16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Lecture g Last time: Linearized error propagation trajectory surface Integrate the errors at deployment to find the error at the surface e=ee See s E SE,S Or p can be integrated from: d=Fd, whereΦ(0)=1 文=f(x) F where F is the linearized system matrix. But this requires the full a(same number of equations as finite differencing) I, =time when the nominal trajectory impacts e(Ln)=Φ(tn)8 e(n)=旦=Φ,旦 where a, is the upper 3 rows of a(n) Covariance matrix E2=d,EΦ Page 1 of 816.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 8 Lecture 9 Last time: Linearized error propagation s 1 e Se = Integrate the errors at deployment to find the error at the surface. 1 1 1 T s ss T T T E ee See S SE S = = = Or Φ can be integrated from: , where (0) ( ) F I x fx 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). nt = time when the nominal trajectory impacts. 1 2 1 () () ( ) n n rn r et t e et e e = Φ = =Φ where Φr is the upper 3 rows of ( ) n Φ t . Covariance matrix: 2 1 T E E =Φ Φ r r