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11 TITLE (Include Security Classification)

Control of Large Space Structures (Unclassified)

12. PERSONAL AUTHOR(S)Leonard Meirovitch

13a. TYPE OF REPORT 113b. TIME COVERED 114. DATE OF REPORT (Year, Month, Day) 1S. PAGE COUNTFinal Report FROM 2-1-88 TO 1-31-8 1990, February 19 13+ 3 Appendixes

16. SUPPLEMENTARY NOTATION

17 COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number)FIELD I GROUP SUB-GROUP State equations of motion for the control of flexible

spacecraft and control of spacecraft with maneuveringflexible appendages in terms of quasi-coordinates.

19. ABSTRACT (Continue on reverse if necessary and identify by block number)

Work during this period was concerned with 1) development of a new method for the

derivation of the state equations of motion for the control of flexible spacecraft

in terms of quasi-coordinates and 2) development of a method for the control of

spacecraft in the form of articulated flexible multi-bodies.

S ELECTE 0L APR 30 1990i

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Dr. Spencer Wu 202/ 767-4937 NA

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VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY

DEPARTMENT OF ENGINEERING SCIENCE AND MECHANICS

Final Scientific Report on

CONTROL OF I- \RGE SPACE STRUCTURES

Covering the Period

I February 1988 - 31 January 1989

on the

Research Grant F49620-88-C-0044

Submitted to

Air Force Office of Scientific Research ,.

February 19, 1990 C-' '-ds

Principal Investigator: Leonard MeirovitchUniversity Distinguished Professor

Abstract

Work during this period was concerned with two aspects: I) development of a new

method for the derivation of the state equations of motion for control of flexible

spacecraft and 2) development of a method for the control of spacecraft in the form of

articulated flexible multi-bodies.

In deriving the equations of motion for the control of flexible spacecraft, it is common

practice to express the rigid-body rotational motions in terms of nonorthogonal inertial

components, such as Euler's angles. On the other hand, in controlling the same motions

of the spacecraft, the control law is generally in terms of angular motions components

along the orthogonal body axes. The elastic motions are ordinarily expressed in terms

of components along body axes. In implementing the control laws, it is desirable that

both the equations of motion and the control laws be in terms of the same type of var-

iables, and in particular in terms of the body axes variables. Reference I presents the

development of state equations of motion for flexible bodies in terms of quasi-

coordinates, which represent this very type of variables. Indeed, the angular velocity

components about the orthogonal body axes are time derivatives of quasi-coordinates.

The state equations of motion for flexible bodies in terms of quasi-coordinates render

the control implementation task considerably easier.

In many space applications, it becomes necessary to reorient the line of sight. If the line

of sight is fixed in the spacecraft, such as in the space telescope, this implies reorien-

tation of the whole spacecraft. In other space applications, the spacecraft consists of a

rigid platform and a number of flexible appendages, such as antennas. If the mission,

requires the reorientation of the line of sight of these antennas relative to the inertial

space, then the preferred strategy is to stabilize the platform relative to the inertial space

and to reorient the flexible antennas relative to the platform. Reference 2 provides a

derivation of the state equations of motion for the task described above, based on the

equations of motion in terms of quasi-coordinates derived in Ref. I. Because the

motions defining the maneuvering of the antennas are known a priori, the state

equations contain time-dependcnt coefficients and persistent disturbances.

The problem of maneuvering an articulated flexible spacecraft and controlling its vi-

bration at the same time is a very complex task. Reference 3 shows an approach to the

problem, in which the maneuvering of the appendages is carried out open-loop using a

bang-bang control law. On the othcr hand, the vibration suppression is carried out

closed-loop, which amounts to controlling a time-varying system .subjected to persistent

disturbances.

In addition to the above research, work was carried out on several other papers (Refs.

4 - 7) describing research under the preceding AFOSR grant. In particular, we single

out Ref. 5, which was presented during this grant period. The paper is concerned with

control of the perturbations experienced by a flexible spacecraft during a minimum-time

maneuver. The spacecraft is modeled as a flexible appendage attached to a rigid hub.

The perturbed model can be divided into a rigid-body part and an elastic part. The

model is described by a linear, time-varying set or ordinary differential equations sub-

jected to piecewise-constant disturbances caused by inertial forces resulting from the

minimum-time maneuver. The control consists of a reduced-order compensator de-

signed such that the perturbed model is finite-time stable.

References (Enclosed)

I. Meirovitch, L., "State Equations of Motion for Flexible Bodies in Terms of Quasi-

Coordinates," Proceedings of the IUTAM/IFAC" Symposium on Dynamics of Con-

trolled Systems, Zurich, Switzerland, 1989 (G. Schweitzer and M. Mansour,

Editors), Springer-Verlag, Berlin, 1989.

2

2. Meirovitch, L. and Kwak, M. K., "State Equations for a Spacecraft with Maneu-

vering Flexible Appendages in Terms of Quasi-Coordinates," Applied Mechanics

Reviews, Vol. 42, No. 1I, 1989, pp. 5161-5170.

3. Meirovitch, L. and Kwak, M. K., "Control of Spacecraft with Multi-Targeted Flex-

ible Antennas." Paper AIAA-88-4268-CP, AIAA/AAS Astrodynamics Conference,

Minneapolis, Minnesota, August 15-17, 1980. To appear in the Journal of Guid-

ance, Control, and Dynamics.

Other References (Not Enclosed)

4. Sharony, Y. and Meirovitch, L. "Accommodation of Kinematic Disturbances Dur-

ing a Minimum-'rime Maneuver of a Flexible Spacecraft," Paper AIAA-88-4253-CP,

AIAA/AAS Astrodynamics Conference, Minneapolis, Minnesota, August 15-17,

1988. To appear in the Journal of Guidance, Control, and Dynamics.

5. Sharony, Y. and Meirovitch, L., "A Perturbation Approach to the Minimum-Time

Slewing of a Flexible Spacecraft," IEEE International Confcrence on Control and

Applications, Jerusalem, Israel, April 3-6, 1989.

6. Meirovitch, L. and Kwak, M. K., "I)ynamics and Control or Spacecraft with Re-

targeting Flexible Antennas," AlAA/ASME/ASCE/AHS/ASC 29th Structures,

Structural Dynamics, and Materials Conference, Williamsburg, Virginia, April

18-20, 1988. To appear in the Journal of Guidance, Control, and Dynamics.

7. Meirovitch, L. and Kwak, M. K., "Some Thoughts on the Convergence of the

Classical Rayleigh-Ritz Method and the Finite Element Method,"

AIAA/ASMrE/ASCE/AIIS/ASC 29th Structures, Structural Dynamics, and Materi-

als Conference, Williamsburg, Virginia, April 18-20, 1988. To appear in the AIAA

Journal.

State Equations of Motion for Flexible Bodiesin Terms of Quasi-Coordinates

LEONARD MEIROVITCH**

Department of Engineering Science and MechanicsVirginia Polytechnic Institute and State UniversityBlacksburg, VA 24061

Summary

This paper is concerned with the general motion of a flexiblebody in space. Using the extended Hamilton's principle for dis-tributed systems, standard Lagrange's equations for hybrid sys-tems are first derived. Then, the equations for the rigid-bodymotions are transformed into a symbolic vector form of Lagrange'sequations in terms of general quasi-coordinates. The hybridLagrange's equations of motion in terms of general quasi-coordi-nates are subsequently expressed in terms of quasi-coordinatesrepresenting rigid-body motions. Finally, the second-orderLagrange's equations for hybrid systems are transformed into aset of state equations suitable for control. An illustrativeexample is presented.

Introduction

The derivation of the equations of motion has preoccupied dynam-

icists for many years, as can be concluded from the texts by

Whittaker (11, Pars [2] and Meirovitch [3]. References 1-3 con-

sider the motion of systems of particles and rigid bodies, and

the equations of motion are presented in a large variety of

forms. In this paper, we concentrate on a certain formulation,

namely, Lagrange's equations. For an n-degree-of-freedom system,

Lagrange's equations consist of n second-order ordinary differen-

tial equations for the system displacements.

In the control of dynamical systems, it is often convenient to

work with first-order rather than second-order differential equa-

* Sponsored in part by the AFOSR Research Grant F49620-88-C-0044 monitored by Dr. A. K. Amos, whose support is greatlyappreciated.

** University Distinguished Professor.

(i ii~nwrtzervi m insnur

D, namics o0 Controlled MechAnical SvslemsI1L TAM/1FAC S~mposurn Zurich/Switzerland 1988

Springer.%erlot Berlin Heidelberg iSQ

38

tions. Introducing the velocities as auxiliary variables, it is

possible to transform the n second-order equations into 2n first-

order state equations. The state equations are widely used in

modern control theory [4].

With the advent cf man-made satellites, there has been a renewed

interest in the derivation of the equations of motion. The

motion of rigid spacecraft can be defined in terms of transla-

tions and rotations of a reference set of axes embedded in the

body and known as body axes. The equations of motion for such

systems can be obtained with ease by means of Lagrange's equa-

tions. It is common practice to define the orientation of the

body relative to an inertial space in terms of a set of rotations

about nonorthogonal axes [3]. However, the kinetic energy has a

simpler form when expressed in terms of angular velocity compo-

nents about the orthogonal body axes than in terms of angular

velocities about nonorthogonal axes. Moreover, for feedback

control, it is more convenient to work with angular velocity

components about the body axes, as sensors measure angular

motions and actuators apply torques in terms of components about

the body axes. In such cases, it is often advantageous to work

not with standard Lagrange's equations but with Lagrange's equa-

tions in terms of quasi-coordinates (1,3]. If the body contains

discrete parts, such as lumped masses connected to a main rigid

body by massless springs, it is convenient to work with a set of

axes embedded in the undeformed body. The equations of motion

consist entirely of ordinary differential equations and can be

obtained by a variety of approaches, including the standard

Lagrange's equations and Lagrange's equations in terms of quasi-

coordinates [53*.

In the more general case, the body can be regarded as being

either entirely flexible with distributed mass and stiffness

properties or as consisting of a main rigid body with distributed

elastic appendages. Unlike the previous case, the equations of

motion are hybrid, in the sense that the equations for the rigid-

* Note that Ref. 5 refers to Lagrange's equations in terms ofquasi-coordinates as Boltzmann-Hamel equations.

39

body motions are ordinary differential equations and those for

the elastic motions are partial differential equations. Hybrid

equations were obtained for the first time in Ref. 6. Moreover,

the formulation of Ref. 6 was obtained by using Lagrange's equa-

tions in terms of quasi-coordinates, but some generality was lost

in that the body considered was assumed to be symmetric and to

undergo antisymmetric elastic motion. As a result, the rigid-

body translations were zero.

This paper is concerned with the general motiun of a flexible

body in space. Using the extended Hamilton's principle for dis-

tributed systems (7], standard Lagrange's equations for hybrid

systems are first derived. Then, using the approach of Ref. 3,

the equations for the rigid-body motions are transformed into a

symbolic vector form of Lagrange's equations in terms of general

quasi-coordinates. The hybrid Lagrange's equations of motion in

terms of general quasi-coordinates are subsequently expressed in

terms of quasi-coordinates representing rigid-body motions. This.

is a very important step, as the latter form permits the derivat-

ion of the hybrid equations of motion with relative ease, thus

eliminating a great deal of tedious work. These hybrid equations

represent an extension to flexible bodies of Lagrange's differen-

tial equations in terms of quasi-coordinates derived in Ref. 3

for rigid bodies. The second-order equations are then used to

derive the hybrid state equations.

As an illustration, the hybrid equations of motion of a space-

craft consisting of a rigid hub with a flexible appendage simu-

lating an antenna are derived.

Standard Lagrange's Equations for Hybrid Systems

Let us consider a flexible body and assume that the Lagrangian L

= T - V, in which T is the kinetic energy and V is the potential

energy, can be written in the general form L :L(q ,uj uju=,..

.,U(P)), where qj = qi(t) (i = 1,2,....m) are generalized coordinates

describing rigid-body motions of the body and uj(Pt) (j =

1.2,...,n) are generalized coordinates describing elastic motions

relative to the rigid-body motions of a typical point in the body

40

identified by the spatial position P. Dots designate derivatives

with respect to time and primes derivatives with respect to the

spatial position. For convenience, we express the Lagrangian in

terms of the Lagrangian density L in the form L = 0 L dO, where 0

is the domain of extension of the body.

we propose to derive Lagrange's equations by means of the

extended Hamilton's principle (7), which can be stated as

t2ftI ( L + 6W)dOdt = 0. 5qi = 6uj 0 at t Z ti, t2 (1)

where 6W is the nonconservative virtual work density, which is

related to the virtual work by W SW dO. The virtual work can

be written in the form

m n6W= Oisq + , U 6u dO (2)

where Qi are nonconservative generalized forces associated with

the rigid body motions and U. are nonconservative generalized

force densities associated with the elastic motions; 6qi and 6uJ

are associated virtual displacements. Following the usual steps

(7], we obtain Lagrange's equations of motion, which can be

expressed in the symbolic vector form

d (L) _ IL = -0 a - a- + LU = (3ab)d- ag a , t aA -

where g and Q are m-vectors, u and 0 are n-vectors and L is an

n xn operator matrix. Because of the mixed nature of the differ-

ential equations, we refer to the set (3) as hybrid. The elastic

displacements are subject to given boundary conditions.

Equations in Terms of Quasi-Coordinates for the Rigid-BodyMotions

Quite often it is convenient to express the Lagrangian not in

terms of the velocities 4, but in terms of linear combinations

wt (t=1,2,...,m) of qi. The difference between q. and w, is that the

former represent time derivatives dqi/dt, which can be integrated

with respect to time to obtain the displacements qi, whereas

wt cannot be integrated to obtain displacements. It is customary

41

to refer to wL as derivatives of quasi-coordinates (3]. The

relation between w, and qi can be expressed in the compact matrix

form w - A Tq, where the notation is obvious. Similarly, we

express the velocities qi in terms of the variables w, as B 6w,

from which it follows that the m m matrices A and B are related

by AT8 = 8TA = 1, where I is the identity matrix of order m.

Our object is to derive Lagrange's equations in terms of w

instead of qi. Using the relations indicated above, it can be

shown [3) that Eqs. (3a) can be replaced by

d L ST

-tA)+8 L (4)

whereC B T - T T aA N = BTQ (5a,b)

and we note that the first matrix in E is obtained by first

carrying out a triple matrix product for every one of the m2

entries in A and then arranging the resulting scalars in a square

matrix. On the other hand, the second matrix in E is obtained by

first generating a row matrix for every generalized coordinate qk

(k - 1,2,.... m) and then arranging the row matrices in a square

matrix. Equation (4) represents a symbolic vector form of the

Langrange equations for quasi-coordinates. The complete formula-

tion is obtained by adjoining to Eq. (4), the equations for the

elastic motion, Eq. (3b), as well as the associated boundary

conditions.

General Equations in Terms of Quasi-Coordinates for aTranslating and Rotating Flexible Body.

Let us consider the body depicted in Fig. 1. The motion of the

body can be described by attaching a set of body axes xyz to the

body in undeformed state. The origin of the body axes coincides

with an arbitrary point 0. Then, the motion can be defined in

terms of the translation of point 0, and the rotation of the body

axes xyz relative to the inertial axes XYZ. The position of 0

relative to XYZ is given by the radius vector R = R(Rx, Ry, RZ).

The rotation can be defined in terms of a set of angles l1 2

and e3 (Fig. 2). Hence, the generalized coordinates are

q, . RX0 q2 = Ry, q3 = RZ, q4 = 81, q5= '2' q6 = e3" In addition, there

42

are the elastic displacement components ux(P,t), Uy(Pt), Uz(Pt).

The displacements RX, Ry, R7 are measured relative to the inertial

axes XYZ. On the other hand, the displacements u. , u, uz are mea-

sured relative to the body axes xyz. Moreover, the components

RX, Ry, RZ of the velocity vector R are also measured relative to

XYZ. On the other nand, the angular velocity vector has compo-

nents iX' , w , measured -elative to the body axes xyz. It will

prove convenient to express all motions in terms of components

along the body axes. To this end, if we denote the velocity of

point 0 in terms of components along the body axes by V, then it

can be shown that V = CR, where C = C(91 ,0 2 ,9 2 ) is a rotation matrix.

Moreover, the angular velocity vector w can be expressed in terms

of the angular velocities 31's 2 and a3 in the form ,= 0, where

0 = 0(81, 3) is a transformation matrix. We note that the angular

velocity components x, y and ., cannot be integrated with respect

to time to yield angular displacements a X ay and az about axes x, y

and z, respectively. Hence, X, y, WZ can be regarded as time

derivatives of quasi-coordinates and treated by the procedure

presented in the preceding section. Although it is not very

common to regard the velor7ty components Vx, Vy and Vz as time

derivatives of quasi-coordinates, they can still be treated as

such. In view of this, if we introduce the generalized velocity

vector = R z 1 a2 63 1 , as well as the "quasi-velocity"

vector w [V x Vy V Z Wy z Y we conclude that the coefficient

matrices are defined by

AT i , 1 (6a,b)

where we recognized that C-1 = CT , because rotation matrices

are orthonormal. It can be shown, after lengthy algebraic manip-

ulations, that

BTE : (7)

where . and V are skew-symmetric matrices corresponding to

and V [31, respectively.

Using Eqs. (3b) and (4) in conjunction with the above relations,

we obtain the hybrid Lagrange's equations in terms Df quasi-coor-

43

dinates

d fL) L- C -LF 8a)t '3V 3V jR -d L) V L L 1 ( T l )L

Tt _ ( _L. -- _ ( (8b)dt h 3V , )-J

_ L 1 (8c))t 3u

where F and M are external nonconservative force and torqie,

respectively, in terms of components about the body axes,

3L/)e = [)L/38 I L/ia 2 L/3e3T and i = u. Note that e does not

really represent a vector and must be interpreted as a mere sym-

bolic notation. We recall tnat the components of u Are still

subject to given boundary conditions.

It should be pointed out that, in deriving Eqs. (8), no explicit

use was made of the angles a,.3 2 and 3' so that Eqs. (8) are

valid for any set of angles describing the rotation of the body

axes, such as Euler's angles, and they are not restricted to the

angles used here. Moreover, point 0 is an arbitrary point, not

necessarily the mass center of the undeformed body, and axes xyz

are not necessarily principal axes of the undeformed body. Clear-

ly, if xyz are chosen as the principal axes with the origin at the

mass center, then the equations of motion can be simplified.

State Equations in Terms of Quasi-Coordinates

Equations (8), and in particular Eqs. (8a) and (8b), can be

expressed in more detailed form. To this end, we write the

velocity vector of a typical point P in the body in terms of com-

ponents along the body axes as follows:

vp =.. w ( + U)+ v = V + ( + + V (9)

where r is the nominal position of P relative to 0. Moreover,

F and u represent skew-symmetric matrices associated with the

vectors r and u, respectively. Then, denoting by o the mass den-

sity, the kinetic energy can be shown to have the expression

T -pp dD vrnv_ . IT S' + r OV dO2 ~o0 pp dO- 2 i VD >.

44

~.T~+ W' )1 dO ,'J- fo dD (10)

2T 2

where S - :D P( + u)dO, J =0 + 6)(F + ) Tdo, in which S is

recognized as a skew-symmetric matrix of first moments and J as a

symmetric matrix of mass moments of inertia, both corresponding

to the deformed body. Moreover, we assume that the potential

energy has the functional form V = V(R, 9, U, u, .... u(P)).

Inserting Eq. (10) into Eqs. (8) and rearranging, we obtain the

explicit Lagrange's equations in terms of hybrid coordinates

mV + T+ dO = (2S, + mV + :S)w - C - + F (a)

SV + J, + f0 o(r + dDO = 12 I' o(r + u)v dO + SV - GJjw

(O ) - I 3 + M (11b)36T) - clb

T- = T 2 TOV + D( )W+ Ov OV Ow- (r + u) - 2o 4-Lu + U (1J1c)

where SV 0 f P dD. The state equations are completed byD

adjoining the kinematical relations

R = CTv, = D-, u = v (lld,e,f)

Illustrative Example

As an illustration, we consider a spacecraft consisting of a

rigid hub and a flexible appendage, as shown in Fig. 3. From

the figure, we can write

C- , i.u-Uyj+Uzk, J+vzk (12)

so that- 0 -JOu Zdx fPU dx

S = dx -i x (13a)

-fu dx mx 0

where o is the mass density of the appendage, m is the total mass and x is the

position of the mass center of the appendage. Moreover,

45

JXX+.r0NY(U. 4.)dx _![Qxu dx -fPXu zdX

j = -,foxu dx J +fiou2dx -,pu u dx] 13b)y yy z y zL joxu dx -- ~udx +1 u2d

where J ,X J yyand J zzare the mass moments of inertia of the

spacecraft regarded as rigid.

Using Eqs. (12) and (13), the state equations, Eqs. (11), can bewritten in the explicit forms

RX ' (ce 2 C 3 + se1 se2se3)Vx - (ce 2c Co - so 1soe2 co3 )V + coiso 2 v (14a)

y . e1s v + Co 1'@3 vy - Se1vz(14b)

Rz = -(se 2 co3 - seICae2 soe3 )VX (so 2 S63 +so C c 2 ce 3)Vy co e s 2 v z (14c)

81 C83 so = s83 WY 663 ~ ~ 4 , ( 14d, e)

so1o3 So1o3

3V + x - dx =x +Vw -coZW + mwizw +y -y Wx 1 f~u ,dg,

2 2V

- ~dx + 2wziOv~ dx - 2wJ*z dx - (ce ce3 +SleS 3) so

JOV CG S8 ' 3 1283 E

- ce so3 L + (s e so - o 8ce-se3) + F. (14i)

2 2mV~ - W XfPuz dx + mixw =~ .mV zW - mVxwz - mlxWXW y + 2W 2 ~p dx

WW dx +2wxJt~ dx +(ce so3 V S8S 2Ce L -

y

-2 2mV z+ wxypu y d-m1 W= x y -vyx W lxxz+ W+W y TOz d

W wf[u dx +2wJXv 2ax -~h cosz o oc V

(14k)

46

- (!Qu z dx)V y+ (IOU dx)V z + (J x +- fP(U + U z)dxhll - (%xu dx)tl

- (fOXU z dx)wj + I",(" v z- u v )dx -(V f oU dx + V Z'%.U dx)

" V x W Ylu ydx +V A . u zdx + *-, yOxu z dx - w w . .;xu ydx

" (W2 _ u, )u u dx + Y , l - Q~ 2 _ u2 )dxjy z y Z y Y ZZ y Z

-2w fQ(u v' + u v )dx -ca 5 1 v oI 3 3V. + M (141)x y y z z3 3 51 Co1a 2 ce1 30 3 x

-2(fpuzdx)Vx - m1xV X-* (Jxu dx)w'x (- yy + Jouz dx)w - (> u yu dx

+ ,( Z xU - xVz )dx = mI xV .y - (V %"uz dx + mI x OW + V y u dx'

- Wx (J - + ;Q 2dx) - 2 _ 2-)"x dx + wyzfx dxz xx -zz z x z z )yx~ Z~u

- WX~OYZdx + 2,X x dx 2w yfPu z Vz dx

+ 2w f,)u v dx + so 3 V co 1 V ceol co 3-a + M (14m)Z zy 3a1 1ae2 ce1 a@ 3 y

-(IOU dx)V x+ m 1 v - (fOxU dx)w x- (JOU yU dx)w y+ (J z + IOU dx)w

y x ly z x yz y z y x.

+ W Wxfo- uv dx 2wfov d +2w OU dx - 2w fu vdx +Vm)w

x z y z x z y y z dx-2fi u ae 3 z

( 14n)

O y -QzWx+ Pxwz+ vy V zwx Ox z -P xW y+ ( +Wz)y

- Owy WzUz zx y + 2ovw - LIyU u (14o)

0 V + 0U ywx - Oxw y + OVz Ov yx +vx Wy - xz - WWy

+O 2 + W2 )U- 2ov - LIu + U (1p+ ~x y)u z y (14p)

where m, is the mass of the appendage, se1 sin oi, cei CO co e(i

1.2.3) and

47

32 32 - L 24Y : ±- (El _L_) _ ±_ I ( , . d )!-- ( 15a)

2 (.L 2

E I. z -4-) , c115b)

in which E is the modulus of elasticity and Ij and Iz are area

moments of inertia. The operators 4y and £ z include the

effects of bending and of the axial force on the appendage [7J.

Summary and Conclusions

In deriving the equations of motion for flexible bodies by the

Lagrangian approach, it is common practice to express the rota-

tional motion in terms of angular velocities about nonorthogonal

axes, which tends to complicate the equations. Moreover, this

creates difficulties in feedback control, in which the torque

actuators apply moments about body axes and the output of sensors

measuring angular motion is also expressed in terms of components

about the body axes. The same can be said about force actuators

and translational motion sensors. It turns out that the equat-

ions of motion are appreciably simpler when the rigid-body trans-

lations and rotations are expressed in terms of components about

the body axes. Such equations can be obtained by introducing the

concept of quasi-coordinates. The concept of quasi-coordinates

was used earlier by this author to derive equations of motion of

rotating bodies with flexible appendages, but never in the

general context considered here. Indeed, in this paper,

Lagrange's equations in terms of quasi-coordinates are derived

for a distributed flexible body undergoing arbitrary rigid-body

translations and rotations, in addition to elastic deforma-

tions. The second-order differential equations in time for the

hybrid system are then transformed into a set of hybrid state

equations suitable for control design. The approach is demon-

strated by deriving the hybrid state equations of motion for a

spacecraft consisting of a rigid body with a flexible appendage

in the form of a beam.

References

1. Whittaker, E. T., A Treatise on the Analytical Dynamics ofParticles and Rigid Bodies, 4th Edition, Cambridge UniversityPress, London, 1937.

48

2. Pars, L. A., A Treatise on Analytical Dynamics, WilliamHeinemann, Ltd., London, 1965.

3. Meirovitch, L., Methods of Analytical Dynamics, McGraw-HillBook Co., New York, 1970.

4. Brogan, W. L., Modern CDntrol Theory, Quantum Publishers,Inc., New York, 971.

5. Kane, T. R. and Levin~sn, D. A., "Formulation of Equaticns ofMotion for Complex 3pacecraft," Journal of Guidance and Con-trol, Vol. 3, No. 2, 1980, pp. 99-112.

6. Meirovitch, L. and Nelson, H. D., "On the High-Spin Motion ofa Satellite Containing Elastic Parts," Journal of Spacecraftand Rockets, Vol. 3, No. 1I, !966, pp. 1597-1602.

7. Meirovitch, L., Computational Methods in Structural Dynamics,Sijthoff & Noordhoff, rne Netherlands, 1980.

0 v

Figure 1. The Flexible BodyY in Space

Z.

x~ ez

3 y ,

- 0 1 ;'2 2 Figure 2. The Angular Motions

X' /

Figure 3. A Rigid Spacecraft with a Flexible Appendage

State equations for a spacecraft with maneuvering flexibleappendages in terms of quasi-coordinates

L Meirovitch and M K Kwak

Depzrtnmert of Engineering tcienrce nd th'chan .r, Virginia Po/ytrehnic Institute andState University. Blackvburg. VA 2d06 I

This paper is concerned with the derivation of the state equations of motion fora spacecraft consisting of a main rigid platform and a given number of flexibleappendages changing the orientation relative to the main body. The equationsare derived by means of lagrange's equations in terms of quasi-coordinates.Assuming that the appendages represent distributed-parameter members, thestate equations of motion are hybrid. Moreover, they are nonlinear. Followingspatial discretization and truncation, the hybrid equations reduce to a systemof nonlinear discretiZed state equations, which are more practical for numericalcalculations and control design. To illustrate the effect of nonlinearity on thedynamic response during reorientation, a numerical example involvingspacecraft with a membrane-like antenna is presented.

1. INTRODUCTION very complicated expressions fGrote. McMunn and Gluck(1970) and Likins (1972)]. so that new methods for deriving

In many space applications, it becomes neccssarv to reorient equations of motion have been proposed [Ho (1977) and Kanea certain line of sight in a spacecraft. Examples o( this are the and Levinson (1980)1. Kane and Levinson compared sevenreerientation of a space telescope or of an antenna in a different methods. Lagrange's equations of motion in termsspacecraft. in some cases, such as in the space telescope, the of quasi-coordinates for a hybrid system were derived first byline of sight can be regarded as being fixed relative to the \leirovitch (1966) and then by Williams (1976) and Brownundeformed structure, in which cases reorientation of the line (1981). Recently. Meirovitch (1988) and Meirovitch andof sight implies maneuvering of the whole spacecraft [Turner Kwak (1989) showed that Lagrange's equations of motion inand Junkins (1980), Dreakwell (1981), llaruh and Silverberg terms of quasi-coordinates are quite useful for deriving the(1984), Meirovitch and Quinn (1987), Meirovitch and Sharony equations of molion for the maneuvering and control of(1987) and Quinn and Meirovitch (1988)1. However, many flexible spacecraft. Because the derived equations of motionspacecraft can be represented by mathematical models are based on body-fixed coordinates, control design based onconsisting of a rigid platform with one or more flexible such equations is very convenient.appendages, such as flexible antennas, so that the missioninvolves the maneuvering of a hybrid (lumped and distributed The mathematical formulation for a spacecraft including aflexible) system. Quite often, the line of sight coincides with rigid platform and several flexible antennas consists of a hybridan axis fixed in a small component of the spacecraft. such as set of equations of motion, in the sense that there are sixan antenna, in which case it may be more advisable to retarget ordinary differential equations for the rigid-body translationsonly the antenna and not the entire spacecraft. This is and rotations of the platform and partial differential equationsparticularly true when the inertia of the antenna is much for the elastic motion of each antenna. The equations of,maller than the inertia of the spacecraft. The argument motion are not only hybrid, but the maneuvering of thebecomes even stronger when several antennas must be antennas relative to the platform according to some prescribedretargeted independently in space. in such cases, it appears function of time introduces time-dependent coefficients into themore sensible to conceive of a spacecraft consisting of a equations. Moreover, the equations contain terms reflectingplatform stabilized in an inertial space with several persistent disturbances caused by inertial forces. Because bothappendages. rigid or flexible, hinged to the platform and numerical simulation and control design of systems governedcapable of pivoting about two orthogonal axes relative to the by sets of hybrid differential equations are not feasible, it isplatform [Meirovitch and Kwak (1988a. 1988b) and necessary to discretize the partial differential equations inMeirovitch and France (1989)1. In this case, reorientation space, which can be carried out by the classical Rayleigh-Ritzrelative to the stabilized platform is equivalent to retargeting in method or the finite element method [Meirovitch (1980)1.an inertial space. Note that such a maneuvering spacecraft ischaracterized by the fact that its configuration varies with time.This paper is concerned with the mission of independent it. GENERAL LAGRANGE'S EQUATIONS INretargeting of the line of sight of each antenna relative to the TERNIS OF QUASI-COORDINATESinertial space.

Ote problem encountered in the dynamics of a complex Let us consider a system consisting of a main rigid body,syqtem is how to derive the equations of motion efficiently. In acting as a platform. and a certain number of flexiblegeneral, the equations of motion for a flexible spacecraft have appendages hinged to the main rigid body. The interest ties in

.PpI Mech Pay vol 42. no 11, Pa'n 2, Nov 1989 S161 Copyright 1969 Anetican Soct of Mechankal BElre

$162 MECHANICS PAN-AMERICA 1989 AppI Mach Rey 1M Supleen

reorienting the flexible appendages independently so as to point and that of a point in the typical appendage e in terms ofin different preselected directions in the inertial space. The components along xyyz isobject is to derive the equations of motion capable ofdescrihing this task. + x r, ) + (ET + ce) x r + tt) + v

To describe the motion of the platform, we introduce a set e = 1,2.N (4b)of inertial axes XYZ and a set of body axes xyz attached to therigid platform. Then, the motion of the platform can be where w. is the angular velocity vector of axes xyZ, E. isdefined in terms of three translations and three rotations of the a matrix or direction cosines between the xyz, and gz andbody axes xyz relative to the inertial axes XYZ. To describe - is the elastic velocity of the point in the appendage reltivethe motion of the flexible appendages, we consider a typical to xj,z.,, = y.. In the maneuver proposed, the angularappendage hinged at point e and regard e as the origin of a set velocity vectors cu of xpza relative to xyz are given, so thatof body axes xyz, embedded in the appendage in it the rotational motions of the appendages relative to theundeformed state. Then, the motion of a nominal point of the platform do not add degrees of freedom. The only degrees ofapendgeonistse. of the motion of a noineloton of te. freedom arise from the rigid-body translations and rotationsappendage consists of the motion of xyz. the motion of x'The of the platform. and the elastic displacements of therelative to xyz and the elastic motion relative to x ',z, . "lheapedgs

system and the various reference frames are shown in Fig. 1. appendages.

From Fig. 1, the position vector of a point in the rigid body The equations of motion can be obtained by means ofand in the appendage can be written as y Lagrange's equations in terms of quasi -coordinates [Meirovitch

(19ig)].-~-0 - d 8L ~-8L O.eLand a)d ( ) + Z - - C=-- F (Sa)

R,=R r + + , e= 1.2 ..... V (lb) d 8L 8L -BL _ L

where R is the radius vector from 0 to o, r is the position -i ( + V0 a,+ "- (T) - = M (Sb)

vector of a nominal point in the rigid body relative to xyz, A A

,r is the radius vector from o to e. r is the position vector A L, 8T, A

of a nominal point in undeformed appendage relative to -C) -)- + "fP ' , - U,, e = 12,..., N (Sc)x,,., and u, is the elastic displacement of that point. VectorR. is given in terms of components along XYZ, r and r. wherein terms of components along xyz, and r, and u. in terms ofcomponents along xyz°. L = T- V (6)

The velocity vector of o can be written in terms of is the Lagrangian in which T is the kinetic energy and V is thecomponents along xyz in the form A

potential energy, L, is the Lagrangian density and T is theK, = CR0 (2a) kinetic energy density. both for appendage e. and . is a

matrix of homogenous differential stiffness operators. Thewhere C is the matrix of direction cosines between xyz and terms F and U on the right side of Eqs. (5a) and (5b),XYZ and R0 is the velocity vector of o in terms of components respectively, are the force and torque vectors on the platform,along XYZ. Matrix C depends on the angular displacements both in terms of components along axes xyz, and the term U,O, (i = 1,2.3) defining the orientation of axes xyz relative to on the right side of Eq. (5c) is a distributed force vector onaxes XYZ. Furthermore, the angular velocity vector of axes appendage e in terms of components along x.Yz . . Equationsxyz in terms of components along xyz is given by (5) are hybrid in the sense that Eqs. (Sa) and (5b) are ordinary

differential equations and Eqs. (5c) are partial differential= DO (2b) equations. It should be noted that the tilde over d symbol

indicates a skew symmetric matrix with entries correspondingwhere 0 is a vector of angular velocities 0, and D is a matrix to the components of the associated vector [Meirovitch anddepending on the angular displacements 0, (- 1,2,3). Figure Kwak (1988a)J. For the system of Fig. 1. we write the kinetic2 shows a set of such angular displacements. For this choice energyof angles, the matrices C and D are as follows:[C02CO3 C01503 + St9 1S02cO3 SOIS03 - cO1sOT-0 3] T'+ FT'J dD, + f + P, TdD

C = - c62SO3 cOc0 3 - sOi0sO5O3 sOic0 3 + cOISO2sO 3 (3a)I I NS- 2 SOiC0 2 cOIc0 2 J 1 r Q fT + I

c02,c0 3 sO3 0 .-

,.P "v iS;D +, Q0 0@c02S0 3 c03 0 (3b) + pTy. + (i dDo

s02 0 1I fpi D

where cO, = cos 0, and sO, = sin 0,. Note that this choice of + wTE/,o,, + (ET, + 0 ,)rf pQ- + ,j.,dD.I (7)angles helps us avoid singularities at the initial stage of the D,motion, 0, = 0. where

In view of the above, the velocity vector of a point in the Nrigid body in terms of components along xyz is simply mp = mD + mf p a D"

= + x (4a) + m,, m- = , m, = D (Sbc)

Appl MR Amv vol 42 no 11, Pait 2, Nov 1U Me0rovitcl and Kwak: State equations for a apacearft S163

N N pt%

= il + E(m + EriE,) (84) -Zp + Z oI - E[(2j,4 - tti.,) + 2r,E, + j,,)

S r S, p7 + J D, + "2 +LTdp,(8f)++24 D, ) E,

T + ±T 1eF - Fa,, -rTS,E E, - fP pFETPe + Ef We( + u1)]ydD,e= 1 -,P, S ,) (g) . .

4, I pjr dD,. 1, = f p,(r- _ + ;)Td1, (RW) +roeE.(weslOe +S4 , +

D,- eDi' q + 41', + 4 ,k't)} + M (I le)

in which p, and p, are mass densities and D, and Do are thedomain of the rigid platform and of a typical appendage, p.(E1 Vo + [eF e + (r1 + )')E1J a 4-}

respectively. m, is the total mass of the system. S, is a skewsymmetric matrix of first moments of inertia for the system andI, is the inertia matrix. For simplicity, we assume that the = P -E. ] E1(Vo- r06 w)potential energy is due entirely to elastic effects, in which case

it ,an be written in the form + 2[ ' - (' + ) + w*(r, + u,)ET - 2 ; vN Y

v 2T E .n] = (9) A - A

,*- (1FI +w 1 +w 1 )( + )} - .', + U

where [, ] denotes an energy inner product [Meirovitch e=1,2.N (ll/)(1980)1. We note that, in deriving Eq. (9). the boundaryconditions were fully considered [Meirovitch (1980)1. where tr denotes a trace of a matrix and

i"= f p..dD. (I 2a)Ill. HYBRID NONLINEAR STATE EQUATIONS OFNIOTIONf

MOTION + U,) + (r, + )v jdD, (1 2b)For control purposes, or for mere integration of the D,equations of motion, it is convenient to work with state

equations. As the state vector, we consider I. THE DISCREIZED NONLINEAR STATEVT T T T DSTRTT T ... LEAT STAT

. -o 41 12 /... M N 1 _2 r ... v (10) EQUATIONS OF MOFION

which represents a unique combination of inertial coordinates, The equations of motion are hybrid, in the sense that theangular displacements, elastic deformations, translational equations for the rigid-body translations and rotations of thevelocities, angular velocities and elastic velocities, the last four platform are ordinary differential equations and those for thebeing in terms or components about the body axes. In view of elastic motions of the appendages are partial differentialthe definition of the state vector, one half of the state equations equations. Moreover, because or the maneuver angularconsists of the kinematical relations velocity vector _y, which is a given function of time, they

possess time-dependent coefiients. Control design of systemsR = CTV 0 D' w = _u , e = 1,2.N (I Ia,b,c) described by hybrid equations is not feasible, so that we wish

~ ~to discretize the partial differential equations in space, leavingInserting Eqs. (7) and (9) into Eqs. (5) and considering Eqs. (2) us with only ordinary differential equations. To this end, we

I express the elastic displacements as linear combinations ofand (3). we obtain the hybrid nonlinear Lagrange's equations space-dependent admissible functions multiplied byfor the other half of the state equations time-dependent generalized coordinates, or

-r N(r,. 1) = t,"r)q,(t) . e - 1.2 ...... V (13)Y. +- i + F.T p &d 1. - Y. +Dwhere 0 is a matrix of admissible functions and q. is a

vector of generali7ed coordinates.

N T ~ -The Lagrangian equations in terms of quasi-coordinates for+ E{(2([,$,I + S.)E., - 2 py dD, + 7- V.Tw. the rigid body motions of the platform remain in the form of

f DEqs. (5a) and (5b). On the other hand, inserting Eq. (13) intoDo * Eqs. (5c), we obtain the ordinary differential equations for the

discretized elastic motionsN 4 L -

+L F .e= d.-,...L (14)

Sty,, + m + D.PE' E. where

.- ~-MA:j6 App Moen Peov 1989 Supplornent

T,/ONWET +1 (I,) + + ,'~ 0d.p

are corresponding vectors of generalized forces. - r- - , 4-r--f p,

The Lagrangian remains in the form (6) hut the kinetic D.energy and potentilW energy change. Indeed. introducing Eq. N(13) into Eq. (7). we obtain the discretized kinetic energy - to 4- / = " r -(, + , +

T=I m VVT TT IT (fIWA -e .e ei,r=-m.,Vo + - + I- e

N - E+(Jee + 'We ev(d I (21b)T, 1t r T'r_

2'T, 7 +r PT e -T -T -T22 f +D) E,Q + (<D, F, r + 0, Ee

T .T -TT O

+ (F. + E) TE-q fP]TedDEpe)

TE.T- _ + (Fe + ,cJ p,(ro 4 ['?.qe]) odDTJ_ 1 (17)

+ 21' ,e + /, - -

Many of the quantities in Eq. (17) are defined by Eqs. (9), with f/)

the exception of

= f. po(', + e]o(IRa) - 1 4- M, + 2H(.)p.De IP7(( , +I, ED )dD .(I __)

+ [K, 4- "e(T!) + + e(r,) (,)1

= p,(r, + +)q, ,+ . r[, ,.hT)n / (I11 8b) -

D aeJ~' + L~'p~- e 1"-' =, ee + Le Pt!DTC411 ,,9?.dD.

Moreover, inserting Eq. (13) into Eq. (9), the discretizedpotential energy has the form

N N + f + E. E.; (21 )V~~~~ ~ 2:.r be(e ZqKe q, (19)

2 2o -# -i-l el where

where

K [0. . ,] (20) .f = f" p,0,TO. (22a)

rollowing the same procedure as used earlier, the discretizednonlinear state equations can be written as follows: = ,Pe(edd0 (22b)

iv /q.O N

• -() f p.. d),, (22d)N

.- + ( + Sw (a'(a) = j P.(DTa-e, di) (22d)

F -+- L F-:(S,-=e + Z,,';,C. (21a)C-, H,(a) = p. De(,, di), (22)

N The state vector is redefined as

irk.+ + + E. kz.1C - VrIC)+2..4e, S1 ) T r 0 T T T T T r T T(3_0" = - ~1 "' ~ ~2 ~ ~ "

In addition, Eq. (I Ic) is replaced by

- 2E. , p*(; + [CD,qe])C',p ,]dO} ., q1, P. e = 1.2.-N (24)

I I J l / lI Iql PC•I II I

Appi Mecl Rv val 42, no 11. Part , Nov 19 mslMl o wid K Sot squa111cm fo a -0an $ia

Equations (1la.b). (24) and (21) represent the discretized Two cases with the same numerical data tot the elasticnonlinear state equations. appendage but with different inertia terms for the rigid body

are tested. The data for the elastic appendage is as follows:

V. NUMERICAL EXAMPLE m, = 0.303 slugs. & = (0 0 1.415)T slugs.fi

The effect of nonlinearity on the system response is = 7.7 12 0.0 Slugs .f, 2

illustrated by means of a spacecraft consising of a rigid 0.0 0.0 1.294platform with a single membrane-type flexible appendage (Fig.3). The maneuver of the appendage relative to the platform ;Y = 0.01 slugs/ft a = 3ft, c = 20 ft/swas carried out by means of a smoothed bang-bang[Meirovitch and Quinn (1987)1 for the angular acceleration, . (0 0 1.0)Trwhere the smoothing of the bang-bang was done to reduce theelastic deformations of the appendage. The elastic vibration For the rigid body, we consider:of the appendage, treated as a circular membrane clamped atr = a, was represented by ten degrees of freedom in the Case 1z-direction, i.e., by ten admissible functions in thediscretization(-in-space) process, so that the matrix (D. in Eq. nr = 134.15 slugs, S, = 0 slugs.ft(13) is 3 x 10, or 186.021 0.0 0.0 1

[0 0 .. 014 0.0 186.02 1 0.0a slugs ft2e 0 0 ... j 0 0.0 0.0 357.733

4 1 ' 2 ... o and Case 2

in which= 21.464 slugs. .S = Q slugs.ft= 0(fo) (for) [ 8,943 0.0 0.0 1 .

46f JI2l(a ) 02 J0( 2) 4t= 0.0 8.943 0.0 slugs.f|

=,3 J (0 1ii) cos Jd/4 1 12") cos 0 0.0 0.0 14.309

J(7iJAa) sin0 N = J 12 ) sin0 As seen above, the rigid-body model of Case 1 has large____J02___ cos 20 06 12 22r) cos 20 inertias relative to those of the flexible body. On the other

01= 2 i 20 6 - J02 2( ) s 20 hand, the mass moment of inertia of the rigid body of Case 22)sin 2 3( 22a) sin 20 is almost the same as that of the flexible body, although the

where mass of the rigid body is sulficiendy large. This is due to thefact that the mass moment of inertia of the rigid body is about

l the center of mass and the mass moment of inertia of theK1 - - flexible body is about the hinge, which is far removed from the

mass center of the flexible body.

In the above, 01 and 02 are recognized as axisymmetric modes "Ihe two cases are compared with results obtained byand the other as antisymmetric modes, respectively [Meirovitch Meirovitch and Kwak (1988a) using linearized state equations.(1967)1. The vector q. is ten-dimensional. The arguments of Figures 4 through 7 show time histories of the rigid-bodythe Bessel functions -of the first kind can be obtained from translations and rolation and the elastic displacements of thefl0a = 2.405. #, 2a = 5.520, /#,a = 3.832, #,,a 7.016. membrane at the center and the point defined byfl21a = 5.136 and fl22a = 8.417. The mass matrix, Eq. (22a), x, = 0 and r = 1.5 ft for Case 1. The figures contain responsesand the stiffness matrix, Eq. (20), are 10 x 10 and have the for the uncontrolled nonlinear system and linearized system.block -diagonal form Figures 8 through I 1 show the responses for Case 2.

M, , K, A The elastic displacements at the center and at

where I is the 10 x 10 identity matrix and A is a diagonal r = 0.5 a, 0 = 900 are calculated by using the followingmatrix with the diagonal entries formulas:

A(l.l) = 2(Polal , A(2.2) = -- (f o2a)2 W, 0 1 .08684q, - 1.6580 7 q2)

2 2A(3,3) = A(7,7) = -- _ (fla) (0.73006q, +0.2991992

A(4,4) = A(R.8) = _ 2 (flt,1)2 1.0692 6q 5 +0.905 74q6 +*1.15061q 7 -0.3563Sqg)a A discretized nonlinear state equation is solved by using IMSL

.2 routine DIVPAG.A(S,5) = A(9,9) = - ( 21a)

a

A(6,6) = A(10,10) = 2(fl22a) 2 VI. SUMMARY AND CONCLUSIONa Slate equations of motion for a spacecraft consisting of a

where c , in which T. is the membrane tension and rigid body and a given number of flexible appendages arei, represents the mass per unit area of the membrane. derived. To this end, Lagrange's equations of motion in termsMoreover, the other matrices given hy Eqs. (1 9) and (22) are of quasi-coordinates proved to be most effective. In general,given in the Appendix. the resulting hybrid equations are nonlinear and time-varying.

8166 MECHANICS PAN-AMEJkCA 1969 ApO Mach Pev 19M Supplnem

In addition, the equations contain persistent disturbances due where I is I x IO identity matrix. Moreover,

to inertial loading. Because numerical simulation by meansor the hybrid state equations is not feasible. discretization and H' (a) - 0 1truncation are carried out. The discretized state equation canbe used for the maneuver and control or the spacecraft, suchas when the object is to change the reorientation of theappendages while suppressing their vibration. A numerical S p [

(V"q" ] )

D , = [0)

example of a spacecraft with a membrane-like antennaillustrates the dilTerences between the responses obtained by where we note thai the above null matices are 10 x 0 and 3 x JO, reqicvely.

using the nonlinear and linear state equations.In addition,

ACKNOWLEDGEMENT 0

Thi study was sponsored by the AFOSR Research Grant F49620-88-C-00440montored by Dr. A. K. Amos whoe support is greatly appreciated, where

ell= 2ah ,I- e 0a'' l

T0 ,'

APPENDIX

P1 P4For the gven configuraton, the position vetor of a nominal point in the 12. __ -

membrane appendage can be expressed as 01 la 012a

r = [rsine rcos6 h1]r (A-I) 01I 4 PPt a

Moreover,

Inserting the admissible functions given by Eqs, (26). together with Eq. (A-1), 1 0 01into Eqs. (22), we obtain pe . vS dO - qp, [0 I 0

0 0 0

0 0 __t "I

r ~ - - W02 °*2q2

0 0 - P u l a -V0'9- d ) 0

o o a COO oqo

0 0 0-ad

0_ 0

0 0 0 0,

a 0 0 JD.od = afa 0 0

1 0 h 0

0 o #I Io 2a 0 2#II

0:0 aGd,

0 0 0 0 0 0

a 0 0 0'02

2 2 0 0 0

-aad 0 00

d2)

Appl Mav~ Ma o 4Z. no 11. Part Z. Nov 1969 MekirO and Kw*r Sbt equadon ftr a Wpccrf S16T

z

Elsi ApgendgO Apedg

E I. .Th&igd ltfr wt Fpeanib9e Apege

zr

fe

YY

K0

FIG. 2. ngular isplaceentS an Veoi i .fThe Rigid Platform wih Fle i . 3.pe paes rf ihaMmrn nen

S168 MECHANICS PAN-.AM CA 1989 Appi Moh RMy 198 Suppemwt

0.008 0.01lit (Linear, Uncontrolled)

0.006 (Nonlinear, Uncontrolled)

-0.01

0.004 CR, (Nonlinear, Uncontrolled)

-0.02/ .-'[-----=-==:...........0.002 / Rz (Unear. Uncogrlld

-0.03 Linear. Uncontrolled

Nonlinear, Uncortroiled

0.000 -. ' .. ..- 0.041.. .. ,. ... ,....

0 1 2 3 4 0 2 3 4 5

t(s) (S)

FIG. 4. Time History of the Rigid-Body Translations (Case 1) FIG. 5. Time History of the Rigid-Body Rotations (Case 1)

0.009 0'0C T

0.008 " U r 0.007 Nonlinear, UncontrolledNonlinear, Uncontrlled/

0.007 0 006

ocs o.sLinear. Unconrle

0.005 Linear, Uncontrolled 0.004,0 00

S0.00 0.003

0.002 0 ,001roieC U

0.001 _ 0.000

0.0001 A -0.00 1

-0.001 .-- - -- ' - v-0.0020 1 2 3 4 5 0 2 3 4

t(e) t(s)

FIG. 6. Elastic Displacements at the Center (Case 1) FIG. 7. Elastic Displacement at the r = 0.Sa and 0 = 90" (Cae 1)

A4PPI Mach Pav vol 4Z~ no 11. Pan 2Z Nov 1960 Madovlth anld Kwak, Stat. equadona ftr a aecraf Slag

R(Linear, Uncontrolled)

0.02'1R, (Nonlinear. Uncontrolled) .i

~ Non 'near, U ncontrolled)__

Cr -0.2

0-000

-002 0 ~R2(Linear, Unconroledr"' o olna. n-told.~

0 1 2 3 4 501 2 34

FIG. 8. Time History of the Rigid-Body Translations (Ca~ze 2) FIG. 9. Time I Iistory of the Rigid-Body Rotations (Cuse 2)

0.003 -0.002'

Nonlinear. UncontrolledNonlinear. Uncontrolled

0.002'

0.001-

220,0000

q-0.001,

Linear. Uncontrolled Liner, Uncontrolled

-0.002 0.011

0 2 3 4 5 0 1 2 .3 4 5

t1(11

V1G. 10. Elastic Displacements at the Center (Case 2) FIG. 11. Elastic Displacements at the r 0.5a and e = 90' (Case 2)

S170 MECHANICS PAN-AMERICA 1969 AppI Mach Rov IM3 &Vp~nun

REFER EN CES

Baruki. If. and Silverberg. L. (1994), "Maneuver of Distributed Spaceci'8A. Meirovitch. L. and Francie. M. E. B. (1989). "Discrete-Time control of AProceedings of the. AIAA Guidanre and Conrrol Conference, Seattle, WA.. Aug. Spacecraft with Retargetable Flexible Antennas." presented at 12th Annual20-22. pp. 637-647. AAS Guidance and Control Conference. Keystone, Colorado.

Dreskwell, 1. A. (1991), "Optintil Feedback Sitwing or flexible Spacecraft" J Mieirovitch, L. and Kwak. M. K. (1998a). "Dynanis and Con" ro aGuianc 4 ontol,4 (). 72-79.Spacecraft with Retargeting Flemxble Antenna&*" esented at

Browen. F. T. (1981). "Energy-Based Modeling and Quaii.Coordinatei.- I Djn AIAAiASME/ASCE/AHS/ASC 29th Structures, Struccwral bynamics andSjsr Mfeat 4 Control, 10.%, 5- 13. Materials Conference. Williamsburg. Va. 1594-1592, also)J Guidance Control

G3rote. p. B., McMunnt, J. C., and Gluck. R, (1970), "Equations of Motion of & Dynamics (to appear).rlexii'te Spacecraft" AIAA 4th7 AeroVpace Scienreye Meetng. New York, 'Jew Meirovitch. L. and Kwak, M. K. (1988b), "Control of a Spacecraft withYork- Mui-TArgeted flexible Antennas." presented at AIAA/AAS Astrodynamiucs

Ho. 1. Y. L. (1977), "Direct Path Method ror flexihle Mulbbody Spacecraft Conference. Minneapolis. Minnesota. also J Astronautical Scimnce.s (toDynartsscs," I Spacecraft & Rockers, 14, 102-110. appear).

IMSL, MATH,tihrary tjter's Manual. V. 10, Fortran Subrourines for Meirovitch. L. and Kvak, M. K. (1989), "State Equations fot a Spacecraft withMathematicol Applications., IMSL Inc. Maneuvering Flexible Appensdages in terms of QusCoordinates." presented

Kane, T. R. and Levinson. D. A. (1980), "Formulation of Equations of Motion at the Pan-American Connes of Applied Mechanics, Rio de Jamero. Brazl.for Complex Spacecraft." J Guidance & Control, 3 (2), 99-112. Meirovitch. L. and Quinn. R.D .(1917. "Maneuvering and Vibration Control

Levinson, D. A. and Kane. T. R. (1981), "Simulation of Large Motions of of Flexible Spacecra&-.I Aktronauicd Sciences, 35 (3), 301-330.Nonunifoifrm B~eams in Orbit. Part I - the Cantilever Beam," I Astronoutical Meirrovitch, L and Shas'oy Y. (1987), "Optimal Vibration Control ofaSciences, 29 (3), 213-264. Flexible Spacecraft During a Minimwn-Tisie Maneuver," Pr'oceedings of the

Likins, P. W. (1972). -rinite Element Appendage Equations for Hlybrid 6th VPl&SU/IAAA Syrpauni an Dynamics and Control of Large Structures.,Coordinate Dynamnic Analyni,"_ Int J Solids Structures, 3, 709-731. Blacksburg. Va., 579-601.

Metrovitch, L. (1966). "On the High-Spin Motion of a Satellite Containing Qwi. R. D. and Meirovitch. L (1993), "Maneuver and Vibration Control ofElastic Parts,"' J Spacecraft & Rockers. 3 (11), 1597-1602. Scle.")I Guidon=a Control it Dynamics. 11 (6). 542.553.

Meirovitch. L. (1967), Analytical Methods in Vibrations. The Macmillan Co., Turner, J. D. and Junkins. J. L. (19P0. "Optimal LArge-Angle Single-AxisNew York. Rotational Maneuvers of Flexible Spacecaft." I GwddancA & Control, 3 (6),

Meirovitch. L. (1980), Computational Methods in Structural Dynamies.. Sijthoff 578-585.& Noordhoff Co., the Netheriands. Williams. C. .1. N. (1976), '*Dynamics Modeiling and Formulation Techniques

Meirovitcis. L. (1988), "State Equations of Motion for Flexible Bodies in Term., for Non-Rigid Spaciecraft.- P'roceedings of a Synryoulum on Dynamics andof Quasi -Coordi nates. -IlUTAMIIFAC Symposium on Dynampcjs of Controlled Control of Non- Rigid Spacecraft, Frascati, Italy, 53-70.Mechanical Systenir, ETH Zurich. Switzerland.

AAIA-88-4268-CP

Control of Spacecraft with Multi-TargetedFlexible Antennas

Leonard Meirovitch and Moon K. KwakDepartment of Engineering Science and MechanicsVirginia Polytechnic Institute and State University

Blacksburg, VA 24061

AIAA/AAS Astrodynamics ConferenceAugust 15-17, 1988, Minneapolis, Minnesota

-e~rmi n O 1 P w , lS. Ameia n , d Aaussa AuuIlII1ILCUP S. W., uismD.C. 2M

Control of Spacecraft with tlutl-TargetedFlexible Antennas-

_o'ar o 4e- iivitZ a'j Oon <. (witeoar!,ient )f lnqineeri - Sc'ence and Mecani:s

Virglnla z,)ytec-t :nstitjte and State iniversity3licvsourg, dirginia 24061

Abstract ;e',jr-rations '- :'e aitform. lence, tle control3aSK amounts to simultaneous stabilization of :ne

This paper is concerned with the problem of platform relative to the inertial space and viora-reorienting the line of sight of a given number of tion suppression In the retargeting antennas.flexible antennas in a spacecraft. The maneuverof the antennas is carried out according to A min- The mathematical formulation consists of aimum-time policy, which implies bang-bang control. hybrid set of equations of motion, In the senseRegarding the maneuver angular motion of the an- that there are six ordinary differential equationstennas as known, the equations of motion contain for the rigid-body translations and rotations oftime-dependent terms In the form of coefficients the platform and partial differential equationsand persistent disturbances. The control of the for the elastic motion of each antenna. The equa-elastic vibration and of the rigid-body motions of tions of motion are not only hbrid, but the m-the spacecraft caused by the maneuver is imple- neuvering of the antennas relative to the platformmented by means of a proportional-plus-integral Introduces time-dependent coefficients into thecontrol. The approach is demonstrated by means of equations. Moreover, the equations contain termsa numerical example in which a spacecraft consist- reflecting persistent disturbances caused by iner.ing of a rigid platform and two maneuvering '1exi- :'al forces. If the mass of the antennas is smallble antennas is controlled. elative to the mass of the platform, then the

equations of motion can be regarded as linear.1. Introduction 3ecause control of systems governed by sets of

hy0rd differential equations cannot be readilyCertain space missions involve the reorlen- lesigned, even when the equations are linear, it

tation of the line of signt in a f'lexoe soace- is iecessary to discretize the partial differen-craft. In many cases, the line of signt can oe lial equations in space, which can be carried outregarded as being fixed relative to the undeformed by the classical Rayleign-Ritz method or thestructjre, in wnich cases the reorientation of tne finite element method (Ref. 7). Reference 6 pre-line of signt implies the reorientation of the sents the mathematical formulation, as well as anwhole soacecraft (Ref. 1-5). )uite often, now- example illistrating the maneuvering of a singleever, t 1e 4ie of si;nt oincides witn an axis flexible antenna.fixed "a small component of tie soacec-aft, suchas a flexbie antenna. In such cases, it nay be Tnis paper extends the work of Ref. 6 inmore advisaole to retarget only the flexible an- several respects. In the first place, it treatstenna and not the entire spacecraft. Retargeting the problem of retargeting several antennas simul-of the antennas can be achieved by attaching the taneously, and not just of a single antenna. Inanrtina to t e platfor tm"ough a nige so as to aidition, it addresses the problem of persistentpermit o,' t'-g aoout two ortiogoal axes. etar- -'stjroances by it:emotnmg to nitigate their;eting of tne antennas, instead if the wnole .'ect luring tme "aneuver. To cope wltn Knownspacecraft, is a virtual necessity when there are listirbances. disturbance-minimization control isseveral antennas that must be retargeted simulta- effected. The retargeting is carried out open-neously, and the line of sight of each antenna loop using a bang-bang control law. This impliesmust be reoriented in a different direction, that the inertial forces arising from the maneuver

angular accelerations are almost constant, exceptRefeence 6 considers a spacecraft consisting for a sign change at one half of the maneuver

of a 'gii oltfori with a number of flexible period. If the naneuver is not very fast comparedantennas i:ig. 11, and the mission is to retarget to tne lowest natural frequency of the antennas,independently the line of sight of each antenna then the control gains can be determined by ignor-relatide to the inertial space. The maneuvering ing the time-dependent terms in the coefficientstratey developed in Ref. 6 consists of stabiliz- natrices. This permits the use of proportional-n'g -'e 3t'oirm 'e'ative to the ''e-ta1 soace o'js-i-tegral feeabacK control for disturbance

a-, rec-'e-ti-; -e line of sight )f each antenna iC:Omrocat'on ef. J, 9 and 10). Of course, in

relative to tie platform. For g9ien tar;et iirec- te conoite" siulation of the naneuver and con-

tions of the antennas, the maneuvers can be Je- trol, the full time-varying system is considered.signed as if tie a-tennas were rigid. Of course,in actuality the antennas are flexible. so that This paper contains the procedure for design-the maneuvers are liely to cause elastic viora- ing the feedbacK control in the presence of dis-:ion of tne antennas, onicn in trn will i-dJce tirbances. The proportional-plus-integral control

Sponsored oy t'e AFOSR Research Grant F4962)-d3-C-0044 monitored by Dr. A. K. Amos, whose support is

gratefjIlly appreclated.

iniversity Qistinguished Professor. cellow A;AA.

t"Graduate Research Assistant. Student Member AIAA.

Copyright 1988 by the American Institute of Aeronautics and Astronautics Inc. All rights reserved.

r I

procedure described above is demonstrated by means displacements sI (1 1.2,3). In view of theof a numerical example invol n"; a soacecri't oove, t'e ver:.:y dectr Of I Poit in t"ezonsisting of A -. j ,i z.~- in -2-'- )f ilf1 y !!

nearns, eaci wli :ne e"~-- "e te 0 .1 . ," t'.dt Of d Point in tneand tne otner 1ee :l 2. , .nere : e oeams are Ply -r -o0 - - d f

original'y parallel to tre z-axis of :ne Pst- typical appendage e in terms of zomoonents alongform. The maneuver consists of slewing eacn o 'e~eze is V 4r I -the beams through a 45* angle, one ioout tne x- e e e .o - e) Ze, -:e)

axis and the other about the y-axis of tne plat- r {r * e ee 1,2.NI, where j is theform. angj' r ielocity of axes xejeze, Ee 's a iatrx of

2. Equations of Mtion direction cosines between axes xe~ez e and xyzand v is the elastic velocity of the point In the-e

We consider the motion of a spacecraft con- and*ie latve ocityeOf e pe" in the

sisting of a rigid platform with a given number of ppendage relative to XeY.ZQ, 4 0 . In theflexible appendages hinged to the platform as maneuver proposed, the angular velocity vectors ueshown in Fig. 1. The rotation of each individual of relative to xyz are given, so that theappendage relative to the platform is given, so xeyeze

that the motion of the system consists of the rotational motions of the appendages relative torigid body otions of the platform and the elastic the platform do not add degrees of freedom.motion of the appendages. To describe the motion,we introduce a set of inertial axes XYZ, a set of The equations of motion are hybrid, in thebody axes xyz attached to the platform. Then, the sense that the equations for the rigid-body trans-motion of the rigid body can be defined in terms lations and rotations of the platform are ordinaryof three translations and three rotations of axes differential equations and those for the elasticxyz relative to the inertial axes XYZ. To -otions of the appendages are partial differentialdescrioe the motion of the flexiole appendages, we equations. Moreover, because of tne maneuverconsider a typical appendage hinged at point e aria angular velocities e they possess time-dependentregard e as the origin of a set of ody axes oeiint. -coefficients. Control design of systems described'eyeze embedded in the appendage in indeformed by hybrid equations is not feasible, so that we

state. Then, the motion of a nomiial :oint )f -'e wish to .iscretize tue system in space. To thisappendage consists of the motion of <yz, -ne eno, we express tne elastic displacements asmotion of xeeze relative to xyz and the elastic linear combinations of space-dependent admissible

motion relative to xey~z.. The system and the functions multiplied by time-dependent generalizedcoordinates, or

various reference frames are shown in Fig. I.

com Fig. 1, vhe position vector of a Point e e e e e

in trhe rigid :ody and ii tne apPendage :an %e wriee te is a atrix of admissible functions andwritten as r * . r , * * e-r o - Re -o -e - e is a vector of generalized coordinates.

(e a 12,...N), where R 0 is the radius vector e The equations of motion can be obtained byfrom 0 to o, r is the position vector of a point leans of Lagr3nges equations in terms of quasi-

,n tne rigid oody "elative to xyz, -, 's '"e :ooro'nates jej. I. 4 linearized version ofsucn ecuations -as .erved in Ref. 6 for tne sameradius vector from o to e, r e s t-e position mathematical model is tne one considered here.

vector of a nominal point in undeformed appendage From Ref. 6, we obtain the state equations ofrelative to 'eyeze and u e is the elastic dis- motion

placement of that point. Vector R is given in-o (t) A(t)x(t) 6 (t)f(t) 0 (t)d(t) (2)

terms of components along XYZ, r and r in terms 7 T T T T ,T T T T. Tr a oe here x(t) [ 3 . T 3 0 PT

of components along gyz, and r and u in terms of w 1 2 N o - 2 N-e e is a state vector, in wich 0 is a symbolic vectorcomponents along xeYe e The velocity vector of o-

can be written in terms of components along xyz in of angular displacements of the platform,

the form 4 - CAo, where C is the matrix of direc- 2e e(e 1 ,2,..., N) and0 0 1

ton :,sines 3etween xyz and XYZ arid 's tie ... ------ ----------• -o !~~~A t) : - M-1,~ t i " ' ( ) 3a)

velocity vector of o in terms of components along M $t,t) t (tXYZ. Furthermore, the angular velocity vector of

axes xyz in terms of components along xzis given ()*3bc"'t ... :" (t) •(3b,c)

y 01, where i is a vector of angular veloci- 6(t) 4'-,t)B (t,1ties and 0 is a matrix epening on tne angulari are coefficient matrices, in which

--T " -: ~T ... M14~' 141, )

NE N 'N V'" Nr3

IS a IlaSs matrix,

2 T E [Sge 2 E WTI 2ET.

e I re

G 222G0'2 r 23 .. "2"G3

G(t) 0 LII - J (i!)1E1 2H (W1 ) ... 0 (5)

0 ., - JN(_gN)JEN 0 ... 2HNl0N)

is a matrix in which

G22 e e ee e e e -roe;e Se..e e) Ia)

e = : ' _ - , T I T-eo e - EJ' e (5b)

' ~ ~ E ee Je ee-e)

and

<(2 T. 2O ]..3. KN

K(") : '] <. 7" H,- , .. . 7)

LO 0 0 ... KN + +r N H HN(;N)-

J ays :-e r'le )f s;if''es -i'- , neee 11, TC2~ + e ~~) "' 8

K23 oeEe' e We e e eeee e'

Moreover,

2 cL 2 ... 3

B*(t) - 0 0 c2 ... 0 (9)

.i ) ,) ..

relates the discrete force vectors to te modal force vectors, in vhicn

- r ..T 7-

oe e e el oe e e e2 .. e e en e

ce r 1 r r E 10"

0e I Ir ''r (r k1be-el. e-e2) $e-en'in addition,

, ]

d(t) •* r *T~i e T-2~~a Q.# e a e do e 13m)

2 a) Z., r -D'13o)

e, e l r2' o

E ( -- e

e;1 e Se eii e :'ee' 1 oricn a is a vector representing 4e or Le. Irand : e re ass ens ities of each o y, Or and De

7- -Tare tne domain of tne rigid platform and of a" .typical appendage, respectively, mt Is the total

mass of the system, St is a ske% matrix of first

moment of inertia for the system and It is the

T- ~inertia matrix. For simplicity, we assumed that" + f o NNNrkNdDN the potential energy is due entirely to elastic0 effects, in wtich case it could be written in the

form(•1T (14)

is a vector of disturbances and e a

f(t) f 0T M'T fT FT f J fT where Ke is the stiffness matrix (Ref. 7).

-31 Oisturbance-Accomodattng Control

plays the role of a control force vector, onere in The maneuver consists of retargeting antennas* * so as to point in given directions in the inertialthe latter Fo and are actuator rte ano soace. 3y stabilizing the platform in an inertial

torque vectors acting on tne o;tor-i I. ire ipace, the task reduces to reorienting theaoe aantennas relative to the platform. For a minimum-actuator forces acting on appendages a at Qoonts . time maneuver, the control law is bang-bang, which

Other quantities entering into te aoove matrices implies that the angular acceleration of anare as follows: antenna relative to the platform is constant, with

t.e sign cianging at half the maneuver period.,-a1eally, the maneuver sould not cause elastic

, efornations n the flexinle appendages. This isi~ely to rec .- e a long maneuver time, which isin conflict &:n the minimum-time requirement. To

me U, e e13c) reduce elastic vibration, a smoothed bang-bang canDe be used. Still, elastic deformations are likely

to occur, which in turn implies perturbation oft-e olatfcrm cm a 9i(ea :osition in the inertial

ce r2 - 3oace.

o r~dr F ~ fThe otion of the system is governed by Eq.Sr - rr , Se 0 e e edoe (13e,f) (2). The system is characterized by two factors

r e that distinguish it from most commonly encounteredsystems: it is time-varying and it is subjected

- e to persistent disturbances. Both factors arisel e2 je be e 'e-e rom the retarget-g maneuver angular velocities

ee' angular accelerations Le and the matrices Ee

E T of direction cosines (e - 1,2,...,N), all quanti-oeSeEe E eSeEeroe) (13g) ties being known functions of time. Clearly, the

eisturbance term D(t)d(t) in Eq. (2) depends on

D•TD r . " tIe -1aeiver o' :v. :n tne case of 'nimum-timer r ari"aneuver, tIe o!ll:J 's oang-bang, wnich implies

]rh e ) 'nat t"e ianeuier anqular acceleration is constantover both halves of the maneuver period. If themaneuver is relatively slow, then the disturbance

eO is constant over both halves of the maneuverMe e e e 13Je oeriod. in this case, we can use proportional-

ae plus-integral 'eeadacK control.

*U-- ~ Introducing the notation

"e 3 ,l(t) I 3(t)f(t) 1 D(t)d(t) :15)

H e U T ed~ee- :3m) E. ,2) can be rewritten asH ea) Dee#ea ed 1m

;(t) • A(t)x(t) *8(t)u(t) (16) Goa G t , G1 a G, - G28tA (25ab)

Assuming tn~at ;'a I na:y '25 1 eoresen~ts .,e Ot -a ~no a,that O(t'i(t) jS aimost :.)ns:an: : rtie t-ee-lvar'nt sstem to ec,rnstant 1istjrnances, and IS KnOwn as ropor-control interval, e can write 'onal-plus-integra control.

!(t) " (t) - fd(t) In general, the above control law cannot beused for the type of problem considered here.

Introducing a new state vector defined oy .nen the maneuver is relatively slow, however, so

z x u JT , Eqs. (16) and (17) can e combined that the matrices A and B are nearly constant, thecontrol law (25) can be used with satisfactory

into the expanded state equation results. In this case, the control must beregarded suboptimal, but close to being optimal.

A(t) • A(t)z(t) * B(t)td(t) (18) We recall that, to reduce vibration, it isadvisable to use a smoothed bang-bang for the

where rigid-body maneuver of the appendages, instead ofA(t ) . 8(t an ideal bang-bang.

[o ,() (1ab)

01 4. Nmw1cal Exleare coefficient matrices. Note that if A(t) and The mathematical model consists of a rigidB(t) art a controllable pair, then A(t) and i(t) platform and two flexible beam, each one with onear also a controllable pair (Ref. 8). end hinged to the platform and the other end free

(Fig. 2), where the beams are originally parallel

We consider an optimal contro ooll:, - te to the z-axis of the platform. The maneuver

sense that f (t) minimizes tme performance rasire consists of slewing each of the beams through ad p450 angle, one about the x-axis and the other

tf about the y-axis of the platform. The beams areaiz(t,)-z 1 . [j (t) J(t((tzt iscretized in space by using three admissible

- ( f fnctions for each component of displacement. SixtJators are iseI for the rigid platform and

ttnR(t)f ~t):dt ,]) three actuators for each displacement component ofd both beams. The latter actuators are located at

The optimal control law is ft, 7 ft and 10 ft from the pivot point, the thirdcoinciding with the tip of the beam. Figure 3

? (t) - (t)B T(t)K(t)z(t' Gt tz~t) shows the maneuver time histories, in which theingular acceleration represents a modified bang-

2,) oang. Figures 4a a-! 4D iisplay both the un-:ontrolled and cont-)lled translational and

where G(t) represents a control gain matrix in angular displacements of the platform, respect-

which K(t) satisfies the matrix Ricatti equation Ively, and Figs. Sa and Sb show the tip dis-placement of the two beams. As can be verified,

. - KA - AK " * R 3 - '22) the maneuver and control of the spacecraft is1uite satisfactory. ",e listirbance-accommodating

,(17) and (21). it is easy to ont'ol is carried out )y the proportional-,lus-verifg that1 integral control approach.

df t t. In obtaining the numerical results, the" (G1 G2 8 AB) * G.B x (23) following data was used:

where 11 and G, represents submatrices of G mr z 134.15 slugs , me * 0.1373 slugs

corresponding to x and j, correspondingly, and r a 0 J' slugsft

B 1 (aT 8)8 s the pseudo-inverse of B.

Integrating Eq. (23), w obtain the optimal e a [O 0 0.9365]T slugsft

control law t '8-2 02 '0 1 - : '.j6.)21 slugs. ft 2

m "r

+ ' A dr (24) 0 357.73

assumed to be constant, then the gain matrix is le = 5.243

constant. Moreover, if x()) and f(0) are zero, 0then Eq. (24) yields

t)) G dr (25) l) 12 1 0 ft, *: 3028.9 lb. ft 2!(t) x + G i . x at1

0 twhere roI " [3 -12.3 .5] ft

E02 1 ED 1.0 0.5)T ft Large-Angle Single-Axls Rotational Maneuversof lexible Spacecraft," joural of uicance

ooreove-, op, ntn s infl -ortrol, Vol. 3, so. T, ;3p, pp. o12-pere is te iceni 7-1. * ? i s

ost d 2. 3rerKwell, J. A., "Optimal Feedback Slewingmr.t i e sof F"exiohe Socecraft,' journal of ;uidance

and Control , O. 4. 40. 5, 181, pp. 47-.

woere I is the identity matri. Iconsistent with a steady-state olajtion of the 3. i4elrovtcn, L. and Quinn, R. 0., "Maneuvering

matrixuratin eqution. sp asaken as erour, nand Vibration Control of Flexible Space-matrx Rccai euaton, wa taen S zro.craft.* The Journal Of AstronauticalFinally, for control design purposes, theGuidac onr 1-328.coefficient matrices A and 8 were taken as S9 . Aconstant and corresponding to the preaneuvero , . and hro , L.. "Mne

and Vibration Control of SCOLE.' Journal ofimplementing the control. the time-varying Guidance, Control and Dynamics (to appar).matrices At and B(t) were used.

S. Sinr.' Mi Ccdstons 5. Meirovitch, L. and Sharony, Y., noptimlVibration Control of a Flexible Spacecraft

Certain space missions involve the reorient- During a Minlimuim-Time Maneuver,Q Proceedingsation in an inertial space of the line of sight of of the 6th VPI A SUIAIAA Si osium oncertain small flexible components of a spacecraft, Dynamicsr VControl Of Larse structuressuch as flexible antennas. :n such cases, -egard- Blacksburg, VA. 19W, Pp. b79-601.lig the main spacecraft as a ri,,, 313tform,asensible strategy s to stabilize tne ,latform 6. 4eirovitch, L. and Kwak, M. K., "Dynamics andreiatlve to the inertial space and reorient Ine Control of a Spacecraft with Retargeting

Flexible Antennas," Proceedings of the 29thline of sight of the various antennas relative to AIAA/ASME/ASCE/ AHSASC Structures, Structuralthe platform. Dynamics and Materials Conference,

'his paper is concerned 'tn t'ie )r:oD'en of oilliamsourg, VA, April 13-2. 19 88, pp.

retargeting several antennas simultaneously, wnile 1564-1592.

suppressing any rigid-body and elastic perturbat- 7. Meirovitch, L. Computational !ethods inions caused by the retargeting naneuver. The Structjral Dynamics. Sijthoff & Noordhoff,retargeting was :arried out ocen-looo using a The etnerans, 180.

smoothed oang-bang control law. This inolies thatt-e inertial firces arising f-om the 'nanever 3. Jonnson, C. 0., "Optimal Control of theangular accelerations are almost constant, except Linea- Regulator with Constant Disturbances,"for a sign cnange at one nalf of tne maneuver IEEE -ransactions on Automatic Control, Vol.period. If the maneuver is not very fast compared AC-13, No. 4, Aug. 1968, pp. 416-4Z1.to the lowest natural frequency of the nonmaneu-vering antennas, then the control gains can be 3. Johnso", C. 1, "Theory of Disturbance-etermiied oy i nor,ng tie t- e-Jecendent terms in "

ie coef'~i~ent natrices. -s~ ce"ltted tne ise A ,::c-o1ating Controllers," chapter in

of proportional-plus-irtegral 'eedoacK control $or AdV112es 'I :ntrol and Dynamic Systems, Vol.

distirbance accommodation. Of course, in the 12 (Ed: C. T. Leondes, Academic Press, New

computer simulation of the maneuver and control, York, 1976, pp. 387-489.

the full time-varying system was considered. 10. Anderson. B. 0. 0. and Moore, J. B., Linear

A numerical example, in which a spacecraft Otl Clif, Prentice-Hall, Inc.,

consisting of j rigid olatform and two flexible Englewood Cliffs, NJ, 1971.

antennas j.neo'rg reorientation in different 11. Meirovitcn, L., "Equations of Motion forplanes was controlled, jemonstrates the approach. Flexible Bodies in Terms of Quasi-

6. References Coordinates," IUTAM/IFAC S mposium on

Dynamics of Controlled mechanical Systems,

' rner, J. 0. and JunknS . ., 'Cptina) :ricn, Soitzer~and, May 30 - June 3, 1988.

jrne , . . an Jun ils.-,pt-na

Elxi Elasticg Apedg

Figure 1. The Rigid Platforin with~ Flexible Appendages

o.2

-02S

02 3

Figuire 3. maneuverTim 1411 PtOfY

OO S 00' mota~ a4-- - - - - - - - - -

0o oo' umcontnor;* A -----

00 1 1j000 41,o . . -0

; ooij --- - ~ ~, - 3ntr~ C:Jstral led I

-0.0014

-0 00' -2

0 ~ )3 40 1 2 cs

Ppiot 44. tit1 it of tiw Platlove realatlapa? Mhugeuf49 ieNsey ftwpate mlrmf

0 06 00 3 1

0 02, 02 09t7O"I1@4d w

0 03 , un o tr l e 1A - .07 3~ ~ l g~ \ A02i Controlled U -.

-00-O

31 -0.0' ' ' fl1il@

-0 02 - .- 0021

-0 03 1 1

- Mofri4 -003

:;A -0041

02 2 3

t(I) tWs

1:?ure Sa. Th. 01saoiment of SsAIM I :.;urs Sb. Tio 0jjl,a fm t of S44 2