We present a novel formulation for motion planning under uncertainties based on variational inference where the optimal motion plan is modeled as a posterior distribution. We propose a Gaussian variational inference-based framework, termed Gaussian Variational Inference Motion Planning (GVI-MP), to approximate this posterior by a Gaussian distribution over the trajectories. We show that the GVI-MP framework is dual to a special class of stochastic control problems and brings robustness into the decision-making in motion planning. We develop two algorithms to numerically solve this variational inference and the equivalent control formulations for motion planning. The first algorithm uses a natural gradient paradigm to iteratively update a Gaussian proposal distribution on the sparse motion planning factor graph. We propose a second algorithm, the Proximal Covariance Steering Motion Planner (PCS-MP), to solve the same inference problem in its stochastic control form with an additional terminal constraint. We leverage a proximal gradient paradigm where, at each iteration, we quadratically approximate nonlinear state costs and solve a linear covariance steering problem in closed form. The efficacy of the proposed algorithms is demonstrated through extensive experiments on various robot models. An implementation is provided in https://github.com/hzyu17/VIMP.

翻译：我们提出了一种基于变分推断的新型不确定性运动规划框架，其中最优运动规划被建模为后验分布。我们提出了一个基于高斯变分推断的框架，称为高斯变分推断运动规划（GVI-MP），通过轨迹上的高斯分布来近似此后验分布。我们证明了GVI-MP框架与一类特殊的随机控制问题对偶，并为运动规划的决策过程引入了鲁棒性。我们开发了两种算法来数值求解该变分推断及其等效的运动规划控制公式。第一种算法采用自然梯度范式，在稀疏运动规划因子图上迭代更新高斯提议分布。我们提出了第二种算法——近端协方差导向运动规划器（PCS-MP），以随机控制形式求解具有附加终端约束的同一推断问题。我们利用近端梯度范式，在每次迭代中对非线性状态成本进行二次近似，并以闭式解求解线性协方差导向问题。通过对多种机器人模型的大量实验，验证了所提算法的有效性。实现代码发布于https://github.com/hzyu17/VIMP。