Closed-form forward kinematics of a novel class of 3-RRR spherical parallel mechanisms with coplanar intermediate-joint axes and coplanar distal-joint axes

Abstract

The forward kinematics (FK) of 3-RRR spherical parallel mechanisms (SPMs) has been studied extensively, which is essential for their calibration and control. These studies indicate that most 3-RRR SPMs are associated with octic characteristic equations, which must be solved using iterative approaches. This paper introduces a new class of 3-RRR SPMs featuring simplified FK formulations, characterized by three coplanar intermediate-joint axes and three coplanar distal-joint axes. Unlike most 3-RRR SPMs, this class admits only up to four distinct assembly modes. Moreover, four distinct formulations are proposed, all yielding quartic characteristic equations, thereby enabling closed-form FK solutions. Furthermore, one special subclass of the robot class is found to admit quadratic characteristic equations with simpler coefficient expressions. Beyond simplified FK relations, this robot class also offers unlimited rotation capability about certain directions, making it potentially attractive for practical applications. It is worth noting that this robot class provides flexibility in selecting architecture parameters, thereby allowing for a wide range of designs. Additionally, the proposed formulations can also be applied to certain parallel mechanisms with alternative topologies.