An algebraic formulation of configuration-space obstacles for spatial robots

An algebraic formulation of the boundaries of configuration-space obstacles for wrist-partitioned spatial robots is presented. When the end-effector of the robot moves in contact with an obstacle, it is constrained not only by the robot reachability constraints but also by the link-obstacle contact constraints. The reachability constraint is modeled by a chain of two spherical joints, and the contact constraint is a combination of spherical joint and planar joint. Each of these constraints defines a manifold in the 6D space of rigid displacements. Parameterized and algebraic expressions defining these manifolds are obtained using dual quaternions. The obstacle boundary is obtained from the intersection of the manifolds associated with two types of constraints. An example is provided to show how this formulation leads to equations for the boundary of a joint obstacle for a PUMA robot