Kinetostatics of S-(nS)PU-SPU and S-(nS)PU-2SPU Nonholonomic Parallel Wrists

S-(nS)PU-SPU and S-(nS)PU-2SPU are two types of nonholonomic wrists that are generated from the “ordinary” wrists of type S-3SPU (fully parallel wrists (FPW)), by replacing a spherical pair (S) with a nonholonomic spherical pair (nS) according to the rules stated by Grosch et al. (2010, “Generation of Under-Actuated Manipulators With Nonholonomic Joints From Ordinary Manipulators,” ASME J. Mech. Rob., 2(1), p. 011005). Position analysis, controllability, and path planning of these two wrist types have been addressed and solved in two previous papers (Di Gregorio, R., 2012, “Type Synthesis of Underactuated Wrists Generated From Fully-Parallel Wrists,” ASME J. Mech. Des., 134(12), p. 124501 and Di Gregorio, R., 2012, “Position Analysis and Path Planning of the S-(nS)PU-SPU and S-(nS)PU-2SPU Underactuated Wrists,” ASME J. Mech. Rob., 4(2), p. 021006) of this author, which demonstrated that simple closed-form formulas are sufficient to control their configuration and to implement their path planning. Their kinetostatics and singularity analysis have not been addressed, yet; and they are studied in this paper. Here, the singularity analysis will reveal, for the first time, the existence of a somehow novel type of singularities, here named “jamming singularity,” that jams the platform motion in some directions and that is also present in all the parallel manipulators with SPU limbs (e.g., Gough-Stewart platforms) where it can be considered a particular type of “leg singularity.” Moreover, the static analysis will demonstrate that the reaction forces due to the static friction, in the nonholonomic constraint, can be controlled in the same way as the generalized forces exerted by the actuators, and that the possible slippage, in the same constraint, can be easily monitored and compensated.