Uncertainty Propagation for General Stochastic Hybrid Systems on Compact Lie Groups

TL;DR

This paper introduces a spectral operator splitting method for uncertainty propagation in stochastic hybrid systems on compact Lie groups, enabling efficient density evolution analysis with applications to rigid body dynamics.

Contribution

It presents a novel computational framework combining spectral methods and operator splitting for solving the Fokker-Planck equation on Lie groups, specifically addressing hybrid systems.

Findings

Method produces results consistent with Monte Carlo simulations.

Explicitly computes the probability density function of the hybrid state.

Applicable to complex systems like rigid body pendulums.

Abstract

This paper deals with uncertainty propagation of general stochastic hybrid systems (GSHS) where the continuous state space is a compact Lie group. A computational framework is proposed to solve the Fokker-Planck (FP) equation that describes the time evolution of the probability density function for the state of GSHS. The FP equation is split into two parts: the partial differential operator corresponding to the continuous dynamics, and the integral operator arising from the discrete dynamics. These two parts are solved alternatively using the operator splitting technique. Specifically, the partial differential equation is solved by the spectral method where the density function is decomposed into a linear combination of a complete orthonormal function basis brought forth by the Peter-Weyl theorem, thereby resulting an ordinary differential equation. Next, the integral equation is solved by approximating the integral by a finite summation using a quadrature rule. The proposed method is then applied to a three-dimensional rigid body pendulum colliding with a wall, evolving on the product of the three-dimensional special orthogonal group and the Euclidean space. It is illustrated that the proposed method exhibits numerical results consistent with a Monte Carlo simulation, while explicitly generating the density function that carries the complete stochastic information of the hybrid state.