Optimal Estimation of Dynamic Systems, 2nd Edition

396 Optimal Estimation of Dynamic Systems

E



xx

T



−E {x}E

T

{x}. Substituting Equation (6.11) into Equation (6.7), and after

taking the appropriate partials, the following optimal error covariance can be derived:

P =

*

−

N

∑

j= 1

σ

−2

j

[Ar

j

×]

2

+

−1

(6.12)

The attitude A is evaluated at its respective true value. In practice, though, Ar

j

is

often replaced with the measurement

˜

b

j

, which allows a calculation of the covariance

without computing an attitude! Equation (6.12) gives the Cram´er-Rao lower bound

(any estimator whose error covariance is equivalent to Equation (6.12) is an efficient,

i.e., optimal, estimator). The Fisher information matrix is nonsingular only if ...