A Classification of Configuration Spaces of Planar Robot Arms with Application to a Continuous Inverse Kinematics Problem

Using results on the topology of moduli space of polygons [Jaggi, 92; Kapovich and Millson, 94], it can be shown that for a planar robot arm with $n$ segments there are some values of the base-length, $z$, at which the configuration space of the constrained arm (arm with its end effector fixed) has two disconnected components, while at other values the constrained configuration space has one connected component. We first review some of these known results. Then the main design problem addressed in this paper is the construction of pairs of continuous inverse kinematics for arbitrary robot arms, with the property that the two inverse kinematics agree when the constrained configuration space has a single connected component, but they give distinct configurations (one in each connected component) when the configuration space of the constrained arm has two components. This design is made possible by a fundamental theoretical contribution in this paper -- a classification of configuration spaces of robot arms such that the type of path that the system (robot arm) takes through certain critical values of the forward kinematics function is completely determined by the class to which the configuration space of the arm belongs. This classification result makes the aforesaid design problem tractable, making it sufficient to design a pair of inverse kinematics for each class of configuration spaces (three of them in total). We discuss the motivation for this work, which comes from a more extensive problem of motion planning for the end effector of a robot arm requiring us to continuously sample one configuration from each connected component of the constrained configuration spaces. We demonstrate the low complexity of the presented algorithm through a Javascript + HTML5 based implementation available at http://hans.math.upenn.edu/~subhrabh/nowiki/robot_arm_JS-HTML5/arm.html

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

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.