Precise Orbit Determination of Global Navigation Satellite System of ...
Precise Orbit Determination of Global Navigation Satellite System of ...
Precise Orbit Determination of Global Navigation Satellite System of ...
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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 />
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sm<br />
sm ,<br />
2<br />
+<br />
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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 />
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�<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 />
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�<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 />
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5<br />
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(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 />
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(6-77)