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.

翻译：高维运动规划问题通常可以通过利用多层抽象来显著加快求解速度。尽管有多种方式可以形式化地描述多层抽象，我们采用纤维丛对其进行建模。纤维丛本质上利用局部积空间描述状态空间的低维投影，从而使我们能够通过束缚限制和束缚截面对新算法进行简洁描述与推导。基于此类结构及相应的可行约束函数，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。这些方法通过束缚原语利用纤维丛结构，进而构建新型束缚规划器：快速探索商空间树（QRRT*）和商空间路线图规划器（QMP*）。两种规划器均被证明具有概率完备性和几乎必然渐近最优性。为评估我们的束缚规划器，我们将其与经典基于采样的规划器在四个低维场景和八个高维场景（自由度范围从21到100，涵盖多机器人系统与非完整约束）的基准测试中进行比较。研究结果表明，改进幅度高达2至6个数量级，充分彰显了多层运动规划器的效率优势，以及利用纤维丛术语挖掘多层抽象价值的实用性。