We consider the motion planning problem for stochastic nonlinear systems in uncertain environments. More precisely, in this problem the robot has stochastic nonlinear dynamics and uncertain initial locations, and the environment contains multiple dynamic uncertain obstacles. Obstacles can be of arbitrary shape, can deform, and can move. All uncertainties do not necessarily have Gaussian distribution. This general setting has been considered and solved in [1]. In addition to the assumptions above, in this paper, we consider long-term tasks, where the planning method in [1] would fail, as the uncertainty of the system states grows too large over a long time horizon. Unlike [1], we present a real-time online motion planning algorithm. We build discrete-time motion primitives and their corresponding continuous-time tubes offline, so that almost all system states of each motion primitive are guaranteed to stay inside the corresponding tube. We convert probabilistic safety constraints into a set of deterministic constraints called risk contours. During online execution, we verify the safety of the tubes against deterministic risk contours using sum-of-squares (SOS) programming. The provided SOS-based method verifies the safety of the tube in the presence of uncertain obstacles without the need for uncertainty samples and time discretization in real-time. By bounding the probability the system states staying inside the tube and bounding the probability of the tube colliding with obstacles, our approach guarantees bounded probability of system states colliding with obstacles. We demonstrate our approach on several long-term robotics tasks.

翻译：本文考虑不确定环境中随机非线性系统的运动规划问题。具体而言，该问题中机器人具有随机非线性动力学与不确定初始位置，环境包含多个动态不确定障碍物。障碍物可具有任意形状、可形变并可移动。所有不确定性不必服从高斯分布。文献[1]已对该通用设定进行了解答。除上述假设外，本文进一步考虑长期任务场景——由于系统状态不确定性在长时间范围内增长过大，[1]中的规划方法将失效。不同于[1]，我们提出一种实时在线运动规划算法。离线构建离散时间运动基元及其对应的连续时间管状域，使得每个运动基元的几乎所有系统状态都能被保证保持在对应管状域内。我们将概率安全约束转化为一组称为风险轮廓的确定性约束。在线执行阶段，利用平方和（SOS）规划验证管状域相较于确定性风险轮廓的安全性。所提供的基于SOS的方法无需实时不确定性采样与时间离散化，即可在存在不确定障碍物的条件下验证管状域安全性。通过约束系统状态保持在管状域内的概率以及管状域与障碍物碰撞的概率，我们的方法可保证系统状态与障碍物碰撞的概率有界。我们在多个长期机器人任务上验证了所提方法的有效性。