Motion Planning of Planar Closed Chains Based on Structural Sets

This paper investigates the motion planning problem of planar m-link ( $m \geq 4$ ) closed chains among point obstacles with extension to arbitrary convex 2-D obstacles. The configuration space (C-space) of closed chains is embedded into two copies of m-3 dimensional tori. Two structural sets, the C-boundaries and the C-obstacles, are analyzed based upon the C-spaces of recursively constructed lower-dimensional closed chains. They contain essential structural information about the connectivity of the collision-free portion (C-free) of the C-space. By approximating each workspace obstacle by a set of points on the boundary after dilation, its corresponding C-obstacle is guaranteed to be covered by the C-obstacle of the convex hull of the point set. This permits a resolution-complete roadmap algorithm that puts specific bias for sampling the structural sets. Several benchmark examples are presented that compare the performance between our algorithm and the traditional algorithms. Animation videos and source codes are also provided which demonstrate the effectiveness of our method for closed chains of up to 20 links.