Hybrid Unscented Kalman Filter: Application to the Simplest Walker

State estimation of hybrid dynamic systems, such as legged robots, is challenging because of the presence of non-smooth dynamics. This paper applies the Unscented Kalman Filter (UKF) state estimator and two novel hybrid extensions (HUKF) to a hybrid system, the simplest walking model. These estimators are identical far from the switching boundary, which partitions dynamic domains, but apply different time update algorithms at the switching boundary. (1) UKF permits sigma points to propagate through the system's hybrid dynamics. (2) HUKF-SPG (Sigma Point Generation) generates new sigma points when the weighted mean of the initial sigma points is on the switching boundary. (3) HUKF-SPT (Sigma Point Transformation) transforms the sigma points forward and backward in time through the system's hybrid dynamics only when the weighted mean of the initial sigma points is on the switching boundary. Results here shows that HUKF-SPG and HUKF-SPT have a lower absolute error but modestly more computations compared to UKF. A caveat of HUKF-SPT is it can only apply to conservative systems. HUKF-SPG is more general and could be applied to any hybrid system.