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Chapter 6 Algorithms <strong>of</strong> <strong>Orbit</strong> <strong>Determination</strong> <strong>of</strong> IGSO, GEO and MEO <strong>Satellite</strong>s<br />

2<br />

+<br />

∂ U ,<br />

∂∂ yz<br />

2<br />

∂ U<br />

∂z<br />

��<br />

�<br />

sm<br />

sm ,<br />

2<br />

+<br />

��<br />

�<br />

2( yi−y) 3(<br />

yi−y) − 3<br />

2 2 2<br />

( x − ) + ( − ) + ( − ) �� �<br />

i x yi y zi z �<br />

� �<br />

( xi − x) + ( yi − y) + ( zi −z)<br />

xz<br />

= 3 −<br />

r<br />

,<br />

�<br />

�<br />

µ i �<br />

�<br />

��<br />

( xi − x) zi−z + ( yi − y) + ( zi −z)<br />

�<br />

5 2 2 2<br />

i= s m<br />

z 1<br />

=−( 1−3 ) − 2 3<br />

r r<br />

2<br />

�<br />

i= s, m<br />

�<br />

�<br />

µ i �<br />

���<br />

��<br />

�<br />

z − z<br />

i<br />

76<br />

2 2 2<br />

−<br />

��<br />

�<br />

2 2 2<br />

i − + i − + i −<br />

( x x) ( y y) ( z z)<br />

3<br />

� � �<br />

5<br />

2<br />

i i<br />

�<br />

�<br />

�<br />

�<br />

��<br />

( y −y) ( z −z)<br />

2 2 2<br />

i − + i − + i −<br />

( x x) ( y y) ( z z)<br />

2( zi−z) 3(<br />

zi−z) − 3<br />

2 2 2<br />

( x − ) + ( − ) + ( − ) �� �<br />

i x yi y zi z �<br />

� �<br />

( xi − x) + ( yi − y) + ( zi −z)<br />

3<br />

2 2 2<br />

� � �<br />

3<br />

� � �<br />

5<br />

�<br />

�<br />

�<br />

�<br />

��<br />

� � �<br />

5<br />

�<br />

�<br />

�<br />

�<br />

��<br />

(6-68)<br />

(6-69)<br />

(6-70)<br />

3) No Perturbations<br />

If in the state transit matrix the perturbation influence on the satellite dynamic model is not considered, the<br />

derivative components in Eq.(6-47) may be written as<br />

2<br />

2<br />

∂ U x 1<br />

=−( 1−3 )<br />

2<br />

2 3<br />

∂x<br />

r r<br />

(6-71)<br />

2<br />

∂ U xy<br />

= 3<br />

5 ∂∂ xy r<br />

(6-72)<br />

2<br />

∂ U xz<br />

= 3<br />

5 ∂∂ xz r<br />

(6-73)<br />

2<br />

∂ U y 1<br />

=−( 1−3 ) 2 2 3<br />

∂y<br />

r r<br />

(6-74)<br />

2<br />

∂ U yz<br />

= 3<br />

(6-75)<br />

5 ∂∂ yz r<br />

2<br />

2<br />

∂ U z 1<br />

=−( 1−3 ) (6-76)<br />

2<br />

2 3<br />

∂z<br />

r r<br />

Now Kalman filter algorithm for dynamic orbit determination can be summarized as follows<br />

ϖ ϖ<br />

x& k′ = f( xk′ , t)<br />

ϖ ϖ ϖ<br />

yk = Hkxk′ + Gkzk P′= Φ P<br />

T<br />

Φ + Γ Q Γ<br />

k k, k−1 k−1 k, k−1<br />

kk , −1 k−1 T<br />

kk , −1<br />

T<br />

k xk k<br />

k<br />

T<br />

T −1 −1 T −1 −1<br />

{ k x k k[ k k( k k k) k k ] }<br />

K = P′ H H P′ H + R I −G<br />

G R G G R<br />

~ ϖ<br />

xk ϖ ϖ<br />

= xk′ + Kk( yk − Hkxk′ )<br />

−1<br />

Px = ( I− K H ) P′<br />

k k k xk<br />

~ ϖ<br />

−1<br />

−1 −1<br />

z =−(<br />

G R G ) G R ( H K<br />

ϖ ϖ<br />

−I)( y − H x′<br />

)<br />

k k T<br />

k k k<br />

= (<br />

−1 −1 ) +<br />

−1 ( − ) ′<br />

−1 (<br />

−1 −1<br />

)<br />

T<br />

k k k k k k<br />

z k T<br />

k k k T<br />

k k k k x k T<br />

k k k T<br />

k k<br />

k k<br />

−1<br />

[ ]<br />

P G R G I G R H I K H P H R G G R G<br />

Eq.(6-77) is also called Extended Kalman Filter (EKF).<br />

�<br />

�<br />

�<br />

�<br />

�<br />

�<br />

�<br />

�<br />

�<br />

�<br />

�<br />

�<br />

�<br />

�<br />

�<br />

��<br />

(6-77)

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