Nonlinear optimal control of planet-pointing space vehicle, basic methods for attitude control, and satellite orbit theorie

Cannon, R. H., Jr.

Flugge-Lotz, I.

English

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en 
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Tenth Semiannual S t a t u s  Report 
June 1966 
on 
BASIC STUDIES I N  SPACE VEHICLE ATTITUDE CONTROL 
i n  t h e  
Department of Aeronautics and Astronaut ics  
Stanford University 
under 
Research Grant NsG-133-61 
from the  
National Aeronautics and Space Administration 
This r epor t  summarizes progress  during the pas t  s i x  
months under a continuing research  grant  f o r  the  per iod 
beginning December 1965. 
on Ref. 1, and i t s  cont inuat ion on Ref. 2, The research  
is  supervised by Prof. I. Fl i igge-btz  and Prof.  R. H. 
Cannon, Jr., Pr inc ipa l  Inves t iga tors .  
A separa te  f inanc ia l  accounting w i l l  be forwarded by 
t h e  University.  
The i n i t i a l  g ran t  is  based 
SUMMARY 
- .  During t h e  present  report per iod,  Ph.D. degrees w e r e  received by 
four  members of t h i s  program, W. G. Eppler ,  R. Busch, J. L. Almuzara, 
and K. Hales. The advanced degree of Aeronautical  Engineer was 
received by R. D. Hensley. 
M r .  Eppler 's  thesis [Ref. 41 was reported a t  l ength  i n  the pre- 
ceding s t a t u s  r epor t .  
Cannon, who i s  on sabba t i ca l  leave t h i s  year .  
I t  was the  only research  supervised by Professor 
The research on nonl inear  and optimal con t ro l  of planet-pointing 
space veh ic l e s ,  begun by Wessrs. Busch, Almuzara, and Hales,  continues 
with new Ph.D. candidates  under Professor  Flugge-Lotz (Sec. A).  Basic 
s t u d i e s  of optimal c o n t r o l ,  under Professor Frankl in ,  and s a t e l l i t e  
t r a j e c t o r y  s t u d i e s ,  under Professor Breakwell, a l s o  cont inue (Secs. A 
and B). 
- 1 -  
. .  
. 
A. NONLINEAR STUDIES, OPTIMAL CONTROL 
(Studies  Supervised by Professor  Fliigge-Lotz) 
1. Optimum and Suboptimal Control of t h e  Pitch Motion of a S a t e l l i t e  
i n  E l l i p t i c  Orb i t ;  Preliminary Consideration of t he  Linearized 
Roll-Yaw Motion (Ph.D. Research of R. Busch) 
The r epor t  on the  work done by Ronald Busch has  been f in i shed  
[Ref. 51. 
r epor t  copies  a r e  made, t he  repor t  has not  y e t  been copied, but i t  is  
expected t h a t  it can be mailed i n  about t h r e e  weeks. 
Due t o  an overload i n  the  publ ica t ion  o f f i c e ,  where the  
For a s a t e l l i t e  i n  an e l l i p t i c  o r b i t  i n  opera t ion  f o r  a year o r  
more, the  suggested s teady-state  con t ro l  of t h e  p i t c h  motion causes 
a very s l o w l y  growing deviat ion of t h e  roll-yaw a t t i t u d e  from its 
des i red  s ta te  because of the  bas ic  non l inea r i ty  of t he  system of 
d i f f e r e n t i a l  equations descr ibing the a t t i t u d e  of the  s a t e l l i t e .  I n  
add i t ion ,  one may consider  t h i s  a l s o  as a consequence of t h e  fact  t h a t  
t he  des i r ed  state i s  not  an equi l ibr ium s t a t e ,  Since the missions of 
many s a t e l l i t e s  r equ i r e  very accurqte  a t t i t u d e  keeping, t h i s  problem 
w i l l  be s tud ied  f u r t h e r .  M r .  W. Boykin, a new Ph.D. candidate ,  is 
p a r t i c u l a r l y  i n t e r e s t e d  i n  t h i s  problem. 
a very accura te  a t t i t u d e  only on p a r t  of each o r b i t  and re lax ing  i n  
between, w i l l  a l s o  be considered. 
Such suggestions as keeping 
2. The Complete At t i tude  Control Problem f o r  an Ea r th  S a t e l l i t e  i n  
E l l i p t i c  Orbit { F i . D .  h5eiirCh Sf E. & I P S )  
The r e p o r t  on t h i s  problem is  f in i shed  [Ref. 61. Copies w i l l  soon 
be ready (see remarks under Sec. 1 ) .  The i t e r a t i o n  procedure used by 
I(. Hales does not  ye t  seem t o  be f a s t  enough. 
p o s s i b i l i t y  of improving it. 
computing equipment, which a t  t h i s  t i m e  would be too heavy and 
voluminous t o  be c a r r i e d  on board, t he  ques t ion  arises whether one can 
hope t o  minia tur ize  the  equipment o r  whether one should consider  
co r rec t ing  a t t i t u d e  error by occasional  s i g n a l s  from an e a r t h  s t a t i o n .  
W e  w i l l  s tudy the 
Since t h i s  procedure r equ i r e s  d i g i t a l  
- 2 -  
I n  t h a t  case, the  f a c t  of delay i n  t h e  s i g n a l  transmission has t o  be 
considered. A new s tuden t  w i l l  continue t h i s  work i n  t h e  very near 
f u t u r e .  
3.  The Validity of Linearization i n  Attitude Control (Ph.D. Research 
of J. L. Almuzara) 
A report on t h i s  work was f inished i n  the l a s t  few weeks [Ref. 71. 
Particularly interest ing are the comparisons of the systems 
and with IuI < A > 1 - - 
x + s i n  x = u(t )  
X + X  = u( t )  
Diagrams showing t h e  d i f f e r e n c e s  i n  minimum s e t t l i n g  t i m e  f o r  zeroing 
a d is turbance  are very i n s t r u c t i v e .  
M r .  Frank C u r t i s '  s t u d i e s  of t h e  same problem f o r  minimum-fuel 
c r i t e r i o n  continue. H i s  s t u d i e s  b e n e f i t  from t h e  bas i c  s t u d i e s  of 
M r .  Almuzara, but  may l a t e r  become more i n t e r e s t i n g  f o r  p r a c t i c a l  
des igns .  
(Studies Supervised by Professor  Frankl in)  
4.  Computation of O p t i m a l  Cont ro ls  by a Method Based on Second 
Var ia t ions  (Ph.D. Research of T. E. Bullock) 
T h i s  work i s  concerned wi th  developing e f f i c i e n t  numerical techniques 
f o r  genera t ing  optimal c o n t r o l s  w i th  a s p e c i a l  emphasis on second-order 
d i r e c t  methods. A cont inua t ion  of t he  work repor ted  i n  t h e  l a s t  S t a t u s  
r e p o r t  has l e d  t o  answers t o  t h e  t h e o r e t i c a l  problems posed. The 
t h e o r e t i c a l  work has  been j u s t i f i e d  from a pragmatic po in t  of view i n  
p a r t  by t h e  successfu l  a p p l i c a t i o n  of t h e  method to  s e v e r a l  numerical 
examples, w i th  the a i d  of a d i g i t a l  computer, t h e  7090. 
For t h e  purpose of t h i s  s t a t u s  r e p o r t ,  t h e  numerical r e s u l t s  of 
one example w i l l  be given. The problem chosen f o r  t h i s  s tudy  i s  found 
i n  Merriam [Ref. 81 on opt imiza t ion  techniques.  I n  us ing  t h i s  problem 
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. .  
- 3 -  
. .  
t o  i l l u s t r a t e  a second-order method described i n  h i s  book, Merriam 
states t h a t  f o r  a p a r t i c u l a r  con t ro l  i n i t i a l i z a t i o n  the  app l i ca t ion  of 
... t he  method based on second v a r i a t i o n s  r e s u l t s  i n  complete f a i l u r e .  (1 I9 
The d i f f i c u l t y  encountered here is a conjugate poin t  i n  the accessory 
problem. Application of t he  theory developed i n  t h i s  research  has 
circumvented these d i f f i c u l t i e s ,  as shown i n  the  numerical r e s u l t s  
presented here. 
The problem given by Merriam may be descr ibed b r i e f l y  a s  follows. 
A driven nonl inear  oscil lator of the type s tudied  by van de r  Pol may 
be descr ibed i n  s ta te-space nota t ion  by 
2 x = x  1 
2 x = -x + (1  - x,) x2 -k u 
2 1 
For the  i n i t i a l  condi t ions  in s ide  the  s t a b l e  l i m i t  cycle 
x2(0) = 0 ,  
xl(0) = 1, 
u ( t )  i s  t o  be chosen to  minimize the cost 
5 - 
J = -  1 / P 2 ( x 1 + x 2 + u ) d u  2 
2 i: 2 
0 
The f i n a l  s t a t e s  a r e  unspecified or f r e e  i n  t h i s  vers ion  of the problem. 
The neighboring extremal con t ro l  law, a s  descr ibed by Breakwell, 
Speyer, and Bryson [Ref. 101 and K e l l e y  [Ref. 113, i s  optimal f o r  the 
given problem and is  the best to  second order  f o r  changes i n  t h e  s i a t e  
o r  terminal cons t r a in t s .  T h i s  c o n t r o l  law i s  given i n  the  feedback 
form 
u ( t )  = c ( t )  x p  + c2(t)  x2( t )  + c p  1 
4 
The neighboring extremal cont ro l  law is an automatic by-product of t h e  
computing method based on second va r i a t ions .  
q u a n t i t i e s  c , ( t ) ,  c 2 ( t ) ,  and c g ( t ) ,  which descr ibe  the  con t ro l  
law f o r  the  given example, a r e  shown i n  Fig.  1. The corresponding 
optimal t r a j e c t o r i e s  a r e  shown i n  Fig. 2. 
The time-varying 
2VVP37 
U.C, r,+c2x2tc3 
0 
I 2 / 3  
FIG. 1. THE NEIGHBORING EXTREMAL CONTROL LAW. 
2VVP37 
. 
FIG. 2. THE OPTIMAL TRAJECTORIES FOR THE 
VAN DER POL PROBLEM WITH QUADRATIC U S S .  
- 5 -  
. 
The r e l a t i v e  convergence of the method based on second v a r i a t i o n s ,  
computed wi th  the  program e n t i t l e d  2wP,  and the method of s t eepes t  
descent [Ref. 1 1 3 ,  computed with the  program SDW, is  shown by comparing 
Fig. 3 w i t h  Fig. 4.  Figure 3 shows seve ra l  successive i t e r a t i o n s  on 
the  con t ro l  by SDVP and the  corresponding optimal con t ro l .  The cor-  
responding r e s u l t s  f o r  2VVP a re  shown i n  Fig. 4 .  Since the  change i n  
the  shape of t h e  c o n t r o l  funct ion is q u i t e  l a rge  from one i t e r a t i o n  t o  
the next ,  some add i t iona l  information i s  he lp fu l  t o  d i s t ingu i sh  the 
var ious i t e r a t i o n s .  A t  the  bottom of Fig. 4 ,  a sequence of small  
numbered p l o t s  shows the general  t rend of each i t e r a t i o n .  These small  
f i gu res  may be used t o  help t r ace  out  each corresponding curve i n  the  
l a rge  p l o t  which shows a l l  of the i t e r a t i o n s  superimposed. 
The method based on second v a r i a t i o n s  has a c l e a r  advantage i n  t h i s  
example. "his is i l l u s t r a t e d  graphica l ly  i n  Figs .  3 and 4 and i n  the  
following da ta .  For u = 0 ,  SDVP obtained a c o s t  which agreed w i t h  
t he  optimal i n  only t w o  s i g n i f i c a n t  f i g u r e s  a f t e r  18 i t e r a t i o n s .  In  
c o n t r a s t ,  2 W P  converged t o  5 s i g n i f i c a n t  f i g u r e s  i n  the  c o s t  i n  only 
5 i t e r a t i o n s .  The r e l a t i v e  convergence r a t e s  may be f u r t h e r  compared 
w i t h  re fe rence  t o  Fig.  5. T h i s  p lo t  shows the  equivalent  number of 
s i g n i f i c a n t  f i g u r e s  i n  J which do not  agree w i t h  t h e  optimal c o s t ,  
J', 
(0) 
f o r  each of the  methods. 
A numerical example of t h e  type discussed i n  the l a s t  s t a t u s  r epor t  
[ R e f .  31,  which has the  dimension of the  cons t r a in t  less than the 
dimension of the  s t a t e  but not equal t o  zero ,  i s  being worked on the  
computer. The successful  running of t h i s  program and the  subsequent 
completion of the  f i n a l  repor t  w i l l  terminate  t h i s  pro jec t .  
- 6 -  
FIG, 3. SEVERAL SUCCESSIVE ITERATIONS ON 
CONTROL BY STEEPEST DESCENT (SDVP) FOR u 
. Y  
j 
.I 
I 
I 
i 1 
FIG, 4 .  SEVERAL SUCCESSIVE ITERATIONS ON THE 
CONTROL BY THE ME 
TIONS (2WP) FOR yo)= 0.OD BASED ON SECOND VARIA- 
- 7 -  
FIG. 5 .  A CObWARISON OF THE CONVERGENCE RATES OF 
STEEPEST DESCENT (SDVP) AND SECOND VARIATIONS 
(ZVVP). 
- 8 -  
e .  
B.  SATELLITE ORBIT STUDIES 
(Studies  Supervised by Professor  Breakwell) 
1. Rigorous E r r o r  Bounds on Posi t ion and Veloc i ty  i n  S a t e l l i t e  Orbi t  
Theories (Ph.D. Research of J. Vagners) 
By u t i l i z i n g  r e s u l t s  of Hamiltonian theory and the von Zeipel method 
f o r  t r e a t i n g  a r t i f i c i a l  sa te l l i te  o r b i t s ,  e r r o r  bounds have been der ived 
f o r  a general  c l a s s  of o r b i t s  with e c c e n t r i c i t y  less than one. I n  
order  t o  extend the  e r r o r  bounds f o r  the general  axisymmetric problem 
t o  t i m e  i n t e r v a l s  of the  order  1/J2, where J2 is  the  obla teness  
parameter, t he  known i n t e g r a l  of energy has been u t i l i z e d  t o  c a l i b r a t e  
the  governing d i f f e r e n t i a l  equation f o r  t he  r ap id ly  r o t a t i n g  phase. 
The nonsingular rap id  phase i n  the ana lys i s  was taken t o  be the  sum of 
the  mean anomaly, argument of p e r i a p s i s  and the  r i g h t  ascension of the 
ascending node. A corresponding ana lys i s  f o r  the  general  asymmetric 
problem ( including the  f e s s e r a l  harmonics) has a l s o  been given. From 
the  genera l  e r r o r  a n a l y s i s ,  an algorithm was der ived f o r  t he  computation 
of  the c o r r e c t  i n i t i a l  condi t ions cons i s t en t  w i t h  the  expected accuracy 
of the  theory. Numerical r e s u l t s  ve r i fy ing  t h e  conclusions of t he  
theory w e r e  a l s o  presented. 
This  work i s  reported i n  f u l l  i n  Ref. 12 ,  and w i l l  be presented a t  
t h e  Space F l i g h t  Mechanics S p e c i a l i s t  Conference i n  Denver, Colorado, 
July 6 ,  1966. 
2. Libration-Point S a t e l l i t e s  (Ph.D. Research of R. W. Farquhar) 
A comprehensive study of t he  c o n t r o l  and use of l ib ra t ion-poin t  
s a t e l l i t e s  has been i n i t i a t e d .  The completed work w i l l  eventual ly  
appear a s  a SUJMAR r epor t .  
w i l l  be published i n  b r i e f  papers a s  soon a s  possible. 
However, s i g n i f i c a n t  r e s u l t s  of t h i s  study 
The f i r s t  paper has been f in i shed  [Ref. 131 and w i l l  be presented 
a t  the  AAS Space F l igh t  Mechanics S p e c i a l i s t  Conference a t  Denver, 
Colorado, Ju ly  8 ,  1966. Six copies  of t h i s  paper have already been 
s e n t  t o  NASA Elec t ronics  Research Center. This paper is  e n t i t l e d  
- 9 -  
Station-Keeping in the Vicinity of Collinear Libration Points with an 1 )  
Application to a Lunar Communications Problem." 
marized in the following abstract. 
The content is sum- 
A feedback control system is used to position a satellite in the 
I t  is  found tha t  v ic in i ty  of an unstable co l l inear  l ib ra t ion  point. 
s t a b i l i t y  can be obtained w i t h  a radial-axis control using only range 
and range-rate measurements. Since the s a t e l l i t e  is controlled about 
an equilibrium path, which is determined by the perturbing accelerations,  
the station-keeping cost  is c h i e f l y  a function of the measurement noise. 
If earth-based measurements are employed, t h i s  cost  is very small. 
A novel method fo r  a direct communications l ink  w i t h  the far side 
of the  moon is offered  as a possible application for the earth-moon 
exterior point. 
point is nearly resonant, the amplitude of the forced oscillation in 
the moon's orbital plane and perpendicular to the earth-moon axis is 
only 4,965 km. By controlling the out-of-plane motion of the com- 
munications satellite, it is possible, at moderate cost, to take 
advantage of the aforementioned forced oscillation and obtain a tra- 
jectory where the satellite is never hidden from the earth. 
I 
Although the solar perturbation on a satellite at this 
I 
The second phase of this study is concerned with sun-planet 
libration points and should be finished in time to be included in the 
next status report. 
- 10 - 
- -  C. OTHER ACTIVITIES 
. .  The four professors participating in this grant will all attend 
the Third Congress of the International Federation of Automatic Control 
(IFAC) in London this month. Five Stanford papers will be given. Two 
of them are based on work sponsored in part by this grant [Refs. 14 
and 151. 
session on Deterministic Optimal Control (3), and Professor Franklin 
will be Rapporteur of the technical session on Deterministic Optimal 
Control (1). A special report on the conference will be forwarded. 
Professor Fliigge-Lotz will be chairman of the technical 
Professor Cannon continued his participation with the NASA Research 
Advisory Committee on Control, Guidance, and Navigation, serving as 
acting chairman of its meeting at ERC in May. He also continues as 
chairman of the AIAA Technical Committee on Guidance and Control, which 
will sponsor both its own Specialist Conference and the Joint Automatic 
Control Conference in August at the University of Washington. 
Professor Breakwell contributed to a six-week summer Institute in 
Dynamical Astronomy, held at Stanford University, July-August 1966. 
He presented two papers at the IAF Congress in Athens, Greece, 
September 1965, one of which was simultaneously presented by the co- 
author in Monterey, California. He co-authored a third paper also 
presented at Monterey. He attended a conference on Guidance Theory 
az:! Trijecfnry Analysis, NASA-ERC, November 1965. He attended an AIAA 
Technical Committee Meeting in Astrodynamics, New York, January 1966, 
followed by the fourth seminar in celestial mechanics at Yale. 
presented some NASA-sponsored Stanford research at a contractors 
meeting--Guidance Theory and Trajectory Analysis--at NASA-ERC, April 
1966. He attended, as co-author of a paper, a seminar at MIT in 
Guidance Theory and Trajectory Analysis, May 31-June 1, 1966. 
He 
- 11 - 
- -  
REFERENCES 
. ,  
1. ?Proposal for Basic Studies on the Dynamics of Space Vehicle 
Attitude Control Systems, submitted to NASA by Stanford University, 
Jan 1960. 
It 
2. 'kequest for Continuation of Research Grant for Basic Studies on 
Space Vehicle Attitude Control Systems," 
Stanford University, AERD 12-65, Jul 1965. 
submitted to NASA by 
*I ' 1  3. Ninth Semiannual Status Report, submitted to NASA by Stanford 
University, Dec 1965. 
' 1  ?I 
4. W. G. Eppler, Jr., Analytical Design of Manned Control Systems, 
SUDAAR No. 280, Department of Aeronautics and Astronautics, 
Stanford University, Stanford, Calif., May 1966. 
5. R. Busch and I. Fliigge-htz, '?he Attitude Control of a Satellite 
in an Elliptic Orbit, SUDAAR No. 261, Department of Aeronautics 
and Astronautics, Stanford University, Stanford, Calif., Apr 1966. 
1 )  
6 .  K. A. Hales and I. Flugge-Lotz, 'bdinimum Fuel Attitude Control of 
a Rigid Body in Orbit by an Extended Method of Steepest Descent, 
SUDAAR No. 257, Department of Aeronautics and Astronautics, 
Stanford University, Stanford, Calif., Jan 1966. 
91  
t t  
7. J. L. Almuzara , 'hinimum Time Control of Nonlinear System, 
SUDAAR No. 271, Department of Aeronautics and Astronautics, 
Stanford University, Stanford, Calif., May 1966. 
8. C. W. Merriam, 111, Optimization Theory and the Design of Feedback 
Control Svstems. McGraw-Hill Book Co.. New York. 1964, pp. 267-270. 
1 )  
9. J. V. Breakwell, J. L. Speyer, and A. E. Bryson, Optimization and 
Control of Nonlinear Syste~s !!sing the Second Variation," SIAM J. 
on Control, Ser. A, Vol. 1, No. 2, 1963. 
11 ' 1  
10. H. J. Kelly, Guidance Theory and Extrema1 Field, Trans. IRE, 
Professional Group on Automatic Control, 1962, pp. 75-82. 
'* 
11. A. E. Bryson and W. F. Denham, A Steepest Ascent Method for Solving 
t' 
Optimal Programming Problems, J. Appl. Mech., 29, 2, 1962, 
pp. 247-257. 
r* 
12. J. V. Breakwell and J. Vagners, Rigorous Error Bounds on Position 
and Velocity in Satellite Orbit Theories," SuDAAR No. 279, 
Department of Aeronautics and Astronautics, Stanford University, 
Stanford, Calif., Jun 1966. 
- 12 - 
13. 
14. 
15. 
I* R. W. Farquhar, Station-Keeping in the Vicinity of Collinear 
Libration Points with an Application to a Lunar Communications 
Problem, Space Flight Mechanics Specialist Conference, University 
of Denver, Colorado, Jul 1966. AAS Preprint No. 66-132. 
J. F. Schaefer and R. H. Cannon, Jr., On the Control of Unstable 
Mechanical Systems, 
London, England, Jun 20-25, 1966. To be published in the 
Proceedings of the Conference. 
?I 
tt 
tt International Federation of Automatic Control, 
. .  
, .  
J. V. Breakwell and F. Tung, "An Optimal Information and Control 
Policy In Interplanetary Guidance, 
Automatic Control, London, England, Jun 20-25, 1966. To be 
published in the Proceedings of the Conference. 
t* International Federation of 
- 13 - 


Basic Studies in Space Vehicle Attitude Control Tenth Semiannual Status Report, Period Ending Jun. 1966

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