A Complete Kinematic Analysis of the SNU 3-UPU Parallel Robot

Several articles have been published about the SNU 3-UPU parallel robot, since the prototype built at the Seoul National University (SNU) showed a rather unexpected behavior, being completely mobile although none of the prismatic joints was actuated. The main goal of this work is to describe all possible poses of the robot by a system of algebraic equations using Study parameters, such that theoretical questions concerning assembly modes and mobility can be answered on the basis of the solutions of this system. We study the number of possible assembly modes for fixed limb lengths, also including the case where all lengths are equal. For the first time a complete analysis of the forward kinematics is given showing that the manipulator has theoretically up to 78 assembly modes, most of them being always complex. Investigating the Jacobian of the system of equations we show that for equal limb lengths the manipulator has some highly singular poses. Furthermore we discuss possible operation modes of the manipulator when the prismatic joints are actuated. To obtain these modes methods from algebraic geometry prove to be very useful. Moreover it is examined for which fixed design and joint parameters the mechanism allows self-motion. It is shown that there are only two such mobile robots. Their operation mode has no similarity with the pathologically mobile prototype.