Large, flexible orbiting systems proposed for possible use in communications, electronic orbital based mail systems, and solar energy collection are discussed. The size and low weight to area ratio of such systems indicate that system flexibility is now the main consideration in the dynamics and control problem. For such large, flexible systems, both orientation and surface shape control will often be required. A conceptual development plan of a system software capability for use in analysis of the dynamics and control of large space structures technology (LSST) systems is discussed. This concept can be subdivided into four different stages: (1) system dynamics; (2) structural dynamics; (3) application of control algorithms; and (4) simulation of environmental disturbances. Modeling the system dynamics of such systems in orbit is the most fundamental component. Solar radiation pressure effects and orbital and gravity gradient effects are discussed

Bainum, P. M.

Diarra, C. M.

Krishna, R.

Kumar, V. K.

Reddy, A. S. S. R.

English

NASA Technical Reports Server

THE DYNAHICS AND CONTROL OF 
LARGE FLEXIBLE SPACE STRUCTURES 
Peter M. Bainum, V. K. Kumar, R. Krishna, 
A. S. S. R. Reddy, and C. M. Diarra 
Howard University 
Washington, D. C. 
91 
https://ntrs.nasa.gov/search.jsp?R=19830013992 2020-03-21T03:44:56+00:00Z
INTRODUCTION 
Large, flexible orbiting systems have been proposed for possible use in com- 
munications, electronic orbital-based mail systems, and solar energy collection. The 
size and low weight-to-area ratio of such systems indicate that system flexibility is 
now the main consideration in the dynamics and control problem. For such large, 
flexible systems, both orientation and surface shape control will often be required. 
Figure 1 illustrates a conceptual development plan of a system software capa- 
bility for use in the analysis of the dynamics and control of large space structures 
technology (LSST) systems. This concept can be subdivided into four different 
stages: (1) system dynamics; (2) structural dynamics; (3) application of control 
algorithms; and (4) simulation of environmental disturbances. Modeling the system 
dynamics of such systems in orbit is the most fundamental component. 
SOLAR RADIATION PRESSURE EFFECTS 
The equations for determining the effects of solar radiation pressure on a 
flexible beam are summarized below. 
Forces: 
Fa = -ho; 
J 
('; - ii) ds 
S 
(absorbing surface) 
FV = -2ho 
s 
c;<; - d A 2 ds (reflecting surface) 
S 
= Fa + E(Fy - Fa) (surface with reflectivity E) 
Moments: 
Ma = ho.;x 
s 
R(T * ii) ds (absorbing surface) 
S 
i$ = 2ho 
J 
f; x R(T * f;)2 ds (reflecting surface) 
S 
ME = Ma + E’iq - Ma> (surface with reflectivity E) 
92 
where 
; = unit vector in the direction of solar radiation 
; = unit vector normal to the surface ds 
h 
0 
= 4.64 X 10m6 N/m2 
Results for a flexible beam: 
f, = -aoh c a,(zl - zo) - b, 2 + h,bo~,(zl - zo) - bo]g 3 
(zl + z,)/2 - 2/S-& ~(COS hR, - cos bE,g f sin CZ, + sin hQn} 
F = -2ho f z' 
(a,z’ - bo12 
Y 2 
S (1 + z') 
dx 2 - 2h, f 
(a,z ’ - b,)’ 
dx i; 
S (1 + z'>2 
Mu 0 = 2h 
(a,z ’ - bo12 
z'z l\ f 
S (1 + z') 
2 - x + a dx k 
where 
aO 
= sin 0 bO = cos 0 
z (4 = flexural deflection 
Rn = nth modal frequency 
93 
Figure 2shows the variation of the resultant horizontal and normal force com- 
ponents of a beam with a completely absorbing surface as the solar incidence angle 8 
is varied from 0' to 90°. Here, 0 represents the angle between the normal to the 
undeflected beam and ?. The horizontal and normal force components are measured 
relative to the beam's undeflected axes. As expected, for small tip deflections of 
the beam, the resultant horizontal absorbing force component becomes zero for inci- 
dence angles of O0 and 90°, while the normal component has a maximum amplitude at 
zero incidence angle. In figures 2 and 3 the individual effect of each mode is 
illustrated, with the assumed beam tip deflection as indicated in the figures. 
The magnitude of the resultant moments as the solar incidence angle is varied is 
shown in figure 3 for the assumed tip deflection of 0.01%. Large moments can result 
for larger deflections, whereas these moments would be zero for a rigid beam. For 
small pitch angle displacements, the moment due to solar radiation pressure may 
become greater than the moment due to the gravity-gradient forces, as shown in fig- 
ure 4. It is seen that at geosynchronous altitudes, the moment due to solar radia- 
tion may become predominant even for deflections on the order of O.OlR. With the aid 
of moment diagrams such as those in figure 3, it is possible to determine the dis- 
turbance torques due to solar radiation pressure once the number of modes and the 
associated modal deflections are specified for a model. 
MODELING ERRORS - ORBITAL AND GRAVITY-GRADIENT EFFECTS 
One of the principal sources of (disturbance) torques acting on an orbiting 
space structure is the orbital (gyroscopic) and gravity-gradient effects. Such 
effects associated with the orbital (angular) motion do not need to be considered 
when developing a system model for an Earth-based large flexible system. Many 
investigators, however, model the pitch, roll, and yaw modes (rigid body motion) of 
large, flexible orbiting systems as double integrator plants (two poles at the 
origin), and the subsequent control system design is based on these models. It is 
the purpose of this section to evaluate the effects of omitting the orbital and 
gravity-gradient effects when designing shape and orientation control laws for 
flexible systems in orbit. Models of flexible square plates and shallow spherical 
shells in orbit are selected as examples. 
The effects of designing control laws without the orbital and gravity-gradient 
torques included in the system models of square plates and shallow spherical shells 
in low Earth orbit (250 nautical miles) are illustrated in figures 5 to 10. A square 
plate was also considered in a geosynchronous orbit. The analysis was performed by 
first calculating the control law, which was in the form u = Fx, for the case where 
the orbital and gravity-gradient effects are not included in the model. The same 
control law is then applied to the model that includes these effects. 
For figure 4, the control law was selected such that the overall response time 
constant of the system is 2.22 hours (which may be reasonable for a large space 
structure). The shift in the closed loop poles of the plate model due to the pres- 
ence of the orbital and gravity-gradient effects is illustrated in this figure. It 
can be seen that some of the poles move to the right half plane, leading to insta- 
bility and thus emphasizing the importance of including the orbital and gravity- 
gradient effects in the model. The poles due to the rigid body modes are shifted 
considerably, but the flexible modes remain virtually unaffected. This can be 
attributed to the high frequency of the flexible modes. (Note that the orbital and 
gravity-gradient effects are of a relatively low frequency.) This result gives an 
94 
indication that by designing a more robust (faster response) control system, the 
shift of the actual closed loop poles would be relatively less pronounced. 
This phenomenon can be demonstrated by designing the gain matrix, F, such that 
the desired response time constant is reduced to 460 seconds. The shift in the 
closed loop poles for this case is shown in figure 6. In figure 7 the control forces 
are shown for the second (more robust) control law, where the closed loop response 
with orbital and gravity-gradient effects is degraded but does not become unstable. 
(It should be noted that time has been nondimensionalized with respect to the orbital 
frequency of the 250-nautical-mile low Earth orbit, in order to provide a basis for 
comparison.) The difference in the total control force impulse as applied to the two 
models (a) and (b) is minimal because of the robustness of the controller. The 
slowly varying orbital and gravity-gradient torques have a relatively greater impact 
on the less robust systems, and can even lead to possible instabilities, as was 
illustrated in figure 5. 
As expected, the orbital and gravity-gradient effects are less pronounced in the 
case of a structure in geosynchronous orbit than in the case of a structure in low 
Earth orbit. However, if the control systems are designed with response times com- 
parable to the orbital periods, under the influence of orbital and gravity-gradient 
effects the closed loop systems may become unstable. 
The shift in the closed loop poles of the spherical shell model due to the pres- 
ence of the orbital and gravity-gradient effects is shown in figure 8. One of the 
closed loop poles is moved to the right-hand side of the S-plane, causing instability. 
As compared to the case of the plate, the effect of the orbital and gravity-gradient 
torques on the shell is more pronounced, as the instability due to movement of the 
poles occurs at the relatively fast designed response time constant of 615 seconds 
(compared to 8000 seconds in the case of plate). When the control is redesigned for 
a response time constant of 400 seconds, the shift in the poles is as shown in fig- 
ure 9. A general shift in the rigid body motion poles is observed, but the system 
remains stable. 
The control forces associated with both models (a) and (b) of figure 9 are com- 
pared in figure 10. A considerable increase in the control effort is observed when 
the model includes the effect of the orbital gyroscope and gravity-gradient torques. 
This may be explained by the fact that the mass distribution of the shell is more 
complex than that of the plate, resulting in relatively greater dynamic coupling when 
the gyroscopic and gravity-gradient effects are included in the shell model. 
THE DEVELOPmNT OF AN ALGORITHM TO EVALUATE COUPLING COEFFICIENTS 
FOR A LARGE FLEXIBLE ANTENNA 
The generic mode equations and the equations of rotational motion of a flexible 
orbiting body contain both coupling terms between the rigid and flexible modes and 
terms due to the coupling within the flexible modes that are assumed to be small and 
thus are usually neglected when a finite element analysis of the dynamics of the 
system is undertaken. In this section a computational algorithm is developed which 
permits the evaluation of the coefficients in these coupling terms in the equations 
of motion as applied to a finite element model of a hoop/column antenna system 
(ref. 1). 
95 
Using a Newton-Euler approach, one can express the equations of motion of an 
elemental mass of the system, in the frame moving with the body, as: 
. . . 
‘: cm+i+ 2ij+r+i xsi+w x (ix i) r,dv={ T + e + L(;)/& dv (1) 
where 
p =,mass per unit volume 
e = external ‘forces per unit mass 
+ elastic transverse displacements of the element of volume 
'I = force due to the gravity on the unit mass 
L= the linear operator which when applied to G yields the elastic forces 
acting on the element of volume considered 
r = position vector of element dv 
w = inertial angular velocity of the body frame 
acm = acceleration of the center of mass 
Equations of Rotational Motion 
The equations of rotational motion of the body are obtained by taking the 
moments of all the external, internal, and inertial forces acting on the body; i.e., 
from equation (1): 
J c 
. . ix a cm + : + (2; x ;) + (w x ;) f (; x (ii x ;)jj p dv = j-F x b(p>/p + 7 + z]odv 
V 
One can obtain the following form for the equations of rotational motion: 
co co co 
g + c p + c ,Cn) = ER + c -,Cn) + c 
n=l n=l n=l 
where 
(3) 
EC 
x (5 x io> - (i. l w)(ii x iofl p dv 
96 
OJ 
c q(n) = s( ‘* i o x ‘q + Zi, X ( wxq ') + go x '(5 x 4) + 4 x (ii x io) 
n=l V 
- (i. : W) c<il X 4) - <‘i l ;d (i x i,)}p dv 
co 
c -,(d = f {p dv x (acm - To) + 2 miAn $ Go x +‘p dv 
n=l V n=l V 
ER = 
s i. x Mgo P dv 
00 
c 
-,h) = i. x MS + i X Mio)P dv 
n=l V 
z= s ?xepdv 
V 
i=io+q 
M= matrix operator which when applied to 7 yields gravity-gradient forces 
a cm = acceleration of the center of mass 
To = force/mass due to gravity at the undeformed center of mass 
m(n) = modal shape vector for the nth mode 
w n = frequency of the nth mode 
An = time-dependent modal amplitude function 
Generic Mode Equations 
The generic mode equation is obtained by taking the modal components of all 
internal, external, and inertial forces acting on the body, i.e., 
s '. l . cm+i+2;xi+Gxi+Wx (ixi) pdv 3 
V = s 3(n) . + z + ; p dv 1 
V 
(4) 
97 
The generic mode equation is obtained in the following form: 
;i, + a;An + '&/Mn + 2 $m/Mn = 
m=l 
+ 2 gm + En + 
m=l 
where 
74, = SC 
;h> 
l w X (w X io) p dv 1 V 
l ; x (ii x ii> p dv 1 m=l V 
gn = s 
;(n) 
l Mio p dv 
03 
c gm = / ;(n> - M<p dv m=l V 
En = s icn) l :p dv 
D; = s 
;<n> p dv l (acm - io) 
V 
Here I/I, is the inertia coupling between rigid body modes and the nth structural 
mode and $mn is the inertia coupling between the mth and nth structural modes. 
Cartesian Components of the Different Coupling Terms 
The expressions for R, 0 Cd , ER, C), $ n3 1cI,, gny ad gm in 
Cartesian components are presented in this section. The following vectors can be 
expressed in their Cartesian component form as: 
(5) 
r 0 = 5,: + 5,: + 5,G; w = wxz + WY3 + wzi; 
98 
G(n) = Q(n)A 
x i+ 9, 9 + QY 
E(n) = ,h>^ x i+G (n>3 + G(n)i; 
Y Z 
where 2, 3, and i; are unit vectors along 
the undeformed state, and 5 and 5 
undeformed state. X’ 5Y* Z 
the body principal axes of inertia in 
are the coordinates of a point in the 
With the use of the component forms of the vectors given above, one can expand 
the various vector expressions given in equations (3) and (5) to obtain 
Jy)wywz] ^i + p,", + (Jx - Jz)~z~x] ; 
Y 
- Jx)w w i; 1 XY 
H Cd - J-+)u - Hcn)a 
YY X 1 yx y 2x z 
- Hi;)) - wxwy(H;;) + Hi:))+ u~u~(H($ + Hi;)) 
ER = (Jz - Jy)M23: + (Jx - Jz)M31; + (Jy - Jx)M2$ 
(6) 
(7) 
(8) 
99 
G(n) = 
X An (M33 c 
- M22)(H;;) + Hi;') - M21(Hz) + Hi;') 
+M + Hz') + 2M23(H;) - H;;' (9) 
where 
100 
a, B = X, y, z or 1, 2, 3 
When a is 
In a similar 
when a is 
for 8. 
x in H? or 
aB 
manner, when a 
z in H$' or 
L($n' , 
is Y 
L(=d 
af3 ' 
the 
in 
a 
corresponding value of a in M is 1. 
,(n) or ,hd aB 
a8 aB ' 
a is 2 in M 
aB' 
and 
is 3 in M 
aB' 
The same reasoning holds 
The expressions for b-9 
Qy 
and Q Cd 
of x, Ys z in the expression for Q$ 
are obtained by the cyclic permutation 
in equation (7), and the expressions for 
Gcn) and Gcn) 
Y z 
are obtained by the cyclic permutation of x3 Ys = in the 
expression for x Gtn) in equation (9). 
For a discretized model, the expressions for the volume integrals are replaced 
by the following summations: 
H(n) = 
aB i=l 
L(m) = 2 (Qqi (4p)p, a6 
i=l 
(12) 
b,B = X,Y,Z) 
(13) 
where 
k = total number of discrete masses 
i = index identifying a nodal point 
m. = mass concentrated at the ith node 
1 
5, = coordinates of mi in the undeformed state 
REFERENCE 
1. Bainum, Peter M.; Reddy, A. S. S. R.; Krishna, R.; Diarra, Cheick M.; and Kumar, 
V. K.: The Dynamics and Control of Large Flexible Space Structures - V. NASA 
CR-169360, 1982. 
101 
SYSTEM DYNAMICS STRUCTURAL ANALYSIS 
Differential Equations - LSST Orbiting Systems STRUDL-II* 
Beam, incl. control (2-D)* 
. 
Det. frequencies, mode shapes 
Plate, incl. control (3-D)* NASTRAN? 
Shallow spherical shell, incl. control (3-D)* 
4 
More complicated system (3-D)* 
I 
I 
Hoop-column+ I 
t 
ENVIRONMENTAL 
Solar radiation 
forces/torques+ 
Thermal effects+ 
CONTROL ALGORITHMS 
Jones & Melsa" - opt. control 
ORACLS* - opt. control 
decoupling 
pole placement 
Hybrid systems.+ - passive/active 
Bang-bang 
*Operational. 
+In progress. 
Figure l.- Development of system software for LSST dynamics analysis. 
102 
- Rigid Beam and 
Symmetric Modes 
Tip Deflection (a) O.Ola 
30 45 60 75 90 
Incidence angle, 8, deg. 
(a) Horizontal component. 
-ia 
E 
ti 
15 30 45 60 
Incidence 
angl:, 90 
8, deg. 
(b) Normal component. 
Figure 2.- Variation of solar force components with incidence angle. 
Totally absorbing surface - free-free beam (length R = 100 m). 
Rigid Beam and Symmetric 
3xlo4; tisymmetric Modes 
, 
Tip Deflection(a) O.OlL 
lxlod 
103 
Tip Deflection O.OlR 
15 30 45 60 75 90 
Incidence angle, 0, deg 
Figure 3.- Pitch moment due to solar radiation pressure (completely absorbing 
surface). Effect of individual modes in the system - free-free beam. 
1o-4 
I lo-5 
a 
Y 
ri 
3 lo+ 
lo-7 
0.01 
(b) geosynchronous orbit 
0 - 0; Corresponds to local 
horizontal orientation 
0.1 1.0 
Pitch angle, 8, deg 
2.0 
Figure 4.- Moment due to gravity-gradient force as a function of 
pitch angle (100-m rigid beam). 
104 
(4 
27 
18 lb) 
9 
Design closed loop 
poles. Model does 
not include orbital 
and gravity-gradient 
effects. 
Closed loop poles 
resulting from the 
control law of (a) 
when applied to a 
model which includes 
orbital and gravity- 
gradient effects. 
3 rigid + 3 flexible modes 
Figure 5.- Shift in closed loop poles due to orbital and gravity-gradient 
torques. Square plate in 250-n.-mi. orbit. Overall designed 
response time constant of the system = 8000 sec. 
105 
t4J 
33 
27 
18 
9 
?’ 
8 
L 
l 
I 
. 
-9 
-18 
-27 
-33 
(a) Designed closed loop poles.. 
Model does not include orbi- 
tal and gravity-gradient 
effects. 
(b) Closed loop poles resulting 
from the control law of (a) 
when applied to a model which 
includes orbital and gravity- 
gradient effects. 
. . . 410 
Real axis 
3 rigid + 3 flexible modes 
Figure 6.- Shift in closed loop poles due to orbital and gravity-gradient 
torques. Square plate in 250-n.-mi. orbit. Overall designed 
response time constant of the system = 460 sec. 
106 
2 
m 
L 
z 
rz 
-10.0 . 
-20.0 
Max. Force Amplitudes 
f; - fl - 14.60 N 
f2 = f2 * 8.88 N 
* 
f3 = f3 = 22.50 N 
f* - f 4 4 - 0.087 N 
(There is no appreciable difference in forces between 
model with G.G. and orbital effects and model without 
them. ) 
XI: f (t) * 261.5 N-set 
1 1 
\ ! 
(Non-Dimensionalized Time) 
Figure 7.- Time history of control forces. Square plate. 
107 
L 
,270 (a) Designed closed loop poles. 
Model doesn't include orbital 
and ,gravity-gradient effects. 
180 
(b) Closed loop poles resulting 
from the control law of (a> 
when applied to a model which 
includes orbital and gratity- 
gradient effects. 
90 
3 rigid + 6 flexible modes 
-270 
Figure 8.- Shift in closed loop poles due to orbital and gravity-gradient 
torques. Shallow spherical shell in 250-n.-mi. orbit. Overall 
designed response time constant of the system = 615.0 sec. 
108 
(a) Designed closed loop poles. 
Model does not include orbital 
and gravity-gradient effects. 
m 
(b) Closed loop poles resulting from 
the control law of (a) when 
applied to a model which includes 
srbital and gravity-gradient 
effects. 
.e modes 
Figure 9. -  Shift in  closed loop poles due to orbital and gravity-gradient 
torques. Shallow spherical shell  i n  250-n.-mi. orbit. Overall 
designed response time constant of the system = 400 sec.  
600 
500 
400 
300 
200 
h 
z 
2 100 
8 2; 
s 
: k 
-100 
-200 
-300 
-400 
-500 
PEAKFORCE AMPLITUDES-CONTROLU4WDEVELOPEDWITEOUT 
ORBITAL AND GRAVITY-GRADIENT EFFECTS IN TEEMODEL AND 
TEEN APPLIED TO A MODEL WITH THEM (Shell in orbit). 
Without Orbital With Orbital and 
and G.G. Effects G.G. Effects* 
fl: 516.40 N 
f2: 73.30 N 
f3: 239.50 N 
f4: 117.54 N 
f5: 132.45 N 
fg: 146.82 N 
fl*: 565.63 N 
f2*: 164.76 N 
f3*: 321.11 N 
f4*: 117.07 N 
f5*: 153.84 N 
f$ 431.61 N 
EE fi - 7020.8 N-set EE f? - 19373.0 N-set 
T (- wet) 
(Non-Dimensionalized Time) 
Figure lO.- Time history of control forces. Shallow spherical shell. 
110 


The dynamics and control of large flexible space structures

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