Efficient motion planning for an L-shaped object

An algorithm that solves the following motion planning problem is presented, Given an L-shaped body L and a two-dimensional region with n point obstacles whether there is a continuous motion connecting two given positions and orientations of L during which L avoids collision with the obstacles. The algorithm requires O(n² log² n) time and O(n²) storage. The algorithm is a variant of the cell-decomposition technique of the configuration space [D. Leven and M. Sharir, J. Algorithms, 8 (1987), pp, 192-215], [J.T. Schwartz and M. Sharir, Comm. Pure Appl. Math, 36 (1983), pp.345-398[, but it employs a new and efficient technique for obtaining a compact representation of the free space, which results in a saving of nearly an order of magnitude. The approach used in our algorithm is also applicable to motion planning of certain robotic arms whose spaces of free placements have a structure similar to that of the L-shaped body.