Path Integral Control in Gaussian Belief Space for Partially Observed Systems

This paper extends path integral control (PIC) to partially observed systems by formulating the problem in Gaussian belief space. PIC uses the matching condition, which requires the diffusion and control channels to be proportional, to linearize the Hamilton--Jacobi--Bellman equation through the Cole--Hopf transform. We show that the belief diffusion satisfies this condition only when the observation function is affine. Thus, the standard Cole--Hopf linearization does not apply to non-affine observation models. Restricting the problem to Gaussian beliefs provides a finite-dimensional approximation with deterministic covariance evolution and reduces the problem to stochastic control of the belief mean. We derive necessary and sufficient conditions for matching in this reduced space, obtain an exact Cole--Hopf linearization and a Feynman--Kac representation, and develop the MPPI-Belief algorithm. Numerical experiments on a navigation task with state-dependent observation noise demonstrate the effectiveness of MPPI-Belief compared with certainty-equivalent and particle-filter-based methods.