The pitch-reciprocal screw relating a twist and an actuating wrench

The screw theory is a powerful tool for the kinematic and static analyses of robotic mechanisms, and the operators in the screw theory include addition, subtraction, dot product, reciprocal product, etc, where the reciprocal product is important since a reciprocal screw relates a twist and a constraint wrench. However, the relation between a twist and an actuating wrench has not been profoundly investigated, and there is no operator that implements the bijection between a twist and an actuating wrench so far. Hence, the concept of the pitch-reciprocal screw is originally proposed in this paper, and the pitch-reciprocal operation and its physical significance are presented. The actuating wrench obtained through the pitch-reciprocal operation is adopted to much simplify the Jacobian analysis of parallel robots with the special limb structures. A novel concept of the actuating angle is put forward based on the actuating wrench and the additional wrench, which is utilized to explore both the kinematic singularity for the actuated joint and the transmission efficiency due to joint frictions. The pitch-reciprocal screw proposed in this paper is a bridge relating a twist and an actuating wrench, and is a novel and effective tool for the robotic performance analysis.

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