Singularity analysis of three-legged parallel robots based on passive-joint velocities

The closed-loop structure of a parallel robot results in complex kinematic singularities in the workspace of the mobile platform. Singularity analysis become important in design, motion planning, and control of parallel robots. Focusing on the instantaneous velocities of passive joints, a new formulation approach is proposed for the instantaneous kinematics and singularity analysis of a class of three-legged parallel robots. Excluding the passive spherical joints at the leg ends, the number of I-dof passive joints in a three-legged, 6-dof, parallel robot is three, which is only half of the number of active joints (six). Consequently, the complexity, of the singularity analysis is significantly reduced because only four 3 x 3 matrices need to be analyzed. Using the product-of-exponential formula, the kinematic equations possess well-defined algebraic structures so that the instantaneous kinematics and singularity analysis algorithms can be readily and systematically formulated. Three types singularities, i.e., the forward, inverse, and combined singularities, have been identified. A unified condition for various singularities is proposed. Significant geometric conditions are also presented for identifying singularity configurations that requires simple computations.