Position analysis, path planning, and kinetostatics of single-loop RU-(nS)PU wrists

Differently from previously presented underactuated parallel wrists (PWs), RU-(nS)PU2 wrists feature a single-loop architecture and only one nonholonomic constraint. The presence of a nonholonomic constraint makes these PWs underactuated: they are able to control the platform orientation in a three-dimensional workspace by employing only two actuated pairs, one prismatic (P) and the other revolute (R), but they cannot perform tracking tasks. Their position analysis, path planning, and kinetostatics are studied. All their relevant position-analysis problems are solved in closed form, and, based on these closed-form solutions, a path-planning algorithm is built. Their instantaneous kinematics and singularity analysis are addressed, too: all their singular configurations are identified through analytic and geometric conditions, whose static interpretation is given. The presented results are relevant for designing this type of underactuated PWs.