16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde e= Fe E(1)=e(l)e(t) E()=(n)e()y+e()() =Fe(n)e(1)+e()e()F You can integrate this differential equation to t, from E(O)=E. This requires the full6×6 Ematrix E eeee ee ee E,= upper left 3 x3 partition of E(t,) perturbed trajectory e (t) For small times around t e(n=e(t,)+v(,(t-L) e(,)+(v,(t,)+e()(t-L) =e(t,)+v((t-t) 1e(1)=1g+1(n)(t-Ln)=016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 8 () () () () () () () () () () () () () () T T T T TT T e Fe Et etet Et etet etet Fe t e t e t e t F FE t E t F = = = + = + = + & & & & You can integrate this differential equation to tn from 1 E E (0) = . This requires the full 6 6 × E matrix. 2 ( ) upper left 3 3 partition of ( ) T T rr rv n T T vr vv n ee ee E t ee ee E Et ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ = × For small times around tn, ( ) ( ) ( )( ) ( ) ( ( ) ( ))( ) ( ) ( )( ) n nn n nn vn n n nn n et et vt t t et v t e t t t et v t t t =+ − =+ + − =+ − 2 2 1 ( ) 1 1 ( )( ) 0 1 ( ) 1 T TT v v v nn n T v i n T v n et e v t t t e t t v = + −= − =−