Topological and Geometrical Synthesis of Three-Degree-of-Freedom Fully Parallel Manipulators by Instantaneous Kinematics Xiaoyu Wang, ∗ Luc Baron, † and Guy Cloutier ‡ D´ epartement de g´ enie m´ ecanique, ´ Ecole polytechnique de montr´ eal P.O. 6079, station Centre-Ville, Montr´ eal, Qu´ ebec, Canada, H3C 3A7 (Dated: May 1, 2007) This paper presents a new synthesis procedure of fully parallel manipulators (PMs) of 3 degrees of freedom (DOFs) that could be implemented in a computer-aided synthesis process. Possible designs of PMs are represented by a set of unit joint-twists at an initial configuration, called here topological and geometric parameters (TGPs). This makes it possible to represent PMs of all topologies and geometries in an easy and consistent way. The kinematic bond between the end- effector (EE) and the base is then formulated as a set of equations involving TGPs, actuated-joint variables and non-actuated joint variables (passive joints). To achieve the required type of EE motion, possible topologies are first derived from tangent space analysis, and then the feasible topologies are retained by further displacement analysis. The geometries are determined such that the set of equations should be isoconstrained when passive-joint variables are taken as unknowns. The synthesis procedure of 3-DOF PMs is illustrated with three numerical examples: one producing a new architecture of one translation and two rotations, while the other two producing existing architectures of translational PMs. PACS numbers: Valid PACS appear here I. INTRODUCTION A parallel manipulator (PM) is a closed-loop mecha- nism in which the end-effector (EE) is connected to the base through at least two independent kinematic chains. A fully PM is a PM with an n-DOF EE connected to the base by n independent kinematic chains, each having a single actuated joint [1]. Applications of PMs can be found in motion simulators, high-precision surgical tools, precision assembly tools, machine tools, and a number of industrial equipments because of their high load-carrying capacity, accurate positioning, high speed, and high ca- pacity of acceleration [2]. Despite of the high potential of performance offered by PMs, many applications are not yet as successful as expected [3]. The closed-loop na- ture of PMs limits the motion of the EE and involves very complex kinematic singularities within the workspace [4]. Moreover, it is difficult to find 6-DOF PMs with orienta- tion performance comparable to the one of serial manip- ulators [5]. To overcome these drawbacks, authors of [6] and [7] employed the modular design concept. An inge- nious design was proposed in [8] which makes a 6-DOF PM’s orienting decoupled from positioning and yields un- limited rotation. Another strategy is to connect in se- ries two PMs of 3-DOF (the two together producing the 6-DOF mobility of the EE) in the aim to improve the overall performance and make the design easier [5]. The advantages of this kind of hybrid manipulator are illus- ∗ Electronic address: xiaoyu.wang@polymtl.ca † Electronic address: luc.baron@polymtl.ca ‡ Electronic address: guy.cloutier@polymtl.ca trated by the hybrid kinematic machine [9]. Therefore, the synthesis of PMs of 3 DOFs has become an important design issue. In the last two decades, numerous topologies of 3- DOF PMs have been published or disclosed in patent files. These topologies can be divided into two large cat- egories: spherical PMs (SPMs) and translational PMs (TPMs) [10, 11]. The SPM was proposed and systemati- cally studied by many authors, e.g. [12]. For TPMs, they can be further divided into two families. The first family is characterized by 3 parallelogram submechanisms which constrain the EE to a constant orientation. Many of the first family have been extensively studied and prototypes built, for example: Delta [13], Y-Star [14], Orthoglide [15], and a TPM proposed by the authors of [16]. The representative design of the second family is the 3-UPU proposed by the author of [17], whose singularity prob- lem has been the focus of a great deal of effort for several years. With the same design principle as the 3-UPU, several TPM topologies were derived [18]. In the mean- while, a significant effort has also been made in 3-DOF PMs which produce displacement in both translation and rotation. PMs proposed in [19–21] are two examples of that. For each of these topologies, different geometric synthesis and kinematic analysis approaches have been proposed. Some recent works in this regard can be found in [22–24]. The topological and geometrical synthesis of the above PMs have been accomplished mostly on a case- by-case basis [25]. On the design methodology side, group theory has al- ready been applied to the topological synthesis of TPMs [26]. Successful examples of this application are the syn- thesis of Y-Star, the establishment of the displacement subgroup inventory [27] and the synthesis of TPMs with