Optimal design of parallel manipulators via LMI approach

This paper deals with the problem of optimal design of parallel manipulators which are singularityless, of high stiffness and manipulability and the most economic. By observing that those requirements can be cast into Linear Matrix Inequalities (LMIs), we formulate the design problem as a convex optimization problem subject to LMIs with either a linear function or a max-det function as its objective function. The variables x associated with LMIs are nonlinear functions of some key kinematic parameters α. If the dimension of x, t, is equal to the number of kinematic parameters, l₀, a two-level algorithm can be applied to solve for a set of optimal kinematic parameters: (1) Applying the interior point algorithm for solving of x; (2) Applying Newton method to a set of nonlinear algebraic equations for solving of α. If the dimension of x is greater than the number of kinematic parameters (i.e., x are not linearly independent), we consider the constrained semi-definite programming problems and the constrained max-det problems by taking account of an additional set of nonlinear constraints. We propose a simplified constrained gradient algorithm for solving of x in such cases, α derives from x using Newton method. Simulation results verify the effectiveness of the proposed algorithms.