The aim of this paper is to present a systematic method for verifying the force-closure condition for general 3-DOF fully-constrained cable manipulators with four cables as based on the CAD (Cylindrical Algebraic Decomposition). A fundamental requirement for a cable manipulator to be fully controllable is that all its cables must be in tension at any working configurations. In other words, all the cable forces must be positive (assuming a positive cable force representing a tension and a negative cable force being a compression). Such a force feasibility problem is indeed referred to a force-closure problem (also called vector-closure problem assuming that the vectors of interest are the row vectors of the Jacobian matrix of the manipulator). The boundaries of the workspace can be obtained by the study of the Jacobian matrix of the manipulator. Therefore, this is equivalent to study the singularity conditions of four 3-RPR parallel robots. By using algebraic tools, it is possible to determine the singularity surfaces and their intersections yielding the workspace. Thus, it will be shown that the use of the CAD allows to get an implicit representation of the workspace as a set of cells. A comparative workspace analysis of three designs of mobile platforms, a line, a square and a triangle will be presented and discussed in this paper for a planar 4-cable fully-constrained robot.

### A comparative study of 4-cable planar manipulators based on Cylindrical Algebraic Decomposition

#### Abstract

The aim of this paper is to present a systematic method for verifying the force-closure condition for general 3-DOF fully-constrained cable manipulators with four cables as based on the CAD (Cylindrical Algebraic Decomposition). A fundamental requirement for a cable manipulator to be fully controllable is that all its cables must be in tension at any working configurations. In other words, all the cable forces must be positive (assuming a positive cable force representing a tension and a negative cable force being a compression). Such a force feasibility problem is indeed referred to a force-closure problem (also called vector-closure problem assuming that the vectors of interest are the row vectors of the Jacobian matrix of the manipulator). The boundaries of the workspace can be obtained by the study of the Jacobian matrix of the manipulator. Therefore, this is equivalent to study the singularity conditions of four 3-RPR parallel robots. By using algebraic tools, it is possible to determine the singularity surfaces and their intersections yielding the workspace. Thus, it will be shown that the use of the CAD allows to get an implicit representation of the workspace as a set of cells. A comparative workspace analysis of three designs of mobile platforms, a line, a square and a triangle will be presented and discussed in this paper for a planar 4-cable fully-constrained robot.
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2011
978-0-7918-5483-9
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11580/28600`
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