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On the approximated and maximal singularity-free workspace of 6-UPS parallel mechanisms using convex optimization

This paper investigates the approximated workspace and the maximal singularity-free ellipsoid of a class of 6-UPS parallel mechanisms referred to as quadratic Gough-Stewart platforms. More emphasis is placed on finding the optimum ellipsoid, by taking into account the stroke of actuators, in which the mechanism exhibits no singularity, a definite asset in practice. Convex optimization is adopted for the mathematical framework of this paper which requires a matrix reformulation for the kinematic properties. For obtaining the approximated workspace, an exact approach which is composed of two convex optimization methods, called exact methods I & II, are proposed. For the maximal singularity-free ellipsoid, in order to find the optimal solution, a judicious iterative procedure, referred to as iterative method, is proposed which is an extension to a recent algorithm leading to a suboptimal solution. The computational time for the proposed algorithms are considerably low compared to other methods proposed in the literature for obtaining the singularity-free workspace which opens an avenue to use them as systematic algorithms for real-time purposes.