arXiv 1208.05732012-08-02math.ATcs.RO

同调类的不变量及其在机器人最优搜索和规划问题中的应用

Invariants for Homology Classes with Application to Optimal Search and Planning Problem in Robotics

Subhrajit Bhattacharya, David Lipsky, Robert Ghrist, Vijay Kumar

备注：: 28 pages

摘要：

我们考虑打孔欧几里得空间上的规划问题，$\mathbb{R}^D - \widetilde{\mathcal{O}}$，其中$\widetilde{\mathcal{O}}$是一个障碍物的集合。这样的空间作为机器人的配置空间频繁出现，其中$\widetilde{\mathcal{O}}$表示机器人需要避开的物理障碍物（例如，墙、其他机器人等）或非法状态（例如，所有腿离开地面）。由于状态规划转化为路径规划的配置空间，我们整理等价规划通过拓扑等价路径。这促使在这样的环境中找到或探索不同的同源类，并在每个这样的类中找到代表性的最佳轨迹。在本文中，我们开始考虑的问题，找到一个完整的一组易于计算的$(N-1)$ -循环$(\mathbb{R}^D - \widetilde{\mathcal{O}})$的同调类不变量。我们实现这一点，通过寻找明确的生成器的$(N-1)^{st}$德拉姆上同调群，这个打孔欧几里得空间，并使用它们的积分来定义上循环。这些对偶上循环在$(N-1)$ -循环上的作用给出了所需的不变量的完备集。我们通过例子来说明计算。我们进一步表明，由于积分的方法，这套完整的不变量是非常适合有效的搜索为基础的规划与拓扑约束的最佳机器人轨迹。最后，我们将这种方法扩展到计算空间中的不变量来自$(\mathbb{R}^D - \widetilde{\mathcal{O}})$折叠子空间，从而允许应用到更广泛的一类非欧环境空间。

英文摘要：

We consider planning problems on a punctured Euclidean spaces, $\mathbb{R}^D - \widetilde{\mathcal{O}}$, where $\widetilde{\mathcal{O}}$ is a collection of obstacles. Such spaces are of frequent occurrence as configuration spaces of robots, where $\widetilde{\mathcal{O}}$ represent either physical obstacles that the robots need to avoid (e.g., walls, other robots, etc.) or illegal states (e.g., all legs off-the-ground). As state-planning is translated to path-planning on a configuration space, we collate equivalent plannings via topologically-equivalent paths. This prompts finding or exploring the different homology classes in such environments and finding representative optimal trajectories in each such class. In this paper we start by considering the problem of finding a complete set of easily computable homology class invariants for $(N-1)$-cycles in $(\mathbb{R}^D - \widetilde{\mathcal{O}})$. We achieve this by finding explicit generators of the $(N-1)^{st}$ de Rham cohomology group of this punctured Euclidean space, and using their integrals to define cocycles. The action of those dual cocycles on $(N-1)$-cycles gives the desired complete set of invariants. We illustrate the computation through examples. We further show that, due to the integral approach, this complete set of invariants is well-suited for efficient search-based planning of optimal robot trajectories with topological constraints. Finally we extend this approach to computation of invariants in spaces derived from $(\mathbb{R}^D - \widetilde{\mathcal{O}})$ by collapsing subspace, thereby permitting application to a wider class of non-Euclidean ambient spaces.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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