Unified kinematics and singularity analysis of a metamorphic parallel mechanism with bifurcated motion

This paper introduces a new metamorphic parallel mechanism consisting of four reconfigurable rTPS limbs. Based on the reconfigurability of the reconfigurable Hooke (rT) joint, the rTPS limb has two phases while in one phase the limb has no constraint to the platform, in the other it constrains the spherical joint center to lie on a plane. This results in the mechanism to have ability of reconfiguration between different topologies with variable mobility. Geometric constraint equations of the platform rotation matrix and translation vector are set up based on the point-plane constraint, which reveals the bifurcated motion property in the topology with mobility 2 and the geometric condition with mobility change in altering to other mechanism topologies. Following this, a unified kinematics limb modeling is proposed considering the difference between the two phases of the reconfigurable rTPS limb. This is further applied for the mechanism modeling and both the inverse and forward kinematics is analytically solved by combining phases of the four limbs covering all the mechanism topologies. Based on these, a unified singularity modeling is proposed by defining the geometric constraint forces and actuation forces in the Jacobian matrix with their change in the variable topologies in terms of constraint screws. Analysis of workspace with singularity distribution is carried out using this model and corresponding singularity loci are obtained with special singular configurations illustrated.