Research Papers

Position Analysis and Path Planning of the S-(nS)P U-SP U and S-(nS)PU-2SP U Underactuated Wrists

[+] Author and Article Information
Raffaele Di Gregorio

Department of Engineering,  University of Ferrara, Via Saragat 1, 44122 Ferrara, Italyrdigregorio@ing.unife.it

Hereafter, P and U stand for prismatic pair and universal joint, respectively, and the underscore indicates the actuated pair.

In this case, the resulting SPU limb is a passive limb with connectivity six, and can be eliminated because it does not add any constraint to the motion of the platform with respect to the base.

The configuration of a manipulator is controllable in a connected subset of its configuration space if, given two any configurations of that subset, it can carry out at least one path, which join the two configurations, by acting on its actuators [(9),10].

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

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.

J. Mechanisms Robotics 4(2), 021006 (Apr 12, 2012) (6 pages) doi:10.1115/1.4006191 History: Received May 19, 2011; Revised December 29, 2011; Published April 06, 2012; Online April 12, 2012

In a previous work, this author showed that ten topologies for underactuated parallel wrists can be generated from a fully parallel wrist (FPW). Three of them are obtained by simply replacing a spherical pair (S) with a nonholonomic spherical pair (nS). The S-(nS)P U-SP U and S-(nS)PU-2SP U wrists are two among these three. The position analysis of these two wrists is studied here. In particular, all the four position-analysis problems, which are necessary for implementing their path planning, are addressed and solved in closed form. Despite their different topology, the position analysis of these two wrists can be practically solved by using the same formulas and algorithms. Based on the deduced formulas, a path-planning algorithm is proposed. The obtained results make the studied wrist topologies able to replace “ordinary” wrists in the manipulation tasks which do not require tracking.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

A manufacturing scheme for passive nS pairs, proposed in Ref. [3] (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

The (nS)-2SP U underactuated parallel wrist

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Figure 3

The S-(nS)P U-SP U underactuated parallel wrist

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Figure 4

The S-(nS)PU-2SP U underactuated parallel wrist

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Figure 5

The S-(nS)P U-SP U underactuated parallel wrist: notations

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Figure 6

Geometric determination of the two possible positions of B1 compatible with an assigned direction, h1 , of the (nS)P U limb’s axis

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Figure 7

Particular platform geometry with w1 parallel to b1 . In this case, even though h1 is assigned, the platform can rotate around the line through O and B1 .

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Figure 8

The S-(nS)PU-2SP U underactuated parallel wrist: notations

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Figure 9

Implementation of a finite rotation, γ, around an axis (A1 , n) with a sequence of three finite rotations around coplanar axes lying on the AP




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