Linear dimensionality reduction in random motion planning

The paper presents a method to control random sampling in motion planning algorithms. The principle of the method is to use on line the results of a probabilistic planner to describe the free space in which the planning takes place, by computing a Principal Component Analysis (PCA). This method identifies the locally free directions of the free space. Given that description, our algorithm accelerates the progression along these favored directions. That way, if the free space appears as a small volume around a submanifold of a high-dimensional configuration space, the method overcomes the usual limitations of probabilistic motion planning algorithms and finds a solution quickly. The presented method is theoretically analyzed and experimentally compared to known motion planners. 1 Problem statement, related work and contribution 1.1 General Framework Motion planning problems have been intensively studied in the last decades, with applications in many diverse areas, such as robotics, part disassembly problems in Product Lifecycle Management (PLM), digital actors in computer animation, or even protein folding and drug design. For comprehensive overviews of motion planning problems and methods, one can refer to (Latombe, 1991), (Choset et al., 2005) and (LaValle, 2006). In the past fifteen years, probabilistic algorithms exploring the configuration space (CS) have been developed with success. The sampling approach, first introduced in (Kavraki et al., 1996) as probabilistic roadmaps (PRM), consists in computing a graph, or a roadmap, whose vertices are collision free configurations, sampled at random in the free space and whose edges reflect the existence of a collision free elementary path between two configurations. PRMs aim at capturing the topology of the collision free space (CS free) in a learning phase in order to handle multiple planning queries in a solving phase. † This work is partly supported by the French ANR-RNTL project PerfRV2. A preliminary version of this paper appeared in WAFR’08. 2 Sébastien Dalibard and Jean-Paul Laumond One class of these algorithms has received special attention in the literature for its ability to rapidly solve single-query problems: tree expansion strategies, introduced in both (Hsu et al., 1999) and (Kuffner & LaValle, 2000), which include RRT planners, consist in growing a tree rooted at the start configuration towards the goal configuration, by repetitively expanding it in random directions. These methods have been proved to be efficient and suitable for a large class of motion planning problems. Work has been done to analyze and validate theoretically these algorithms. Probabilistic completeness has been studied and proved for PRM (Kavraki et al., 1998), as well as for RRT (Kuffner & LaValle, 2000).