AI 中文总结
本文指出现有基于采样的运动规划算法渐近近优性证明的隐含假设不成立，提出无需δ-相似性即可实现渐近近优性，并通过示例验证了排挤导致无法采样δ-相似轨迹的情况。
AI 中文摘要
基于采样的运动规划算法是一类流行的轨迹规划算法，其优势在于在复杂高维环境中速度快，且能处理运动动力学约束，具体通过前向动力学传播实现。许多此类规划器声称通过证明对状态空间中接近最优轨迹的轨迹（即δ-相似轨迹）进行几乎必然采样，来实现渐近近优性。本文表明，渐近δ-相似性背后的证明依赖于一个未明确说明的假设：即δ-相似轨迹段一旦被采样就会始终保留，而该假设在一般情况下不成立。文中描述了一种称为“排挤”的问题情形：局部低成本路径会阻止与最优轨迹δ-相似的轨迹被添加到树中。然而，研究表明，当恰当考虑排挤问题时，即便没有δ-相似解轨迹的保证，仍可实现渐近近优性保证。文中提供了一个会发生排挤的示例环境和系统，展示了无法归纳采样到δ-相似解轨迹的场景。
英文摘要
Sampling-based motion planning algorithms are a popular class of trajectory planning algorithm due to their speed in complex, high-dimensional environments and ability to handle kinodynamic constraints, specifically through the use of forward dynamics propagation. Many such planners claim to achieve asymptotic near-optimality by proving the almost sure sampling of trajectories that are close to an optimal trajectory in the state space, known as $δ$-similar trajectories. This paper shows that the proof behind asymptotic $δ$-similarity relies on an unstated assumption that $δ$-similar trajectory segments will always be kept once sampled. This assumption does not hold in general. A problematic case, referred to as ``crowding out,'' is described, where locally low-cost paths prevent trajectories that are $δ$-similar to the optimal trajectory from being added to the tree. It is shown, however, that asymptotic near-optimality guarantees can still be achieved without guarantees of $δ$-similar solution trajectories when crowding out is properly accounted for. An example environment and system are provided where crowding out is shown to occur, demonstrating a scenario where inductively sampling a $δ$-similar solution trajectory is impossible.
CommentsSubmitted to IEEE RA-L
