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Research Papers

Direct Geometrico-Static Problem of Underconstrained Cable-Driven Parallel Robots With Three Cables1

[+] Author and Article Information
Marco Carricato

Department of Industrial Engineering
and Interdepartmental Center
for Health Sciences and Technologies,
University of Bologna,
Viale Risorgimento 2,
Bologna 40136, Italy
e-mail: marco.carricato@unibo.it

In very special cases, it may happen that Eq. (5) is fulfilled because L1, L2 and L3 become linearly dependent. In these configurations, equilibrium is not possible if rank(M) = 3, since the external wrench would not belong to the screw subspace generated by the cable lines. Configurations like these need to be discarded from the solution set.

The notation Mhij,klm denotes the block matrix obtained from rows h, i and j, and columns k, l and m, of M. When all columns of M are used, the corresponding subscripts are omitted.

In a computation performed on a more powerful workstation, Maple estimated a memory usage of about 12GB, in order to derive J5 from J4.

1A preliminary version of this paper was presented at the 2011 IEEE International Conference on Robotics and Automation, Shanghai, China, 2011.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the Journal of Mechanisms and Robotics. Manuscript received March 24, 2012; final manuscript received April 17, 2013; published online June 20, 2013. Assoc. Editor: Vijay Kumar.

J. Mechanisms Robotics 5(3), 031008 (Jun 24, 2013) (10 pages) Paper No: JMR-12-1150; doi: 10.1115/1.4024293 History: Received September 26, 2012; Revised April 17, 2013

This paper studies the direct geometrico-static problem (DGP) of underconstrained cable-driven parallel robots (CDPRs) with three cables. The task consists in determining the end-effector pose and the cable tensile forces when the cable lengths are assigned. The problem is challenging, because kinematics and statics are coupled, and they must be tackled simultaneously. An effective elimination procedure is proposed and a least-degree univariate polynomial free of spurious factors is obtained in the ideal governing the problem. This is proven to admit 156 solutions in the complex field. Several approaches for the efficient computation of the complete solution set are presented, including an eigenproblem formulation and homotopy continuation.

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Figures

Grahic Jump Location
Fig. 1

Model of a cable-driven parallel robot with 3 cables

Grahic Jump Location
Fig. 2

DGP of the 3-3 CDPR presented in Table 2: equilibrium configurations with nonnegative tension in all cables

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