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Research Papers: Mechanisms and Robotics

# Singularity Equations of Gough–Stewart Platforms Using a Minimal Set of Geometric Parameters

[+] Author and Article Information
Qimi Jiang

Department of Mechanical Engineering, Laval University, Quebec, QC, G1V 0A6, Canadaqimi.jiang.1@ulaval.ca

Clément M. Gosselin

Department of Mechanical Engineering, Laval University, Quebec, QC, G1V 0A6, Canadagosselin@gmc.ulaval.ca

J. Mech. Des 130(11), 112303 (Sep 23, 2008) (8 pages) doi:10.1115/1.2976450 History: Received September 13, 2007; Revised May 27, 2008; Published September 23, 2008

## Abstract

So far, in the derivation of the singularity equations of Gough–Stewart platforms, all researchers defined the mobile frame by making its origin coincide with the considered point on the platform. One problem can be that the obtained singularity equation contains too many geometric parameters and is not convenient for singularity analysis, especially not convenient for geometric optimization. Another problem can be that the obtained singularity equation cannot be used directly in practice. To solve these problems, this work presents a new approach to derive the singularity equation of the Gough–Stewart platform. The main point is that the origin of the mobile frame is separated from the considered point and chosen to coincide with a special point on the platform in order to minimize the geometric parameters defining the platform. Similarly, by defining a proper fixed frame, the geometric parameters defining the base can also be minimized. In this way, no matter which practical point of the platform is chosen as the considered point, the obtained singularity equation contains only a minimal set of geometric parameters and becomes a solid foundation for the geometric optimization based on singularity analysis.

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## Figures

Figure 14

Singularity locus with a given position (x=0, y=234∕3, and z=1)

Figure 7

TSSM architecture (top view)

Figure 8

Semiregular hexagons (top view)

Figure 9

Similar symmetric hexagons (top view)

Figure 10

Regular hexagons (top view)

Figure 11

General 3-3 Gough–Stewart platform

Figure 12

MSSM architecture (top view)

Figure 13

Singularity locus with a given orientation (ϕ=30 deg, θ=45 deg, and ψ=0 deg)

Figure 15

The maximal singularity-free workspace around P0(0,234∕3,54) with ϕ=θ=ψ=0 deg

Figure 16

The maximal singularity-free workspace around P0(0,234∕3,54) with ϕ=30 deg, θ=45 deg, and ψ=0 deg

Figure 1

General Gough–Stewart platform

Figure 2

General Gough–Stewart platform with new defined frames

Figure 3

Classification of Gough–Stewart platforms

Figure 4

Irregular hexagons (top view)

Figure 5

Similar irregular hexagons (top view)

Figure 6

SSM architecture (top view)

## Errata

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