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

Abstract

This paper investigates the approximated workspace and the maximal singularity-free ellipsoid of a class of 6-UrS 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.

A new neural gas network approach for obtaining the singularity-free workspace of 3-DOF planar parallel manipulators

“…Consequently, it is of paramount of importance to investigate the singularity-free workspace of such mechanisms which can be regarded as one of the challenges in kinematic analysis of PMs. [9][10][11] Since obtaining singularity-free workspace of PMs can be made equivalent to an optimization problem, the problem has thus been investigated with four different approaches in the literature: (1) interval analysis; [12][13][14] (2) convex optimization; 15,16 (3) Lagrangian multipliers; 17,18 and (4) numerical approaches. 4,5,[19][20][21][22] .…”

“…Convex optimization has previously been used 15,16 as the main mathematical framework for obtaining the maximal singularity-free ellipse within the workspace of the 3-RPR PM and the Gough-Stewart platform. As mentioned, the latter approach could not be extended to other planar PMs with complex singularity expressions.…”