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.

翻译：我们考虑不确定条件下的运动规划问题，并利用概率推断方法进行求解。具有线性随机动力学的无碰撞运动规划由后验分布建模。高斯变分推断是在高斯分布范围内对路径分布进行优化以推断该后验分布。我们提出高斯变分推断运动规划器（GVI-MP）算法来求解该高斯推断问题，其中采用自然梯度范式迭代更新高斯分布，并利用联合分布的因子化结构。我们证明，在GVI-MP中对状态分布的直接优化等价于求解一个具有闭式解的随机控制问题。基于这一观察，我们提出第二个算法——近端梯度协方差操控运动规划器（PGCS-MP），以求解具有终端约束的同一推断问题在随机控制形式下的表达。我们采用近端梯度范式处理具有非线性碰撞代价的线性随机控制问题，其中非线性代价通过二次函数迭代逼近，并在每次迭代中通过求解线性协方差操控获得闭式解。通过在多种机器人模型上的大量实验，我们评估了所提方法的有效性和性能。本文代码可在https://github.com/hzyu17/VIMP获取。