Global motion planning algorithm, based on high-order discretisation and on hierarchies of singularities

Let B be a system comprising a collect of rigid subparts, of which some might be attached to each other at certain joints and some might move independently. Suppose that B has a total of k degrees of freedom. Suppose further that B is free to move in a 2D or 3D space amid a collection of obstacles whose geometry is known. Typical values of k range from 2 (for a rigid object translating on a planar floor without rotating, for example) to 6 (the typical number of joints for a manipulator arm). The values can also be much larger, for example, when it is necessary to coordinate the motion of several independent systems in same workspace. The motion-planning problem for B is: given an initial placement Z1 and a desired target placement Z2 of B, determine whether there exists a continuous obstacle-avoiding motion of B from Z1 to Z2, and, if so, plan such a motion. An algorithm is presented for a fast solution to this problem. This algorithm is deterministic, involves approximations, and hence solves the motion planning problem up to a certain (prescribed) degree of accuracy