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The purpose of the next subsection is to show that the formulation of eqns. (5.57-5.58) is equivalent to the formulation of eqns. (5.61-5.62). Then Subsection 5.4.2 will derive two additional forms of the covariance update equation. The various implementations are summarized in Table 5.5. Additional alternative implementations, including square-root implementations, are presented in [60]. It is important that the reader remember that although the Kalman lter can be implemented by a variety of techniques, the techniques are equivalent theoretically. However, some of the implementation techniques require less computation while others have better numeric properties. It is up to the designer to select the most appropriate form for a given application. For example, the form of the measurement update in eqns. (5.57-5.58) is often not convenient for real-time implementations due to the required matrix inversion. The alternative formulation of eqns. (5.61-5.62) also involves a matrix inversion; however, the inverted matrix has dimensions corresponding to the number of measurements instead of the number of states. Since the number of measurements is often signi cantly less than the number of states, the update of eqns. (5.61-5.62) is often preferred. In fact, when the problem can be formulated with only scalar measurements, as discussed in Section 5.6.1, then the matrix inversion in eqn. (5.61) reduces to scalar division. Such issues are discussed in Section 5.6. Table 5.5 summarizes the discrete-time Kalman lter equations. The Kalman lter could be implemented by one of (at least) four techniques. All the approaches would use the same time update (eqns. (5.54-5.56)). For the measurement covariance update, four methods have been presented. The user must select one approach. In comparing the various covariance measurement update equations, eqn. (5.62) is the simplest formula and requires the least computation. Although eqn. (5.57) results naturally from the derivation presented in Section 5.3.4, it is not as e cient to implement as the alternative, equivalent solutions. Eqns. (5.68) and (5.69) involve only symmetric operations. Of the two, eqn. (5.69) is more numerically stable. Note that eqn. (5.69) is the same as eqn. (5.12) which is valid for
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The main sources for the material in this appendix were [23, 45, 50, 103, 115, 118, 137]. Attitude representations are compared in [120].
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