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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 eS e = Integrate the errors at deployment to find the error at the surface. 11 1 T ss s TT T Ee e See S SE S = = = Or Φ can be integrated from: , where (0) () FI xf 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 21 nn rn r et t e e e == Φ where r Φ is the upper 3 rows of n t Φ . Covariance matrix: T rr EE Φ
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 8 () () () T TT T T eF e Et etet Fe 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) EE = . This requires the full 66 × E matrix. 2 upper left 3 3 partition of ( ) rr rv n vr vv n ee t ⎡⎤ = ⎢⎥ ⎣⎦ For small times around t n , ( ) ( ) () ( () ) ( ) ( ) nn n n v n n n n et vt t t v t e t t t v t t t + 2 2 1( )1 1 ( ) ( )0 1 1 T v n n n T v in T vn e e tt v = −=
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 8 3 2 2 2 position error at impact 1 1 1 "projection matrix" 1 T v n T vn T nv T e e ev v v Ie v = =− ⎡⎤ ⎢⎥
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This note was uploaded on 11/07/2011 for the course AERO 16.322 taught by Professor Wallacevandervelde during the Fall '04 term at MIT.

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

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