VRM: A Unified Framework for Closed-form Solutions of a Special Class of Serial Manipulators

This paper proposes the virtual reconfiguration method (VRM) to construct the unified framework for closed-form solutions of a special class of serial manipulators. Central to the research is the inverse kinematics problem (IKP) of 6- and 7-DOF serial manipulators, which contain either the Pieper's geometry or the Duffy's geometry. Given the desired end-effector pose of the manipulator, a virtual single chain (SLC) is developed by connecting the base and the end-effector with a hypothetical link. The equivalent single open chain (SOC) with different configurations can be obtained by cutting open the virtual SLC at one link between adjacent joints. Kinematic equivalence between the original manipulator and the new SOC is proven. Closed-form solutions of the original manipulator can be determined by solving the IKP of the equivalent SOC. The VRM is further developed on the basis of the relationship between the manipulator and its equivalent SOC. In this paper, the IKPs of 6-DOF manipulators with the spherical wrist and manipulators with the three-axis parallel shoulder joint are analysed. Principles and applications of the VRM are proposed. Finally, the validity and efficiency of the VRM are demonstrated by kinematics simulations of four different manipulators. Unlike traditional approaches, the VRM simplifies the computation of IKPs and establishes a unified framework for closed-form solutions of the special class of 6-and 7-DOF serial manipulators, irrespective of the allocation of either the Pieper's geometry or the Duffy's geometry.