Abstract. We present a simple and efficient paradigm for computing the exact solution of the motion planning problem in environments with a low obstacle density. Such environments frequently occur in practical instances of the motion planning problem. The complexity of the free space for such environments is known to be linear in the number of obstacles. Our paradigm is a new cell decomposition approach to motion planning and exploits properties that follow from the low density of the obstacles in the robot's workspace. These properties allow us to decompose the workspace, subject to some constraints, rather than to decompose the higher-dimensional free configuration space directly. A sequence of uniform steps transforms the workspace decomposition into a free space decomposition of asymptotically the same size. The approach leads to nearly optimal O(n log n) motion planning algorithms for free-flying robots with any fixed number of degrees of freedom in workspaces with low obstacle density.
We study the motion-planning problem for pairs and triples of robots operating in a shared workspace containing n obstacles. A standard way to solve such problems is to view the collection of robots as one composite robot, whose number of degrees of freedom is d, the sum of the numbers of degrees of freedom of the individual robots. We show that it is sufficient to consider a constant number of robot systems whose number of degrees of freedom is at most d − 1 for pairs of robots, and d − 2 for triples. (The result for a pair assumes that the sum of the number of degrees of freedom of the robots constituting the pair reduces by at least one if the robots are required to stay in contact; for triples a similar assumption is made. Moreover, for triples we need to assume that a solution with positive clearance exists.)We use this to obtain an O(n d ) time algorithm to solve the motion-planning problem for a pair of robots; this is one order of magnitude faster than what the standard method would give. For a triple of robots the running time becomes O(n d−1 ), which is two orders of magnitude faster than the standard method. We also apply our method to the case of a collection of bounded-reach robots moving in a low-density environment. Here the running time of our algorithm becomes O(n log n) both for pairs and triples.
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