Optimal structural design for a planar parallel platform for machining

Abstract

Parallel manipulators have many advantages over traditional serial manipulators. These advantages include high accuracy, high stiffness and high load-to-weight ratio, which make parallel manipulators ideal for machining operations where a high accuracy is required to meet the requirements that modern standards demand. A number of previous workers have determined the stiffness of parallel platforms using the duality between instantaneous kinematics and statics in parallel mechanisms. For the aforementioned analysis, compliance is introduced in the actuators, resulting in a platform stiffness matrix. This methods furthermore predicts when the platform approaches a singular or ill-conditioned configuration. However, this idealized estimate of the stiffness is not accurate enough to determine how an actual platform assembly will react to an externally applied force. For a planar parallel platform, the out of plane stiffness is not included in the resulting stiffness matrix since the kinematics equations are derived only in the plane in which the platform operates. Recently, the finite element method (FEM) has been used by some workers to determine the stiffness of spatial manipulators. These models are mainly used to verify stiffness predicted using kinematic equations, and are restricted to relatively simple truss-like models. In this study, state-of-the-art finite elements are used to determine the out of plane stiffness for parallel manipulators. Beam elements that make use of Timoshenko beam theory and flat shell elements with drilling degrees of freedom are used to model the platform assembly. The main objective of this study is to quantify the stiffness, particularly the out of plane stiffness, of a planar parallel platform to be used for machining operations. The aim is to suggest a design that is able to carry out machining operations to an accuracy of 10 µm for a given tool force. Reducing the weight of a parallel manipulator used in machining applications has many advantages, e.g. increased maneuverability, resulting in faster material removal rates. Therefore the resulting proposed design is optimized with respect to weight, subject to displace¬ment and stress constraints to ensure feasible stiffness and structural integrity. This optimization is carried out with both gradient-based methods and a genetic algorithm (GA). The gradient-based methods include LFOPC and Dynamic-Q. A binary GA, imple¬mented as both a micro GA and full GA, is used to provide for the future inclusion of discrete design variables. Stiffness maps, as proposed by Gosselin, are drawn for the optimal design. These stiffness maps can aid in determining the best toolpath inside a feasible workspace. It is envisaged that this work, together with current work at the University of Pretoria, will result in a feasible design for a planar parallel platform to be used in industry. An application of such a planar parallel platform lies in retro-fitting existing, relatively inexpensive 3-axis milling equipment. This increases their capability at a lower cost than the of the alternative of purchasing a traditional 5-axis milling center.