Position Analysis of the RRP-3(SS) Multi-Loop Spatial Structure

The paper presents the position analysis of a spatial structure composed of two platforms mutually connected by one RRP and three SS serial kinematic chains, where R, P, and S stand for revolute, prismatic, and spherical kinematic pair respectively. A set of three compatibility equations is laid down that, following algebraic elimination, results in a 28 th -order univariate algebraic equation, which in turn provides the addressed problem with 28 solutions in the complex domain. Among the applications of the results presented in this paper is the solution to the forward kinematics of the Tricept, a well-known in-parallel-actuated spatial manipulator. Numerical examples show adoption of the proposed method in dealing with two case studies. INTRODUCTION Parallel kinematics machines have gained more and more interest in research as well as in industry. In effect, their high stiffness and low inertia make them attractive alternative designs for demanding tasks such as high speed machining [1]. Parallel kinematics machines may feature a fully-parallel kinematic architecture, like the Gough-Stewart platform, or a hybrid parallel-serial kinematic architecture, like the Tricept machine, in which a parallel positioning device carries a serial wrist [2]. With reference to Fig.1, a Tricept manipulator can be thought of as obtained by serially connecting two elemental manipulators, precisely a three-degree-of-freedom (3-dof) inparallel-actuated manipulator and – depending on the specific Tricept model – a 2-dof or 3-dof serial manipulator (Fig. 1 shows a 6-dof Tricept). The in-parallel-actuated elemental manipulator of a Tricept consists of a fixed base connected to a movable platform by four serial kinematic sub-chains. One of these sub-chains is of type UP (U stands for universal – or Hooke – joint, P means prismatic kinematic pair), whereas the remaining three subchains are of type UPS (S signifies spherical kinematic pair). The serial elemental manipulator connects the alreadymentioned movable platform to the end-effector of the Tricept via a universal joint (5-dof Tricept) or a spherical wrist realized by three revolute pairs having mutually-intersecting axes (6-dof Tricept, see Fig. 1). In any Tricept, the centers of the U-joints in the UPS subchains, as well as the centers of the S-joints, are at the vertices of equilateral triangles fixed to the base and to the movable platform respectively. Moreover, the center of the U-joint in the UP sub-chain is equidistant from the centers of the remaining U-joints; in the U-joint of the UP sub-chain, the axis of the revolute kinematic pair next to the manipulator base is either orthogonal or parallel to a side of the aforementioned fixed equilateral triangle; the line through the center of the U-joint of the UP sub-chain and parallel to the sliding movement of this sub-chain’s P-joint intersects the movable triangle at its center; the UP sub-chain is so arranged as to make – for a subset of manipulator configurations – the sides of the movable equilateral triangles parallel to and equidistant from the sides of the fixed equilateral triangle. The Tricept end-effector is moved in space by actuating the P pairs in the three UPS sub-chains of the in-parallelactuated elemental manipulator, as well as the revolute pairs of the serial elemental manipulator (in Fig. 1 all actuated kinematic pairs are highlighted by asterisks). In the sequel, only the case of a 6-dof Tricept will be taken into account, the differences with respect to the case of a 5-dof Tricept being manifest and easily manageable. Position analysis is a necessary step for the control of a manipulator. The position analysis problem consists in determining the relationship between the parameters of motion of the actuated joints and the rigid-body position (location) of the manipulator’s end-effector. The position analysis problem comprises two dual problems, namely, the inverse kinematics and the forward kinematics. The inverse kinematics of a manipulator is the search for the parameters of motions of all actuated kinematic pairs in the UPS sub-chains once the location of the end-effector has been assigned. It is not difficult to recognize that the inverse kinematics of the in-parallel-actuated elemental manipulator of a Tricept is equivalent to the inverse kinematics of a serial UP regional manipulator endowed with a spherical wrist [3]. Addressing exhaustively this problem leads to eight real sets of parameters of motion that stem from solving firstand secondorder univariate algebraic equations only. ha l-0 04 54 56 3, v er si on 1 8 Fe b 20 10 Author manuscript, published in "Journal of Mechanical Design 128, 1 (2006) 272-278"