- ----- Proceedings of the IEEE SMC UK-RI Chapter Conference 2007 on Cybernetic Systems, September 6-7, Dublin, Ireland Modeling and Solving Strategies for the Forward Kinematics Problem of the translation parallel manipulators, application to high speed manufacturing and high accuracy Luc Rolland University of Central Lancashire Email: Luc.Rolland@gmail.comorlrolland@uclan.ac.uk Abstract- This article investigates the forward kinematics problem formu- lation and resolution specifically applied to spatial parallel manip- ulators which allow only for translationnal displacements of the end-effector. The translation manipulator is usually constituted of three to six kinematics chains with three actuators usually located on the base. These mechanisms are designed to prevent any rotation. This article shall focus on the Delta manipulator, but the analyses can be extrapollated to any other translation manipulators. To acheive FKP computations, it is relevant to ask wheather it is possible to improve computations by proper modeling and algorithms. The goal is to minimize the real solution number along with response times. ' An algebraic method is applied to solve the FKP for parallel robots. It is implementing proven Grobner bases from which an equivalent univariate polynomial system is deduced. This univariate system resolution leads to all the system real solutions which exactly corresponds to the manipulator postures. The Delta manipulator can be described by two equivalent models: the 3-RCC and the 6-6 from which four different kinematics formu- lations are determined. Finally, using the second formulation, a solution reduction algortithm is introduced. A discussion follows which allows to select the best approach in typical applications such as material handling and high speed milling. Index Terms- Parallel robot, translation manipulator, forward kinematics, position based model, Grobner basis, rational uni- variate representation, real root isolation, exact algebraic method, certified results. 1. INTRODUCTION Since their successful application as flight simulators, [1], parallel manipulators have attracted academic and industrial interest. Over the years, the industry has been trying to implement robots with better performance for increased pro- ductivity. In order to maintain stiffness to appropriate levels, serial robots have been constructed with high mass bodies and large joints. Furthermore, their serial mechanical architecture accounts for their inherent flexibility. To insure controllability, their accelerations were limited to no more than 10~. In the mid eighties, the Delta manipulator was introduced as a translational robot which happens to be the first parallel robot to meet commercial success in fast material handling, fig. (I), [2], [3]. Parallel robots have capitalized on their parallel mechanical architecture with better dynamics insuring for higher accelerations (up to 500;:;, [4]) and larger payloads. ISSN 1744-9170 Copyright IEEE 2007 Fig. 1. The Delta and its application in industry, (photos courtesy of ABB and Demaurex SA) Seizing on the occasion, many other translational robots were imagined such as Herve's Star [5], Reboulet's Speed- R-Man [6], Schonflies based [7], Tsai's 3UPU [8], Kong's 3CRR [9], Shen's 3CCR [10], 3RCC [11], etc. This article shall focus on the Delta manipulator which is the most widely used in industry, but our analyses can be extrapolated to most other cited manipulators. From the Delta, evolved a certain number of linear manipulators replacing the rotation motors and forearms by prismatic drives with guiding rails (Star, linear Delta, Linapod, Triaglide, etc), fig. (2). More recently, these linear manipulators were also implemented as high speed milling machine, [12], [13]. The typical translational parallel manipulator is constituted 106 ~ ~~- -- ~~~=- .,. -.. _.~ ~~~~