Error equation for inertial navigation systems formulated and solved in state space notatio

Britting, K. R.

English

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https://ntrs.nasa.gov/search.jsp?R=19700008298 2020-03-12T01:43:08+00:00Z
RE-53 
STATE TRANSITION J!lZATRIX 
FOR 
INERTIAL NAVIGATION SYSTEMS 
Kenneth R. B r i t t i n g  
March, 1 9 6 9  
Approved : 
Direc tor  
Measurement Systems Laboratory . 
ii 
ABSTRACT 
This  r e p o r t  formulates  and so lves  i n  s ta te  space 
n o t a t i o n  t h e  error equat ion for  i n e r t i a l  naviga t ion  sys- 
t e m s .  The system i s  assumed t o  be moving a t  a cons t an t  
celestial  longi tude  r a t e .  The s t a t e  t r a n s i t i o n  mat r ix  
i s  e x p l i c i t l y  de r ived  both fo r  long-term and short-term 
opera t ion .  Examples a r e  included t o  demonstrate t h e  ease  
with which t h e  s ta te  t r a n s i t i o n  mat r ix  can be used for  
error a n a l y s i s .  
- 
. 
iii 
ACKNOWLEDGPENTS 
This r e p o r t  was prepared under DSR P r o j e c t  7 0 3 4 3  
sponsored by t h e  Nat ional  Aeronautics and Space Admin- 
i s t r a t i o n  E l e c t r o n i c  Research Center,  Cambridge, Mass., 
through NASA Grant number NGR 22-009-229. 
The p u b l i c a t i o n  of t h i s  r e p o r t  does n o t  c o n s t i t u t e  
approval by t h e  Nat iona l  Aeronautics and Space Adminis- 
t r a t i o n  o r  by t h e  MIT Measurement Systems Laboratory of 
t h e  f i n d i n g s  or t h e  conclusions contained t h e r e i n .  
i s  publ ished only f o r  t h e  exchange and s t imu la t ion  of 
ideas .  
It 
. 
S ec'ti on 
1. 
2. 
3 .  
4.  
5. 
5 .1  
5 . 2  
i v  
TAELE OF CONTENTS 
Page 
In t roduc t ion  
Formulation i n  S t a t e  Space Notat ion 
S t a t e  T rans i t i on  Matrix 
S t a t e  T r a n s i t i o n  Matrix f o r  Shor t  
Sampling T i m e s  
Examples 
I n i t i a l  Condition E r r o r s  
Accelerometer B i a s  E r r p r s  
-2- 
g 'L g r a v i t y  magnitude 
&Nf ED 'L error angles  r e l a t i n g  t h e  computed o r  instrumented 
geographic frame t o  t h e  geographic T r a m e .  These 
error angles  r e s u l t  from p o s i t i v e  r o t a t i o n s  of 
t h e  computed o r  instrument  axes with r e s p e c t  t o  
t h e  geographic axes.  
6L k l a t i t u d e  error 
6~ 'L longi tude  error 
Equation (1) is  v a l i d  f o r  a naviga t ion  system which i s  assumed t o  
be moving a t  a cons t an t  celest ia l  Zongitude rate ( f ixed  base navi- 
g a t i o n  i s  inc luded  as a s p e c i a l  c a s e ) .  I t  has  been assumed t h a t  
t h e  Coriolis compensations are supplied. e i t h e r  from e x t e r n a l  i n -  
formation o r  wi thout  e r r o r .  
2. Formulation i n  S t a t e  Space Notat ion 
Since t h i s  equat ion  arises so f r equen t ly ,  it i s  advantageous 
t o  use s ta te  space methods t o  ob ta in  a s o l u t i o n  which i s  v a l i d  
fo r  an a r b i t r a r y  f o r c i n g  vec to r .  This i s  accomplished by w r i t i n g  
0 
equat ion (1) as follows: 
where 
, 
-3- 
- A =  
* 
- A  s inL 0 -i s inL 0 0 COSL 
i sinL 0 i COSL 0 0 -1 0 
0 -i COSL 0 4 COSL 0 0 -sinL 
0 0 0 0 0 1 0 
0 0 0 0 0 0 1 
0 0 0 0 0 - gr 0 
0 0 0 0 0 0 -- secL g r - 
I B ( t )  7 x gth order  mat r ix  r o t a t i n g  t h e  fo rc ing  vec to r  t o  t he  
sta te  
superdot  t i m e  d i f f e r e n t i a t i o n  
3 .  State  Trans i t i on  Matrix 
The solutLon t o  equat ion ( 2 )  i s  given i n  t e r m s  of t h e  s t a t e  
A t  t r a n s i t i o n  matr ix ,  - @(t) = e- as: 
The s t a t e  t r a n s i t i o n  mat r ix  sat isf ies  the  mat r ix  d i f f e r e n t i a l  equa- 
tion: 
-4- 
where - I is the identity matrix, and the composition law: 
- @(t) = - @(t-to) ?(to) 
from which it follows that: 
Q 
3 @--"t) = - @(-t) 
The transition matrix is found from the relationship: 
-1 -1 @(t) = L Q S - A )  - 
where 
S % Laplace operator 
L - ~  inverse Laplace transformation 
0 - l  i, matrix inversion opbrator 
Applying equation ( 5 ) ,  the state transition matrix is found to be 
given by: 
_. 
-5- 
cos w s t  --sinL i sin w t 
W S S 
i -sinL(sin w t- 
-sin At) 
w S " i  . cos w s t  
w 
S 
tanL(cos it- -secL (sin it- 
cos wst) i -sin2L w sin wst) 
S 
sinL(sin it- 
i I ( t )  = - -sin w t) w S 
S 
secL (cos wst- tanL(sin it- 
i ;;;-sin w t) os2L--sin2L cos it) 
S 
S 
w sin w t isinL(cos it- S S 
cos wst) 
- w  secL(sin w s t -  i tanL(cos it- S 
-sin2L i sin it) cos wst) w 
S 
where 
w = 3 % Schuler frequency squared. 
S r 
I 
I 
I 1 I 
i -cosL(sin w e- w S 
S * 
cos L t) s i  -sin w At) S 
S 
0 --sin 1 w t S 0 
I 
-- 
sinL(l-cos it) 
icosL (cos it- -i(sin it- 
---sin w t) i w S cos wst) S 
0 cos w s t  0 
isinL(sin it- i tanL ( cos it- 
i -sin w t) 
S 
cos w s t )  
0 I 0 cos wst 
x 
-6- 
_- - - 
1 -it sinL 0 -it sinL o 0 t COSL 
it sinL 1 it COSL 0 0 -t 0 
0 -it COSL 1 -it COSL o o -t sinL 
@(t)  = 0 0 0 1 O t  0 
0 0 0 0 1 0  t 
0 us- 0 0 0 1  0 
- u s 2 t  secL 0 0 0 0 0  1 
- 
I- - 
4. S t a t e  T r a n s i t i o n  Matrix f o r  Shor t  Sampling Time's  
(7 )  
The s t a t e  space approach is  used when opt imal  f i l t e r i n g  tech- 
niques a r e  appl ied  t o  t h e  i n e r t i a l  naviga t ion  system. I n  t h i s  s i t u -  
a t i o n ,  t h e  s ta te  t r a n s i t i o n  mat r ix  i s  used t o  model t h e  system's  
behavior over t h e  sampling t i m e ,  T. Thus s m a l l  angle  assumptions 
can be made i n  t h e  above expression:  
e < 10% f o r  T < 6 min % cos  w s t  = 1, 
, .os2T2 
e < 1 0 %  f o r  T < 1 6  min 2 ,  2 1 -  
s i n  w s t  2 wST, e < 1 0 %  f o r  T < 1 2  min 
w 3 T 3  
6 
S 2 wsT - ,, e < 10% f o r  T < 2 3  min 
cos i t  2 1, ie e < 10% f o r  T < 100 min, i = w 
ie s i n  it 2 i ~ ,  e < 10% for T < 190 min, i = w 
where e i s  t h e  maximum e r r o r  a s soc ia t ed  w i t h  t h e  approximation. 
Thus f o r  update t i m e s  of less than  s i x  minutes (T < 6 min) ,  t h e  
following s ta te  t r a n s i t i o n  mat r ix  should give adequate r e s u l t s :  
-7- 
- 
0 
0 
0 
0 
0 
0 
s e c L  
r - 
. 
5. Examples 
5.1 I n i t i a l  Condition E r r o r s  
The s o l u t i o n  f o r  t h e  i n i t i a l  condi t ion  e r r o r s  i s  made by in-  
spec t ion  of equat ions  ( 4 )  and ( 6 ) .  Thus: 
8r 
where - @(t) i s given by equat ion ( 6 )  and I x ( t )  i s  given by equat ion 
(3 )  
5.2 Accelerometer B i a s  Errors 
If w e  t ake  accelerometer b i a s  t o  be t h e  s o l e  source of e r r o r ,  
then it can be shown t h a t : *  
- F ( t )  = - F = (0, O, ( ')fN, ( U ) f E }  
where ( u ) f N  and ( u ) f E  are t h e  n o r t h  and e a s t  accelerometer b i a s ' ,  
r e spec t ive ly .  Thus, i n  state space no ta t ion ,  
, 
"Br i t t i ng ,  K.R. ;  "Analysis of Local V e r t i c a l  I n e r t i a l  Navigation 
Systems," M.S.L. Report RE-52, February, 1 9 6 9 .  
-8- 
Thus, from equat ion ( 4 ) ,  t h e  e r r o r  response t o  accelerometer b i a s  
s t a r t i n g  a t  to = 0 ,  i s  given by: 
@ ( t - a )  B u da 
, Io - -I x ( t )  = - 
or: 
x ( t )  = - 
t . (.u) .f E I, s i n  ws ( t -o)  do 
g 
( U ) f E  t 
-tanL j0 s i n  ws(t-a) do 
4 
( u ) f N  
r I, COS ws(t-o) do 
.(U).fE 
s e c L  r I ,  COS w s ( t - C f )  do 
-9- 
d I n t e g r a t i o n  y i e l d s  t h e  r e s u l t :  
x ( t )  = - 
N E 
E E 
D 
6L 
E 
B X  
BiJ  
B i  
- cos ust) (u)fE/g 
-(1 - cos u s t )  (U)fN/9 
-(1 - cos u s t )  tanL (u) fE/g  
(1 - cos (u ) fN/g  
(1 - cos u s t )  s e c L  (u) fE/g  
s i n  u s t  (u)fN/rus  
s i n  w s t  secL (u)fE/rus  
b 
Thus it i s  seen t h a t  t h i s  method y i e l d s  r e s u l t s  very quick ly  
and e f f i c i e n t l y  compared with so lv ing  equat ion  (1) v i a  C r a m e r ' s  
Rule. 


State transition matrix for inertial navigation systems

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