On motion planning with six degrees of freedom: Solving the intersection problems in configuration space

The Movers´ problem is to find a continuous, collision-free path for a moving object through an environment containing obstacles. The classical formulation of the three-dimensional Movers´ problem is as fellows: given an arbitrary rigid polyhedral moving object P with three translational and three rotational degrees of freedom, find a contineous, collision-free path taking P from some initial configuration to a desired goal configuration. The six degree or freedom Movers´ problem may be transformed into a point, navigation problem in a six-dimensional configuration space (called C-Space). The C-Space obstacles, which characterize the physically unachievable configurations, are directly represented by six-dimensional manifolds whose boundaries are five dimensional C-surfaces. By characterizing these surfaces and their intersections, collision-free paths may be found by the closure of three operators which (i) slide along 5-dimensional level C-surfaces parallel to C-Space obstacles; (ii) slide along 1- to 4-dimensional intersections of level C-surfaces; and (iii) jump between 6-dimensional obstacles. We show how to construct and represent C-surfaces and their intersection manifolds. We also demonstrate how to intersect trajectories with the boundaries of C</-Space obstacles. The theory and representations we develop extend to Cartesian manipulators with six degrees of freedom.