Application of Lagrangian Formulation to Predict the Positioning Error of Parallel Manipulator Under External Loads

Abstract

In last few decades, parallel manipulators have become an essential tool for many industries due to their high speed, accuracy, payload, stiffness and precision. These features make parallel manipulator ideal for jobs like, robotic surgery, precise manufacturing, missile launching and flight simulation, but precise part positioning on parallel manipulator is a challenging task in industry. With errors in position, one can never achieve the required quality of the part. In this work, our main focus is to find out positioning errors of a hexapod platform that are caused while machining. Legs will be considered elastic while the platform, part and leg-platform contacts will be considered rigid bodies. All legs are assumed to be 3D massless elastic trusses. When the load is applied on the part placed on the platform, its position may change due to elasticity of the manipulator. Lagrange formulation is used to calculate the positioning error of the platform as a result of machining forces and torques. A generalized algorithm is established, using Mathematica®, which gives us the positioning error of “n” legged system and for any initial orientation of the platform. In the second part, Natural frequencies of Stewart Platform are calculated from results generated in first part. The proposed methodology is applied on H-840 Hexapod designed and manufactured by Physik Instrumente. The Hexapod is designed and modelled in CATIA® and results are validated with Finite Element Analysis, as well as, from the literature.