The optimal design of a planar Stewart platform for prescribed machining tasks

Abstract

Recently parallel platforms, also known as Stewart platforms, have been the subject of much active research because of their distinct advantages over serially linked manipulators. Parallel platforms may have a great impact, especially in the field of machine tools. Parallel platforms are however, not yet commonly used as machine tools. The main reason for this is the lack of a general and rationally based design system that is also easily implementable. The availability of such a system will allow for the set-up of a platform so that, not only will the task be executable, but it will also be performed in an optimum manner according to a criterion specified by the user. This study proposes an easy to use methodology that may by applied to the optimum design of planar parallel platforms for machining applications. The design methodology presented here is based on mathematical optimization. This approach is simple and intuitive, and all the most important design criteria can be implemented, in some mathematical form, in the application of the proposed optimization methodology. Six possible design variables are defined, all related to the physical dimensions and placement of the platform for a prescribed task. Different platform designs are studied by using different combinations of design variables, where some design variables are fixed at specific values while the remaining design variables are allowed to vary. Two types of design constraints are considered, namely geometrical constraints that specify physical bounds on the platform size and placement, and secondly, limits on the maximum and minimum allowable actuator leg lengths. The platform design is optimized according to a prescribed criterion. Two design criteria, also called cost functions, are considered in this study. The first is the minimization of the actuator forces as the manipulator executes a prescribed task. The actuator forces are calculated by means of a dynamical analysis software package, DADS. The other design criterion is the maximization of the so-called quality index over the prescribed tool path. The success of mathematical design optimization depends largely on the optimization algorithm that is used to solve the minimization problem. It is shown in this study that the minimization of the actuator forces is a difficult problem to solve by mathematical optimization. Important reasons for this are that some of the cost functions considered here have discontinuities in their gradients with respect to the design variables, and that they are also highly non-convex and non¬quadratic. Another difficulty is the presence of numerical noise superimposed on the cost functions. For this reasons the robust and reliable LFOPC optimization algorithm, of Snyman was used. This method, although relatively slow to converge, was highly successful in solving the various design optimization problems. Because of the slow convergence of the method and the computational cost of evaluating the actuator force cost function and gradients when many design variables are involved, it was decided to attempt to speed up convergence by the use of an approximation method. The approximation algorithm. Dynamic-Q also proposed by Snyman, was selected and its application illustrated by solving a design problem with many design variables. This algorithm quickly converged to an acceptable optimum design. The main conclusion of this study is that the design methodology proposed here can successfully be applied to the design of planar Stewart platforms and may easily be extended to also apply to spatial Stewart platforms. This methodology solves difficult design problems that are inherent to the design of parallel manipulators. Its future implementation in a more comprehensive design and operating system for Stewart platforms to be used for machining tasks is therefore imperative.