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RESEARCH PAPERS

# Degeneracy Study of the Forward Kinematics of Planar $3-RP̱R$ Parallel Manipulators

[+] Author and Article Information
P. Wenger, D. Chablat, M. Zein

Institut de Recherche en Communications et Cybernétique de Nantes, 1 rue la Noe, BP 92101, 44321 Nantes Cedex 3, France

The underlined letter refers to the actuated joint.

J. Mech. Des 129(12), 1265-1268 (Dec 13, 2006) (4 pages) doi:10.1115/1.2779893 History: Received September 19, 2006; Revised December 13, 2006

## Abstract

This paper investigates two situations in which the forward kinematics of planar $3-RP̱R$ parallel manipulators degenerates. These situations have not been addressed before. The first degeneracy arises when the three input joint variables $ρ1$, $ρ2$, and $ρ3$ satisfy a certain relationship. This degeneracy yields a double root of the characteristic polynomial in $t=tan(φ∕2)$, which could be erroneously interpreted as two coalesce assembly modes. However, unlike what arises in nondegenerate cases, this double root yields two sets of solutions for the position coordinates $(x,y)$ of the platform. In the second situation, we show that the forward kinematics degenerates over the whole joint space if the base and platform triangles are congruent and the platform triangle is rotated by $180deg$ about one of its sides. For these “degenerate” manipulators, which are defined here for the first time, the forward kinematics is reduced to the solution of a third-degree polynomial and a quadratic in sequence. Such manipulators constitute, in turn, a new family of analytic planar manipulators that would be more suitable for industrial applications.

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## Figures

Figure 1

A 3-RPR parallel manipulator

Figure 2

The six assembly modes corresponding to Table 1. The last two ones correspond to the degenerate root.

Figure 3

The six assembly modes corresponding to Table 2.

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