Singularity-free workspace analysis of general 6-UPS parallel mechanisms via convex optimization

• The maximum singularity-free ellipsoid of 6-UPS parallel mechanism is obtained.
• Two algorithms are proposed: improved lower bound SDP and sum of squares method.
• The proposed algorithms solve the problem in the low computational times.
• Convex optimization is applied for the kinematics of parallel mechanisms.
• The proposed methods open an avenue to use them for on-line purposes.

This paper investigates the singularity-free zones in the workspace of general 6-UPS parallel mechanisms. The emphasis is placed on obtaining the maximum volume ellipsoid or sphere in the singularity-free subregions, while taking into account the actuator strokes. The proposed algorithms are based on the convex optimization and have several advantages over the existing methods such as Lagrange multipliers approach, which makes it appropriate for other applicable optimization problems. For determining the maximal singularity-free zones, in order to find the maximum volume ellipsoid, a judicious iterative procedure, referred to as Improved Lower Bound Semidefinite Programming, is proposed. Additionally, different reformulations of the problem are proposed in order to solve other interesting problems in the robotics community such as obtaining the maximum volume sphere. Furthermore, an approach based on the sum of squares method is proposed to solve this problem for a general Gough–Stewart platform with any arbitrary geometric parameter which is conducive to a polynomial optimization problem. 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 on-line purposes.