On the maximal singularity-free ellipse of planar 3-RPRRPR parallel mechanisms via convex optimization

• The maximal singularity-free ellipse of 3-RPRRPR parallel mechanisms is obtained by means of convex optimization.
• Two algorithms are proposed, namely direct approach for convex problem and iterative approach for non-convex problem.
• The algorithms proposed in this work can be distinguished from the others reported in the literature due to its computational time and robustness.
• Compared to other works reported in the literature, one more constraint is taken into account for the optimization problem, namely, the constant-orientation workspace.
• The paper lays down the essential for applying convex optimization in the context of optimization of kinematic properties of parallel mechanisms.

This paper investigates the maximal singularity-free ellipse of 3-RPRRPR planar parallel mechanisms. The paper aims at finding the optimum ellipse, by taking into account the stroke of actuators, in which the mechanism exhibits no singularity, which is a definite asset in practice. Convex optimization is adopted for the mathematical framework of this paper which requires a matrix representation for the kinematic properties of the mechanism under study in order to solve the latter optimization problem. Based on the nature of the expressions involved in the problem, two situations may arise: dealing with either convex or non-convex expressions. Both situations are treated separately with two very fast and systematic algorithms. For the first situation, an exact method is applied while for the second one, which is a general form of the first situation, convex optimization is accompanied with an iterative procedure. The computational time for the two proposed algorithms are considerably low compared with other methods proposed in the literature which opens an avenue to use the proposed algorithms for real-time purposes.