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Research Papers

# Singularity Free Revolute–Prismatic–Revolute and Spherical–Prismatic–Spherical Chains for Actuating Planar and Spherical Single Degree of Freedom Mechanisms

[+] Author and Article Information
David A. Perkins

Mechanical and Aerospace Engineering, University of Dayton, Dayton, OH 45469dperkins1@notes.udayton.edu

Andrew P. Murray

Mechanical and Aerospace Engineering, University of Dayton, Dayton, OH 45469

J. Mechanisms Robotics 4(1), 011007 (Feb 03, 2012) (6 pages) doi:10.1115/1.4005530 History: Received July 16, 2010; Revised October 20, 2011; Published February 03, 2012; Online February 03, 2012

## Abstract

Given a single degree of freedom mechanism, a moving reference frame attached to any link has a motion that can be described with a single parameter. A point relative to this moving frame is sought such that it either continually increases or decreases in distance from a point in the fixed frame over the entire motion. These points can be used to define a revolute–prismatic–revolute (RPR) chain for a planar mechanism or a spherical–prismatic–spherical (SPS) chain for a spherical mechanism capable of actuating the device over its entire range of motion. Moreover, the singularities relative to the joints in the original mechanism are not a concern and the dimensional synthesis can focus on creating the set of circuit-defect free solutions. From this analysis, a unique fixed point is determined in the planar case relative to two positions and their velocities with the following characteristic. All points in the moving reference frame that are moving away from it in the first position are approaching it in the second position, and vice versa. This point is as critical to the identification of singularity-free driving chains as the centrodes or the poles.

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## Figures

Figure 1

The motion of frame M is defined by a single parameter, φ, driven by the RPR chain, represented by the vector l→, and shown in the initial and final positions

Figure 2

The point P1 moves toward G→ in position i and away from it in position j. Point P2 moves away from the fixed pivot in position i and then approaches it in position j.

Figure 3

The region of universally bad fixed joints whose locations are found in Eq. 6

Figure 4

Lines sweeping out the region of moving joints that cannot form a strictly monotonic chain with the chosen G→

Figure 5

A universally bad fixed joint results in no moving joints with which to form a strictly monotonic chain

Figure 6

The example reorientation task depicted as motion on the surface of a sphere

Figure 7

The region of universally bad fixed joints is indicated by the triangles

Figure 8

The region of moving joints that cannot form a strictly monotonic chain with the chosen fixed joint

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