A new approach for the identification of reciprocal screw systems and its application to the kinematics analysis of limited-DOF parallel manipulators

The theory of reciprocal screws plays an important role in the kinematics and statics analysis of robot manipulators. Thus, efficient algorithms are usually required to identify the reciprocal screw system in closed-form. Inspired by the spatial stiffness/compliance, this paper presents a new approach for the identification of reciprocal screw system for any given one. The concept of symmetric screw matrix is introduced to uniformly represent the screw systems with 6 × 6 symmetric positive semidefinite matrices. By means of the change of coordinates and the recombination of base elements, any screw system can be decomposed into the direct sum of a general subsystem and a special one, which comprise only finite- and infinite-pitch elements belonging to the original system, respectively. Hence, for an arbitrary screw system, the corresponding reciprocal system can be achieved as the direct sum of those reciprocal to its subsystems. In the proposed framework, the decomposed subsystems can be uniquely identified by a diagonal and a symmetric 3 × 3 matrices, respectively, with respect to a particular choice of coordinate frame. Based on the theory of orthogonal complement spaces, the reciprocal subsystems can then be obtained in a straightforward manner with intuitive geometric interpretation. To verify the effectiveness of the proposed method in the kinematics analysis of limited degree-of-freedom (DOF) parallel manipulators, several typical candidates are investigated as examples. In all these examples, the screw systems can be identified in closed-form and the instantaneous kinematics can be characterized according to the properties of the obtained subsystems.