Equations for boundaries of joint obstacles for planar robots

Parameterized equations for the boundaries of joint space obstacles for planar robots are derived. The boundary of a joint obstacle prescribes the link motion of the closed chain formed by the robot in contact with a given obstacle. The theory of kinematic mappings is used to represent the link motion as a set of points in a projective three space. The link positions that are feasible to the arm define a manifold in this space, termed its reachable manifold. The link positions satisfying contact conditions define another manifold in the same space called the contact manifold. Their intersection is mapped to joint space to obtain equations defining the obstacle boundary. While the theory applies to general planar robots, the authors focus on the 3R robot arm. An example is also provided in which the boundaries of joint obstacles are computed.