NASA Contractor Report 3350 

NASA 

CR 

3350 

C.1 

Satellite Power System (SPS) 

Antenna Pointing Control 

CONTRACT NASS-33604 

NOVEMBER 1980 
TECH LIBRARY KAFB. NM 

NASA Contractor Report 3350 

Satellite Power System (SPS) 

Antenna Pointing Control 

The University of Tennessee 

Knoxville, Tennessee 

Prepared for 

Marshall Space Flight Center 

under Contract NASS-3 3604 

National Aeronautics 

and Space Administration 

Scientific and Technical 

Information Branch 

1980 
ACKNOWLEDGMENT 

This report documents the work done under contract NAS8-33604 which 

was sponsored by the National Aeronautics and Space Administration (NASA), 

Marshall Space Flight Center (MSFC). The work was performed during the 

period from 1 August 1979 to 29 February 1980. 

Dr. P. Z. Peebles was responsible for the task concerning the pointing 

error! sensing method, while Drs. J. C. Hung and C. W. Lee were joint1.y 

responsible for tasks involving the rigid body pointing control and the 

modeling approaches for the flexible collector. The Contracting Officer's 

Technical Representative at NASA/MSFC was Yr. J. T. Howell. 

ii 

I I 
- - 

Table of Contents 

Section 1. Introduction ..................... 1 

Section 2. Error Sensing Method for Antenna Pointing ...... 3 

2.1 Sensing method ................... 3 

2.2 Received signals .................. 5 

2.3 Subarray signal processing ............. 7 

2.4 Plrray signal processing ............... 10 

2.5 Effect of noise ................... 13 

2.6 Effect of vibration errors ............. 19 

Section 3. Rigid Body Antenna Pointing Control ......... 29 

3.1 The baseline satellite configuration ........ 29 

3.2 Slip-ring friction characteristics ......... 32 

3.3 Estimation of motor parameters ........... 32 

3.4 Effect of centrifugal force induced torque ..... 33 

3.5 Effect of variation in antenna's moment of inertia . 41 

3.6 Pointing control system modeling .......... 44 

3.7 Pointing control system design ........... 51 

3.3 Effect of noise in sensed pointing error signal ... 67 

3.9 Effect of collector vibration ............ 67 

Section 4. Approaches for Modeling the Hexible Collector Body . . 69 

Section 5. Conclusion ....................... 75 

References ............................ 77 

Appendices 

A. CSFIP for the first rotary joint control system .... 73 

B. CSNP for the second rotary joint control system ... 79 

iii 
List of Figures 

Figure 1. Subarrays applicable to gimbal servo error generation . . . 4 

Figure 2. Antenna-centered coordinates . . . . . . . , . . . . . . . 6 

Figure 3. Subarray locations . , . . . . . . . . . . . , . . . . . . 8 

Figure 4. Signal processor at: (a) a typical subarray, and (b) the array 

Figure 5. 

Figure 6. 

Figure 7. 

Figure 3. 

Figure 9. 

Figure 10. 

Figure 11. 

Figure 12. 

processor......................... 

Array pattern . . . . . . . . . . . . . . . . . . . . . . . 

Sumpattern........................ 

Difference pattern . . . . . . . . . . . . . . . . . . . . 

Slope of difference pattern and gain loss of sum pattern on 

boresight......................... 

Normalized difference pattern . . . . . . . . . . . . . . . 

System noise powers . . . . . , . . . . . . . . . . . . . . 

Displacement on the antenna over a line through its center 

(origin).......................... 

Error 0 due to vibration as a percentage on array basic 

beamwidyh @B. . . . . . . . . . . . . . . . . . . . . . . . 

of 

. . . 

Figure 13. Array basic beamwidth o as a function of the number M 

subarrays in a linear a P ray . . . . . . . . . . . . . 

Figure 14. Error 0 due to vibration as a percentage of array bas 

beamwidyh oB , . . . . . . . . . . . , . , . . . . . 

ic 

Figure 15. 

Figure 16. 

Figure 17. 

Figure 13. 

Figure 19. 

Figure 20. 

Figure 21. 

Signal-to-noise ratio loss due to vibration with E = l.Om 

inallcurves....................... 

Baseline configuration for SPS satellite . . , . . . . . . 

Slip-ring friction characteristics exhibits shifting 

hysteresis . . . . . . . . , . . . . . . . . . . . . . . . 

Analog circuits for the nonlinear slip-ring friction . . . 

Digital simulation for slip-ring friction . . . . . . . . . 

Centrifugal force induced torque . . . . , . . . . . . . . 

Variation in antenna's moment of inertia about ZY axis . . 

9 

12 

14 

15 

16 

17 

20 

21 

24 

25 

26 

28 

30 

33 

34 

35 

40 

42 

iv 
Figure 22. Two-axis antenna pointing control ............ 45 

Figure 23. Block diagram for controlled plants ........... 48 

Figure 24. Moment of inertia about XA axis ............. 50 

Figure 25. Antenna pointing control for the first rotary axis ... 53 

Figure 26. The root-square locus plot ............... 55 

Figure 27. Compensation schemes .................. 53 

Figure 28. Closed-loop control system simulation diagram ...... 59 

Figure 29. Step response of the nominal system ........... 60 

Figure 30. System with constant drag torque ............ 63 

Figure 31. System with nonlinear friction torque .......... 65 

Figure 32. Transformation to an equivalent orthotropic plate .... 70 

Figure 33. definition of the geometry of a grid .......... 71 

V 
1. INTRODUCTION 

This report documents the results of a six-month study on the pointing 

control of a microwave antenna of the Satellite Power System (SPS). The study, 

performed under NASA contract NAS3-33604, may be considered as the continuation 

of an early effort on the accuracy analysis of pointing control system of 

SPS, which was conducted under NASA contract NASH-33065. Primary goals of 

the study encompass three areas, namely, the SPS antenna pointing error sensing 

method, a rigid body pointing control design study, and approaches for modeling 

the flexible body characteristics of the solar collector. 

Accuracy requirements for the antenna pointing control consist of a 

mechanical pointing control accuracy of three (3) arc-minutes and an electronic 

phased array pointing accuracy of three (3) arc-seconds. Results of the 

study, based on the factors considered in current analysis, show that the 

three arc-minute overall pointing control accuracy can be achieved in practice. 

Section 2 of this report deals with the antenna pointing error sensing 

method and its performance. Among the topics discussed are the pointing error 

sensing method, received signals, subarray signal processing, array signal 

processing, effect of noise, and effect of vibrations. 

Section 3 presents the details of a mechanical pointing control design 

for the antenna. Included in the section are the baseline satellite configuration and a set of baseline parameters, an estimation of motor rating 

and motor parameters, slip-ring friction characteristics, the effect of 

centrifugal force-induced torque on antenna due to gimbal rotation, the 

effect of variation of antenna moment of inertia due to its changing attitude, 

control system modeling, control system design, system simulation, and the 

effect of noise in sensed pointing error signal. Also included in the section 

is the effect of collector vibration on the accuracy of pointing control. 
Shortly after the start of the study, it was found that the required 

effort for an analytical modeling of the flexible structure of the solar 

collector was immense, way beyond the planned-effort.. As a result, work 

in this area was redirected into two directions. The first direction is 

to explore the approaches for obtaining an analytic flexible model, and 

the second direction is to estimate the effect of the collector's flexible 

body vibration on the antenna pointing control accuracy. Section 3 is 

devoted to the discussion of the approaches for the flexible collector 

modeling. 

A list of references and two appendices are included for the convenience of readers. 
3 

2. ERROR SENSING METHOD FOR ANTENNA POINTING 

In the final report of an earlier effort [l] a method was proposed 

for developing angle error signals to be used by angle servomechanisms 

to point the antenna toward the ground station. In this effort, the 

proposed method has been shown to be feasible. In addition, a normalization 

technique, similar to that used in angle tracking radar, has been added 

that produces error signals that are independent of received signal magnitude. 

Finally, some results related to affects of vibration and noise have been 

derived. 

2.1 Sensing Method ---_ 

The basic concept is illustrated in Figure 1. Two intersecting linear 

arrays formed from subarrays of the main array are used to develop error 

signals. The array normal to axis 1 is used to develop an error signal 

proportional to the error that the antenna pointing direction makes relative 

to the desired direction (toward the earth site) in a plane for which axis 

1 is normal (error about axis 1). The other array generates a signal 

proportional to the (space) orthogonal angle in the plane for which axis 

2 is normal. The two error signals are used by angle servomechanisms to 

turn the antenna about axes 1 and 2 to cause the errors to approach zero. 

Each array is presumed to have 2f4 + 1 = M subarrays with the common 

subarray at the antenna's center serving as the "reference" for the two 

arrays. In general, the two arrays do not have to have equal numbers of 

subarrays; however, since accuracy requirements are likely to be the same 

in both coordinates, there seems to be no reason to not make the two 

identical, as is assumed in following discussions. 
4 

Axis 2 

Axis 1 

Subarrays applicable 

to coordinate 2 

rrays applicable 

to coordinate 1 

Figure 1. Subarrays applicable to gimbal servo error generation. 
2.2 Received Signals 

We define antenna-centered coordinates as shown in Figure 2 where 

x and y are axes in the plane of the antenna's face. Axis z is the 

antenna's true pointing direction. A wave arriving from the earth site 

makes an angle e from the z (boresight axis) direction; it has an 

"orientation" defined by an angle 0 in the xy plane. Using these definitions 

we find the signal received by a typical subarray located along the x 

direqtion. Only the x axis array is analyzed since the other behaves in 

a similar manner. 

The error angles sensed by the system in Figure 2 are ex (in the xz 

plane) and ey (in the yz plane). The various angles are related by 

tan(ax) = tan(e) cos($) 

tan(ey) = tan(e) sin(@) 

from geometry. 

A typical subarray pattern in the SPS system will 

illumination function, have a rectangu lar shape (sidehave a uniform 

lengths X and Y) and 

a one-way voltage pattern, denoted G(e,g) by 

(1) 

(2) 

G(e,4) = Sa [y sin(e)cos($)]Sa[F sin(a)s (3) 

where 

Sa(<) = si:(S) (4) 

and x is wavelenght. If we now recognize that lel<<l for any reasonable 

SPS system, then lexl<<l and leyl<<l, so (1) through (3) give 

ex % e cos(o) (5) 

ey % 0 sin 4 (6) 
I \ 

Figure 2. Antenna-centered coordinates, 
7 

G(e,d % SaC(nX/x)e cos(s)l SaC(W~)e sin(o)] 

= Sa[sXax/A]Sa[aYey/h] = G(ex.ey) (7) 

Now consider the array of subarrays along the x axis as shown in 

Figure 3. Subarray locations along x are nX, -Na<N, and they have dis- 

- - 

placement in the z direction of zn and tilt angles e,, (tilt in the other 

coordinate is eyn). If vn denotes the voltage produced by the nth subarray by a wave having error angles 8, and e 

Y' 

then 

2lTz 

V = A0 G(ex+exn, a +e 

n 

y yn)cos[~pilot+ F sin(ex)+ -$ cos(ax)] (3) 

where A0 is the voltage produced when the subarray points toward the 

pilot source without tilt error and $pilot is the phase of the arriving 

pilot signal at the reference (n=O) subarray. Since jexl<<l, 

V n % A0 Gbx+exn, 8 y yn)cos[+pilot+ +e 

2mXex 27rz 

t-- 

x x" I. 

2.3 Subarray Signal Processing 

We next show that desirable error sensing occurs if the signal processing of Figure 4(a) is available at each subarray in the line array. 

Straightforward analysis gives 

e (4 

AO 

I 

27mXe 2*(zn-zo) 

= 2 G(ex+gxn, eyfeyn) EcosC x ’ + x + neql 

n 

2mXe 2T( Zn-ZO) 

tcos[ x x + x neql I. 

(9) 

(10) 
8 

3 

_. . 

WAVE 

vrREGTIOM 

/ 

rt 

/ 

0 

0 

0 

/ -1 

8 

wxn ’ 

T ------ 

I - . . f . . . n . . . NX 8 6 .-. _ 

-NNX 0 

I 

a % 

Figure 3. Subarray locations. 
9 

p c-1 c+1 

-w e -N+I e ‘U p 

0 

c c+’ e c+> 

N-l u 

. . . . . l 

. . l . . . 

z 

I I 

L: 

I 

e 

C-) c+> 

e 

Figure 4. Signal processor at : (a) a typical subarray, 

and (b) the array processor. 
10 

The signals en t-1 and en(') are assumed conveyed to a central processor for 

the array, as shown in Figure 4(b). 

2.4 Array Siqnal Processing - r?lo Errors 

Suppose no tilt or displacement errors are present. Then exn=O, 

eYn 

=0, and z,=O for all n. Now (10) becomes 

x, ey) {cos[n(F ex+eq)] 

+cos[n(zF 0 -a )]I. 

x 9 

The signals e (+) and e(-) in Figure 4(b) become 

,(+I 

AO sin[M(2"X 

I= 2 Gbx, By) C 

2 h ex+e )I 

,(-1 sin[$+$ ex+eq)] 

M 27x 

t 

sin[$T ax-e >] 

- sin[$G+ ex-eq)]' 

where 

M=2Pl+l 

and we have used the known series 

! 

sin[T 2Wl Bl 

cos(n6) = 

n=-N sin(a/2) * 

(11) 

(12) 

(13) 

(14) 
11 

Next, define a subarray bearMdth (in rad) between -392dB points 

as es,b, given by 

0 sub : x/x, 

and 

B 

9 

:e 0 /2il. sub q 

Then (12 becomes 

sin++- 

sub 

bx+FqH 

1. 

- sin[+ 

sub 

(?/%)I 

(15) 

(16) 

(17) 

Now if we define an "array pattern" Ga(ax) by 

sin[F ex] 

M 7T El 

sin( 

esub 

x) 

Ga(ex) = 

sub 

ll e % y 

M sin(+) (' e:";x) 

for IeXI'<eSUb/n (18) 

sub 

then 

e(+) A II 

,(-1 

I=+ G(e,, ey) G,(a,+sq) 2 Gab,-dq)L 

A sketch of Ga(sx) is shown in Figure 5 for M=3,5, and 9. The function 

is symmetric about z=O and + TT and about z=3/2. Thus, there is another 

peak in the function at z=fsr. For ?4 l/x large Ga(gx) approaches a sin(x 

(19) 
12 

GJ 8,) = 

S/N(M t) 

M SIN(t) 

Figure 5. Array pattern. 
13 

function for small magnitudes of x. The responses (peaks) at +n never 

show up in the response (17), however, because G(ex, ey)=O for z=*Xax/I = 

Ire /e 

x sub = TT from (7). 

Next, define array sum and d ifference patterns by 

G,(e,) 0 ; [PJ,(e,+eg) + Ga(ex-~q)l 

G&e,) 4 i [G, ( ex+cq ) - Ga(ex-~q)l, 

respectively. Equations of (19) become 

(20) 

(21) 

,(+I = A0 M G(ex, ey) G,bx) 

,(-I = A0 M G(ex, ey) G,(ex). 1 

I 

(22) 

(23) 

The response ex in Figure 4(b) becomes 

eX 

= e(-)/e(+) = GA(ex)/Gz(ex). (24) 

The behaviors of the sum, difference, and "normalized difference 

pattern" GA(Rx)/Gz(ex) are shown in Figures 6-9. In these figures we 

define an "array basic beamwidth" by 

4 eB - esub/M = x/MX. (25) 

Figure 6 shows that the sum pattern can be of the usual single-peak form 

or double-peaked. The transition takes place when es/es = 0.6625. Figure 

7 shows typical difference channel responses with Figure 8 showing that the 

slope of the response at the origin is maximum when s/e 

q 3 

is about 0.66. 

Figure 8 also shows the loss in gain of the sum channel as a function of 
14 

I.0 

0.8 

Figure 6. Sum pattern. 
15 

Figure 7. Difference pattern, 
16 

0 0.2 0.4 0.6 0.8 1.0 

xl* /OB 

0 

-16 

-20 

Figure 8. Slope of difference pattern and gain loss 

of sum pattern on boresight. 
17 

-I 

-2 

2 

I 

‘li 

Figure 9. Normalized difference pattern. 
.----,-. . .-.._-. 

18 

S /e 

q B' 

Finally, Figure 9 shows the normalized difference pattern. From 

these data, which apply for M larger than about 9, it is clear that Fq/6B 

near 0.5 should be a good system choice based on small sum channel gain 

loss and beamwidth broadening, large error slope, and linearity of the 

normalized difference pattern. 

For Tq/eB = 0.5 the slope of the difference pattern on boresight is 

d G,bx) 1 

slope= de -4 -4 M E-Z-. 

T 8 

X B T 8 

ex=O sub 

The slope of the normalized difference pattern on boresight is 

d[: 'lope = d 8, 

Gn(ex), 1 -2 -2 M 

G,(ex) 

=- =-* 

ex=O 'B 'sub 

(26) 

(27) 

2.5 Effects of Noise 

Assume each subarray output contains a bandpass gaussian noise N,(t). 

Such noises can be described by [2] 

r4,(t) = Nan(t) C”[wpilot tl - $,,.,(t) Sin[wpilot tl (28) 

where !jan(t) and Nbn(t) are independent, gaussian random variables for any 

-t and a given value of n. They are also independent between subarrays 

(different values of n). Furthermore E[Nn2(t)] = E[Nan2(t)] = E[Nbn2(t)] i oN2 

is the 

the ful 

noise power (assumed 

Straightforward ana 

1 noise bandwidth, g 

the same at each subarray). 

ysis of Figure 4(a), assuming the LPF's pass 

ves the noises present in e (+I 

n 

and e(-), 

n 

denoted Nn(+) and N,(-I, respectively. 

N,(+)(t) = [rj,,(t) cos(eT) + rlbn(t) sin( cos(neq) (29) 
N,(-)(t) = [Nan(t) sin(eT) - Nbn(t) cos(.eT)] sin(neq) 

19 

(30) 

Here $pilot(t) = Wpilot t + eT, where aT iS an arbitrary pilot source 

phase angle. 

After summing the various noises as implied in Figure 4(b) the 

noises on the signals e (4 and e(+) , denoted N(-)(t) and N(+)(t), respectively, 

are found. The powers in these noises follow taking the mean-squared 

values: 

24 

UC = E[d+)(t)+] = u; [iu; 2 E[{N %)12] = u; [; 

(31) 

(32) 

Figure 10 plots (31) and (32) versus $/eB = M eq/2n. Clearly, for squint 

angles of interest (B 

9 

= eB/2) we have oz2= oA2 so output noise powers 

are equal: 

2 2 

2 = u 

A 

= M oN2/2, G,/eB = 0.5. (33) 

2.6 Effect of Vibration (Displacement) Errors 

We assume that there is symmetry in the vibration of the overall 

array so that the geometry of Figure 11 applies. Since subarray positions 

are 

then 

'n = n X, -N <n< N, (34) 

'n = z(x,) = AZ cos[ (2K-1) TT X n/D]. (35) 

For small displacements AZ << D, as is the case in practice, it is readily 

shown that the vibration induced tilt is 

9 = -Az(zK-1)~ 

xn D sin[(2K-1) TF X n/D]. (36) 
20 

2 

I 

0 

Figure 10. System noise powers. 
21 

K= ~t0RA7toN MODE = I,2, --- 

Figure 11. Displacement on the antenna over a line 

through its center (origin), 
22 

Numerically, if AZ = 1 m, D = 1010 m, X = 11.64 m, 

then jexnj % 0.89 esub, so ti It errors may be near ly as large as the 

and x = 0.1224 m, 

subarray's beamwidth. 

lrle evaluate vibration in 

when 

AZ = i-? COS(w"t) 

the x direction when 

eY = 0, eyn = 0, and 

(37) 

where wV/2a is the frequency of the vibrations. From (lo), using (7), 

the signals e (+I and e(-) in Figure 4(b) are 

,(+I 

Ao N 

,(-1 

1 = T 1 Sa[aX(ex+exn)/x] 

n=- N 

2i7Xe 2dz,-zo) 

- {cos[n(++ eq) + x ] 

237Xe, 

+ cos[n(T - Bq) + 

2.rr(zn-zo) 

A 'I. (38) 

For relatively small values of ex the system's response will be 

eX 

= ,G),,(+). However, the true response without vibration will very 

nearly equal the slope when 8, = D times a,, so that the "error" in radians 

produced by vibration, denoted eV, will be approximately 

,(-I l-1 

eV = 

- [.$-JT-) + ex] = - [--- 2;x > + ax'5 (39) 

after (27) is used. If normalized to the array basic beamwidth eB we have 

eV ,(-1 ex 

-=-- -- 

eB 2 e(+) 'B 

(40) 

while (38) becomes (for s = 0.5 3B) 

9 
23 

27~ ex 

a{cos[n(r +$+ 

2dz,-zo) 

B 

x 1 

2~ ex 

+ cosL+j --;, + 

2dzn-zo) 

eB x II. 

An examination of (41) shows that e (-) = 0 if 8 = 0. Thus, for X 

no antenna pointing error (ex - = 0) -, the oresence of vibration produces - 

error, regardless of the vibration mode, so long as it is symmetric. -- -- --- 

(41) 

no - 

To study the error 5,,/e8 when ex # 0, (40) was computed on a digital 

computer for various values of 11, K, and ox when peak vibration displacement in each case is 1.0 m. Figure 12 illustrates curves of ev as 

a percentage of array basic beamwidth eB versus the number M of subarrays 

in the array. Curves are shown for pointing error magnitudes /ex/eB/ of 

0.05 and 0.1 for two vibration modes (K=1,2). The curves assume X=11.64 m, 

D=1012.68 m, and X=0.1224 m. It is clear that some combinationsof vibration 

mode and pointing error give rise to large errors due to vibration. 

For example, with M=7 subarrays in the linear array and mode 2 vibration (K=Z), when pointing error ex is 5% of the array's basic beamwidth 

eB, the maximum error due to vibration of peak amplitude 1.0 m is 7.32% of 

eB. Figure 13 plots eB versus M and gives eB = 5.16myn for M=7. Thus, 

maximum error due to vibration is 0.0732 (5.16) = 0.38nfn. Since the 

pointing error is 0.05 (5.16) = 0.268m?n. we find the vibration errcr even 

exceeds the pointing error to be corrected by the servo system. The vibration error cycles with time, and, for the example taken here, is illustrated 

in Figure 14. 
24 

K=2 

K= I 

t 

M 

I I I I I I 

Ii30 2 + 6 8 IO 12 f 

Figure 12. Error o,, due to vibration as a percentage 

on array basic beamwidth GB. 
25 

. I I 

4= 8 

M 

12 16 

Figure 13. Array basic beamwidth OB as a function of the 

number M of subarrays in a linear array. 
26 

4 TT 277 

*Ve 

Figure 14. Error ov due to vibration as a percentage 

of array basic beamwidth QB. 

I .- 
27 

Finally, we note that vibration will have an affect on signal-to-noise 

ratio of the composite signals e (+I and e(-). The affect on e(+) is 

of great concern since signal-to-noise ratio for this signal should be 

maintained above 10 to 100 for good performance of practical normalizing 

circuits. Figure 15 shows the maximum loss in this signal-to-noise ratio 

for various M, K, and ex/eg when E= 1.0 m. 
28 

10 

I 

id’ 

-a 

JO 

Figure 15. Signal-to-noise ratio loss due to vibration 

with E = l.Om in all curves. 
3. RIGID BODY ANTENNA POINTING CONTROL 

In. this section, an antenna pointing control system is modeled and 

then designed assuming that both the solar collector and antenna are rigid 

bodies. The purpose of this design is two-fold. First, the design will 

generate technical data such as motor requirements, system response time, 

effect of noise and system parameter changes, and the stiffness of the 

control loop. I Second, by examining time constant of the control system 

and the mode frequencies of the flexible collector, the effect of the 

collector vibration mode on antenna pointing accuracy can be estimated. 

The goal of the mechanical pointing control is to maintain a mechanical 

pointing accuracy of three arc-minutes. 

3.1 The Baseline Satellite Configuration 

Figure 16 depicts the baseline satellite configuration. The dimension 

of the solar energy collector and the position of the microwave antenna 

are shown in Figure 16(a). The size and the general shape of the yokeantenna combination is shown in Figure 16(b). Figure 16(c) is included 

to show the size of the rotary joint connecting the yoke to the solar 

collector. 

The mechanical control of the antenna pointing is done by rotational 

motions about two axes. One rotational motion is about the Zy axis which 

is along the longitudinal center line of the collector as shown in Figure 

16(a). A second rotational motion is about the XA axis which is along 

the direction of hinges joining the antenna and the yoke as shown in 

Figure 16(b). 

Two coordinate frames are used in the sequel, one is the antenna 

frame (XA,YA,ZA) and the other is the yoke frame (Xy,Yy,Zy). 

29 
(a) SPS satellite 

--I I-35OM JOINT 

” 

(c) The rotary joint 

"A 

(b) Antenna and yoke 

Figure 16. Baseline configuration for SPS satellite. 
31 

The physical parameters of the satellite are collected and listed 

here. For the collector: 

Mass: MC = 16~10~ kg 

Dimension = R = 17,000 m 

a = 4,200 m 

b = 600 m 

moments of inertia = 

(Vzy = $- (a2+b2) = 19.5x1012 kg-m2 

('c)x, = $ ML2 = 1.54x1o15 kg-m2 

For the combination of antenna and yoke: 

Mass = MA = 13~10~ kg 

Diameter: D = 1,100 m 

Moments of inertia: 

('A)Z, z ; MR2 = & MD2 = .983x1012 kg-m2 

(1,&o 2 ; MR2 = .983x 1 

Other phys ical data include: 

slip-ring diameter = 35 

motor shaft diameter = 

012 kg-m2 

Om 

.l m 

g$an-train ratio CL = 

Antenna speed = lo-4 

motor speed 

The gear-train is used to connect the motor shaft to the rotary joint. Since 

the rotary joint has a diameter of 350 m and the motor diameter is .l m, 

a gear-train ratio of 10m4 is reasonable. 
32 

3.2 Slip-ring Friction Characteristics 

Conduction of electric power from the solar collector to the antenna 

yoke and from the yoke to the antenna is done by using slip-rings.- The 

friction torque at each rotary joint is mostly due to the slip-ring friction 

torque which is a nonlinear function of speed of rotation. The nonlinear 

slip-ring friction characteristics, as provided by YASA/MSFC, are shown 

in Figure 17. In the figure e represents the relative angular position 

between two sides of the rotary joint. The arrowheads indicate directions 

of motion. Notice that the friction torque has the saturation values of 

C Tmax = + lo6 N-M. In the region between the saturation values, the 

friction troque depends on e in a way resembling the compliance torque. 

The compliance is the slope KF which is -lo6 N-M per degree. The nonlinear 

friction torque exhibits the nature of shifting hysteresis. 

Figure 18 gives two analog electronic circuits which can be used to 

simulate the above friction characteristics on an analog computer. For 

system simulation using a digital computer, however, neither one of these 

circuits are convenient to use. To provide a convenient way for later 

digital simulation, a digital simulation flow-diagram is developed as 

shown in Figure 19. 

3.3 Estimation of Motor Parameters 

It is assumed that d-c motors are used as torques for antenna pointing 

control. The rating for the d-c motors are estimated here. Assume a maximum 

perturbation angle for the antenna to be [3] 

JAelmax = 1 degree = .0175 radian 

The moment of inertia of the yoke-antenna load is 

IA = .983x1012 kg-m2 
t 

l/o6 N-m 

Slop 

- -- 

lo6 N-m 

/ 

Figure 17. Slip-ring friction characteristics exhibits shifting hysteresis. 

W 

W 
34 

-7;nax 

l I-P- 

-Sax 

(4 

+T 

max 

b) 

Figure 18. Analog circuits for the nonlinear slip-ring friction. 
35 

OUTPUT TK 

TX+ = TK 

Figure 19. Digital simulation for slip-ring friction. 
36 

Given a gear-train ratio of a=10 -4 , the reflected maximum angle perturbation 

and the moment of inertia of the load at the motor shaft is 

Ii 

2 =a1 A = 9830 kg-m2 

lAe'/max = $ 1~8 Imax = 175 radians 

(see discussion in Section 3.6). 

The maximum slip-ring friction torque is 

lrFlmax = lo6 N-M at the ring 

= axlO = 100 N-M at the motor shaft 

The initial motor torque for a maximum step perturbation is 

Iho1 

max 

= IA l’lmax 

= I/i lAe'lmax 

T2 

= 1 .72x1@ N-f., 

T2 

where T is the system time constant. Therefore the total maximum torque is 

ITI max = ITFlmax + ITOlmax = 100 + l*7:,T06 

An estimate of motor power P is now given 

P = Average of [A;(t) T(t)] 

< l”Imax ITlmax 

= lAe’lmax l’lmax 

T 

= y (lo~+~7.;;106) 

= 1.75~10~ T2+3x10S watts 

T3 

(42) 

(43) 
= 23.46 T2+4.021x105Hp 

T3 

Under the nominal condition the steady-state motor power is to overcome 

the slip-ring friction torque. This power is 

pO 

= +!!!fi HP (45) 

37 

(44) 

A list of P and T versus T is shown in Table 1. A baseline rating is 

chosen to be: 

P = 1681 watts = 2.253 HP 

T = 578 N-M 

T= 60 seconds 

Table 1. Motor Ratings 

T (set) -r (N-M) P(!?latts) 

10 17293 301750 

30 2010 11694 

60 578 1681 

100 272 475 

300 119 69 

600 105 31 

1200 101 15 

1800 100 10 

Adopting the baseline motor rating, motor parameters are now estimated. 

Take the effective d-c voltage to be 

"dc = 120 v. 

The average motor amature current is then 
38 

m= 

120 14 A 

The motor torque constant is 

578 

KT=t 14 

= -= 41.3 !q 

Using the MKS system of units, the motor emf constant equals the motor 

torque constant in value, giving 

KE = 41.3 V-set 

Assuming a 5% motor power loss, the motor armature resistance is computed 

by 

Ra 

.05P 

= - = .43n 

I 2 

a 

The result is summarized in Table 2 for the ease of future reference. 

Table 2. Flotor Parameters 

Rated Power 1681 w 

Rated Voltage 120 v 

Torque constant, KT 41.3 y 

Emf constant, KE 41.3 V-set 

Armature resistance, Ra .43 sl 

3.4 Effect of Centrifugal Force Induced Torque 

Under the nominal condition the yoke-antenna assembly is rotating 

about the Zy axis of the SPS satellite at nearly constant speed. The 

rotation generates on each mass element of the antenna a centrifugal 

force which is directed perpendicularly to and away from the Zy axis. 

This force produces a torque about the XA axis of the antenna, which may 
39 

be viewed as a load disturbance torque for the pointing control system. 

This torque is analyzed with the help of Figure 20. 

Let &I be the incremental strip of antenna mass as shown in Figure 

20. Let "E be the nominal angular velocity of the yoke-antenna assembly 

about the Zy axis, which is very close to -the Earth's rate of rotation. 

The incremental torque about the XA axis due to the centrifugal force on 

A/d i s 

ATC 

= (AM)(acceleration)(torque arm) 

= (oXA AZ,)(i$ ZA sine)(ZA case) (46) 

where u is the area mass density of the antenna, which is assumed constant. 

Integrating over the entire antenna plane gives centirfugal torque as 

Tc = J dT C =4 J R 22 CT "E ZA XA sine COST dZA 

area 0 

rR - 

= 2an 2 E sin2e 

J 

JR2-Z2 72 dZ A-A A 

0 

where R is the radius of the antenna. After the integration the total 

centrifugal torque is obtained as 

'c 8 = E u ai R4 sin20 

Maximum torque occurs when e=45". The Earth's rotation rate is 

a, = 15 deg/hour = 7.27~10~~ rad/sec 

us ing the base1 ine parameter values, 4 = 550 m and 

u = 

tiA = 13x106 

2 = 13.68 kg/m2 

7lR ax5502 

(47) 
LAntenna disc 1 yY 

(a) Front view (b) Side view 

Figure 20. Centrifugal force induced torque. 
Therefore the maximum torque is 

lrclmax = 2598.2 N-m 

41 

(48) 

Comparing this torque to the maximum slip-ring friction torque, 

which is 106N-m, the former is less than .3% of the latter. The effect 

of the centrifugal force induced torque therefore needs no special 

attention, but may be lumped into the slip-ring friction torque as a 

perturbation. It is expected that such a small perturbation will not 

cause problems for the antenna pointing control. Computer simulation 

will be used to verify this fact. 

3.5 Effect of Variation in Antenna's Moment of Inerita -~ 

Since the antenna disc may rotate about the X A axis as indicated in 

Figure 21, its angular position e about this axis may change. As a result, 

the antenna's moment of inertia about the Zy axis may change. It is 

desired that such change will not have adverse effects on the accuracy 

of the pointing control. 

Let the moment of inertia matrix of the antenna expressed along the 

antenna frame (XA,YA,ZA) be 

I = i 

IX 

0 

0 

L 

0 

I 

Y 

0 

- 

0 

0 

IZ 

- 

Referring to Figure 21, the antenna frame and the yoke frame are related 

by an angle 8. The antenna's moment of inertia expressed along the yoke 

frame (X,,YY,Z,) is 
- To collector 

/ 

I 

Figure 21. Variation in antenna's moment of inertia about Zy axis. 
43 1 1 0 0 - ’r 7 

I’ -L 0 ce se 

i 

0 Ix 0 IY O 0 0 1 C8 0-se 0 1 

LO 

-se C0 0 O 5 

- -I L O se ce- 

- 

Ix 0 

= 0 Iy ce2+Iz se* 

1 0 

t 

-Iy cese+Izcese 

0 7 

-Iycose+IZsece 

Iyse2+Iz ce2 

(4p) 

where ce = case and se = sine. In particular, the moment of inertia about 

the Zy axis is 

I; = Iy se2+Iz ce* (50) 

Note 

12 Iy = T MAR = max, ant. mom. of inertia 

Iz = &R* = min. ant. mom. of inertia 

Taking the derivative of (50) with respect to 8 

J& 1; = 2 se ce I 

Y 

- 2 se ce I, = (I,,-Iz)sin 29 

Therefore the variation of the moment of inertia normalized with respect 

to I, is 

AI Iz 

-= 

5 

(4 - 1) sin 28 A8 = sin 29 ae (51) 

2 

For e=45", (51) is at its maximum. For As=1 degree = .0175 radian, the 
44 

specified attitude control accuracy for the SPS satellite, 

r - AI; 

-7 

L - max 

= .0175 = 1.75% (52) 

A plant parameter variation shown in (52) is too small to cause adverse 

effects in pointing control, and can be ignored in the design of pointing 

control. However, computer simulation will be performed to confirm this 

observation. 

3.6 Pointing Control System Nodeling 

An all-attitude pointing control capability of the SPS antenna is made 

possible by allowing two degrees of rotational freedom for the antenna disc 

as shown in Figure 22. The first rotary joint allows the rotation of the 

antenna-yoke assembly with respect to the collector about the ZY axis, while 

the second rotary joint gives rotational freedom for the antenna with 

respect to the yoke about the XA axis. It is assumed that the coupling of 

the antenna's motion about these two axis is negligible. This is due to 

the fact that the motion of the antenna is very slow and that the angular 

perturbations about the two axes are very small. Therefore the dynamics of 

the pointing control system for each rotary joint may be separately modeled. 

The First Rotary Joint Let eA and ec be the absolute angular positions 

of the antenna-yoke assembly and the collector, respectively, about the ZY 

axis. Their dynamics are given by 

IA s* 9A = T' - -cF (53) 

Ic s* ec = --c ‘+T F (54) 

I 
Rotation 

about 

Zy axis 

Antenna and yoke 

Figure 22. Two-axis antenna pointing control. P 

ul 
46 

where IA and I, are moments of inertia of the antenna-yoke assembly and 

the collector, respectively; T' isthe control torque originated from the 

motor; and 'F is the frictional torque of the slip-ring. Rotational 

compliance and other kinds of friction are assumed negligible. Adding 

(53) and (54) gives 

IAeA+Icec = 0 

Define the ratio of moments of inertia 

IA 

B =T 

Then (55) gives 

ei = - 6eA (57) 

The relationship among the generated torque T, applied voltage V, and 

the shaft speed se for a separately excited d-c motor is 

T = 

V-KEse 

KT Ra (58) 

whel-e K T, KE and Ra are defined in Section 3.3. Noting the presence of 

r 

a gear-train with a ratio CI, the following relat ionships ex ist. 

1 8 = F (eA-e,) = t (1+6)eA (59) 

T = a T’ (60) 

Ignore the nonlinear friction torque ~~ in (53) for the time being. Then 

(55) 

(56) 

IA s2 eA = ~1 

Eliminating 9 ,T and T' from (58) 

KT 

R v = [("*IA 

a 

(61) 

(59), (60).and (61), yield 

S2 

KTKE 9A 

+ R (l+B)sl a 

a 

(62) 
Let 

I/i 

2 =a1 A 

which represents the reflected moment of inertia of the antenna-yoke 

assembly seen at the motor shaft. The transfer function of the first 

rotary system including the motor is therefore 

aK 

For later convenience, define 

eA eA = F 

47 

(63) 

(64) 

(65) 

which is the reflected absolute attitute of the antenna-yoke as seen by 

the motor. Using (64) in (65) gives 

KT 

e;\ 

I 

GP=T = 

'A Ra 

KTKE 

Sb+IiRa (l+dl (66) 

Eq. (66) is the transfer function of the plant at the first rotary joint. 

Notice that the effect of the reaction motion of the finite mass collector 

is represented by B in the equation. For an infinite mass collector, B=O. 

A block diagram for this plant is given in Figure 23(a). 

In sections 3.1 and 3.3, values of various parameters have been 

obtained as: 

IA = 9830 kg-m* 

KT = 41.3 y 

KE = 41.3 V-s 

Ra = .43 s-2 

a = 10-4 
48 

c 

Ia 7' 1 

KT 

c 

. 

Ii s* 

(a) First rotary joint 

1 

(b) Second rotary joint 

Figure 23. Block diagram for controlled plants. 
49 

Quantity B as defined by (56) has a value of 

B = .983x10'* = .o5 

19.5x107* 

With numerical values inserted, (66) becomes 

G =- "A = .Ol 

P v s(s+.424) (67) 

In this model the nonlinear friction was not included. The effect of 

the nonlinear friction will be treated as a torque disturbance in the control 

system design. 

The Second Rotary Joint For the control about the XA axis at the 

second rotary joint, the collector's moment of inertia may be assumed 

infinite. This is demonstrated as follows. Referring to Figure 24 and 

using the baseline data given in Section 3.1, moments of inertia about 

XA axis are: 

For the antenna: 

('A)X A 

= ;MAR2 

For the collector 

(VX A=5 c 

' M ,* = ;(l.231MA)(31R)* 

P-3 

= 394.3 MARL 

= 1577 (IA)x 

A 

Thus, a rotation of antenna about XA by 1" causes a rotation of collector 

about XA less than 3 arc-seconds. For a 3 arc-minute mechanical pointing 

accuracy, the assumption of infinite moment of inertia contributes only 

.ll arc-second error. 
Figure 24. Moment of inertia about XA axis. 
The following are further assumed: (7) the same type of d-c motor 

will be used at the second rotary joint as the type used at the first 

rotary joint. (2) Same gear-ratio. (3) Mass of yoke is small as compared 

to that of antenna. Under this condition the transfer function for pointing 

control at the second rotary joint can be obtained from (66) by setting s=O, 

yeilding 

(68) 

where eA is the reflected antenna angle about 

the motor. The actual antenna angle is 

@A = a$A, 

lrlith numerical values inserted, (68) gives 

a;\ 

Go = v = s(si”;04) . 

A block diagram of the plant representing the 

in Figure 23(b). 

3.7 Pointing Control System Design 

Design Background 

Under the assumption of negligib e dynamical coup ling between 

the two axes of rotation, the two associated control channels can be 

separately designed. The need of pointing control is to counter-act the 

effect of external disturbances and changes in plant parameters. External 

disturbances include gravity gradient torque, solar pressure torque, and 

collector motion coupled through the rotary joint. Change in plant parameters include the change in the antenna's moment of inertia due to change 

the XA axis as seen by 

(69) 

(70) 

second rotary axis is given 
52 

in antenna attitude, change in motor armature ressistance caused by thermal 

conditions, aging effects, change of structural shapes and dimensions 

due to temperature changes, and others. Most of the disturbances are constant or very low frequency cyclic. Parameter changes are also very low 

frequency in nature. It is assumed that the frequency range for disturbances and parameter changes is below .OOl hertz (15 minutes per cycle). 

Effects of instrument noise will be ignored because they will be well 

attenuated by the low-pass characteristics of the system. 

Under the nominal condition, the antenna-yoke is rotating at the 

first rotary joint with a constant rate of one.revolution per 24 hours. 

Magnitudes of disturbances at this joint are assumed small so the direction 

of rotation is not reversed. The second rotary joint is stationary under 

the nominal condition. With disturbances added, it may rotate in either 

direction. 

Comparing (67) and (70) shows that they differ only in time constant 

of an amount of 1 ess than 5%. Many well designed control loop for one plant 

should therefore perform well for the other. In view of this, a control 

system design wil 1 be made for (67), the plant for the first rotary joint. 

Figure 25 depicts an antenna pointing control loop for the first rotary 

joint. An ideal gear-train is assumed. The pointing error is sensed 

by the microwave error sensing method presented in Section 2.1. An 

algorithm is used to generate, from the error signal, a command signal 

for pointing control. Three ways of generating command signal have been 

developed previously. [l] For the convenience of design the transfer 

function between error e and command signal may be taken to be unity. 

The present control system design is to determine the block labeled "network 

and amplifier gain". 
Network 

and 

amplifier 

gain 

--L-- 

, 

U 

Plant G 

P 

Antenna 

Motor 

I 

I 

I I Comand -I 

signal 

generation e, the pointing error 

L 

Figure 25. Antenna pointing control for the first rotary axis. 
54 

The Design Except for the 3 arc-minute mechanical pointing accuracy 

requirement, no other system performance specifications were given. 

Linear quadratic optimal control design will be adopted. Since the plant, 

represented by (67), has single-input and single output, frequency domain 

optimization by way of the root-square-locus technique will be used. [4]. 

Because of the constant rate rotation under the nominal conditions, the 

control system for the first rotary joint should be a follow-up system. 

Follow-up design will also be used for the second rotary joint to provide 

flexibility for the possible need of follow-up operations. 

Begin with (67), the plant transfer function* 

G = 

.Ol 

P s(s+.424) ($1 

which has poles at p,=O, p2=0, and p3=-.424. Open-loop poles for the 

root-square locus plot are 

Pl = A p1 e O 

p* =- p* = 0 

p3 = - p3 = - .18 

There are no open-loop finite zeros. The root-square loci are shown in 

Figure 26. Choose the closed-loop root-square poles at 

'1, & Q, = .8 + j 1.48 

9, = -.18-9,-Q, = - 1.78 

Then 

-Q, & -4, = -.8 + _ 

j 1.48 

-9, = 1.78 

t 

An integrator is a dded to sake the system tv;Je-2. 
55 

jl . 

j1.C 

j.E 

j.6 

j.4 

j.2 

-4 .18 ' 

Im Q 

I 

.2 

I 

.4 

I 

.6 

Re Q 

.8 

Figure 26. The root-square locus plot. 
56 

The closed loop poles for the-control system are given by 

9, = - J-Q, = 1.297 1120.8 

92 = - J-92 = 1.297 I-120.8 

93 = - J-a3 = -1.33 

Compute 

9192 = 1.682 

91+92 = -1.328 

41+92 

-= - .79 

9192 

The reciprocal of the closed loop zero is 

= .79 + .75 = 1.54 

The closed-loop transfer function is 

H eli 

=-r= 

(l+Ts) 

(l-CM- 

= l+Ts 

(1 - q(1 - z s + -J- s2) 

L 9192 

1 f 1.54 s = 

(1+.75s)(l+.79s + .595s2) 

= 1 + 1.54s 

1 + 1.54s + 1.18~~ + .45 s3 

(71) 

(72) 
57 

Step response simulation was performed to test (72). The overshoot 

was found to be more than 40%: not a good response. Repeated trialand-error adjustment led to the following closed-loop transfer function. 

ei H=R= 1 + 3.164s 

1 + 3.164s + 3.082~~ + .595s3 

This system had a step responce overshoot of 18.7% and a time-to-peak 

of 3.4 seconds. The corresponding open-loop transfer function Go was 

obtained from 

1 -= ,+& 

H 0 

giving 

Go = 

.3245(1 + 3.164s) 

s2(l + .193s) 

(73) 

(74) 

(75) 

Two compensation configurations were used as shown in Fiqure 27; 

the first is a series compensation and the second is a series compensation 

with rate feedback. The second configuration offers better counter-action 

for load disturbance and plant parameter variation. 

For series compensation: 

GO G, = r = 13.75 (1 + 3.164s)(l + 2.36s) 

P s(1 + .193s) 

For series canpensation with rate feedback: 

choose %= 475.76 = 476 

then Gc = 

168.1(1 + 3.164s) 

S 

(76) 

(77) 

(78) 

A block diagram for the simulation of the designed closed-loop 

system is given in Figure 28. The dashed line part is used only for 

series compensation with rate feedback. Figure 29 shows the step response 
58 

+ 

-7 =: 

Gc - G 

P 

Compensator Plant 

(a) Series compensation 

Plant 

* 

- Gc GP I - 

Compensator 

KtS 

Rate feedback 

(b) Series compensation with rate feedback 

Figure 27. Compensation schemes. 
b = 175 rad.) i3i 

Gc 

_, 1 

.S 

I 

I 

I 

I 4 

I 

I 

l- ---- 

I 

I 

‘) 

'Es - 

Figure 28. Closed-loop control system simulation diagram. 
aa 

0 

,wt,,&- /7s.13fd 

Error = . IS tad ct. ~o+or 

=. o+ arc-min d.t 

1 a&enNL 

I 

c 

I 

i 

I 

I 

I 

i I I 

i i i 

11s. ed. ---- ( I 

I 

‘I 

Ii 

il 

j / 

II 

i I 

I i 

!J 

CL 

I 

I 

i 

I 

I 

I 

I 

I 

i 

I 

i 

1 

I 

L 

I 

i I 

I I 

I 

I 

i 

i 

I 

i 

i L 

TI 

I 

i 

I 

1 

T 

iI II 

Figure 29. Step response of the nominal system. 
61 

of the system under the nominal conditions. The reference input is 175 

radians for motor, corresponding to 1" for the antenna. It should be 

mentioned that the system is not expected to be subjected to a step 

function type of command. Step response simulation is used merely to 

reveal the system's transient characteristics. 

System Sensitivity The response sensitivity of the designed system 

was studied by way of computer simulation. Results are given in Table 

3. As a whole, the response did not change much for parameter change 

as high as 20%. Also, the superiority of the series compensation with 

rate feedback is evident. 

The Constant Drag Torque Since the first rotary joint does not change 

its direction of rotation the nonlinear slip-ring friction torque appears 

as a constant drag torque with a magnitude of ~~~~ entering the system 

at the point as shown in Figure 30. Results of a step response simulation 

are given in Table 4. One sees that the control is capable of nulling 

the effect of constant drag torque very effectively. A CSMP listing of the 

simulation is attached as Appendix A. 

Table 4. Response of System with Drig Friciton 

Overshoot n 21 % 

Time to peak tp 3.3 set 

Error at 4 t : motor angle 

P 

.13 rad 

antenna angle .04 arc-min 

Theoretically, the steady-state pointing error due to a constant 

drag torque should be zero. From Figure 30, the response of the system 

A output due to a constant disturbance torque TF = s is 
62 

Table. 3. System Sensitivity. 

System Overshoot Time Settling 

to peak time 

condition n t P 3 

Nominal 19 % 3.4 s 3.0 s 

KT/Ra 10 % up 16 % 3.2 s 9.6 s 

Series 

compensation 1' 10 % down 21 % 3.6 s 8.8 s 

II 20 % up 15 % 3.2 s 9.8 s 

II 20 % down 24 % 3.8 s 8.6 s 

Nominal 19 % 3.2 s 8.8 s 

Series II 10 % up 19 % 3.2 s 8.8 s 

compensation 

with rate II 10 % down 20 % 3.2 s 8.8 s 

feedback 

II 20 % up 18 % 3.2 s 8.8 s 

II 20 % down 20 % 3.2 s 8.6 s 
Constant drag torque 

rF = qmax = 100 N-m 

rad.) 9 

V 

KT'Ra ; 

1 "i 

p 

> I _ 

t S 

Figure 30. System with constant drag torque. 
64 

e;\(s) = 

Ii s2 + &s) + K (1 + B)sl ' 

a 

E 

The-steady-state value of e;\ is 

ed, (t-4 = se~(s+O) = 0 (80) 

where the property Gc(s+O) = m has been used. 

The Nonlinear Friction Torque The second rotary joint has little 

rotation under the nominal conditions. To counter anamalous effects, 

it may rotate in either direction. Thus the shifting hysteresis type of 

nonlinear friction torque, as portrayed in Figure 17, may come into play. 

This friction torque is a nonlinear function of the speed of the motor, 

and it enters the system in a way as shown in Figure 31. The nonlinear 

function is designated GF. GF can be simulated according to the flow 

diagram of Figure 19. Note that all load quantities have been reflected 

into quantities as seen by the motor; the value for KF in Figure 19 should 

be handled similarly. Using the data in Figure 17, 

KF = 

-106N-F4 

De3 

at antenna 

57.3x106 ;+;y " 

10' 

lo4 

= - .573 s at motor 

rad 

'max = lo6 ;d-M at antenna 

lo6 

= - = 100 N-M 

lo4 

at motor 
350 rad. 

Centrifugal force 

induced torque 

7c 

Nonlinear friction torque 

GF L 1 

I 

r --- -l 

I Kts I 

L --- -I 

1 

I 

I 

Figure 31. System with nonlinear friction torque. 
66 

To reveal the effect of nonlinear friction torque, a square-wave may be 

used as reference input for system simulation since rate reversal is 

needed to exhibit the shi-fting hysteresis characteristic of the nonlinearity. 

In order to drive the nonlinear friction torque to its saturation level 

the amplitude of the square-wave should be greater than 1" in terms of 

the actual antenna angle. An amplitude of 2" is chosen. The corresponding 

amplitude for the motor angle is (2x104)/57.3 = 349.04 radians. A simulation 

was performed in the manner described. Results of simulation are listed in 

Table 5, which shows the adequancy of the control. Appendix B contains 

the CSMP listing of the simulation. 

Table 5. Square-Wave Response of the System having 

Shifting Hysteresis Type of Friction Torque 

Overshoot n 20% 

Time to peak tp 3.19 set 

Error at 4 tp: Motor angle .14 rad 

Antenna angle .05 arc-min 

The Centrifugal Force Induced Torque Tr Recall from the discussion 

in Section 3.4 that the maximum torque on antenna caused by the centrifugal 

force is less than -3% of Tmay which is 100 N-M as seen by the motor. Insert 

, 

TC = 1 N-11, which is 1% of TmaX, into the last simulation program and repeat 

the simulation. The results are nearly the same as those in Table 5, 

showing that the pointing control i,s capable of nulling the effect of 7C 

satisfactorily. 
67 

3.5 Effect of Noise in Sensed Pointing Error Siqnal 

It has been shown that, based on the set of given assumptions, the 

designed mechanical pointing control system more than adequately meets 

the 3-arc-minute requirement for pointing accuracy. If the microwave 

sensed pointing error is noise contaminated, however, the situation 

could be quite different. As discussed in Section 2, the noise in sensed 

error is low frequency in nature, which can be as low as d-c. Under this 

condition the noise directly dictates the accuracy of the overall pointing 

control system. At this moment, it is not practical to arrive at a 

numerical value for the noise level. The remark here serves to emphasize 

the possible adverse effect of the noise. 

3.9 Effect of Collector Vibrations 

It has been discussed in Section 3.7 that the frequencies of vibration 

for the collector is less than .OOl hertz which is considerably lower than 

.2 hertz, the approximate bandwidth of the pointing control system. Therefore 

one expects errors caused by the collector vibration be effectively 

diminished by the pointing control. To support this claim, computer simulations were performed as follows. 

Pointing error caused by collector vibration is equivalent to the 

error of the closed-loop control system excited by a sinusoidal reference 

input. Reference inputs having frequencies of .OOl hertz (about 16.6 

minute/cycle) and .Ol hertz (about 1.66 minutes/cycle) were used in the 

simulations. Results are shown in Table 6, which confirm the claim. 
68 

Table 6. Tracking Error Due to Collector Vibrations 

First simulation 

vibration frequency = .OOl hertz (= 16.6 min/cycle) 

vibration amplitude = 1 degree 

maximum steady-state error = .0055 22. at motor 

,002 arc-min at antenna 

Second simulation 

vibration frequency = .Ol hertz (= 1.66 min/cycle) 

vibration amplitude = 1 degree 

maximum steady-state error = .55 rad. at motor 

.19 arc-min at antenna 
4. APPROACHES FOR MODELING THE FLEXIBLE COLLECTOR BODY 

It is recognized that the structural dynamics of the flexible collector 

body may be modeled by transforming it to an equivalent orthotropic plate. 

Then the wealth of well known theory of plates can be applied readily. 

However, a thorough study of the structural responses of the collector 

plate requires a major effort. The discussion here will focus on the 

two approaches which may be used to obtain the desired model. They are the 

analytical approach and the simulation approach. 

The Analy_tical Approach In this approach the structure of the collector 

truss is first transformed to an equivalent grid, which is then transformed to an equivalent orthotropic plate as shown in Figure 32. The 

differential equation for an orthotropic plate, including the inertialforce term, has the form [5] 

+&&+Dy$= 

2 

q+ph% 

w atL 

where w = w(x,y,t) is the deflection in the z-direction as shown in 

Figure 32(c), q is the external loading per unit area, h is the thickness, 

and p is the density of the plate mater ial. The problem of modeling, simp 

stated, is the determination of constants Dx, D , and H. 

Y 

Transformation from grid work to an orthotropic plate has been 

(81) 

lY 

thoroughly studied by Timoshenko and his cohort [l]. Let the geometry 

of a grid be defhned in Figure 33, then the constants for an equivalent 

orthotropic plate are given by 

D,=F3 ? 

D E h3 t2 

Y =12 q 

69 
u (a) The collector truss 

(b) An equivalent grid 

Figure 32. Transformation to an equivalent orthotropic plate. 

_ -.___. . ,...-.. -. _.. -_.. _.,__ _ ,,_, _ .,, . ., . ._ ._ ______.__ --_-.-----.. -. -- 
Figure 33. Definition of the geometry of a grid. 
72 

H= ;$++ (82) 

where E is the modulus of elasticity, and cl and c2 are torsional 

rigidities of the beams parallel to the x and y axes, respectively. When 

the beams have rectangular cross-section as shown, cl and c2 are given by [6] 

c1 = ;t; h[l - 

lT n=1,3,5--- n 

(83) 

192 

c2 

+;h[l _- 

l-3 

tanh (!!F$ 

2)' 

For most practical cases the infinite series converges rapidly; usually 

two or three terms are sufficient to give accurate result. For example, if 

h 

- = 1, then 

? 

Tm --$-tanh y 

n=1,3,5--- h 

= tanh > + 1 

35 

tanh + + 1 tanh 

55 ?+--- 

= 0.91715 + i!-- 243 (0.9998) + - 3:25(l) 

+--- 

= 0.91715(1 + .004486 + -0003489 + - - - ) 

= 0.9216 

Similarly, the results of the series become 1.00073 and 1.00414 for 

3 equals 2 and 4 respectively. h Using the result for - = 2 in (83) gives 

1 tl 
73 

cl=31 G t3 h[l - q (;I (1.00073)] 

Tr 

5 3 4 

= $ (-,-) h (0.6861) 

= ; (;, h4 (0.6861) 

= 0.0286 G h4 

Clearly c2=c1 if t2=tl. The bending moments and twisting moments are 

related to the deflection w by the following expressions. 

Eh3 

12 

5 

bl 

a2w 

2X2 

My = 

M = 9 a2w 

XY b 1 axay 

(84) 

M = c2 a2Lu 

YX al axay 

Analytic methods for transforming a truss structure as shown in Fiqure 

32(a) to an equivalent grid, or directly to an equivalent orthotropic 

plate have not been developed. It is this part of the study which requires 

a major effort. However, some related cases have been studied, such as 

plates with many holes [7-IO] and corrugated pipes. [7-117. The 

equivalent solid plate or smooth pipe will produce the same strain, deformation, or strain energy as the actual perforated plate or corrugated 
74 

pipe. The results for the actual structure have been carried out by testing, 

numerical calculation through point-matching or collocaticn technique, 

and other appropriate analysis such as energy method. 

The Simulation ApEroach Constant Dx, D 

Y' and H of Eq. (8C') for a 

given truss may be obtained by a parameter matching technique with the 

help of computer simulation. Well known computer programs exist, such 

as NASTRAN and STRg'lN, which use finite el.+ment approximations for 

flexible structure. 
I 

5. COr~lCLUSION 

A study of the pointing control system for SPS was accomplished. The 

study included two major areas, namely, the analysis of a pointing error 

sensing method for the SPS antenna, and the design of a mechanical pointing 

control system for the antenna. In addition, approaches for modeling the 

flexible body collector dynamics were given. 

In the area of point error sensing, an analysis was made on a previously proposed method for developing pointing angle signals to be used 

by a servomechanism to point the antenna toward a rectenna on ground. From 

the analysis result, the method was found to be feasible. A normalized 

technique was added to the method to produce error signals that are independent of the received signal magnitude. Effects of vibration and noise 

on pointing control accuracy were analyzed and presented in the form of 

charts. 

In the area of mechanical pointing control of the antenna, a baseline 

satellite configuration was selected as shown in Figure 16 (p. 30) together 

with a set of baseline parameters. .4nalysis were made on various phenomena 

which might influence the accuracy of the pointing control. They include 

the nonlinear slip-ring friction of rotary joints, motor characteristics, 

effect of centrifugal force induced torque, effect of variation in antenna's 

moment of inertia, effect of finite collector mass on pointing control, and 

the effect of vibration modes of the collector. The friction and backlash 

characteristics of the gear train was not available for analysis and were 

ignored. A mathematical model of the plant, whose output is the antenna 

pointing direction, was developed. The pointing control system was designed 

using an optimal control approach with the help of the root-square-locus method. 

75 
76 

The mechanical pointing accuracy, with the various anomalies present, was found 

to be well within 1 arc-minute. 

From the results of this study it appears that the desired pointing 

accuracy of + 3 min is possible. Suppose a reasonable error budget is 

t 1 min for each of the major error sources of (1) noise, (2) vibration, 

and (3) mechanical effects (gear backlash, stricktion, etc.). Suppose 

further that we require the + 3 min to correspond to at most 50% of the 

beamwidth eg so that operation is always in the linear part of the error 

pattern. Then from Figure 13 and eg = 12 min we require +I<3 subarrays in 

the linear array. The allowed vibration error is therefore 1 min/l2 min = 

0.083 or 8.3% of eB maximum. From Figure 12 this allowance is not exceeded 

for ?I=3 even for mode 2 (k=2) vibration (E = 1 m). The noise error of 

+ 1 min can be achieved by providing sufficient power in the ground station 

pilot transmitter. 
77 

6. REFERENCES 

I 

El1 

PI 

c31 

c43 

El 

C61 

c71 

C81 

I33 

cw 

Cl11 

Hung, J.C. and P.Z. Peebles, Jr., "Final Report, Accuracy Analysis 

of Pointing Control System of So'lar.Power Station," NASA Contract 

NAS8-33065 with the Electrical Engineering Department, University 

of Tennessee, December 23, 1978. 

Peebles, Peyton Z., Jr., Probability, Random Variab_!es, and RandomSiqnal Principles, McGraw-Hill Book Co., 1980. 

"Satellite Power Systems (SPS) Concept Definition Study", Final 

Report, Vol. 3, SD-78-AP-0023-3, Rockwell International, Space 

Division, April 1978, p. 3-201. 

Chang, S.S.L., Synthesis of Optimum Control Systems, McGraw-Hill 

Book Co., 1961. 

Timoshenko, S., and S. Woinowsky - Krieger, Theory of Plates and 

Shells, second edition, McGraw-Hill Book Co., 1959. 

Timoshenko, S., and Goodier, Theory of Elasticity, Thrid edition, 

McGraw-Hill Book Co., 1970. 

Goldberg, J.E., and K.N. Jabbour, "Stresses and Displacements in 

Perforated Plates," Nuclear Structural Engineering, Vol. 2, pp. 360- 

331., 1965. 

Hulbert, L.E., and F.W. Niedenfuhr, "Accurate Calculation of Stress 

Distributions in Multiholed Plates," J. Eng. for Ind. Trans. ASME, 

Vol. 87, pp. 331-343, 1965. 

Meijers, P., "Doubly-Periodic Stress Distributions in Perforated 

Plates," Ph.D. Thesis, Technological University of Delft, the 

Netherlands, 1967. 

Slot, T., and N.J. O'Donnell, "Effective Elastic Constants for Thick 

Perforated Plates with Square and Triangular Penetration Patterns," 

J. Eng. for Ind., Trans. AWE, Vol. 93, pp. 935-942, 1971. 

Donnell, L.H., "The Flexibility of Corrugated Pipes Under Longitudiual 

Forces and Bending," Trans. ASME, Vol. 54, pp. 69-75. 
78 

APPENDIX A. CSMP FOR THE FIRST ROTARY JOINT CONTROL SYSTEM. 

****CONTINUOUS SYSTEbl avQDELING PROGRAM**** 

~*Jz:PRC~~LEM IPJPUT STATEMENTS*.** 

* STEP RESPONSE kiITH DRAG DISTURBANCE 

R=175. 

TD=lOO.O 

E=R-C 

X1=13.75;4E 

XZ=INTGRLIO.O,X11 

X3=LEDLAG(3.164,0.193rXZ) 

X4=DfRIVIO.O,X31 

X9=2.36*X4 

v-=x9+x3 

E3=V-C2 

C2=43.4=Cl 

cl=DERIV(o.o,c1 

C2=43.4*Cl 

X5=96.05*E3 

Xb=TD+X5 

X7=0.00010173*X6 

XB=INTGRL(O.O,X7) 

C-INTGRL(3.0,XB) 

* 

TIMER FINTII’I=~O,, GUTDEL=.2, OELT=.OOl; PRDEL=.2 

PRTPLT E, V, C 

LASEL STEP RESPOF:SE hITH ORAG DISTURBANCE 

END 

STCP 
79 

APPENDIX B. CSMP FOR THE SECOND ROTARY JOINT CONTROL SYSTEM. 

*+**CONTINlJOlJS SYSTEM MODELING ?RDGRbM  

** *PRGBLEE/ INPUT STATEMENTS+** 

* POINTING CCNTRUL WITH SHIFTING HYSTERESIS TYPE 

* OF NCNLINEAR FRICTION TORQUE 

d KF=-. 573, DELT=.OOl, D=KF*DELT 

INITIAL 

CONSTANT SGNP=0.~T:~l=l00.rTDP=O.,TD=O.,D=-.OOO573 

DYNAMIC 

SORT 

R=350.*STEP(O.O)-200.*STEP(lO.OJ 

E=R-C 

Xl=25.ld*E 

X2=INTGRL(O.O,Xl) 

X3=LEDLAG(2.20,0.270,X21 

X4=DERIVIO.O,X3) 

X9=2.36*X4 

v=x9+x3 

E3=V-C2 

C2=43.4*Cl 

Cl=DERIV(O.O,C1 

C2=43.4*Cl 

X5=96.05*E3 

X6=TD+X5 

X7=0.00010173*X6 

RATE=INTGRL(O.O,X71 

C=INTGRLLO.O,RATEI 

NOS0R.T 

SGN=SIGN(L.O,RATEJ 

IF iSGN .EQ. SGNPI GOT0 301 

TDP=O. 

302 TD=TDP+D*RATE 

IF (ABS(TDI .GT, TM) TD=SIGN(TM,TD) 

GO TO 304 

301 IF (ABS(TDP) .EQ. TYI GOT0 303 

GOT0 302 

303 TD=TDP 

304 TDP=TD 

SGNP=SGN 

SORT 

* 

TERM1 NAL 

TIMER FINTIM-20.r OUTDEl=.2, DELT=.OOlr PRDEL=.2 

METHOD RKSFX 

PRTPLT C, E, RATE, TD 

LABEL STEP RESPDNSE 

END 

STOP 
1. REPORT NO. 2. GOVERNMENT ACCESSION NO. 3. RECIPIENT’S CATALOG NO. 

- 

4. TITLE AND SUBTITLE 5. R%PORT DATE 

November 1980 

Satellite Power System (SPS) Antenna Pointing Control 6. PERFORMING ORGANIZATION Cl 

7. AUTHOR(S) 6. PERFORMING ORGANIZATION REI 

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. WORK UNIT, NO. 

The University of Tennessee 

Electrical Engineering Department, Systems & Radar Lab 

Knoxville, Tennessee 3’7916 

2. SPONSORING AGENCY NAME ANDiD= 

National Aeronautics and Space Administration 

Washington, DC 20546 

15. SUPPLEMENTARY NOTES 

NASA Marshall Technical Monitor: Joe Howell 

Final Report 

6. ABSTRACT 

Accuracy requirements for the SPS antenna pointing control consists of a mechanic 

pointing accuracy of three arc-minutes and an electronic phased array pointing accuracy o 

three arc-seconds. Results of this study, based on the factors considered in current anal; 

show that the three arc-minute overall pointing control accuracy can be achieved for the S: 

in practice. 

7. KEY WORDS 

SPS antenna pointing error 

antenna pointing error 

mechanical pointing control design 

error sensing 

9. SECURITY CLASSIF. (of this r*portl 20. SECURITY CLA! 

16. DISTRIBUTION STATEMENT 

Unclassified - Unlimited 

Subject Category 

IF. (of this page) 

Unclassified 

21. NO. OF PAGES 

I--- ---1 

22.CE 

Unclassified 84 A05 

For sale by Nationrl Technical Information Service. Springfield, Virginia : 

NASA-Langley, 1980