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lecture09

# lecture09 - 16.322 Stochastic Estimation and Control Fall...

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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 8 Lecture 9 Last time: Linearized error propagation 1 s e Se = Integrate the errors at deployment to find the error at the surface. 1 1 1 T s s s T T T E e e S e e S SE S = = = Or Φ can be integrated from: , where (0) ( ) F I x f x df F dx Φ = Φ Φ = = = & & where F is the linearized system matrix. But this requires the full Φ (same number of equations as finite differencing). n t = time when the nominal trajectory impacts. 1 2 1 ( ) ( ) ( ) n n r n r e t t e e t e e = Φ = = Φ where r Φ is the upper 3 rows of ( ) n t Φ . Covariance matrix: 2 1 T r r E E = Φ Φ

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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 8 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) T T T T T T T e Fe E t e t e t E t e t e t e t e t F e t e t e t e t F FE t E t F = = = + = + = + & & & & You can integrate this differential equation to t n from 1 (0) E E = . This requires the full 6 6 × E matrix. 2 ( ) upper left 3 3 partition of ( ) T T r r r v n T T v r v v n e e e e E t e e e e E E t = ⎢ = × For small times around t n , ( ) ( ) ( )( ) ( ) ( ( ) ( ))( ) ( ) ( )( ) n n n n n n v n n n n n n e t e t v t t t e t v t e t t t e t v t t t = + = + + = + 2 2 1 ( ) 1 1 ( )( ) 0 1 ( ) 1 T T T v v v n n n T v i n T v n e t e v t t t e t t v = + = = −
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 8 3 2 2 2

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lecture09 - 16.322 Stochastic Estimation and Control Fall...

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