The derivation of the equations of motion for the Spacecraft Control Laboratory Experiment (SCOLE) is reported and the equations of motion of a similar structure orbiting the earth are also derived. The structure is assumed to undergo large rigid-body maneuvers and small elastic deformations. A perturbation approach is proposed whereby the quantities defining the rigid-body maneuver are assumed to be relatively large, with the elastic deformations and deviations from the rigid-body maneuver being relatively small. The perturbation equations have the form of linear equations with time-dependent coefficients. An active control technique can then be formulated to permit maneuvering of the spacecraft and simultaneously suppressing the elastic vibration

Meirovitch, L.

Norris, M. A.

Quinn, R. D.

English

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https://ntrs.nasa.gov/search.jsp?R=19850018539 2020-03-20T19:30:07+00:00Z
VIRGINIA POLYTECHNIC, INSTITUTE AND STATE UNIVERSITY
ENGINEERING SCIENCE AND MECHANICS DEPARTMENT
BLACKSBURG, VIRGINIA 24061
(NASA-CR-175790) IDENTIFICATION AND CONTROL
	 N85-26850
OF STRUCTURES IN SPACE Progress Report, 1
Jul. - 31 Dec. 1984 (Virginia Polytechnic
Iust. and State Univ.) 18 p BC L02/HP A01
	 Unclas
CSCL 22B G3/18 21282
A Progress Report on
IDENTIFICATION AND CONTROL OF STRUCTURES IN SPACE
NASA Research Grant NAG-1-225
Covering the Period 	 ^1
,luly 1, 1984 - December 31, 1984
Principal Investigator: Leonard Meirovitch
University Distinguished Professor
Graduate Research Assistants: Roger D. Quinn
Mark A. Norris
ri
A
V	 '^
4
ev
Abstract
This report is concerned with the derivation of the equations of
motion for the Spacecraft Control Laboratory Experiment (SCOLE). Ior
future reference, the equations of motion of a similar structure
orbiting the earth are also derived. The structure is assumed to
undergo large rigid-body maneuvers and small elastic deformations. A
perturbation approach is proposed whereby the quantities defining the
rigid-body ma.ieuver are assumed to be relatively large, with the elastic
deformations and deviations from the rigid-body maneuver being
relatively small. The perturbation equations have the form of linear
equations with time-dependent coefficients. An active control technique
can then be formulated to permit maneuvering of the spacecraft and
simultaneously suppressing the elastic vibration.
(M
Contents
1. Introduction .....................................................1
2.^ Equations of Motion of the Spacecraft ............................ 2
3. Equations of Motion for the Laboratory Experiment ................7
A.	 Simulation and Control ..........................................12
5. Conclusions .....................................................13
6. Reference .......................................................13
Figures
rr.
1.	 Introduction
We shall first derive Lagrange's equations of motion for the
spacecraft of Fig. 1 regarding the structure as orbiting about the earth
and then modify these equations so as to describe the laboratory
experiment.	 In the derivation, the shuttle is treated as a rigid body
and the beam and antenna as flexible, distributed parameter systems.
The equations of motion can be further modified for the case of a rigid
antenna.
The equations describing a maneuver of a rigid space structure,
consist of nonlinear ordinary differential equations.
	 On the other
i
+	 ")
hand, the equations describing the small elastic displacements of a I	 J
flexible structure, relative to the rigid-body maneuver are linear
I
j
partial differential equations.	 Hence, the complete equations of motion
describing a flexible bogy during a maneuver represent a set of
I	 '1
nonlinear hybrid differential equations.
Hybrid systems possess an inf i . ,, i. n
 number of degrees of freedom. j
In practice, however, it is necessary to reduce the number of degrees of
freedom to a finite one, which implies spatial discretization and
truncation.
	 Substructure synthesis often proves
	 p	 s 
	 as a method of i
discretization and	 truncation,
	 particularly	 in the case of di-•'•ibuted
substructures.
	 Even in the case of discrete substructures,
	 et of
linearly independent vectors can be used as admissible vectors to reduce
the number of equations of motion.
In this report, we propose a perturbation technique whereby the
flexible spacecraft maneuver is assumed to consist of a combination of a
rigid-body maneuver and small motions including rigid-body deviations
from the rigid-body maneuver and elastic vibrations.
	 Regarding the
1
rigid-body maneuver as known, the perturbation equations for the
vibration control reduce to a set of linear ordinary differential
equations with known time-varying coefficients.
2. Equations of Motion of the Spacecraft
It is convenient to refer the motion of the spacecraft to a given
reference frame xOyOz0+ where the frame can be regarded as being
i
	
embedded in the rigid shuttle. The reference frame has six degrees of
	
!,
freedom, three rigid-body translations and three rigid-body rotations.
We propose to derive the equations of motion by means of the
Lagrangian approach. To this end, we must first obtain expressions for
the kinetic energy, the potential energy and the virtual work.
Considering Fig. 1 and denoting the position of the origin 0 of the
frame xOyOz0
 by the vector R and the position of a point S in the
shuttle relative to 0 by r, the position of S relative to the inertial
frame XYZ is RS
 = R + r. Moreover, denoting by a the vector from 0 to a
nominal point A on the appendage and by u the elastic displacement
vector of the point, the position of A in the displaced configuration
is RA
 = R + a + u. It must be noted that the vectors r, a and u are
likely to be measured relative to axes x0y0zo. In view of the above,
the velocity of a point S in the shuttle is
R S = R + w x r
	
(1)
where R is the translational velocity and w is the angular velocity of
the frame x O yOz0
 with respect to the inertial frame. Similarly, the
velocity of a point A in the appendage is
RA =R+w x (a+u)+u
	
(2)	 1
2
where u is the elastic velocity of the point relative to the xOyOz0
frame. Hence, the kinetic energy of the spacecraft is
T.	 m	 mI f IRS 12dmS + ^ f IRA I'dmA
	
S	 A
=2 f 1R+wx rl 2dmS +2 f IR+wx (a+u)+ul2dmA
	
m
s
	MA
	
2 mIRI2 +
	 J,	 + R•(w 
x S.	 + 2 
f 1u12dmA
	
"	
MA "
+ R•[f u dmA + w x f u dmA ] + f ll • (w x a)dmA
	
" MA"	
"	
MA"	 MA"
+ f (w x a) • (w x u)dmA + z f Iw x u12dmA
	
MA "
	 "	 "	 "	
MA "
+ f u • (w x u)dmA
	(3)
MA"
where
SO = f 
ms 
r dmS + f a dmA
	(4)
	
"	 mA
and mS , mA and m are the masses of the shuttle, appendage and entire
spacecraft, respectively. Also, I 0 is the total mass moment of inertia
matrix of the undeformed structure about point 0. Note that 1x12
denotes the inner product x•x.
The potential energy is due to the combined effects of gravity and
strain energy. Assuming that the origin of the inertial coordinate
system coincides with the center of the gravitational field, the
gravitational potential can be expressed as
Vg
 = - Gme [f 1R + rJ -1dmS + f 1R + a + uj -1dmA ]	 (5)ms "	 ..	
MA
3
l
-..: ,	 -
where me is the mass of the earth and G is the gravitational constant.
The strain energy can be expressed as an energy inner product
symbolized by [ , i (Ref. 1). The total potential energy then becomes
V = 2[u,ul + V9	 (6)
The virtual work is due to external forces, including control
forces. Denoting by f S the force vector per unit volume of the shuttle
and by fA the force vector per unit volume of the appendage, we can
write the virtual work as
6W = f  fS •6RSdDS + f0 EA' 6RAdDA	 (7)
S	 A
where DS and DA are the domains of the shuttle and appendage,
respectively.
Before deriving the equations of motion, we consider certain
simplifying assumptions. To this end, we estimate the maximum possible
angular velocities by ignoring the effects of the appendage and
examining the prescribed maneuvers of the shuttle alone. For the 200
maneuver about the x 0 axis, applying maximum torque, the maximum angular
velocity is approximately .06 rad/s. For the 90 0 maneuver about the z0
axis the maximum angular velocity is .047 rad/s. If the elastic
displacements are small, then the last two terms of Eq. (3) are of
higher order and can be neglected. The third to last term of Eq. (3)
will be retained, despite leading to nonlinear terms, because the
magnitude of the factor multiplying the independent variables u and
w tends to offset the smallness of the independent variables. Next, we
express the elastic displacements in the form of linear combinations of
admissible functions, or
4
s
6
i
rkV
u=o9 (8)
where o is a matrix of space-dependent admissible functions and 9 is a
vector of time-dependent generalized coordinates. Introducing Eq. (8)
into Eq. (3) and neglecting the last two terms in Eq. (3), the kinetic
energy takes the matrix form
T 
= 2 mRTR + 2 wTI0 + RTCTSOw + 2 9TMA9
+ RTCToq + RTCTwTog + jTm Tw + u,Tf a%To dmA9
MA
where
(9)
m = f o dm A' $T = f 01'a dm 
M
A	 A
and
(10a,b)
MA= f oTo dm 	 (10c)
mA
is the mass matrix of the appendage. The symbol C represents a rotation
matrix from the inertial frame to the x 0y0z 0 frame and its elements are
nonlinear functions of a set of Euler angles a. The tilde over a given
vector such as v denotes a skew symmetric matrix of the form
0	 vz -vy
v = - v z	0	
v 
	 (11)
vy -vx	0
Recognizing that the magnitude of R is large and u is small in
comparison with the other vectors in Eq. (5) and ignoring terms of order
higher than three, a binomial expansion permits us to write
Vg
 = - Gme (mIRI -1 - R•(SO + f u dmA)I R I -3 J	 (12)MA-
5
V,.
Introducing Eq. (8) into Eq. (6) and considering Eq. (12), the potential
energy can be written in the matrix form
Gm m Gm
V 2 9TKA9 -	 + e3 RTCT(Sp+^ ^)	 (13)
i	 IRI
where
KA = 10, 01	 (14)
is the stiffness matrix of the appendage. The virtual work can be shown
to have the expression
dW = FTCsR + MTsa + QT69	 (15)
where
F = f  f S dDS + f0 EA do 
S	 A
M = f jfS dDS + f aTfA dDA + f EA -D dDA 	(16)
-	
DS _
	
DA	 DA
Q = 1  j fA dDA
A
o.re generalized force vectors in terms of components about x 0 ,y0 and z0.
Without loss of generality, we let point 0 correspond to the center
of mass of the spacecraft'in its undeformed state, so that the vector
SO is zero. Then, Lagrange's equations of motion can be written in the
symbolic form
dt (,T) + aR = CT F	 (17a)
aR
d ( aT ) _ aT + 9V= M
	
(17b)
dt 
as	
2a	 as
Ft ( a^ ) - ag + ag = Q	
(17c)
6
S
i	
1
t
^	 i
so that, considering Eqs. (9), (13) and (15), the equations of motion
for the spacecraft in orbit are
_„	 mR	 CT—^ R(3RTCTog)	 T
mR + CT Mg + Gme [R3 
+ IRI3 -
	 IRIS	 [ = C F
	
(18a)
IGw + w IGw + [CR[mg + icy + wT$g + J(w)q + J(w)9
Gm	 _
+ wTJ(w)g + IRi3 (CR	 (18b)
—T "	 T•	 T-T-	 Gme —TMAg + CR + m w + KAY + f	 w a dmAw + 3 m CR = Q	 (18c)
M 	
IRI
where
N
J(w) = f (K + [aw])m dmA	 (18d)
MA
Higher-order terms have been neglected in Eq. (18b), consistent with the
preceding discussion concerning the magnitude of the maneuver angular
velocities. However, as before, nonlinear terms with 1%.rge
coefficients, such as Io and am 
A' 
have been retained. The position
vector R, its time derivatives, and the Euler angles vector a have been
considered to be of arbitrary magnitude, with the result that many other
nonlinear terms have been retained in Eqs. (18).
3. Equations of Motion for the Laboratory Experiment
In the laboratory experiment, the spacecraft is not actually free
In space, but suspended from the ceiling by means of a cable or a
beam. The following analysis applies to either case. The support is
likely to affect the dynamic characteristics of the system. Hence, in
the sequel, the support is added to the free model in the form of an
elastic member.
7
i
i
1
i
!	 ra
I
i
i
i
I
r .,
Considering Fig. 2 1
 the position vector fur an arbitrary point C on
the cable is RC - c + w, where c is a position vector and w is the
elastic displacement of the cable, both of which are measured with
respect to the inertial frame. The position vector for the point 0 is
R=c8 +w8 +e	 (19)
where the subscript B denotes evaluation at the point B and a is the
vector from point B (ball joint) to the point 0 fixed on the "shuttle",
measured with respect to the x Oy0Z0 frame. The velocity vector of an
arbitrary point C on the cable is then
RC =w
	(20)
and the velocity of point 0 is
I
R =wB +w x e	 (21)
The kinematics for the shuttle body and appendage remain the same as for
the unrestrained spacecraft in space. Hence, the kinetic energy for the
entire structure is
T = Z f IRCl 2dmC + p fI RS I 2dm s + z f IRg12dmA
mC	 S	 mA
= 2 f 1w12 dm + 2 f Iw8 + w x (e + r)I2dmS
m 
	
m 
+ 1
 f IwB +w x (e+a+u) +612dmA
m 
	 "	 "
= z f 1w12 dm+ 2 mli1
2
 + 2 wTIBw
mC
+ wOT x S B ) + (w x e)-(w x S B ) + 2 f 161 2dmA"	 "	 "	 mA "
+ wB ' f u dmA + (w x e) -f u dmA
m 
	 MA"
8
+ j U • (w x a)dmA + f (w x a)-(a x u) dmA	(22)
MA"
	
MA
where
I  Y I D + meTe, SB - SO + me	 (23)
The elastic displacement vectors u and w have been assumed to be
small. Also, the angular velocity was assumed to be, at most, of the
same order of magnitude as in the case of unrestrained spacecraft, so
that terms of higher order can once again be neglected.
The expression for the virtual work is given by Eq. (7), but the
potential energy must be modified. The acceleration of gravity will be
assumed to be constant and can be expressed as 2 _ - gu Z so that the
gravitational potential is
V9 = f (c + w)•9 dmE + f (R + r)•2 dmS + f (R + a + u)-9 dmA
mC _	 -	 mS	 -	 mA 	
(24)
where R is defined by Eq. (19) and u Z is a unit vector in the Z
direction. The elastic potential energy for the system can be expressed
as
VE = 2 [u,u] + 2 (w,w] 	 (25)
where [w,w] is the energy inner product for the cable, which includes a
stiffening effect due to the weight of the "spacecraft". Because a
cable has little inherent bending stiffness, this effect can be
significant.
As with the appendage in the preceding section, the elastic
displacement of the cable can be approximated by a linear combination of
admissible functions, or
9
v
PW • in	 (26)
where ^ is a matrix of space-dependent admissible functions and n is a
vector of time dependent generalized coordinates.' Introducing Eqs. (Q)
and (26) into Eq. (22) 9 the kinetic energy takes the matrix form
T n 1 ;TM - +	 B-1 wT I w + nTVTC B- 	 0-TS w wT-T- w f 
I TM + ry^T JC Tog
'
C- '^ -	 B	 -	 A9	 B
T-T	 T-T	 T	 T-T
+ w e og + 9 v w+ w f mAa w 0 dm 	 (27)
where	 r
MC = f jo dmC + m^VBV B 	(27a)
mC
is the mass matrix of the cable, including the mass of the structure
t
lumped at the end of the cable.
	
y 8 denotes the matrix 0 evaluated at
b. Introducing Eqs. (8) and (26) into Eqs. (24) and (25) the potential
energy can be written in the matrix form
k	 ^
T—T	 T	 T T	 7 T	 1 T	 1 T
n	 + SBC^ + mn ^B^ +	 C2 + 2 g KAg + 2 n KC n	 (28)
where
= f i dmC	(29a)
mC
and
KC	 (29b)
is the stiffness matrix of the support. Considering Eqs. (19) and (25),
the virtual work can be expressed as
6W = FTC 6n + MTda + QTsq
	
(30)
10
O'er
R
where all the terms have been defined previously. The effect of the
friction of the ball ,joint can be assumed in the form of an external
torque. Hence, we let M = M c
 + M  in Eq. (30), where M c is a vector of
control moments and Mf
 is a vector of frictional moments caused by the
ball Joint.
Without loss of generality, we assume that point 0 is the center of
mass of the spacecraft, so that S O
 is zero. For the laboratory
experiment to be successful the vector a must be very small, so that the
center of mass 0 tends to coincide with point B. This is difficult to
accomplish in practice due to the friction of the ball Joint and the
difficulties of dynamically balancing a large flexible structure.
Realistically, we can expect a to be small but nonzero. Hence, for
simulation purposes, we choose a to be small, so that the kinetic energy
of Eq. (27) can be approximated as follows:
T = 2 nTMC n + 2 wT I Ow + mn T^UBCTew + 2 9TMAq
gT*BCTOS + t§Tj. + wT f JT To dmA	 (31)
"
	
MA
Lagrange's equations remain in the symbolic form of Eqs. (17), with
the exception of Eq. (17a) which must be replaced by
d ( DT ) + IV = O
TC TF	 (32)dt 
an	
aq	 B
This results in the following equations of motion for the laboratory
experiment:
MCn + mUBC%Tew + mjCTew + ^ TCTo- + (m UT + T)^ + Kcn = JCTF"	 "	 B "
(33a)
I Ow + wT I Ow + 09 + Jjj + J(w)q + J(w)§ + wTJ(w)q
F	 '.
11
.. ^" -; ,'^-sue- w•-,--?t---^.
+ J%qm + (C2 1(em + mg) = M
	 (33b)
Mg + mTw + 'TC*Bn + f 0TwTa dmA w + mTC^ + K^ = Q	 (33c)
- MA
Higher-order terms have been neglected and some nonlinear terms retained
in Eq. (33b), consistent with the case of the spacecraft in space. In
this case, the Euler angles vector a is assumed to be of arbitrary
magnitude, which is responsible for many nonlinear terms in Eqs. (33).
4. Simulation and Contr ol
The nonlinear equations of motion take the same basic form for the
orbiting spacecraft as for the laboratory experiment, as can be seen
above. Hence, the'solution techniques suggested here apply for both
situations.
Consider a first-order perturbation in the vectors R and w.
R=Ro +R 1 , w=w0 +w 1 	 (34)
where the first-order terms R 1 and w 1 are small compared to the zero-
order terms Rot wp. introducting Eqs. (34) into the nonlinear equations
of motion, Eqs. (18) or (33), we obtain zero-order and first-order
equations. The zero-order solutions RO and wo can be obtained from an
open-loop, rigid body maneuver strategy. These solutions can then be
inserted into the first-order equations yielding linear equations with
known time-varying coefficients. Simulation requires numerical
integration. The equations can be put into their most compact form by
approximating the first-order motion by means of a linear combination of
eigenvectors corresponding to the lower frequencies. A control
technique suppressing elastic vibration as well as rigid-body motion
12
V "
that deviates from the desired maneuver, can also be formulated
considering the equations in this compact form.
5. Conclusions
The equations of motion for the structure both in orbit about the
earth and in the laboratory are nonlinear, even when the elastic
deformations are small. The nonlinear terms result from the large
rigid-body maneuver. For instance, centrifugal and Coriolis terms
affecting the elastic displacements can be significant in a minimum-time
maneuver, so that they must be retained in the equations. Through a
perturbation approach, the nonlinear equations of motion can be
transformed into a set of equations governing the rigid-body motions and
a set of time-varying, linear equations governing small deviations from
the prescribed rigid-body maneuver and elastic motion. Future work will
Include applying this perturbation approach to the above equations of
motion for use in both simulation and control of the structure.
6. Reference
1.	 L. Meirovitch, Computational Methods in Structural Dynamics,
Sijthoff-Noordhoff, The Netherlands, 1980.
13
w
P
a3
Figure 1 - SCOLE Configuration
Xi
N
r y
I
S
7
Figure 2 - Laboratory Model


Identification and control of structures in space

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