Geometric Unscented Particle Filters on Lie Groups for State Estimation

This article proposes two types of unscented particle filters (UPFs) that leverage unscented transformation (UT) from a geometric perspective to compute the proposal distribution. An UPF on Lie groups is first developed. Specifically, both the propagation of the sigma points and the computation of the mean and covariance are performed on the Lie groups, while the weight update and resampling are conducted on the Lie algebra. Second, we introduce the log-linear property of group elements to streamline particle propagation by reducing redundant operations, thereby optimizing the proposed UPF framework. In the update process, intermittent measurements that are caused by factors such as packet dropouts and stochastic sensor scheduling are considered. While lowering computational demands, these measurements pose challenges to filter stability. To this end, the introduced property is used to prove that the estimation error remains bounded under certain assumptions. We further establish a critical threshold for the arrival rate of intermittent measurements and derive an upper bound for the expected state error covariance. Moreover, a detailed computational complexity analysis is conducted to evaluate the efficiency of the proposed method. Finally, with the original method serving as a benchmark, simulation and real-world GNSS/INS integrated navigation experiments confirm that the redesigned approach delivers comparable performance and significantly improved computational efficiency.