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Attitude control of a spinning flexible spacecraft
Comput. &Elect. Engng,Vol. 1, pp. 323-339.PergamonPress, 1973.Printedin GreatBritain.
ATTITUDE CONTROL OF A SPINNING FLEXIBLE SPACECRAFTt S. M . SELTZERS, J. S. PATEL§ a n d G . SCHWEITZERII
(Received 18 July 1973)
A~traet--The dynamics of rotational motion of a spinning orbiting spacecraft consisting of two rigid bodies connected by a flexiblejoint and arbitrary number of flexible appendages (two of which are flexible massless booms having masses on their tips) is analyzed. Active attitude control is provided by momentum exchange devices (e.g. control moment gyroscopes) or a mass expulsion system. The linearized equations of motion describing the vehicle are presented, and a large scale digital simulation that has been developed at the Marshall Space Flight Center is presented. A simplified model of the geometrically complex vehicle is selected to make it analytically tractable. The simplified model consists of a single rigid core body with two attached flexible massless booms having tip masses. The states of the vehicle are defined as small perturbations about its steady-state spin. An analysis is performed to determine the domain of stability.
INTRODUCTION
In 1970, NASA initiated a set of studies of several Skylab configurations, including a spinning vehicle configuration (the presently approved Skylab program does not include a spinning vehicle). The spin would provide an artificial-gravity environment to assess and compare the physiological benefits and problems of prolonged zero-gravity and artificialgravity environments. The results of the study were applied to a spinning Skylab configuration[i-3]. They have been extended so they are now applicable to a more general class of spinning spacecraft. Specifically, the study analyzes the equations of motion associated with a geometricallycomplex spinning spacecraft. The spacecraft consists of a primary rigid core body to which are attached a second rigid body, an arbitrary number of flexible appendages, and two flexible massless booms having masses on their tips. Also included in the equations of motion are a number of equations describing the applied torque, such as would be attainable from spinning momentum exchange devices or mass expulsion systems used to provide active attitude control of the vehicle. The numerical values used in the examples are those associated with a spinning Skylab configuration. tThe coauthors gratefully acknowledge the support of the National Research Council which made it possible for Dr. Schweitzer to work on this research as a NASA Postdoctoral Resident Research Associate during 19701971. They also appreciate the support and advice of Professor Peter W. Likins, University of California, Los Angeles, and that of Professor Bernard Asner, Jr., National Academy of Sciences Visiting Scientist with the Institute of Mathematics, Bucharest University, under auspices of the Academy of Romania, during his summer tenure at the Marshall Space Flight Center in 1971. An earlier version of portions of this paper was presented at the 5th IFAC World Congress, Paris, France, June 12, 1972. :l:Senior Research Scientist, Astrionics Laboratory, George C. Marshall Space Flight Center, Alabama 35812, U.S.A. §Engineering Mechanics Staff, Naval Underwater Systems Center, New London, Connecticut 06320, U.S.A. IrAkademischer Oberrat, Institute B fuer Mechanics, Technical University Munich, Munich, Germany. 323
324
S . M . SEt.TZER. J. S. PAFEL and G. SCHWEITZER
The method of approach is first to define a realistic model and then derive or state the applicable equations of motion. The problem of preserving realism while remaining within the physical constraints (e.g. memory capacity) of a digital computer is solved by casting the equations in hybrid coordinates[@ A simplified model is used to obtain an insight into the dynamic behavior of the vehicle and to help select numerical values to be used in the digital simulation. A control law is postulated to provide a satisfactory dynamic response to initial errors in attitude and attitude rate, considering the dynamics of the attached rigid and flexible members. (A practical constraint is that the control law should be easy to implement.) Applicability of the analysis is not restricted to a spinning vehicle since the development of the equations of motion is adaptable to non-spin cases. In spinning the spacecraft, it is assumed that the vehicle may be required to spin about a predetermined axis. Typical is the spinning Skylab example of this paper, where it is necessary to keep the solar panels pointed nominally toward the sun. As a consequence, the spin axis would have to be an axis nearly coaxial with the principal axis of intermediate moment of inertia of the original nonspinning configuration [5]. To cope with this possibility and still achieve passive stability, masses are deployed either on cables or extended booms so that the spin axis would become a principal axis of maximum moment of inertia. If not needed, the booms and tip masses are easily removed from the equations of motion. Also, geometric asymmetries of the original vehicle may require that the extendable booms be attached in two different planes normal to the desired rotation axis to make it an axis of principal inertia. Although the out-of-plane distance of the booms is usually small compared with their lengths, its effect is considered. EQUATIONS OF MOTION Modern space vehicles can be described as a combination of essentially rigid bodies and relatively flexible bodies. The analytical modeling of extremely flexible appendages is usually performed in such a way that they are considered to consist of small rigid bodies interconnected by elastic massless bodies. At present, two different methods of writing the equations of motion for spinning vehicles with appendages are widely used : the discrete parameter approach and the hybrid coordinate method. In the discrete parameter approach, equations of motion for various subsets of bodies or individual bodies are obtained by the direct use of Newton-Euler equations of translation and rotation. In modeling the flexible appendage as interconnected rigid bodies, a large number of them has to be taken into account to make the digital simulation effective. This will yield a large number of first- and second-order differential equations of motion. Since there is no way of reducing the number of variables, it is difficult to numerically integrate these equations. However, the hybrid coordinate method removes this difficulty by transforming a set of discrete coordinates into a smaller set of distributed coordinates, each of which is associated with the normal mode of a spinning or nonspinning appendage. In the hybrid coordinate approach, first the equations of motion of the flexible appendages are written in discrete coordinates, separately from those of the entire vehicle. Then the discrete motion of idealized bodies of the appendage is transformed into a fewer number of modal coordinates, thereby reducing the total number of equations of motion to be numerically integrated. The effect of spin is induced in the development of equations of motion despite the great added complexity introduced. This is necessary because the results of several cases studied here indicate the noticeable and possibly severe effect of spin on the modes of vibration.
Attitude control of a spinning flexible spacecraft
325
The complete set of equations used to describe the entire vehicle consists of a set of n equations describing n modes of the flexible appendage(s), a set of three equations describing the entire vehicle, a set of three equations describing the effect of relative rotation of the second rigid body elastically attached to the rigid core body, a set of three equations describing the skew symmetric mode of the attached booms, and a set of control torque equations• These are described in the following subsections•
Flexible appendage equations The equations of motion of a spinning structure are derived• Although they are used herein to describe the dynamics of satellite flexible appendages attached to the rigid core body, they apply equally to the important class of problems where structures are spinning at nominally constant angular velocities. Rotating shafts, blades of spinning turbines, rotating linkages, and spin stabilized satellites are examples of problems falling within this class. The direct use of the Newton-Euler equations give F s = roSA s
T s = zd H s
dt
(1)
for the sth rigid body of a flexible appendage; where ms is the mass, A s is the absolute translational acceleration vector. F s and T s are the sum of the external and connection force and torque vectors, respectively. H s is the angular momentum vector, and superscript i denotes differentiation in the inertial frame of reference. The expressions for the force and torque acting on the sth rigid body of an appendage spinning nominally in the steady state with an angular velocity co (Fig. 1) are written as
[bd2
bd
Fs = m s t ~ t 2 ( c + u s+r s)+2ta x ~ ( c + u
+tax
id
(ta x r , ) + ~ t a
TS = ~~d( r .
s+r
x (c+u
s)+ta
x Eta x ( c + u g ]
id2 }
s+r s)+~-~R
(2)
id
tas) = ~ [ r . (co + #s)]
p . r~ tda + ~ bd~#s ~ + t a x ~bdpAs7 j +taxP.ta+taxl
s.
p s + ~bd / I As x V.ta
(3)
where c is a vector representing the instantaneous location of the mass center of the entire vehicle with respect to its steady state position• Vectors u s and fls represent the instantaneous displacement and small rotation, respectively, from the steady state of the sth rigid body to its instantaneous deformed state. Vector r s represents the location of the sth rigid body of the appendage in its steady state configuration measured from the steady state mass center location, vector R is the distance from the origin of the inertial reference frame to the actual center of mass, and P is the inertia dyadic of the sth rigid body• Superscript b denotes
326
S . M . SELTZER, J. S. PATEL and G. SCHWEITZER C e n t e r o f mass (In s~eady s t a t e )
Central rigid b o d y ~
~
/
. b1
C e n t e r of m o Configuration of an (instantaneous)-- - ~ ~ / J ( I n s t a n t o n e ° u s )
appendage
Us
/'2 .///a F ~
-
'
u of an appendage (in s t e a d y s~rate)
Fig. I. Geometry of spinning flexible appendage and central body.
differentiation in the reference frame b imbedded in the rigid body with the origin at the steady state mass center location. The equations of motion describing rotating flexible structures, differ from those associated with nonspinning structures in several significant ways. The accelerations of the masses of a nonspinning structure are represented by the second time derivatives of the spatial variables. In the case of a structure spinning at a constant angular velocity co, accelerations are represented in equation (2) by the second time derivatives of the spatial variables with respect to a frame rotating with co; in addition, they contain terms caused by coriolis accelerations, which are proportional to the velocities of the masses in the rotating frame. Further, these expressions reflect the variations in steady-state centripetal accelerations caused by the small displacements of the masses in the rotating frame. For zero spin (co = 0), frame b is equivalent to frame i and equations (2) and (3) reduce to the familiar form
rid2 1 F ~ = rn~L~ (R + c + r ~ + u~)
(2at
T ~ = I ~ id2 fl~.
(3a)
In matrix notation the second term on the right hand side of equation (2), which is due to coriolis acceleration, gives rise, in general, to a fully populated skew-symmetric matrix G'. The third term, which is due to the centripetal acceleration, yields a symmetric matrix K , . The fourth term in equations (2) and (3) represents a steady-state centripetal force which leads to the steady-state configuration. Stretching forces and moments and rotations obtained therefrom give rise to the second order geometric (differential or kinetic) stiffness as represented for example by the stiffness of a rotating rope. In the absence of angular acceleration, the fifth term of equation (2) vanishes. If rotational dynamics are the primary concern, the effect of translation of the orbit is disregarded and the last term of equation (2) also vanishes. The last two terms in equation (3) will cancel each other if the inertia matrix is diagonal with all the terms having the same magnitude. In the computer program to be described, no such restriction is imposed on the P matrix.
Attitude control of a spinning flexible spacecraft
327
Conservation of linear momentum provides the relation 1 e -
P ~', m ~ u ~
(4)
~¢s=l
where ~ is the cumulative mass of all appendages and the central rigid body, and p is the total number of masses representing all of the appendages. Conservation of angular momentum is not imposed. As a result, the central rigid body is restricted against variations in rotations. Conservation of linear momentum permits the translation of the central rigid body, thus allowing the coupling of the vibrations of all the appendages attached to the central rigid body. The set of equations representing the motion of all the rigid bodies comprising the flexible appendage or appendages attached to the rigid body about the steady-state configuration is obtained by substituting equation (4) into equation (2) and writing the resulting equations (2) and (3), for all rigid bodies, in matrix form : M'ii + D'fi + G'fi + K ' u = F,
(5)
where U
Eu~ , u ~ . . . . . u ~ l ~, uJ = [ui~,/~J~l,
K ' = EKe + Ke + Ks], T TT F = [F1, F2T . . . . . Fp] ,
and F~ = [Fi r , F ~ r ] r , ( s = 1 , . . . , p )
Fs1 = m'[~w - ( ~ F 2 = -Ply
+ ~)r~],
- ~I~fl - ~ P w .
The steady-state equation in matrix form is : [Ke + Kc]u = P.
(5a)
The use of equation (4) eliminates the remaining translational rigid body degrees-offreedom. As a result, the mass matrix M' is a symmetric non-diagonal matrix. Matrix D' is defined as a symmetric matrix that describes structural damping. It is omitted in the ensuing analysis and added in the digital simulation of the system dynamics. K e and Kg are elastic and geometric (differential) stiffness matrices, respectively (and may be obtained from the NASTRAN programt) with the latter showing the effect of stretching caused by spin. Vector u represents the discrete generalized displacements about the steady-state configuration of the masses of the appendage(s) with respect to the core rigid body. In the absence of spin, matrices G', Kc, and K~ will all be identically zero; and the eigenvalue problem reduces to the standard eigenvalue problem of a free-free structure or a cantilever. The steady-state force matrix/~ is used to obtain the geometric matrix, K,. The matrix F represents a function of the angular velocities of the central rigid body and their time derivatives and couples the motion of the flexible appendages to the motion of the rigid body. Structural damping is added after truncation has taken place. t N A S T R A N : NASA Structura ! Analysis--A general purpose digital computer program for the analysis of large complex structures.
328
S . M . SELTZER, J. S. PATEL and G. SCHWEITZER
The transformation y=
Tu
(Ot
GT, K,. = K ''r.
(7}
leads to the relations M'=
MT, G'=
The transformation matrix Trelates the displacements of nodal masses in the body frame with the origin at steady-state mass center to the displacements in another body frame obtained by translating the above frame to the instantaneous mass center. Matrices Tand T-1 can be written explicitly[6]. In the absence of vibrations both above body frames coincide. If the axis of rotation and the origin of the body frame are both fixed in inertial space, Tand T - 1 become identity matrices. Substitution of transformation (6) into equation (5) gives m): + G.~, + K'y = 0
[8)
where K ' = K " + [ K e, + K g ] T
(8a)
1
and y is the vector representing the displacements of the masses with respect to the instantaneous center of mass of the system. M is a diagonal mass matrix, G is the pentadiagonal (i.e. five diagonals) skew symmetric matrix, and K" is a pentadiagonal symmetric matrix. The set of equation (8) is used because it has the advantage of requiring less computer storage core than the set of equations (5). However, this approach has the disadvantage that matrix [K e + K ~ ] T - ~ is non-symmetric. The eigenvector u i of equation (5) is obtained from the eigenvector Yi of equation (8) by using the relation (6). Since it is not possible to uncouple either the set of equations (5) or (8), they are reduced to the state space formulation. This leads to an uncoupled set of equations, which are amenable to truncation. The first step is to transform equations (5) into state space formulation (9)
P(I + Qq = L,
where P=
[: 0]
K' '
Q=
[
o]
-K'
,
L=
[ol E:] , q=
.
(10)
The eigenvectors (I)j of equation (9) are constructed from those of equation (5) in the following manner: S~bt = [ i a j u j ] , ( jl
L
IAj j
where i = x / - 1 and aj is thejth eigenvalue, and ~j is the jth eigenvector of the adjoint of equation (9). Eigenvector ~j is the complex conjugate of eigenvector O~, if P is symmetric and Q is skew symmetric. This is the case if either particle masses or spherical rigid bodies are used at each node point of the flexible body model resulting in a symmetric K' matrix. However, Van Ness has developed a program which has been incorporated into the digital program described herein that has the capability of easily (in terms of computer time)
Attitudecontrol of a spinningflexiblespacecraft
329
obtaining the eigenvalues and eigenvectors of equation (9) and its adjoint equation for small nodal rigid bodies of any shape (which results in a non-symmetric K' matrix)[7]. Using the transformation q = Oz and premultiplying equation (9) by the matrix ~,r, one finally obtains a set of uncoupled equations,
pD~ + Qoz = ~ , T L ,
(11)
where pO and QO are diagonal matrices defined by pO = ~,Tp~, QO = ~,TQ~, ~ = [ ~ a l % l . . . I~,], ~ ' = E~'11%1... I~'r3, where r denotes the number of eigenvectors retained after truncation, using the criterion described below. Normally for structural dynamical analysis, the first few lower frequency modes and the modes whose frequencies are near the frequencies of applied loads are selected for simulation. However, for stability analysis, only those modes which might cause instability are retained. A criterion for such a selection described by Likins[4] is obtained by constructing a square matrix of the order three for each mode shape. This matrix has the dimensions of the moment of inertia. If the matrix contains relatively large terms, it indicates that the corresponding modes may contribute to rotational motion instability, and only such modes are retained. This may eliminate the consideration of some low-frequency modes that otherwise would have been retained.
Vehicle equations For the entire vehicle, equations of deplacement and rotation are written using NewtonEuler relations. Retaining the rotational relations and neglecting the effect of the center of mass motion with respect to the body, one obtains 0
T=I.tb+to
x I. to + I . t o +
pxpdm,
(19)
where I is the inertia dyadic of the entire vehicle, and to is the rotation vector. Since the flexible appendages vibrate, I varies with time. Superscript o indicates differentiation with respect to time in body coordinates. The generic vector p joins the steady-state mass center of the entire system to each of the mass points of the flexible appendages and the attached rigid bodies. The vehicle equations can be linearized by first defining the variables to be small variations from the spinning steady-state configuration, i.e. substituting to = 12 + w, where fl = [f~l, t22, f~s]T is the nominal spin rate and w is the variation in spin speed, and then removing all second order terms. The contribution of mass center motion due to flexible appendage motion and orbital dynamics is ignored. The equations are based on the assumption that the axis of maximum moment of inertia of the overall vehicle does not coincide with the desired spin axis--usually an axis normal to the solar panels and to be pointed toward the sun. In that case, the overall mass distribution of the system must be altered to align the two axes--hence, the inclusion of booms (assumed to be massless flexible booms) with tip masses. It is assumed that the booms are attached to the core rigid body at the axis of rotation of the overall system. All flexible appendages, except for the flexible booms, are accumulated into one set of flexible appendage equations whose complex normal modes at nominal spin rate f~ are used. Each of the local coordinate frames in which the equations of motion are written is centered at the mass center of each of the rigid bodies, i.e. the core body and the attached rigid body, boom tip masses, and each of the
S.M. SELTZER,J. S. PATELand G. SCHWE|TZER
330
assumed rigid bodies comprising the flexible appendage(s). The basic coordinate system's origin is at the steady-state mass center of the overall vehicle, and the axes coincide with the principal axes of inertia of the steady-state configuration of the vehicle. (In steady state, the vehicle rotates about the major principal axis.) The resulting three linearized equations of motion are written explicitly in matrix notation : T = Ivi' + ~ l w + v~Iff2+
1~0)2112~"~ -
I20~21~'~ + ~"~120921 + 12([) 21
+ f i [ ~ I 2 - I 2 ~ + m2(eZr T ~ 2dE - ~ 2d 2rr + 2r 2dr ~]f2 + m2[(2Zrrt5 21 2dE - u521 2d2 r ~ + 2r 2dr e521)f~
+ 2 ~ 2 1 2d + fi 2r~21 2d] } + [ ~ ~ m~(2r ~r u~E - r ~ u~r _ u ~ r~r)~ s
+ ~" m~(2r sr fi~E - r ~ fi~r _ fis r~7")~ s
s
s
+ [mofi(2ror u " E - rou """ -- u" rr)fl + mo(2r~ilnE _ itBr~ _ rou.nT )f~ + mo?oii 8 + m o ~ o f ~ ~]
(20)
where q) = [61, ~2, ~33 r is the relative velocity of the attached second rigid body with respect to the core body (for small angular rotations, 0) 21 ~ (J), 12 is the moment of inertia matrix of the attached second rigid body about its center of mass, m 2 represents the mass of the second rigid body, 2r represents the distance between the center of mass of the second rigid body and the overall system, 2d is the distance between the center of mass of the second rigid body and the joint connecting it to the core body, E is an identity matrix, m o is the mass of each flexible boom's tip mass, and u s = u 1 - u 2, where u I and u 2 represent the displacements of the tip masses of each of the two flexible booms. Since only the skew symmetric motion of the boom affects rotational stability, u B describes that motion and matrix equation (22) describes only three (rather than six) equations of motion.
M o t i o n between connected rigid bodies
A set of three equations describes the effect of relative rotation between the core rigid body and the second rigid body which are elastically connected to each other: --K6' - C~ = I°(w + (~21 + ~'~C021)-t- •IOw + ffvlOf~ + ~ 2 1 1 0 ~ + f i ( ~ I ° -- 10~)n
+~,-~i00)21 + mz(2~2p _ 2~0~21~2 p + ~-~2~2p +
~2d~'~2
p
(2l)
"4-(~)21 2 ~ 2 p ) where matrices K and C represent the stiffness and damping, respectively, of the joint connecting the second rigid body to the core body and 2p = 2 r _ 2 d is the distance between the connecting joint and the steady-state location of the overall system's center of mass. I ° is the moment of inertia matrix of the second rigid body about its point of connection to the central rigid body.
Attitude control of a spinning flexible spacecraft
331
Boom equations A set of three equations describes the skew symmetric mode of the attached booms and tip masses : moilB + 2mo~tiB + (KBe + moO-O + K~u B - mo~oW+ m o ( ~ + ~h)r0 = 0
(22)
where K~ and K~ represent the elastic and geometric matrices, respectively, of the attached flexible booms (represented as massless cantilever beams with attached discrete tip masses). Matrix K~ results from the stretching of the boom due to centrifugal forces. Equations (9, 20-22) form a set from which passive stability (see section on that subject) can be determined.
Attitude control equations Active closed loop control is implemented by applying control torques to the vehicle. It is assumed that the 3-axis of the body-fixed b-frame must be held fixed in inertial space. This is achieved by applying control torques to the vehicle to compensate for the effect of disturbance torques. These control torques must in some manner depend on error signals that are proportional both to the angle between the 3-axis and the inertial reference vector and to its time derivative. This information can be generated if the spacecraft hardware includes, for example, sun sensors and rate gyros and the sun is used in place of an inertial reference. The sun sensors measure the angular rotations ~1, ~2 (as also indicated for the simplified model of Fig. 2), and rate gyros measure the angular velocities wl, w2. The control torques 7"1, T2 may be provided by momentum exchange devices such as control moment gyroscopes and augmented if necessary by a mass expulsion system; for simplicity it is assumed that Ta = 0. (It is observed that the existing Skylab, which is used as a model to obtain numerical results herein, has onboard sun sensors, rate gyros, CMG's, and a mass expulsion system [5].) Axis of Ins~'anfaneous /angular momenl'urn
-,
"'"
Fig. 2. Simplified model.
2
332
S.M.
SELTZER,J. S. PATEL and G. SCHWEITZER
A linear control postulate can be formulated as
0{12] ~{22J
I~121 ~- L/]21 [-/Jl1
tiff::]
-011
I~:]+
~1
[,w,; 1
(23)
where, from kinematic relations
[w,]:[_, w2
0
[ -O
[~:]
(24)
An estimate of numerical values for the control gains may be obtained by an analysis of a simplified model of the vehicle. In that case as shown subsequently--control gains for the output feedback of equation (23) may be determined by application of D-decomposition and parameter mapping techniques[8]. Alternately, optimal gains may be determined for complete state feedback by a linear quadratic loss program [9]. Digital program The linearized equations of motion of this section have been incorporated into a program called NASTRANt. It was developed for all the NASA Centers and has been modified by the authors to obtain eigenvalues and eigenvectors of any given spinning structure [6, 10]. The original NASTRAN Program can generate the elastic and geometric stiffness matrices, Ke and Kg, respectively, and either the diagonal mass matrix M or the nondiagonal consistent mass matrix for any structure. Matrices D', G, K" are generated for any arbitrary structure in the empty modules available in NASTRAN. The complex eigenvalue subroutine now available in NASTRAN can solve either of the sets of equations (5) or (8). The NASTRAN Program has been modified to accommodate the described equations of motion and is now documented[11]. SIMPLIFIED
MODEL
Equations of motion The general model described above is simplified to make it analytically tractable. It is modeled as a single rigid-core body with two attached flexible massless booms having tip masses (Fig. 2). The translational and rotational equations of motion associated with the simplified model may be obtained by setting the quantities co21, ~ and their derivatives and rns equal to zero in equations (20, 22). The elastic deformation characterizing the skewsymmetric mode of the flexible booms is represented by p~, where #i = (u~ -
u~)/2F2.
(25)
The resulting equations describe the attitude rates of the vehicle : Mz'~' + Dz'~ + Kz~ = r,
(26)
where ' refers to differentiation with respect to r = ~t, Zr = [W1/~"~, W2/~'~, #3, W3/~'~'
#1, #2] T'
(27)
U = [U 1 ,'U2, O, U3, O, O] T
= [T,/11~ 2, T2/I,f~ 2, O, T3/I1 n2, 0, 0] T, tNASTRAN
NASA general-purpose finite element structural analysis program.
(28)
Attitude control of a spinning flexible spacecraft
333
and matrices M,, D,, and K, are given by equation (29).
M~=
0
0
0
0
~ I 0 L 0 i0
0
0
710
0
0
0
0
0
0
0
0
0
0
0
0
-TFa/F2-
7Fa/F2
0
0
0
-7
0
7
(29)
0
0
0 ~I
0
1
I
0
7 0
- 27Fa/F2
0
I
D~=
0
(1 + KO/(1 - K2)
O t
0
0
- 27Fa/F 3
7 0
0
7A311 I-
0
0
0
0
(1 + K~K2)/(1 - K2)
0
27
0
7F3/F3
0
--7
7A1
-2~
- 7F3/F2
0
0
0
0
2~
7A2
(30)
K~ =
0
-K1
7
0
0
-- 7 F 3 / F a -
- - K 2 ( 1 + K1)/(1 - K2)
0
0
0
-- 7Fa/F2
0
0
7
7(a 2 + 1)
0
0
0
0
0
0
0
0
0
7Fa/F2
0
0
0
0
yF3/F 2
0
-27
~ 0
0 y(a 2 - 1) (31)
The symbols are defined as K1 = (12 - 13)/11, K2 = (Ia - 11)/12, ~ = 2moF2/I1, A i = dffmof~,a 2 = k!/mofl 2, (i = 1, 2, 3).
(32)
In the steady state, the principal axes of inertia of the total vehicle coincide with the 1, 2, 3 axes and the principal moments of inertia are 11, 12, I3, respectively, with 11 < 12 < I3. The stiffness of the nonrotating booms in the directions of the 1, 2, 3 axes is represented by ki and the structural damping by d i. The effect of F a is to couple the wobble motion, described by wfff~, w2/~, #a, and the in-plane motion, described by Wa/F~, Pl, #2. If F a is sufficiently small it may be ignored, and the six equations of motion of equation (26) become uncoupled into two sets of three equations each : one set describing wobble motion, and the other describing in-plane motion. Similarly, equations describing the attitude dynamics of the simplified model may be obtained from equation (20), (22), (24) and written in terms of attitude angles : M.z~ + Dag'a + K . z . = - v ,
(33)
334
S. M. SELTZER,J. S. PATEL and G. SCHWEITZER
where Za =
1
[(]1)1, 4/)2,123, 4 3 , P l , / 2 2 ] r,
(34)
0
-7
0
0
7F3/['e
I+Ka K 2
0
0
- 7F3/['2
0
- 7
0
";
0
0
0
0
0
0
112/13
7
0
0
- yF3/F 2
0
7
7
0
7F3/F2
0
0
0
0
0
- (1 + K1)
0
0
27Fa/F 2
0
0
0
0
0
2)'F3/F 2
0
0
7A3
0
0
0
0
0
0
0
0
-j
_ 2~/F3/1-2
0
0
0
7A I
_ ").,
0
-- 2"?F3/F 2
0
,'~7
-~"
)'A 2
-K 1
0
- i'
0
0
- T F 3 / F 2-
0
0
}'i~3/1-'2
0
o
~;(a~ + l)
o
o
0
0
0
0
0
0
0
0
)'F3/F 2
0
0
41 ~a 2I
0
- 7F3/F2
0
0
0
0
~,,(o2 - l
0
1
M. =
I+K Da =
1
0
K. =
-
K [I + K I I 2 1 ~ ]
(35)
,
(36)
(37)
Again, if F 3 = 0 the six equations represented in matrix equation (33) b e c o m e uncoupled into two sets of three equations each: the wobble motion, described by ¢1, ¢2,/23, and the in-plane m o t i o n , described by ¢3,/21, and #2. A new matrix e q u a t i o n for the wobble m o t i o n is written as m
wYIt + Dwy t + K w y =
-Vw
(38)
where Y = [ ¢ 1 , ¢ 2 , / 2 s ] r,
vw = [T1/Ilfl z, T z / I i ~ z, 0] r, and the new Mw, Dw, Kw matrices are obtained from equations (35-37) by setting and taking the upper left 3 x 3 matrices.
(39) (40) F 3 =
0
Passive stability The m o t i o n of the vehicle can be described by a nutation about the axis of angular
Attitude control of a spinning flexiblespacecraft
335
momentum which, in the absence of external torques, is inertially fixed. This motion is termed passively stable if the nutation damps out and the vehicle rotates only about the axis of angular momentum (the effect of applied torques T~is to change the attitude of this axis). In re [2], the conditions under which the simplified model of the spinning vehicle can be passively stabilized are prescribed, assuming F 3 = 0. The stability of the wobble motion (the motion associated with variables wl, w2, #3) and the in-plane motion (associated with variables w 3, P l, #2) is found by analyzing each of the two uncoupled sets of three equations each that result from setting F 3 = 0 in equation (26). The stability conditions are summarized below : - - K 1 K 2 < K 2 < - ( t r 2 + 1)K1K2/7,
(41)
0 < KIK 2 < -1,
(42)
0 < 7 < 1,
(43)
0 < 73 < 1, where 73 = 712/13,
(44)
A~ > 0
(45)
(j = 1, 3),
o.22 > 1 - @3,
(46)
tr2 > 0.
(47)
However, from the definitions of K1, K 2, o.22, and 7, they only have physical meaning in the r a n g e s - 1 < K 2 < 1, - 1 < K t K 2 < 1, o.2 >~ 0, 7 >i 0, and 73 >/0. These results are presented in graphical form (Figs. 3 and 4)[2], where v represents the frequency of the root responsible for changes in stability and is defined below. The corresponding boundaries for a rigid body may be found by letting o.~ ~ oo. Active attitude control
T o simplify the analysis, control torque will be applied only about a single axis, i.e. the 1-axis. However, it is observed that control is thereby achieved about both the 1- and 2-axes. K2
z,.I /
-'
K I K2
-I
Notes. (1) Cross-hatchedregionis stability region lost becauseof flexibility.(2) Encirclednumbers refer to number of stable roots of characteristic equation. (3) Additional constraints: 0 < ~ < 1, A3 >0. Fig. 3. Stabilityregion for wobble motion, passivecase.
336
S . M . SELTZER, J. S. PATEL and G. SCHWEITZER
2k22
Stable
2
Notes. (l) Assume k~ = k 3, d 1 = d 3 . (2) See Note (2), Fig. 3. (3) Additional constraint : A~, A.~ > t). Fig. 4. Stability region for in-plane motion, passive case.
This requires retention of the control gains cql, e12, f i ~ , and fl~ 2 of equation (23). Only cq2 and fit1 are used in the following development and are redefined for subsequent convenience as g = ~12/I1~"~ 2, 3 = fl11/'I1~'-~.
(48)
Stable regions in terms of control parameters 6 and c may be determined by the Ddecomposition method. The resulting stability boundaries are found to be equations (41-44) and the line c = O.
(49)
The remaining stability boundaries are found by writing the real and imaginary parts of the characteristic equation, P(2) = 0, that is associated with equation (38) as two separate equations and solving them simultaneously for the value of v2, where ). is the operator, 2 = t 1 + iv, and v is a frequency normalized with respect to (i.e. divided by) ~. This yields a cubic equation in v2 : (1 -- )')v ¢' + [A~ + K I K 2 + )'(K 2 + 1) -to3 +
+
1)(2 -
),)Iv 4
IA3K1K2-
+ ( ~ + II(G~ + I - 2K,K~)Iv'- + ta~ + 1)K2 b' + Kt(o~ + 1)] = 0.
150)
"~ v2, v32 are found by solving equation (50) for them. They are The three values of v2 --v~, then substituted, one at a time, into either the real or the imaginary part of the characteristic equation to result in three linear relationships between 6 and ~. F o r illustrative purposes, the
Attitude control of a spinningflexiblespacecraft
337
imaginary part of the characteristic equation is shown : (1 -
v2)6 + (1 - K2)e
= Aa(K1K 2 + v2)(1 - vZ)/(v 2 - a 2 - 1).
(51)
The resulting stability diagram is shown in Fig. 5. The corresponding boundaries for a rigid system may again be found by letting a,2. -+ o0. In addition to lines of zero damping (stability boundaries), contours of constant damping ratio can be shown on the control parameter plane, and gains leading to specified damping can be chosen there. Alternately, optimal control gains for complete state feedback can be determined by minimizing the performance index PI =
f/
( x r Q x + rrRr) dt,
(52)
where x = [yry, r ] r and v = vw represent the state and control vectors in the state space equation corresponding to equation (38)[9]. The feedback control is r = C x with the optimal gain matrix C. Elements of the cost-state weighting matrix Q and cost-control weighting matrix R are chosen to emphasize various aspects of the desired vehicle response. An analysis of attitude control dynamics is presented in [3] to include a study of the nonlinear effect of limited available control torque. EXAMPLE
The digital computer program described in the Equations of Motion Section was used to simulate a spinning S k y l a b vehicle (Fig. 6). The initial model of the system of solar panels and beam fairing mounted on each side of the orbital workshop had 2100 degrees-of-freedom and was reduced to 90 degrees-of-freedom for dynamic analysis. This 90 degrees-of-freedom model was further reduced to 27 degrees-of-freedom in such a way that the first five natural
.gl
w
"VI ,-8
~ - ~ Flexible body stable region
Rigid body stable region
Fig. 5. Stability boundaries for active attitude control.
C.A.E.E.. Vol. I, No. 3 B
S.M. SELTZER,J. S. PATELand G. SCHWEITZER
338
5
Obital w~shop .
~ F l e x i b l e Solar rays Fig. 6. Spinning Skylah.
frequencies of both the models closely matched. Each of the four solar panels on the Apollo Telescope Mount (ATM) was idealized as 40 degrees-of-freedom models and subsequently reduced to a 12 degrees-of-freedom system. The final stiffness matrix of the entire system of solar panels was regarded as one flexible appendage with 102 degrees-of-freedom. The detailed spinning Skylab simulation is programmed on a Univac 1108 digital computer with a 64,000 word core storage. Whereas equations (5) would have required 31,000 words of storage, the equations of motion expressed as equation (8) require 11,000. The simplified model was used in two ways. First, it was used to determine suitable numerical values for Skylab design parameters (such as boom characteristics and control gains). These values were then used in the digital program. Next, the system response to input disturbances was calculated using the simplified model. These responses were then used to check digital program responses during the time-consuming program "de-bugging" procedure. The agreement between responses determined by each of the two methods is extremely close, showing that in the Skylab example the system of solar panels does not contribute to instability. In this example, it is the boom flexibility which determines the stability boundary. CONCLUSIONS
A large scale digital simulation of a complex spinning structure has been developed. The effect of spin has been induced, despite the attendant complexity, by the use of hybrid coordinates. It is shown analytically the potentially incorrect results that might occur if spin is neglected, such as the potentially severe effect on the structure's modes of vibration. The resulting simulation is applicable to the class of problems where elastic structures spin at nominally constant angular velocities, such as rotating machinery of various types and spin stabilized satellites. Simultaneously, a mathematical analysis of a simplified model is presented to determine regions of stability in terms of design parameters available to the engineer. The two models
Attitude control of a spinning flexible spacecraft
339
--the mathematical model and the digital simulation--are used to complement each other in the design phase. Both models were used in the design of a spinning Skylab. The responses of each of the models compared favorably[3]. REFERENCES I. S. M. Seltzer, D. W. Justice, J. S. Patel and G. Schweitzer, Stablizing a spinning Skylab, Proc. 5th IFAC World Congr. Paris, France (12 June 1972). 2. S. M. Seltzer, Passive stability of a spinning Skylab, J. Spacecraft Rockets 9 (9) (1972), 3. S. M. Seltzer, G. Schweitzer and B. Asner, Attitude control of a spinning Skylab, J. Spacecrqft Rockets, accepted for publication (1973). 4. P. W. Likins, Dynamics and control of flexible space vehicles, NASA TR 32-1329, Rev. 1 (January 1970). 5. W. B. Chubb and S. M. Seltzer, Skylab attitude and pointing control system, NASA TN D-6068 (February 1971). 6. J. S. Patel and S. M. Seltzer, Complex eigenvalue solution to a spinning Skylab program, NASTRAN: Users' Experiences, NASA TM X-2378, II, 439-449 (September 1971). 7. J. E. Van Ness, Inverse iteration method for finding eigenvectors, 1EEE Trans. Automatic Control AC-14 (1), (1969). 8. D. D. Siljak, Nonlinear Systems. Wiley, New York (1969). 9. T. E. Bullock and C. E. Fosha, A general purpose FORTRAN program for estimation, control, and simulation, Proc. 8th Ann. IEEE Region III Convention, Huntsville, Alabama (19-21 November 1969). I0. J. S. Patel and S. M. Seltzer, Complex eigenvalue analysis of rotating structure, NASTRAN: Users' Experiences, NASA T M 35-2637, 197-234 (11-12 September 1972). 11. H. T. Franklin, Spinning structural analysis, Computation Laboratory, George C. Marshall Space Flight Center, NASA, Job. No. 310261 (25 April 1972).
ATTITUDE CONTROL OF A SPINNING FLEXIBLE SPACECRAFTt S. M . SELTZERS, J. S. PATEL§ a n d G . SCHWEITZERII
(Received 18 July 1973)
A~traet--The dynamics of rotational motion of a spinning orbiting spacecraft consisting of two rigid bodies connected by a flexiblejoint and arbitrary number of flexible appendages (two of which are flexible massless booms having masses on their tips) is analyzed. Active attitude control is provided by momentum exchange devices (e.g. control moment gyroscopes) or a mass expulsion system. The linearized equations of motion describing the vehicle are presented, and a large scale digital simulation that has been developed at the Marshall Space Flight Center is presented. A simplified model of the geometrically complex vehicle is selected to make it analytically tractable. The simplified model consists of a single rigid core body with two attached flexible massless booms having tip masses. The states of the vehicle are defined as small perturbations about its steady-state spin. An analysis is performed to determine the domain of stability.
INTRODUCTION
In 1970, NASA initiated a set of studies of several Skylab configurations, including a spinning vehicle configuration (the presently approved Skylab program does not include a spinning vehicle). The spin would provide an artificial-gravity environment to assess and compare the physiological benefits and problems of prolonged zero-gravity and artificialgravity environments. The results of the study were applied to a spinning Skylab configuration[i-3]. They have been extended so they are now applicable to a more general class of spinning spacecraft. Specifically, the study analyzes the equations of motion associated with a geometricallycomplex spinning spacecraft. The spacecraft consists of a primary rigid core body to which are attached a second rigid body, an arbitrary number of flexible appendages, and two flexible massless booms having masses on their tips. Also included in the equations of motion are a number of equations describing the applied torque, such as would be attainable from spinning momentum exchange devices or mass expulsion systems used to provide active attitude control of the vehicle. The numerical values used in the examples are those associated with a spinning Skylab configuration. tThe coauthors gratefully acknowledge the support of the National Research Council which made it possible for Dr. Schweitzer to work on this research as a NASA Postdoctoral Resident Research Associate during 19701971. They also appreciate the support and advice of Professor Peter W. Likins, University of California, Los Angeles, and that of Professor Bernard Asner, Jr., National Academy of Sciences Visiting Scientist with the Institute of Mathematics, Bucharest University, under auspices of the Academy of Romania, during his summer tenure at the Marshall Space Flight Center in 1971. An earlier version of portions of this paper was presented at the 5th IFAC World Congress, Paris, France, June 12, 1972. :l:Senior Research Scientist, Astrionics Laboratory, George C. Marshall Space Flight Center, Alabama 35812, U.S.A. §Engineering Mechanics Staff, Naval Underwater Systems Center, New London, Connecticut 06320, U.S.A. IrAkademischer Oberrat, Institute B fuer Mechanics, Technical University Munich, Munich, Germany. 323
324
S . M . SEt.TZER. J. S. PAFEL and G. SCHWEITZER
The method of approach is first to define a realistic model and then derive or state the applicable equations of motion. The problem of preserving realism while remaining within the physical constraints (e.g. memory capacity) of a digital computer is solved by casting the equations in hybrid coordinates[@ A simplified model is used to obtain an insight into the dynamic behavior of the vehicle and to help select numerical values to be used in the digital simulation. A control law is postulated to provide a satisfactory dynamic response to initial errors in attitude and attitude rate, considering the dynamics of the attached rigid and flexible members. (A practical constraint is that the control law should be easy to implement.) Applicability of the analysis is not restricted to a spinning vehicle since the development of the equations of motion is adaptable to non-spin cases. In spinning the spacecraft, it is assumed that the vehicle may be required to spin about a predetermined axis. Typical is the spinning Skylab example of this paper, where it is necessary to keep the solar panels pointed nominally toward the sun. As a consequence, the spin axis would have to be an axis nearly coaxial with the principal axis of intermediate moment of inertia of the original nonspinning configuration [5]. To cope with this possibility and still achieve passive stability, masses are deployed either on cables or extended booms so that the spin axis would become a principal axis of maximum moment of inertia. If not needed, the booms and tip masses are easily removed from the equations of motion. Also, geometric asymmetries of the original vehicle may require that the extendable booms be attached in two different planes normal to the desired rotation axis to make it an axis of principal inertia. Although the out-of-plane distance of the booms is usually small compared with their lengths, its effect is considered. EQUATIONS OF MOTION Modern space vehicles can be described as a combination of essentially rigid bodies and relatively flexible bodies. The analytical modeling of extremely flexible appendages is usually performed in such a way that they are considered to consist of small rigid bodies interconnected by elastic massless bodies. At present, two different methods of writing the equations of motion for spinning vehicles with appendages are widely used : the discrete parameter approach and the hybrid coordinate method. In the discrete parameter approach, equations of motion for various subsets of bodies or individual bodies are obtained by the direct use of Newton-Euler equations of translation and rotation. In modeling the flexible appendage as interconnected rigid bodies, a large number of them has to be taken into account to make the digital simulation effective. This will yield a large number of first- and second-order differential equations of motion. Since there is no way of reducing the number of variables, it is difficult to numerically integrate these equations. However, the hybrid coordinate method removes this difficulty by transforming a set of discrete coordinates into a smaller set of distributed coordinates, each of which is associated with the normal mode of a spinning or nonspinning appendage. In the hybrid coordinate approach, first the equations of motion of the flexible appendages are written in discrete coordinates, separately from those of the entire vehicle. Then the discrete motion of idealized bodies of the appendage is transformed into a fewer number of modal coordinates, thereby reducing the total number of equations of motion to be numerically integrated. The effect of spin is induced in the development of equations of motion despite the great added complexity introduced. This is necessary because the results of several cases studied here indicate the noticeable and possibly severe effect of spin on the modes of vibration.
Attitude control of a spinning flexible spacecraft
325
The complete set of equations used to describe the entire vehicle consists of a set of n equations describing n modes of the flexible appendage(s), a set of three equations describing the entire vehicle, a set of three equations describing the effect of relative rotation of the second rigid body elastically attached to the rigid core body, a set of three equations describing the skew symmetric mode of the attached booms, and a set of control torque equations• These are described in the following subsections•
Flexible appendage equations The equations of motion of a spinning structure are derived• Although they are used herein to describe the dynamics of satellite flexible appendages attached to the rigid core body, they apply equally to the important class of problems where structures are spinning at nominally constant angular velocities. Rotating shafts, blades of spinning turbines, rotating linkages, and spin stabilized satellites are examples of problems falling within this class. The direct use of the Newton-Euler equations give F s = roSA s
T s = zd H s
dt
(1)
for the sth rigid body of a flexible appendage; where ms is the mass, A s is the absolute translational acceleration vector. F s and T s are the sum of the external and connection force and torque vectors, respectively. H s is the angular momentum vector, and superscript i denotes differentiation in the inertial frame of reference. The expressions for the force and torque acting on the sth rigid body of an appendage spinning nominally in the steady state with an angular velocity co (Fig. 1) are written as
[bd2
bd
Fs = m s t ~ t 2 ( c + u s+r s)+2ta x ~ ( c + u
+tax
id
(ta x r , ) + ~ t a
TS = ~~d( r .
s+r
x (c+u
s)+ta
x Eta x ( c + u g ]
id2 }
s+r s)+~-~R
(2)
id
tas) = ~ [ r . (co + #s)]
p . r~ tda + ~ bd~#s ~ + t a x ~bdpAs7 j +taxP.ta+taxl
s.
p s + ~bd / I As x V.ta
(3)
where c is a vector representing the instantaneous location of the mass center of the entire vehicle with respect to its steady state position• Vectors u s and fls represent the instantaneous displacement and small rotation, respectively, from the steady state of the sth rigid body to its instantaneous deformed state. Vector r s represents the location of the sth rigid body of the appendage in its steady state configuration measured from the steady state mass center location, vector R is the distance from the origin of the inertial reference frame to the actual center of mass, and P is the inertia dyadic of the sth rigid body• Superscript b denotes
326
S . M . SELTZER, J. S. PATEL and G. SCHWEITZER C e n t e r o f mass (In s~eady s t a t e )
Central rigid b o d y ~
~
/
. b1
C e n t e r of m o Configuration of an (instantaneous)-- - ~ ~ / J ( I n s t a n t o n e ° u s )
appendage
Us
/'2 .///a F ~
-
'
u of an appendage (in s t e a d y s~rate)
Fig. I. Geometry of spinning flexible appendage and central body.
differentiation in the reference frame b imbedded in the rigid body with the origin at the steady state mass center location. The equations of motion describing rotating flexible structures, differ from those associated with nonspinning structures in several significant ways. The accelerations of the masses of a nonspinning structure are represented by the second time derivatives of the spatial variables. In the case of a structure spinning at a constant angular velocity co, accelerations are represented in equation (2) by the second time derivatives of the spatial variables with respect to a frame rotating with co; in addition, they contain terms caused by coriolis accelerations, which are proportional to the velocities of the masses in the rotating frame. Further, these expressions reflect the variations in steady-state centripetal accelerations caused by the small displacements of the masses in the rotating frame. For zero spin (co = 0), frame b is equivalent to frame i and equations (2) and (3) reduce to the familiar form
rid2 1 F ~ = rn~L~ (R + c + r ~ + u~)
(2at
T ~ = I ~ id2 fl~.
(3a)
In matrix notation the second term on the right hand side of equation (2), which is due to coriolis acceleration, gives rise, in general, to a fully populated skew-symmetric matrix G'. The third term, which is due to the centripetal acceleration, yields a symmetric matrix K , . The fourth term in equations (2) and (3) represents a steady-state centripetal force which leads to the steady-state configuration. Stretching forces and moments and rotations obtained therefrom give rise to the second order geometric (differential or kinetic) stiffness as represented for example by the stiffness of a rotating rope. In the absence of angular acceleration, the fifth term of equation (2) vanishes. If rotational dynamics are the primary concern, the effect of translation of the orbit is disregarded and the last term of equation (2) also vanishes. The last two terms in equation (3) will cancel each other if the inertia matrix is diagonal with all the terms having the same magnitude. In the computer program to be described, no such restriction is imposed on the P matrix.
Attitude control of a spinning flexible spacecraft
327
Conservation of linear momentum provides the relation 1 e -
P ~', m ~ u ~
(4)
~¢s=l
where ~ is the cumulative mass of all appendages and the central rigid body, and p is the total number of masses representing all of the appendages. Conservation of angular momentum is not imposed. As a result, the central rigid body is restricted against variations in rotations. Conservation of linear momentum permits the translation of the central rigid body, thus allowing the coupling of the vibrations of all the appendages attached to the central rigid body. The set of equations representing the motion of all the rigid bodies comprising the flexible appendage or appendages attached to the rigid body about the steady-state configuration is obtained by substituting equation (4) into equation (2) and writing the resulting equations (2) and (3), for all rigid bodies, in matrix form : M'ii + D'fi + G'fi + K ' u = F,
(5)
where U
Eu~ , u ~ . . . . . u ~ l ~, uJ = [ui~,/~J~l,
K ' = EKe + Ke + Ks], T TT F = [F1, F2T . . . . . Fp] ,
and F~ = [Fi r , F ~ r ] r , ( s = 1 , . . . , p )
Fs1 = m'[~w - ( ~ F 2 = -Ply
+ ~)r~],
- ~I~fl - ~ P w .
The steady-state equation in matrix form is : [Ke + Kc]u = P.
(5a)
The use of equation (4) eliminates the remaining translational rigid body degrees-offreedom. As a result, the mass matrix M' is a symmetric non-diagonal matrix. Matrix D' is defined as a symmetric matrix that describes structural damping. It is omitted in the ensuing analysis and added in the digital simulation of the system dynamics. K e and Kg are elastic and geometric (differential) stiffness matrices, respectively (and may be obtained from the NASTRAN programt) with the latter showing the effect of stretching caused by spin. Vector u represents the discrete generalized displacements about the steady-state configuration of the masses of the appendage(s) with respect to the core rigid body. In the absence of spin, matrices G', Kc, and K~ will all be identically zero; and the eigenvalue problem reduces to the standard eigenvalue problem of a free-free structure or a cantilever. The steady-state force matrix/~ is used to obtain the geometric matrix, K,. The matrix F represents a function of the angular velocities of the central rigid body and their time derivatives and couples the motion of the flexible appendages to the motion of the rigid body. Structural damping is added after truncation has taken place. t N A S T R A N : NASA Structura ! Analysis--A general purpose digital computer program for the analysis of large complex structures.
328
S . M . SELTZER, J. S. PATEL and G. SCHWEITZER
The transformation y=
Tu
(Ot
GT, K,. = K ''r.
(7}
leads to the relations M'=
MT, G'=
The transformation matrix Trelates the displacements of nodal masses in the body frame with the origin at steady-state mass center to the displacements in another body frame obtained by translating the above frame to the instantaneous mass center. Matrices Tand T-1 can be written explicitly[6]. In the absence of vibrations both above body frames coincide. If the axis of rotation and the origin of the body frame are both fixed in inertial space, Tand T - 1 become identity matrices. Substitution of transformation (6) into equation (5) gives m): + G.~, + K'y = 0
[8)
where K ' = K " + [ K e, + K g ] T
(8a)
1
and y is the vector representing the displacements of the masses with respect to the instantaneous center of mass of the system. M is a diagonal mass matrix, G is the pentadiagonal (i.e. five diagonals) skew symmetric matrix, and K" is a pentadiagonal symmetric matrix. The set of equation (8) is used because it has the advantage of requiring less computer storage core than the set of equations (5). However, this approach has the disadvantage that matrix [K e + K ~ ] T - ~ is non-symmetric. The eigenvector u i of equation (5) is obtained from the eigenvector Yi of equation (8) by using the relation (6). Since it is not possible to uncouple either the set of equations (5) or (8), they are reduced to the state space formulation. This leads to an uncoupled set of equations, which are amenable to truncation. The first step is to transform equations (5) into state space formulation (9)
P(I + Qq = L,
where P=
[: 0]
K' '
Q=
[
o]
-K'
,
L=
[ol E:] , q=
.
(10)
The eigenvectors (I)j of equation (9) are constructed from those of equation (5) in the following manner: S~bt = [ i a j u j ] , ( jl
L
IAj j
where i = x / - 1 and aj is thejth eigenvalue, and ~j is the jth eigenvector of the adjoint of equation (9). Eigenvector ~j is the complex conjugate of eigenvector O~, if P is symmetric and Q is skew symmetric. This is the case if either particle masses or spherical rigid bodies are used at each node point of the flexible body model resulting in a symmetric K' matrix. However, Van Ness has developed a program which has been incorporated into the digital program described herein that has the capability of easily (in terms of computer time)
Attitudecontrol of a spinningflexiblespacecraft
329
obtaining the eigenvalues and eigenvectors of equation (9) and its adjoint equation for small nodal rigid bodies of any shape (which results in a non-symmetric K' matrix)[7]. Using the transformation q = Oz and premultiplying equation (9) by the matrix ~,r, one finally obtains a set of uncoupled equations,
pD~ + Qoz = ~ , T L ,
(11)
where pO and QO are diagonal matrices defined by pO = ~,Tp~, QO = ~,TQ~, ~ = [ ~ a l % l . . . I~,], ~ ' = E~'11%1... I~'r3, where r denotes the number of eigenvectors retained after truncation, using the criterion described below. Normally for structural dynamical analysis, the first few lower frequency modes and the modes whose frequencies are near the frequencies of applied loads are selected for simulation. However, for stability analysis, only those modes which might cause instability are retained. A criterion for such a selection described by Likins[4] is obtained by constructing a square matrix of the order three for each mode shape. This matrix has the dimensions of the moment of inertia. If the matrix contains relatively large terms, it indicates that the corresponding modes may contribute to rotational motion instability, and only such modes are retained. This may eliminate the consideration of some low-frequency modes that otherwise would have been retained.
Vehicle equations For the entire vehicle, equations of deplacement and rotation are written using NewtonEuler relations. Retaining the rotational relations and neglecting the effect of the center of mass motion with respect to the body, one obtains 0
T=I.tb+to
x I. to + I . t o +
pxpdm,
(19)
where I is the inertia dyadic of the entire vehicle, and to is the rotation vector. Since the flexible appendages vibrate, I varies with time. Superscript o indicates differentiation with respect to time in body coordinates. The generic vector p joins the steady-state mass center of the entire system to each of the mass points of the flexible appendages and the attached rigid bodies. The vehicle equations can be linearized by first defining the variables to be small variations from the spinning steady-state configuration, i.e. substituting to = 12 + w, where fl = [f~l, t22, f~s]T is the nominal spin rate and w is the variation in spin speed, and then removing all second order terms. The contribution of mass center motion due to flexible appendage motion and orbital dynamics is ignored. The equations are based on the assumption that the axis of maximum moment of inertia of the overall vehicle does not coincide with the desired spin axis--usually an axis normal to the solar panels and to be pointed toward the sun. In that case, the overall mass distribution of the system must be altered to align the two axes--hence, the inclusion of booms (assumed to be massless flexible booms) with tip masses. It is assumed that the booms are attached to the core rigid body at the axis of rotation of the overall system. All flexible appendages, except for the flexible booms, are accumulated into one set of flexible appendage equations whose complex normal modes at nominal spin rate f~ are used. Each of the local coordinate frames in which the equations of motion are written is centered at the mass center of each of the rigid bodies, i.e. the core body and the attached rigid body, boom tip masses, and each of the
S.M. SELTZER,J. S. PATELand G. SCHWE|TZER
330
assumed rigid bodies comprising the flexible appendage(s). The basic coordinate system's origin is at the steady-state mass center of the overall vehicle, and the axes coincide with the principal axes of inertia of the steady-state configuration of the vehicle. (In steady state, the vehicle rotates about the major principal axis.) The resulting three linearized equations of motion are written explicitly in matrix notation : T = Ivi' + ~ l w + v~Iff2+
1~0)2112~"~ -
I20~21~'~ + ~"~120921 + 12([) 21
+ f i [ ~ I 2 - I 2 ~ + m2(eZr T ~ 2dE - ~ 2d 2rr + 2r 2dr ~]f2 + m2[(2Zrrt5 21 2dE - u521 2d2 r ~ + 2r 2dr e521)f~
+ 2 ~ 2 1 2d + fi 2r~21 2d] } + [ ~ ~ m~(2r ~r u~E - r ~ u~r _ u ~ r~r)~ s
+ ~" m~(2r sr fi~E - r ~ fi~r _ fis r~7")~ s
s
s
+ [mofi(2ror u " E - rou """ -- u" rr)fl + mo(2r~ilnE _ itBr~ _ rou.nT )f~ + mo?oii 8 + m o ~ o f ~ ~]
(20)
where q) = [61, ~2, ~33 r is the relative velocity of the attached second rigid body with respect to the core body (for small angular rotations, 0) 21 ~ (J), 12 is the moment of inertia matrix of the attached second rigid body about its center of mass, m 2 represents the mass of the second rigid body, 2r represents the distance between the center of mass of the second rigid body and the overall system, 2d is the distance between the center of mass of the second rigid body and the joint connecting it to the core body, E is an identity matrix, m o is the mass of each flexible boom's tip mass, and u s = u 1 - u 2, where u I and u 2 represent the displacements of the tip masses of each of the two flexible booms. Since only the skew symmetric motion of the boom affects rotational stability, u B describes that motion and matrix equation (22) describes only three (rather than six) equations of motion.
M o t i o n between connected rigid bodies
A set of three equations describes the effect of relative rotation between the core rigid body and the second rigid body which are elastically connected to each other: --K6' - C~ = I°(w + (~21 + ~'~C021)-t- •IOw + ffvlOf~ + ~ 2 1 1 0 ~ + f i ( ~ I ° -- 10~)n
+~,-~i00)21 + mz(2~2p _ 2~0~21~2 p + ~-~2~2p +
~2d~'~2
p
(2l)
"4-(~)21 2 ~ 2 p ) where matrices K and C represent the stiffness and damping, respectively, of the joint connecting the second rigid body to the core body and 2p = 2 r _ 2 d is the distance between the connecting joint and the steady-state location of the overall system's center of mass. I ° is the moment of inertia matrix of the second rigid body about its point of connection to the central rigid body.
Attitude control of a spinning flexible spacecraft
331
Boom equations A set of three equations describes the skew symmetric mode of the attached booms and tip masses : moilB + 2mo~tiB + (KBe + moO-O + K~u B - mo~oW+ m o ( ~ + ~h)r0 = 0
(22)
where K~ and K~ represent the elastic and geometric matrices, respectively, of the attached flexible booms (represented as massless cantilever beams with attached discrete tip masses). Matrix K~ results from the stretching of the boom due to centrifugal forces. Equations (9, 20-22) form a set from which passive stability (see section on that subject) can be determined.
Attitude control equations Active closed loop control is implemented by applying control torques to the vehicle. It is assumed that the 3-axis of the body-fixed b-frame must be held fixed in inertial space. This is achieved by applying control torques to the vehicle to compensate for the effect of disturbance torques. These control torques must in some manner depend on error signals that are proportional both to the angle between the 3-axis and the inertial reference vector and to its time derivative. This information can be generated if the spacecraft hardware includes, for example, sun sensors and rate gyros and the sun is used in place of an inertial reference. The sun sensors measure the angular rotations ~1, ~2 (as also indicated for the simplified model of Fig. 2), and rate gyros measure the angular velocities wl, w2. The control torques 7"1, T2 may be provided by momentum exchange devices such as control moment gyroscopes and augmented if necessary by a mass expulsion system; for simplicity it is assumed that Ta = 0. (It is observed that the existing Skylab, which is used as a model to obtain numerical results herein, has onboard sun sensors, rate gyros, CMG's, and a mass expulsion system [5].) Axis of Ins~'anfaneous /angular momenl'urn
-,
"'"
Fig. 2. Simplified model.
2
332
S.M.
SELTZER,J. S. PATEL and G. SCHWEITZER
A linear control postulate can be formulated as
0{12] ~{22J
I~121 ~- L/]21 [-/Jl1
tiff::]
-011
I~:]+
~1
[,w,; 1
(23)
where, from kinematic relations
[w,]:[_, w2
0
[ -O
[~:]
(24)
An estimate of numerical values for the control gains may be obtained by an analysis of a simplified model of the vehicle. In that case as shown subsequently--control gains for the output feedback of equation (23) may be determined by application of D-decomposition and parameter mapping techniques[8]. Alternately, optimal gains may be determined for complete state feedback by a linear quadratic loss program [9]. Digital program The linearized equations of motion of this section have been incorporated into a program called NASTRANt. It was developed for all the NASA Centers and has been modified by the authors to obtain eigenvalues and eigenvectors of any given spinning structure [6, 10]. The original NASTRAN Program can generate the elastic and geometric stiffness matrices, Ke and Kg, respectively, and either the diagonal mass matrix M or the nondiagonal consistent mass matrix for any structure. Matrices D', G, K" are generated for any arbitrary structure in the empty modules available in NASTRAN. The complex eigenvalue subroutine now available in NASTRAN can solve either of the sets of equations (5) or (8). The NASTRAN Program has been modified to accommodate the described equations of motion and is now documented[11]. SIMPLIFIED
MODEL
Equations of motion The general model described above is simplified to make it analytically tractable. It is modeled as a single rigid-core body with two attached flexible massless booms having tip masses (Fig. 2). The translational and rotational equations of motion associated with the simplified model may be obtained by setting the quantities co21, ~ and their derivatives and rns equal to zero in equations (20, 22). The elastic deformation characterizing the skewsymmetric mode of the flexible booms is represented by p~, where #i = (u~ -
u~)/2F2.
(25)
The resulting equations describe the attitude rates of the vehicle : Mz'~' + Dz'~ + Kz~ = r,
(26)
where ' refers to differentiation with respect to r = ~t, Zr = [W1/~"~, W2/~'~, #3, W3/~'~'
#1, #2] T'
(27)
U = [U 1 ,'U2, O, U3, O, O] T
= [T,/11~ 2, T2/I,f~ 2, O, T3/I1 n2, 0, 0] T, tNASTRAN
NASA general-purpose finite element structural analysis program.
(28)
Attitude control of a spinning flexible spacecraft
333
and matrices M,, D,, and K, are given by equation (29).
M~=
0
0
0
0
~ I 0 L 0 i0
0
0
710
0
0
0
0
0
0
0
0
0
0
0
0
-TFa/F2-
7Fa/F2
0
0
0
-7
0
7
(29)
0
0
0 ~I
0
1
I
0
7 0
- 27Fa/F2
0
I
D~=
0
(1 + KO/(1 - K2)
O t
0
0
- 27Fa/F 3
7 0
0
7A311 I-
0
0
0
0
(1 + K~K2)/(1 - K2)
0
27
0
7F3/F3
0
--7
7A1
-2~
- 7F3/F2
0
0
0
0
2~
7A2
(30)
K~ =
0
-K1
7
0
0
-- 7 F 3 / F a -
- - K 2 ( 1 + K1)/(1 - K2)
0
0
0
-- 7Fa/F2
0
0
7
7(a 2 + 1)
0
0
0
0
0
0
0
0
0
7Fa/F2
0
0
0
0
yF3/F 2
0
-27
~ 0
0 y(a 2 - 1) (31)
The symbols are defined as K1 = (12 - 13)/11, K2 = (Ia - 11)/12, ~ = 2moF2/I1, A i = dffmof~,a 2 = k!/mofl 2, (i = 1, 2, 3).
(32)
In the steady state, the principal axes of inertia of the total vehicle coincide with the 1, 2, 3 axes and the principal moments of inertia are 11, 12, I3, respectively, with 11 < 12 < I3. The stiffness of the nonrotating booms in the directions of the 1, 2, 3 axes is represented by ki and the structural damping by d i. The effect of F a is to couple the wobble motion, described by wfff~, w2/~, #a, and the in-plane motion, described by Wa/F~, Pl, #2. If F a is sufficiently small it may be ignored, and the six equations of motion of equation (26) become uncoupled into two sets of three equations each : one set describing wobble motion, and the other describing in-plane motion. Similarly, equations describing the attitude dynamics of the simplified model may be obtained from equation (20), (22), (24) and written in terms of attitude angles : M.z~ + Dag'a + K . z . = - v ,
(33)
334
S. M. SELTZER,J. S. PATEL and G. SCHWEITZER
where Za =
1
[(]1)1, 4/)2,123, 4 3 , P l , / 2 2 ] r,
(34)
0
-7
0
0
7F3/['e
I+Ka K 2
0
0
- 7F3/['2
0
- 7
0
";
0
0
0
0
0
0
112/13
7
0
0
- yF3/F 2
0
7
7
0
7F3/F2
0
0
0
0
0
- (1 + K1)
0
0
27Fa/F 2
0
0
0
0
0
2)'F3/F 2
0
0
7A3
0
0
0
0
0
0
0
0
-j
_ 2~/F3/1-2
0
0
0
7A I
_ ").,
0
-- 2"?F3/F 2
0
,'~7
-~"
)'A 2
-K 1
0
- i'
0
0
- T F 3 / F 2-
0
0
}'i~3/1-'2
0
o
~;(a~ + l)
o
o
0
0
0
0
0
0
0
0
)'F3/F 2
0
0
41 ~a 2I
0
- 7F3/F2
0
0
0
0
~,,(o2 - l
0
1
M. =
I+K Da =
1
0
K. =
-
K [I + K I I 2 1 ~ ]
(35)
,
(36)
(37)
Again, if F 3 = 0 the six equations represented in matrix equation (33) b e c o m e uncoupled into two sets of three equations each: the wobble motion, described by ¢1, ¢2,/23, and the in-plane m o t i o n , described by ¢3,/21, and #2. A new matrix e q u a t i o n for the wobble m o t i o n is written as m
wYIt + Dwy t + K w y =
-Vw
(38)
where Y = [ ¢ 1 , ¢ 2 , / 2 s ] r,
vw = [T1/Ilfl z, T z / I i ~ z, 0] r, and the new Mw, Dw, Kw matrices are obtained from equations (35-37) by setting and taking the upper left 3 x 3 matrices.
(39) (40) F 3 =
0
Passive stability The m o t i o n of the vehicle can be described by a nutation about the axis of angular
Attitude control of a spinning flexiblespacecraft
335
momentum which, in the absence of external torques, is inertially fixed. This motion is termed passively stable if the nutation damps out and the vehicle rotates only about the axis of angular momentum (the effect of applied torques T~is to change the attitude of this axis). In re [2], the conditions under which the simplified model of the spinning vehicle can be passively stabilized are prescribed, assuming F 3 = 0. The stability of the wobble motion (the motion associated with variables wl, w2, #3) and the in-plane motion (associated with variables w 3, P l, #2) is found by analyzing each of the two uncoupled sets of three equations each that result from setting F 3 = 0 in equation (26). The stability conditions are summarized below : - - K 1 K 2 < K 2 < - ( t r 2 + 1)K1K2/7,
(41)
0 < KIK 2 < -1,
(42)
0 < 7 < 1,
(43)
0 < 73 < 1, where 73 = 712/13,
(44)
A~ > 0
(45)
(j = 1, 3),
o.22 > 1 - @3,
(46)
tr2 > 0.
(47)
However, from the definitions of K1, K 2, o.22, and 7, they only have physical meaning in the r a n g e s - 1 < K 2 < 1, - 1 < K t K 2 < 1, o.2 >~ 0, 7 >i 0, and 73 >/0. These results are presented in graphical form (Figs. 3 and 4)[2], where v represents the frequency of the root responsible for changes in stability and is defined below. The corresponding boundaries for a rigid body may be found by letting o.~ ~ oo. Active attitude control
T o simplify the analysis, control torque will be applied only about a single axis, i.e. the 1-axis. However, it is observed that control is thereby achieved about both the 1- and 2-axes. K2
z,.I /
-'
K I K2
-I
Notes. (1) Cross-hatchedregionis stability region lost becauseof flexibility.(2) Encirclednumbers refer to number of stable roots of characteristic equation. (3) Additional constraints: 0 < ~ < 1, A3 >0. Fig. 3. Stabilityregion for wobble motion, passivecase.
336
S . M . SELTZER, J. S. PATEL and G. SCHWEITZER
2k22
Stable
2
Notes. (l) Assume k~ = k 3, d 1 = d 3 . (2) See Note (2), Fig. 3. (3) Additional constraint : A~, A.~ > t). Fig. 4. Stability region for in-plane motion, passive case.
This requires retention of the control gains cql, e12, f i ~ , and fl~ 2 of equation (23). Only cq2 and fit1 are used in the following development and are redefined for subsequent convenience as g = ~12/I1~"~ 2, 3 = fl11/'I1~'-~.
(48)
Stable regions in terms of control parameters 6 and c may be determined by the Ddecomposition method. The resulting stability boundaries are found to be equations (41-44) and the line c = O.
(49)
The remaining stability boundaries are found by writing the real and imaginary parts of the characteristic equation, P(2) = 0, that is associated with equation (38) as two separate equations and solving them simultaneously for the value of v2, where ). is the operator, 2 = t 1 + iv, and v is a frequency normalized with respect to (i.e. divided by) ~. This yields a cubic equation in v2 : (1 -- )')v ¢' + [A~ + K I K 2 + )'(K 2 + 1) -to3 +
+
1)(2 -
),)Iv 4
IA3K1K2-
+ ( ~ + II(G~ + I - 2K,K~)Iv'- + ta~ + 1)K2 b' + Kt(o~ + 1)] = 0.
150)
"~ v2, v32 are found by solving equation (50) for them. They are The three values of v2 --v~, then substituted, one at a time, into either the real or the imaginary part of the characteristic equation to result in three linear relationships between 6 and ~. F o r illustrative purposes, the
Attitude control of a spinningflexiblespacecraft
337
imaginary part of the characteristic equation is shown : (1 -
v2)6 + (1 - K2)e
= Aa(K1K 2 + v2)(1 - vZ)/(v 2 - a 2 - 1).
(51)
The resulting stability diagram is shown in Fig. 5. The corresponding boundaries for a rigid system may again be found by letting a,2. -+ o0. In addition to lines of zero damping (stability boundaries), contours of constant damping ratio can be shown on the control parameter plane, and gains leading to specified damping can be chosen there. Alternately, optimal control gains for complete state feedback can be determined by minimizing the performance index PI =
f/
( x r Q x + rrRr) dt,
(52)
where x = [yry, r ] r and v = vw represent the state and control vectors in the state space equation corresponding to equation (38)[9]. The feedback control is r = C x with the optimal gain matrix C. Elements of the cost-state weighting matrix Q and cost-control weighting matrix R are chosen to emphasize various aspects of the desired vehicle response. An analysis of attitude control dynamics is presented in [3] to include a study of the nonlinear effect of limited available control torque. EXAMPLE
The digital computer program described in the Equations of Motion Section was used to simulate a spinning S k y l a b vehicle (Fig. 6). The initial model of the system of solar panels and beam fairing mounted on each side of the orbital workshop had 2100 degrees-of-freedom and was reduced to 90 degrees-of-freedom for dynamic analysis. This 90 degrees-of-freedom model was further reduced to 27 degrees-of-freedom in such a way that the first five natural
.gl
w
"VI ,-8
~ - ~ Flexible body stable region
Rigid body stable region
Fig. 5. Stability boundaries for active attitude control.
C.A.E.E.. Vol. I, No. 3 B
S.M. SELTZER,J. S. PATELand G. SCHWEITZER
338
5
Obital w~shop .
~ F l e x i b l e Solar rays Fig. 6. Spinning Skylah.
frequencies of both the models closely matched. Each of the four solar panels on the Apollo Telescope Mount (ATM) was idealized as 40 degrees-of-freedom models and subsequently reduced to a 12 degrees-of-freedom system. The final stiffness matrix of the entire system of solar panels was regarded as one flexible appendage with 102 degrees-of-freedom. The detailed spinning Skylab simulation is programmed on a Univac 1108 digital computer with a 64,000 word core storage. Whereas equations (5) would have required 31,000 words of storage, the equations of motion expressed as equation (8) require 11,000. The simplified model was used in two ways. First, it was used to determine suitable numerical values for Skylab design parameters (such as boom characteristics and control gains). These values were then used in the digital program. Next, the system response to input disturbances was calculated using the simplified model. These responses were then used to check digital program responses during the time-consuming program "de-bugging" procedure. The agreement between responses determined by each of the two methods is extremely close, showing that in the Skylab example the system of solar panels does not contribute to instability. In this example, it is the boom flexibility which determines the stability boundary. CONCLUSIONS
A large scale digital simulation of a complex spinning structure has been developed. The effect of spin has been induced, despite the attendant complexity, by the use of hybrid coordinates. It is shown analytically the potentially incorrect results that might occur if spin is neglected, such as the potentially severe effect on the structure's modes of vibration. The resulting simulation is applicable to the class of problems where elastic structures spin at nominally constant angular velocities, such as rotating machinery of various types and spin stabilized satellites. Simultaneously, a mathematical analysis of a simplified model is presented to determine regions of stability in terms of design parameters available to the engineer. The two models
Attitude control of a spinning flexible spacecraft
339
--the mathematical model and the digital simulation--are used to complement each other in the design phase. Both models were used in the design of a spinning Skylab. The responses of each of the models compared favorably[3]. REFERENCES I. S. M. Seltzer, D. W. Justice, J. S. Patel and G. Schweitzer, Stablizing a spinning Skylab, Proc. 5th IFAC World Congr. Paris, France (12 June 1972). 2. S. M. Seltzer, Passive stability of a spinning Skylab, J. Spacecraft Rockets 9 (9) (1972), 3. S. M. Seltzer, G. Schweitzer and B. Asner, Attitude control of a spinning Skylab, J. Spacecrqft Rockets, accepted for publication (1973). 4. P. W. Likins, Dynamics and control of flexible space vehicles, NASA TR 32-1329, Rev. 1 (January 1970). 5. W. B. Chubb and S. M. Seltzer, Skylab attitude and pointing control system, NASA TN D-6068 (February 1971). 6. J. S. Patel and S. M. Seltzer, Complex eigenvalue solution to a spinning Skylab program, NASTRAN: Users' Experiences, NASA TM X-2378, II, 439-449 (September 1971). 7. J. E. Van Ness, Inverse iteration method for finding eigenvectors, 1EEE Trans. Automatic Control AC-14 (1), (1969). 8. D. D. Siljak, Nonlinear Systems. Wiley, New York (1969). 9. T. E. Bullock and C. E. Fosha, A general purpose FORTRAN program for estimation, control, and simulation, Proc. 8th Ann. IEEE Region III Convention, Huntsville, Alabama (19-21 November 1969). I0. J. S. Patel and S. M. Seltzer, Complex eigenvalue analysis of rotating structure, NASTRAN: Users' Experiences, NASA T M 35-2637, 197-234 (11-12 September 1972). 11. H. T. Franklin, Spinning structural analysis, Computation Laboratory, George C. Marshall Space Flight Center, NASA, Job. No. 310261 (25 April 1972).