Geometric design of RRP, RPR and PRR serial chains

This paper formulates and solves the design equations for three degree-of-freedom spatial serial chains constructed with two revolute (R) joints and one prismatic (P) joint. Previous work has shown that equating the kinematics equations of the chain to a set of end-effector positions yields 24 equations in 25 unknowns, which means one of the design parameters can be chosen arbitrarily. Here it is shown that by specifying one of the directional parameters of the R-joints the design equations can be partitioned and solved separately. The derivation is presented in detail for the RRP chain, and the calculations are essentially identical for the RPR and PRR chains. The special case in which the axes of the two R-joints are perpendicular and intersect, forming a T-joint, is also presented. This chain has a spherical workspace and is often used as the shoulder joint of a robot. Example calculations are presented.