Efficient algorithms for exact motion planning amidst fat obstacles

The complexity of exact motion planning algorithms greatly depends on the complexity of the robot´s free space, i.e., the set of all collision-free placements of the robot. Theoretically, the complexity of the free space can be very high. In practice, the complexity of the free space tends to be much smaller. It is shown that, under some realistic assumptions, the complexity of the free space of a robot moving amid fat obstacles is linear in the number of obstacles. The complexity results lead to efficient algorithms for motion planning amidst fat obstacles; it is shown that the O(n⁵) motion planning algorithm by J.T. Schwartz and M. Sharir (1983) runs in O(n ²) time if the obstacles are fat. The algorithm is modified to improve the running time to O(n log² n)