Attitude Uncertainty Analysis of a Three-Vehicle Constrained Formation

The uncertainty analysis of attitude estimates enables the comparison between different methods, and, thus, it is important for practical applications. This work studies the uncertainty for the attitude determination of a three-vehicle constrained formation. Moreover, the existing solution is improved by including the uncertainty results in a weighted orthogonal Procrustes problem. In the formation considered herein, the vehicles measure inertial references and relative line-of-sight vectors. Nonetheless, the line of sight between two elements of the formation is restricted. The uncertainty analysis uses perturbation theory and, consequently, considers a small first-order perturbation in the measurements. The covariance matrices are obtained for all relative and inertial attitude candidates from the linearization of the solution using a first-order Taylor expansion. Then, the uncertainty is completed by considering the covariance for the weighted orthogonal Procrustes problem, from the literature, and the definition of covariance for the remaining attitudes. The uncertainty characterization is valid for configurations with a unique solution. Finally, the theoretical results are validated by applying Monte Carlo simulations, which show that the predicted errors are statistically consistent with the numerical implementation of the solution with noise. Furthermore, the theoretical uncertainty predicts the accuracy changes near special configurations where there is loss of information.