Motion Planning in Environments with Low Obstacle Density

We present a simple and e cient 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 con guration space directly. A sequence of uniform steps transforms the workspace decomposition into a free space decomposition of asymptotically the same size. The approach applies to robots with any xed number of degrees of freedom and turns out to be e cient in many cases: it leads to nearly optimal O(n logn) algorithms for motion planning in 2D, for motion planning in 3D amidst obstacles of comparable size, and for motion planning on a planar work oor in 3D. In addition, we obtain algorithms for planning 3D motions among polyhedral obstacles, running in O(n logn) time, and among arbitrary obstacles, running in time O(n). Several other interesting instances of the motion planning seem adequately solvable by the paradigm as well.