Parameterized descriptions of the joint space obstacles for a 5R closed chain robot

An examination is made of the equations for the boundaries of the joint-space obstacles in the workspace of a 5R closed-chain robot. This two-degree-of-freedom chain appears in many robotics applications. The equations describing the joint-space obstacle boundaries are derived, using three different sets of parameters, and the obstacles are graphically displayed. The boundaries of polygonal obstacles in contact with the polygonal links of a robot arm are studied using the 4D image space of planar displacements. The obstacles are mapped into the image space as the surface representing the position and orientation that a body assumes while maintaining contact with the obstacle. The intersection of the available robot positions and the obstacle contact positions is a representation of the obstacles in the robot´s workspace. Three parameterizations of joint-space obstacles are determined from the image-space representation: the joint angles of one of the 2R robots, the end effector position, and the base joint angles of the 5R chain. The first and third parameter sets define two possible configurations for the 5R chain, and the second defines four possible configurations