The inverse kinematics of lobster arms

Abstract

The roots of the closure polynomial associated with a given mechanism determine its assembly modes. In the case of 6R closed-loop mechanisms, these polynomials are usually expressed in the half-angle tangent of one of its joints. In this paper, we derive closure polynomials of 6R robots in terms of distances, not angles. The use of a distance-based formulation provides, in general, a fundamental advantage since it leads to closure conditions without requiring neither variable eliminations nor variable substitutions. We restrict our attention, though, to robots with coplanar consecutive joint axes, i.e., robots whose consecutive axes intersect at either proper or improper points. We show that this particular arrangement of joints does not result in a reduction in the maximum number of the inverse kinematic solutions with respect to the general case. Moreover, this family of robots include broadly used offset-wrist arms. For instance, in this paper, we obtain closure polynomials for robots such as the FANUC CRX-10iA/L, the UR10e, and the KUKA LBR iiwa R800 robot in generic form (i.e., as a function of their end-effector locations).