Complete path planning for closed kinematic chains with spherical joints

We study the path planning problem, without obstacles, for closed kinematic chains with n links connected by spherical joints in space or revolute joints in the plane. The configuration space of such systems is a real algebraic variety whose structure is fully determined using techniques from algebraic geometry and differential topology. This structure is then exploited to design a complete path planning algorithm that produces a sequence of compliant moves, each of which monotonically increases the number of links in their goal configurations. The average running time of this algorithm is proportional to n(3). While less efficient than the O(n) algorithm of Lenhart and Whitesides, our algorithm produces paths that are considerably smoother More importantly, our analysis serves as a demonstration of how to apply advanced mathematical techniques to path planning problems. Theoretically, our results can be extended to produce collision-free paths, paths avoiding both link-obstacle and link-link collisions. An approach to such an extension is sketched in Section 4.5, but the details are beyond the scope of this paper Practically, link-obstacle collision avoidance will impact the complexity of our algorithm, forcing us to allow only small numbers of obstacles with "nice" geometry, such as spheres. Link-link collision avoidance appears to be considerably more complex. Despite these concerns, the global structural information obtained in this paper is fundamental to closed kinematic chains with spherical joints and can easily be incorporated into probabilistic planning algorithms that plan collision-free motions. This is also described in Section 4.5.