Estimation using quaternion probability densities on the unit hypersphere

Techniques have been developed for carrying out manipulations of quaternion probability densities that are defined only on the unit hypersphere. The goal is to generalize standard probabilistic concepts so that Monte-Carlo-type methods can be used with quaternion attitude representations. The optimal quaternion estimate is not the simple expectation value. Instead, it is the unit-normalized minimum mean-squared error estimate, with squared error defined as the square of the sine of half of the total attitude error. This definition causes the optimal quaternion estimate to equal the normalized eigenvector that is associated with the largest eigenvalue of the second moment of the quaternion distribution. The other three eigenvectors and eigenvalues of the second moment model the covariance of the first three components of the multiplicative error quaternion. Thus, the quaternion second moment contains both the optimal quaternion estimate and its error covariance. This estimation technique is used to solve a simulated batch GPS attitude determination problem using a particle computation. Accuracies commensurate with traditional methods can be achieved by performing a Monte-Carlo calculation of the quaternion second moment. An additional contribution is a new quaternion probability density function whose formula is defined solely in terms of its second moment.