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Research Papers

# On the Ability of a Cable-Driven Robot to Generate a Prescribed Set of Wrenches

[+] Author and Article Information
Samuel Bouchard

Département de Génie Mécanique, Université Laval, 1065, Avenue de la Médecine,Québec, QC, G1V 0A6, Canadasamuel@robotiq.com

Clément Gosselin1

Département de Génie Mécanique, Université Laval, 1065, Avenue de la Médecine,Québec, QC, G1V 0A6, Canadagosselin@gmc.ulaval.ca

Brian Moore

Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69, A-4040 Linz, Austriabrian.moore@ricam.oeaw.ac.at

1

Corresponding author.

J. Mechanisms Robotics 2(1), 011010 (Dec 18, 2009) (10 pages) doi:10.1115/1.4000558 History: Received August 15, 2008; Revised August 02, 2009; Published December 18, 2009

## Abstract

This paper presents a new geometry-based method to determine if a cable-driven robot operating in a $d$-degree-of-freedom workspace $(2≤d≤6)$ with $n≥d$ cables can generate a given set of wrenches in a given pose, considering acceptable minimum and maximum tensions in the cables. To this end, the fundamental nature of the available wrench set is studied. The latter concept, defined here, is closely related to similar sets introduced by Ebert-Uphoff and co-workers (2004, “Force-Feasible Workspace Analysis for Underconstrained, Point-Mass Cable Robots,” IEEE Trans. Rob. Autom., 5, pp. 4956–4962; 2007, “Workspace Optimization of a Very Large Cable-Driven Parallel Mechanism for a Radiotelescope Application,” Proceedings of the ASME IDETC/CIE Mechanics and Robotics Conference, Las Vegas, NV). It is shown that the available wrench set can be represented mathematically by a zonotope, a special class of convex polytopes. Using the properties of zonotopes, two methods to construct the available wrench set are discussed. From the representation of the available wrench set, computationally efficient and noniterative tests are presented to verify if this set includes the task wrench set, the set of wrenches needed for a given task.

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## Figures

Figure 1

Planar mechanism n=3, d=2 with its associated A drawn (bottom) and example of three different T

Figure 2

Minkowski sum of four line segments

Figure 3

Illustration of the four steps in the mathematical construction of a zonotope for n=3 and d=2. The top image shows the mechanism.

Figure 4

Mechanism d=3, n=4 in three different poses. The associated base zonotopes are shown on the right.

Figure 5

Mechanism d=2, n=5 in three different poses. The associated base zonotopes are shown on the right. The diamonds are the projected vertices of the orthotope.

Figure 6

Three belt zones around a zonotope

Figure 7

Example of a wrench-closed pose and its corresponding base zonotope

Figure 8

Shifting of the four initial hyperplanes for a d=2, n=4 mechanism

Figure 9

The intersection of all the half-spaces defined by the hyperplanes form the zonotope

Figure 10

Different possible shapes of task wrench set

Figure 11

Ellipse crossing a support plane of A

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