CORE Metadata, citation and similar papers at core.ac.uk Provided by Texas A&M Repository GENERALIZATION OF ROTATIONAL MECHANICS AND APPLICATION TO AEROSPACE SYSTEMS A Dissertation by ANDREW JAMES SINCLAIR Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY May 2005 Major Subject: Aerospace Engineering GENERALIZATION OF ROTATIONAL MECHANICS AND APPLICATION TO AEROSPACE SYSTEMS A Dissertation by ANDREW JAMES SINCLAIR Submitted to Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Approved as to style and content by: John L. Junkins John E. Hurtado (Co-Chair of Committee) (Co-Chair of Committee) Srinivas R. Vadali John H. Painter (Member) (Member) Helen L. Reed (Head of Department) May 2005 Major Subject: Aerospace Engineering iii ABSTRACT Generalization of Rotational Mechanics and Application to Aerospace Systems. (May 2005) Andrew James Sinclair, B.S., University of Florida; M.S., University of Florida Co–Chairs of Advisory Committee: Dr. John L. Junkins Dr. John E. Hurtado This dissertation addresses the generalization of rigid-body attitude kinematics, dynamics, and control to higher dimensions. A new result is developed that demon- strates the kinematic relationship between the angular velocity in N-dimensions and the derivative of the principal-rotation parameters. A new minimum-parameter de- scription of N-dimensional orientation is directly related to the principal-rotation parameters. The mapping of arbitrary dynamical systems into N-dimensional rotations and the merits of new quasi velocities associated with the rotational motion are studied. A Lagrangian viewpoint is used to investigate the rotational dynamics of N-dimensional rigid bodies through Poincar´e’s equations. The N-dimensional, orthogonal angular- velocity components are considered as quasi velocities, creating the Hamel coefficients. Introducing a new numerical relative tensor provides a new expression for these co- efficients. This allows the development of a new vector form of the generalized Euler rotational equations. An N-dimensional rigid body is defined as a system whose configuration can be completely described by an N×N proper orthogonal matrix. This matrix can be related to an N×N skew-symmetric orientation matrix. These Cayley orientation variables and the angular-velocity matrix in N-dimensions provide a new connection iv between general mechanical-system motion and abstract higher-dimensional rigid- body rotation. The resulting representation is named the Cayley form. Several applications of this form are presented, including relating the combined attitude and orbital motion of a spacecraft to a four-dimensional rotational motion. A second example involves the attitude motion of a satellite containing three momentum wheels, which is also related to the rotation of a four-dimensional body. The control of systems using the Cayley form is also covered. The wealth of work on three-dimensional attitude control and the ability to apply the Cayley form motivates the idea of generalizing some of the three-dimensional results to N- dimensions. Some investigations for extending Lyapunov and optimal control results to N-dimensional rotations are presented, and the application of these results to dynamical systems is discussed. Finally, the nonlinearity of the Cayley form is investigated through computing the nonlinearity index for an elastic spherical pendulum. It is shown that whereas the Cayley form is mildly nonlinear, it is much less nonlinear than traditional spherical coordinates. v ACKNOWLEDGMENTS Special thanks to my advisors Dr. John L. Junkins and Dr. John E. Hurtado. Their deep insight into dynamics and control and their collegial philosophy has bene- fited me greatly here at Texas A&M and will continue to do so throughout my career. I thank my committee members Dr. Rao Vadali and Dr. John Painter for their guid- ance. I am also grateful to Dr. Daniele Mortari for many educational, constructive, and entertaining conversations on the subject of generalized rotational mechanics. Thanks to Lisa Willingham and Karen Knabe for all of their assistance throughout my degree program. I wish to thank F. Landis Markley, Robert Bauer, Jackie Schandua, and Michael Swanzy for their helpful comments and suggestions in the course of preparing this work. Thanks also to Christian Bruccoleri and Puneet Singla for use of their Mat- lab code to select initial conditions for computing nonlinearity indices. I gratefully acknowledge the support of the National Defense Science and Engineering Graduate Fellowship. I feel deeply fortunate to have wonderful friends who have made my time in Col- lege Station a special part of my life: Roshawn Bowers, Eddie Caicedo, Todd Griffith, Bjoern Kiefer, Luciano Machado, Josh O’Neil, Gary Seidel, Lesley Weitz, and Matt Wilkins. Finally, I offer my gratitude to my family for their advice, encouragement, and belief in me. vi TABLE OF CONTENTS CHAPTER Page I INTRODUCTION .......................... 1 II KINEMATICS OF N-DIMENSIONAL PRINCIPAL ROTATIONS 5 A.Introduction.......................... 5 B. Review of N-DimensionalRotations............. 6 1.RotationMatrix..................... 6 2.PrincipalRotationMatrices............... 9 3.ExtendedRodriguesParameters............ 10 4.EulerMatrix....................... 14 C.KinematicsofPrincipalRotations.............. 17 D.OptimalKinematicManeuvers................ 26 E.Conclusion........................... 31 III MINIMUM-PARAMETER REPRESENTATIONS OF N-DI- MENSIONAL PRINCIPAL ROTATIONS ............. 33 A.Introduction.......................... 33 B. Review of N-DimensionalOrientations........... 34 C. Minimal Representations of Principal Rotations . 39 1. Numeric Analysis for N =4............... 45 2. Numeric Analysis for N =5............... 49 D.Discussion........................... 55 IV HAMEL COEFFICIENTS FOR THE ROTATIONAL MO- TION OF AN N-DIMENSIONAL RIGID BODY ........ 57 A.Introduction.......................... 57 B. Review of N-DimensionalKinematics............ 59 j C. Definition of the Numerical Relative Tensor χik ...... 62 D. N-DimensionalHamelCoefficients.............. 71 E. Lagrange’s Equations for N-Dimensional Angular Velocities 77 F. The Lax Pair Form Via the Lagrangian Method . 79 G.Conclusions.......................... 82 vii CHAPTER Page V CAYLEY KINEMATICS AND THE CAYLEY FORM OF DYNAMIC EQUATIONS ..................... 84 A.Introduction.......................... 84 B.CayleyKinematics...................... 86 C. Tensor Form of Lagrange’s Equations . 92 D. Cayley Quasi Velocities and the Cayley Form . 101 E.PlanarMotionExample................... 104 F.Discussion........................... 109 VI APPLICATION OF THE CAYLEY FORM TO GENERAL SPACECRAFT MOTION ..................... 111 A.Introduction.......................... 111 B.CayleyKinematics...................... 113 C. N-DimensionalRigidBodyDynamics............ 114 D. General Spacecraft Motion . 116 E. Satellite with Three Momentum Wheels . 124 F.Discussion........................... 127 G.Conclusions.......................... 130 VII STABILIZATION AND CONTROL OF DYNAMICAL SYS- TEMS IN THE CAYLEY FORM ................. 131 A.Introduction.......................... 131 B.DefinitionofCayleyQuasiVelocities............ 132 C.LinearRodrigues-ParameterFeedback............ 135 D. Work/Energy-Rate Expression for N-Dimensional Dynamics 137 E. Feedback Control for N-Dimensional Rotations . 140 F.QuasiVelocitiesforLinearFeedback............ 148 G.OptimalityResultsforRegulationTerms.......... 151 H. Stabilization Using Velocity Feedback . 154 I.Conclusion........................... 158 VIII NONLINEARITY INDEX OF THE CAYLEY FORM ...... 160 A.Introduction.......................... 160 B.NonlinearityIndex...................... 161 C. Elastic Spherical Pendulum . 163 D.NumericalResults....................... 167 E.Discussion........................... 172 viii CHAPTER Page IX SUMMARY ............................. 174 REFERENCES ................................... 176 APPENDIX A ................................... 182 APPENDIX B ................................... 183 APPENDIX C ................................... 185 VITA ........................................ 190 ix LIST OF TABLES TABLE Page I EXAMPLE OF ORTHOGONAL PLANES FOR N =6 ....... 61 II CORRESPONDING VALUES OF i, j, AND k ............ 64 j III SUMMARY OF PROPERTIES RELATED TO χik .......... 71 IV THE CAYLEY FORM OF DYNAMIC EQUATIONS ......... 103 V ELASTIC SPHERICAL PENDULUM REPRESENTATIONS .... 165 VI NUMERICAL RESULTS FOR NONLINEARITY ........... 169 x LIST OF FIGURES FIGURE Page 1 Coordinatization of the principal plane by p(1) and p(2) frames, which are related by a flipping about the axis a. ........... 44 2 Relationships between four of the eight solutions for N =4:f- flip, s - swap, fs - flip and swap. ..................... 50 3 Relationships between four of the sixteen solutions for N =5:fr - flip-rotate, rf - rotate-flip, ff - flip-flip. ................. 54 4 Planar rigid body. ............................ 106 5 Generalized coordinates. ......................... 107 6 Generalized velocities. .......................... 108 7 Cayley quasi velocities. .......................... 108 8 Example planar motion. ......................... 109 9 Attitude and orbital motion variables. ................. 123 10 Attitude and orbital motion Cayley quasi velocities. .......... 123 11 Satellite system motion variables. .................... 128 12 Satellite system