DTIC ADA441056: Motion Control for Nonholonomic Systems on Matrix Lie Groups

129 Pages·0.56 MB·English

by  Defense Technical Information Center

P .D. T
H HESIS
Motion Control for Nonholonomic Systems on Matrix Lie
Groups
by Herbert Karl Struemper
Advisor: P.S. Krishnaprasad
CDCSS Ph.D. 98-1
(ISR Ph.D. 98-1)
+ CENTER FOR DYNAMICS
C D
- AND CONTROL OF
S SMART STRUCTURES
The Center for Dynamics and Control of Smart Structures (CDCSS) is a joint Harvard University, Boston University, University of Maryland center,
supported by the Army Research Office under the ODDR&E MURI97 Program Grant No. DAAG55-97-1-0114 (through Harvard University). This
document is a technical report in the CDCSS series originating at the University of Maryland.
Web site http://www.isr.umd.edu/CDCSS/cdcss.html
Report Documentation Page Form Approved
OMB No. 0704-0188
Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and
maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information,
including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington
VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it

does not display a currently valid OMB control number.
1. REPORT DATE 3. DATES COVERED
1998 2. REPORT TYPE -
4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
Motion Control for Nonholonomic Systems on Matrix Lie Groups
5b. GRANT NUMBER
5c. PROGRAM ELEMENT NUMBER
6. AUTHOR(S) 5d. PROJECT NUMBER
5e. TASK NUMBER
5f. WORK UNIT NUMBER
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION
Army Research Office,PO Box 12211,Research Triangle Park,NC,27709 REPORT NUMBER
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S)
11. SPONSOR/MONITOR’S REPORT
NUMBER(S)
12. DISTRIBUTION/AVAILABILITY STATEMENT
Approved for public release; distribution unlimited
13. SUPPLEMENTARY NOTES
The original document contains color images.
14. ABSTRACT
see report
15. SUBJECT TERMS
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF
ABSTRACT OF PAGES RESPONSIBLE PERSON

a. REPORT b. ABSTRACT c. THIS PAGE 128
unclassified unclassified unclassified
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
Abstract
Title of Dissertation: Motion Control for Nonholonomic Systems
on Matrix Lie Groups
Herbert Karl Struemper, Doctor of Philosophy, 1997
Dissertation directed by: Professor P. S. Krishnaprasad
Department of Electrical Engineering
In this dissertation we study the control of nonholonomic systems de(cid:12)ned by
invariant vector (cid:12)elds on matrix Lie groups. We make use of canonical construc-
tions of coordinates and other mathematical tools provided by the Lie group
setting. An approximate tracking control law is derived for so-called chained
form systems which arise as local representations of systems on a certain nilpo-
tentmatrixgroup. Afterstudying thetechnique ofnilpotentizationinthesetting
of systems on matrix Lie groups we show how motion control laws derived for
nilpotent systems can beextended to nilpotentizable systems using feedback and
state transformations. The proposed control laws exhibit highly oscillatory com-
ponents both for tracking and feedback stabilization of local representations of
nonholonomic systems on Lie groups. Applications to the control and analysis
of the kinematics of mechanical systems are discussed and numerical simulations
are presented.
Motion Control for Nonholonomic Systems
on Matrix Lie Groups
by
Herbert Karl Struemper
Dissertation submitted to the Faculty of the Graduate School
of The University of Maryland in partial ful(cid:12)llment
of the requirements for the degree of
Doctor of Philosophy
1997
Advisory Committee:
Professor P. S. Krishnaprasad, Chairman/Advisor
Professor John J. Benedetto
Professor Steven I. Marcus
Professor Shihab A. Shamma
Professor Andre L. Tits
(cid:13)c Copyright by
Herbert Karl Struemper
1997
Dedication
To Andrea
ii
Acknowledgements
In this dissertation we study thecontrol of nonholonomic systems de-
(cid:12)ned by invariant vector (cid:12)elds on matrix Lie groups. We make use of
canonical constructions of coordinates and other mathematical tools
provided by the Lie group setting. An approximate tracking control
law is derived for so-called chained form systems which arise as local
representations of systems on a certain nilpotent matrix group. After
studying the technique of nilpotentization in the setting of systems
on matrix Lie groups we show how motion control laws derived for
nilpotent systems can be extended to nilpotentizable systems using
feedback and state transformations. The proposed control laws ex-
hibit highly oscillatory components both for tracking and feedback
stabilization of local representations of nonholonomic systems on Lie
groups. Applications to the control and analysis of the kinematics
of mechanical systems are discussed and numerical simulations are
presented.
iii
In this dissertation we study thecontrol of nonholonomic systems de-
(cid:12)ned by invariant vector (cid:12)elds on matrix Lie groups. We make use of
canonical constructions of coordinates and other mathematical tools
provided by the Lie group setting. An approximate tracking control
law is derived for so-called chained form systems which arise as local
representations of systems on a certain nilpotent matrix group. After
studying the technique of nilpotentization in the setting of systems
on matrix Lie groups we show how motion control laws derived for
nilpotent systems can be extended to nilpotentizable systems using
feedback and state transformations. The proposed control laws ex-
hibit highly oscillatory components both for tracking and feedback
stabilization of local representations of nonholonomic systems on Lie
groups. Applications to the control and analysis of the kinematics
of mechanical systems are discussed and numerical simulations are
presented.
In this dissertation we study thecontrol of nonholonomic systems de-
(cid:12)ned by invariant vector (cid:12)elds on matrix Lie groups. We make use of
canonical constructions of coordinates and other mathematical tools
provided by the Lie group setting. An approximate tracking control
law is derived for so-called chained form systems which arise as local
representations of systems on a certain nilpotent matrix group. After
studying the technique of nilpotentization in the setting of systems
on matrix Lie groups we show how motion control laws derived for
nilpotent systems can be extended to nilpotentizable systems using
feedback and state transformations. The proposed control laws ex-
iv
hibit highly oscillatory components both for tracking and feedback
stabilization of local representations of nonholonomic systems on Lie
groups. Applications to the control and analysis of the kinematics
of mechanical systems are discussed and numerical simulations are
presented.
v
Table of Contents
Section Page
1 Introduction 1
2 Preliminaries 9
2.1 Lie groups and Lie algebras : : : : : : : : : : : : : : : : : : : : : 9
2.2 Matrix Lie groups : : : : : : : : : : : : : : : : : : : : : : : : : : : 15
2.3 Systems on matrix Lie groups : : : : : : : : : : : : : : : : : : : : 19
2.4 Local representations of systems on Lie groups : : : : : : : : : : : 21
2.4.1 Single exponential representation : : : : : : : : : : : : : : 22
2.4.2 Product of exponentials representation : : : : : : : : : : : 25
2.5 Example Systems on Matrix Lie Groups : : : : : : : : : : : : : : 27
2.5.1 Rigid motions on n : : : : : : : : : : : : : : : : : : : : : 27
R
2.5.2 Unicycle : : : : : : : : : : : : : : : : : : : : : : : : : : : : 29
2.5.3 Spacecraft : : : : : : : : : : : : : : : : : : : : : : : : : : : 31
2.5.4 Underwater Vehicle : : : : : : : : : : : : : : : : : : : : : : 33
3 Approximate Inversion 35
3.1 Problem De(cid:12)nition : : : : : : : : : : : : : : : : : : : : : : : : : : 35
vi