We present an algorithm that solves the following motionplanning problem. Given an L-shaped body L and a 2-dimensional region with R point obstacles, decide 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 log n) time and O(n2) storage. The algorithm is a variant of the cell-decomposition technique of the configuration space ([SS, LS]) but it employs a new and efficient technique for obtaining a compact representation of the free space, which results in a saving of an order of magnitude. The approach used in our algorithm seems applicable to motion-planning of certain robotic arms whose spaces of free placements have a structure similar to that of the L-shaped body.