Robotica (2005) volume 23, pp. 521–526. © 2005 Cambridge University Press doi:10.1017/S0263574704001109 Printed in the United Kingdom Direct position analysis of parallel manipulators which generate SP-2PS structures Raffaele Di Gregorio Department of Engineering, University of Ferrara, Via Saragat, 1-44100 FERRARA; Italy E-mail: rdigregorio@ing.unife.it (Received in Final Form: September 30, 2004) SUMMARY The determination of the assembly modes of the parallel structures with three legs of type PS or SP (P and S stand for prismatic pair and spherical pair, respectively) consists of solving the direct position analysis of all the three-legged parallel manipulators which have, in each leg, one not actuated prismatic pair, one not actuated spherical pair and one or two one-dof actuated pairs of any type, placed along the leg in any order. There are two types of such structures: (i) 3PS structures and (ii) SP-2PS structures. The procedure to determine the assembly modes of the SP-2PS structures has not been presented yet, in the literature. This paper presents the analytic form determination of the assembly modes of the SP-2PS structures. In particular, the closure equations of a generic SP-2PS structure will be written and their solution will be reduced to the solution of an eight-degree univariate polynomial equation with real coefficients. Finally, the proposed algorithm will be applied to a real case. The result of this study is that the assembly modes of any SP-2PS structure are at most eight, and the end-effector poses, which solve the direct position analysis of the parallel manipulators that generate those structures, are also eight. KEYWORDS: Kinematics; Position analysis; Parallel mechanisms; Parallel structure. I. INTRODUCTION Parallel manipulators (PMs) are closed-loop mechanisms where the end effector is connected to the frame through a number of kinematic chains (legs). Nowadays, PMs are used in nearly all the fields that, once, were reserved to serial manipulators. This increase in the PM applications was possible because of the progressive solution of a set of theoretical problems regarding the spatial closed-loop mechanisms. One out of these problems is the position analysis. The position analysis of a manipulator involves the solution of two sub-problems: the direct position analysis (DPA) and the inverse position analysis (IPA). The DPA is the determination of the end-effector poses (positions and orient- ations) for given values of the actuated-joint variables. The IPA is the determination of the values of the actuated-joint variables for a given end-effector pose. Both these problems are involved in the design of the manipulator controller. In the study of the PMs’ position analysis, great attention 1−5 has been paid to the Stewart platforms that are PMs with six degrees of freedom (dof) and six legs of type UPS (U, P and S stand for universal joint, prismatic pair and spherical pair respectively) where the prismatic pair is actuated. The interest for these types of PMs is due to the fact that the first PM applications 6,7 used them. The inverse position analysis of the Stewart platforms is straightforward (the unknowns can be immediately written as explicit functions of the data), whereas their direct position analyses involve the solution of non-linear equation systems that are very difficult to solve. 2−5 Actually, many parallel manipulators do not belong to the family of the Stewart platforms and, often, both the direct and the inverse position analyses of these other PMs involve the solution of non-linear equation systems 8,9 whose solution techniques are not presented yet. A set of PMs which do not belong to the Stewart-platforms’ family is the one collecting all the PMs that, when the actuators are locked, become parallel structures constituted of two rigid bodies (platform and base) connected by three legs of type PS or SP (Fig. 1). Such structures can have two topologies: (i) 3PS topology (3PS structures), where the three spherical pairs lie on the platform (base) (Fig. 1a), and (ii) SP-2PS topology (SP-2PS structures) where only two spherical pairs lie on the platform (base) (Fig. 1b). The assembly modes of a structure, without link permutations, correspond one-to-one to the end-effector poses that are solutions of the DPA of the manipulator which generates such a structure when the actuators are locked. Different manipulators can generate structures that are topologically equal. Therefore, the determination of the assembly modes of a structure with a given topology brings to find the solutions of the DPA of a wide family of manipulators. In particular, the determination of the assembly modes of the parallel structures with three legs of type PS correspond to solve the DPA of all the three-legged PMs which have, in each leg, one not actuated prismatic pair, one not actuated spherical pair and one or two one-dof actuated pairs of any type placed in any order along the leg. The method to determine the assembly modes of the 3PS structures has been presented by Parenti-Castelli and Innocenti, 10 whereas the determination of the assembly modes of the SP-2PS structures has not been presented yet, in the literature.