Jul 20, 1983 - ... and Astronautics. 1633 Broadway, New York, NY 10019 ... has been suppressed in Equation (2) for brevity. 3. The dynamics of the two ...
AIAA-84-1987 Autonomous Navigation Using Landmark and Intersatellite Data F. L. Markley, U.S. Naval Research Laboratory, Washington, DC
A I AA/ AAS Ast rody na tnics Conference August 20-22, 1984/Seattle, Washington ~
For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1633 Broadway, New York, NY 10019
90TONOYO'JS URVTGATTON U S l N G LRNDMARK ANI) JNTLRSkTFLLITE DRTA
U.S.
F.L. Y l a ~ k l ~ e + Naval Resawnh L?boratorY 'dashington, 9.C.
Abstract
An
autonomous n a v i g a t i o n system employing t h r e e - a % i s gyros, an e a r t h s e n s o r , sun s e n s o r 5 , and i n t e r s a t e l l i t e r a n g e and a n g l e d a t a f o r ntr.it.ode and o r b i t e s t i m a t i o n i s i n v e s t i g a t e d v i a c o v a r i a n c e analysis. Of p r i m a r y i n t k r e s t i s t h e a b i l i t y o f t h e syStem t o determine t h e l o c a t i o n i n earth-fixed c o o r d i n a t e s o f f e a t u r e s of i n t e r e s t observer) b y t h e e a r t h s e n s o r , i . e . , t h e g e o p o ~ i t i o n i n g accuracy. Known landmarks observed by t h e e a r t h sensor a r e e x c e l l e n t r e f e r e n c e s for g e o p o s i t i o n i n g s i n c e they p r o v i d e h o t h o r b i t and a t t i t u d e i n f o r m a t i o n and a l s o a v o i d p r o b l e m s a P i S i n K f r o m d y n a m i c misa l i g n m e n t s between t h i s p r i m a r y sensor a n d t h e secondary a t t i t u d e sensors. The a d d i t i o n o f i n t e r s a t e l l i t e data s i g n i f i c s n t l y reduces t h e root-sum-square s a t e l l i t e position estimation e r r o r , b u t p r o v i d e s o n l y R modest i n p o v e n m t i n geopositioning accuracy.
3
r e s u l t s of t h i s s t u d y h a v e been r e p o r t e d
Sole
.
p r a v i o u s l y 3.4 m i s p a p e r r e p o r t s on t h r e e e x t e n s i o n s t o t h e model: t h e developnent o f a m o d e l o f a l i n e a r a r r a y e a r t h Sensor o p e r a t e d i n t h e pushbroom mode: t h e i n c l u s i o n o f an a t m o s p h e r i c d r a g parameter i n t h e s t a t e e z t i m a t i o n model, and t h e i i ~ eo f S a t , e l l i t e - t o - s a t e l l i t e r a n e e and a n g l e M !e o n l y con side^ t w o - s a t e l l i t e systems i n data. t h i s work, and s i m u l t a n e o u s l y e s t i m a t e t h e o r b i t and a t t i t u d e of b o t h S a t e l l i t e s , i n c l u d i n g a l l c o - r e l a t i o n s , t,hus a v o i d i n g s t a b i l i t y problems t h a t can a r i s e i n reduced-order community navigation
5
schemes
In t h e f o l l o w i n g s e c t i o n s , we d i s c u s s t h e f i l t e r f o r m u l a t i o n and dynamic models, t h e measwement models, t h e p x a m e t e r s o f t h e t e s t , S y s t e m , and t h e t e s t results. F i l t e r F o r m u l a t i o n and Dynamic Models We employ a s t a n d a r d Kalman f i l t e r c o v a r i a n c e a n a l y s i s i n t h i s study. The 26x26 c o v a r i a n c e m a t r i x P i s propagated arid updated a c c o r d i n g t o
Introduction
P - ( t + A t ) = mPc(t)m T + 4'
(1)
mere h a s been a g r e a t d e a l of i n t e r e s t and r e s e a r c h i n autonomous s p i l c e c r a f t n a v i g a t i o n h i s a c t i v i t y h a s i n t e n s i f i e d i n rec e n t y e a r s due i n p a r t , t o t h e a v a i l a b i l i t y o f h i g h - r e s o l u t i o n imaging sensors and t o t h e i n c r e a s e d power a n d f l e x i b i l i t y o f o n b o a r d c o m p u t e r s y s t e m s , which make autonomous i d e n t i f i c a t i o n of s t a r p a t t e r n s and e a r t h landmarks f e a s i b l e . other factors favoring t h i s a c t i v i t y include t h e d e s i r a b i l i t y of d e c r e a s i n g t h e v o l m e o f s p a c e c r a f t t e l e m e t r y and e l i m i n a t i n g t r a c k i n e s t a t i o n s . Many o p e r a t i o n a l s p a c e c r a f t a r e c a p a b l e o f autonomous a t t i t u d e d e t e r m i n a t i o n . which i s used for a t t i t u d e cant-01; but autonomous o r b i t d e t e r m i n a t i o n i s s t i l l i n t h e experimental s t a g e . For t h i s reason, a s t u d y o f autonomous n a v i g a t i o n schemes i s b e i n g conducted a t NRL. h e aim of this s t u d y i s t o determine t h e o r b i t and a t t i t u d e d e t e r m i n a t i o n a c c w a c y a t t a i n a b l e with c a n d i d a t e autonomous n a v i g a t i o n Systems, and t h e r e s u l t i - 8 p r e c i s i o n i n t h e S p a c e c r a f t ' s a b i l i t y end t o l o c a t e f e a t u r e s o f i n t e r e s t . on t h e s u r f a c e o f t h e e a - t h . me o r b i t and a t t i t u d e e s t i m a t i o n ppoblnns are t r e a t e d i n a coupled f a s h i o n , so t h a t t h e c f f e c t s of o r h i t / a t t i t u d e correlations are not obsclred.
In t h e above and t h e f o l l p w i n g , a s u p e r s c r i p t T d e n o t e s t h e m a t r i x t r a n s p o s e and a s u b s c r i p t (+) d e n o t e s a q u a n t i t y h e f o r e ( a f t e r ) an u p d a t e . ln Fquation ( 1 ) N i s t h e p-ocess n o i s e i n t e g r a t e d from t t,o t + A t and m i s t h e 25x26 S t a t e t r a n s i t i o n matrix. me measurement s e n s i t i v i t y m a t r i x H and t h e measurement, 8rro'- c o v a r i a n c e m a t r i x R, which a p p e a r i n Equation (Z), w i l l be d i s c u s s e d i n t h e s e c t i o n on measurement models. me time p a r a m e t e r , which is t h e Same an b o t h s i d e s o f t h e e q u a t i o n , h a s been s u p p r e s s e d i n Equation ( 2 ) f o r b r e v i t y .
-
3
The d y n a m i c s o f t h e t w o s p a c e c r a f t a r e uncoupled, and wp i g n o r e a n y c o u p l i n g o f t h e o r b i t and a t t i t u d d g y r o - d r i f t dynamics ( e x p r e s s i n g , f o r example, t h e a t t i t u d e dependence o f t h e aerodynamic d r a g c o e f f i c i e n t ) , so t h e s t a t e t r a n s i t i o n m a t r i x can be f w i t t e n i n block-diagonal form
L Where and mRi
mn i
J i s t h e 7x7 o r b i t s t a t ? t r a n s i t i o n matrix
i s t h e 6x6 a t t i t u d e l g y r o - d r i f t s t a t e trans-
i t i o n m a t r i x f o r s p a c e c r a f t i = 1.2. The p r o c e s s n o i s e m a t r i x N can a l s o be w i t t e n i n blockd i a g o n a l form.
*Research P h y s i c i s t , Astrodynamics a n d Space A p p l i c a t i o n s S e c t i o n . Systems Research R a n c h , Member A I A A This paper is declared B work of the U.S. Government and therefore is in the public domain.
The seven-component orbit state vector comprises s i x n o n s i n g u l a r o r b i t elements and a d i m e n s i o n l e a s d r a g parameter d:
L
Q R , and Q
B t o the integrated o r b i t process noise N were c o n s i d e r e d i n d e t a i l i n Reference (3) and W i l l
n o t be r e p e a t e d h e r e .
For low-earth orbits, the principal c o n t r i b u t i o n t o t h e o r b i t p r o c e s s n o i s e w i l l come from unmcdeled terms i n t h e e a r t h ' s g r a v i t y f i e l d , and we can g e t an order-of-magnitude e s t i m a t e o f 0 as follows. Rssume t h a t t h e s t a t e p-opagation a l g o r i t h m uses a g e o p o t e n t i a l o f o r d e r a - 1 , so t h a t t h e main c o n t r i b u t i o n t o t h e p e r t u - b a t i o n a c c e l e r a t i o n s c a n e s from t h e d e g r e e a p o t e n t i a l
L J L 1 where n i s t h e s p a c e c r a f t mean motion, II i s t h e g r a v i t a t i o n c o n s t a n t o f t h e e a r t h , a , e , i , Q, w, M a r e t h e K e p l e r i a n o r b i t e l e m e n t s (semimajor a x i s , e c c e n t r i c i t y , i n c l i n a t i o n , r i g h t ascension of t h e a s c e n d i n g n o d e , a r g u m e n t o f p e r i g e e , a n d mean ananaly a t epoch), P is t h e atmospheric d e n s i t y , a n d CD, A , a n d m a r e t h e S p a c e c r a f t d r a g
where
c o e f f i c i e n t , f r o n t a l a r e a , and mass. r e s p e c t i v e l y . T h i s s t a t e v e c t o r i s chosen t o be n o n s i n g u l a r i n t h e o a s e of z e r o e c c e n t r i c i t y .
a s s u n i n g a s p h e r i c a l e a r t h and c o n s t a n t a t m o s p h e r i c dpag. g i v i n g
r
n
1
0
0
0
0
-sinU
0
0
1
0
0
0
COSU
0
~
-
0
0
1
0
0
0
0
0
0
1
0
A t
0
0
0
0
1
0
0
0
0
0
0
+ sinU
-
and
fS a p e t h e n o r m a l i z e d with c o e f f i c i e n t s
Spherical
Eem
and
Sem
r e s p e c t i v e l y , and P , i s t h e e a r t h e q u a t o r i a l radius. me a c c e l e r & i o n s are a p p c o x i m a t e l y e q u a l t o t h e p o t e n t i a l d i v i d e d by a c h a r a c t e r i s t i c l e n g t h of a/¶.. Then a p p r o x i m a t i n g t h e D i r a c d e l t a f u n c t i o n i n Equation ( 5 ) by an inverse C o r r e l a t i o n t i m e , for which we use n a , and r e p l a c i n g t h e e x p e c t a t i o n v a l u e by an a v e r a g e over t h e u n i t sphere give
The o r b i t s t a t e t r a n s i t i o n m a t r i x i s c a l c u l a t e d
0
fC
harmonic f u n c t i o n s
7
cosu,
(5)
0
n ( 3 / 4 ) ("at)' 1
where we have used K a u l a ' s e s t i m a t e f o r t h e d e g r e e
-
v a r i a n c e ( t h e sum over m) i n t h e second l i n e
7
.
U n c e r t a i n t i e s i n t h e d r a g parameter are modeled a s a randowwalk process
E [ d ( t ) ;I ( t ' ) ] = Qd 6 ( t - t ' )
(9 I
where d o t s denote t i m e d e r i v a t i v e s , and Qd i s a s c a l a r w i t h t h e dimensions o f i n v e r s e time, l i k e Q,,, Q R , and QB. me c o n t r i b u t i o n o f Qd t o t h e i n t e g r a t e d pPoceSS n o i s e N h a s been c a l c u l a t e d . b u t s i n c e t h e r e s u l t i n g f o r m u l a s a r e a l g e b r a i c a l l y con!p l e x and u n e n l i g h t e n i n g , and s i n c e d r a g i s r e l a t i v e l y unimportant f o r t h e t e s t c a s e s c o n s i d e r e d i n t h i s p s p e ~ , t h e e q u a t i o n s w i l l n o t be p r e s e n t e d here.
mA
The a t t i t u d e dynamics models used t o c a l c u l a t e and t h e Corresponding t e r m s i n t h e i n t e g r a t e d
ape i d e n t i c a l t o t h o s e employed i n References 3 and 4 , which a r e d i s c u s s e d a t sqme l e n g t h i n Reference 8. The gyros a r e employed i n t h e model replacement mode, so t h e gyros appear a s p a r t o f t h e system dynamics, w i t h p r o c e s s - n o i s e c o n t r i b u t i o n s Q, a n d Q, ( d e n o t e d Q2 a n d Q 1 ,
process n o i s e
respectively, i n measurements.
Reference 8 ) .
rather
than as
Measurement Models Models o f sun s e n s o r s , fixed-head star t r a c k e r s . l i n e a r a r r a y e a r t h sensors, and i n t e r s a t e l l i t e r a n g e and a n g l e s e n s o r 3 a r e provided i n t h e cova*iance a n a l y s i s program. The measwements o f t h e sun sensops, s t a r t r a o k e r s , and i n t e r s a t e l l i t e a n g l e s e n s o r s h a v e t w o components, which are modeled a s t h e d i s p l a c e m e n t s o f an image i n t h e Sensor f o c a l p l a n e . These measurements have 2x2 error c o v a r i a n c e m a t r i c e s R , which a r e t a k e n t o be 2
m u l t i p l e s o f t h e i d e n t i t y m a t r i x for t h i s s t u d y . h e c o r v e s p o n d i n g measurement s e n s i t i v i t y m a t r i c e s H h a v e 2x26 d i m e n s i o n s , b u t most o f t h e e1ement.s a r e i d e n t i c a l l y z e r o . mus, t h e o n l y nonvanisbing p a r t o f H f o r a s t a r or sun a n g l e measurement i s t h e 2 x 3 p a r t expressing t h e s e n s i t i v i t y t o t h e a t t i t u d e e r r o r angles o f the spacecraft bearing the s e n s o r , w h i l e t h e 14 m a t r i x fo- an i n t e r s a t e l l i t e a n g l e measurement ( t h e a n g l e s o f t h e l i n e - o f - s i g h t t o one s p a c e c r a f t a s measured i n t h e body c a o r d i n a t e s o f t h e s e c o n d ) has a s i m i l a r n o n v a n i s h i i g 2 x 3 p a r t i t i o n , and t w o ? x 6 p s r t s e x p r e s s i n g t h e s e n s i t i v i t y o f t h e measurement t o t h e o r b i t elements o f t h e two s p a c e c r a f t . me s t a r and sun s e n ~ o rmodels a p e i d e n t i c a l t o t h o s e employed p r e v i o u s l y 3 ' " , and t h e i n t e r s a t e l l i t e a n g l e model i s a s t r a i g h t f o - w a r d
extension. 4n i n t e r s a t e l l i t e r a n g e aessurement 'has o n l y one component, so t h e R "matrix" i s a s c a l a r and t h e H m a t r i x i s 1x26, n o n v a n i s h i n g o n l y f o r t h e two 1 x 6 D a r t i t i o n s c o r r e s o o n d i n n- t o t h e s n a ~ e c r a f t o r h i t e l e m e n t s , s i n c e t h e r a n g e measurement i s independent of a t t i t u d e . C i t h e r 0- b o t h o f t h e t,wo s p 3 c e c r a f t may h a v e -1 l i n e a r - a r r a y e a r t h sensor. me f i e l d o f v i e w o f a l i n e a r - a r r a y sensor d e f i n e s a p l a n e , WhiCh i s c h a r a c t e r i z e d by t h r e e o r t h o g o n a l u n i t vect.ors:
R3
d i r e c t e d from t h e s p a c e c r a f t a l o n g t h e cint,er
of
t h e sensor and 0,
= 0
f i e l d o f view.
3
x
0,.
O1 normal
Before a
for
t h i s study)
the
covariance run,
vect.ors
a
.;
from t h e
s p x e c r a f t t o a l l t h e landmarks above t h e h o r i z o n (indexed by i ) a r e coaputed, a s a r e t h e dot products
;i
* G1.
A change
n i s computed
as
the arcsin
o f 6,
91. .
Note t h a t t h e observed value o f t h e f i r s t component o f z . i s always zero. Tbe p a r t i a l d e r i v a t i v e s o f 2.
1
w i t h r e s p e c t t o t h e t h r e e components o f
Fi form
a 7x3 matrix B
v
l?)
The n o n v a n i s h i n g p a r t s o f t h e ?x?6 S e n s i t i v i t y m a t r i x f o r a landmar,L m e a s u r e n m t a p e a 2 x 3 p a r t expressing t h e s e n s i t i v i t y t o the a t t i t u d e error a n g l e s and a ? x 6 p a r t f o r t h e o r b i t elements. In a d d i t i o n , t h e r e i s a 2x3 matrix e x p r e s s i n g t h e measurement S e n s i t i v i t y t o t h e landmark l o c a t i o n u n c e r t a i n t i e s , which a r e t r e a t e d a s " c o n s i d e r " lt i s s t r a i g h t f o r w a r d t o compute t h e s e parameters. u s i n g E q u a t i o n s ( 1 0 ) and ( 1 2 ) a n d a l g o r i t h m s c o n t s i o e d i n Reference ( 3 ) . Note t h a t we do n o t ass~mcne a small field-of-view, so t h e s e n s i t i v i t y m a t r i c e s depend on a. A geopositioning observation i s a s p e c i a l case o f a lsndmar'd measurement, u s i n g t h e same e a r t h sensor, i n t h e l i m i t t h a t t h e h o r i z o n t a l l o c a t i o n
o f t,he "landmark" i s c m p l e t e l y unk,,0m394. me d i s c u s s i o n o f t h e d e t a i l s o f t h i s l i m i t w i l l n o t be repeated here. Test Results
t o t h e plane,
reduced f i l e o f landmarks for each s p a c e c r a f t havi n g a n e a r t h sensor i s s e l e c t e d from a v a s t e r f i l e o f 479 c a n d i d a t e landmarks ( F i g w e l ) , t h u s f a c i l i t a t i n g compsrison runs w i t h t h e same o r b i t elements, b u t with v a r i a t i o n s i n othe. pararnoters. The master f i l e i s a s e t o f r e l a t i v e l y e v e n l y spaced ( m i n i m u m 9 d e g r e e a r c l e n g t h s e p a r a t i o n ) s h o r e l i n e . l a k e , and i s l a n d p o i n t s ; n o a t t e m p t h a s been made i n t h i s s t u d y t o l o c a t e a c t u a l l y wellsurveyed l a n d n s r k s . For t h e r e d u c t i o n p-ocildwe, t h e s p a c e c r ? f t o r b i t i s propagated a r o u n d t h e r o t a t i n g e a r t h . and an e a r t h - r e f e r e n c e d a t t i t u d e i s maintained. A t evenly-spaced time i n t e r v a l s (one minute
where
h e p a r a m e t e r s (used f o r t h e t e s t runs are g i v e n i n T a b l e s 1 a n d ?. Both s p a c e c r a f t o r b i t s a r e sunsynchronous c i r c u l a r o r b i t s a t l a n d s a t - 4 a l t i t u d e . The o r b i t s o f s p a c e c r a f t 1 and 2 a r e i d e n t i c a l e x c e p t f o r r i g h t a s c e n s i o n o f a s c e n d i n g node a and mean anomaly a t epoch M. The p a r a m e t e r s o f t h e e a r t h s e n s o r are based on t h e l a n d s a t h e m a t i c Yapper. The o r b i t p r o c e s s n o i s e l s v e l s a r e e s t i m a t e d u s i n g E q u a t i o n ( 8 ) w i t h L = 3 ; we i m p l i c i t l y assume t h a t a 7x7 g e o p o t e n t i s l mcdel i s used i n t h e onboard n a v i g a t i o n a l g o r i t h m . me d r a e p r o c e s s n o i s e l e v e l i s e s t i m a t e d a s s u m i n g 510 percent v a r i a t i o n s i n atmospheric d e n s i t y ; t h e exact, v a l u e i s n o t c r i t i c a l s i i c e d r a g i s n o t important a t i a n d s a t a l t i t u d e s . Other c o n t r i b u t i o n s t o t h e o r b i t p r o c e s s noise a r e n e g l i g i b l e . T y p i c a l v a l u e s were chosen f o r t h e gyro p r o c e s s n o i s e l~svels4
and 0".
L/
9
o f s i g n of any d o t
p r o d u c t means t h a t t h e pushbroom h a s swept over landmark i d l s i n g t h e time i n t e r v a l i n which t h e s i g n c h a n g e s . L i n e a r i n t e r p o l a t i o n i s used t o f i n d t h e time o f s i g h t i n g and t h e v e c t o r
a t t h e t i m e o f s i g h t i n g . me reduced contains t h e times and a values. f u r t h e r reduced a t t h e b e g i n n i n g o f run t o r e s t , r i c t (L t o he 1.ess t h a n t h e w i d t h o f t h e sensor field-of-view.
landmark f i l e This f i l e is a covariance angular half-
For t h e c o v a r i a n c e a n a l y s i s we assume a t.wocomponent landmark measurement
Sun a n g l e s and i n t e r s a t e l l i t e r a n g e and a n g l e s interva1.s i n t h e run8 i n Which t h e y a r e u s e d . S t a r Sensors a r e n o t tused i n t h e s e '.ests b e c a u s e p r e v i o u s r e s u l t s showed e q u a l l y a r e ohserved a t one-minute
good r e s u l t s w i t h o u t them. ' 9 ' Spacecraft 1 uses landmarks f o r n a v i g a t i o n i n a11 t h e t e s t runs: t h e landmark l o c a t i o n s are assumed t o have' u n c o r r e l s t e d u n c e r t a i n t i e s o f 230 meters ( l o ) i n t h e N/S. E N , and v e r t i c a l d i r e c t i o n s . V e r t i c a l u n c e r t a i n t i e s o f t3O m e t e r s a r e a l s o a s s m e d f o r g e o p o s i t i o n i n g , i h i c h i s s l r n y s c a l c u l a t e d f o r o b j e c t s a t t h e suhs a t e l l i t e p o i n t ( a 3 0). h e presence o f observat i o n s o t h e r t h a n landmarks d i f f e r s among t,he v a r i ous - u n s a s i n d i c a t e d i n Table 2. h i s t a b l e a l s o n o t e s whether s p a c e c r a f t 2 i s a c t i v e or p a s s i v e . An a c t i v e s p a c e c r a f t 2 makes a11 t h e same measmements a s s p a c e c r a f t 1. i n c l u d i n s landmarks. A p a s s i v e s p a c e c r a f t 2 makes no measurements a t a l l , and i t s a t t i t u d e i s i r r e l e v a n t .
L
90' N
v
60' N Figure 1
30' N
Locations of 479 Landmarks T h e 8 landmarks seen by SC 1 in the sample runs are numbered sequentially.
O0
30' S
60's goa s
-I
180' W
goo w
O0
goo E
I
180' E
Figure 2 RSS Spacecraft Position Uncertainties f o r Test Run 6
T h e numbered points a r e landmark sightings.
Figure 3 Geopositioning Uncertainties for Test Run 6 T h e numbered points are landmark sightings. D/N and N/D are day/night and night/day terminator crossings.
An i n i t i a l r u n o f o n e d a y was made w i t h landmark and sun o b s e r v a t i o n s b y b o t h s p a c e c r a f t m e converged and w i t h no i n t e r s a t e l l i t e d a t a . value of t h e covariance matrix. uncorrelated between t h e t h w s p a c e c r a f t , was used as a s t a r t i n g h e v a l u e f o r t h e 19 runs s u m a r i z e d i n Table 7. a v e r a g e i n t e r v a l between landmark s i g h t i n g 3 for t h e one-day r u n was a b o u t 11 m i n u t e s . Each o f t h e 19 r u m s p r e s e n t e d i n Table 2 i s 200 m i n u t e s i n l e n g t h , d u r i n g which t i m e 8 landmarks h e s e landmarks are a r e observed by s p a c e c r a f t 1. numbered s e q u e n t i a l l y i n Figure 1. and t h e l a s t 6 a r e i n d i c a t e d on Figwes 2 and 3 . which p r e s e n t t h e (RSS) s p a c e c r a f t l o c a t i o n e r r o r s Foot-sum-square and g e o p o s i t i o n i n g e r r o r s f o r rum 6 , chosen a s a n example. h e f i r s t two landmarks are observed d w i n g t h e f i r s t 20 minutes o f t h e run, which are n o t p l o t t e d because t h e y a r e d m i n a t e d by m i n t s f e s t i n g t r a n s i e n t behavior. It is i n t e r e z t i n g t o n o t e t h a t t h e s p a c i n g o f t h e landmark o b s e r v a t i o n s i s q u i t e (meven; t h e r e a r e gaps o f g r e a t e r t h a n 40 m i n u t e s between landmarks 2 and 3, between 3 a n d 4 , and a f t e r 8 , b u t o n l y a t o t a l of 50 minutes between t h e r e l a t i v e l y e v e n l y spaced l a n d n a r k s 4, 5, 6 , 7, and 8. Some o f t h e s e gaps c o u l d be e l i m i n a t e d by c h o o s i n g t h e landmarks more d e n s e l y a l o n g c o a s t l i n e s ; t h e sensor s w a t h w i d t h o f 1 8 5 k m i s e q u i v a l e n t t o 1.66 d e g r e e s a r c l e n g t h and can s l i p betheen t h e landmarks spaced a t 4 d e g r e e s . %her g a p s . due t o f l i g h t O Y ~ Plandmark-free ocean a r e a s , are u n a v o i d a b l e . Examination o f F i g w e 2 r e v e a l s t h a t t h e RSS p o s i t i o n e r r o r s o f s p a c e c r a f t 1 and 2 a r e approxim a t e l y e q u a l ; t h i s h o l d s f o r a l l t h e r u n s i n Table 2. Cases with a p a s s i v e s p a c e c r a f t 2 and n o i n t e r s a t e l l i t e range d a t a s h o w d d i v e r g e n c e i n t h e r e l a t i v e RSS and s p a c e c r a f t 2 RSS p o s i t i o n e r r o r s ; t h e s e c a s e s a r e n o t l i s t e d i n Table 2 because a r e a l i s t i c system r e q u i r e s o b s e r v a b i l i t y o f t h e r e l a t i v e position. The r e l a t i v e p o s i t i o n i s observable without i n t e r s a t e l l i t e range d a t a i f b o t h s p a c e c r a f t are a c t i v e ; rums 13, 15, and 1 6 i n Table 2 r e p r e s e n t t h i s case.
We c a n a l s o see i n F i g w e 2 t h a t t h e a b s o l u t e RSS e r r o r s ( t h e p o s i t i o n e r r o r s of s p a c e c r a f t 1 or s p a c e c r a f t 2 ) grow r a p i d l y between landmark observ a t i o n s and drop s h a r p l y a t t h e landmark observations. l h e y r e a c h t h e i r l a r g e s t v a l u e s a t t h e end o f l o n e g a p s i n landmark d a t a , b e f o r e landmarks 3 and 4 a n d a t t h e end o f t h e rum, and t a k e on t h e i r minimm v a l u e s i n t h e 120-160 minute time i n t e r v a l . niis behavior a l s o h o l d s f o r a l l t h e rms i n 3ibl.e 2, e x c e p t t h a t t h e curves are much smoother. a n d t h e magnitudes much l a r g e r , f o r rrns w i t h o u t i n t e r s a t e l l i t e a n g l e d a t a . It is more d i f f i c u l t t o make g e n e r a l s t a t e m e n t s a b o u t t h e r e l a t i v e RSS e r r o r s , s i n c e even t h e q u a l i t a t i v e behavior v a v i e s among t h e d i f f e r e n t rums. h e maxirnm r e l a t i v e RSS v a l u e s i n Table 2 a r e c m p u t e d f o r t i m e s g r e a t e r t h a n 40 m i n u t e s , i n an a t t e m p t t o minimize t h e e f f e c t s o f t r a n s i e n t s , b u t are s t i l l n o t a s r e l i a b l e as t h e o t h e r maxima and minima i n t h e t a b l e . h e g e o p o s i t i o n i n g e r r o r c u r v e s i n Figure 3 show more complex behavior t h a n F i g w e 2. Some s i m p l i c a t i o n r e s u l t s from r e s t r i c t i n g t h e d i s c u s s i o n t o t h e maximim and m i n i m m e r r o r c w v e s , which r e p r e s e n t t h e major and minor s e m i a x i s l e n g t h s of t h e g e o p o s i t i o n i n g e r r o r e l l i p s e . and i g n o r i n g t h e l e s s i n t e r e s t i n g e a s t / w e s t and n o r t h / south e r r o r curves. The maximum a n d m i n i m u m
geopositioning e r r o r s a r e a l s o the e r r o r s i n the i n t r a c k ( a l o n g t h e s p a c e c r a f t ground t r a c k ) and crosstraok directions, respectively. h i s can be Seen from t h e r e l a t i o n s h i p of t h e two s e t s of c u t v e s i n Figure 3, e s p e a i a l l y a t 26, 75, 124, and 173 m i n u t e s , when t h e g r a m d t r a c k r e a c h e s maximun n o r t h o r s o u t h l a t i t u d e , and t h e e a s t / w e s t and intrack directions coincide. The i n t r a c k a n d c r o s s t r a c k g e o p o s i t i o n i n g e r r o r s grow r a p i d l y between landmark o b s e r v a t i o n s , d r o p s h a r p l y a t landmark o b s e r v a t i o n s , aod t a k e on t h e i r m i n i m u m v a l u e s i n t h e 120-160 minute i n t , e r v a l , l i k e t h e lhe geoa b s o l u t e RSS e r r w c u r v e 9 i n Figure 2. p o s i t i o n i n g error curves a l s o e x h i b i t j m p s a t t h e d a y / n i g h t ( D i N ) and n i g h t l d a y (Nil)) t r a n s i t i o n s when t h e S p a c e c r a f t g r o u n d t r a c k crosses t h e t e r m i n a t o r ; t h e s e jumps a r e due s o l e l y t o t h e d i f f e r e n t daytime and n i g h t t i m e r e s o l u t i o n s assumed for t h e e a r t h Senmr. nie maximum i n t r a c k geop o s i t i o n i n g e r r o r s occur a t t h e Same t i m e s a s t h e maximum a b s o l u t e RSS e r r o r s . b u t t h e maximum c r o s s t r a c k g e o p o s i t i o n i n g e r r o r s a r e found n e a r 9 0 A l l these r e s u l t s generalize t o a l l the minutes. r u n s i n Table 2, e x c e p t t h a t t h e maximum c r o s s t r a c k e r r o r s occur a t c l o s e r t @70 m i n u t e s f o r t h e nonc o p l a n a r (Q, = 3.8 d e g r e e ) rums with i n t e r s a t e l l i t e
'd
angle data. T a b l e 2 summarizes t n e r e s u l t s o f 19 o f t h e ? u n a rnade i n t h i s s t , u d y . h e f i r s t thing t o note i n t h i s t a b l e i s t h a t t h e v a r i a t i o n s of t h e RSS s p a c e c r a f t p o s i t i o n e r r o r s ( b o t h a b s o l u t e and r e l a t i v e ) among t h e v a r i o u s rums a r e much g r e a t e r than t h e v a r i a t i o n s o f the eeopositioning e r r o r s . S i n c e t h e T i n c t p a l emphasis of t h i s s t u d y is on t h e g e o p o s i t i o n i n g problem, we w i l l u s e t h e eeop o s i t i o n i n g a c c u r a c i e s a s a b a s i s f o r comparison o f t h e m e r i t s of d i f f e r e n t systems. lhe RSS p o s i t i o n e r r o r s may be o f more i n t e r e s t t o o t h e r s who h a v e d i f f e r e n t o b j e c t i v e s f o r an autonomous n a v i g a t i o n system.
'd
G s o p o s i t i o n i n e e r r o r s a r e much l e s s s e n s i t i v e than RSS p o s i t i o n e r r o r s t o t h e a d d i t i o n o f o t h e r d a t a t o landmark o b s e r v a t i o n s because n a v i g a t i o n w i t h landmark d a t a r e s u l t s i n h i g h l y c o r r e l a t e d a t t i . t u d e and o r b i t e r r o r s t h a t t e n d t o c a n c e l o u t i n a g e o p o s i t i o n i n g measurement. An i n d e p e n d e n t a n g l e measurement r e d u c e s both t h e a t t i t u d e and o r b i t e r r o r s , b u t a l s o reduces t h e c o r r e l a t i o n s among t h e e r r o r s . h u s t h e r e i s l e s s c a n c e l l a t i o n o f a t t i t u d e and o r b i t e r r o r s i n a g e o p o s i t i o n i n g d e t e r m i n a t i o n , and g e o p o s i t i o n i n g a c c w a c i e s a r e n o t reduced as much a s m u l d be e x p e c t e d from t h e r e d u c t i o n i n RSS p o s i t i o n e r r o r s . For example, one can s e e i n Table 2 t h a t t h e g e o p o s i t i o n i n g e r v o r s a r e much s m a l l e r t h a n t h e a b s o l u t e RSS p o s i t i o n e r r o r s i n r u n s without i n t e r s a t e l l i t e angle d a t a , but t h e t h w t y p e s of e r r o r s a r e roughly equal f o r r u n s with i n t e r s a t e l l i t e a n g l e d a t a . I n t e r s a t e l l i t e a n g l e and r a n g e d a t a are v e r y powerful; i t i s shown i n t h e appendix t h a t f o r a l m o s t a l l t w o - s a t e l l i t e c o n f i g u r a t i o n s , measurement o f t h e r e l a t i v e p o s i t i o n v e c t o r i n i n e r t i a l s p a c e g i v e s complete o b s e r v a b i l i t y o f t h e o r b i t s o f both s p a c e c r a f t w i t h o u t r e q u i r i n g any o t h e r d a t a . One o f t h e t w o s p a c e c r a f t must have an independent a t t i t u d e r e f e r e n c e f o r an i n t e r s a t e l l i t e a n g l e measurement t o p r o v i d e i n f o r m a t i o n a b o u t t h e i n t e r s a t e l l i t e d i r e c t i o n i n i n e r t i a l space. rn t h i s s t u d y , t h e a t t i t u d e r e f e r e n c e i s provided i n p a r t by landmark o b s e r v a t i o n s , which a l s o c o n t a i n
kd
Spacecraft
a = 7083.14 km ( a l t i t u d e = 705 km) e = 0 i = 98.208' (sun-synchronous)
(SC) 1 O r b i t
n=
0
w =
0
M = epoch =
W
0 9 : 4 5 AM GMT, J u l y 20, 1983
QN = QR = QB = 10-16sec-l
Process N o i s e L e v e l s
0 . = 10-20 sec-1
Swath w i d t h = 5 7.5 degrees from n a d i i = 185 krn a t e a r t h
E a r t h Sensor v i s i b l e (day)
resolution R
1 . 5 ~ 1 0 - ' r~a d 2
=
= ( 3 0 m e t e r s / r e ) 2 /12 2
2 . 4 ~ 1 0 . ~r a d = (120 m e t e r s / r e ) /21 2
I R (night) resolution R =
orientation
G
= spacecraft v e l o c i t y -u1 = n e g a t i v e o r b i t normal -2 u3 = n a d i r
rad2 = ( 1 arcmin) 2
R =
Sun Sensor
I n t e r s a t e l l i t e A n q l e Sensor R = 1 0 - l '
rad 2
2
= ( 2 arcsec)
2
I n t e r s a t e l l i t e Range Sensor R =: 2 meters
Table 1
Common Parameters for A1 I T e s t Runs
RUN
SC 2
1 2
NONE NONE
3
4
5
6
7 8 9 10 I 1
12
13 14 15
I6
17 18 1
u
DATA':
...
M2
n2
1 RSS ERRORS ( d c g r c c s ) (rndcr.s) MlN MAX SC
_ - - _._ 4.50
550
,370
460 430
S _.... I'ASSTVE R 6.6 0 I'.\SSlVE R S 6.6 0 6.6 0 P A S S I V E AR PASSlVF ARS 6.6 0 PASSIVE RS 6 . 6 3.8 PASSTVE AR 6.6 3.8 PASSIVE ARS 6.6 7 . X P A S S T V E AR 30.0 0 PASSTVC AR 0.1 n ACTIVE RS 6.6 0 ACTIVE AS 6.6 0 ACTTVE ARS 6.6 0 A ACTTVE 6.6 3.8 AS ACTTVI; 6.6 7.8 ACl'llVE RS 3.8 6.6 A C T T V I ? AR 6.6 3.8 AC'rlVr; A R S 6.6 3 . 8
350 310
A6 44 290 44 43 44
REl.AT'IVE RSS E R R O R S (mct,crs) M I N MIAX
-----
GEOPOSIlTONING E R R O R S (soters) M1.N MAX MAX CROSS'PR I N T R
~ - - 33 --32
140
--
120 120 100
A. 0,
110 100 100
60
80
33
370 120
32
77
31
79 63
12
Ad ~,
- I
70
6A
120
i0
370
30 9
28 31
54 59 49
110 110
.
32 67
EO 20
25
140
81 63
inn
120
9 52
87
25 28
62
170
1
7
270 35 35 38
370
49 50 14 44
.,7 7
7 ,11 -
140
26
93
26 27
53 53
120
21
51
29
8. 6 6fl
9 0
19 I7
36
~~
120 100 110 100
17 2kb
370 84 84
34 03
6
A, 7"
6q
31
57
63
7
100 I10 100
86
2c
_ I
A0
110 100
j8
100
22 22
47 46
73 72
information about t h e o r b i t , d e p e n d on i n t e r s a t e l l i t e In fact, a l l observability. o r b i t runs i n Table 2 ( t h o s e
so we do n o t h a v e t o data for f u l l ovbit t h e c i r c u l a r coplanar with Q 0 ) are cases
-
2 -
f o r which i n t e r s a t e l l i t e d a t a a l o n e would n o t g i v e o b s e r v a b i l i t y , a s i s a h o m i n t h e appendix. Corn p a r i s o n o f Otherwise i d e n t i c a l p a i r s of c o p l a n a r and non-coplanar r u n s ( 5 w i t h 8, 6 w i t h 9, 1 3 with 16, 14 w i t h 1 9 ) shows s l i g h t l y b e t t e r performance with "on-coplanar o r b i t s , a s would be e x p e c t e d . Comparing r u n s w i t h and w i t h o u t sun d a t a i n Table 2 shows t h a t t h e presence o f sun d a t a h a s a v e r y s m a l l e f f e c t i n t h e non-coplanar r u n s , and o n l y s i g n f i c a n t l y improves t h e maximum c r o s s t r a c k g e o p o s i t i o n i n g p e r f o r m a n c e i n t h e c o p l a n a r and s i n g l e - s a t e l l i t e runs; b u t sun s e n s o r s a r e so e a s y t o p r o v i d e compared t o t h e o t h e r o p t i o n s c o n s i d e r e d t h a t we w i l l always i n c l u d e them i n t h e comparison o f d i f f e r e n t sensor o p t i o n s t h a t f o l l o w s . m e improvement i n g e o p o s i t i o n i n g v o v i d e d by l i n k i n g two a c t i v e s a t e l l i t e s i s n o t g r e a t enough t o j u s t i f y t h e expense of providing a second Landsat t y p e s a t e l l i t e , u n l e s s t h e r e i s Some o t h e r r e a s o n t o have t w o such s a t e l l i t e s i n o r b i t s i m u l taneously. I n t h e l a t t e r c a s e , t h e two s a t e l l i t e s w i l l e i t h e r be i n c o p l a n a r or non-coplanar o r b i t s f o r reasons n o t d i r e c t l y r e l a t e d t o n a v i g a t i o n . mus we can r e s t r i c t o w o m p a r i s o n s t o t h r e e The f i r s t WOUping (runs 2, 4. groupings o f TMS. 6, 7, 9 ) shows t h e r e s u l t s o f a d d i n g a p a s s i v e second s a t e l l i t e , e i t h e r c o p l a n a r or "on-coplanar, t h e second (runs 2. 12. 13, 1 4 ) shows t h e r e s u l t s o f a d d i n g i n t e r s a t e l l i t e d a t a between t u o c o p l a n a r a c t i v e s a t e l l i t e s , and t h e t h i r d (runs 2. 16. 17, 1 9 ) shows t h e r e s u l t s o f adding i n t e r s a t e l l i t e d a t a between t w o non-coplanar a c t i v e s a t e l l i t e s .
I n t h e runs w i t h t w o a c t i v e s p a c e c r a f t , we a s s u n e t h e g e o p a s i t i o n i n g t o be "on-cooperative. That i s , we assume t h a t t h e t w o S p a c e c r a f t do n o t This i s a a t t e m p t t o g e o l o c a t e t h e same o b j e c t . r e a s o n a b l e assumption f o r p r a c t i c a l a p p l i c a t i o n s , s i n c e t h e o r b i t s o f t h e t w o s p a c e c r a f t would p r o b a b l y be chosen t o maximize e a r t h c o v e r a g e , which p - e c l u d e s having t h e o v e r l a p p i n g f i e l d s - o f view r e q u i r e d f o r c o o p e r a t i v e g e o p o s i t i o n i n g . Canparing runs 2, 4 , a n d 7, among t h e group w i t h a s i n g l e s a t e l l i t e OP a p a s s i v e s e c o n d s a t e l l i t e , shows t h a t v e r y l i t t l e i m p r o v e m e n t ~ e s u l t sf r o m a d d i n g i n t e r s a t e l l i t e r a n g e d a t a without adding i n t e r s a t e l l i t e angle d a t a . In f a c t , run 6 w i t h r a n g e and a n g l e d a t a b e t w e n c o p l a n a r s a t e l l i t e s o u t p e r f o r m s r u n 7, w i t h r a n g e Only b e t w e n "on-coplanar satellites. Thus, t h e most i n t e r e s t i n g r u n s a r e 6 a n d 9, with i n t e r s a t e l l i t e r a n g e and a n g l e d a t a . I n s p e c t i o n o f t h e s e shows t h a t a non-coplanar S a t e l l i t e p a i r o u t p e r forms a coplanar S a t e l l i t e p a i r , but t h e d i f f e r e n c e i n performance may n o t be g r e a t enough t o j u s t i f y t h e added c o m p l e x i t y o f t h e non-coplanar c o n f i g r a t i o n . The c o p l a n a r case i s e a s i e r t o implement because t h e i n t e r s a t e l l i t e a n g l e s e n s o r can be o r i e n t e d i n a fixed direction i n t h i s c o n f i g v a t i o n . In t h e non-coplanar c a s e , t h e a n g l e S e n s o r m u s t be designed t o Cover sane r a n g e o f y a w a n g l e s . '30 d e g r e e s f o r Q2 = 3.8 d e g r e e s and % I: 6.6 d e g r e e s .
Thus t h e c o p l a n a r c o n f i g v a t i o n with b o t h a n g l e and range d a t a is probably p r e f e r r e d for a p a s s i v e second s a t e l l i t e .
I f b o t h s a t e l l i t e s a r e a c t i v e , we h a v e a c h o i o e o f r a n g e o n l y , a n g l e o n l y , or r a n g e and a n g l e s y s t e n s , s i n c e i n t e r s a t e l l i t e range d a t a a r e not r e q u i r e d for o b s e r v a b i l i t y of t h e r e l a t i v e s a t e l l i t e p o s i t i o n . Comparing t h e c o p l a n a r runs 12, 13, and 1 4 and t h e non-coplanar r u n s 16, 17, and 19 with t h e s i n g l e s a t e l l i t e run 2 r e v e a l s t h a t a n g l e measurements provide 31moSt a l l o f t h e imvovement i n t h e minimun g e o p o s i t i o n i n g e r r o r and t h e maximum o r o s s t r a c k g e o p o s i t i o n i n g erPor and Sane of t h e improvement i n t h e maximun i n t r a c k g e o p o s i t i o n i n g e r r o ? , w h i l e r a n g e d a t a provide a l a r g e r s h a r e o f t h e maximum i n t r a c k improvement, b u t a l m o s t none o f t h e imvovement i n t h e maximun c r o s s t r a c k and minimum g e o p o s i t i o n i n g BPPOPS. Thus we c e r t a i n l y want t o i n c l u d e i n t e r s a t e l l i t e a n g l e d a t a a n d probably want t o i n c l u d e i n t e r s a t e l l i t e r a n g e d a t a a s w e l l , so t h e v e f e r r e d coplanar c o n f i g u r a t i o n i s t h a t c o n s i d e r e d i n pull 14 (or p o s s i b l y 1 3 ) , and t h e pref e r r e d non-coplanar c o n f i g u r a t i o n i s t h a t of r u n 1 9 ( p o s s i b l y 1 7 ) . The u l t i m a t e c h o i c e w i l l depend on t h e added c m p l e x i t y of a n i n t e r s a t e l l i t e r a n g i n g system.
L/
F i n a l l y , i t can be seen from t h e r e s u l t s o f r u n s 5, 10, a n d 1 1 t h a t l o n g e r b a z e l i n e s g i v e g r e a t e r improvements i n g e o p o s i t i o n i n g , b u t t h a t t h e improvement b e t w e e n a 6.6 d e g r e e a n g u l a r s e p a r a t i o n (815 km r a n g e ) and a 30 degree a n g u l a r s e p a r a t i o n ( 3 6 6 7 km r a n g e ) i s l e s s t h a n t h a t b e t E e n 6.6 d e g r e e s and 0.1 d e g r e e s ( 1 2 km r a n g e ) . Conclusions Covariance a n a l y s i s i n d i c a t e s t h a t landmarkbased a u t o n m o u s n a v i g a t i o n o f a s a t e l l i t e with o r b i t and s e n s o r P a r a m e t e r s chosen t o r e s e m b l e Landsat can r e s u l t i n 30-140 meter g e o p o s i t i o n i n g accupacy. t h e a c c u r a c y o f l o c a t i n g f e a t u r e s o f i n t e r e s t on t h e s t r f a c e of t h e e a r t h . me minimum g e o w s i t i o n i n g e r r o r s occur i m m e d i a t e l y a f t e r a landmark u p d a t e : and t h e maximm g e o p o s i t i o n i n g e r r o r s , which a r e i n t h e d i r e c t i o n o f t h e spaaec r a f t ' s ground t r a c k , OCCUT a f t e r g a p s i n t h e landmark d a t a , which can exceed 40 minutes w i t h t h e f i l e of 479 s i m u l a t e d landmarks used i n t h i s s t u d y . The a d d i t i o n o f a n g l e and r a n g e measurements t o p a s s i v e cc-orbiting S a t e l l i t e (with i d e n t i c a l o r b i t p a r a m e t e r s e x c e p t for mean anomaly) r e d u c e s t h e maximm g e o p o s i t i o n i n g e r r o r s t o 100 m e t e r s , w i t h o u t r e d u c i n g t h e minimun W P O P S s i g n i f i c a n t l y . Although it is i n t e r e s t i n g t h a t t h e s e measwements a r e e f f e c t i v e i n p r o v i d i n g t h e above r e d u c t i o n i n t h e g e o p o s i t i o n i n g e r r o r s , and a 7 5 p e r c e n t r e d u c t i o n i n t h e root-sum-square l o c a t i o n e r r o r s o f t h e s p a c e c r a f t , t h e modest improvement probably d o e s n o t j u s t i f y t h e a d d i t i o n a l complexity o f t h e s u b s a t e l l i t e and s e n s i n g system. a
I f a p a i r o f L a n d s a t - t y p e s a t e l l i t e s were continUOUSly i n view o f one a n o t h e r , t h e p r o v i s i o n o f r a n g e and a n g l e measurements between t h e two s a t e l l i t e s would r e d u c e g e o p o s i t i o n i n g e r P o r s t o 25-90 m e t e r s f o r c o p l a n a r o r b i t s and 20-70 m e t e r s for "on-coplanar o r b i t s , and a l s o r e d u c e root-sums q u a r e l o c a t i o n e r r o r s by 80 p e r c e n t . P r o v i s i o n o f t h e s e measwements would s t i l l i n v o l v e a s i g n i f i cant amotmt of comulexitv. e s o e c i a l l v i n t h e nonc o p l a n a r c a s e , s i n c e S m e S o r t o f onboard t r a c k i n g System would be r e q u i r e d . \.-
W
+
h e o v e r a l l performance i n d i c a t e d for landmarkbased autonomous n a v i g a t i o n s y s t e m s i s p r o m i s i n g , b u t t h e onboard d a t a s t o r a g e and p r o c e s s i n g requirements are s i g n i f i c a n t , including a f i l e o f hundreds o f lendmark images. me v o v i s i o n of i n t e r s a t e l l i t e d a t a g i v e s o n l y a modest improvement i n geopositioning accwacy, especially considering t h e i n c r e a s e d c o m p l e x i t y o f t h e system.
C , ~ Y . t I0 ax
- I
01
(A.6)
h e f u l l s t a t e i s observable only i f there i s sane i n t e g e r k f o r which t h e 3k x 1 2 o b s e r v a b i l i t y matrix
Appendix 10
.
rank
m e following statement i s equivalent t o t h e c o n d i t i o n on a perturbation x is
flk:
u n o b s e r v a b l e i f and o n l y i f
*
=
(A.9)
t
(A.10)
6Y
= Ax
(A.11)
2
Equation ( A . 9 ) and (A.10) unobservable o n l y i f 6 ( r l
(A.2)
-$owr 2 ) =t h1a t andt h ea ( vp, , -e v2)i s
= 0 : t h u s t h e f u l l r e l a t i v e s t a t e i s observable, t h e o n l y u n o b s e r v a b i l i t y can be i n t h e determination of the absolute state. Equation ( A . l l ) , a l o n g with Eq. ( A . 4 ) . is a n e c e s s a r y c o n d i t i o n f o r absolute s t a t e unobservability; t h e reference o r b i t s m u s t be s u c h t h a t t h e s a t e l l i t e s w i l l e x p e r i e n c e e q u a l f o r c e s i f given equal d i s p l a c e ments. It i s n o t d i f f i c u l t t o show from Eq. ( A . 4 ) t h a t t h i s c o n d i t i o n can be s a t i s f i e d o n l y i f
with
(A.3)
Here I i s t h e 3 x 3 i d e n t i t y m a t r i x , matrix o f z e r o s , and
= i) 6;
8 6; 1
2
(A.12)
= 72
'1
0 is the 3x3
for a l l time. h i s demands t h a t t h e two s p a c e c r a f t always be a t t h e Same d i s t a n c e from t h e C e n t r a l body; t h u s t h e y must be i n o r b i t s w i t h t h e same semimajor a x i s and e c c e n t r i c i t y . and t h e y m u s t be phased so t h a t t h e y r e a c h p e r i g e e s i m u l t a n e o u s l y . I n t h i s case we a l s o have
* where r . i s e v a l u a t e d on t h e r e f e r e n c e o r b i t .
v
A s i m p l i f i e d measurement model i s also adopted i n t h e a n a l y t i c apvoach. h e three-vector
-
k.
a r e t h e Cartesian p o s i t i o n and
v e l o c i t y p e r t u r b a t i o n s o f s p a c e c r a f t i L 1 , 2, e x p r e s s e d i n an i n e r t i a l c o o r d i n a t e s y s t e m . h e time-dependence o f t h e s t a t e v e c t o r i s
i s assmed
0 for a l l
three necessary
t
2 -
+ + * y = r1 - r 2
I
6r
6r2
1 -
+ + 6 v 1 = 6"
Y
flkx
From ( A . 1 ) a n d ( R . 9 ) we f i n d c o n d i t i o n s for u n o b s e r v a b i l i t y :
(A.1)
and 6;.
It i s e a s y t o show, u s i n g and ( A . 7 ) . t h a t for k = 3
(A.6)
(A.8)
6;
where
.
i s of r a n k 1 2 Eqs. ( A . 3 ) . (A.4),
Observability is discussed lbsing standard r e s u l t s o f l i n e a r systems t h e o r y me system dynamics must be e x p r e s s e d a s l i n e a r f i r s t - o r d e r A s i m p l i f i e d model i s d i f f e r e n t i a l equations. needed f o r t h i s d e r i v a t i o n so t h e s a t e l l i t e s are assuned t o be i n Kepler o r b i t s a b o u t a s p h e r i c a l l y symnetric e a r t h . Since t h e e q u a t i o n s f o r t h e s p a c e c r a f t wsitions and v e l o c i t i e s are n o n l i n e a r , i t i s c o n v e n i e n t t o expand t h e e q u a t i o c s a s T a y l o r ' s s e r i e s a b o u t some r e f e r e n c e o r b i t s , and take t h e f i r s t - o r d e r p e r t u r b a t i o n s as t h e elements o f t h e s t a t e v e c t o r . Thus t h e stat.e v e c t o r h a s 1 2 canponents:
1
=
"2
* and I
+
v1
* 1
r2
+
* v2
(A.13)
f o r a l l time.
I n a d d i t i o n t o Eq. ( A . 1 2 ) . either
(A.5)
t o be
c o n t i n u o u s l y measured. his v e c t o r i n c l u d e s t h e i n t e r s a t e l l i t e r a n g e and t h e d i r e c t i o n o f t h e i n t e r s a t e l l i t e vector i n i n e r t i a l space. In order for t h e d i r e c t i o n t o be measured, one o f t h e t w o s p a c e c r a f t m u s t h a v e an a c c 6 a t e inertial attitude. m e measurement model o f Fq. (A.5) g i v e s t h e 3 x 1 2 measurement s e n s i t i v i t y matr i x .
i,
= 2
F1
.
it i s n e c e s s a r y t h a t
F2
or
for
8
Eq.
+ + 6r = P 2 * 6;
(A.11)
= 0
t o hold.
(A. 14)
Since
= ;2
is
i m p s s i b l e p h y s i c a l l y and
T1
=
- T2
means t h a t t h e
two s p a c e c r a f t are on a p p o s i t e s i d e s o f t h e c a r t h and t h u s unable t o s e e each o t h e r , t h e former case i s o f n o p r a c t i c a l importance. Thus a b s o l u t e s t a t e u n o b s e r v a b i l i t y a r i s e s o n l y i f Eqs. (11.12) and (A.14) a r e s a t i s f i e d . Equation ( A . 1 4 ) shows t h a t t h e unobservable a b s o l u t e p o s i t i o n i s i n t h e o u t of-plane d i r e c t i o n . t h e p l a n e b e i n g t h a t c o n t a i n i n g t h e t w o s p a c e c r a f t and t h e c e n t r a l body. F u r t h e r r e s u l t s f o l l o w from r e q u i r i n g &x
The q u a n t i t i e s d ( j ) and e ( j ) a r e s c a l a r f u n c t i o n s k k + * of r i , v . a n d v; and with t h e r e s t r i c t i o n of Eqs. 1’
(A.12) and (A.13) t h e y a r e a l l independent o,f t h e s p a c e c r a f t index. S u b s t i t u t i o n of Eqs. (A.22) and (11.23) i n t o Eqs. (A.16) (A.19) l e a d s , a f t e r some algebra, t o the conditions
= 0
-
for k > 3. me general o b s e r v a b i l i t y m a t r i x can be w r i t t e n i n t h e form
3
(j) = e ( 3 5
(A.24)
tA.15)
... .. ..
f o r a l l j . and t o r e c u r s i o n r e l a t i o n s fop a l l t h e d i ( j ) and e i ( j ) c o e f f i c i e n t s .
It i s e a s i l y seen from Eqs. (A.3). t h a t t h e 3x3 m a t r i c e s D i ( J )
(A.6). and (A.7)
and E i ( j )
Sinoe f a n d f, a r e l i n e a r l y i n d e p e n d e n t 1 v e c t o r s ( t h i s was assuned i n r e j e c t i n g T1 = T2 i n
a r e given
r e c w s i v e l y by Di(l)
r
f
I
(A.16)
+ t 6r = g r
Ei(l)
E
t
+
1
OF
a?,
Since 6;
i s t h e time d e r i v a t i v e
we also h a v e +
6V
S,x
(A.26)
r2
for Sane scalar E.
(A.17)
0
+
f a v o r of Eq. ( A . l U ) ) , t h e t r i a d f , , r2. r 1 x r 2 forms a b a s i s ; and we can w r i t e v i t h Eq. (A.14)
.*
= W
x
f* +
g ( t1 x
T2 +
;1
x
T2)
(A.27)
Now t h e r e q u i r e m e n t f o r u n o b s e r v a b i l i t y , t h a t = o for a l l k , g i v e s w i t h ~ q s . ( A . I ) , (A.9),
( A . 1 0 ) and (A.15) D1 ( ” 6 ;
+
El(j)6;
D2 ( ” 6 ;
+ E2‘j’6;
(A.28)
The e q u a t i o n s of motion a l o n g t h e r e f e r e n c e o r b i t are S u b s t i t u t i o n o f Eqs. for a l l j. g i v e s t h e e q u i v a l e n t form
df. 1 dt =
- ti
(A.27)
and
(A.23)
(A.20)
It f o l l o w s from Eqs. ( A . 4 ) the matrices D i ( j )
-
(A.22)
and (A.16)
and E i ( j )
-
(A.21) t h a t
can be r e w i t t e n
Comparison
o f Eqs.
(A.8),
shows t h a t ek ( J ) = 0 for j
5
(A.15).
3 and k > 1.
n u s Eq.
This (A.29) is i d e n t i c a l l y s a t i s f i e d for j 5 3. c l e a r l y h o l d s because Eqs. (A.9) (A.12) and ( A . 1 4 ) were d e r i v e d by r e q u i r i n g t h a t = 0. For j I: 4 we have e (4) (4) = 0, so no new 3 = e 4 unobservability conditions a r i s e i n t h i s order. For j = 5 we have
-
9
4
L/
;, - F2
direction of
?2 i s
and t h u s t h e d i r e c t i o n of
T1 -
lt i s not d i f f i c u l t t o show t h e
constant.*
has a either + ?i or F1 - 2; 1 2 c o n s t a n t d i r e c t i o n , t h e n F4. ( A . 3 3 ) h o l d s and t h e st,ate i s unobservable. lhis i n c l u l e s t h e c a s e o f
converse; i f
coplanar o r b i t s t r a v e r s e d i n o p p o s i t e senses. which t h e d i r e c t i o n s o f b o t h
so r e q u i - i n g u n o b s e r v a b i l i t y for j = 5 g i v e s t w o
.
*
= +" 2
6"
.
n
=
6;
-
(A.30)
6%
and
witing
result
the
+
i n t h e c a s e t h a t 1;
It i s c l e a r from Eq. ( A . 2 9 ) t h a t t h e s e condit.ions = 0 far a l l k. S u b s t i t u t i n g Fq. ( A . 2 7 ) give i n t o these equations m a t r i x form g i v e s
for
- F2
and 1;
are constant,. For a l l u n o b s e r v a b l e non-copl.anar o r b i t C a s e s , t h e two s p a c e c r a f t cross t h e l i n e o f i n t e r s e c t i o n o f t h e o r b i t p l a n e s SirnultaneOuSlY: * + + * and r 1 + r 2 = 0 (or r 1 r 2 = 0) a t these crossings
additional conditions :
jl
F1 + i2
;2)
F2
+
(or c o r r e s p o n d i n g l y r
has a constant direction.
-
1 h e s e last results
are b e s t Seen u s i n g an e x p l i c i t r e p r e s e n t a t i o n o f t h e Keplerian 0rbit.s. It i s worth n o t i n g t h a t + * c a s e s where t h e d i r e c t i o n of r l - r 2 i s c o n s t a n t ,
in
and o n l y t h o s e c a s e s , a r e c o l l i s i o n c o w s e s f o r t h e
two s p a c e c r a f t . For t h e c o p l a n a r c a s e s , Fq. ( A . 3 1 ) i s t r i v i a l l y s a t i s f i e d and t h e r e are two a r b i t r a r y c o n s t a n t s i n I n order for a Don-tvivial s o l u t i o n t o e x i s t , t h a t is i n o r d e r f o r t h e s t a t e t o be unobservahle, t,he d e t e r m i n a n t o f t h e matrix must v a n i s h . After s t r a i g h t f o r w a r d a l g e b r a maHng U S E o f Eqs. ( A . 1 2 ) and ( A . 1 3 ) .
det =
*v2
(;
. (F2
x
-
r*2
;l)[(?l
i,, - (T 1
1
6f(t) = c
C1)
x
x $ )2 I 1
or
;.: 1
i2x ; 2
(A.32)
2 = r 2 - *v1
(A.33)
+
and
thus
that
orthogonal t o
(C1 + C2)
+ T~
=
either x
+
r2.
. G1 - F2) + r+2 ,
Now;
r+ 1
-
*
c1
6,
Y
+
;2
(A.141,
(A.311)
and ( A . 3 0 ) .
are dimensionless constant vectors
El
or
b,
h a s two i n d e p e n d e n t canpon-
*
In t h e s e c a s e s , e i t h e r r
* + r2 i s
l:
+
"2
Or
" +1
; 2
1
- :2) +
r+2 , and r
. x
( ; ,
+
; ) 2
i s constant,
or
;l
is
*
x t;,(t)
i ;,(t)l
(A.35) where c i s a
Analytic o b s e r v a b i l i t y c o n s i d e r a t i o n s g i v e a yes-no answcr t o t h e q u e s t i o n o f o b s e r v a b i l i t y . C o n f i g w a t i o n s t h a t are t h e o r e t i c a l l y o b s e r v a b l e ,
- 0
-
+
r2 ~ o n s t i t u t ea
- G2
= c ti
To S m a r i z e , t h e c a n p l e t e state i s o b s e r v a b l e w i t h t h e measweement m d e 1 o f Eq. ( A . 5 ) u n l e s s t h e two s p a c e c r a f t are i n o r b i t s with t h e same semimajor a x i s and e c c e n t . r i c i t y , phased so a s t o r e a c h p e r i g e e s i m u l t a n e o u s l y : and i n a d d i t i o n , e i t h e r t h e o r b i t s a p e c o p l a n a r o r , i f non-coplanar, t h e y are o r i e n t e d so t h a t t h e two s p a c e c r a f t cross t h e l i n e O f i n t e r s e c t i o n o f t h e two p l a n e s s i m u l t a n e o u s l y .
' h e Same e q u a t i o n s g i v e
= ($
or
Corresponding t o t h e s e two cases.
obeys Eqs. ( A . 2 ) . ( A . 1 4 ) , and (A.30). dimensionless constant scalar.
*
To avoid awkward p h r a s e o l o g y , we s a y t h a t t m v e c t o r s are i n t h e same d i r e c t i o n i f t h e y a r e e i t h e r p a r a l l e l or a n t i p a r a l l e l . a n d t h a t t h e d i r e c t i o n o f a v e c t o r i s c o n s t a n t i f i t is e i t h e r p a r a l l e l 01' a n t i p a r a l l e l t o a c o n s t a n t v e c t o r .
i m p l i e s t h a t e i t h e r G1 + ;2 i s + * i n t h e d i r e c t i o n o f r 1 + r 2 , and t h u s t h e d i r e c t i o n
F1
x ;2(t)
r2 + i s i n the direction of a constant m i t
6;(tl
r,)l 2
+
1
-
v e c t o r 0-.
b a s i s , so Eq. ( A . 3 3 )
Of
z2
i n t h e dit-ection o f sane c o n s t a n t u n i t v e c t o r 0 r
Equation ( A . 3 2 ) j u s t S t a t e s t h e e q u a l i t y o f t h e o r b i t a l a n g u l a r rnanenta o f t h e t m s p a c e c r a f t : t h i s h o l d s i f t h e o r b i t s are c o p l a n a r and are tiaYerSed i n t h e Same sense. lhe c o n d i t i o n s under which Eq. t A . 3 3 ) h o l d s a r e n o t so o b v i o u s . T t can be shorn from Eqs. ( A . 1 2 1 , ( A . 1 3 ) . and ( A . 3 3 ) t h a t x ;91*
d2
and
c o n s t a n t i n 6;.
t h e r e f e r e n c e o r b i t guarantee t h a t t h e y w i l l hold for a l l time.
(Tl
i
e n t s , e a c h r e r e s e n t a t i o n h a s t h e c o r r e c t nunber o f i n d e p e n d e n t c o n s t a n t h . For t h e non-coplanar c a s e s , Eq. (A.31) gives a first-order differential e q u a t i o n f o r g , so t h e r e i s o n l y one i n d e p e n d e n t
r 2 for a l l t i m e , t h e n t h e e q u a t i o n s o f motion a l o n g
*
G1
since either
for u n o b s e r v a b i l i t y . lhese c o n d i t i o n s a r e t i m e i n d e w n d e n t : i f t h e y h o l d a t one time and i f r 1 =
r;,
x r (t) 1
see t h a t t h e t w o
i n t h e common o r b i t plane. It i s a l s o e a s y t o see t h a t t h e s e t w o r e r e s e n t a t i o n s a r e e q u i v a l e n t , and
J x 1;
,
1
a r e consistent with Eqs. ( A . 2 ) . where
n u s we must have e i t h e r
F1
difficult to
6;. It i s n o t representations,
is in the
in
b u t a r e " c l o s e " t o unobservable c o n f i g u r a t i o n s i n Some s e n s e , w i l l u s u a l l y g i v e m a r g i n a l observab i l i t y i n t h e p-esence of measuement n o i s e and Process noise. Numerical computations a r e i n d i s p e n s a b l e for r e s o l v i n g t h e s e q u e s t i o n s . References
1. E r o g a n , W.L. a n d LeMay. J.L. (1973), "Autonomous S a t e l l i t e N a v i g a t i o n An l i i s t o r i c Sunmary and C u r e n t S t a t u s , " i n Roc. 1913 J o i n t Automatic C o n t r o l Conference, June 1973.
-
"Autonmnous ., Navigation Systems 2. L o w i e . .I.# A I A A Paper 79-0056, hiew Technology Assessment Cr l e a n s , Loui s i a n a , January, 1979.
3.
Markley, F.L., "Autonomous S 3 t e l l i t e Navigation Using landmarks", AAS Paper 81-205, lak Tahoa, Nevada, August 1981.
4. Markley, F.L.. " G e o p o s i t i o n i n g Accllracy o f an Autonomous N a v i g a t i o n S y s t e m U s i n g Landmarks", Proceedings of the 1982 American Control Conference, A r l i n g t o n . VA, June, 1982. pp. 628-673.
5.
M a j w , C.S..
"Autonomous S p a c e c r a f t Navigation L o c a l l y Optimal F i l t e r s ,'* A I A A pp. 352-358, G a t l i n b w g , TN, August.
Using D i s t r i b u t e d , Paper 83-2210, 198 3.
6. May, J a n e t A.. " A S t u d y o f t h e E f f e c t s o f S t a t e l i - a n s i t i o n M a t r i x Approximations," NASA/GSFC F l i g h t Mechanics/Estimation m e o r y Symposium, G r e e n b e l t , MD. October. 1979.
7. K a u l a , W i l l i a m A.. Theory of S a t e l l i t e Geodesy, B l a i s d e l l , Waltham, M A , 1966, p. 98. 8. L e f f e r t s , E.J.. Markley, F.L.. and m u s t e r , M.D., "Kalman F i l t e r i n g for % a c e c r a f t A t t i t u d e E s t i m a t i o n " , J o u r n a l if G u i d a n c e , C o n t r o l , a n d Dynamics, Vol. 5, Sept-Oct, 1982. pp. 417-429. Also A I A A paper 82-0070, m l a n d o , FL, J a n u a r y . 1982. 9. F a r r e n k o p f . R.L., "Analytic Steady-State A c c u r a c y S o l u t i o n s f o v Two Common S p a c e c r a f t A t t i t u d e E s t i m a t o r s , " J o u r n a l of G u i d a n c e and V O 1 . 1. July-August 1978, pp. 282-284.
-.
10. K a i l a t h , 'Inomas, &tall, Inc.. Fnglewood C
L i n e a r S stems, l i f e
Prentice-
11
