ABSTRACT
The Kalman filter [1,2] is a widely used algorithm in spacecraft navigation. Although the Kalman filter is usually employed to estimate vector quantities, such as position or velocity, modifications to the standard algorithm exist to estimate attitude. One classic proof of the optimality of the Kalman filter is the orthogonality principle [3]. This principle establishes that the optimal estimator (in a minimum mean square error [MMSE] sense) results in an estimation error that is orthogonal to all possible estimators. Let be a set of functions closed under addition (i.e., ∀ ∈ + ∈g g g g1 2 1 2, , ) and scalar multiplication (i.e., ∀ ∈ℜ ∈ ∈α α, ,g g ). Given two random variables X and Y with joint distribution, the orthogonality principle establishes that the following two statements are equivalent:
1. g* ∈ minimizes E {[X − g(Y)]2} out of all g∈ 2. E [ ( )] ( )*X g Y g Y g−{ } = ∀ ∈0 It is assumed that the above two expected values exist. A proof of the orthogonality principle can be found in the Appendix A together with a derivation of the associated Kalman update.
