16.322 Stochastic Estimation and Control, Fall 2004 Prof. VanderⅤelde Errors are zero mean ≌=xm-x Covariance matrix EoL=ee Lander deployment is impulsive(assume short burn, negligible change in DOS =+△ +6y-(M+△) e,+d1 日+J5y, where J E,=ee [e+Joye+8yJ'] Eo+JD/, if eSy=8ve,=0 Eo D=Vov: covariance matrix for velocity correction errors Now these errors must be propagated to the surface. A linearized description of error propagation is given by a transition matrix which relates perturbations in position and velocity components at the initial point to perturbations in position and velocity at any later point. For the present purpose, we may be interested only in the position perturbation at the end point. Direct sensitivity analysis or i=1.2 ax Page 5 of 616.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Errors are zero mean: e0 = x0actual − x0est. Covariance matrix: T × = e0 e [E0 ]66 0 Lander deployment is impulsive (assume short burn, negligible change in position): e = er0 r 1 = − v1nom ev1 v1true = v + ∆v + − δ v (v + ∆v) 0true 0est. = e + δ v v0 ⎡ er0 ⎤ e1 = ⎢ ⎥ ⎣ev0 + δ v ⎢ ⎦⎥ ⎡ ⎤ 0 = e0 + J v, where J = ⎢ ⎥ δ I ⎣ ⎦63 × E1 = e e T 1 1 T T T = [e0 + J v e ⎦ δ ]⎣ ⎡ 0 + δ v J ⎤ T T δ δ ve = E 0 = 0 0 + JDJ T , if e v T 0 = ⎡0 0 ⎤ = E0 + ⎢ ⎥ ⎣0 D⎦6 6× T D = δ v δ v : covariance matrix for velocity correction errors. Now these errors must be propagated to the surface. A linearized description of error propagation is given by a transition matrix which relates perturbations in position and velocity components at the initial point to perturbations in position and velocity at any later point. For the present purpose, we may be interested only in the position perturbation at the end point. Direct sensitivity analysis 6 ∂rsi δ r = ∑ δ xij , i = 1, 2 si j=1 ∂xij Page 5 of 6