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AIAA-80-1642 Solution of Euler’s Equations of Motion and Eulerian Angles for Near Symmetric Rigid Bodies Subject to Constant Moments J. M . Longuski, Jet Propulsion Lab, Pasadena, Ca.

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AIAA/AAS ASTRODYNAMICS CONFERENCE

i/ August 11 -1 3, 1980/Danvers, Massachusetts

For permission to copy or republish, contact the American instilute of Aeronautics and Aslronautics.1290 Avenue of the Americas. New York, N.Y. 10019

SOLUTION OF EULER'S EQUATIONS OF MOTION AND EULERIAN ANGLES FOR NEAR SYMMETRIC RIGID BODIES SUBJECT TO CONSTANT MOMENTS*

James M. Longuski** Jet Propulsion Laboratory

Pasadena, California 'L

Abstract

Analytic expressions are found for Euler's Equations of Motion and for the Eulerian Angles f o r both symmetric and near symmetric rigid bodies under the influence of arbitrary constant body- fixed torques. These solutions provide the body- fixed angular velocities and the attitude of the body, respectively, as functions of time. They are of special interest in applications to spin- ning spacecraft (such as the Galileo Spacecraft to be launched in 1984) because they include the effect of time-varying spin rate. Thus they can be applied to spin-up and spin-down maneuvers as well as to error analysis for thruster misalignments. The solutions are given for arbitrary initial con- ditions in terms of Fresnel, Sine and Cosine Integrals. Numerical integration of the governing differential equations has verified that the approximate analytic solutions are very accurate in many physical situations of interest.

I. Introduction

The rotational motion of a rigid body is gov- erned by Euler's equations of motion which are, in general, nonlinear. When a solution of these equations can be found it provides the body fixed angular velocities ux(t), wy(t) and w,(t). order to determine the attitude of the body as a function of time, a second set of nonlinear differ- entia1 equations involving the particular Eulerian angles chosen and wx(t), wy(t) and w,(t) must also be solved.

In

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The main purpose in solving these equations is found in their applications to satellites and deep space probes. are easily found by computer simulations, analytic solutions can provide deeper insight and understand- ing, and can be used in obtaining quick solutions, error analyses and compact algorithms for onboard computations. In space applications, certain sim- plifying, yet realistic assumptions can be made so that valuable approximate analytic solutions can be found.

Even though numerical solutions

In the first step, Euler's equations of motion are solved for near symmetric rigid bodies subject to constant moments. The assumption of near sym- metry allows the heuristic assumption that w,(t) varies linearly with time to be used to convert the nonlinear differential equations into linear differential equations with time varying coeffi- cients. By a proper change of the independent variable the equations are transformed into linear differential equations with constant coefficients, but with time varying forcing functions. From the impulse response, a particular solution can be generated in terms of Fresnel Integrals. The par- ticular solution p l u s the easily obtained homoge-

neous solution provide the total heuristic solution. When the body is symmetric this becomes an exact analytic solution of ux(t), wy(t) and u,(t).

solved to determine the attitude of the body. Although a particular set of Eulerian Angles are defined in the text (the Type 1: 3-1-2 Eulerian Angles), the method can be applied to others. At this point, the most limiting assumption must be made: the spin axis must be small. Thus, the body must be initially spinning about the z axis or the torque about the z axis must be large relative to the other two torques. This assumption applies to many actual spacecraft and allows the nonlinear differ- ential equations to be reduced to linear differen- tial equations with time varying coefficients. Changing the independent variable by the same transformation used in the Euler equations of motion results in a set of linear differential equations with constant coefficients and time vary- ing forcing functions. By expressing the Fresnel Integrals, which appear in the forcing functions, in terms of auxiliary functions, the impulse Iesponse can be obtained by series and asymptotic expansions. Thus, an approximate analytic solution is found for the Eulerian Angles.

approximate analytic solutions, the ACSL (Advanced Continuous Simulation Language) was used to numeri- cally integrate the governing differential equations and compare with the results given by the analytic solutions. Specific parameters were taken from the Galileo spacecraft, which is a dual spinner sched- uled to be launched for an orbital mission of Jupiter in 1984. occur when the stator and rotor are locked together (when the Galileo acts as a single spinner) so that the approximate analytic solution can be extremely useful if it is accurate. The computer simulations reveal that the heuristic solutions are very accurate in describing the rotational motion of the spacecraft. Hence, these solutions which have not been previ.ously published, will find important applications not only in the maneuver analysis of the Galileo spacecraft, but also in the analysis of many other satellites and deep space probes of the future.

In the second step the Eulerian A n g l ~ s are

the two angles defining the direction of

In order ta test the accuracy of these

A number of important maneuvers

11. Euler's Equations of Motion

The rotational equations of motion of a rigid body with principal axes at the center of mass are

M x = I A x + ( I z - I ) u w Y Y =

M = I 6 + (Ix - Iz) w z wx (1) Y Y Y

Y M, = Iz hz + (I - Ix) wx w Y

*This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by the National Aeronautics and Space Administration.

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**Senior Engineer, Guidance and Control Section, Jet Propulsion Laboratory.

Copyright 0 American lntlilule of Aeronaulics and A t l r ~ n a a l i ~ s , Inc.. 1980. All riehlsrererved. 1

These a r e known as Euler's equa t ions of motion. Then Eqs. ( 4 ) are w r i t t e n as They are n o n l i n e a r i n gene ra l and considered t o be i n t r a c t a b l e . I n t h e cases where a s o l u t i o n e x i s t s i t should be noted t h a t t h e t ime i n t e g r a l s of wx, wy, would s o e c i f v t h e o r i e n t a t i o n of t h e body. To o b t a i n

+ hl ( a t + b) y = c

j' - h 2 ( a t + b) x = d and w z do no t provide any p h y s i c a l a n g l e s which

t h e o r i e n t a t i o n t h e d i f f e r e n t i a l equa t ions spec i fy - i n g t h e E u l e r i a n Angle r a t e s i n terms of wx, wy and w z are i n t e g r a t e d . I n t h i s pape r , Eqs. (1) are i n t e g r a t e d f o r a s p e c i a l case corresponding to t h e s p i n up and s p i n down maneuvers of t h e G a l i l e o Spacec ra f t . The s o l u t i o n of t h e Eu le r i an Angle5 i s d i s c u s s e d i n S e c t i o n V.

111. I n t e g r a t i o n of E u l e r ' s Equat ions of Motion f o r Near SymmeLric Body

A h e u r i s t i c s o l u t i o n of E q s . (1) is avai lab le by assuming

Computer s i m u l a t i o n s have i n d i c a t e d t h a t Eq. (2 ) prov ides a very a c c u r a t e approximate SOlutiOn when

Ix 1 Iy ( 3 )

T h i s i s e s p e c i a l l y true f o r t h e s p i n up maneuver of t h e G a l i l e o Spacec ra f t s i n c e wx and wy are b a t h small (of t h e o r d e r and o s c i l l a t e about a mean n e a r zero. I n t h i s case t h e discrepancy between t h e approximation f o r w z and t h e exac t s o l u t i o n i s i n t h e f o u r t h or f i f t h d i g i t .

S u b s t i t u t i n g Eq. ( 2 ) i n Eqs. (1) g ives

wzo b =

M

I d - 2

Y

( 5 ) x = w

y = w Y

(6) V

Note t h a t Eqs. ( 6 ) are coupled l i n e a r d i f f e r e n t i a l equa t ions w i t h t ime va ry ing c o e f f i c i e n t s and con- s t a n t f o r c i n g f u n c t i o n s . The t ime va ry ing c o e f f i - c i e n t s can b e removed by an a p p r o p r i a t e change of t h e independent v a r i a b l e .

Eq. ( 7 ) i s used t o pu t Eqs. ( 6 ) i n t h e fo l lowing form

Now che o r i g i n a l independent v a r i a b l e t i n Eqs. (8) m u s t be wri t ten i n term of the new v a r i a b l e T. Solu r ion of t h e q u a d r a t i c equa t ion from Eq. ( 7 ) g ives

Great care must be exe rc i sed i n choosing t h e c o r r e c t r o o t when t h e s i g n s o f a and b are a r b i t r a r y . I n t h e case of a and b bo th p o s i t i v e , t h e p o s i t i v e

r e p r e s e n t s t h e pure s p i n up maneuver i n which t h e o r i g i n a l s p i n i s p o s i t i v e and t h e app l i ed torque i s p o s i t i v e . S i m i l a r l y , when bo th a and b are negarive, t h e n e g a t i v e s i g n on t h e squa re r o o t term is used. When a i s p o s i t i v e and b is n e g a t i v e , t h e s i g n o f t h e square r o o t term i s a t f i r s t nega t ive but swi t ches t o p o s i t i v e f o r t 1 - bla . r e p r e s e n t s t h e s p i n down case u n t i l t = -b / a a t which p o i n t w z = 0. Beyond t h i s p o i n t , any addi- t i o n a l torque serves t o s p i n UP t h e s p a c e c r a f t ; hence, t h e s i g n becomes p o s i t i v e . I n t h e same manner, when a i s nega t ive and b is p o s i t i v e , t h e s i g n of t h e square m o t is p o s i t i v e f o r O z t L - b l a and nega t ive f o r t - b l a . With t h e s e considera- t i o n s i n mind, Eqs. ( 9 ) becomes

Sign on t h e squa re r o o t term always a p p l i e s . Th i s L/

This

(sa) t = - -b + u smfa] Jb2+2ar

a

and Eqs. (8) can be r e w r i t t e n as

where

u = 1 f o r s p i n up (a and b same s i g n ) L

u = -1 f o r Spin down (a and b oppos i t e s i g n ) and only f o r 0 5 t 5 - b / a .

2

Thus, the solution of Eqs. (10) will be given for a constant value of u which corresponds to a pure spin up or a pure spin down maneuver. If the spin down maneuver solution is desired for t > -b/a then the final conditions wx(-b/a), wy(-b/a) and

= 0 should be used as initial conditions with the spin up solution (~1).

I w,(-b/a) L'

Differentiation of Eqs. (10) provides the decoupled equations

The solutions of Eqs. (14) take the form

- x(7) = Ax COST

-

f,(+) sin(?- +) d+

(15)

y(7) = A cos? Y

f2t+) sin6- ad+ s' + B sin? + Y

where f,(T) and f2(f) are forcing functions of Eqs. (14)

The following integrals have been obtained for

J -ad(b2 + Z ~ T ) - ~ / *

The solution of Eqs. (11) will be obtained for all cases in which

arbitrary signs on a and b.

h1 h2 > 0 (12)

so that the equations are stable. This implies that I, is the largest or the smallest of the principal moments o f inertia. For the Galileo Spacecraft, Iz is the largest of the principal moments of inertia. When I, is the intermediate moment of inertia, then AI h2 0 and the Eqs. (11) are unstable. The solution of Eqs. (11) for the unstable case will not be investigated in this work.

L,,

ii

At this point it is convenient to make a second change of variables. Let

T G T -

a = JhlhZa

- b = G b

Then E q s . (11) become

2 0 + y = u sgna di 2

+ s sin [sg"z (7 + $)]I

3

b- - sgna

2a Solv ing E q s . (17) i n terms of t h e o r i g i n a l -. .

independent v a r i a b l e , t he t i m e t , for t h e i n i t i a l cond i t ions g i v e s t h e f i n a l r e s u l t : (16)

4

where These functions are interrelated through a change of the arzument:

L

ii

and

u = 1 for pure spin up (z and b same sign)

From the definition of 5 and 5 in Eqs. (18) it is apparent that

c = Ji;, [c2($+& 2 +") - c2($> ] 2:

= 42-i [ c (@nj)- 2.3 .(E,] na

Z=G [ s2 (.:'+Ft+Z) - 2a - s 2 ($)I

= & [ s ( q & q ) - s ( @ ) 2a Tla

( 2 2 ) = -1 for pure spin down (a and 6 opposite

s i g n and 0 5 t 5 - b/:) There are many ways of obtaining accurate

expressions for the Fresnel Integrals. case of the Galileo Spacecraft, for maneuvers in which the spacecraft acts as a_2single spinner, the initial spin rate parameter, b , IS large compared to the rotational acceleration parameter, 2a, so that the following asymptotic expansions are adequate far the maneuver analys is

For the The integrals 2 and in Eqs. (18) are closely

related to the Fresnel Integrals2:

C ( z ) = J z cos (; t') dt 0

J o

Other functions that are in use are - (0.19147-v) + ~ ( x )

X

0.3989423-7 __ x

X

c,(x) = E I cos t2 dt , 0

X

dt s in t s (x) = - 2

Thus E q s . ( 2 ) . (le), (22) and (23) completely specify the heuristic solutions for w,(t), wy(t) and ~,(t) for the s p i n up and spin down maneuver analysis of the Galileo Spacecraft.

5

For t h e sake of completeness , t h e F r e s n e l I n t e - g r a l s w i l l be eva lua ted f o r s m a l l arguments. The F r e s n e l I n t e g r a l s can be expressed i n terms of t h e a u x i l i a r y f u n c t i o n s f ( x ) and g(x)

-l.w

s(x) = - 1 - f ( x ) cos (;x~) - g ( x ) s i n (; x z j 2

( 2 4 ) The fo l lowing r a t i o n a l approximations f o r f ( x ) and

- 0.M) 0.80 I .to 2.40 3.20

g(x ) can be used2

1 + 0 . 9 2 6 ~ + c(x) f ( x ) = -_ 2 C 1 . 7 9 2 ~ + 3 . 1 0 4 ~ '

g(.) = 1- 3 + €(XI 2 +4 .142x+ 3.492x2+ 6 . 6 7 0 ~

Note t h a t t h e approximations apply t o all p o s i t i v e values of X . For l a r g e v a l u e s of x, however, Eqs. ( 2 3 ) are recommended s i n c e t h e asymptot ic expansions are much more a c c u r a t e w h i l e the f i x e d error i n Eqs. (25) allows a f a i r l y l a r g e f r a c t i o n a l e r ror t o occur .

I V . Numerical R e s u l t s f o r t h e S o l u t i o n of E u l e r ' s Equat ions of Motion-

Eqs. (18) and ( 2 ) p rov ide t h e h e u r i s t i c so lu - t i o n s f o r wx( t ) , wy(t) and w,(t) f o r near symmetric r i g i d bod ies . t h e e x a c t a n a l y t i c e x p r e s s m n s f o r t h e symmetric r i g i d body. The accuracy of t h e h e u r i s t i c s o l u t i o n s i s t e s t e d by comparison wi th t h e numerical i n t e g r a - t i o n of Eqs . (1) by ACSL (Advanced Continuous S imula t ion Language).

When I, = Iy, t h e s o l u t i o n s become

I n o r d e r t o t e s t t h e s o l u t i o n f o r t h e s p i n up maneuver, the f o l l o w i n g pa rame te r s were chosen from t h e f u l l y loaded, f u l l y deployed c o n f i g u r a t i o n of t h e Galileo S p a c e c r a f t .

2 Ix = 2985 ki logram - meters

I = 2729 Y

Iz = 4183

Mx = -1.253 newton - meters

M = -1.494

Mz = 13.5 (26 )

Y

I n t h e case of che s p i n down maneuver two t h r u s t e r s are used, forming a couple , so t h a t

and only t h e homogeneous s o l u t i o n s remain. S ince t h e p a r t i c u l a r s o l u t i o n s are the main area of i n t e r e s t , t h e case f o r Eq. ( 2 7 ) was no t si.mulated. Rather , t h e h y p o t h e t i c a l case of s p i n down wi th a s i n g l e t h r u s t e r w a s s imulated wi th reverse s i g n s

of p o s i t i v e and nega t ive s i g n s f o r a and b were t e s t e d t o ensu re t h e c o r r e c t n e s s of t h e h e u r i s t i c s o l u t i o n s (Eqs. ( 2 ) and (18)).

on t h e moments i n Eqs. ( 2 6 ) . A 1 1 f o u r combinations W

For t h e sake of b r e v i t y , only t h e s p i n up maneuver s i m u l a t i o n r e s u l t s w i l l be included i n t h i s r e p o r t w i th t h e assumption t h a t w x ( 0 ) = wy(0) = 0. It should be noted, however, t h a t t h e h e u r i s t i c solu- t i o n s given by E q s . ( 2 ) and (18) have been v e r i f i e d and t h a t t h e g e n e r a l conc lus ions f o r t h e s p i n up maneuver apply t o t h e s p i n down maneuvers as w e l l .

Case 1 - Spin Up from 3.15 t o 10 RPM

Th i s case d i r e c t l y a p p l i e s t o t h e s p i n up maneuver of t h e G a l i l e o Spacec ra f t from 3.15 rpm t o 10 rpm. The h e u r i s t i c s o l u t i o n is taken from Eqs. (18) , ( 2 2 ) and t h e asymptot ic expansions oi t h e F r e s n e l I n t e g r a l s of E q s . (23 ) . The exac t s o l u t i o n i s ob ta ined by t h e numerical i n t e g r a t i o n o f Eqs. (1) by ACSL. The discrepancy between t h e h e u r i s t i c and exac t s o l u t i o n s is i n d i s c e r n i b l e i n F igs . 1, 3 and 5. The accuracy of t h e h e u r i s t i c s o l u t i o n is shown i n F igs . 2 , 4 and 6 i n which t h e d i f f e r e n c e s between t h e exac t and t h e h e u r i s t i c s o l u t i o n s have been p l o t t e d . From t h e s e p l o t s i t i s clear t h a t t h e h e u r i s t i c soluCions f o r w x and wy d e v i a t e from t h e exac t s o l u t i o n by only 0 . 1 pe rcen t . The h e u r i s t i c

0.60

0.20 7 E!

x

-0.20 3x

-0.60 I

7

i

1.00

0.60

0.20

M w

x" 3 -0.20

-0.60

-1.00 (

I I 1

I I I K) 0.80 0.a 2.40

2 T x IO

Fig 2 Case 1: w = w - w Xerr exact "neuristic

(See Fig. 1).

I

-1.04 0.00 0.80 1.60 2.40 3.

T x IO 2

Fig 3 Case 1: Heuristic and Exact Solutions of o y ( t ) for the Near Symmetric Case with Spin Up from 3.15 rpm to 10 rpm. The heuristic solution for aiy(t) is determined from Eqs. 18, 22 and 13 while the exact solution is found by integrating Eqs. 1 with ACSL.

i,

-1.001 I I I 0.00 0.80 1.M) 2.40

T x 10'

Fig 4 Case 1: w = u - - w "err "exact Yheuristic

(See Fig. 3).

T x I02

Fig 5 Case 1: Heuristic and Exact Solutions for the Near Symmetric Case with Spin Up from 3.15 rpm to 10 rpm. The heuristic solution for w,(t) is from Eq. 2 while the exact Solution is found by integrating E q s . 1 with ACSL.

0.00

-0.20

-0.40 0 - x

z ”, 3 H -0.M)

-0.80

-1.00 C

! LY\

i ”:

I I I

I 0.80 1.M) 2.40 2 T x 10

0

w ZeXaCt - h e u r i s t i c

F i g 6 Case 1: wz = w err

(See Fig. 5 ) .

assumption t h a t w z varies l i n e a r l y w i t h t i m e is time is j u s t i f i e d i n F ig . 6 which i n d i c a t e s a d i sc repancy of only 0.01 pe rcen t from t h e exac t s o l u t i o n .

Case 2 - Spin up from 0 t o 3.15 RPM

While t h i s case does no t apply t o t h e G a l i l e o Spacec ra f t i n t h e p r e s e n t p l a n , i t is inc luded f o r t h e sake of completeness . Here i t is emphasized t h a t t h e asymptot ic expansions f o r t h e F r e s n e l I n t e g r a l s cannot b e used s i n c e t h e r e i s a singu- l a r i t y when ozO = 0. I n s t e a d , t h e r a t i o n a l approximations f o r f ( x ) and g(x) are used (Eqs. 2 5 ) a long w i t h Eqs. ( 2 4 ) , ( 2 2 ) and (18) t o p rov ide t h e h e u r i s t i c s o l u t i o n s . F i g s . 7 through 12 d i s p l a y t h e s i m u l a t i o n resul ts . A l l the remarks o f Case 1 app ly t o Case 2 except t h a t t h e l a t te r i s l e s s a c c u r a t e than t h e former. T h i s i s probably due t o t h e less a c c u r a t e e x p r e s s i o n s used f o r t h e F r e s n e l I n t e g r a l s .

V. Eu le r Angle Represen ta t ion

A Type 1: 3-1-2 Eu le r Angle R o t a t i o n i s used f o r t h e k inemat i c e q u a t i o n s of motion.3 t h a t t h e E u l e r i a n Angles (+x, +y, +,) are def ined by s u c c e s s i v e r o t a t i o n s by angles m Z , bX and +y about t h e 2, X‘ and Y” c o o r d i n a t e axes. The r e s u l t - i n g k inemat i c e q u a t i o n s of motion are

Th i s means

+x = wx cos+

iy = w - (wz c ~ s + ~ - w , s i n + ) tan+x

+ w z s i n + Y Y

Y Y

+= = (oz cos+ Y X - w s i n 4 Y ) set+, (28)

8

V I . S o l u t i o n of Eu le r i an Angles f o r a Near Symmetric Rigid Body Sub jec t t o

Constant Moments - The s o l u t i o n of Eqs. (28) p rov ides t h e a t t i t u d e

W’ of t h e body as a f u n c t i o n o f t i m e . t h e e q u a t i o n s are n o n l i n e a r i n general and are v e r y d i f f i c u l t ( i f no t impossible) t o s o l v e . However, f o r many s p a c e c r a f t a p p l i c a t i o n s , small a n g l e . approximiations f o r +x and eY are a p p r o p r i a t e and

Unfo r tuna te ly ,

4.N

2.oc

0.W 7 0 x

-2.00 3”

-4.00

-6.00 (

I I I

I I I ) 0.40 0.80 1.20

2 T x 10

Fig 7 CUSP 2: H e u r i s t i c and Exact So lu t ions of w,(t) f o r t h e Near Syormetric Case w i t h Spin Up from 0 rpm t o 3.15 rpm. The h e u i i s t i c s o l u t i o n f o r u X ( t ) i s determined from Eqs. 18, 2 2 , 24 and 25 w h i l e t h e exac t s o l u t i o n is found by i n t e g r a t i n g Eqs. 1 w i t h ACSL.

Io L

4.00

2.00

P - 0.00 0

x s I,,

3x -2.00

-4.00 c

I I I

f

I I I

T x 10’

I 0.40 0.80 1.20

F i g 8 Case 2: w = w - - w err exac t X h e u r i s t i c X

(See Fig . 7 ) .

I =..

-4.00 0.w 0.40 0.80 1.20

T x 10'

F i g 9 Case 2: H e u r i s t i c and Exact So lu t ions of wy(t) f o r t h e Near Symmetric Case and Spin Up from 0 rpm t o 3.15 rpm. The h e u r i s t i c s o l u t i o n f o r wy(t) is determined from Eqs. 18 , 2 2 , 24 and 15 wh i l e t h e exac t s o l u t i o n i s found by i n t e g r a t i n g E q s . 1 w i t h ACSL.

-2.00 , 0.00 0.40 0.80 1.20

2 T x 10

F i g 10 Case 2: w = w - w 'err 'exact ' h e u r i s t i c

(see F ig . 9 ) .

useful assumptions s i n c e they apply t o r i g i d bod ies w i t h h igh i n i t i a l s p i n rates o r w i t h a l a r g e to rque about a s i n g l e a x i s . Thus, s o l u t i o n of t h e l i n e a r - i z e d form of E q s . (28) can b e used i n t h e a n a l y s i s of spin-up and spin-down maneuvers of s p a c e c r a f t and i n t h e error a n a l y s i s of t h r u s t e r misal ignments i n sp inn ing s p a c e c r a f t .

c

0.4

0.3;

0.24

0.16 3"

0.ot

0.7 AI 0 0.40 0.W 1.a

2 T x IO

d

F i g 11 C a s e 2: H e u r i s t i c and Exact S o l u t i o n s f o r t h e Near Symmetric C a s e w i t h Sp in Up from 0 rpm t o 3.15 rpm. The h e u r i s t i c s o l u t i o n f o r o , ( t ) is from Eq. 2 wh i l e t h e exac t s o l u t i o n i s found by i n t e g r a t i n g E q s . 1 w i t h ACSL.

1.00 I I

bo

2 TxlO

F i g 12 Case 2: w = w - w =err = e x a c t ' h e u r i s t i c

(See F ig . 11).

9

1- 2 - Assuming m X and 4 are small we obtain from Eqs. ( 2 8 ) Y Ft +bt

+ " sgna m , = w , + m w - Jo Y Z

4 = wy - m x w z

4z = w z - my wx

Y

It is also appropriate to assume that + w Y X Equation continued is small compared to w z so that

( 3 0 )

The solution for wz is given in Eq. ( 2 ) . Integrat- ing

% --at +bt

-

+ z = w z = - Iz + wzo

+ u sgnZ

W

Following the method of Section 111, the inde- pendent variable is changed to 7 as given by Eqs. (7) and (13), so that the first two equations of Eqs. (29) become (after differentiating and rearranging) : - u sgna

(32)

where

1 a =- a Solving Eqs. (32) as functions of 7, then

applvina initial conditions and outtine the result .. . . ~

in terms of t gives the form of the solutions m,(t) and + (t): Y

L

--- dF, b + 2 a <

( 3 3 )

To obtain the solutions of mX(t) and $,,(t) the integrals of Eqs. ( 3 3 ) must be evaluated. Unfortunately, certain terms appear which cannot be directly integrated and so they are evaluated by asymptotic expansions. labor systematic the problem is divided into those integrals that are known and those that are unknown. This is seen most clearly by replacing the Fresnel Integrals of the solutions of a,(<) and coy(<) in Eqs. (18) with the f and g functions of Eqs! ( 2 4 ) :

In order to make the

W

10

c

k y l = wyo *

k = u sgn? Y3

ky6 = -U s g n i JI“ [@-I la1 la1

c1 = Cos 6, S 1 = sin<

so = s ( p q na

(34)

It is convenient to work with the independent variable 7. defined as follows.

Let the integrals to be evaluated be

- w ( 5 ) sin(a7 - “ 5 )

w (3 = dF, Ye

(35)

- T w X ( c ) sin(u7- a<)

rn wxs(i) - S, dS

Inspection of Eqs. (34) and (35) reveals that the following integrals must be solved:

11

c

Define the i n t e g r a l s L and L :

- G s ( r ) = - d i

0 c o s ( h l < + h2) Lc(y, hl, h2) = dF,

( 3 6 )

- F r m Eqs. (341, (35) and (36), t h e WYs, Wxc, Wyc and W,, i n t e g r a l s can be uritten Ls(;, hl, h l ) = d F

sin(hli; + h2)

- ( 3 8 ) W (T) = k J (7, 1, 0, -a , u?)+k J (7, 1, 0 , -a, a?) YS y l cs y2 ss

'Then, by well-known trigonometric identicies:

J (7, k13 k2, k3, k ) = L (7 , k, - k3, k2 - k4) Only two terms of the expansions will be used ss c

- L c ( i , kl + k3, k2 + k4) 1 3 5

T I Z

( 4 2 ) 1 15

(39) g(.) = - - - 2 3 .4=7

l r z The integrals L, and Ls can be expressed solely in terms of Fresnel Integrals C2 and S 2

Substitution of Eqs. ( 4 2 ) into the last four equa- tions of Eqs. ( 3 6 ) yields the approximations for

Fs, Gc and G : Le(<, hl, h2) =- Fc ,

Ul)

2 + sinul sinl(uO. ul)] - 3 (4)

- sint2 [s2(tl) - s 2 ( t o ) ] I 1 . [cosulcos 3 (u 0' u 1 ) = sin"

L S 6 , hl, h2) = -

L

Thus the integrals Jcc. Jcs and J,, are known integrals since they are explicit functions of the Fresnel Integrals.

- c o s y sin2(u0. ul)] -15 (f)

Next, the unknown integrals Fc, F,, 6, and 6, must be evaluated by asymptotic expansion Since they cannot be expressed explicitly in terms of known functions. The asymptotic expansions of the where f and g functions are

and the definitions have been used

(43)

s i n m ( t o , el) and cos,,(to, t l ) can be reduced t o s i n l ( t 0 , t i ) and c o s l ( t O , t i ) by r e p e a t e d a p p l i c a - t i o n of t h e formulas:

r _I

(44)

Thus

1 2 1 0' - ~ cos ("

The S ine and Cosine Integrals cnn-be wrLtten i n terms of the a u x i l i a r y f u n c t i o n s f and g :

u - ~~~

c . ( z ) = f ( z ) s i n z - g ( z ) c o s z

T I - s i ( z ) = 7 - f ( z ) cos% - i ( z ) s i n z

( 4 7 ) - - R a t i o n a l approximations f o r f and g are: '

+ c ( x )

[ < 5 for lZx<-

a = 38.027264 bl = 40.021432 1

aZ = 265.187033 bZ = 322.624911

a3 = 335.677320 b3 = 570.236280

ah = 38.102495 b4 = 157.105423

v

14

c

I .. al = 42.242855 bl = 48,196927

a2 = 302.757865 b2 = 482.L85984

a3 = 352.018498 b3 1114.978885

a4 = 21,821899 b4 = 449.690326

(48)

When t h e argument is l a r g e (,IO) the asymptot ic expansions are a c c u r a t e

2! I+! 6! 6 1 - - + - - - - + . . .

z 2 E 4 z

. . .) - &,(.) -1, + 2; - % +

z z z z

(49)

Thus, t h e approximate a n a l y t i c s o l u t i o n of Eqs . ( 2 8 ) when mX, m y and $ (u are small is

Y X

$,(t) - $,, COS [e [: at2 + XC) ] + $yo

where

1 a =-

+A Iz - I

I X

h =J I

- a =+T- HZ

IZ

and t h e symbols W s , W x s , W and Wxs are def ined i n E q s . (37) , (38y, (391, d b ) , ( 4 3 ) , (45), (461, (47 ) , and ( 2 0 ) . Useful approximations for t h e Fresnel, Sine and Cosine I n t e g r a l s are given i n E q s . (23) - (25) and (47) - (49) .

V I I . Conclusion

Ana ly t i c expres s ions have been found f o r Euler's Equat ions of Motion and for t h e E u l e r i a n Angles f o r bo th symmetric and near symmetric r i g i d b o d i e s under t h e i n f l u e n c e o f a r b i t r a r y c o n s t a n t body-fixed to rques . s o l u t i o n of E u l e r ' s Equat ions o f Motion is e x a c t . Numerical s t u d i e s have shown t h a t for a t y p i c a l near symmetric s p a c e c r a f t t h e near symmetric so lu - t i o n is v e r y accu ra t e . The s o l u t i o n of the E u l e r i a n Angles is much more r e s t r i c t e d . I n t h i s c a s e t h e two ang le s d e s c r i b i n g t h e o r i e n t a t i o n of t h e s p i n a x i s must be small. F u r t h e r r e s e a r c h w i l l be done t o determine t h e accuracy and range of a p p l i c a t i o n of t h e s e s o l u t i o n s .

When t h e body is symmetric t h e

VIII. References

(1.) Greenwood, D. T., P r i n c i p l e s of D p a m i c s , Englewood C l i f f s , New J e r s e y , P r e n t i c e - H a l l , Inc . , 1965.

(2 ) Abramowitz, M. and Stegun, I. A . , Handbook of Mathematical Funct ions, New York: Dover P u b l i c a t i o n s , I n c . , 1972.

(3) Wertz, J . R . , S p a c e c r a f t A t t i t u d e Determinat ion and Cont ro l , Boston: D. Reide l Pub l i sh ing Co., 1.978.

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