Motion planning can be cast as a trajectory optimisation problem where a cost is minimised as a function of the trajectory being generated. In complex environments with several obstacles and complicated geometry, this optimisation problem is usually difficult to solve and prone to local minima. However, recent advancements in computing hardware allow for parallel trajectory optimisation where multiple solutions are obtained simultaneously, each initialised from a different starting point. Unfortunately, without a strategy preventing two solutions to collapse on each other, naive parallel optimisation can suffer from mode collapse diminishing the efficiency of the approach and the likelihood of finding a global solution. In this paper we leverage on recent advances in the theory of rough paths to devise an algorithm for parallel trajectory optimisation that promotes diversity over the range of solutions, therefore avoiding mode collapses and achieving better global properties. Our approach builds on path signatures and Hilbert space representations of trajectories, and connects parallel variational inference for trajectory estimation with diversity promoting kernels. We empirically demonstrate that this strategy achieves lower average costs than competing alternatives on a range of problems, from 2D navigation to robotic manipulators operating in cluttered environments.

翻译：运动规划可被构建为轨迹优化问题，其中成本作为生成轨迹的函数被最小化。在存在多个障碍物且几何结构复杂的场景中，该优化问题通常难以求解且易陷入局部极小值。然而，近年来计算硬件的发展使得并行轨迹优化成为可能，即同时获取多个从不同初始点出发的解决方案。不幸的是，若无防止两个解相互坍缩的策略，朴素并行优化可能遭受模式坍缩问题，从而降低方法的效率及找到全局解的可能性。本文利用粗糙路径理论的最新进展，设计了一种促进解集多样性的并行轨迹优化算法，从而避免模式坍缩并实现更优的全局性能。我们的方法基于路径签名与轨迹的希尔伯特空间表示，将轨迹估计的并行变分推断与多样性促进核相结合。实验表明，从二维导航到在杂乱环境中操作的机械臂等一系列问题中，该策略相比竞争方法实现了更低的平均成本。