Spacecraft Dynamics 9780070378438, 0070378436

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SPACECRAFT DYNAMICS

Thomas R. Kane, Peter W Likins,. and David A. Levinson

Spacecraft Dynamics Thomas R. Kane, Stanford University Peter W. Likins, Lehigh University David A. Levinson, Lockheed Palo Alto Research Laboratory

Copyright 2005 by the authors Previously published by McGraw-Hill Book Company, 1993

Now published by The Internet-First University Press

This manuscript is among the initial offerings being published as part of a new approach to scholarly publishing. The manuscript is freely available from the Internet-First University Press repository within DSpace at Cornell University at: http://dspace.library.cornell.edu/handle/1813/62 The online version of this work is available on an open access basis, without fees or restrictions on personal use. A professionally printed and bound version may be purchased through Cornell Business Services by contacting: [email protected] All mass reproduction, even for educational or not-forprofit use, requires permission and license. For more information, please contact [email protected]. Ithaca, N.Y. January 2005

£NGJN.

MOV 1 1982

CORNELL UNIVERSITY LIBRARIES ITHACA, N. Y. 14853

Engineering Library Carpenter Hall

SPACECRAFT DYNAMICS

Thomas R. Kane Stanford University

Peter W. Likins Lehigh University

David A. Levinson Lockheed Palo Alto Research Laboratory

McGraw-Hill Book Company New York St. Louis San Francisco Auckland Bogota Hamburg Johannesburg London Madrid Mexico Montreal New Delhi Panama Paris Sao Paulo Singapore Sydney Tokyo Toronto

This book was set in Times Roman by Andover Graphics. The editors were Diane D. Heiberg and Madelaine Eichberg; the production supervisor was Leroy A. Young. New drawings were done by Danmark & Michaels, Inc. The cover was designed by Mark Wieboldt. Halliday Lithograph Corporation was printer and binder.

SPACECRAFT DYNAMICS Copyright © 1983 by McGraw-Hill, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher.

1234567890 H D H D 898765432

ISBN 0-Q7-Q37flM3-b Library of Congress Cataloging in Publication Data Kane, Thomas R. Spacecraft dynamics. Includes index. 1. Space vehicles—Dynamics. I. Likins, Peter W. II. Levinson, David A. III. Title. TL1050.K36 629.47'0r53111 81-8431 ISBN 0-07-037843-6 AACR2

j>

CONTENTS

Preface To the Reader Chapter 1 Kinematics 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 l .21

Simple Rotation Direction Cosines Euler Parameters Rodrigues Parameters Indirect Determination of Orientation Successive Rotations Orientation Angles Small Rotations Screw Motion Angular Velocity Matrix Angular Velocity Vector Angular Velocity Components Angular Velocity and Euler Parameters Angular Velocity and Rodrigues Parameters Indirect Determination of Angular Velocity Auxiliary Reference Frames Angular Velocity and Orientation Angles Slow, Small Rotational Motions Instantaneous Axis Angular Acceleration Partial Angular Velocities and Partial Velocities

vii xi 1 1 4 12 16 20 23 30 38 42 47 49 53 58 62 68 71 73 78 81 84 87

iii

iv CONTENTS

Chapter 2 Gravitational Forces 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18

Gravitational Interaction of Two Particles Force Exerted on a Body by a Particle Force Exerted on a Small Body by a Particle Force Exerted on a Small Body by a Small Body Centrobaric Bodies Moment Exerted on a Body by a Particle Moment Exerted on a Small Body by a Small Body Proximate Bodies Differentiation with Respect to a Vector Force Function for Two Particles Force Function for a Body and a Particle Force Function for a Small Body and a Particle Force Function in Terms of Legendre Polynomials Force Function for Two Small Bodies Force Function for a Centrobaric Body Force Function for a Body and a Small Body Force Function Expressions for the Moment Exerted on a Body by a Particle Force Function Expression for the Moment Exerted on a Small Body by a Body

Chapter 3 Simple Spacecraft 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11

Rotational Motion of a Torque-Free, Axisymmetric Rigid Body Effect of a Gravitational Moment on an Axisymmetric Rigid Body in a Circular Orbit Influence of Orbit Eccentricity on the Rotational Motion of an Axisymmetric Rigid Body Torque-Free Motion of an Unsymmetric Rigid Body Effect of a Gravitational Moment on an Unsymmetric Rigid Body in a Circular Orbit Angular Momentum, Inertia Torque, and Kinetic Energy of a Gyrostat Dynamical Equations for a Simple Gyrostat Rotational Motion of a Torque-Free, Axisymmetric Gyrostat Reorientation of a Torque-Free Gyrostat Initially at Rest Effect of a Gravitational Moment on an Axisymmetric Gyrostat in a Circular Orbit Effect of a Gravitational Moment on an Unsymmetric Gyrostat in a Circular Orbit

Chapter 4 Complex Spacecraft 4.1 4.2 4.3 4.4

Generalized Active Forces Potential Energy Generalized Inertia Forces Kinetic Energy

CONTENTS

4.5 4.6 4.7 4.8 4.9 4.10 4.11

I II

V

Dynamical Equations Linearized Dynamical Equations Computerization of Symbol Manipulation Discrete Multi-Degree-of-Freedom Systems Lumped Mass Models of Spacecraft Spacecraft with Continuous Elastic Components Use of the Finite Element Method for the Construction of Modal Functions

275 277 279 284 296 318

Problem Sets

344

Problem Problem Problem Problem

344 358 378 395

Set 1 Set 2 Set 3 Set 4

332

Appendixes

422

Direction Cosines as Functions of Orientation Angles Kinematical Differential Equations in Terms Orientation Angles

422 427

Index

433

PREFACE

This book is the outgrowth of courses taught at Stanford University and at the University of California, Los Angeles, and of the authors' professional activities in the field of spacecraft dynamics. It is intended both for use as a textbook in courses of instruction at the graduate level and as a reference work for engineers engaged in research, design, and development in this field. The choice and arrangement of topics was dictated by the following considerations. The process of solving a spacecraft dynamics problem generally necessitates the construction of a mathematical model, the use of principles of mechanics to formulate equations governing the quantities appearing in the mathematical model, and the extraction of useful information from the equations. Skill in constructing mathematical models of spacecraft is acquired best through experience and cannot be transmitted easily from one individual to another, particularly by means of the printed word. Hence, this subject is not treated formally in the book. However, through examples, the reader is brought into contact with a considerable number of mathematical models of spacecraft and, by working with the book, he can gain much experience of the kind required. By way of contrast, the formulation of equations of motion is a subject that can be presented formally, and it is essential that this topic be treated effectively, for there is no point in attempting to extract information from incorrect equations of motion. Now, every spacecraft dynamics analysis necessitates use of various kinematical relationships, some of which have played such a small role in the development of technology prior to the space age that they have been treated only cursorily, if at all, in the general mechanics literature. Accordingly, the book begins with what is meant vii

Vill PREFACE

to be a unified, modern treatment of the kinematical ideas that are most useful in dealing with spacecraft dynamics problems. To place the topics to be treated in the book into perspective, we turn to the familiar relationship F=ma, here regarding it as a conceptual guideline rather than as the statement of a law of physics. Seen in this light, the a represents all kinematical quantities, the F all forces that come into play, the m all inertia properties, and the sign of equality the assertion that kinematical quantities, forces, and inertia properties are related to each other. It is then clear that one should deal with the topics of kinematics, forces, and inertia properties before taking up the study of a technique for formulating equations of motion. The subject of inertia properties, that is, the finding of mass centers, moments and products of inertia, principal axes of inertia, and so on, is treated extensively in available textbooks and acquires no new facets in connection with spacecraft. Hence, we presume that the reader knows this material. Detailed information regarding forces that affect the behavior of spacecraft is not so readily accessible. Therefore, we address this topic in Chapter 2, confining attention to gravitational forces, which play a preeminent role in spacecraft dynamics. This brings us into position to attack specific problems in Chapters 3 and 4, these chapters differing from each other in one important respect: throughout Chapter 3, which deals with relatively simple spacecraft, we rely solely upon the angular momentum principle for the formulation of dynamical equations of motion, whereas in Chapter 4, where we are concerned with complex spacecraft, we first develop and then use a more powerful method for formulating equations of motion, one that is particularly well suited for problems involving multi-degrees-of-freedom spacecraft. Most sections of the book are arranged as follows. They begin with the presentation of theoretical material in the form of categorical statements that either constitute definitions or that can be shown to be valid by means of deductive arguments. In the latter case, the arguments are presented under the heading "Derivations"; each section contains, in addition, an example in the application of the theory set forth in the section. In the classroom, an instructor can, therefore, focus attention wherever he pleases, that is, on the presentation of theory, on formal proofs, or on examples, leaving it to his students to devote time outside of the classroom to the matters not pursued in depth in class. In addition to the examples appearing in the text, the book contains four sets of problems, each set associated intimately with one of the four chapters. To acquire mastery of the material in the book, a student should solve every one of these problems, for each covers material not covered by any other. Results are given wherever possible, so that one always can determine whether or not one has solved a problem correctly. To cover all of the material in the book is rather difficult in the time generally available in a graduate curriculum. However, the material lends itself well to presentation in two courses dealing, respectively, with Sections

PREFACE ix

1.1-1.20, 2.1-2.8, 3.1-3.11, and Sections 1.21, 2.9^-2.18, and 4.1-4.11. The first course, covering kinematical fundamentals, the vectorial treatment of gravitational forces and moments, and simple spacecraft, can prepare a student to work effectively on spacecraft dynamics problems and to pursue further studies in this field; successful completion of the second course, which concentrates upon new kinematical ideas, gravitational potential theory, a new approach to the formulation of equations of motion, and complex spacecraft problems, enables one to undertake work at the forefront of spacecraft dynamics. The use of computers in the solution of spacecraft dynamics problems is so pervasive that familiarity of a worker in this field with programming concepts must be presumed. The writing and running of computer programs are meant to take place in connection with courses based on this book, but are not absolutely essential; that is, most problems are set up in such a way that considerable benefit can be obtained by studying them even without the aid of a computer. Since it was our intention to produce a self-contained book, we have omitted all references to the extensive literature on spacecraft dynamics. However, we gratefully acknowledge our debt to the many authors whose work has influenced our thinking, and we wish to express our thanks to students who, with comments and suggestions, have aided us in writing this book, as well as to Stanford University and the University of California at Los Angeles, which supported this undertaking by generously making time and computational resources available to us. Thomas R. Kane Peter W. Likins David A. Levinson

TO THE READER

Each of the four chapters of this book is divided into sections. A section is identified by two numbers separated by a decimal point, the first number referring to the chapter in which the section appears, and the second identifying the section within the chapter. Thus, the identifier 2.13 refers to the thirteenth section of the second chapter. A section identifier appears at the top of each page. Equations are numbered serially within sections. For example, the equations in Sees. 2.12 and 2.13 are numbered (1M46) and (1H31), respectively. References to an equation may be made both within the section in which the equation appears and in other sections. In the first case, the equation number is cited as a single number, whereas, in the second case, the section number is included as part of a three-number designation. Thus, within Sec. 2.12, Eq. (1) of Sec. 2.12 is referred to as Eq. (1), whereas, in Sec. 2.13, the same equation is referred to as Eq. (2.12.1). To locate an equation cited in this manner, one may make use of the section identifiers appearing at the tops of the pages. Figures appearing in the chapters are numbered so as to identify the sections in which the figures appear. For example, the two figures in Sec. 3.9 are designated Fig. 3.9.1 and Fig. 3.9.2. To avoid confusion of these figures with those in problem sets, the figure number is preceded by the letter P in the case of the latter; the double number that follows this letter refers to the problem statement in which the figure is introduced. For example, Fig. P3.8 is introduced in Prob. 3.8. Similarly, Table 1.15.2 is the designation for a table in Sec. 1.15, whereas Table PI.21 is associated with Prob. 1.21. Thomas R. Kane Peter W. Likins David A. Levinson xi

CHAPTER

ONE KINEMATICS

Every spacecraft dynamics problem involves considerations of kinematics. Indeed, the solution of most such problems begins with the formulation of kinematical relationships. Hence, this topic requires detailed attention. The first nine sections of this chapter deal with changes in the orientation of a rigid body in a reference frame, without regard to the time that necessarily elapses when a real body experiences a change in orientation. Time enters the discussion in the remaining sections, in all of which the concept of angular velocity plays a central role. The concluding section contains the definitions of quantities employed to great advantage, in Chap. 4, in connection with the formulation of dynamical equations of motion of complex spacecraft. This section (and Probs. 1.27-1.31) may be omitted by readers concerned primarily with simple spacecraft, such as those treated in Chap. 3. 1.1 SIMPLE ROTATION A motion of a rigid body or reference frame B relative to a rigid body or reference frame A is called a simple rotation of B in A if there exists a line L, called an axis of rotation, whose orientation relative to both A and B remains unaltered throughout the motion. This sort of motion is important because, as will be shown in Sec. 1.3, every change in the relative orientation of A and B can be produced by means of a simple rotation of B in A. If a is any vector fixed in A (see Fig. 1.1.1), and b is a vector fixed in B and equal to a prior to the motion of B in A, then, when B has performed a simple rotation in A, b can be expressed in terms of the vector a, a unit vector k parallel to L, and the radian measure 6 of the angle between two lines, LA and LB, which are fixed in A and B, respectively, are perpendicular to L, and are parallel to each other initially. Specifically, if 6 is regarded as

2

KINEMATICS 1.1

Figure 1.1.1

positive when the angle between LA and LB is generated by a X-rotation of L B relative to LA, that is, by a rotation during which a right-handed screw fixed in B with its axis parallel to X advances in the direction of X when B rotates relative to A, then b = a cos0 - a x X sin0 + a • XX(l - cos0)

(1)

Equivalently, if a dyadic C is defined as C = U cos0 - U x X sin0 + XX(l - cos0)

(2)

where U is the unit (or identity) dyadic, then (3)

b = a •C

Derivations Let ax and a2 be unit vectors fixed in A, with ax parallel to LA and a 2 = A. x a,; and let 0, and fi2 be unit vectors fixed in B, with 0, parallel to LB and /32 = X x fix, as shown in Fig. 1.1.1. Then, if a and b are resolved into components parallel to a!, a 2 , X and p{, #>, K respectively, corresponding coefficients are equal to each other because ax = 0,, a2 = fa, and a = b when 0 = 0. In other words, a and b can be expressed as a = pa.\ + qa2 + rX

(4)

b = pfi, + 4/32 + rX

(5)

and where /?, q, and r are constants. Expressed in terms of a, and a2, the unit vectors 0, and /32 are given by = cos0

sin0 OL2

(6)

1.1 SIMPLE ROTATION

3

Figure 1.1.2

and

p2 = - sin0 a! + cos0 a2

O)

so that, substituting into Eq. (5), one finds that b = (p cos0 - q sin0)a t + (p sin0 + q cos0)«2 + rk

(8)

The right-hand member of Eq. (8) is precisely what one obtains by carrying out the operations indicated in the right-hand member of Eq. (1), using Eq. (4), and making use of the relationships k x a{ = a2 and X x a2 = —ax. Thus the validity of Eq. (1) is established, and Eq. (3) follows directly from Eqs. (1) and (2). Example A rectangular block B having the dimensions shown in Fig. 1.1.2 forms a portion of an antenna structure mounted in a spacecraft A. This block is subjected to a simple rotation in A about a diagonal of one face of B, the sense and amount of the rotation being those indicated in the sketch. The angle / between the line OP in its initial and final positions is to be determined. If a,, a2, and a3 are unit vectors fixed in A and parallel to the edges of B prior to ZTs rotation, then a unit vector k directed as shown in the sketch can be expressed as 3a2 + 4a3 (9) k=

4

KINEMATICS

1.2

And if a denotes the position vector of P relative to O prior to 5 ' s rotation, then a = - 2a! + 4a3

(10)

- 12a! + 8a2 - 6a3

axX = and

(11)

48a2 + 64a3

aAX =

(12)

Consequently, if b is the position vector of P relative to O subsequent to fTs rotation,t , * ft 12ai — 8a2 + 6a3 . TT A . b = ( - 2at + 4a 3 ) cos — + sin — (i)

6

+

5

6

48a2 + 64a:3 / IT l COS 25 25 { 6

= - 0.532a, - 0.543a2 + 4.407a3

(13)

Since is the angle between a and b, cos

*

=

where la I and Ibl denote the (equal) magnitudes of a and b. Hence (-2K-0.532) + 4(4.407) (4 + 16)1/2 (4 + 16)1

=

0_935

and = 20.77°.

1.2 DIRECTION COSINES If a,, a 2 , a3 and bi, l>2, b 3 are two dextral sets of orthogonal unit vectors, and nine quantities Q (i,j = 1, 2, 3), called direction cosines, are defined as QAa.,b, then the two row matrices [ai other as follows: [bi

b2

a2

(/

^.=

2 , 3)

lf

a3] and [bj

b3] = [a!

a2

(1)

b2 b3J are related to each

a3] C

(2)

t Numbers beneath signs of equality are intended to direct attention to equations numbered correspondingly.

1.2 DIRECTION COSINES 5

where C is a square matrix defined as Cn

C\2 C13

C21

C22

C23

C31

C32

C33

(3)

If a superscript T is used to denote transposition, that is, if CT is defined as -11

C21 C31

(4) C13

C23

C*33_

then Eq. (2) can be replaced with the equivalent relationship [ai a2 a3] = [bj b2 b3] CT

(5)

The matrix C, called a direction cosine matrix, can be employed to describe the relative orientation of two reference frames or rigid bodies A and B. In that context, it can be advantageous to replace the symbol C with the more elaborate symbol ACB. In view of Eqs. (2) and (5), one must then regard the interchanging of superscripts as signifying transposition; that is, B A

C = (ACBf

(6)

The direction cosine matrix C plays a role in a number of useful relationships. For example, if v is any vector and Av, and *v, (/ = 1, 2, 3) are defined as V-va,.

(i = l , 2 , 3 )

,=vb,

0 = 1,2,3)

(7)

and A

(8) A

B

while v and v denote the row matrices having the elements vl9 and Bvl9 Bv2, *v3, respectively, then By = A

A

A

A

v2, v3 (9)

VC B

Similarly, if D is any dyadic, and A, and DU (i9j = 1, 2, 3) are defined as (10) and B

Dij = b, • D • b,

A

B

(1, j = 1, 2, 3) A

(11) B

while D and D denote square matrices having Dtj and Dijy respectively, as the elements in the ith rows and yth columns, then B

D = CTADC (12) Use of the summation convention for repeated subscripts frequently makes it possible to formulate important relationships rather concisely. For example, if 80- is defined as

6

KINEMATICS

1.2

«• a 2 *b,3 . l1 ^- .0 . . 1 2 2 4(0 a, • b 3 — a3 (16)

b,

-1 - 0

2

(34) (35)

1 2

(36)

Figure 1.3.1

=

a2 • bi - ai • tfc

=

-1 - 0

=

__1

(37)

(17)

Then

x=

\ ai - \ a2 - \ as _ at - a2 - a3 X1/2

(38)

(19)

and . !

2

TT

,

0 = 2 cos - = — rad (20)

2

(39)

3

1.4 RODRIGUES PARAMETERS A vector p , called the Rodrigues vector, and three scalar quantities, p\, p 2 , p 3 , called Rodrigues parameters, can be associated with a simple rotation of a rigid body B in a reference frame A (see S e c . 1.1) by letting 6 A w p = X tan ~

(1)

and p, = p • a, = p • b,

(i = 1, 2, 3)

(2)

where X and 6 have the same meaning as in Sec. 1.1, and a,, a:, a3 and bi, b2, b3 are dextral sets of orthogonal unit vectors fixed in A and By respec-

1.4

RODRIGUES PARAMETERS

17

tively, with a, = b, (i = 1, 2, 3) prior to the rotation. (When a discussion involves more than two bodies or reference frames, notations such as ApB and A B Pt will be used.) The Rodrigues parameters are intimately related to the Euler parameters (see Sec. 1.3):

p. = J

(/ = 1,2,3)

(3)

An advantage of the Rodrigues parameters over the Euler parameters is that they are fewer in number; but this advantage is at times offset by the fact that the Rodrigues parameters can become infinite, whereas the absolute value of any Euler parameter cannot exceed unity. Expressed in terms of Rodrigues parameters, the direction cosine matrix C (see Sec. 1.2) assumes the form 1 + p.2 —pi 2(p,

Pi

>- 2(p3

Pi

P32

+ - PI)

2(pi 1 +P1

2

2(P2

Pi — Pi) - P32 -

Pi

+

2(P3 Pi + Pi)

P.2

Pd

2(P2 P3 " P i ) 1+P32 -

C =

P22.

(4) 1 + Pi' + P22 + P32

The Rodrigues vector can be used to establish a simple relationship between the difference and the sum of the vectors a and b defined in Sec. 1.1: a - b = (a + b) x p (5) This relationship will be found useful in connection with a number of derivations, such as the one showing that the following is an expression for a Rodrigues vector that characterizes a simple rotation by means of which a specified change in the relative orientation of A and B can be produced: 9

(< («2 ~ ft)

=

(a, + ft) • (a2 -

U

ft)

where a, and ft (/ = 1, 2) are vectors fixed in A and B, respectively, and «/ = ft (i = 1, 2) prior to the change in relative orientation. Rodrigues parameters for such a rotation can be expressed as 1 + d i + C22 + C33 =

91

y )

1 + C,i + C22 + C33 -

P3

C13 ~ C31

C21 ~ C12

1 + Cn + C22 + C33

l

;

Derivations The equality of p • a, and p • b, [see Eq. (2)] follows from Eqs. (1) and (1.2.22); and Eqs. (3) are obtained by noting that

18

KINEMATICS

1.4

-

= X tan - = p

€ 4 (1.3.1, 1.3.3)

(10)

2 (1)

so that Pi==±U±=ti (2)

e4

(11)

(f-,,2,3) (1.3.2) € 4

From Eq. (1.3.4), 1 = €, 2 + €22 + €32 + e42 = (Pi 2 + p 2 2 + P32 + 1)€42

(12)

(3)

Hence, and 1

n

— *r 2 — - c — €1 — €2 (1.3.6)

2

- - c 2 4 - ^ 2 — €3 + e 4

— r> 2 ^ 2 — ^ 2 - r — p i € 4 — P2 e 4 (3)

2

^ 2 - r 2 - U ^ 2 — P3 € 4 -1- € 4

(14) == (P. (P22 " P, 2 " P32 + D C 2 = **' ~ ??' ~ ^^2 + ++ ''22 2 1 + P, + P 2 + P 3 in agreement with E q . (4), and the remaining elements of C are found similarly. As for E q . (5), note that cross-multiplication of E q . (1.3.21) with e yields € X b = € X a+ 2 { €

4

e x ( € X a ) + € X [ € X ( € X

a)]}

(15)

Hence, = 2{c x a + e4 € x (c x a) + € x [c x (c x a)]} (16)

Now € X [€ X (€ X a)] = ~ € 2 € X a =

~(€i 2 + €22 + € 3 2 )e X a

^ - I X x ' a

(17)

Consequently, e x (a + b) = 2€4[e4 € x a + e x ( € x a)] = €4(b - a) (16,17)

(18)

(1.3.21)

and a - b = (a + b) x (18)

= (a + b) x k tan ^ = (a + b) x p

€ 4 (1.3.1, 1.3.2)

2 (1)

r

(19) V

which is E q . (5). Equation (6) c a n n o w b e obtained b y observing that a , - 0 , = ( a , +Pi)xp

(20)