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Research Papers

Kinematic Analysis of the (nS)-2SP U Underactuated Parallel Wrist

[+] Author and Article Information
Raffaele Di Gregorio

Department of Engineering,  University of Ferrara, Via Saragat, 1, 44122 Ferrara, Italyraffaele.digregorio@unife.it

Here, instantaneous dof (or velocity dof) stands for “minimum number of independent rates necessary to uniquely determine the velocities of all the points belonging to the mechanical system.” It, in general, depends on the configuration assumed by the system.

P and U stand for prismatic pair and universal joint, respectively. The underline indicates the actuated pair.

The configuration of a manipulator is controllable in a connected subset of its configuration space iff, given any two configurations of that subset, it can carry out at least one path, which joins the two configurations, by acting on its actuators [5-6].

In this case, the elimination of the actuator in one SP U limb makes that limb a passive SPU kinemaric chain, which can be completely eliminated because a passive limb with connectivity six does not constrain the relative motion between platform and base.

Hereafter, (P, a ) will denote the oriented line passing through the point P and with the direction of the unit vector a . Moreover, the rotation angles are meant counterclockwise with respect to the oriented line given as rotation axis.

It is worth reminding that SO(3) is the operational space for a wrist.

The correctness of this analytic expression can be demonstrated by simply verifying that the matrix product N N −1 is equal to the identity matrix.

A singular configuration of a mechanism is named “uncertainty configuration,” if the mechanism’s instantaneous dof locally increases [32-33] when the mechanism assumes that configuration.

J. Mechanisms Robotics 4(3), 031006 (Jun 08, 2012) (7 pages) doi:10.1115/1.4006832 History: Received January 21, 2011; Revised April 30, 2012; Published June 07, 2012; Online June 08, 2012

The majority of the manipulation tasks require the motion of the end-effector from an initial pose to a final one without any particular condition on the path. Thus, the reduction of the practicable paths between any two poses is a possible choice, exploitable at the design stage, for simplifying the manipulator hardware. This choice is adopted in underactuated manipulators. The (nS)-2SP U wrist is one out of the underactuated parallel wrists this author proposed in a previous paper. Here, the kinematic analysis of this wrist is studied: both its finite and its elementary kinematics are considered. It is shown that its control algorithms can be written by using simple closed-form formulas, which can take advantage from the wide literature on the spherical four-bar linkages. Moreover, the demonstration that its singular configurations can be avoided more easily than the ones of the fully parallel wrist is provided.

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Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Manufacturing scheme of a passive nS pair [17] (R stands for revolute pair): the nS center is the sphere center, and the axes’ plane (AP) of this nS pair is the plane containing the roller axis and the sphere center

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Figure 2

An (nS)-2SP U underactuated parallel wrist: the sphere and the frame of the nS pair (Fig. 1) are fixed to the platform and the base, respectively

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Figure 3

The S-3SP U ordinary wrist

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Figure 4

Notation: σ and O are the AP and the center of the nS pair, respectively; (O, n ) and θ are the axis and the angle of the finite rotation to carry out

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Figure 5

The nS pair and the ith SP U limb: O is the center of the nS pair; Oxb yb zb and Oxp yp zp are two Cartesian reference systems embedded in the base and the platform, respectively. The xb yb -plane coincides with the axes’ plane (AP) of the nS pair.

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Figure 6

Single-dof spherical mechanism of type S-2SU whose constraint equations are the ones of system 4

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Figure 7

Spherical four-bar linkage kinematically equivalent to the mechanism of Fig. 6

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Figure 8

Free-body diagram of the platform

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