the connection between
c
Q and
,
T
c
A Gz
Q , our aim being to obtain
c
Q
explicitly from
,
T
c
A Gz
Q . We now claim that this can indeed be done by appropriately
choosing the m-vector z which has so far been left arbitrary.
To show this, we begin by considering only the third member on the right-hand side of
the first equality of equation (4.3.30). Expanding it, we have
1/2 1/2 1
,
1/2 1
1/2 1/2
( ) ( [ ])
()
()
TT T TT T
TT T T
TT T TT T
T
AG AG AG z AG A G AG
T
AG AG AG AG
AG AG AG AG A G AG
M B b Aa M B b AM Q A G z
M B b Aa AM A G z
M B b Aa M B AM
+ +−
+−
+ +−
−= − +
= −−
= −−
1
1/2
( ).
TT T
T
T
AG AG AG
A Gz
M B b Aa A G z
+
= −−
(4.3.32)
Notice that we have denoted
1
:
TT
AG AG
a MQ
−
= in the second equality, and used relation
(4.3.27) in the last equality of the above equation. Using equation (4.3.32) in the third
member on the right-hand side of equation (4.3.30) yields
1/2 1/2 1/2
( ) ( ): ,
T TT T T TT T T
c
S
AG AG AG AG A G A G A G AG AG
M q Q M B b Aa M I B B M C Q Q
+ + −
=+ − + − =+
(4.3.33)
where
1
,
TT
AG AG
a MQ
−
=
(4.3.34)
and
1/2 1/2 1/2
( ) ( ).
T TT T T TT T
c
AG AG AG AG AG AG AG AG
Q M B b Aa M I B B M C
+ + −
= −+ −
(4.3.35)
93
Equation (4.3.33) shows that the acceleration
S
q of the constrained mechanical system S
is given by
1
,
TT
c
S
AG AG
q M QQ
−
= +
(4.3.36)
and that it is indeed independent of the arbitrary m-vector () zt as remarked in the
previous sub-section. Furthermore, equating the right-hand sides of equations (4.3.30)
with (4.3.33) (both of which equal
T
S
AG
Mq ), we get
,, ,
,
TT T T
c Tc c
AG z AG z AG z AG
Q QQ A G z QQ Q +=++=+
(4.3.37)
so that from the last equality we have
,
.
TT
cc T
AG AG z
Q Q A Gz = +
(4.3.38)
We now prove the following result:
Result 4.2: When the unconstrained mechanical system, S, and the unconstrained
auxiliary system,
T
AG
S , have the same initial conditions and when they are each subjected
to the same (ideal and non-ideal) constraints, with the choice of the m-vector,
2
() () ( , , ), z t t b q qt α = (4.3.39)
94
where () t α is a nowhere-zero, sufficiently smooth function of time, the constraint force
acting on the unconstrained auxiliary system
T
AG
S is the same as the constraint force
acting on the unconstrained mechanical system S at each instant of time. In short,
2
,
.
T
cc
AG b
QQ
α
=
(4.3.40)
Proof: From equation (4.3.33), we know that at each instant of time
2
2
2
()
.
TT
c
S
AG A G
T
S
T
S
cT
Q Mq Q
M A GA q Q
Mq Q A GAq
Q A Gb
α
α
α
= −
=+−
= −+
= +
(4.3.41)
In the last equality above, we have used equations (4.3.29) and (4.2.4).
Substituting equation (4.3.41) in equation (4.3.38), we get
2
,
,
T
cc T T
A Gz
Q Q A G z A G b α = +−
(4.3.42)
which is the general result that relates the constraint force
c
Q acting on the unconstrained
mechanical system S to the constraint force
,
T
c
A Gz
Q acting on the unconstrained auxiliary
system
T
AG
S , at each instant of time. Note that in equation (4.3.42) () t α is any arbitrary
nowhere-zero scalar function, the m-vector () zt is any arbitrary sufficiently smooth
95
function, and ( ,) G qt is any positive definite matrix whose elements are continuous
functions of its arguments.
Finally, using equation (4.3.42), when
2
zb α = , the result follows .
Therefore, the force of constraint
c
Q acting on the unconstrained mechanical system S
can also directly be obtained from the force of constraint
,
T
c
A Gz
Q acting on the
unconstrained auxiliary system
T
AG
S by the appropriate selection of the m-vector
2
() () ( , , ) z t t b q qt α = .
We have now shown that if one would like to derive the constrained equations of motion
of a general mechanical system S that has either a positive semi-definite or positive
definite mass matrix, which is subjected to the kinematical constraints Aq b =
and the
non-ideal dynamical constraints described by the n-vector ( , , ) C q qt (under the proviso
that matrix
ˆ
M has rank n), one could obtain the (explicit) constrained equation of
motion of the mechanical system S by following the three-step conceptualization of
constrained motion as follows in terms of a new unconstrained auxiliary system [33]:
96
1) Description of the unconstrained auxiliary system:
(a) Replace the mass matrix 0 M ≥ of the actual unconstrained
mechanical system S as given in equation (4.2.1) with the augmented
mass matrix
T
AG
M as given in equation (4.3.8);
(b) Choose
2
zb α =
and replace the ‘given’ force Q of the actual
unconstrained mechanical system S with the augmented ‘given’ force
2
,
T
AG b
Q
α
as defined in equation (4.3.9);
(c) Use the augmented mass matrix described in (a) and the augmented
‘given’ force described in (b) to obtain equation (4.3.10) which
describes the unconstrained auxiliary system
T
AG
S ;
2) Description of the constraints: Subject this unconstrained auxiliary system
to the same set of constraints (both ideal and non-ideal) and initial
conditions as the actual unconstrained mechanical system S;
3) Description of the constrained auxiliary system: Apply the fundamental
equation [43], [47] (see equation (4.3.30)) to the unconstrained auxiliary
system described in (1) above, which is subjected to the constraints
described in (2).
97
The resulting equation of motion of this constrained auxiliary system has the following
two important features:
(1) The explicit acceleration of the constrained auxiliary system
T
AG
S , obtained by using
equation (4.3.18), is the same, at each instant of time, as the explicit acceleration of
the constrained mechanical system S, obtained by using equation (4.2.9), and
(2) at each instant of time, the constraint force
c
Q (see equation (4.3.29)) acting on the
unconstrained mechanical system S, because of the presence of the constraints
imposed on it, is the same as the constraint force
2
,
T
c
AG b
Q
α
(see equation (4.3.31))
acting on the unconstrained auxiliary system
T
AG
S , which is described by equation
(4.3.10).
We are thus led to the somewhat surprising conclusion: the dynamics of the actual
constrained mechanical system S are completely mimicked by the dynamics of the above-
mentioned constrained auxiliary system
T
AG
S .
Lastly, we point out that if one were interested only in obtaining the acceleration at each
instant of time of the constrained mechanical system S, one can use any arbitrary m-
vector z(t) in equation (4.3.20) to obtain the augmented ‘given’ force
,
T
A Gz
Q
and then use
equation (4.3.18); for simplicity, Occam’s razor would then suggest that we might prefer
98
to take () 0 zt ≡ . As pointed out in item (1), part (b), above, if one were, in addition, also
interested in finding the correct constraint force acting on the unconstrained mechanical
system S, one would need to choose
2
() () ( , , ) z t t bqq t α = and use equations (4.3.21) and
(4.3.31). Clearly then, when the constraints are such that ( , ,) 0 bqq t ≡ , the choice of
() 0 zt ≡ in the augmented ‘given’ force (4.3.9) in the description of the unconstrained
auxiliary system (4.3.10) is automatically selected. In that case, the use of equation
(4.3.30) yields, at each instant of time, the correct acceleration of the constrained
mechanical system S as well as the correct force of constraint acting on the unconstrained
mechanical system S.
The approach of the above three-step conceptualization of constrained motion by
utilizing the auxiliary system can be summarized as in Table 4.1. This table schematically
shows how one generates the auxiliary system
T
AG
S from the actual given mechanical
system S. Step 1 deals with the description of the unconstrained system S and the
corresponding unconstrained auxiliary system
T
AG
S . Instead of using the mass matrix M
of an actual mechanical system S that may or may not be singular, we use the mass
matrix
T
AG
M for the auxiliary system
T
AG
S which is positive definite under the proviso
that
ˆ
M has full rank. In addition, we also augment the given force Q of the actual
mechanical system S with the term
T
A Gz in defining the unconstrained auxiliary system.
99
Then the unconstrained acceleration of the auxiliary system can be written as
1
,,
T TT
AG z AG AG z
a MQ
−
= while the unconstrained acceleration is undefined, as shown, in the
case where the mass matrix is singular for the unconstrained mechanical system. In Step
2—the description of the constraints—while describing the constraints we apply the same
set of (ideal and non-ideal) constraints to the auxiliary system
T
AG
S
as applied to the
actual mechanical system S.
In Step 3—the description of the constrained system—we can obtain the explicit equation
of the constrained acceleration of the actual mechanical system by using the
unconstrained auxiliary system and the constraints defined in the previous two steps and
applying the so-called fundamental equation [43], [47] (see equation (4.3.18)). The
fundamental equation also explicitly gives the constraint force on the actual mechanical
system when using the m-vector
2
zb α = in our definition of the unconstrained auxiliary
system (see equations (4.3.21) and (4.3.31)).
We also note that even though both the actual mechanical system and the auxiliary
system approach yield the same dynamical results, as mentioned earlier, in order to obtain
a unified explicit equation of constrained motion, we prefer using the auxiliary system
approach.
100
System
Descriptions
Actual Mechanical System, S
Auxiliary System,
T
AG
S
Step 1: Description of Unconstrained System
Mass Matrix
( ,) 0 M qt ≥ ,
M is an n by n matrix
2
( ,) ( ) ( , ,) ( ,) ( , ,) 0,
T
T
A G
M M qt t A q qt G qt A q qt α=+>
( ) 0, is an arbitrary function of time t α ≠
( ,) 0 G qt > are arbitrary m by m matrix
Given Force ( , ,) Qq q t
,
( , ,) ( , ,) ( ,) ( )
T
T
A Gz
Q Q q qt A q qt G qt z t = + ,
m-vector () zt is an arbitrary m-vector
Equation of motion Mq Q =
,
T T
T
S
AG AG z
A G
Mq Q =
Unconstrained
Acceleration
1
a MQ
−
= or Undefined
1
, ,
T TT
AG z AG AG z
a MQ
−
=
Step 2: Description of Constraints
Description of
Kinematic
Constraints
( , ,) ( , ,) A q qt q b q qt = ,
A is an m by n matrix
( , ,) ( , ,) A q qt q b q qt =
Description of
Non-ideal
Constraints
( , ,) Cqq t ( , ,) Cqq t
Step 3: Description of Constrained System
Equation of
motion
c
Mq Q Q = +
,,
T TT
T
c
S
AG AG z AG z
A G
Mq Q Q = +
Constrained
Acceleration
() QC I A A M
q
b A
+
+
+ −
=
1/2
1/2 1/2
,,
()
( ),
T TT T
T
T TT T
S
AG z AG AG AG z
A G
AG AG AG AG
q q a M B b Aa
M I B B M C
−+
− + −
= = + −+
−
1/2
TT
AG AG
B AM
−
=
( ) 0, is an arbitrary function of time t α ≠
( ,) 0 G qt > is an arbitrary m by m matrix
() zt is an arbitrary m-vector,
Constraint Force on
Unconstrained
System
c
Q Mq Q = −
1/2
2
1/2 1/2
2
, ,
()
( ),
TT T
T TT T
cc
T
AG AG AG b A G b
AG AG AG AG
Q Q M B b Aa
M I B B M C
α α
+
+ −
= = − +
−
1/2
TT
AG AG
B AM
−
=
( ) 0, is an arbitrary function of time t α ≠
( ,) 0 G qt > are arbitrary m by m matrix
2
() () ( , , ) zt t bq q t α =
Table 4.1: The three-step conceptualization of constrained motion by utilizing the auxiliary system
101
4.4 Illustrative Examples
In this section, five examples are provided to illustrate the results developed and to show
their usefulness. The first three examples are purposely chosen to be very simple so that
the central ideas in the chapter are better understood. The fourth example is substantive
because in it we show how our results can be directly used to obtain in a simple and
straight forward manner the nonlinear equations describing the rotational dynamics of a
rigid body in terms of quaternions. The last example takes the reader even further
showing that there can be an unlimited number of mechanical systems which, when
subjected to the same constraints, the same externally applied ‘given’ forces, and the
same initial conditions, exhibit identical dynamical behavior, making them completely
indistinguishable from one another.
4.4.1 Example 1
Consider a wheel of mass m and radius R rolling on an inclined surface without slipping,
as shown in Figure 4.1, with the gravitational acceleration g downwards. The angle of the
inclined surface is β , where 0 / 2 β π < < .
The system clearly has just one degree of freedom, which can be described by θ .
102
Figure 4.1: A wheel rolling down an inclined plane under gravity
If y is the vertical displacement of the center of the wheel as it rolls down the inclined
plane, the wheel’s potential energy can be simply expressed as
. V mgy = −
(4.4.1)
Were we to take θ and y as the independent generalized coordinates and use the
Lagrangian
22
11
() ,
22
c
L T V m R I mgy θθ = − = + +
(4.4.2)
where
c
I is the moment of inertia around the center of the wheel, to obtain Lagrange’s
equations of motion for the unconstrained system (since we are assuming that the
coordinates are independent), we would get the relations of the unconstrained system in
the form of the equation (4.2.1) as
θ
x
y
g
R
β
m
103
2
0 0
.
00
c
mR I
mg y
θ +
=
(4.4.3)
Note that the mass matrix M describing the unconstrained motion is now singular. This
singularity is a consequence of the fact that in reality the system has only one degree of
freedom and we are using more than the minimum number of generalized coordinates to
describe the system, and pretending that these coordinates are independent. The
advantage of doing this is that the unconstrained equations of motion, (4.4.3), can be
trivially written down. However, the two coordinates y and θ are in reality not
independent of one another, since the wheel rolls down without slipping. They are related
through the equation of constraint
sin yRθβ =
(4.4.4)
that must be added to the formulation in order to model the dynamics of the physical
situation properly. Differentiating equation (4.4.4) twice with respect to time, we obtain
the constraint equation in the form of equation (4.2.4), Aq b = , where
[ ] sin 1 and 0. A R b β = −=
(4.4.5)
Since in this problem the system is ideal, 0 C = . The next step is usually done through
the use of the fundamental equation (4.2.8), but in this problem the mass matrix is now
104
positive semi-definite ( 0) M ≥ , so the unconstrained acceleration a is undefined and the
fundamental equation (4.2.8) cannot be applied. Thus, we have to use instead the relation
(4.2.9) that can deal with the system when the mass matrix is positive semi-definite.
However, in order to use the relation (4.2.9) we have to make sure that we model the
system correctly. We can do that by checking the condition of the uniqueness of the
equation of motion—the matrix |
T
MA
has full rank [50]—and so it is. Thus using the
relation (4.2.9), the constrained mechanical system becomes
22 2
sin
1
.
sin
c
mgR
mgR mR I y
β θ
β
=
+
(4.4.6)
We again note that the structure of equation (4.2.9) differs widely from the so-called
fundamental equation (4.2.8); equation (4.2.9) does not readily lend itself to physical
interpretation as does equation (4.2.8) [43], [47] but the drawback of equation (4.2.8) is
that it cannot be directly applied when the mass matrix of the system is singular.
However, if we use the unconstrained auxiliary system which replaces the mechanical
system with its positive semi-definite mass matrix by one that is positive definite, then
the general form to obtain the equations of motion of the constrained system, the
fundamental equation [43], [47], can be handily applied. By choosing ( ) 3, t α =
2
() 0 zt b α = = and ( , ) () 0 G qt g t = >
where () gt is an arbitrary positive function of time,
105
we obtain the unconstrained auxiliary system (4.3.10) where the augmented mass matrix
becomes
2 2 2
2
9 ( ) sin 9 ( ) sin
0,
9 () sin 9 ()
T
T c
AG
mR I gt R gt R
M M A GA
gt R gt
ββ
α
β
+ + −
=+= >
−
(4.4.7)
and the augmented ‘given’ force is
2
2
,
0
.
T
T
AG b
Q Q A Gb
mg
α
α
=+=
(4.4.8)
Then using the augmented mass matrix from equation (4.4.7) and the augmented ‘given’
force from equation (4.4.8) in the fundamental equation (4.3.30), the equation of motion
of the constrained auxiliary system yields the same equation of motion, (4.4.6), as that of
the constrained mechanical system. Furthermore since 0 b = , equation (4.3.30) also gives
the correct constraint force required in the application to the unconstrained mechanical
system. This constraint force is given by equation (4.3.31) as
sin
.
c
c
y
Q mgR
Q mg
θ
β
=
−
(4.4.9)
The advantage of using the auxiliary system is that the so-called fundamental equation,
the general form of obtaining the equations for constrained motion, can directly be
applied and yields the correct equations of motion for the constrained mechanical system.
106
4.4.2 Example 2
Consider a system of two masses,
1
m
and
2
m , connected with springs,
1
k and
2
k , as
shown in Figure 4.2.
Figure 4.2: A two degree-of-freedom multi-body system
Defining
1 1 10
x xl = − and
2 2 20
x xl = − , where
10
l and
20
l are the unstretched lengths of the
springs
1
k and
2
k respectively and using the Lagrangian to obtain Lagrange’s equations
of motion for the system, we get the dynamics of the unconstrained system as the form of
equation (4.2.1) as
12 12 2 2 10 1 11
2 1 2 10 2 22
( ) () 0
: :,
() 0
k k x k x k l d m xx
MQ
k x x l d m xx
−+ + − +
= = =
−+ +
(4.4.10)
where d is the length of the mass
1
m .
Assuming that
1
m and
2
m are both positive, we then obtain the acceleration
107
12 12 2 2 10
1 1
2 1 2 10 2
2
( ) ()
.
()
k k x kx k l d
m x
kx x l d x
m
−+ + − +
=
−+ +
(4.4.11)
We now assume that
1
0 m = . The mass matrix on the left-hand side of equation (4.4.10)
now becomes singular. However, the physical system that has now just one spring
connecting mass
2
m , has a unique acceleration. In order to obtain the equation of motion,
our attention is therefore drawn to the need for an additional constraint, so that the
condition for determining the acceleration uniquely—namely, that the matrix |
T
MA
has full rank—is satisfied.
Considering the location of the mass
1
0 m = , we obtain the following constraint relation:
11 2 2 1
( ). kx k x x d = −−
(4.4.12)
Differentiating equation (4.4.12) twice with respect to time t and then putting it in the
form of equation (4.2.4), we get the equation of the constraint Aq b = , where
[ ]
12 2
and 0. A kk k b =+− =
(4.4.13)
We assume that the constraint is ideal so that 0 C = .
108
To obtain the equations of motion of this constrained system, which now has a singular
mass matrix, we replace the mass matrix of the unconstrained system 0 M ≥ with the
augmented mass matrix 0
T
AG
M > . Since the system is subjected to only one constraint,
an arbitrary positive definite matrix G is now an arbitrary positive function ( , ) g qt
which can be combined with the no-where zero function () t α .
Thus we choose
2
gg α =
where ( , ) g qt is an arbitrary positive function. Furthermore, since
0, b =
we choose
2
() 0 zt b α = = . The augmented mass matrix
T
AG
M of the auxiliary system in relation
(4.3.8) can be then expressed as
2
2 12 12 2
2
1 22 2 2
( ) ( , ) ( ) ( , )
0.
( ) ( , ) ( , )
T
T
AG
k k g qt k k k g qt
M M A GA
k k k g qt m k g qt
α
+ − +
=+= >
−+ +
(4.4.14)
The augmented ‘given’ force for the unconstrained auxiliary system is
,
T
A Gz
QQ = , since
() zt is chosen to be zero. Using this new augmented mass matrix
T
AG
M
instead of M in
the fundamental equation (4.3.18), we obtain the acceleration of the constrained motion
2
12
2
1 12 2 10
2 2 12
12
() ()
,
()
kk
x k k xl d
m x kk
k k
+ − −
= −
+
(4.4.15)
which is the correct result. We note that when
1
0 m = , one can imagine that the system is
now composed of only the mass
2
m , with the springs
1
k and
2
k connected to it in series.
109
As anticipated, the acceleration
2
x of the mass
2
m is given by the second row of equation
(4.4.15), which is the correct equation, since
2 10
() xl d −− is the total extension of both
the springs
1
k and
2
k . Again, this example illustrates how the actual constrained
mechanical system is mimicked by the constrained auxiliary system.
In the next example, we present another approach to deriving the equations of motion of
the constrained system in Example 2. This gives us an idea about the advantage of using
the auxiliary system that can ‘transform’ any system S with a positive semi-definite mass
matrix to a system
T
AG
S
with a positive definite mass matrix. The derivation of the
unconstrained system is simpler and at the same time the so-called fundamental equation
can be readily utilized.
4.4.3 Example 3
Consider the same system of two masses,
1
m and
2
m , connected with springs,
1
k and
2
k ,
that is shown in Figure 4.2. But for this example, we shall model this system by
decomposing it into two separate sub-systems—that is, we consider it as a multi-body
system—as shown in Figure 4.3. The two sub-systems are then connected together by the
‘connection constraint’,
11
, q xd = +
where d is the length of the mass
1
m . We use the
coordinates
1
x ,
1
q and
2
q to describe the configuration of the two sub-systems, and first
treat these coordinates as being independent in order to get the unconstrained equations
110
of motion. We then ‘connect’ the two sub-systems by imposing the constraint
11
q xd = +
to obtain the equations of motion of the composite system shown in Figure 4.2.
Figure 4.3: Decomposition of the multi-body system shown in Figure 4.2 using more than two coordinates
Again defining
1 1 10
x xl = −
and
2 2 20
, q ql = − where
10
l and
20
l are the unstretched lengths
of the springs
1
k and
2
k respectively and using Lagrange’s equation, the equations of
motion for the unconstrained system can be written as the form of equation (4.2.1) as
1 1 1 11
1 2 21
2 2 2 2 22
00
: 0 0 : .
0
x m x kx
Mq m m q Q
q m m q kq
−
= = =
−
(4.4.16)
Note that these equations for the unconstrained system are almost trivial to obtain. To
model the system shown in Figure 4.2 using these two separate sub-systems, we connect
the two sub-systems by using the constraint
1 1 1 10
q x d x l d = + = + + . Differentiating this
constraint twice with respect to time we get the equation of the constraint in the form of
equation (4.2.4), Aq b = , where
111
[ ] 1 1 0 and 0. A b =−=
(4.4.17)
Again by choosing more than the minimum number of coordinates to describe the
configuration of the system, and treating them as being independent, we obtain a mass
matrix, M, that is singular (positive semi-definite). The description of the constrained
mechanical system S thus is provided by the description of the unconstrained system
given by equation (4.4.16), which has a singular mass matrix, and the constraint Aq b = ,
where A and b are given by relations (4.4.17) , that is imposed on it.
We assume that the constraint is ideal so that 0 C = . To obtain the equations of motion
for the constrained mechanical system S we replace the mass matrix of the unconstrained
mechanical system M that is positive semi-definite with the augmented mass matrix
T
AG
M that is positive definite of an unconstrained auxiliary system. This augmented mass
matrix (see equation (4.3.8)) is obtained by again choosing
2
gg α = , where g is an
arbitrary positive function, as
1
2
22
22
0
0.
0
T
T
AG
mg g
M M A GA g m g m
mm
α
+−
=+ =−+ >
(4.4.18)
112
Since
0, b =
we choose
2
() 0 zt b α = = , and thus
,
T
A Gz
QQ = . The augmented mass matrix
(4.4.18) is positive definite, so we use the fundamental equation (4.3.18), to obtain the
acceleration of the constrained auxiliary system as
11 2 2
1
1
11 2 2
1
1
2
1 1 22 22
12
,
kx k q
m
x
kx k q
q
m
q
k x kq kq
mm
−+
−+
=
−+
− −
(4.4.19)
which is the correct result. This verifies that the dynamic of the constrained mechanical
system S is completely mimicked by the dynamic of the constrained auxiliary system
T
AG
S . Although both systems have different mass matrices, the resulting accelerations of
the constrained motion of both systems are exactly the same.
In the next example we show how the results obtained in this chapter can be directly
applied to obtaining the quaternion equations of rotational motion for rigid bodies in a
simple and direct manner. When considering the rotational dynamics of rigid bodies, the
use of quaternions removes singularity problems that inevitably arise when using Euler
angles. However, the quaternion 4-vector describing a physical rotation is constrained to
have unit norm, and hence the equations of motion in terms of quaternions can be
considered constrained equations of motion.
113
4.4.4 Example 4
Consider a rigid body that has an absolute angular velocity,
3
ω ∈ , with respect to an
inertial coordinate frame. The components of this angular velocity with respect to its
body-fixed coordinate frame whose origin is located at the body’s center of mass are
denoted by
1
ω ,
2
ω , and
3
ω . Let us assume, without loss of generality, that the body-
fixed coordinate axes attached to the rigid body are aligned along its principal axes of
inertia, where the principal moments of inertia are given by 0
i
J > , 1, 2, 3 i = . The
rotational kinetic energy of the rigid body is then simply
T
1
= 2 2,
2
TT TT
T J u E JEu u E JEu ωω = =
(4.4.20)
where
T
12 3
=[ , , ] ω ωω ω , ) , , ( =
3 2 1
J J J diag J , and
0 1 2 3
[ ,, , ]
T
u u uu u = is the unit
quaternion 4-vector that describes the rotation such that = 2 = 2 Eu Eu ω −
, where
10 3 2
2 30 1
3 2 10
=.
uu u u
E u u u u
u u uu
−−
−−
−−
(4.4.21)
We note that the components of u are not independent and are constrained since the
quaternion u must have unit norm to represent a physical rotation. Under the assumption,
however, that these components are independent, one obtains the unconstrained equations
of motion of the system using Lagrange’s equations as
114
T T
:= 4 = 8 := ,
u
Mu E JEu E JEu Q Γ − +
(4.4.22)
where the 4-vector
u
Γ in equation (4.4.22) represents the generalized impressed
quaternion torque. The connection between the generalized torque 4-vector
u
Γ and the
physically applied torque 3-vector
[ ]
T
12 3
,,
B
Γ ΓΓ Γ = , whose components
i
Γ , i =1, 2, 3
are about the body-fixed axes of the rotating body, is known to be given by the relation
T
2.
uB
E Γ Γ = (4.4.23)
We note now that the 4 by 4 matrix
T
= 4 M E JE in relation (4.4.22) of this unconstrained
system is singular, since its rank is 3.
The unit norm constraint on the quaternion u requires that
2222
01 2 3
1
T
u u uuuu = +++ = ,
which yields
T
A u = and
T
b uu = − . The 4 by 5 matrix
ˆ
[ | ]
TT
M MA = has rank 4 since
2
16 0
[ ]
0 1
T
T
M
E J E
Mu
u
=
(4.4.24)
is a symmteric matrix whose eigenvalues are 0, 1,
2
1
16J ,
2
2
16J , and
2
3
16J . Hence, by
Lemma 4.1, the matrix
2
()
T
T
AG
M M g t uu α = + given in (4.3.8) is positive definite, where
the arbitrary function ( ) 0, gt t >∀ and we can choose α to be any positive constant.
115
Using equation (4.3.18) we obtain using some algebra the generalized acceleration of the
system given by
T1 T1
1 11
= () ,
2 42
B
u EJ J u EJ ωω ω Γ
− −
− −+ (4.4.25)
where ω is the usual skew-symmetric matrix obtained from the 3-vector ω
and () ω
is the norm of ω .
In addition, by virtue of the presence of the unit norm constraint, if we would like to
obtain the constraint torque (
c
Q ) applied to the unconstrained mechanical system (4.4.22)
directly by using the unconstrained auxiliary system, which is obtained by replacing the
mass matrix M in equation (4.4.22) with
T
AG
M , by Result 4.2 and equation (4.3.9), we
would also have to augment the ‘given’ torque Q in equation (4.4.22) as
2
2
,
() ( )
T
AG b
Q Q g tu u
α
α = − , where ()
T
u uu = is the norm of u . Then using equations
(4.3.21) and (4.3.31) the constraint torque for the rotational motion of a rigid body is
explicitly obtained by
2
,
2( ) ,
T
cc T
AG b
Q Q Ju
α
ωω = = −
(4.4.26)
which is the correct constraint torque that the unconstrained system (4.4.22) is subjected
to when the constraint
2222
01 2 3
1
T
u u uuuu = +++ = is imposed on it.
116
Thus far we have been careful to choose examples that contain a positive semi-definite
mass matrix ( 0). M ≥ In the next example we consider a system under the condition that
the mass matrix M is positive definite ( 0). M > In this situation the usual approach for
obtaining the constrained equations of motion, the fundamental equation (4.2.8), is
applicable and there is no need to think of the auxiliary system to obtain the equations of
motion of the constrained system. However, this example will show that two uncoupled
nonlinear systems which can be modeled separately can be altered to a coupled nonlinear
auxiliary system that can be much more complicated. Both coupled and uncoupled
systems will yield exactly the same constrained dynamics; however, the auxiliary system
can hardly be discerned as one that would yield the same dynamics.
4.4.5 Example 5
Consider two particles of masses
1
m and
2
m that have no impressed forces acting on
them and move in the XY-plane. The coordinates of mass
1
m in an inertial frame of
reference are
11
(, ) xy and those of mass
2
m are
22
(, ) xy . The vector of coordinates
describing the configuration of the system is therefore given by
112 2
[, , , ]
T
q x yx y = . The
two masses are independently constrained to
(1) trace elliptical trajectories (with a common focus) with eccentricities
1
ε and
2
ε ,
22
, 1,2,
i i ii i
x y x pi ε += + = (4.4.27)
117
where the constant 0
ii
pl ε = ≠ , and
(2) move along the ellipses so that the sectorial areas which they trace per unit of
time are constants,
, 1, 2,
i ii ii
x yy xc i −= = (4.4.28)
where
i
c is a constant.
To obtain the relation of the constraints in the matrix form as in (4.2.4) we differentiate
equations (4.4.27) and (4.4.28) appropriately with respect to time and obtain the
constraint
, Aq b =
(4.4.29)
where the matrices
22
1 11 1 11
22
2 22 2 22
11
22
00 /
00 /
and .
00 0
00 0
xr y c r
xr y c r
A b
yx
yx
ε
ε
− −
− −
= =
−
−
(4.4.30)
Note that we have used
22
, 1,2.
i ii
r x yi = + = in equation (4.4.30).
Assuming that the mass
1
m moves along the inner ellipse, and the mass
2
m moves along
the outer ellipse, the mass matrix M = Diag{
11 2 2
,, , mmm m } is a constant diagonal matrix.
118
Since there are no ‘given’ forces, the 4-vector {0,0,0,0} Q Diag = . The constraints are
assumed to be ideal so that C = 0.
The dynamical system S comprises the unconstrained mechanical system described by
the equation 0 Mq Q = = , which is subjected to the constraint equations (4.4.29) and
(4.4.30). The mass matrix is invertible, and we can use equation (4.2.8), to directly
obtain the acceleration of the constrained mechanical system S, which is
22 2 2
11 11 2 2 2 2
22 2 2
111 1 1 1 2 2 2 2 2 2
11 1 1
.
T
xc y c x c y c
q
p r rp r rp r rp r r
= − − − −
(4.4.31)
We shall now show that the same equation (4.4.31) of the constrained motion results
from a multitude of other systems that are subjected to the same constraints (described by
equations (4.4.29) and (4.4.30)) as the dynamical system S.
As an illustration, we consider the unconstrained auxiliary system (4.3.10) by replacing
the mass matrix M of the unconstrained mechanical system S with the augmented mass
matrix
T
AG
M as given in equation (4.3.8). We choose, for the sake of simplicity, () 2 t α =
for all time t,
2
zb α = and the positive definite matrix
119
41 0 0
1 2 01
()
0 0 20
0 1 01
Gt
=
(4.4.32)
for all t. We then have the augmented mass matrix of the auxiliary system given by
11 12 13 14
12 22 23 24 2
13 23 33 34
14 24 34 44
2 2
11 1 1 1 1 1 12 1 1 1 1
13 1 1 1 2 2 2 14 2 1 1 1
22 1
0,
where 16( ) 8 , 8 ( 2 )
4( )( ), 4 ( )
8
T
T
AG
M M A GA
m x r y yx r
xr x r y xr
mx
α
εε
εε ε
∆∆ ∆ ∆
∆∆ ∆ ∆
=+= >
∆∆ ∆ ∆
∆∆ ∆ ∆
∆= + − + ∆ = −
∆= − − ∆ = −
∆= +
22
1 1 23 1 2 2 2 24 1 2
22
33 2 2 22 2 22 2
34 2 22 2 2 222 2 2
22
44 2 2 2 2
16 , 4 ( ), 4
4( ) 4( )
4 ( ) 4( )( )
4( ) 4 ,
y y x r yy
m xr xr y
y xr x y xr y
m x y y
ε
ε ε
εε
+ ∆= − ∆ =
∆=+− +− −
∆= − + + − −
∆= + + +
(4.4.33)
and the augmented ‘given’ force described by
2
22 22
1 1 1 1 2 2 1
22 22
2 1 2 21 1
22 22 22 22 ,
12 2 2 2 1 2 21 21 2
22 22 22
21 2 2 1 2 2 1 2
( )(4 )
(4 ) 4
.
( )( 2 )
( )( )
T
T
AG b
x r cr cr
cr cr y
Q Q A Gb
rr x r cr cr cr y
cr x y cr cr y
α
ε
α
ε
−+
+
= += −
− +−
++ +
(4.4.34)
When this unconstrained auxiliary system is subjected to the same constraints as the
system S, namely, the constraints given by equations (4.4.29) and (4.4.30),
using
equations (4.3.21) and (4.3.31) with 0 C = we obtain the explicit constraint force
120
2
,
T
c
AG b
Q
α
acting on the unconstrained auxiliary system, which is, as expected, the same as
the constraint force
c
Q acting on the unconstrained mechanical system S, and which is
given by ((4.3.31))
22
1/2
,,
22 2 2
1 11 1 11 2 2 2 2 2 2
22 2 2
1 1 11 1 1 2 2 22 2 2
()
.
T TT T
cc
AG b AG AG AG b
T
Q Q M B b Aa
m x cm y cm x cm y c
p r rp r rp r rp r r
αα
+
= = −
= − − − −
(4.4.35)
Using equations (4.4.33), (4.4.34) and (4.4.35), the constrained equations of motion of
the system can be obtained from equation (4.3.30) as
2 2
,,
.
T TT
T
A G
c
S
AG AG b AG b
Mq Q Q
αα
= +
(4.4.36)
Pre-multiplying both sides of equation (4.4.36) by
1
T
AG
M
−
, the acceleration of the
constrained auxiliary system
T
AG
S is obtained, which as expected, is the same as that
given by equation (4.4.31) for the constrained mechanical system S.
Clearly, other (positive) functions () t α and other positive definite matrices () Gt would
give other unconstrained auxiliary systems which, when subjected to same constraints as
the system S, would yield motions that would be indistinguishable from those of the
constrained system S, assuming that both systems start with the same initial conditions.
121
We see that these results follow Results 4.1 and 4.2 in Section 4.3, and illustrate that:
(1) The explicit acceleration of the constrained auxiliary system
T
AG
S given in
equation (4.3.18) is the same as the explicit acceleration of the constrained
mechanical system S, and,
(2) the constraint force
c
Q acting on the unconstrained mechanical system S, because
of the presence of the constraints imposed on it, is the same as the constraint
force
2
,
T
c
AG b
Q
α
(see equation (4.3.31)) acting on the unconstrained auxiliary
system
T
AG
S when the unconstrained auxiliary system is subjected to the same
constraints as the unconstrained mechanical system S.
It should be noted that while the unconstrained mechanical system S that we started with
is simple (it has a diagonal mass matrix M, with 0 Q = ), the unconstrained auxiliary
system appears much more complex and has both inertial coupling, as seen from equation
(4.4.33), as well as force coupling, as seen from equation (4.4.34). However, the
constrained motion of both of these unconstrained systems, when subjected to the (same)
constraints given by relations (4.4.29) and (4.4.30), are identical for any given set of
initial conditions!
122
4.5 Summary
The main contributions of this chapter are the following:
(i) In Lagrangian mechanics, describing mechanical systems with more than
the minimum number of required coordinates is helpful in forming the
equations of motion of complex mechanical systems since this often
requires less labor in the modeling process. The reason that we do not
usually use more coordinates than the minimum number is because in
doing so we often encounter singular mass matrices and then standard
methods for handling such constrained mechanical systems become
inapplicable. For example, methods that rely on the invertability of the
mass matrix (such as the use of the fundamental equation given in (4.2.8))
cannot be used.
(ii) Under the proviso that
ˆ
|
TT
M MA =
has rank n, in this chapter a unified
explicit equation of motion for a general constrained mechanical system
has been developed irrespective of whether the mass matrix is positive
definite or positive semi-definite (singular). This is accomplished by
replacing the actual unconstrained mechanical system S with an
unconstrained auxiliary system
T
AG
S , which is obtained by adding
2 T
A GA α to the mass matrix M of the unconstrained mechanical system S,
123
and adding
T
A Gz to the ‘given’ force Q acting on the unconstrained
mechanical system S. The mass matrix,
2
T
T
AG
M M A GA α = + , of this
unconstrained auxiliary system
T
AG
S
is always positive definite
irrespective of whether the mass matrix M is positive semi-definite
( 0) M ≥ or positive definite ( 0) M > . Thus, by applying the fundamental
equation to this unconstrained auxiliary system, which is subjected to the
same constraints (and initial conditions) as those imposed on the
unconstrained mechanical system S, one directly obtains the acceleration
of the constrained mechanical system S.
(iii) The restriction that
ˆ
|
TT
M MA =
has full rank n, is not as significant a
restriction in analytical dynamics as might appear at first sight, because it
is a necessary and sufficient condition that the acceleration of the
constrained system be uniquely determinable—a condition that is always
satisfied in classical mechanics. In fact it provides a useful check on the
modeling being done, especially when dealing with complex multi-body
systems.
(iv) We show that the acceleration of the constrained mechanical system S so
obtained through the use of the auxiliary system
T
AG
S
is independent of
the arbitrarily prescribed: (1) nowhere-zero function () t α ; (2) the m-
124
vector () zt ; and (3) the positive definite matrix ( ,) G qt , provided these
are sufficiently smooth (
2
C ) functions of their arguments.
(v) In the special case, when () 1 t α = , () 0 zt = , and
m
GI = , the
unconstrained auxiliary system simplifies and is the same as the
unconstrained mechanical system S except that the mass matrix of the
auxiliary system is obtained by adding
T
AA to that of the unconstrained
mechanical system S. Under identical constraints and initial conditions,
the accelerations of the constrained auxiliary system and the constrained
mechanical system are identical, and the latter can then be obtained from
the former.
(vi) The constraint force
c
Q acting on the unconstrained mechanical system S
(by virtue of the presence of the constraints) can be obtained directly from
the constraint force
,
T
c
A Gz
Q acting on the unconstrained auxiliary system
from the relation
2
,
T
cc T T
AG z
Q Q AG z AGb α = +− . Furthermore, by
choosing
2
zb α = when describing the unconstrained auxiliary system, we
obtain the simpler result
2
,
T
cc
AG b
Q Q
α
= . Thus, when ( , ,) 0 bqq t ≡ , the
choice of () 0 zt ≡ for the constrained auxiliary system equation (4.3.30)
and application of equation (4.3.31) gives both the acceleration of the
constrained mechanical system S as well as the constraint force directly.
125
(vii) The results (and their derivations) that have been developed in this chapter
differ from those in Ref. [51] in two important respects. (1) They are
simpler, because we do not use the generalized Moore-Penrose (MP)
inverse of the matrix A in the determination of the unconstrained auxiliary
system. Instead, we simply use its transpose. (2) They are more general
because we can incorporate the arbitrary function () t α and the arbitrary
positive definite matrix ( ,) G qt in the creation of our unconstrained
auxiliary systems. Besides the simplicity and the aesthetic value that result
from these differences, there are substantial practical benefits that accrue.
Most importantly, these new results provide a major improvement in terms
of computational costs since the computation of the transpose of a matrix
is near-costless compared to its generalized inverse; this difference in cost
becomes increasingly important as the size of the computational model
increases. Another advantage is that the flexibility in choosing () t α and
( ,) G qt can become important from a numerical conditioning point of
view, especially when dealing with large, complex multi-body systems.
126
(viii) The results in this chapter point to deeper aspects of analytical mechanics
and show that:
(a) given any constrained mechanical system S described by the
matrices ( ,) 0 M qt ≥ and ( , ,) Aq q t , and the column vectors ( , ,) Q q qt ,
( , ,) bqq t , and ( , ,) C q qt ,
(b) there exists a kind of gage invariance whereby there are infinitely
many unconstrained systems with positive definite mass matrices given by
2
( ,) ( ) ( , ,) ( ,) ( , ,)
T
T
AG
M M qt t A q qt G qt A q qt α = +
and ‘impressed’ forces given by
,
( , ,) ( , ,) ( , ,) ( ,) ( , ,)
T
T
A Gz
Q q qt Q q qt A q qt G qt z q qt = +
which,
(c) when subjected to the same constraints (both holonomic and
nonholonomic, ideal and non-ideal) as those on the given mechanical
systems S, and when started with the same initial conditions as the given
mechanical system S,
(d) will be indistinguishable in their motions from those of the given
constrained mechanical system S.
The arbitrariness of the (nowhere-zero) function () t α and that of the
matrix ( ,) 0 G qt > ensures this gage invariance.
127
As already mentioned, one of the central problems in analytical dynamics is the
determination of equations of motion for constrained mechanical systems, in which the
most important feature is how to connect these equations of motion to real world
practices.
The perfect mathematical model, describing those exactly physical motions has yet to be
found, considering all real physical systems have their own uncertainties at various
levels. Some of these uncertainties are mainly derived from modeling errors, while others
originate in outer disturbances. In regards to the modeling error, the structures and
designed parameters of the mathematical model which describes a dynamical system
cannot be known perfectly. For the latter, many disturbances from external environments
are unpredictable and random.
The error, which has the greatest impact in describing mechanical systems and which is
most problematic and critical to arriving at the most accurate result, arises from the
modeling error. Consequently, in order to arrive at constrained equations of motion that
can best describe the actual physical practices, one must take into consideration the
uncertainties that may arise from two general sources: uncertainties in the knowledge of
the physical system and uncertainties in the ‘given’ forces applied to the system.
128
In the next chapter, we further explore and improve the fundamental equation so that its
results, in the presence of those two sources of uncertainties, give us the same dynamics
of the nonlinear constrained mechanical systems as those obtained when no uncertainty
pertaining to the system is present.
129
Chapter 5
METHODOLOGY FOR TRACKING CONTROL OF
NONLINEAR UNCERTAIN SYSTEMS
5.1 Introduction
All real-life physical systems are known only to within some bounds of uncertainty that
may depend on the various levels of their description. Controlling the motion of such
uncertain complex multi-body systems to follow prescribed control requirements has
become a topic of great interest during the past few years. The uncertainties that arise in
them underlie the general approach to modeling such systems, and they stem from two
main sources: uncertainties in our knowledge of the physical system, like uncertainties in
the stiffness and mass distribution, the nature of damping, etc.; and, uncertainties in our
knowledge of the forces acting on the system, like the non-uniform gravitational field of
the earth, solar wind, gravity gradients, etc., when considering precise satellite motion
control. The two sources of uncertainty are simultaneously considered in this chapter, and
in what follows, all these uncertainties are included in what we call the ‘real-life
mechanical system,’ or ‘actual system,’ whose description is known only imprecisely. It is
assumed that though the actual system is not known precisely, we have bounds on the
uncertainties involved. These uncertainties are thus assumed to be in general time varying
130
and unknown, but bounded. Our best assessment of a given actual system will be referred
to as the ‘nominal multi-body system,’ or the nominal system, for short. The description
(model) of the nominal system then naturally includes our best assessment of the
characteristics of the physical system and the nature of the ‘given’ forces acting on it.
The aim in this chapter is to develop a general control methodology for determining the
control, which when applied to an ‘actual system,’ causes this system to track a desired
reference trajectory of, and to satisfy the control requirements imposed on, the
corresponding ‘nominal system.’ The tracking control methodology is developed in a
two-step process. The first step uses the concept of the fundamental equation presented in
Section 2.2 to provide the closed-form control force needed to satisfy the control
requirements imposed on the nominal system model, where, as stated before, the nominal
model is the model adduced from our best assessment of the characteristics of the actual
multi-body system. Once the nominal system model is specified, no
linearizations/approximations are made in the description of the dynamics, and the
nonlinear controller that exactly satisfies the desired control requirements is obtained. In
the next step of the control methodology, this nonlinear controller is augmented by an
additional additive controller. This then provides a general approach to the control of
nonlinear uncertain systems, leading to a closed-form nonlinear controller that can
131
guarantee satisfaction of the prescribed control requirements under uncertainties—a
controlled actual system. This controlled actual system is a general formulation that is
applied by several tracking control laws which are discussed later in Chapter 7 and
Chapter 8 in detail.
The general approach that we shall follow is to view the tracking control problem in the
framework of constrained motion. We shall view the control requirements as constraints
on the nonlinear dynamical system, and obtain closed-form generalized control forces to
satisfy these requirements. In what follows we shall therefore use the interchangeable the
terms ‘requirements’ and ‘constraints,’ the terms ‘control forces’ and ‘constraint forces,’
and the terms ‘controlled system’ and ‘constrained system.’
5.2 The Description of Control Approach
Since descriptions of real-life complex multi-body systems are usually uncertain, an exact
model describes those mechanical system cannot be perfectly obtained. Before
developing a control approach that can deal with these uncertainties, it is better to show
how the response of the assumed nominal system can be altered due to the effect of the
uncertainty in the modeling process. Thus, we begin in this section by considering the
description of the actual real-life mechanical system. We do this by considering the
presence of modeling uncertainties from the two general sources as prescribed earlier
132
based on the dynamical descriptions of the so-called nominal system presented in Section
2.2.
5.2.1 System description of the actual systems
We begin by describing the so-called nominal system—our best assessment of the ‘actual
system,’ whose description is known only imprecisely. This nominal system is modeled
by starting with an unconstrained system to which constraints are then added through the
use of the fundamental equation [50] in order to model the physical multi-body system
appropriately [33] (similar to Section 2.2).
First, we describe the so-called unconstrained multi-body system in which the
coordinates are all taken to be independent of each other. We do this by considering an
unconstrained system whose motion at any time t can be described using Lagrange’s
equation, by
( , ) ( , , ), M qt q Q q qt =
(5.2.1)
with the initial conditions
00
( 0) , ( 0) , qt q qt q = = = = (5.2.2)
where q is the generalized coordinate n-vector, the n by n matrix 0 > M is the mass
matrix which is a function of q and t, and Q
is an n-vector, called the ‘given’ force, which
133
is a known function of q , q , and t. In (5.2.1), the components of the n-vector q are
assumed to be independent of one another.
Second, we impose a set of physical constraints on this unconstrained system. We
suppose that the unconstrained multi-body system is now subjected to the
p
m sufficiently
smooth physical constraints given by
( , , ) 0, 1,2,..., ,
pp
i
q qt i m ϕ = = (5.2.3)
where
pp
rm ≤
equations in the equation set (5.2.3) are functionally independent. The
constraints described by (5.2.3) include all the usual varieties of holonomic and/or
nonholonomic constraints, and then some. We shall assume that the initial conditions
(5.2.2) satisfy these
p
m physical constraints. Therefore, the components of the n-vectors
0
q and
0
q
cannot all independently be assigned.
Differentiating the constraints (5.2.3) with respect to time t, we obtain the relation
( , , ) ( , , ),
pp
A q qt q b q qt =
(5.2.4)
where
p
A
is an
p
m
by n matrix whose rank is
p
r , and
p
b is an
p
m -vector. We note that
each row of
p
A arises by appropriately differentiating one of the
p
m
physical constraint
equations in the set given in relation (5.2.3). In what follows, we shall often suppress the
arguments of the various quantities unless required for clarity.
134
Using relations (5.2.1) and (5.2.4) we obtain the equation of motion of the nominal multi-
body system by applying the fundamental equation to get [50] (see also (2.2.7))
1
( , ) ( ) ( ): .
TT
pp p p p
M q t q Q A A M A b A a Q
− +
=+ −=
(5.2.5)
The superscript “+” in (5.2.5) denotes the Moore-Penrose (MP) inverse of a matrix, and
the unconstrained acceleration is denoted as
1
: a MQ
−
=
. We refer to equation (5.2.5) as
describing the ‘nominal system,’ implying that the equation includes our best assessment
of the information we have regarding the system’s parameters and structure as well as the
nature of the ‘given’ force n-vector Q
acting on it. In short, this nominal system model
includes our best estimate of the actual physical system that we are trying to model.
We again note that equation (5.2.5) is valid (i) whether or not the equality constraints
(5.2.3) are holonomic or nonholonomic; (ii) whether or not they are nonlinear functions
of their arguments, and (iii) whether or not they are functionally dependent.
In order to subject our best estimate of the physical system—the nominal system
described by (5.2.5)—to a set of desired control requirements we again apply the
fundamental equation. We impose a set of
c
m sufficiently smooth control requirements as
constraints on it given by
135
( , , ) 0, 1,2,..., ,
c c
i
q qt i m ϕ = = (5.2.6)
where
cc
rm ≤
equations in the equation set (5.2.6) are functionally independent. We
shall assume that the initial conditions (5.2.2) satisfy the
c
m
control constraints. (If not,
the control constraints can be expressed in an alternative form to ensure that they are
asymptotically satisfied [39].) Thus one obtains on differentiation of (5.2.6) the relation
( , , ) ( , , ).
c c
A q qt q b q qt = (5.2.7)
In addition to satisfying the physical constraints, the controlled system must also satisfy
the control requirements. To ensure that the nominal system satisfies the physical
constraints as well as the control requirements, we augment the equation set (5.2.7) by
the set (5.2.4) to get the set of
pc
mm m = + relations (corresponding to (2.2.5))
( , , ) : : ( , , ),
p p
c c
Ab
A q qt q q b q qt
Ab
= = =
(5.2.8)
where A is now an m by n matrix whose rank is :
pc
rr r = + , and b is an m-vector.
Again using the fundamental equation we obtain the equation of motion of the controlled
nominal system as
1
( ,) ( , ,) ( , ,)( ) ( ( , ,) ( , ,)) : ,
TT c
M qt q Q q qt A q qt AM A b q qt Aa q qt Q Q
− +
=+ −= + (5.2.9)
136
where the acceleration
1
: a MQ
−
= .
We note from equation (5.2.9) that the control force
c
Q , which the nominal system is
subjected to, so that it satisfies the control requirements (5.2.6), can be explicitly
expressed as
1
( ) : ( ( ), ( ), ) ( ) ( ).
cc T T
Q t Q q t q t t A AM A b Aa
− +
= = − (5.2.10)
Pre-multiplying both sides of equation (5.2.9) with
1
M
−
, a constrained acceleration of the
nominal system can be expressed as
11 1
( ) ( ) : ( ),
TT c
q a M A AM A b Aa a M Q t
− −+ −
=+ −=+ (5.2.11)
which represents the acceleration of the controlled nominal system that guarantees that
the control requirements (5.2.6) are satisfied.
As mentioned before, there are always uncertainties in the description of any real-life
dynamical system. They arise from our lack of precise knowledge of the system, and/or
of the given forces acting on it. With the conceptualization of the nominal system given
in (5.2.5), these uncertainties are now assumed to be encapsulated in the elements of the n
by n matrix M and/or the n-vector Q
(see (5.2.1)).
137
We assume that the mass matrix of the actual real-life system, which we do not know
exactly, is :
a
M MM δ = + , where M is the n by n nominal mass matrix—our best
estimate of the mass matrix of the actual system—and M δ is the n by n matrix that
characterizes our uncertainty in the mass matrix M . The subscript ‘a’ denotes the actual,
real-life system about whose parameters we are uncertain. Similarly, the ‘given’ force n-
vector acting on the real-life system is taken to be :
a
Q QQ δ = +
, where the n-vector Q
denotes the nominal ‘given’ force on the nominal system (see (5.2.1)), and Q δ
denotes
the n-vector of uncertainty in Q
. The equation of motion of the actual system that is
subjected to the physical constraints given by (5.2.3) is then given, in a manner similar to
the way (5.2.5) was obtained, by the fundamental equation as
1
( , ) ( ) ( ) : ( , , ),
TT
a a p p a p p p a a
M qt q Q A A M A b A a Q q qt
−+
=+ −=
(5.2.12)
where q is the generalized coordinate n-vector of the actual system, and
1
:
a aa
a MQ
−
=
.
Equation (5.2.12) is then the description of the ‘actual system’ whose parameters are
known only imperfectly, since ( ,) M qt δ and ( , ,) Q q qt δ
are, in general, unknown.
Our aim is to control this ‘actual system’ so that it satisfies the control requirements
(5.2.7). With no exact knowledge of M δ and Q δ
, the only control force that we have at
hand to satisfy the constraints (5.2.7) is the one we have obtained for the nominal
system—our best estimate of the actual system. We then attempt to control the actual
138
system so that it satisfies the trajectory requirements given by the set (5.2.7), by using
this control force
c
Q obtained in (5.2.10). Thus, the equation of motion of the actual
system, so controlled, becomes
: ( , , ) ( ).
c
aa
M q Q q qt Q t = +
(5.2.13)
We note that equation (5.2.13) involves (i) the description of the actual system given by
(5.2.12) whose parameters are only known imperfectly and (ii) the control force ()
c
Qt
given by (5.2.10) which is obtained on the basis of our best estimate of this actual system,
namely on the basis of the corresponding nominal system.
However, the generalized control force ()
c
Qt given in (5.2.10) and used in (5.2.13) is
predicated on perfect knowledge of the system and the use of an accurate model—our
nominal system. By applying this control force to the actual system described by (5.2.12),
one obtains a different state ( q , q
) from that obtained for the controlled nominal system
(q , ) q . This causes an error in satisfying our desired control requirements (5.2.6), and
yields an error in our trajectory of the actual system. One way to see how the response of
the assumed nominal system can be altered due to the effect of the uncertainty in the
modeling process is to observe the response of the system when applying a correct
control force to the actual system (5.2.12). This leads to the description of the actual real-
life uncertain system
139
1
( , , ) ( ) ( ),
TT
a a aaa a a
M q Qq qt A A M A b A a
−+
=+− (5.2.14)
where
a
q is the generalized coordinate n-vector of the actual real-life uncertain system,
whereby uncertainties in the mass matrix and/or the ‘given’ force are presented in the
modeling process; the n by n matrix 0
a
M > is the actual mass matrix which is a function
of
a
q and t; the actual ‘given’ force
a
Q is an n-vector, which is a function of
a
q ,
a
q , and
t; and
1
:
a aa
a MQ
−
= .
We now refer to the system (5.2.14) as an ‘actual real-life uncertain system,’ implying
that it relates to (i) the description of the actual system and (ii) the correct control forces
that the actual system is subjected to because of the presence of the control requirements
(5.2.7). Pre-multiplying both sides of (5.2.14) with
1
a
M
−
, the constrained acceleration of
the actual real-life uncertain system can be expressed as
11
( ) ( ).
TT
aa a a a
q a MA A MA b A a
− −+
= +− (5.2.15)
5.2.2 The effect of uncertainties in mechanical systems
As stated earlier, there are always uncertainties in the description of all actual systems.
To show how the response of the nominal system (see Section 2.3.2) can be altered if
there are uncertainties in its description, we consider for the sake of simplicity only the
uncertainties in the masses
1
m ,
2
m , and
3
m of the same triple pendulum system
140
considered in Section 2.3. We assume that their actual values of these parameters differ
from our nominal (best-estimate) values by a random uncertainty of 10% ±
of the
nominal value chosen for each mass. For illustrative purposes, we choose
12 3
( , , ) (0.1, 0.2,0.3) mm m δδ δ = − (5.2.16)
and perform a simulation using equation (5.2.15), with all other parameter values the
same as those previously prescribed in Section 2.3. The response of the mass
3
m of the
nominal system shown in Figure 5.1(a) is changed to that shown in Figure 5.1(b). The
start and end of the trajectories are indicated by a circle and a square respectively, as
shown in the figure. Figure 5.2 also shows the differences in the trajectories in
12 3
, , and θθ θ of the nominal system (dashed line) and of the actual system (solid line).
As time increases, the angular responses of masses
2
m (
2
θ ) and
3
m (
3
θ ) of the actual
system are diverted from those of the nominal system.
141
Figure 5.1: The difference in trajectory responses of the mass
3
m in the XY-plane over a period of 10 secs. of (a) the
nominal system and (b) the actual system when the uncertainties in masses are prescribed as
1
0.1 kg m δ = ,
2
0.2 kg m δ = − , and
3
0.3 kg m δ =
142
Figure 5.2: Angular responses (no. of revolutions ( 360
)) of the masses
23
( ) and ( ) bm c m of the actual system
move away from those of the nominal system as time increases
143
We note that since the mass matrix ( , ) M qt of this specific example of the triple
pendulum (see equation (2.3.1)) is a function of the coordinate q and time t, the change in
the masses of the system causes the change in the coordinate at each instance of time.
And since the given forces ( , , ) Q q qt is also a function of the coordinate (and its
derivative), uncertainty in our knowledge of the masses will proliferate into the given
force acting on the system.
After seeing that the effect of uncertainties can cause change in the system’s behavior, we
now go further to explore the control approach that can deal with these uncertainties.
Since the control force ()
c
Qt given in (5.2.10) is predicated on perfect knowledge of the
system and on the use of an accurate model, this control force cannot satisfy the control
requirements and cannot maintain the motion of the nominal system when applying it to
the actual system (5.2.12) that assumed uncertainties involved.
The control force given by the second member on the right-hand side of (5.2.13), ()
c
Qt ,
hence needs to be modified to compensate for the fact that it has been calculated on the
basis of the nominal system and is now being applied to the actual unknown system. We
do this by adding another control force
u
Q
from a compensating controller, resulting in a
new state
c
q (see Figure 5.3). We now define the difference between
c
q and q
as a
144
tracking error signal e (see Figure 5.3). We note that the additional force
u
Q
from the
compensating controller depends on the state (, ) qq and the tracking error e. In this work,
we develop this additive controller based on a generalization of the notions of the sliding
surface control and the damping control, which are discussed later in Chapter 7 and
Chapter 8 respectively.
Figure 5.3: The block diagram of the controlled actual system
145
5.2.3 System description of the controlled actual systems
In order to ensure that the actual system, whose model we do not know exactly, track the
trajectory requirements of the nominal system, that is, the requirements of our best-
estimate system, the equation of motion of the controlled actual system becomes
( , ) ( , , ) ( )
cu
ac c ac c
M qt q Q q q t Q t Q = + + (5.2.17)
where
c
q is the generalized coordinate n-vector of the controlled actual system and
u
Q Mu = is the additional control force n-vector, which we shall develop in closed form.
We note that the n by n mass matrix of the actual system :0
a
M MM δ =+ > is a function
of
c
q and t, and the actual ‘given’ force vector of the controlled actual system
:
a
Q QQ δ = +
is a function of
c
q ,
c
q and t.
We now refer to the system (5.2.17) as a ‘controlled actual system,’ or ‘controlled
system,’ for short, implying that in additional to the control force ()
c
Qt given by (5.2.10)
and obtained on the basis of the corresponding nominal system, it is also subjected to the
additional control forces
u
Q . Pre-multiplying both sides of equation (5.2.17) by
1
a
M
−
, the
acceleration of the controlled system can be expressed as
11
() .
c
ca a a
q a M Q t M Mu
−−
=++ (5.2.18)
146
We note that
1
:
a aa
a MQ
−
= and
1
:
u
u M Q
−
= is the additional generalized acceleration
provided by the additional control forces
u
Q to compensate for uncertainties in our
knowledge of the actual system. The determination of this control force
u
Q is discussed
later in Chapter 7 and Chapter 8.
5.3 Uncertainties in the Dynamics of Mechanical Systems
As mentioned in the previous section, we define the tracking error signal (see Figure 5.3)
as
.
c
eq q = − (5.3.1)
Differentiating equation (5.3.1) twice with respect to time, we get
,
c
eq q = − (5.3.2)
which upon use of (5.2.11) and (5.2.18) yields
[ ]
11 1
11 1
1
( , , ) ( , ) ( ) ( , , ) ( , ) ( )
( , , ) ( , , ) ( , ) ( , ) ( )
: :.
cc
a cc a c a
c
a cc a c a
aa
e a q q tM q t Q tM Mu a q q tM q t Q t
a q q t a q qt M q t M qt Q t M Mu
q M Mu q u M u δδ
−− −
− − −
−
= + +− +
= −+ − +
= + = +−
(5.3.3)
In the above equation, we have defined
( )
1
1
:
a
M I IM M δ
−
−
=−+
(5.3.4)
and denoted the acceleration q δ as
147
[ ]
11
( , , , , ) ( , , ) ( , , ) ( , ) ( , ) ( ),
c
cc a cc a c
qqqq q t a q q t aqq t M q t M q t Q t δ
− −
= −+ −
(5.3.5)
where : ( ,) ( ,),
ac c
M Mq t Mq t δ = +
: ( , ,) ( , ,),
a cc cc
Q Qq q t Qq q t δ = + and
1
:
a aa
a MQ
−
= .
We note that in our control approach, we have the assumption that
1
1 MM δ
−
<< . Thus
by Taylor expansion,
a
M in equation (5.3.4) is approximated as
1
.
a
M MM δ
−
≈ (5.3.6)
Similarly, using Taylor expansion, equation (5.3.5) can be expanded as
1 1
,, 1
,, ,,
11
1
,
,
1
( , , , , ) ( , ) ( , , ) ( , ) ( , , )
( , ) () ()
( ) ( , , )
cc a a
nn
ai ai
a q q t cj j q q t cj j
jj
cj cj
n
a ik a
q t cj j a
j
cj
qqqq q t M q t Q qq t M q t Q qq t
QQ
M q t qq qq
qq
MQ
q q Q q qt
q
δ
− −
−
= =
−
=
= −
∂∂
+ −+ −
∂∂
∂∂
+ − +
∂
∑∑
∑
,,
,, ,,
11
1
, 11
,
1
() ()
( , ) ( ) ( , ) ( )
. . ., for 1,..., and 1,..., ,
nn
i ai
q q t cj j q q t cj j
jj
cj cj
n
a ik c
a q t cj j
j
cj
Q
qq qq
qq
M
M qt q q M qt Q t
q
H OT i n k n
= =
−
−−
=
∂
−+ −
∂∂
∂
+ + −−
∂
+= =
∑∑
∑
(5.3.7)
where H.O.T. denotes the higher order terms of ()
c
qq − and ()
c
qq − .
We note that in equation (5.3.7),
, ai
Q ,
cj
q , and
j
q denote the corresponding i-th and j-th
components of the n-vector
a
Q , of the n-vector
c
q , and of the n-vector q. Also
1
, a ik
M
−
represents the element at the i-th row and k-th column of the n by n matrix
1
a
M
−
.
148
The aim is to develop a controller u
such that the motion of the controlled actual system
closely tracks the motion of the nominal system, and thereby satisfies the control
requirements (5.2.6). We assume that the compensating controller u is capable of this
and causes the trajectory of the controlled actual system (, )
cc
qq to sufficiently
approximate that of the nominal system so that ( , ) (, )
cc
q q qq ≈ .
Under this assumption, q δ can be approximated as
1 1 11
( , , ) ( , ) ( , , ) ( , ) ( , , ) ( , ) ( , ) ( ).
c
aa a
q q qt M qt Q q qt M qt Q q qt M qt M qt Q t δ
− − −−
≈ − + −
(5.3.8)
Henderson and Searle [21] show that the inverse of a sum of matrices can be expressed as
1 1 1 11 1
( ) ( ) M M M M I MM MM δ δδ
− − − − − −
+ =−+ (5.3.9)
where I is an n by n identity matrix.
Expanding equation (5.3.8) and utilizing equation (5.3.9), we obtain the combined effect
of uncertainties M δ and Q δ to yield (see Appendix A)
11 1
(,,) () ( )() .
c
qqq t M M M M Q Q M M Q δ δδ δδ
−− −
≈− + + + + (5.3.10)
With the knowledge on the bounds of the uncertainties in the mass matrix, M δ , and the
given force, Q δ , and the assumption that
1
1 MM δ
−
<< , we can now obtain an
estimate of a suitable bound on q δ as
149
( ) ( )
1 11
1 ( ),
c
q M M M M QQ M Q t δ δ δδ
− − −
≤ + + + ≤Γ (5.3.11)
where () t Γ is an arbitrary positive function of time. This bound depends on our bounds
on M δ
and , Q δ which in turn depends on the state of our knowledge about the
actual system. Equation (5.3.11) gives the general form of q δ and is also applicable to
situations in which either M δ
or Q δ may be judged to be so negligibly small as to be
approximated by zero.
Remark 5.1: We note that there are two types of uncertainties involved in our control
approach. The first is the uncertainty
a
M denoted by equation (5.3.4) and the second is
q δ as denoted by equation (5.3.5). The estimates of both uncertainties (equations (5.3.6)
and (5.3.10)) are used later to design a compensating controller in Chapter 7 and Chapter
8. We further note that the approximation of equation (5.3.5) by (5.3.10) is verified using
a Monte Carlo simulation of a given set of random uncertainties as illustrated in Chapter
7. This shows that the estimate of q δ by using (5.3.10) is similar to q δ in equation
(5.3.5), thereby making it a valid choice for use in our control approach.
150
5.4 Summary
In this chapter, we have developed a new methodology for controlling motion of
nonlinear uncertain constrained multi-body systems. The development is done by using
the fundamental equation to provide the closed-form control force required to control the
nonlinear nominal system—the best-estimate system of the actual real-life situation—so
that the desired control requirements are met. No approximations/linearlizations is done
related to the nonlinear nature of the system. In the presence of uncertainties, the control
force obtained from the fundamental equation has been modified by augmenting an
additional force from a compensating controller. This controller is claimed to be able to
rescind the effects of unwanted uncertainties in the system modeling process and also to
allow the actual system’s trajectories to track those obtained from the desired nominal
system. This then leads to a new, simple, and general closed-form equation of motion that
can guarantee tracking of nominal system’s trajectories, which we refer to as ‘the
controlled actual system.’
In the next chapter, the controlled actual system is discussed in detail. Also, the
background idea in developing a compensating controller to use with the proposed
controlled actual system is introduced.
151
Chapter 6
GENERALIZED TRACKING CONTROLLERS FOR
NONLINEAR UNCERTAIN SYSTEMS
6.1 Introduction
In this chapter, we discuss in detail the formulation of the controlled actual system
developed in Chapter 5. The additional control force
u
Q and its generalized acceleration
are investigated. We also introduce an idea in developing a compensating controller to
work with the proposed controlled actual system. We show that our aim is to develop a
compensating controller that permits the use of a large class of control laws that can be
adapted to the specific real-life practical limitations of a given, particular controller being
used.
6.2 Closed-Form Controlled Actual Systems
Consider the controlled actual system as proposed in equation (5.2.17)
( , ) ( , , ) ( ) .
cu
ac c ac c
M qt q Q q qt Q t Q = + + (6.2.1)
The system (6.2.1) consists of (i) the n by n mass matrix of the actual system
: ( , ) ( , ) 0
ac c
M Mq t Mq t δ = +> , and the n-vector ‘given’ force of the actual system
: ( , , ) ( , , )
a cc cc
Q Qq q t Qq q t δ = + , which are uncertain; (ii) the control force ()
c
Qt given by
152
(5.2.10) and obtained on the basis of the corresponding nominal system; (iii) the
additional control forces
u
Q , which we shall develop in closed form and which depends
on the estimates of the uncertainties in
a
M and q δ as discussed by (5.3.6) and (5.3.10)
respectively. The aim is to find the control force :
T c u
Q QQ = + , which when applied to
the ‘actual system,’ namely ( , ) ( , , )
ac c ac c
M qt q Q q q t = , causes the actual system to mimic
the motion of the corresponding ‘nominal system,’ and thereby satisfies the control
requirements imposed on the nominal system.
We claim that we can find the control force
T
Q that is capable of this. Thus, our proposed
closed-form controlled actual system can guarantee satisfaction of control requirements
by using the control force
c
Q . It can also guarantee tracking the nominal system’s
trajectories by applying the additional control force
u
Q . Both control forces are
simultaneously applied to the actual system so that the controlled actual system is able to
satisfy the control requirements imposed on the corresponding nominal system as well as
mimic the motion of the nominal system at the same time.
Pre-multiplying both sides of equation (5.2.17) by
1
a
M
−
, the acceleration of the controlled
actual system can be expressed as
11
() .
c u
ca a a
q a MQ t MQ
−−
=++ (6.2.2)
153
Since in this work we consider the uncertainty in the mass matrix, which is the most
problematic parameter that proliferates throughout the constrained acceleration, we see
that the generalized acceleration due to the additional control force
u
Q contains the
actual mass matrix
a
M which is uncertain. However, as is shown later by the proof of
Lyapunov stability in Chapter 7 and Chapter 8, our proposed controlled actual system can
also take care of the uncertainty involved in the additional controller
1 u
a
MQ
−
itself.
We further note that a compensating controller u (see (5.2.18)) is given in the form of
the acceleration. However, we cannot apply the acceleration to the system. We instead
have to apply the compensating controller in the form of the force to the mass of the
system. Since the mass of the system is uncertain, the only mass that we can apply the
additional control force to is from the nominal mass M . Thus the additional control force
is of the form
u
Q Mu = (see also Figure 5.3).
6.3 Motivation in Developing Compensating Controllers
Many previous studies have considerd a specific compensating controller, which usually
uses a discontinuous control function such as a signum function (see Figure 6.1) or its
continuous-approximation function such as a saturation function (see Figure 6.1).
However, a controller containing these kinds of control functions is considered to be a
high-gain controller, which is sometimes not practical in real-life situations. In real-life
154
practical limitation, a physical controller might not be able to generate a high control
force right after it starts. Thus, it is useful to also consider a general formulation of a
compensating controller that can be used with a control function that can meet practical
real-life constraints. For example, the use of a cubic function (see Figure 6.1) in a
compensating controller obviates the need for a high-gain controller. Thus, this leads to
the idea of developing a set of closed-form controllers that permits the use of a large class
of control laws that can be adapted to meet practical limitations of the specific controller
being used. This idea is explained incorporating a generalization of the concept of several
control schemes as shown later in Chapter 7 and Chapter 8.
Figure 6.1: The control functions
155
6.4 Summary
In this chapter, the advantages of the proposed closed-form controlled actual system are
illustrated. The key idea in developing a compensating controller to work with the
controlled actual system is presented. The main contributions of this chapter are:
(i) The formulation of the proposed closed-form controlled actual system can
simultaneously guarantee satisfaction of control requirements and tracking of
the nominal system’s trajectories under uncertainties.
(ii) Uncertainties involved in the compensating controller are also taken care of
by the use of the proposed controlled actual system.
(iii) The control approach is developed in order to obtain a set of closed-form
controllers that permits the use of a large class of control laws that can be
adapted to the specific real-life practical limitations of a given, particular
controller being used.
The formulation of the controlled actual system is thus used as a base-line equation in
order to develop in the next few chapters an additional additive controller to compensate
for uncertainties.
156
In the next chapter, a compensating controller based on a generalization of the concept of
the sliding surface control is developed. This controller is able to guarantee tracking of
the nominal system’s trajectories within desired error bounds. It also provides for fast
response, good transient behavior and robustness with respect to the system uncertainties.
157
Chapter 7
TRACKING CONTROLLERS BASED ON THE
CONCEPT OF THE GENERALIZED SLIDING
SURFACE CONTROL
7.1 Introduction
This chapter presents a new approach to obtaining a tracking controller for nonlinear
uncertain multi-body mechanical systems. As mentioned earlier, descriptions of real-life
complex multi-body systems are usually uncertain, with the uncertainties arising from
two general sources: uncertainties in the knowledge of the physical system and
uncertainties in the ‘given’ forces applied to the system. Both categories of uncertainties,
which we assume to be time-varying and unknown yet bounded, are considered in this
chapter. The uncertainties being unknown, what is available in hand is therefore just the
so-called ‘nominal system,’ which is simply our best assessment and description of the
actual real-life situation. The aim in this chapter is to develop a set of closed-form
controllers, which when applied to a real-life uncertain system, causes this system to
track a desired reference trajectory that is pre-specified for the nominal system to follow.
In short, the real-life system’s motion is required to be coincident with, and mimics, the
motion desired of the nominal system.
158
In this chapter, we ensure that the unknown, uncertain system tracks the reference
trajectory of the nominal system by using an additional additive controller—based on a
generalization of the concept of the sliding surface control—to the proposed closed-form
controlled actual system proposed in the previous chapter. This then provides a general
approach to the control of nonlinear uncertain systems that can guarantee tracking
nominal system’s trajectories. While based on the idea of the generalized sliding surface,
the approach permits the use of a large class of control laws that can be adapted to the
specific real-life practical limitations of a given, particular controller being used. Thus, in
sum, a methodology is developed to obtain a set of closed-form nonlinear controller that
can guarantee tracking the desired reference trajectories under uncertainties within
desired error bounds. The same example of a triple pendulum as illustratated in Section
2.3 is again used to demonstrate the efficacy of the tracking control, regardless of the
exact knowledge of the real-life system. Numerical results illustrate the effectiveness of
the approach.
7.2 Generalized Sliding Surface Controllers (
SS
G )
Having obtained the bound Γ described in Section 5.3 we now develop a compensating
controller u using a generalization of underlying concept of the sliding surface control.
By using the generalized concept of the sliding surface [11], [24], [38], [52]-[53] (a lower
dimensional surface along which the dynamical system moves) we present a general form
159
for the additional controller needed to mitigate the effect of the uncertainties in our
knowledge of the system. The formulation of this controller permits the use of a large
class of control laws that can be adapted to the practical limitations of the specific
controller being used and the extent we are desire compensating for the uncertainties. The
controller can guarantee tracking the nominal system’s trajectories in the presence of
uncertainties within desired error bounds.
We begin by showing the key structure of the tracking system that will be used many
times from here on in developing our control methodologies.
Definition 7.1: Defining the tracking error as
c
eq q = − (7.2.1)
and differentiating equation (7.2.1) twice with respect to time, we get (see (5.3.3))
1
,
c a a
e q q q M Mu q u M u δδ
−
= − = + = +− (7.2.2)
where
( )
1
1
:
a
M I IM M δ
−
−
=−+ has been used in the last equality.
We next write equation (7.2.2) in the state space form, by defining
1 2 1
and , ee e e = =
(7.2.3)
160
as
( )
11
22
00
.
00
δ
= + +−
a
ee I
q u Mu
ee I
(7.2.4)
We note that this form of the tracking error is useful for the generalzed damping control
design discussed in Chapter 8. We further note that equation (7.2.4) satisfies the matching
condition [9], i.e., the system (7.2.4) is controllable.
We define a sliding surface
() () (), s t Ket et = + (7.2.5)
where
12
: ( , ,..., )
n
K diag k k k = , 0
i
k > for all 1,..., in = are arbitrary small positive
numbers and s is an n-vector. Our aim is to maneuver the system to the sliding surface
s
ε
∈Ω , whereupon by (7.2.5) ideally speaking, when the size of the surface
ε
Ω is zero,
we obtain the relation e Ke = − , whose solution
0
( ) exp( ) e t e Kt = − shows that the
tracking error () et exponentially reduces to zero along this lower dimensional surface in
phase space.
Differentiating equation (7.2.5) with respect to time and using equation (7.2.2), we get
.
a
s Ke e Ke q u M u δ = + = + +− (7.2.6)
161
Since ()
c
qq −
can be measured, to cancel the known term ()
c
Ke K q q = − in equation
(7.2.6), we choose the controller u to be of the form
() (),
SS
u Ke t G t = −+
(7.2.7)
where
SS
G is a generalized sliding surface controller (discussed later), so that
( ).
SS a SS
s G q M Ke G δ = + − −+
(7.2.8)
We note that () qt δ ≤Γ . Here, we have used the bound () t Γ that is related to the
uncertainties involved in the actual system and that is obtained from equations (5.3.10)
and (5.3.11). In what follows we shall denote ⋅ to mean the infinity norm.
We further note that there are two types of uncertainties involved in our control
approach—the uncertainty in
a
M (see (5.3.6)) and the uncertainty in q δ (see (5.3.11)).
The estimates of both uncertainties are used in developing a compensating controller as
shown later in this chapter.
We shall now show that under the following assumption, the system can indeed be
maneuvered to the sliding surface s
ε
∈Ω when
ε
Ω is defined as an appropriate small
surface around 0 s =
(as discussed later in this section).
162
Assumption 7.1: In order for the controller () ut given by (7.2.7) to cause () st to reach
the sliding surface s
ε
∈Ω and remain inside this surface we shall assume that there is a
function () t β such that
0
0
()
( ) > 0,
t
t
β
β
α
′ Γ+
≥
(7.2.9)
with , ′ Γ >Γ
0 a
KM e β > is any arbitrary positive constant, and
( )
0
01 ,
a
M α σ < <− where the positive constant σ is chosen such that
, σγ ≥
(7.2.10)
where (see Appendix B)
()
: 1.
()
T
s fs
s fs
γ = ≤ (7.2.11)
We note that since 1 γ ≤ , the choice 1 σ = would suffice in (7.2.9). The function () fs
will be defined shortly.
Definition 7.2: We now define a generalized sliding surface control n-vector ()
SS
Gt as
() : () ( ).
SS
G t tfs σβ = −
(7.2.12)
The i-th component, ()
i
f s , of the n-vector () fs is defined as
( ) ( ) / , 1, . . .,
i i
f s g s i n ε = =
(7.2.13)
163
where
i
s is the i-th component of the n-vector s, ε is defined as any small positive
number and the function ( ) /
i
gs ε is any arbitrary monotonic increasing odd continuous
function of
i
s on the interval (, ) −∞ +∞ that satisfies
0
( ) ( / ) , if is outside the surface ,
a
ii i
KM e
f s gs s
ε
ε
β
′ Γ+
= ≥Ω
′ Γ+
(7.2.14)
where
ε
Ω
is defined as the surface of the n-dimensional cube around the point 0 s =
each of whose sides has length
( )
1
0
2 /.
a
L g KM
ε
εβ
−
′ ≈ Γ+ Γ +
We note from
(7.2.14) that since
0 a
KM e β > ,
0
/ 1
a
KM e β ′′ Γ+ Γ+ < and the choice
() 1 fs ≥ would suffice in (7.2.14).
Corollary 7.1: The controller (7.2.12) becomes the conventional sliding mode controller
when the control function ( ) sgn( ) fs s = .
Result 7.1: Under the above-mentioned Assumption 7.1, the control law
() () [ () () ( )]
SS
u Ke t G t Ke t t f s σβ = −+ = − +
(7.2.15)
with
12
: ( , ,..., )
n
K diag k k k = , 0
i
k > for all 1,..., in = and ()
SS
Gt defined in (7.2.12) to
(7.2.14) causes () st
ε
→ Ω .
Proof: Consider the Lyapunov function
164
1
.
2
T
V ss = (7.2.16)
Differentiating equation (7.2.16) once with respect to time, we get
.
T
V ss =
(7.2.17)
Substituting equation (7.2.8) in equation (7.2.17), we have
( ).
T TT
SS a SS
V s G s q s M Ke G δ = + − −+
(7.2.18)
Then using equation (7.2.12) in equation (7.2.18), we obtain
() (),
T TT T
aa
V s f s s q sM Ke sM f s σβ δ σβ = − + + +
(7.2.19)
so that
() () .
TT T T
aa
V s fs s q s M K e s M fs σβ δ σβ ≤− + + +
(7.2.20)
Then using relation () qt δ ≤Γ and noting
T
ss = , we obtain
() () .
T
aa
V s fs s s M K e s M fs σβ σβ ≤ − + Γ+ +
(7.2.21)
Since (see (7.2.10) and (7.2.11))
() (),
T
s fs s fs σ ≥ (7.2.22)
relation (7.2.21) becomes
( )
( )
() ()
1 ( ) .
aa
aa
V s fs M fs K M e
s M fs K M e
β σβ
βσ
≤ − − −Γ −
= − − −Γ −
(7.2.23)
165
Since
0
0
+ β
β
α
′ Γ
≥ and
( )
0
01
a
M α σ < <− , we then have
( )
0
() .
a
V s fs K M e β
′ ≤ − Γ + −Γ −
(7.2.24)
Since
0
()
a
KM e
fs
β
′ Γ+
≥
′ Γ+
outside the surface
ε
Ω (see (7.2.14)), we have
( ). Vs ′ ≤ − Γ −Γ
(7.2.25)
Since ′ Γ >Γ, the derivative V
is non-positive and we have attractivity to the region
enclosed by the surface s
ε
Ω .
Noting (7.2.24),
ε
Ω
can be defined as the surface of the n-dimensional cube around the
point 0 s = each of whose sides has length
1
0
2 .
a
KM e
Lg
ε
ε
β
−
Γ+
=
′ Γ+
(7.2.26)
And since we expect () 1 et << , we thus have
1
0
2 .
a
KM
Lg
ε
ε
β
−
Γ+
≈
′ Γ+
(7.2.27)
Noting (7.2.5) and the fact that () st is bounded by / 2 L
ε
, we have the error bound
()
2
L
et
K
ε
<≈ and () et L
ε
<≈ , as t → ∞ .
(7.2.28)
166
Main Result ()
SS
G : The closed-from generalized sliding surface controller (
SS
G ) for the
uncertain system is given by
( ) () () ( ) ,
cc
ac a a
M q Q Qt Mu Q Q t M Ke f s σβ =+ + =+ − +
(7.2.29)
where the control force ()
c
Qt is given by (5.2.10)
1
() ( ) ( )
c T T
Q t A AM A b Aa
−+
= − (7.2.30)
and is obtained on the basis of the nominal system;
12
: ( , ,..., )
n
K diag k k k = , 0
i
k > for all
1,..., in = are arbitrary small positive numbers; the choices of 1 σ =
and () 1 fs ≥ ,
where () fs is any arbitrary monotonic increasing odd continuous function of s on the
interval (, ) −∞ +∞ , would suffice in (7.2.29);
0
0
()
() ,
t
t
β
β
α
′ Γ+
=
(7.2.31)
where ′ Γ >Γ is chosen based on the estimate of q δ from equation (5.3.11),
( ) ( )
1 11
1 ( ),
c
q M M M M QQ M Q t δ δ δδ
− − −
≤ + + + ≤Γ (7.2.32)
where the bound Γ depends on our bounds on M δ
and , Q δ which in turn depends
on the state of our knowledge of the actual system;
0
α is a small positive number that is
defined by using the estimate of
a
M from equation (5.3.6) , which yields
1
1,
a
M MM δ
−
≈ <<
(7.2.33)
so that
167
( )
1
0
1 1 1;
a
M MM α σ δ
−
< − ≈− ≈ (7.2.34)
and again by using the estimate of
a
M (7.2.33) and the expectation that e is small,
0
β
is chosen such that
0
. K β =
(7.2.35)
We note that the uncertainties q δ and
a
M are not known but we have the estimates of
these uncertainties. We further note that the responses of the controlled actual system
(7.2.29) are not sensitive to the estimate of the bound on the uncertainty q δ (7.2.32) as is
shown later in the next section.
7.3 Numerical Results and Simulations of the Generalized Sliding
Surface Control
In this section, we illustrate the control methodology by considering the same example of
the triple pendulum considered in Section 2.3. It is straightforward to extend this example
of application to more general situations.
Under the presence of the uncertainties, in order to control the actual system’s responses
to track those obtained from the nominal system (the best assessment of an actual
system), we would have to use equation (7.2.29) (or (5.2.18)) that has the additional
controller to compensate for uncertainties.
168
We next select the parameters for the controller u given by equation (7.2.15). We choose
2
2
( / ) , where 0
()
( / ) , where 0
c i i
i
c i i
ss
fs
ss
αε
αε
−<
=
≥
(7.3.1)
where ,0
c
αε >
and ε is a suitable small number, we obtain in closed-form the
additional controller needed to compensate for uncertainties in the actual system as
0
0
()
( ) ( ).
ii i
t
u t Ke f s
β
σ
α
′ Γ+
= −−
(7.3.2)
We note that with the choice of ()
i
f s in (7.3.1), the region outside the surface
ε
Ω is the
region outside of the n-dimensional cube around s = 0 whose sides each have length
1/2
0
2 ( )/ ( )
a c
L KM
ε
ε αβ
′ ≈ Γ+ Γ +
(see (7.2.27)). In this region equation (7.2.25)
assures us that the control given by equation (7.3.2) will cause () st
to strictly decrease,
until it reaches the boundary s
ε
∈Ω and remains inside this n-box thereafter.
Thus, using the additional controller, equation (7.3.2) in equation (5.2.18) (or (7.2.29)),
we obtain the closed-form equation of motion of the controlled uncertain constrained
mechanical system
11 0
0
()
() ( )
c
ca a a
t
q a M Q t M M Ke f s
β
σ
α
−−
′ Γ+
=+− +
(7.3.3)
that will cause the actual system to track the trajectory of the nominal system, thereby
compensating for the uncertainty.
169
Equation (7.3.3) might strike some alarm in the reader’s mind, since it contains the
uncertain mass matrix
a
M , which is known imprecisely. However, we note that we use
this equation to verify our control approach with random sampling values of the uncertain
parameters in the system. In our control approach, we actually develop the control force
:
T cu c
Q Q Q Q Mu = + = + that is obtained on the basis of the nominal system and the
estimates of the uncertainties in the system ( q δ and
a
M ), which can be obtained
implicitly (see also Section 6.2). We use equation (7.3.3) to show that when the control
force
T
Q is applied to the actual system, the motion of the actual system mimics the
motion of the nominal system as can be seen from the comparison between the use of
equations (5.2.11) and (7.3.3). This idea is also used for the next two controllers that are
developed in Chapter 8.
We note that the various quantities in (7.3.3) have been defined in Sections 5.2.3 and 7.2.
Again, the numerical integration throughout this chapter is done in the Matlab
environment, using a variable time step integrator with a relative error tolerance of
8
10
and an absolute error tolerance of
12
10
.
170
To perform a simulation, we choose control parameters 10, for 1,..., ,
i
k in = =
0
10, β =
0
0.5, 1, 2,
c
α σα = = = and
2
10 ε
−
=
in equation (7.3.3), with all other parameter values
the same as those previously prescribed in Section 2.3.
As mentioned earlier, while we have no knowledge of the actual parameters, in order to
affect a compensating controller a suitable bound on the uncertainty in q δ is required.
Thus, to estimate Γ , which is the bound on q δ (see equation (5.3.5)) in the presence of
the 10 ± percent uncertainties in each of the masses
1
m ,
2
m , and
3
m as described in
Section 5.2.2, we perform the Monte Carlo simulation of 1014 samplings of the
uncertainty in each mass
1
, m δ
2
m δ , and
3
m δ using equation (5.3.5). The location of the
actual masses of each sampling data can be expressed in Figure 7.1 and the probability
density function of Γ occurring from the 1014 samplings is shown in Figure 7.2.
We note that we also consider a case of 10028 samplings of 10% ± uncertainties in
masses. This case also shows a similar result of probability density function of Γ . Thus
for ease of visualization we represent the case of 1014 Monte Carlo samplings as shown
in Figure 7.1 and Figure 7.2.
171
Figure 7.1: Location of the actual masses , 1,2,3
ii
m mi δ± = of the 1014 Monte Carlo simulation
Figure 7.2: Probability density function of Γ occurring from the 1014 Monte Carlo simulation of 10 ± percent
uncertainties in masses of the triple pendulum when using the estimate of q δ from equation (5.3.5)
172
As mentioned earlier in Remark 5.1, in order to verify equation (5.3.10), we again
perform the Monte Carlo simulation of 1014 samplings of the 10 ± percent uncertainties
in each mass
1
, m δ
2
m δ , and
3
m δ by using equations (5.3.10) and (5.3.11). The location
of the actual masses of each sampling data can be expressed in Figure 7.3 and the
probability density function of Γ occurring from the 1014 samplings is shown in Figure
7.4. We see that the approximation of q δ from equation (5.3.5) by using equation
(5.3.10) is valid. As we can see from Figure 7.2 and Figure 7.4, the probability density
functions of Γ from both figures are approximately the same.
Figure 7.3: Location of the actual masses , 1,2,3
ii
m mi δ± = of the 1014 Monte Carlo simulation
173
Figure 7.4: Probability density function of Γ occurring from the 1014 Monte Carlo simulation of 10 ± percent
uncertainties in masses of the triple pendulum when using the estimate of q δ from equation (5.3.10)
We further note that even for a larger uncertainty, the use of equation (5.3.10) to
approximate equation (5.3.5) is still valid. This can be shown by the Monte Carlo
simulation of 1014 samplings of the 50 ± percent uncertainties in each of the masses
1
, m δ
2
, m δ and
3
m δ of the same triple pendulum. The locations of the actual masses of
each sampling data can be expressed in Figure 7.5 and Figure 7.7 for the simulations
using (5.3.5) and (5.3.10) respectively. Similarly, the probability density functions of Γ
are correspondingly shown in Figure 7.6 and Figure 7.8 when using equations (5.3.5) and
(5.3.10). Both figures show approximately the same probability density functions of Γ .
174
Figure 7.5: Location of the actual masses , 1,2,3
ii
m mi δ± = of the 1014 Monte Carlo simulation
Figure 7.6: Probability density function of Γ occurring from the 1014 Monte Carlo simulation of percent
uncertainties in masses of the triple pendulum when using the estimate of q δ from equation (5.3.5)
50 ±
175
Figure 7.7: Location of the actual masses , 1,2,3
ii
m mi δ± = of the 1014 Monte Carlo simulation
Figure 7.8: Probability density function of Γ occurring from the 1014 Monte Carlo simulation of percent
uncertainties in masses of the triple pendulum when using the estimate of q δ from equation (5.3.10)
50 ±
176
Remark 7.1: The estimate of Γ is not sensitive in our control approach. The tracking
errors ( e and e ) and the additional control forces (
u
Q ) are not much different when we
miss-estimate the size of the bound on the uncertainty Γ. This is shown later in the
simulation results.
For the 10 ± percent uncertainties, we choose 100 ′ Γ = as shown in Figure 7.4 (red line)
for the numerical simulation in equation (7.3.3). This figure confirms that the chosen
value 100 ′ Γ = in the simulation satisfies equation (5.3.11). And thus the additional
controller u (7.3.2) is able to guarantee tracking of the desired nominal system’s
trajectories within desired error bounds as we will see later on in Figure 7.11 and Figure
7.12.
We next perform a simulation by using 100, ′ Γ = 10, for 1,..., ,
i
k in = =
0
10, β =
0
0.5, α =
1, σ = 2,
c
α = and
2
10 ε
−
=
in equation (7.3.3), with all other parameter values the same
as those previously prescribed in Section 2.3. The constrained trajectories of the mass
3
m
in the XY-plane of the nominal, controlled, and actual systems are illustrated in Figure
7.9. In this figure, we again assume that each trajectory starts at a circle and stops at a
square. We see that the controlled system (Figure 7.9(b)) tracks the nominal system
(Figure 7.9(a)) while the actual system (Figure 7.9(c)) deviates from the desired nominal
177
system. Figure 7.10 gives an alternative view of the trajectory responses. In this figure,
the trajectories of
12 3
, , and θθ θ of the nominal (solid red line), the actual (solid blue
line) and the controlled (dotted line) systems are shown. As time increases, the actual
system’s response differs from those of the nominal and the controlled systems while the
controlled system tracks the nominal system very well. Figure 7.11 and Figure 7.12
correspondingly show the tracking error responses in displacement (
c
θ θ − ) and in
velocity (
c
θ θ −
) between the nominal system (5.2.11) and the controlled actual system
(7.3.3). Both figures show that the tracking errors are small. The errors are seen to be of
O(
4
10
−
) for the displacement and of O(
3
10
−
) for the velocity. We see that these errors are
within the error norms
4
( ) / 2 2 10 et L K
ε
−
<≈ ≈ × and
3
( ) 4 10 et L
ε
−
<≈ ≈ ×
as
prescribed by (7.2.27) and (7.2.28). We note that use of the specified quadratic function
eliminates chattering.
178
Figure 7.9:
SS
G - Trajectory responses (in meters) of the mass
3
m
over a period of 10 secs. of (a) the nominal system
and (b) the controlled system are approximately the same while (c) the actual system yields a totally different trajectory
when the uncertainties in masses are prescribed as
1
0.1 kg m δ = ,
2
0.2 kg m δ = − , and
3
0.3 kg m δ = ( 10% ± of
nominal masses) and the uncertainty’s bound in equation (7.3.3) is chosen to be 100 ′ Γ=
179
Figure 7.10
SS
G - Angular responses (no. of revolutions ( 360
)) of the masses
23
( ) and ( ) bm c m of the actual
system move away from those of the nominal system as time increases while those of the controlled system track the
nominal system very well
180
Figure 7.11:
SS
G – Tracking errors ) (
c
θθ − in displacement of the masses
12 3
( ) , ( ) , and ( ) a m bm c m between the
nominal and the controlled systems (in radians)
181
Figure 7.12:
SS
G - Tracking errors ) (
c
θθ −
in velocity of the masses
12 3
( ) , ( ) , and ( ) a m bm c m between the
nominal and the controlled systems (in rad/s)
182
The obtained control forces are shown in Figure 7.13. We note that for ease of
visualization and comparison we represent the magnitude of each control force—the
control forces applied to the nominal system to satisfy the control requirement
c
Q
(Figure 7.13(a)), the control forces applied by the additional compensating controller to
compensate for our inexact knowledge of the actual system
u
Q (Figure 7.13(b)), and the
total control forces applied to the actual system
T cu
Q QQ = + (Figure 7.13(c)). We see
that the magnitude of additional control forces to compensate for uncertainties of the
generalized sliding surface controller (Figure 7.13(b)), which are obtained from
:
u
Q Mu = , where u is defined as in (7.3.2), is small when compared with the magnitude
of the control forces applied to the nominal system
c
Q given in (5.2.10) to satisfy the
energy control,
23
() () () Et E t E t = + , (Figure 7.13(a)).
The guaranteed errors within the desired error bounds in tracking both in displacement
and velocity from Figure 7.11-Figure 7.12 and the small additional control forces to
compensate for uncertainties from Figure 7.13 guarantee that the proposed generalized
sliding surface control design is robust with respect to the uncertainties in the masses of
the triple pendulum.
183
Figure 7.13:
SS
G - (b) Additional force to compensate for uncertainties in masses,
u
Q , is small when compared to (a)
the control force on the nominal system to satisfy the energy control,
c
Q , where (c) shows the sum of
cu
QQ + (in
Newtons)
184
However, in order to further verify that the proposed generalized sliding surface control
approach is valid and works well in the environment of the uncertainties, as a second
example, we consider the same triple pendulum system given in Section 5.2.2 except that
(i) we try a different set of uncertainties from the location of the actual masses as
expressed in Figure 7.3, so that we choose
12 3
0.1, 0.05, and 0.2 mm m δδ δ = −= = − ; and
(ii) we use a different monotonic increasing odd continuous function of s , for which we
choose a cubic function of the form
3
( ) ( / ),
i c i
fs s αε = (7.3.4)
where ,0
c
αε >
and ε is a suitable small number. We now obtain in closed-form, instead
of (7.3.2), the additional controller needed to compensate for uncertainties in the actual
system as
3 0
0
()
() ( / ) .
i i c i
t
u t Ke s
β
σ αε
α
′ Γ+
= −−
(7.3.5)
With this choice of
3
() ( / )
i c i
fs s αε = , the region outside the surface
ε
Ω is the region
outside of the n-dimensional cube around s = 0 whose sides each have length
1/3
0
2 ( )/ ( )
a c
L KM
ε
ε αβ
′ ≈ Γ+ Γ +
(see equation (7.2.27)). We thus use equation
(7.3.5) in equation (5.2.18) (or (7.2.29)), so that we obtain the closed-form equation of
motion of the controlled uncertain mechanical system to be of the form
185
11 3 0
0
()
() ( / ) .
c
ca a a c
t
q a M Q t M M Ke s
β
σ αε
α
−−
′ Γ+
=+− +
(7.3.6)
We then perform a simulation again using equation (7.3.6), with all other parameter
values the same as those prescribed in the previous simulation. The constrained
trajectories of the mass
3
m in the XY-plane of the nominal, the controlled and the actual
systems are illustrated in Figure 7.14(a), (b), and (c) respectively. In this figure, we again
assume that each trajectory starts at a circle and stops at a square. We see that the
controlled system tracks the nominal system while the actual system deviates from the
desired nominal system. Figure 7.15 shows the trajectories of
12 3
, , and θθ θ of the
nominal (solid red line), the actual (solid blue line) and the controlled (dotted line)
systems. As time increases, the actual system’s response differs from those of the
nominal and the controlled systems while the controlled system tracks the nominal
system very well. Figure 7.16 and Figure 7.17 correspondingly show the tracking error
responses in displacement (
c
θ θ − ) and in velocity (
c
θ θ −
) between the nominal system
(5.2.11) and the controlled actual system (7.3.6). Both figures show that the tracking
errors are small (of Q(
4
10
−
) for the displacement and of Q(
3
10
−
) for the velocity). Again
these errors are within the tracking error norms
4
( ) / 2 2.8 10 et L K
ε
−
<≈ ≈ ×
and
3
( ) 5.6 10 et L
ε
−
<≈ ≈ ×
as prescribed by (7.2.27) and (7.2.28) and again there is no
chattering in the tracking results.
186
Figure 7.14:
SS
G - Trajectory responses (in meters) of the mass
3
m
over a period of 10 secs. of (a) the nominal system
and (b) the controlled system are approximately the same while (c) the actual system yields a totally different trajectory
when the uncertainties in masses are prescribed as
1
0.1 kg m δ = − ,
2
0.05 kg m δ = , and
3
0.2 kg m δ = − and the
uncertainty’s bound in equation (7.3.6) is chosen to be 100 ′ Γ=
187
Figure 7.15
SS
G - Angular responses (no. of revolutions ( 360
)) of the masses
23
( ) and ( ) bm c m of the actual
system move away from those of the nominal system as time increases while those of the controlled system track the
nominal system very well
188
Figure 7.16:
SS
G - Tracking errors ) (
c
θθ − in displacement of the masses
12 3
( ) , ( ) , and ( ) a m bm c m between the
nominal and the controlled systems (in radians)
189
Figure 7.17:
SS
G - Tracking errors ) (
c
θθ −
in velocity of the masses
12 3
( ) , ( ) , and ( ) a m bm c m between the
nominal and the controlled systems (in rad/s)
190
The obtained control forces are shown in Figure 7.18. In Figure 7.18(b), the magnitude of
additional control forces applied by the generalized sliding surface controller, :
u
Q Mu = ,
where u is defined as in (7.3.5), is seen to be very small relative to the magnitude of the
control forces on the nominal system
c
Q (see (5.2.10)) to satisfy the energy control,
23
() () () Et E t E t = + , (Figure 7.18(a)).
Again the numerical results shown in Figure 7.16-Figure 7.17 and in Figure 7.18
guarantee that the proposed generalized sliding surface control design works very well in
controlling the motions of the actual system to mimic those of the nominal system of the
triple pendulum, thereby satisfying the requirements of the energy control.
191
Figure 7.18:
SS
G - (b) Additional force to compensate for uncertainties in masses,
u
Q , is small when compared to (a)
the control force on the nominal system to satisfy the energy control,
c
Q , where (c) shows the sum of
cu
QQ + (in
Newtons)
192
As mentioned earlier in Remark 7.1, we next consider the previous example to show that
even if we miss-estimate the bound ′ Γ , for example, 5 times its original value of a 100,
i.e., 500 ′ Γ = , the magnitude of the additional control forces
u
Q that is obtained by using
the generalized sliding surface controller (7.3.6) with 500 ′ Γ = (see Figure 7.21(b)) is
still approximately the same amount as that obtained by using 100 ′ Γ = (see Figure
7.18(b)). Also the tracking errors in both displacement responses (
c
θ θ − ) and in velocity
responses (
c
θ θ −
) between the nominal system (5.2.11) and the controlled actual system
(7.3.6) are approximately in the same order of magnitudes as can be seen from the
comparisons between Figure 7.16 and Figure 7.19 and between Figure 7.17 and Figure
7.20 respectively. These show that the equation of the approximation of the uncertainty’s
bound (5.3.10) is valid to use instead of equation (5.3.5) in our control design.
193
Figure 7.19:
SS
G - Tracking errors ) (
c
θθ − in displacement of the masses
12 3
( ) , ( ) , and ( ) a m bm c m between the
nominal and the controlled systems when using 500 ′ Γ= (in radians)
194
Figure 7.20:
SS
G - Tracking errors ) (
c
θθ −
in velocity of the masses
12 3
( ) , ( ) , and ( ) a m bm c m between the
nominal and the controlled systems when using 500 ′ Γ= (in rad/s)
195
Figure 7.21:
SS
G - (b) Additional force to compensate for uncertainties in masses,
u
Q , is small when compared to (a)
the control force on the nominal system to satisfy the energy control,
c
Q , where (c) shows the sum of
cu
QQ + when
using 500 ′ Γ= (in Newtons)
196
In order to further verify our control approach, we also consider the case of a random
uncertainty of 50% ±
of the nominal value chosen for each mass (see Section 2.3). For
illustrative purposes, we assume
12 3
0.5, 0.8, 1.5 mm m δδ δ == −= and choose 800 ′ Γ =
(see Figure 7.8) to perform a simulation using equation (7.3.6), with all other parameter
values the same as those prescribed in the previous example. The response of the mass
3
m in the XY-plane of the nominal, the controlled, and the actual systems are illustrated
in Figure 7.22. In this figure, we again assume that each trajectory starts at a circle and
stops at a square. We see that the controlled system (Figure 7.22(b)) tracks the nominal
system (Figure 7.22(a)) while the actual system (Figure 7.22(c)) deviates from the
desired nominal system. Figure 7.23 shows the trajectories in
12 3
, , and θθ θ of the
nominal (solid red line), actual (solid blue line) and controlled (dotted line) systems. As
time increases, the actual system’s response differs from those of the nominal and the
controlled systems while the controlled system tracks the nominal system very well.
Figure 7.24 and Figure 7.25 correspondingly show the tracking error responses in
displacement (
c
θ θ − ) and in velocity (
c
θ θ −
) between the nominal system (5.2.11) and
the controlled actual system (7.3.6). Both figures show that the tracking errors are small.
The errors are seen to be of O(
4
10
−
) for the displacement and of O(
3
10
−
) for the velocity.
We see that these errors are within the error norms
4
( ) / 2 3.1 10 et L K
ε
−
<≈ ≈ × and
3
( ) 6.2 10 et L
ε
−
<≈ ≈ ×
as prescribed by (7.2.27) and (7.2.28).
197
Figure 7.22:
SS
G - Trajectory responses (in meters) of the mass
3
m
over a period of 10 secs. of (a) the nominal system
and (b) the controlled system are approximately the same while (c) the actual system yields a totally different trajectory
when the uncertainties in masses are prescribed as
1
0.5 kg m δ = ,
2
0.8 kg m δ = − , and
3
1.5 kg m δ = and the
uncertainty’s bound in equation (7.3.6) is chosen to be 800 ′ Γ=
198
Figure 7.23
SS
G - Angular responses (no. of revolutions ( 360
)) of the masses
23
( ) and ( ) bm c m of the actual
system move away from those of the nominal system as time increases while those of the controlled system track the
nominal system very well
199
Figure 7.24:
SS
G - Tracking errors ) (
c
θθ − in displacement of the masses
12 3
( ) , ( ) , and ( ) a m bm c m between the
nominal and the controlled systems (in radians)
200
Figure 7.25:
SS
G - Tracking errors ) (
c
θθ −
in velocity of the masses
12 3
( ) , ( ) , and ( ) a m bm c m between the
nominal and the controlled systems (in rad/s)
201
The obtained control forces are shown in Figure 7.26. In Figure 7.26(b), the magnitude of
additional control forces applied by the generalized sliding surface controller, :
u
Q Mu = ,
where u is defined as in (7.3.5), is seen to be small relative to the magnitude of the
control forces on the nominal system
c
Q (see (5.2.10)) to satisfy the energy control,
23
() () () Et E t E t = + , (Figure 7.26(a)).
Again the numerical results of the triple pendulum simulation shown in Figure 7.24-
Figure 7.25 and in Figure 7.26 show that even for a large uncertainty (a situation in which
a linear controller is not applicable), the proposed generalized sliding surface control
design is able to guarantee motion control of the actual system to mimic the motions of
the nominal system, thereby satisfying the requirements of the energy control.
202
Figure 7.26:
SS
G - (b) Additional force to compensate for uncertainties in masses,
u
Q , is small when compared to (a)
the control force on the nominal system to satisfy the energy control,
c
Q , where (c) shows the sum of
cu
QQ + (in
Newtons)
203
Remark 7.2: Uncertainty in the unconstrained system (UC)
The closed-form equation of motion of the controlled actual system proposed in equation
(7.3.6) has two separated controllers adding up together—the constraint-following
(nominal) controller and the tracking controller (generalized sliding surface controller) to
compensate for uncertainties in the system. In real-life situation, in order for the uncertain
system to track the unconstrained nominal system, we can use equation (7.3.6) by
ignoring the constraint-following part (the second term in the right-hand side of equation
(7.3.6)). This leads to the closed-form equation of the controlled uncertain unconstrained
system, which can be obtained as
13 0
0
()
(/ ) .
ca a c
t
q a M M Ke s
β
σ αε
α
−
′ Γ+
=−+
(7.3.7)
We again note that the cubic function in the last term of equation (7.3.7) can be replaced
with any arbitrary monotonic increasing odd continuous function on the interval
(, ) −∞ +∞ as mentioned earlier in Section 7.2.
In the case of the same set of the uncertainties assumed as in the previous example
12 3
( 0.1, 0.2, and 0.3) mm m δδ δ == −= , the simulation was conducted by using the same
control parameters described earlier in the previous example except that we choose the
204
initial displacements
1
(0) 0.2 rad, θ =
2
(0) 0.5 rad, θ = and
3
(0) 0.2 rad θ = and the initial
velocities
1
(0) 0.5 rad/s, θ =
2
(0) 1 rad/s, θ =
and
3
(0) 2 rad/s θ =
. The unconstrained
trajectories of the mass
3
m in the XY-plane of the nominal, the controlled, and the actual
systems are obtained as shown in Figure 7.27. Again in this figure, we assume that each
trajectory starts and stops at a circle and a square respectively. Alternatively, Figure 7.28
shows the trajectories in
12 3
, , and θθ θ of the nominal unconstrained system (solid red
line), of the actual unconstrained system (solid blue line), and of the controlled
unconstrained system (dotted line). As time increases, the angular responses of masses
1
m (
1
θ ),
2
m (
2
θ ) and
3
m (
3
θ ) of the controlled system track those of the nominal system
very well while those of the actual system deviated from those of the desired nominal
system. The tracking errors in Figure 7.29 and Figure 7.30 also guarantee the tracking
responses of the nominal unconstrained system,
1
, a MQ
−
= when using the controlled
system equation (7.3.7). These errors are small which can be seen from the figures that
they are of O(
4
10
−
) and of O(
3
10
−
) for the errors in displacement (
c
θ θ − ) and in velocity
(
c
θ θ −
) respectively. Again, the errors are within the prescribed error norm
4
( ) / 2 3.3 10 et L K
ε
−
<≈ ≈ × and
3
( ) 6.6 10 et L
ε
−
<≈ ≈ × . Lastly, the control forces,
:
u
Q Mu = , applied by the generalized sliding surface controller u from equation (7.3.5)
is shown in Figure 7.31.
205
Figure 7.27: UC - Trajectory responses (in meters) of the mass
3
m over a period of 10 secs. of (a) the nominal system
and (b) the controlled system are approximately the same while (c) the actual system yields a totally different trajectory
when the uncertainties in masses are prescribed as
1
0.1 kg m δ = ,
2
0.2 kg m δ = − , and
3
0.3 kg m δ = and the
uncertainty’s bound in equation (7.3.7) is chosen to be 100 ′ Γ=
206
Figure 7.28: UC - Angular responses (in degrees) of three masses
12 3
( ) , ( ) , and ( ) a m bm c m
of the actual system
are different from those of the nominal system as time increases while those of the controlled system track the nominal
system very well
207
Figure 7.29: UC - Tracking errors ) (
c
θθ − in displacement of the masses
12 3
( ) , ( ) , and ( ) a m bm c m between the
nominal and the controlled systems (in radians)
208
Figure 7.30: UC - Tracking errors ) (
c
θθ −
in velocity of the masses
12 3
( ) , ( ) , and ( ) a m bm c m between the
nominal and the controlled systems (in rad/s)
209
Figure 7.31: UC - Additional control forces to compensate for uncertainties in masses,
u
Q (in Newtons)
7.4 Summary
By means of the Lyapunov stability theorem, a controller based on a generalization of the
concept of the sliding surface control is formulated as the additional controller to
compensate for the effects of the uncertainty. The additional controller guarantees that
the actual system, with unknown parameters, tracks the nominal system trajectories
within specified error bounds. Its parameters can be altered to: (i) meet the practical
limitations of a specific controller, and (ii) meet the desired error bounds. The main
contributions of this chapter are:
210
(i) A set of general closed-form controllers for a nonlinear uncertain constrained
mechanical system in the presence of model uncertainties—the generalized
sliding surface controllers (
SS
G )—has been developed. This is obtained by
adding to the nominal controller an additional control that compensates for the
uncertainties. Uncertainties in the two time-dependent dynamical quantities M
and Q that characterize the system and the given forces on the system can be
accommodated.
(ii) The formulation of the closed-from generalized sliding surface controller
()
SS
G is given by
0
0
()
() () ( ) ,
cu c
ac a a
t
M q Q Qt Q Q Q t M Ke f s
β
σ
α
′ Γ+
=+ + =+ − +
where
12
: ( , ,..., )
n
K diag k k k = , 0
i
k > for all 1,..., in = are arbitrary small
positive numbers; the choices of 1 σ = and () 1 fs ≥ , where () fs is any
arbitrary monotonic increasing odd continuous function of s on the interval
(, ) −∞ +∞ , would be sufficient; ′ Γ >Γ is chosen based on the estimate of q δ
from equation (5.3.11);
0
1 α < is a small positive number that is defined by
using the estimate of
a
M from equation (5.3.6); and
0
β is chosen such that
0
. K β =
(iii) We note that the generalized sliding surface controller ()
SS
G involves: (i) the
description of the actual system given by
ac a
Mq Q = , whose parameters are
211
only known imperfectly; (ii) the control force ()
c
Qt given by (5.2.10) that is
obtained on the basis of our best estimate of this actual system, namely on the
basis of the corresponding nominal system; and (iii) the additional control
forces :
u
Q Mu = to compensate for uncertainties, which depend on the
estimates of the uncertainties in
a
M and q δ .
(iv) The uncertainties
a
M and q δ are not known but we have the estimates of
these uncertainties. We further note that the tracking results and the additional
control forces that are obtained by using the generalized sliding surface
controller ()
SS
G
are not sensitive to the estimate of the bound on the
uncertainty q δ .
(v) The control function () fs and the parameters that define the additive
controller can be chosen depending on a practical consideration of the control
environment and on the extent to which the compensation is desired. These
parameters can be adjusted so that when the actual system is required to track
the nominal system, desired error bounds can be guaranteed. Thus, there is
flexibility in using this additional controller, especially when dealing with
large, complex multi-body systems. For example, the uses of the specified
quadratic function and cubic function in the examples considered allow the
additive control to be continuous, thereby preventing chattering.
212
(vi) The general closed-form controllers for nonlinear uncertain constrained
motions proposed in this chapter have two separate controllers, which are the
constraint-following controller and the tracking controller to compensate for
uncertainties. Each controller has an independent response so that we can use
each of them individually. If one is interested only in tracking unconstrained
systems, one can ignore the control force
c
Q from the fundamental equation
and use equation (7.3.7) to obtain the equation of motion of the controlled
uncertain unconstrained system.
(vii) While the only uncertainties demonstrated in this chapter are those related to
the masses of the triple pendulum, the formulation of the current methodology
considers both general sources of uncertainties—uncertainties in the
description of physical systems and uncertainties in the given force on the
system. The set of closed-form controllers developed herein is therefore
general enough to be applicable to more complex dynamical systems in which
the uncertainties in the given force may be important.
(viii) The effectiveness of the design of the additional additive controller to
compensate for the uncertainty is evaluated using numerical solution
comparisons of the tracking errors between the nominal and the controlled
actual systems. The results demonstrate that the proposed controller has good
213
transient behavior and is robust with respect to the uncertainties in the
modeling process. Furthermore, with the simplicity and accuracy obtained, the
control scheme proposed in this chapter can be implemented for various cases
and for even more complex dynamical problems.
In the next chapter, two alternative compensating controllers based on a generalization of
the concept of the damping control are developed. Both controllers can relax the
limitation on the forcing function of a tracking control being used in the general
discontinuous control while they are still able to guarantee tracking of the nominal
system’s trajectories within a specified error bound. The first controller requires the
knowledge of the bound on the uncertainty in order to guarantee tracking nominal
trajectories. Thus this controller has behavior similar to the generalized sliding surface
controller. However, when using the second controller, tracking of the nominal system’s
trajectories can be obtained irrespective of whether the uncertainty’s bound is known.
214
Chapter 8
TRACKING CONTROLLERS BASED ON THE
CONCEPT OF THE GENERALIZED DAMPING
CONTROL
8.1 Introduction
This chapter presents an alternative approach to obtaining tracking controllers for
nonlinear uncertain multi-body systems as an extension of the previous chapter. The
uncertainty in the modeling of the mechanical system which is time-varying, unknown
but bounded, is assumed, whereby this uncertainty again may arise from the lack of
knowledge of the physical system and/or of the ‘given’ forces applied to the system. The
use of the new, simple and general closed-form equation of the controlled actual system
developed in Section 5.2 is illustrated incorporating a generalization of the concept of the
damping control [24]. Two kinds of controllers (one with the requirement of knowledge
of the uncertainty’s bound and one without this requirement) are developed. Both
proposed controllers are able to guarantee a tracking signal within desired error bounds of
the desired nominal system’s trajectory—our best assessment of the actual real-life
situation—regardless of the knowledge of the uncertainty. A comparison of the use of the
proposed controllers with the generalized sliding surface controller (Section 7.2) is
215
illustrated using the example of the triple pendulum problem as shown in Section 2.3. By
using the controllers based on the generalization of the concept of the damping control,
we demonstrate the ease, simplicity and accuracy with which control of such highly
nonlinear systems is achieved.
8.2 Generalized Damping Controllers (
D
G )
Again, we start with the same approach as that adopted in the generalized sliding surface
control design by defining the system tracking error as in equations (7.2.1)-(7.2.4).
From equation (7.2.4), in general, this system can be expressed in the state space form as
[9]
(, , ) (, , ) [ (, , ) (, , ) ], e t e Ae t e B h t e G t e u δ δ δδ = ++
(8.2.1)
where δ is an uncertainty, the argument e in the parenthesis stands for (, ) ee ,
00
and ,
00
= =
I
AB
I
(8.2.2)
also,
(, , ) , ht e q δδ =
(8.2.3)
and
216
( )
(, , ) .
na
Gt e I M δ = −
(8.2.4)
We note again that
( )
1
1
:
a
M I IM M δ
−
−
=−+ and equation (8.2.1) satisfies the matching
condition [9].
Definition 8.1: System
(, ) (, ) et e Aet e =
(8.2.5)
is globally uniformly exponentially stable (GUES) about zero state if there are scalars
,0
εε
αβ >
such that every solution () e ⋅
of (8.2.5) satisfies [9]
[ ]
00
22
( ) ( ) exp ( ) et et t t
εε
βα ≤ −−
(8.2.6)
for all
0
tt ≥ .
In systems containing unknown persistently acting disturbance, one usually has to resort
to discontinuous controllers to obtain GUES. To avoid this, the requirement of GUES can
be relaxed as follows: For any 0 r
ε
≥ , let ( ) Br be the closed ball of radius r
ε
and center
0, i.e., [9]
{ }
2
( ) : : .
n
Br e R e r
ε
=∈ ≤
(8.2.7)
217
Definition 8.2: System (8.2.5) is globally uniformly exponentially convergent (GUEC) to
( ) Br if and only if there are scalars ,0
ε ε
αβ > such that every solution () e ⋅
of (8.2.5)
satisfies [9]
[ ]
00
22
( ) ( ) exp ( ) et r et t t
εε ε
βα ≤+ − −
(8.2.8)
for all
0
tt ≥ . Clearly, GUES is a special case of GUEC with 0 r
ε
= .
Theorem 8.1: Suppose there exists a continuously differentiable function (, ) V t e
with
the following properties:
(i) There are scalars
12
,0 ωω >
such that
22
12
22
(, ) e V t e e ωω ≤≤ (8.2.9)
for all
n
e R ∈ .
(ii) There are scalars 0, 0 V
ε
α
∗
≥>
such that for all tR ∈
( ( ))
2 (, ) .. (, ) .
dV e t
V t e V e s t V t e V
dt
ε
α
∗∗
≤− − ∀ >
(8.2.10)
Then system (8.2.5) is GUEC with rate
ε
α
to the ball of radius
12
1
( / ). rV
ε
ω
∗
= (8.2.11)
A proof of this theorem can be found in [9].
218
Our aim is to ensure that the system (8.2.1) is GUEC with rate
ε
α to the ball of radius r
ε
,
whereupon by (8.2.8) ideally speaking, when the size of the ball 0 r
ε
= , we obtain GUES,
whose solution (8.2.6) shows that the tracking error () et exponentially reduces to zero as
time goes to ∞ .
We shall now show that the system (8.2.1) is GUEC and the tracking error () et can
indeed be converged to the error ball of radius r
ε
.
Definition 8.3: We start by defining a controller, which is of the form [9]
0
(, ) (, ) (, ),
s
u p t e p t e p t e
ε
= + + (8.2.12)
where
0
, , and
s
pp p
ε
are defined as follows:
Design
0
p :
0
(, ) p t e
is chosen so as to eliminate the known term of (, , ) ht e δ , i.e., to
reduce the magnitude of the uncertain term h. In our case (, , ) ht e δ
does not have the
known term (see equation (8.2.3)), thus [9]
0
( , ) 0. p t e = (8.2.13)
219
Design
s
p : The control
s
p
is chosen to guarantee that the system e Ae =
is globally
uniformly exponentially stable (GUES). This is done by using the PID controller, which
is of the form
1 1 2
0
(, ) ( )
t
s
pi d
p t e K e K e d K e τ τ
= − + +
∫
(8.2.14)
where 1
p
K = , 0 2,
i
K
ε
α ≤≤ 0
d
K
ε
α ≤≤ and 0
ε
α > are any small positive numbers.
Design p
ε
: The control p
ε
is a continuous control where ε
is any positive real number
that makes the closed-loop uncertain system to satisfy the hypotheses of Theorem 8.1
with 2 V
ε
εα
∗
= ; hence it is GUEC with rate 0
ε
α >
to the ball of radius [9]
12
2
1
1
or 2 ,
2
r r
ε εε
ε
ε
ε αω
αω
= =
(8.2.15)
where
ε
α is any small positive number. Thus as ε
approaches 0 ( 0) r
ε
→ , the behavior
of the closed-loop system approaches GUES; i.e., its behavior with the discontinuous
controller.
By Theorem 8.1, we choose the Lyapunov candidate function
11 2 2
11
22
TT
V ee e e = + . Thus
1
1
2
ω = . Then from equation (8.2.15) we have
2
. r
εε
εα = (8.2.16)
220
Assumption 8.1: There is a known, non-negative, continuous, bounding function () t ρ
such that
0
0
()
( ) 0,
t
t
ρ
ρ
α
Γ+
≥ >
(8.2.17)
where
0 1 1 2
0
()
t
ai d
M e K e d Ke ρ τ τ ≥+ +
∫
is any arbitrary positive constant and we
again define the parameter
( )
0
01 .
a
M α σ < <− (8.2.18)
The positive constant σ is chosen such that
, σγ ≥
(8.2.19)
where (see also Appendix B)
( )
( )
() ()
: 1.
() ()
T
e fe
ef e
ηη
γ
ηη
= ≤ (8.2.20)
The tracking error parameter is defined as
() : () () , η =
TT
e B e DV e (8.2.21)
where
12
( ):
VV
DV e
ee
∂∂
=
∂∂
. Substituting equation (8.2.2) in equation (8.2.21), we get
2
() . ee η =
(8.2.22)
We note from (8.2.20) that since 1 γ ≤ , the choice 1 σ = would suffice in (8.2.18).
221
We also note that the function () t Γ is the bound on q δ , i.e., ( ), qt δ ≤Γ as defined in
Section 5.3. We again denote ⋅ to mean the infinity norm.
Assumption 8.2: We now consider the continuous control (, ) p t e
ε
as
( )
2
( , ): ( ) . p t e t f e
ε
σρ = −
(8.2.23)
The i-th component, ( )
2 i
fe , of the n-vector
( )
2
fe is defined as
( ) ( )
2 2,
/ , 1, . . ., ,
ii
f e ge i n ε = =
(8.2.24)
where
2,i
e is the i-th component of the n-vector
2
e , ε is defined as in (8.2.16) and the
function
( )
2,
/
i
ge ε is any arbitrary monotonic increasing odd continuous function of
2,i
e
on the interval (, ) −∞ +∞ that satisfies
( ) ( )
2 2, 2,
, 1 if is outside the surface ,
ii i
f e ge e
ε
ε=≥ Ψ
(8.2.25)
where
ε
Ψ
is defined as the surface of the ball of radius
1/2
r
ε
ε
ε
α
=
around the point
2,
0
i
e = .
Corollary 8.1: The controller (8.2.23) becomes the Leitmann-Corless controller [9] when
the control function
22
( ) sgn( ) f e e = .
222
Result 8.1: Under the above-mentioned Assumptions 8.1 and 8.2, combining the controls
0
, , and
s
pp p
ε
, the generalized damping controller
D
G ,
( )
1 1 2 2
0
( ) () ,
t
pi d
u K e K e d Ke t f e τ τ σρ
= − + + −
∫
(8.2.26)
causes () et
ε
→Ψ .
Proof: Consider the Lyapunov candidate function
11 2 2
11
.
22
TT
V ee e e = + (8.2.27)
Differentiating equation (8.2.27) once with respect to time and using (7.2.4), we get
11 2 2
12 2
( ).
TT
TT
a
V ee e e
ee e q u M u δ
= +
= + +−
(8.2.28)
Substituting equation (8.2.26) in equation (8.2.28), we have
( ) ( )
( ) ( )
12 2 2 1 1 2 2
0
11 2 2 1 2 2 2 2 2 2
2 11 2
0
( ) ()
( 1)
2
() .
t
TT T
na p i d
T T TT T T i
dp a
t
T
ap i d
V e e e q e I M K e K e d Ke t f e
K
e e K e e K e e e q e f e e M f e
e M K e K e d Ke
δ τ τ σρ
δ σρ σρ
τ τ
= + + −− + + −
= − − − − + − +
+ ++
∫
∫
(8.2.29)
223
We note that the second equality above follows from the use of
1 2 21
TT
ee e e =
and the fact
that
2 1 11
0
()
2
t
TT i
i
K
Ke e d e e τ τ −= −
∫
in the previous step.
Thus equation (8.2.29) yields
( ) ( )
( ) ( )
11 2 2 1 2 2 2 2 2 2
2 11 2
0
11 2 2 1 2 2 2 2 2 2
2 11 2
0
( 1)
2
()
( 1)
2
() ,
T T TT T T i
dp a
t
T
ap i d
TT T T i
dp a
t
ap i d
K
V e e K e e K e e e q e f e e M f e
e M K e K e d Ke
K
e e K e e K e e e e f e e M f e
e M K e K e d Ke
δ σρ σρ
τ τ
σρ σρ
τ τ
≤− − − − + − +
+ ++
≤ − − − − + Γ− +
+ ++
∫
∫
(8.2.30)
where the last inequality follows from
22
T
ee = and equation (5.3.11).
Since
2
() f e is a monotonic increasing odd continuous function of
2
e , and
2
e and
2
() fe
have the same sign, we have (see also (8.2.19), (8.2.20), and(8.2.22))
22 2 2
() ().
TT
e fe e fe σ ≥ (8.2.31)
Using equation (8.2.31) in equation (8.2.30), we obtain
224
( )
( )
( ) ( )
11 2 2 1 2 2 2 2
2 22 1 1 2
0
11 2 2 1 2
2 2 11 2
0
( 1)
2
()
( 1)
2
1 () .
TT T i
dp
t
a ap i d
TT T i
dp
t
a ap i d
K
V e e K e e K e e e e f e
eM f e eM K e K e d K e
K
e e K e e K e e
e M f e M K e K e d Ke
ρ
σρ τ τ
ρ σ τ τ
≤ − − − − + Γ−
+ + ++
= − − − −
− − −Γ − + +
∫
∫
(8.2.32)
Since 1,
p
K =
0
0
()
()
t
t
ρ
ρ
α
Γ+
≥ and
( )
0
0 1
a
M α σ < <− , we then have
( ) ( )
11 2 2 2 0 2 1 1 2
0
() .
2
t
TT i
d ai d
K
V e e Ke e e f e M e K e d Ke ρ τ τ ≤ − − − Γ+ − Γ− + +
∫
(8.2.33)
And since
2
() 1 f e ≥ outside the surface
ε
Ψ (see relation (8.2.24)-(8.2.25)) and
0 11 2
0
()
t
ai d
M e K e d Ke ρ τ τ ≥+ +
∫
, we have
( )
( )
11 2 2
11 2 2 11 2 2
2
2
2
2 11
22 2
:,
TT i
d
TT T T i
d
K
e e K e e
K
e e e e e e K e e
VV
V
ε
ε
εε
α
α
α
α
∗
− −
−
− −
≤
−
= + + +−
=
(8.2.34)
where
225
11 22
( / 2)
:.
22
TT id
KK
V ee ee
εε
εε
αα
αα
∗
−−
= +
(8.2.35)
Since 0
ε
α > , 0 2,
i
K
ε
α ≤ ≤ and 0
d
K
ε
α ≤≤ , equation (8.2.35) shows that 0 V
∗
≥ and
VV
∗
> . Noting (8.2.34), the Lyapunov derivative V
is non-positive. Thus, equation
(8.2.34) guarantees that we have attractivity to the region
2
e enclosed by the surface
ε
Ψ .
By Theorem 8.1, the solution of the closed-loop system (equation (8.2.1)) is thus GUEC
with rate
ε
α to the ball of radius
( )
12
/ r
εε
εα = . Hence, we can conclude that
2
e
is
bounded by the ball of radius r
ε
, i.e.,
2
2
er
ε
≤ .
Main Result ()
D
G : The closed-from generalized damping controller (
D
G ) for the
uncertain system is given by
( )
11 2 2
0
() () ( ) () ,
t
cc
ac a a i d
M q Q Qt Mu Q Q t M e K e d K e t f e τ τ σρ
=+ + =+ − + + +
∫
(8.2.36)
where the control force ()
c
Qt is given by (5.2.10)
1
() ( ) ( )
c T T
Q t A AM A b Aa
−+
= − (8.2.37)
and is obtained on the basis of the nominal system; and 0
id
K K > are arbitrary small
positive numbers; the choices of 1 σ = and
2
() 1 f e ≥ , where
2
() f e is any arbitrary
monotonic increasing odd continuous function of
2
e on the interval (, ) −∞ +∞ , would
suffice in (8.2.36);
226
0
0
()
() ,
t
t
ρ
ρ
α
Γ+
=
(8.2.38)
where Γ is chosen based on the estimate of q δ from equation (5.3.11),
( ) ( )
1 11
1 ( );
c
q M M M M QQ M Q t δ δ δδ
− − −
≤ + + + ≤Γ (8.2.39)
0
α is a small positive number that is defined by using the estimate of
a
M from equation
(5.3.6), which yields
1
1,
a
M MM δ
−
≈ <<
(8.2.40)
so that
( )
1
0
1 1 1;
a
M MM α σ δ
−
< − ≈− ≈ (8.2.41)
and again by using the estimate of
a
M (equation (8.2.40)) and with the expectation that
11 2
0
()
t
id
e K e d Ke τ τ ++
∫
is small,
0
ρ is chosen such that
0
1. ρ ≥
(8.2.42)
8.2.1 Numerical Results and Simulations of the Generalized Damping Controller
(
D
G )
In order to show the efficacy in tracking the nominal system’s trajectories of the proposed
generalized damping controller
D
G , we again consider the triple pendulum problem as
227
presented in Section 2.3. For simplicity, we consider only the uncertainties in the masses
1
m ,
2
m , and
3
m similar to what we did in Chapter 7. Again for illustrative purpose, we
assume that each mass has a random uncertainty of ±10 percent of its best assessed
(nominal) mass; we assume
12 3
0.1, 0.2, and 0.3 mm m δδ δ == −= (see also (5.2.16)).
We select the parameters for the generalized damping controller
D
G given by equation
(8.2.26) by choosing
2
2, 2,
2 2
2, 2,
( / ) , where 0
()
( / ) , where 0
ci i
i
ci i
ee
fe
ee
α ε
α ε
−<
=
≥
(8.2.43)
where again ,0
c
αε >
and ε is a suitable small number. We thus obtain in closed-form
the additional controller needed to compensate for uncertainties in the actual system,
0
11 2 2
0 0
()
( ( ) ) ( ).
t
id
t
u e K e d Ke f e
ρ
τ τ σ
α
Γ+
= − + + −
∫
(8.2.44)
Using the additional controller equation (8.2.44) in equation (5.2.18) (or (8.2.36)), we
obtain the closed-form equation of motion of the controlled actual system
11 0
11 2 2
0 0
()
() ( ( ) ) ( )
t
c
ca a a i d
t
q a M Q t M M e K e d Ke f e
ρ
τ τ σ
α
−−
Γ+
=+ − + ++
∫
(8.2.45)
that will cause the actual system to track the trajectory of the nominal system, thereby
compensating for the uncertainty.
228
For the simulation, the ball of radius 0.07 r
ε
= is chosen in order to guarantee GUEC
with the rate 2
ε
α =
(see equation (8.2.15)). We also choose the parameters
1
1
2
ω = ,
0,
i
K = 3,
d
K =
1, σ =
0
1 ρ = ,
0
0.5, α = 2
c
α =
and 100 Γ= (similar to what we did
for the generalized sliding surface controller). All other parameter values the same as
those previously prescribed in Section 2.3. Figure 8.1-Figure 8.2 correspondingly plot
the response trajectories of the mass
3
m in the XY-plane of the nominal, the controlled,
and the actual systems and their trajectory responses in
12 3
, , and θθ θ . We notice that
these responses are similar to what we obtained from the generalized sliding surface
controller (see Figure 7.9 and Figure 7.10). The tracking errors in displacement ()
c
θ θ −
and in velocity (
c
θ θ −
) between the nominal system (5.2.11) and the controlled actual
systems (8.2.45) are shown in Figure 8.3 and Figure 8.4 respectively. Again the errors are
seen to be small (of O(
4
10
−
) for the displacement and of O(
3
10
−
) for the velocity) and
they are also within the ball of radius 0.07 r
ε
= as prescribed. Figure 8.5 illustrates the
control forces applied to the system. In Figure 8.5(b), the magnitude of the additional
control forces from the generalized damping controller
D
G to compensate for the
uncertainties, :
u
Q Mu = , where u is defined as in equation (8.2.44), is seen to be small
when compared with the magnitude of the control forces
c
Q given in (5.2.10) (and
shown in Figure 8.5(a)) in order to satisfy the energy control requirement,
23
() () () Et E t E t = + .
229
Figure 8.1:
D
G - Trajectory responses (in meters) of the mass
3
m
over a period of 10 secs. of (a) the nominal system
and (b) the controlled system are approximately the same while (c) the actual system yields a totally different trajectory
when the uncertainties in masses are prescribed as
1
0.1 kg m δ = ,
2
0.2 kg m δ = − , and
3
0.3 kg m δ = ( 10% ± of
nominal masses) and the uncertainty’s bound in equation (8.2.45) is chosen to be 100 Γ =
230
Figure 8.2:
D
G - Angular responses (no. of revolutions ( 360
)) of the masses
23
( ) and ( ) bm c m of the actual
system move away from those of the nominal system as time increases while those of the controlled system track the
nominal system very well
231
Figure 8.3:
D
G - Tracking errors ) (
c
θθ − in displacement of the masses
12 3
( ) , ( ) , and ( ) a m bm c m between the
nominal and the controlled systems (in radians)
232
Figure 8.4:
D
G - Tracking errors ) (
c
θθ −
in velocity of the masses
12 3
( ) , ( ) , and ( ) a m bm c m between the
nominal and the controlled systems (in rad/s)
233
When comparing Figure 7.11 and Figure 7.12 with Figure 8.3 and Figure 8.4
respectively, we observe that the generalized damping controller
D
G shows responses in
tracking trajectories of the nominal system comparable to those of the generalized sliding
surface controller
SS
G . And when considering the control forces we see that the
magnitude of the additional control forces to compensate for uncertainties from the
generalized damping controller
D
G (see Figure 8.5(b)) is the same as that which was
obtained from the generalized sliding surface controller
SS
G (Figure 7.13(b)).
234
Figure 8.5:
D
G - (b) Additional force to compensate for uncertainties in masses,
u
Q , is small when compared to (a)
the control force on the nominal system to satisfy the energy control,
c
Q , where (c) shows the sum of
cu
QQ + (in
Newtons)
235
As a second example, we consider the same system given in the previous example except
that we choose
12 3
0.1, 0.05, and 0.2 mm m δδ δ = −= = − and replace the monotonic
increasing odd continuous function (8.2.43), which is the parameter for the generalized
damping controller
D
G given by equation (8.2.26), with a cubic function
( )
3
2 2,
() / ,
i c i
fe e α ε = (8.2.46)
where again ,0
c
αε >
and ε is a suitable small number. We obtain in closed-form the
additional controller needed to compensate for uncertainties in the actual system,
( )
3
0
1, 1, 2, 2,
0 0
()
( () ) / ,
t
i i i i di c i
t
u e K e d Ke e
ρ
τ τ σ α ε
α
Γ+
= −+ + −
∫
(8.2.47)
which upon use of this equation in equation (5.2.18) (or (8.2.36)), we obtain the closed-
form equation of motion of the controlled actual system as
( )
3
11 0
11 2 2
0 0
()
() ( ( ) ) / .
t
c
ca a a i d c
t
q a M Q t M M e K e d Ke e
ρ
τ τ σ α ε
α
−−
Γ+
=+ − + ++
∫
(8.2.48)
We next perform a simulation again using equation (8.2.48), with all other parameter
values the same as those prescribed in the previous example. Figure 8.6 shows the
response trajectories of the mass
3
m of the nominal system, the controlled system, and
the actual system in the XY-plane for duration of 10 seconds. The angular responses in
236
12 3
, , and θθ θ of these three systems are also illustrated in Figure 8.7. The tracking
errors both in displacement (
c
θ θ − ) and in velocity (
c
θ θ −
) between the nominal system
(5.2.11) and the controlled actual system (8.2.48) are demonstrated in Figure 8.8 and
Figure 8.9 respectively. Lastly, the obtained control forces of the system are shown in
Figure 8.10. Again Figure 8.8 and Figure 8.9 show that the tracking errors are small (of
O(
4
10
−
) for the displacement and of O(
3
10
−
) for the velocity). These errors are within the
ball of radius 0.07 r
ε
= as prescribed. When comparing Figure 7.16 and Figure 7.17 with
Figure 8.8 and Figure 8.9 respectively, we observe that the generalized damping
controller
D
G shows responses in the tracking trajectories of the nominal system
comparable with those of the generalized sliding surface controller. And when
considering the control forces we see that the magnitude of the additional control forces
to compensate for uncertainties from the generalized damping controller
D
G , :
u
Q Mu = ,
where u is defined as in (8.2.47) (see Figure 8.10(b)), is also small when compared with
the magnitude of the control forces applied to the nominal system
c
Q given in (5.2.10)
(and shown in Figure 8.10(a)) to satisfy the energy control,
23
() () () Et E t E t = + . We note
that the magnitude of the control force shown in Figure 8.10(b) is about the same amount
as that which was obtained from the generalized sliding surface controller (Figure
7.18(b)).
237
Figure 8.6:
D
G - Trajectory responses (in meters) of the mass
3
m
over a period of 10 secs. of (a) the nominal system
and (b) the controlled system are approximately the same while (c) the actual system yields a totally different trajectory
when the uncertainties in masses are prescribed as
1
0.1 kg m δ = − ,
2
0.05 kg m δ = , and
3
0.2 kg m δ = − and the
uncertainty’s bound in equation (8.2.48) is chosen to be 100 Γ =
238
Figure 8.7:
D
G - Angular responses (no. of revolutions ( 360
)) of the masses
23
( ) and ( ) bm c m of the actual
system move away from those of the nominal system as time increases while those of the controlled system track the
nominal system very well
239
Figure 8.8:
D
G - Tracking errors ) (
c
θθ − in displacement of the masses
12 3
( ) , ( ) , and ( ) a m bm c m between the
nominal and the controlled systems (in radians)
240
Figure 8.9:
D
G - Tracking errors ) (
c
θθ −
in velocity of the masses
12 3
( ) , ( ) , and ( ) a m bm c m between the
nominal and the controlled systems (in rad/s)
241
Figure 8.10:
D
G - (b) Additional force to compensate for uncertainties in masses,
u
Q , is small when compared to (a)
the control force on the nominal system to satisfy the energy control,
c
Q , where (c) shows the sum of
cu
QQ + (in
Newtons)
242
Similar to the generalized sliding surface controller, the generalized damping control
approach (
D
G ) is further verified by considering the case of a random uncertainty of
50% ±
of the nominal value chosen for each mass of the triple pendulum as described in
Section 2.3. We assume
12 3
0.5, 0.8, 1.5 mm m δδ δ == −= and choose 800 Γ= (see
Figure 7.8) to perform a simulation using equation (8.2.48), with all other parameter
values the same as those prescribed in the previous example. Figure 8.11-Figure 8.12
correspondingly plot the response trajectories of the mass
3
m in the XY-plane of the
nominal, the controlled, and the actual systems and their trajectory responses in
12 3
, , and θθ θ . We notice that these responses are similar to what we obtained from the
generalized sliding surface controller (see Figure 7.22 and Figure 7.23). The tracking
errors in displacement ()
c
θ θ − and in velocity (
c
θ θ −
) between the nominal system
(5.2.11) and the controlled actual systems (8.2.48) are shown in Figure 8.13 and Figure
8.14 respectively. Again the errors are seen to be small (of O(
4
10
−
) for the displacement
and of O(
3
10
−
) for the velocity) and they are also within the ball of radius 0.07 r
ε
= as
prescribed. Thus we see that even for a large uncertainty (a situation in which a linear
controller is not applicable), the proposed generalized damping control design
D
G is able
to guarantee motion control of the actual system to mimic the motions of the nominal
system.
243
Figure 8.11:
D
G - Trajectory responses (in meters) of the mass
3
m
over a period of 10 secs. of (a) the nominal
system and (b) the controlled system are approximately the same while (c) the actual system yields a totally different
trajectory when the uncertainties in masses are prescribed as
1
0.5 kg m δ = ,
2
0.8 kg m δ = − , and
3
1.5 kg m δ = and
the uncertainty’s bound in equation (8.2.48) is chosen to be 800 Γ =
244
Figure 8.12:
D
G - Angular responses (no. of revolutions ( 360
)) of the masses
23
( ) and ( ) bm c m of the actual
system move away from those of the nominal system as time increases while those of the controlled system track the
nominal system very well
245
Figure 8.13:
D
G - Tracking errors ) (
c
θθ − in displacement of the masses
12 3
( ) , ( ) , and ( ) a m bm c m between the
nominal and the controlled systems (in radians)
246
Figure 8.14:
D
G - Tracking errors ) (
c
θθ −
in velocity of the masses
12 3
( ) , ( ) , and ( ) a m bm c m between the
nominal and the controlled systems (in rad/s)
247
Figure 8.15:
D
G - (b) Additional force to compensate for uncertainties in masses,
u
Q , is small when compared to (a)
the control force on the nominal system to satisfy the energy control,
c
Q , where (c) shows the sum of
cu
QQ + (in
Newtons)
248
Figure 8.15 illustrates the control forces applied to the system. In Figure 8.15(b), the
magnitude of the additional control forces from the generalized damping controller
D
G
to compensate for the uncertainties, :
u
Q Mu = , where u is defined as in equation
(8.2.47), is seen to be small when compared with the magnitude of the control forces
c
Q
given in (5.2.10) (and shown in Figure 8.15(a)) in order to satisfy the energy control
requirement,
23
() () () Et E t E t = + .
Again when comparing Figure 7.24 and Figure 7.25 with Figure 8.13 and Figure 8.14
respectively, we observe that the generalized damping controller
D
G shows responses in
tracking trajectories of the nominal system comparable to those of the generalized sliding
surface controller
SS
G . And when considering the control forces we see that the
magnitude of the additional control forces to compensate for uncertainties from the
generalized damping controller
D
G (see Figure 8.15(b)) is the same as that which was
obtained from the generalized sliding surface controller
SS
G (Figure 7.26(b)).
By using the concept of the generalized damping control, we obtain the compensating
controller
D
G that is able to guarantee tracking of the nominal system’s trajectories.
However, this controller requires information about the bound on uncertainties in order to
design the controller. We thus next use a generalization of the concept of the damping
249
control [24] to develop another controller
D
G that when used with the closed-form
controlled actual system (5.2.18), also causes the trajectories of the actual system to track
those obtained from the nominal system. Noting that unlike the two previous proposed
controllers (
SS
G and
D
G ), this control design
D
G does not require knowledge of the
bound on uncertainties.
We note that the subscript D denotes the generalized damping controller that requires
knowledge of the bound on the uncertainty to define the controller, while the subscript D
denotes the generalized damping controller that does not require knowledge of the
uncertainty’s bound in the controller design.
8.3 Generalized Damping Controllers (
D
G )
Similar to the procedure developed in the generalized sliding surface control
SS
G and the
generalized damping control
D
G , we start by defining the system tracking error between
the nominal and the controlled actual system as in equations (7.2.1)-(7.2.4). We obtain
( )
11
22
00
,
00
δ
= + +−
a
ee I
q u Mu
ee I
(8.3.1)
where
( )
1
1
:
a
M I IM M δ
−
−
=−+ .
Then we define the Lyapunov candidate function as
250
11 2 2
11
.
22
TT
V ee e e = + (8.3.2)
Differentiating equation (8.3.2) once with respect to time and using equation (8.3.1), we
get
11 2 2
12 2
( ).
TT
TT
a
V ee e e
e e e q u Mu δ
= +
= + +−
(8.3.3)
Definition 8.4: In order to guarantee the Lyapunov stability of the system (8.3.1), i.e., the
tracking of the nominal system’s trajectories, we shall now show that the derivative V
is
non-positive. We then start by considering the controller, which is of the form
( )
12 2
2
() .
D
G
u e f e
e
σ
∞
= − + +
(8.3.4)
The positive constant σ is chosen such that
, σγ ≥
(8.3.5)
where (see also Appendix B)
22
22
()
: 1.
()
T
e f e
e f e
γ
∞ ∞
= ≤ (8.3.6)
We note that since 1 γ ≤ , the choice 1 σ = would suffice in (8.3.4). The i-th component,
2
()
i
fe , of the n-vector
2
() f e is defined as
251
( )
2 2,
( ) / , 1, . . .,
ii
f e ge i n ε = = (8.3.7)
where
2,i
e is the i-th component of the n-vector
2
e , ε is defined as any small positive
number and the function
2,
( /)
i
ge ε is any arbitrary strictly increasing odd continuous
function of
2,i
e on the interval (, ) −∞ +∞ and it goes to ∞ as
2,i
e goes to ∞ .
Definition 8.5: Similar to the generalized sliding surface control
SS
G and the generalized
damping control
D
G , we define the parameter
( ) 0
01 ,
a
M α σ
∞
< <− (8.3.8)
and assume
1
1. MM δ
−
∞
<< (8.3.9)
Then using the controller (8.3.4) in (8.3.3), we have
( )
{ }
22
2
2 2 2 21 2 2
2
() () .
T
T
Da D
e
V e f e G e q M e e f e G
e
σ δσ
∞ ∞
∞
= − + + + + −
(8.3.10)
The above equation follows from the use of
1 2 21
TT
ee e e =
in the previous steps.
Since (see (8.3.5) and (8.3.6))
252
22 2 2
() () ,
T
e f e e f e σ
∞ ∞
≥
(8.3.11)
using equation (8.3.11) in equation (8.3.10), we obtain
( )
{ }
22
2
2 2 2 21 2 2
2
() () .
T
Da D
e
V e f e G e q M e e f e G
e
δσ
∞ ∞ ∞ ∞
∞
≤− + + + + −
(8.3.12)
Definition 8.6: Define parameters
2
2
2 2
2
: and :
e
we
e
∞
∞
= Ω= , thus equation (8.3.12)
yields
( ) { }
22 1 2
() () .
T
Da D
V e f e w G q M e f e G δσ
∞ ∞
≤ − + +Ω + Ω + −
(8.3.13)
Result 8.2: The generalized damping controller
2
: ,
n
D
G k w =−Ω (8.3.14)
where 0 and 1 k n >≥
are arbitrary small positive constants, guarantees that the
Lyapunov derivative V
is non-positive. Thus, the solution of the closed-loop system
(equation (8.3.1)) is uniformly bounded.
Proof: Substituting equation (8.3.14) in equation (8.3.13), we have
2
2 21
22 1 2
2
() () .
n T T n
a
V e f e k w w q w M e f e k w δσ
−
∞ ∞
≤− − Ω + Ω + Ω + + Ω
(8.3.15)
253
Since
1
1 MM δ
−
∞
<< (see (8.3.9)),
12
0 Me Me δδ≈≈ ,
2
we Ω= , and also k is any
small positive number so that the term
( ) ( ) ( )
( )
1
2 1 1 2 1
11
1 22
2 12
( ) 0,
T nT n
a
Tn
w M ek w w I I M M ek w
e I IM M e k e
δ
δ
−
− − −
− −
Ω + Ω = Ω − + + Ω
≈ − − + Ω ≈
(8.3.16)
thus equation (8.3.15) yields
( )
2
2
22 2 2
2
2
2
22 2 2
2
2
22
() ()
() ()
1 () .
nT T
a
nT T
a
n
a
V e f e k w w q e M f e
e f e k w w e M f e
e M f e k w w
δσ
σ
σ
∞ ∞
∞ ∞ ∞ ∞
∞ ∞∞
∞ ∞ ∞∞
∞
≤− − Ω + Ω +
≤ − − Ω + Ω Γ+
≤− − − Ω + Ω Γ
(8.3.17)
The second inequality above follows from
2
ww
∞
≥ and equation (5.3.11) where Γ is
an unknown upper bound on q δ and the last inequality follows that
22
T
ee
∞
∞
=
and
.
T
ww
∞
∞
=
We note that the term
2
2
n
kw w
∞∞
− Ω + ΩΓ above attains a maximum value
1/(4 3)
4 2
43
4 2 (42)
n
n
n
n nk
−
−
− Γ
− −
at
1/(4 3)
(4 2)
n
w
nk
−
∞
Γ
Ω=
−
. Therefore
( )
1/(4 3)
42
22
43
1 () .
42 (42)
n
n
a
n
V e M f e
n nk
σ
−
−
∞ ∞
∞
−Γ
≤− − +
− −
(8.3.18)
254
Since
( )
10
a
M σ
∞
− > and
2
() f e is a strictly increasing function of
2
e , which goes to
∞ as
2
e goes to ∞, it is always true that V
is negative outside some ball. Thus the
solution of the closed-loop system (equation (8.3.1)) is uniformly bounded [24].
We now propose the generalized damping controller
D
G as (see (8.3.4) and (8.3.14))
( )
2
12 2
2
() ,
n
kw
u e f e
e
σ
∞
Ω
= − + −
(8.3.19)
which upon use of our design parameters
2
2
2 2
2
: and :
e
we
e
∞
∞
= Ω=
, gives us:
44
1 2 2 2
( ) ( ).
n
u e k e e f e σ
−
∞
= −+ +
(8.3.20)
Corollary 8.2: The controller (8.3.19) becomes the Lyapunov redesign controller when
1 n = and the control function
22
( ) sgn( ) f e e = .
Main Result ()
D
G : The closed-from generalized damping controller (
D
G ) for the
uncertain system is given by
44
1 2 2 2
() () ( ) ( ) ,
n
cc
ac a a
M q Q Qt Mu Q Q t M e k e e f e σ
−
∞
=+ + =+ − + +
(8.3.21)
where the control force ()
c
Qt is given by (5.2.10)
255
1
() ( ) ( )
c T T
Q t A AM A b Aa
−+
= − (8.3.22)
and is obtained on the basis of the nominal system; 0 k > is arbitrary small positive
constant; the choices of 1 n = and 1 σ = would suffice in (8.3.21); and
2
() f e is any
arbitrary strictly increasing odd continuous function of
2
e on the interval (, ) −∞ +∞ and
goes to ∞ as
2
e goes to ∞ .
Note that the advantage of this controller is that no information of the bound of the
uncertainty, Γ , (see equation (5.3.11) is needed to design the controller as in the case of
the previous two controls (
SS
G and
D
G ).
Remark 8.1: Even though the generalized damping controller
D
G (8.3.20) does not
require the knowledge of the bound on the uncertainty Γ , we note from equation (8.3.18)
that with the knowledge of the bound Γ , by using the generalized damping controller
,
D
G the tracking error bound can be defined as
1
2
43
.
42
n
eg
n
ε
−
−
≤Γ
−
(8.3.23)
Remark 8.2: We note from (8.3.20) and (8.3.23) that 1 n = gives a smallest tracking
error bound, thereby yielding the best result in tracking the nominal system’s trajectories.
256
8.3.1 Numerical Results and Simulations of the Generalized Damping Controller
()
D
G
In this section, the same example of the triple pendulum used in Section 2.3 is again
introduced to demonstrate the effectiveness of the proposed control methodologies
suggested in the previous section. Again, it is straightforward to extend this example of
application to more general situations. The numerical integration throughout this section
is done in the Matlab environment, using a variable time step integrator with a relative
error tolerance of
8
10
and an absolute error tolerance of
12
10
.
We select the parameters for the generalized damping controller
D
G given by equation
(8.3.20) by choosing
2
2, 2,
2 2
2, 2,
( / ) , where 0
()
( / ) , where 0
ci i
i
ci i
ee
fe
ee
α ε
α ε
−<
=
≥
(8.3.24)
where ,0
c
αε >
and ε is a suitable small number, we obtain in closed-form the
additional controller needed to compensate for uncertainties in the actual system as
44
1, 2 2, 2
( ) ( ).
n
i i ii
u e k e e f e σ
−
∞
= −+ +
(8.3.25)
Thus, using the additional controller, equation (8.3.25) in equation (5.2.18) (or (8.3.21)),
we obtain the closed-form equation of motion of the controlled actual mechanical system
257
( )
44
11
1 2 2 2
() ( ) ( )
n
c
ca a a
q a M Q t M M e k e e f e σ
−
−−
∞
=+− + +
(8.3.26)
that will cause the actual system to track the trajectory of the nominal system, thereby
compensating for the uncertainty.
Using the generalized damping controller (8.3.26) with the parameters 1, n = 10, k =
1, σ = 2,
c
α = and
2
10 ε
−
= , and with all other parameter values the same as those
previously prescribed in Section 2.3 to perform a simulation, we obtain the trajectory
responses of the mass
3
m in the XY-plane as shown in Figure 8.16. The trajectory
responses in
12 3
, , and θθ θ of the nominal, the controlled, and the actual systems are
shown in Figure 8.17. Both Figure 8.16 and Figure 8.17 are correspondingly
approximately the same with Figure 7.9 and Figure 7.10 that are obtained from the
generalized sliding surface controller
SS
G and also with Figure 8.1 and Figure 8.2 that are
obtained from the generalized damping controller
D
G . The tracking errors between the
nominal system (5.2.11) and the controlled actual system (8.3.26) are obtained in Figure
8.18 for the displacement trajectories (
c
θ θ − ) and in Figure 8.19 for the velocity
trajectories ()
c
θ θ −
. Figure 8.20 presents the control forces both from the nominal system
in order to satisfy the control requirement (Figure 8.20(a)) and from the generalized
damping control
D
G to compensate for the uncertainties (Figure 8.20(b)). Again Figure
8.18 and Figure 8.19 correspondingly show small acceptable errors in tracking
258
trajectories of the nominal system (of O(
3
10
−
) for the displacement and of O(
2
10
−
) for the
velocity). We note that when using the generalized damping controller
D
G , no bound of
the uncertainty is required; however, these errors obtained from Figure 8.18-Figure 8.19
are greater in number than those found in the use of the generalized sliding surface
controller
SS
G and the generalized damping controller
D
G . We note that since we have
the knowledge of the bound on the uncertainties, 100 Γ= , and
0
0.5 α = , the tracking
error bound can be defined as
( )
1/2
2
2
/ 2 5 10
c
e εα
−
≤Γ =× (see (8.3.23)). Both Figure
8.18 and Figure 8.19 show that the tracking errors are within this pre-specified error
bound. The magnitude of the additional control forces obtained by applying the
generalized damping controller
D
G , :
u
Q Mu = , where u is defined as in (8.3.25), is
shown in Figure 8.20(b). This force is seen to be small when compared with
c
Q given in
(5.2.10) (and shown in Figure 8.20(a)) to satisfy the energy control,
23
() () () Et E t E t = + .
We see that all three control schemes—the generalized sliding surface controller
SS
G , the
generalized damping controller
D
G and the generalized damping controller
D
G —require
approximately the same amount of the additional control forces to compensate for
uncertainties as seen from Figure 7.13(b), Figure 8.5(b) and Figure 8.20(b) respectively.
259
Figure 8.16: - Trajectory responses (in meters) of the mass
over a period of 10 secs. of (a) the nominal
system and (b) the controlled system are approximately the same while (c) the actual system yields a totally different
trajectory when the uncertainties in masses are prescribed as , , and
( of nominal masses)
D
G
3
m
1
0.1 kg m δ =
2
0.2 kg m δ = −
3
0.3 kg m δ =
10% ±
260
Figure 8.17: - Angular responses (no. of revolutions ( )) of the masses of the actual
system move away from those of the nominal system as time increases while those of the controlled system track the
nominal system very well
D
G 360
23
( ) and ( ) bm c m
261
Figure 8.18: - Tracking errors in displacement of the masses between the
nominal and the controlled systems (in radians)
D
G ) (
c
θθ −
12 3
( ) , ( ) , and ( ) a m bm c m
262
Figure 8.19: - Tracking errors in velocity of the masses between the
nominal and the controlled systems (in rad/s)
D
G ) (
c
θθ −
12 3
( ) , ( ) , and ( ) a m bm c m
263
Figure 8.20: - (b) Additional force to compensate for uncertainties in masses, , is small when compared to (a)
the control force on the nominal system to satisfy the energy control, , where (c) shows the sum of (in
Newtons)
D
G
u
Q
c
Q
cu
QQ +
264
In order to verify the proposed generalized damping controller
D
G , we again choose
12 3
0.1, 0.05, and 0.2 mm m δδ δ = −= = −
as a second example. We also replace the
monotonic increasing odd continuous function (8.3.24) with a cubic function
( )
3
2 2,
() / ,
i c i
fe e α ε = (8.3.27)
where again ,0
c
αε >
and ε is a suitable small number. We thus obtain in closed-form
the additional controller needed to compensate for uncertainties in the actual system,
( )
3
44
1, 2 2, 2,
( ) /,
n
i i i ci
u e k e e e σα ε
−
∞
= −+ +
(8.3.28)
which upon use of this equation in equation (5.2.18) (or (8.3.21)), we obtain the closed-
form equation of motion of the controlled actual system as
( )
( )
44 3
11
1 2 2 2
() ( ) / .
n
c
ca a a c
q a M Q t M M e k e e e σα ε
−
−−
∞
=+− + +
(8.3.29)
Using the generalized damping controller (8.3.29) to perform a simulation with all other
parameter values the same as those prescribed in the previous example. We obtain the
trajectory responses of the mass
3
m in the XY-plane as shown in Figure 8.21. The
trajectory responses in
12 3
, , and θθ θ are shown in Figure 8.22. Again both Figure 8.21
and Figure 8.22 are correspondingly approximately the same with Figure 7.14 and Figure
7.15 that are obtained from the generalized sliding surface controller
SS
G and also with
Figure 8.6 and Figure 8.7 that are obtained from the generalized damping controller
D
G .
265
The tracking errors between the nominal system (5.2.11) and the controlled actual system
(8.3.26) are obtained in Figure 8.23 for the displacement (
c
θ θ − ) and in Figure 8.24 for
the velocity (
c
θ θ −
). Figure 8.25 presents the control forces both from the nominal
system in order to satisfy the requirement of the energy control (Figure 8.25(a)) and from
the generalized damping controller
D
G to compensate for the uncertainties (Figure
8.25(b)). Again Figure 8.23 and Figure 8.24 correspondingly show small acceptable
errors in tracking the nominal system’s trajectories (of O(
3
10
−
) for displacement and of
O(
2
10
−
) for velocity). Both errors are again seen to be within the pre-specified error norm
2
2
2.9 10 e
−
≤× . However, these errors are greater in number than those found in the use
of the generalized sliding surface controller
SS
G (see Figure 7.16 and Figure 7.17) and in
the generalized damping controller
D
G (see Figure 8.8 and Figure 8.9). This is a
consequence of the fact that no knowledge of the bound on the uncertainties is required in
the description of the generalized damping controller
D
G . However, when considering
the control forces we again see that the three controllers require approximately the same
amount of the additional control forces,
u
Q , to compensate for uncertainties as seen from
Figure 7.18(b), Figure 8.10(b) and Figure 8.25(b). We note that in Figure 8.25(b), we plot
the additional control force, :
u
Q Mu = , using u from equation (8.3.25). This control
force is also small when compared with
c
Q given in (5.2.10) (and shown in Figure
8.25(a)) to satisfy the energy control requirement,
23
() () () Et E t E t = + .
266
Figure 8.21: - Trajectory responses (in meters) of the mass
over a period of 10 secs. of (a) the nominal
system and (b) the controlled system are approximately the same while (c) the actual system yields a totally different
trajectory when the uncertainties in masses are prescribed as , , and
D
G
3
m
1
0.1 kg m δ = −
2
0.05 kg m δ =
3
0.2 kg m δ = −
267
Figure 8.22: - Angular responses (no. of revolutions ( )) of the masses of the actual
system move away from those of the nominal system as time increases while those of the controlled system track the
nominal system very well
D
G 360
23
( ) and ( ) bm c m
268
Figure 8.23: - Tracking errors in displacement of the masses between the
nominal and the controlled systems (in radians)
D
G ) (
c
θθ −
12 3
( ) , ( ) , and ( ) a m bm c m
269
Figure 8.24: - Tracking errors in velocity of the masses between the
nominal and the controlled systems (in rad/s)
D
G ) (
c
θθ −
12 3
( ) , ( ) , and ( ) a m bm c m
270
Figure 8.25: - (b) Additional force to compensate for uncertainties in masses, , is small when compared to (a)
the control force on the nominal system to satisfy the energy control, , where (c) shows the sum of (in
Newtons)
D
G
u
Q
c
Q
cu
QQ +
271
Similar to the previous two controllers, our generalized damping control approach (
D
G )
is further verified by considering the case of a random uncertainty of 50% ±
of the
nominal value chosen for each mass of the triple pendulum as described in Section 2.3.
We again assume
12 3
0.5, 0.8, 1.5 mm m δδ δ == −= and perform a simulation using
equation (8.3.29), with all other parameter values the same as those prescribed in the
previous example. The trajectory responses of the mass
3
m in the XY-plane and the
trajectory responses in
12 3
, , and θθ θ of the nominal, the controlled, and the actual
systems are shown in Figure 8.26 and Figure 8.27 respectively. Both Figure 8.26 and
Figure 8.27 are correspondingly approximately the same as Figure 7.22 and Figure 7.23,
which are obtained from the generalized sliding surface controller
SS
G . They are also
correspondingly the same as Figure 8.11 and Figure 8.12, which are obtained from the
generalized damping controller
D
G . The tracking errors between the nominal system
(5.2.11) and the controlled actual system (8.3.29) are obtained in Figure 8.28 for the
displacement trajectories (
c
θ θ − ) and in Figure 8.29 for the velocity trajectories ()
c
θ θ −
.
Figure 8.30 presents the control forces both from the nominal system in order to satisfy
the control requirement (Figure 8.30(a)) and from the generalized damping controller
D
G
to compensate for the uncertainties (Figure 8.30(b)). Again Figure 8.28 and Figure 8.29
correspondingly show small acceptable errors in tracking trajectories of the nominal
system (of O(
3
10
−
) for the displacement and of O(
2
10
−
) for the velocity). The numerical
272
results shown in Figure 8.26-Figure 8.29 show that even for a large uncertainty (a
situation in which a linear controller is not applicable), the proposed generalized damping
control design
D
G is able to guarantee motion control of the actual system to mimic the
motions of the nominal system. With the knowledge of the bound on the uncertainties,
800 Γ= , we can define the tracking error bound by using equation (8.3.23) as
( )
1/3
2
2
/ 2 5.8 10
c
e εα
−
≤Γ ≈ × . Both Figure 8.28 and Figure 8.29 show that the tracking
errors are within this pre-specified error bound. The magnitude of the additional control
forces obtained by applying the generalized damping controller
D
G , :
u
Q Mu = , where u
is defined as in (8.3.28), is shown in Figure 8.30(b). This force is seen to be small when
compared with
c
Q given in equation (5.2.10) (and shown in Figure 8.30(a)) to satisfy the
energy control requirement,
23
() () () Et E t E t = + . We again see that all three control
schemes—the generalized sliding surface controller
SS
G , the generalized damping
controller
D
G , and the generalized damping controller
D
G —require approximately the
same amount of the additional control forces to compensate for uncertainties as seen from
Figure 7.26(b), Figure 8.15(b) and Figure 8.30(b) respectively.
273
Figure 8.26: - Trajectory responses (in meters) of the mass
over a period of 10 secs. of (a) the nominal
system and (b) the controlled system are approximately the same while (c) the actual system yields a totally different
trajectory when the uncertainties in masses are prescribed as , , and
D
G
3
m
1
0.5 kg m δ =
2
0.8 kg m δ = −
3
1.5 kg m δ =
274
Figure 8.27: - Angular responses (no. of revolutions ( )) of the masses of the actual
system move away from those of the nominal system as time increases while those of the controlled system track the
nominal system very well
D
G 360
23
( ) and ( ) bm c m
275
Figure 8.28: - Tracking errors in displacement of the masses between the
nominal and the controlled systems (in radians)
D
G ) (
c
θθ −
12 3
( ) , ( ) , and ( ) a m bm c m
276
Figure 8.29: - Tracking errors in velocity of the masses between the
nominal and the controlled systems (in rad/s)
D
G ) (
c
θθ −
12 3
( ) , ( ) , and ( ) a m bm c m
277
Figure 8.30: - (b) Additional force to compensate for uncertainties in masses, , is small when compared to (a)
the control force on the nominal system to satisfy the energy control, , where (c) shows the sum of (in
Newtons)
D
G
u
Q
c
Q
cu
QQ +
278
Remark 8.3: The guaranteed errors within the desired error bounds in tracking in both
displacement and velocity from Figure 7.11-Figure 7.12 (also Figure 7.16-Figure 7.17 or
Figure 7.24-Figure 7.25) of the generalized sliding surface controller (
SS
G ) and Figure
8.3-Figure 8.4 (also Figure 8.8-Figure 8.9 or Figure 8.13-Figure 8.14) of the generalized
damping controller (
D
G ); the small acceptable errors in tracking trajectories from Figure
8.18-Figure 8.19 (also Figure 8.23-Figure 8.24 or Figure 8.28-Figure 8.29) of the
generalized damping controller (
D
G ); and the small additional control forces to
compensate for uncertainties from the three controllers guarantee that the proposed
control designs are robust with respect to the uncertainties in modeling systems. Both the
generalized sliding surface controller
SS
G and the generalized damping controller
D
G
require knowledge of the bound on uncertainty in order to guarantee a tracking signal of a
desired reference trajectory within the desired error bound. In contrast, when using the
generalized damping controller
D
G , tracking of the nominal system’s trajectory can be
obtained regardless of the knowledge of the uncertainty’s bound. However, the tracking
results from the generalized damping controller
D
G are the least optimal when compared
with those obtained from the other two controllers.
8.4 Summary
Two additional additive controllers (
D
G and
D
G ) based on a generalization of the
concept of the damping control are developed in this chapter. Both controllers are able to
279
rescind the effects of unwanted uncertainties in the system modeling process and also
allow the actual system’s trajectories to track those obtained from the desired nominal
system (the best assessment of the actual system). Their parameters can be altered to
meet the practical limitations of a specific controller being used. The main contributions
of this chapter are:
(i) In the system containing unknown uncertainties, one usually has to resort to
discontinuous controllers to obtain GUES. To avoid this, the requirement of
GUES can be relaxed by using the idea of the generalized damping control.
The amount of the forcing function needed for tracking controls can also be
reduced while the tracking results are still in a fully acceptable region.
(ii) The formulation of the closed-form generalized damping controller ()
D
G is
given by
( )
0
11 2 2
0 0
()
()
() ( ) ,
c
ac a
t
c
a id
M q Q Q t Mu
t
Q Q t M e K e d Ke f e
ρ
τ τ σ
α
=+ +
Γ+
=+ − + + +
∫
where ,0
id
KK > are arbitrary small positive numbers; the choices of 1 σ =
and
2
() 1 f e ≥ , where
2
() f e is any arbitrary monotonic increasing odd
continuous function of
2
e on the interval (, ) −∞ +∞ , would be sufficient; Γ is
chosen based on the estimate of q δ from equation (5.3.11);
0
1 α < is a small
280
positive number that is defined by using the estimate of
a
M from equation
(5.3.6); and
0
ρ is chosen such that
0
1. ρ ≥
(iii) The formulation of the closed-form generalized damping controller ()
D
G is
given by
44
1 2 2 2
() () ( ) ( ) ,
n
cc
ac a a
M q Q Qt Mu Q Q t M e k e e f e σ
−
∞
=+ + =+ − + +
where 0 k > is arbitrary small positive constant; the choices of 1 n = and
1 σ = would be sufficient;
2
() f e is any arbitrary strictly increasing odd
continuous function of
2
e on the interval (, ) −∞ +∞ and goes to ∞ as
2
e goes
to ∞ .
(iv) We note that both generalized damping controllers (
D
G and
D
G ) involve: (i)
the description of the actual system given by
ac a
Mq Q = , whose parameters
are only known imperfectly; (ii) the control force ()
c
Qt given by (5.2.10) that
is obtained on the basis of our best estimate of the actual system, namely on
the basis of the corresponding nominal system; and (iii) the additional control
forces :
u
Q Mu = to compensate for uncertainties.
(v) The generalized damping controller
D
G is able to guarantee tracking of the
nominal system’s trajectories with the requirement of the knowledge of the
bound on the uncertainties. However, when using the generalized damping
controller
D
G , tracking of trajectories of the nominal system can be obtained
281
with no further information about the uncertainty’s bound. Consequently, its
tracking responses are the worst when compared to the generalized sliding
surface controller
SS
G and the generalized damping controller
D
G .
(vi) Since no information about the bound of the uncertainty is needed, the
generalized damping controller
D
G has the simplest structure and the fastest
computational time, while the other two controllers (
SS
G and
D
G ) need more
time to compute the bound on the uncertainties in the system.
(vii) Even though the information about the bound on the uncertainty is not needed
for the generalized damping controller
D
G , a tracking error bound can be pre-
specified if knowledge of the bound on uncertainties is provided.
(viii) The function ( )
2
fe and the parameters that define the generalized damping
controllers both for
D
G and for
D
G can be chosen depending on practical
considerations of the control environment, and on the extent to which the
compensation is desired. Thus, there is flexibility in using these additional
controllers, especially when dealing with large, complex multi-body systems.
The flexibility in choosing the function ( )
2
fe becomes more important when
practical limitations of the control are specified. For example, the uses of the
specified quadratic function and the cubic function in the example considered
obviate the need for a high-gain controller and also allow the additive control
282
to be continuous, thereby preventing chattering.
(ix) While the only uncertainties demonstrated in this chapter are those related to
the masses of the triple pendulum, the formulations of the proposed control
methodologies consider both general sources of uncertainties—uncertainties
in the description of physical systems and uncertainties in the given force on
the system. The closed-form controllers developed herein are therefore
general enough to be applicable to more complex dynamical systems in which
uncertainties in the given force may be important.
(x) The generalized damping control designs (
D
G and
D
G ) are both evaluated
using numerical solution comparisons of the tracking errors between the
nominal and the controlled actual systems. The results demonstrate that both
generalized damping controllers
D
G and
D
G have good transient behavior
and are robust with respect to the uncertainties in the modeling process.
Furthermore, with the simplicity and accuracy obtained, the control schemes
proposed in this chapter can be implemented for various cases and for even
more complex dynamical problems.
In the next chapter, the main conclusions of this work are summarized in greater detail.
283
Chapter 9
GENERAL CONCLUSIONS
The present chapter outlines the fundamental conceptualizations in analytical dynamics
from Chapter 2 and Chapter 3 and provides a summary of the procedures in modeling
constrained mechanical systems utilizing new methodologies from Chapter 4 Chapter 5
and Chapter 6. Some comparisons among the three controllers in Chapter 7 and Chapter
8 are also discussed and summarized.
The main contributions of this dissertation are as follows.
(i) If one needs to conceptualize a constrained mechanical system so that the
equations of motion that are obtained properly reflect physical
observational evidence, one should follow the three-step conceptualization
of constrained motion which involves:
– the description of the unconstrained system in which the coordinates
and the corresponding virtual displacements are all independent of one
another,
– the description of the constraints, and
– the description of the constrained system using the previous two
descriptions.
284
(ii) Conflating the unconstrained and the constrained descriptions, the kinetic
energy of such a system cannot, in general, be considered either as
– an appropriate description of the given constrained system, or
– an appropriate description of the given unconstrained system on which
the constraints then need to be further imposed.
(iii) In the special situation in which the system must satisfy only holonomic
constraints, substitution of the constraint into the kinetic energy of the
unconstrained system represents simply an appropriate change in
coordinates, and the use of the amended kinetic energy will then lead to
the correct equations of motion.
(iv) In Lagrangian mechanics, describing mechanical systems with more than
the minimum number of required coordinates is helpful in forming the
equations of motion of complex multi-body mechanical systems since this
often requires less labor in the modeling process. The reason that we do
not usually use more coordinates than the minimum number is because in
doing so we often encounter singular mass matrices, and then standard
methods for handling such constrained mechanical systems become
inapplicable. For example, methods such as the use of the fundamental
285
equation, and all others that rely on the invertibility of the mass matrix
cannot be used.
(v) If one would like to derive the constrained equations of motion of a
general mechanical system that has either a positive semi-definite or
positive definite mass matrix, which is subjected to the ideal constraints
Aq b =
and the non-ideal constraints described by the n-vector ( , , ) C q qt
(under the proviso that matrix
ˆ
|
TT
M MA =
has rank n), one could
obtain the (explicit) constrained equation of motion of the mechanical
system by following the three-step conceptualization of constrained
motion as follows in terms of a new unconstrained auxiliary system:
(1) Description of the unconstrained auxiliary system:
(a) Replace the mass matrix 0 M ≥ of the actual unconstrained
mechanical system with the augmented mass matrix
2
T
T
AG
M M A GA α = + in which 0
T
AG
M >
and,
(b) choose
2
zb α =
and replace the ‘given’ force Q of the actual
unconstrained mechanical system with the augmented ‘given’
force
2
2
,
T
T
AG b
Q Q A Gb
α
α = + ;
This describes the new unconstrained auxiliary system
2
,
;
TT
AG AG b
M qQ
α
=
286
(2) Description of the constraints: Subject this unconstrained auxiliary
system to the same set of constraints and initial conditions as the actual
unconstrained mechanical system S;
(3) Description of the constrained auxiliary system: Apply the
fundamental equation to the unconstrained auxiliary system described
in (1), which is subjected to the constraints described in (2);
The resulting equation of motion of this constrained auxiliary system has
two important features:
(1) The explicit acceleration of the constrained auxiliary system
T
AG
S is
the same, at each instant in time, as the explicit acceleration of the
constrained mechanical system S, and,
(2) at each instant in time, the constraint force
c
Q acting on the
unconstrained mechanical system S, because of the presence of the
constraints imposed on it, is the same as the constraint force
2
,
T
c
AG b
Q
α
acting on the unconstrained auxiliary system
T
AG
S .
Our findings thus lead to the following conclusion: the derivation of
constrained mechanical systems through our three-step conceptualization
of a constrained auxiliary system results in the dynamics of the actual
287
constrained mechanical system S being completely mimicked by the
dynamics of the aforementioned constrained auxiliary system
T
AG
S .
(vi) Real-life complex multi-body systems are in general highly nonlinear and
intrinsically error-prone due to uncertainties in the modeling system. The
uncertainty arises from two general sources: uncertainty in the knowledge
of the physical system and/or uncertainty in the ‘given’ force applied to
the system. In recent years, the optimal controller that can exactly describe
those highly nonlinear constrained systems, when no uncertainty is
assumed can be analytically obtained with the aid of a recent finding in
analytical dynamics, the so-called fundamental equation (see Chapter 2).
This leads to the exact closed-form equation of motion for nonlinear
constrained mechanical systems that can guarantee constraint-following
for the mechanical system assumed. However, this analytical result is
correct only under the assumption that the modeling of the physical
system has no errors and uncertainties.
(vii) In this work, the general closed-form equation of motion for nonlinear
uncertain multi-body systems—the so-called controlled actual system—
has been developed. This is obtained by adding to the nominal controller
provided by the fundamental equation an additional control that
288
compensates for the uncertainties. Uncertainties in the two time-dependent
dynamical quantities M and Q that characterize the system and the given
forces on the system can be accommodated.
(viii) Control configuration variables subjected to both holonomic and
nonholonomic control requirements, or to a combination of such
requirements, are handled in a uniform manner in the proposed control
methodology.
(ix) The proposed closed-form controlled actual system is illustrated by
incorporating three control designs—the generalized sliding surface
control (
SS
G ), the generalized damping control (
D
G ), and the generalized
damping control (
D
G )— resulting in three sets of closed-form controllers
that can guarantee, regardless of uncertainty, a tracking signal of a desired
reference trajectory of the nominal system model. The nominal model is
the model adduced from our best assessment of the characteristics of the
real-life system.
(x) Among the three control designs, the generalized damping controller
D
G
has the least accurate responses in tracking the nominal system’s
trajectory, while the generalized sliding surface controller
SS
G and the
generalized damping controller
D
G have approximately comparable
289
results. Note that all three controllers require approximately the same
amount of additional forces to compensate for uncertainties.
(xi) Both the generalized sliding surface controller
SS
G and the generalized
damping controller
D
G require knowledge of the bound on uncertainty to
guarantee tracking of a desired reference trajectory. In contrast, when
using the generalized damping controller
D
G , tracking of the nominal
system’s trajectory can be obtained regardless of the knowledge of the
uncertainty’s bound. Consequently, the tracking responses from the
generalized damping controller
D
G are the least optimal when compared
with those obtained from the other two controllers (
SS
G and
D
G ).
Nevertheless, all three controllers have good responsive performance and
robustness for solving control problems of the nonlinear system with
various uncertainties in various levels.
(xii) Even though the information of the bound on the uncertainty is not needed
for the generalized damping controller
D
G , the guaranteed tracking error
bound can be estimated if knowledge of the bound on uncertainties is
provided.
(xiii) Additionally, the control functions and the parameters that define all three
additional additive controllers (
SS
G ,
D
G , and
D
G ) can be chosen
290
depending on the practical considerations of the control environment and
on the extent to which the compensation is desired. For example, in all
three control designs, the parameters can be adjusted so that when the
actual system is required to track the nominal system, desired error bounds
can be guaranteed. Thus, there is flexibility in using these additional
controllers, especially when dealing with large, complex multi-body
systems, for instance, the uses of the specified quadratic function and the
cubic function in the examples considered obviate the need for a high-gain
controller and also allow the additive controls to be continuous, thereby
preventing chattering.
(xiv) While the only uncertainties demonstrated in this work are those related to
the masses of the triple pendulum problem, the formulations of the current
control methodologies consider both general sources of uncertainties,
uncertainties in the description of physical systems and uncertainties in the
given force on the system. The closed-form controllers developed herein
are therefore general enough to be applicable to more complex dynamical
systems in which the uncertainties in the given force or indeed both types
of uncertainties may be important.
291
(xv) Numerical solution comparisons of the tracking errors between the
nominal and the controlled actual systems are used to evaluate the
effectiveness of the designs of the compensating controllers. The results
demonstrate that the three proposed controllers have good transient
behavior and are robust with respect to the uncertainties in the modeling
process. Furthermore, with the simplicity and accuracy obtained, the
control schemes proposed in this work can be implemented for various
cases and for even more complex dynamical problems.
292
Chapter 10
FUTURE WORK
Some of the future directions can be summarized as follows:
While the formulation of the current control methodology of the closed-form equations of
motion for nonlinear uncertain constrained multi-body mechanical systems—the
controlled actual system—considers only a system with positive definite mass matrix, the
proposed control methodology can be implemented using the conceptualization of
constrained motion as follows in terms of an auxiliary constrained system, so that the
closed-form equations of motion for nonlinear uncertain constrained multi-body
mechanical systems can have either a positive semi-definite or positive definite mass
matrix.
In general, the controlled system obtained from the current control methodologies cannot
achieve the same level of accuracy in satisfying the control requirements or constraints as
obtained from the nominal system. The constraints are usually functions of (i) the
physical system’s descriptions such as the mass m, the stiffness k and the damping
coefficient d ; and (ii) the states in the system which are the displacement q and the
velocity q ; i.e., ( ;, ) ( ;, ) A m qqq b m qq = . Under uncertainties, the constraint equations of
293
the controlled system become ( ; , ) ( ; , )
ccc cc
A m m qqq b m m qq δ δ +=+ , which are
functions of the actual system’s properties ( mm δ + ) and the controlled states ( ,
cc
qq ).
The proposed control approaches developed in this work can guarantee tracking of the
nominal system’s trajectories within some error bounds so that ( , ) (, )
cc
q q qq ≈ ; however,
the physical system’s properties are still uncertain ( mm δ + ). As a result, the constraints
of the controlled system yield ( ;, ) ( ;, ) A m m qqq b m m qq δδ + =+ . With the comparison
between the constraints of the nominal system ( ( ,, ) ( ,, ) A m qqq b m qq = ), which are
functions of the nominal system’s properties ( m ) and the nominal states ( , qq ), and the
constraints of the controlled system ( ( ;, ) ( ;, ) A m m qqq b m m qq δδ + =+ ), which are
functions of the actual system’s properties ( mm δ + ) and the nominal states ( , qq ), we
can obtain information on the differences in the system’s properties between the nominal
and actual systems ( m δ ). This leads to the problem of system identification. We can thus
finally reveal the uncertainty, i.e., we are able to turn an uncertain system into a certain
system, in the modeling system, which is by far one of the most problematic factors in
deriving the equations of motion for nonlinear uncertain constrained multi-body
mechanical systems.
294
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Appendix A
SIMULTANEOUS UNCERTAINTIES IN MASSES AND
GIVEN FORCES ON THE SYSTEM
When there are uncertainties in the description of nonlinear uncertain constrained
mechanical systems, can be conceptualized as (see (5.3.8))
(A-1)
Using the relation [45]
(A-2)
in (A-1), we then separate the right-hand side of (A-1) into two parts:
(i) (A-3)
and
(ii) (A-4)
Then using (A-3) and (A-4) in (A-1), we get
(A-5)
q δ
1 1 11
1 1 11
() ( )
( , ) ( , , ) ( , ) ( , , ) ( , ) ( , ) ( ).
() ( ) () .
c
aa a
c
i ii
q M qt Q q qt M qt Q q qt M qt M qt Q t
MM Q Q M Q MM M Q
δ
δ δ δ
− − −−
− − −−
≈ − +−
= + +− + + −
1 1 1 11 1
( ) ( ) , M M M M I MM MM δ δδ
− − − − − −
+ =−+
1 1 1 1 11 1 1
1 1 11 1 1 1
11 1
( )() ( ( ) )()
( ) ( )
() () ,
M M Q Q M Q M M I MM MM Q Q M Q
MQ M I M M M MQ M M Q MQ
MM M M Q MM Q
δ δ δ δ δ
δ δ δδ
δδ δδ
− − − − − − − −
− − − − − − −
−− −
+ +− = − + +−
= − + ++ −
= −+ + +
11 11 1 1 1 1
1 11 1
11
( ) ( ( ) )
( )
() .
cc
c
c
M M M Q M M I MM MM M Q
M I MM MM Q
M M MM Q
δ δδ
δδ
δδ
−− −− −− − −
− − − −
−−
+ − = −+ −
= −+
= −+
11 1
() ( )() .
c
q MM M M Q Q MM Q δ δδ δδ
−− −
≈− + + + +
300
Appendix B
THE PROOF OF
∞ ∞
≥
T
xy x y
Let be any arbitrary n-vectors, where the i-th component, , of the n-vector y
is any monotonic increasing odd continuous function of on the interval ,
where is the i-th component of the n-vector x, i.e., .
Define
(B-1)
(B-2)
and
(B-3)
Since is an odd function of , i.e., and have the same sign, we have
(B-4)
Consider
(B-5)
and x y
i
y
i
x (, ) −∞ +∞
i
x : ()
ii
y fx =
: max ,
i
i
xx
∞
=
: max ,
i
i
yy
∞
=
1
: .
n
T
i i
i
x y xy
=
=
∑
i
y
i
x
i
x
i
y
.
i i i i i i
x y xy xy = =
11
max max .
nn
T
i i i i i i
ii
ii
x y x y x y x y
= =
= = ⋅≥ ⋅
∑∑
301
The last inequality follows that is a monotonic increasing odd function of . Then
using (B-1), (B-2) in (B-5), we obtain
(B-6)
i.e.,
(B-7)
i
y
i
x
,
T
x y x y
∞∞
≥
: 1.
T
x y
x y
γ
∞∞
= ≤
Abstract (if available)
Abstract
This dissertation develops in a unified manner a new and simple approach for the modeling and controlling of general nonlinear constrained mechanical systems. Since, in general, the description of constrained mechanical systems is highly nonlinear, the determination of the equations of motion of such highly nonlinear constrained mechanical systems is considered one of the central problems in analytical dynamics. Even though this problem has been the focus of numerous studies, several questions remain unanswered at the present time. Of particular importance among these questions is an appropriate approach to obtaining the equations of motion for constrained mechanical systems when the mass matrix of the unconstrained mechanical system is singular, or if there are uncertainties in the description of the systems, what should be done to cancel the effects of this uncertainty while the constrained equations of motion are being derived. Moreover, misinterpretations persist concerning the fundamental conceptualizations for deriving these constrained equations of motion. ❧ The theoretical framework developed in this dissertation provides a concrete explanation that deals with those fundamental conceptualizations for deriving the equations of motion for general nonlinear constrained mechanical systems. Utilizing these fundamental conceptualizations, an explicit constrained equation of motion that works with systems containing either positive semi-definite or positive definite mass matrices is also further developed. And when considering uncertainties, the explicit equation of motion for nonlinear constrained mechanical systems has been modified by augmenting additional additive controllers that are able to guarantee tracking of reference trajectories of the system with no uncertainty assumed. The new, simple, general, and closed-form equations of motion for nonlinear constrained uncertain mechanical systems are thus developed. The results herein provide deeper insights into the behavior of constrained motion and open up new approaches to modeling complex, uncertain, constrained mechanical systems, such as those encountered in real-life multi-body dynamics.
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Asset Metadata
Creator
Wanichanon, Thanapat
(author)
Core Title
On the synthesis of controls for general nonlinear constrained mechanical systems
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Aerospace Engineering
Publication Date
04/30/2012
Defense Date
02/27/2012
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
energy control,fundamental equation of constrained motion,Hamel paradox,OAI-PMH Harvest,singular mass matrix,tracking control,uncertain system
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Udwadia, Firdaus E. (
committee chair
), Flashner, Henryk (
committee member
), Safonov, Michael G. (
committee member
), Shiflett, Geoffrey R. (
committee member
)
Creator Email
sumosung@hotmail.com,wanichan@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-17675
Unique identifier
UC11288849
Identifier
usctheses-c3-17675 (legacy record id)
Legacy Identifier
etd-Wanichanon-683.pdf
Dmrecord
17675
Document Type
Dissertation
Rights
Wanichanon, Thanapat
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
energy control
fundamental equation of constrained motion
Hamel paradox
singular mass matrix
tracking control
uncertain system