The Constant-Orientation Dimensional Synthesis of Planar Cable-Driven Parallel Mechanisms Through Convex Relaxations
The wrench-closure workspace (WCW) of cable-driven parallel mechanisms is the set of poses for which any wrench can be produced at the end-effector by a set of positive cable tensions. In this paper, we tackle the dimensional synthesis problem of finding
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Abstract The wrench-closure workspace (WCW) of cable-driven parallel mechanisms is the set of poses for which any wrench can be produced at the end-effector by a set of positive cable tensions. In this paper, we tackle the dimensional synthesis problem of finding a geometry for a planar cable-driven parallel mechanism (PCDPM) whose constant orientation wrench closure workspace (COWCW) contains a prescribed workspace. To this end, we first introduce a linear program to verify whether a given pose is inside or outside the WCW of a given PCDPM. The relaxation of this linear program over a box leads to a nonlinear feasibility problem that can only be satisfied when this box is completely inside the COWCW. We extend this feasibility problem to find a PCDPM geometry whose COWCWs include a given set of boxes. These multiple boxes may represent an estimate of the prescribed workspace, which may be obtained through interval analysis. Finally, we introduce a nonlinear program through which the PCDPM geometry is changed while maximizing the scaling factor of the prescribed set of boxes. When the optimum scaling factor is greater or equal to one, the COWCW of the resulting PCDPM contains the set of boxes. Otherwise, the COWCW generally offers a good coverage of the set of boxes.
1 Introduction A planar parallel cable driven mechanism (PCDPM) generally consists of a moving platform (MP) and a fixed frame, which are connected with multiple cables. Each cable is wound around an actuated reel fixed to the base, and is attached to the moving platform at its other end. The cables and the moving platform are assumed to K. Azizian (B) · P. Cardou Department of Mechanical Engineering, Laval University, Quebec City, QC G1V 0A6, Canada e-mail: [email protected] P. Cardou e-mail: [email protected] T. Bruckmann and A. Pott (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 12, DOI: 10.1007/978-3-642-31988-4_14, © Springer-Verlag Berlin Heidelberg 2013
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be contained in the same plane. The actuated reels control the position and orientation of the moving platform in this plane by controlling the lengths of their corresponding cables. Since the cables can be wound onto reels over long lengths, the workspace of a cable-driven mechanism can be larger than that of a conventional parallel mechanism. This is only a potential advantage, however, as the workspace of a PCDPM is further limited by the inability of cables to push on the moving platform. Indeed, there generally exist many poses inside this workspace for which the cables cannot balance all applied wrenches, because at least one of them would have to push on the platform. More formally, the wrench-closure workspace (WCW) is the set of poses for which all applied wrenches are feasible. An applied wrench is said to be feasible if it can be balanced by a set of a non-negative cable tensions. This is a special case of the wrench-feasible workspace (WFW), which is the set of poses of the moving platform for which the cables ca
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