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Research Papers

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

[+] Author and Article Information
Raffaele Di Gregorio

Department of Engineering,
University of Ferrara,
Via Saragat, 1,
Ferrara 44122, Italy
e-mail: rdigregorio@ing.unife.it

This nomenclature refers to architectures (parallel architectures) featuring two rigid bodies, one fixed (base) and the other mobile (platform), connected to each other by means of a number of kinematic chains (limbs). The symbols S, (nS), P, and U stand for spherical pair, nonholonomic spherical pair, prismatic pair, and universal joint, respectively; the underscore denotes the actuated joint. Hyphens separate strings of symbols which describe limbs' topologies, and the possible numbers ahead of such strings is the number of limbs with those topologies which appear in the architecture (no number means only one limb).

AP of an nS pair is the diametral plane of the sphere where the axes of the rollers lie on. It is worth reminding [10] that, when more than one roller-sphere contact appear in an nS pair, all the roller axes must lie on the same diametral plane of the sphere. In the constructive scheme of Fig. 1, the AP is the plane located by the sphere center and the axis of the single roller. From a kinematic point of view, the instantaneous relative motion of the two links joined by the nS pair must be an elementary rotation around an axis passing through the center of the nS pair and lying on the AP [9,10]. This kinematic condition is obtained by exploiting the friction in the roller-sphere contacts.

Remind [10,11] that both these wrist types can make their platform reach any orientation it assumed in the fully parallel wrist that generated them (i.e., their finite dof are three and are one more than their instantaneous dof).

It is worth reminding that the constraint reactions of a nS pair can be reduced to one force applied on its center together with a torque perpendicular to its AP [9,10], whereas the constraint reactions of an U joint can be reduced to a force applied on its center together with a torque perpendicular to the plane the axes of its two revolute pairs lie on.

The fully parallel wrist [10,22], which generated the two studied wrists, is an S-3SPU wrist that is obtained from the S-(nS)PU-2SPU of Fig. 3 by replacing the passive (nS)PU limb with an actuated SPU limb whose endings still are A1 and B1.

1Any passive spherical (S) pair can be transformed into a nonholonomic spherical (nS) pair by introducing, in parallel with the S pair, a roller-sphere contact, whose sphere is a spherical shell, fixed to one out of the two joined links and having the center coincident with the center of the S pair, whereas the roller is hinged on the other link. In Fig. 1, the S pair is constituted by three revolute (R) pairs in series whose axes intersect one another at the same point; such a point is the center both of the S pair and of the nS pair.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received March 9, 2013; final manuscript received July 31, 2013; published online October 10, 2013. Assoc. Editor: J.M. Selig.

J. Mechanisms Robotics 5(4), 041018 (Oct 10, 2013) (8 pages) Paper No: JMR-13-1052; doi: 10.1115/1.4025220 History: Received March 09, 2013; Revised July 31, 2013

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.

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References

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Figures

Grahic Jump Location
Fig. 1

A manufacturing scheme for nonholonomic spherical (nS) pairs [10] (R stands for revolute pair): the nS center is the sphere center, and the AP of this nS pair is the plane containing the roller axis and the sphere center

Grahic Jump Location
Fig. 2

The S-(nS)PU-SPU underactuated parallel wrist: (a) kinematic scheme and (b) notations

Grahic Jump Location
Fig. 3

The S-(nS)PU-2SPU underactuated parallel wrist: (a) kinematic scheme and (b) notations

Grahic Jump Location
Fig. 4

Free-body diagram of the actuated (nS)PU limb, which, by setting τ1 = 0, becomes the free-body diagram of the passive (nS)PU limb, and, by setting σ = 0, becomes the free-body diagram of an SPU limb

Grahic Jump Location
Fig. 5

Platform's free-body diagram: (a) in the S-(nS)PU-SPU wrists and (b) in the S-(nS)PU-2SPU wrists

Grahic Jump Location
Fig. 6

Free-body diagrams of (a) the (nS)PU and of (b) the SPU limbs at the jamming singularity that makes h1·(v1 × w1) null. If the axial force τ1h1 is canceled (i.e., if τ1 = 0) the free-body diagrams of (a) the passive (nS)PU and of (b) the passive SPU limbs are obtained

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