Stochastic Kinematic Optimal Control on SO(3)

In this paper, we develop a novel method for deriving a global optimal control strategy for stochastic attitude kinematics on the special orthogonal group SO(3). We first introduce a stochastic Lie-Hamilton-Jacobi-Bellman (SL-HJB) equation on SO(3), which theoretically provides an optimality condition for the global optimal control strategy of the stochastic attitude kinematics. Then we propose a novel numerical method, the Successive Wigner-Galerkin Approximation (SWGA) method, to solve the SL-HJB equation on SO(3). The SWGA method leverages the Wigner-D functions to represent the Galerkin solution of the SL-HJB equation in a policy iteration framework, providing a computationally efficient approach to derive a global optimal control strategy for systems on SO(3). We demonstrate the effectiveness of the SWGA method through numerical simulation on stochastic attitude stabilization.