多层次运动规划：一种纤维丛的表述
Multilevel Motion Planning: A Fiber Bundle Formulation
AI总结：
本文基于纤维丛形式化多层次抽象，开发了渐近最优的采样运动规划方法QRRT*与QMP*，在21至100自由度场景下比经典规划器快2至6个数量级。
摘要：
高维运动规划问题通常可以通过使用多层次抽象显著加快求解速度。尽管有多种方式可以形式化地描述多层次抽象，我们采用纤维丛来表述它们。纤维丛本质上利用局部乘积空间描述状态空间的低维投影，这使我们能够基于丛限制和丛截面简明地描述并推导出新型算法。给定这样的结构以及相应的容许约束函数，我们为高维状态空间开发了高效且渐近最优的基于采样的运动规划方法。这些方法通过使用丛原语来利用纤维丛的结构。这些原语被用于创建新型丛规划器，即快速探索商空间树（QRRT*）和商空间路线图规划器（QMP*）。这两个规划器均被证明是概率完备且几乎必然渐近最优的。为了评估我们的丛规划器，我们在包含四个低维场景和八个高维场景的基准测试上将它们与经典的基于采样的规划器进行了比较，这些场景的自由度从21到100不等，包含多个机器人和非完整约束。我们的结果显示性能提升高达2到6个数量级，并强调了多层次运动规划器的效率以及利用纤维丛术语来开发多层次抽象的益处。
英文摘要：
High-dimensional motion planning problems can often be solved significantly faster by using multilevel abstractions. While there are various ways to formally capture multilevel abstractions, we formulate them in terms of fiber bundles. Fiber bundles essentially describe lower-dimensional projections of the state space using local product spaces, which allows us to concisely describe and derive novel algorithms in terms of bundle restrictions and bundle sections. Given such a structure and a corresponding admissible constraint function, we develop highly efficient and asymptotically-optimal sampling-based motion planning methods for high-dimensional state spaces. Those methods exploit the structure of fiber bundles through the use of bundle primitives. Those primitives are used to create novel bundle planners, the rapidly-exploring quotient-space trees (QRRT*), and the quotient-space roadmap planner (QMP*). Both planners are shown to be probabilistically complete and almost-surely asymptotically optimal. To evaluate our bundle planners, we compare them against classical sampling-based planners on benchmarks of four low-dimensional scenarios, and eight high-dimensional scenarios, ranging from 21 to 100 degrees of freedom, including multiple robots and nonholonomic constraints. Our findings show improvements up to 2 to 6 orders of magnitude and underline the efficiency of multilevel motion planners and the benefit of exploiting multilevel abstractions using the terminology of fiber bundles.
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