We consider the motion planning problem under uncertainty and address it using probabilistic inference. A collision-free motion plan with linear stochastic dynamics is modeled by a posterior distribution. Gaussian variational inference is an optimization over the path distributions to infer this posterior within the scope of Gaussian distributions. We propose Gaussian Variational Inference Motion Planner (GVI-MP) algorithm to solve this Gaussian inference, where a natural gradient paradigm is used to iteratively update the Gaussian distribution, and the factorized structure of the joint distribution is leveraged. We show that the direct optimization over the state distributions in GVI-MP is equivalent to solving a stochastic control that has a closed-form solution. Starting from this observation, we propose our second algorithm, Proximal Gradient Covariance Steering Motion Planner (PGCS-MP), to solve the same inference problem in its stochastic control form with terminal constraints. We use a proximal gradient paradigm to solve the linear stochastic control with nonlinear collision cost, where the nonlinear cost is iteratively approximated using quadratic functions and a closed-form solution can be obtained by solving a linear covariance steering at each iteration. We evaluate the effectiveness and the performance of the proposed approaches through extensive experiments on various robot models. The code for this paper can be found in https://github.com/hzyu17/VIMP.

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