Effective direction measurements for spacecraft attitude: I. Equivalent directions

My derivation was from ancestors Who stood equivalent to mighty kings

William Shakespeare (1564–1616) and George Wilkins (fl. 1607) Pericles, Prince of Tyre, Act V , scene i

Abstract

The equivalent directions are a set of three unit vectors, conforming to the QUEST measurement model, which can lead in maximum-likelihood estimation to any given attitude estimate and attitude covariance matrix. Since they conform to the QUEST measurement model (i.e., are “QUEST-like”), the equivalent directions can be used not only in general estimation problems but also directly in the Wahba problem. It can be shown that three equivalent directions can always be found, and that they are unique within signs if the eigenvalues of the attitude co-information matrix are non-degenerate, but that the equivalent inverse variances for these equivalent directions may not always be non-negative, hence, not always physically meaningful. This can occur, for example, for attitude covariance matrices computed by the TRIAD algorithm for QUEST-like inputs. The equivalent inverse variances for attitude covariance matrices computed from original QUEST-like measurements will always be non-negative. The connection of the equivalent directions and inverse variances to Markley’s SVD algorithm is presented. While not a practical vehicle for data fusion, the equivalent directions and their equivalent inverse variances can provide a useful tool for the analysis of attitude systems in data fusion problems and in mission design.