Comparison of 3-PP R Parallel Planar Manipulators Based on their Sensitivity to Joint Clearances Nicolas Binaud, St´ ephane Caro, Shaoping Bai and Philippe Wenger Abstract— In this paper, 3-PP R planar parallel manipulators with Δ- or U-shape base are compared with respect to their workspace size and kinematic sensitivity to joint clearances. First, the singularities and workspace of a general 3-PP R planar parallel manipulator are analyzed. Then, an error prediction model applicable to both serial and parallel manipulators is developed. As a result, two nonconvex quadratically constrained quadratic programs are formulated in order to find the max- imum reference-point position error and the maximum orien- tation error of the moving-platform for given joint clearances. Finally, the contributions of the paper are highlighted by means of a comparative study of two manipulators. I. INTRODUCTION Parallel manipulators are mechanisms that consist of two platforms, one fixed and the other movable, connected by multiple kinematic chains. Compared with serial industrial manipulators, parallel manipulators have advantages of high stiffness, high accuracy, high payload-mass ratio. However, an obvious drawback is the small workspace. The kinematics and design of Planar Parallel Manip- ulators (PPMs) have been extensively studied. Gosselin and Angeles studied the optimum kinematic design of 3- RRR PPMs [1]. Ur-Rehman et al. focused on the multiobjec- tive design optimization of 3PRR PPMs [2]. The singularities of PPMs were analyzed by means of screw theory in [3]. The actuation with a redundant degree of freedom was investigated in [4]. Merlet reported the direct kinematics of planar robot [5]. In the design of parallel manipulators, most designs adapt a symmetric topology. In the case of planar parallel ma- nipulators with three legs, a symmetric topology implies that the base and mobile platforms are equilateral and the three legs are identical. The design with a symmetric topology simplifies the manufacture and assembly. However, a symmetrical design may not be optimal in terms of some kinematic performance. A 3-PP R PPM with a U-shape base was proposed in [6] and a prototype is shown in Fig. 1(a). Among the most important sources of errors, we find manufacturing errors, assembly errors, compliance in the mechanical architecture, resolution of the servoactuators, backlash in the reductors, and clearances in the joints. As indicated in [7], [8], the errors due to manufacturing, assembly and compliance can be compensated through cal- ibration and model-based control. Joint clearances, on the N. Binaud, S. Caro and P. Wenger are with the Institut de Recherche en Communications et Cybern´ etique de Nantes, UMR CNRS 6597, 1 rue de la No¨ e, 44321 Nantes, France, nicolas.binaud@irccyn.ec-nantes.fr S. Bai is with the Department of Mechanical Engineering, Aalborg University, Denmark, shb@ime.aau.dk contrary, exhibit low repeatability, which generally makes their compensation difficult. For this reason, the focus of this paper is the impact of joint clearances on the pose errors of the moving platform of 3-PP R PPMs with Δ- or U-shape base. The paper is organized as follows. First, the two ma- nipulators under study are presented. Then, their kinematic model is derived in order to analyze their workspace and singularities. Finally, an error prediction model is developed and the contributions of the paper are highlighted by means of a comparative study of two 3-PP R PPMs with a Δ- and a U-shape base, respectively. II. MANIPULATORS UNDER STUDY Here and throughout this paper, R, P and P denote rev- olute, prismatic and actuated prismatic joints, respectively. Figure 1(a) illustrates the prototype of a 3-PP R PPM with a U-shape base introduced in [6] while Fig. 1(b) shows a 3-PP R PPM with a Δ-shape base. Both the manipulators contain a Δ-shape moving platform (MP) connected to the base by means of three identical kinematic chains, each one being composed of two orthogonal prismatic joints and a revolute joint. Notice that the second prismatic joint of each chain is actuated. The parameterization of the 3-PP R PPM (a) (b) Fig. 1. 3-PP R PPMs with (a) a U- and (b) a Δ-shape base is illustrated with Fig. 2. F b and F p are the base and the moving platform frames of the manipulator. In the scope of this paper, F b and F p are supposed to be orthogonal. F b is defined with the orthogonal dihedron (  Ox,  Oy), point O being its center and  Ox parallel to segment A 1 A 2 . Likewise, F p is defined with the orthogonal dihedron (  PX ,  PY ), point P being its center and  PX parallel to segment C 1 C 2 . The manipulator MP pose, i.e., its position and its orientation, is determined by means of the Cartesian coordinates vector p =[ p x , p y ] T of operation point P expressed in frame F b and angle φ , namely, the angle between frames F b and F p . The 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems October 18-22, 2010, Taipei, Taiwan 978-1-4244-6676-4/10/$25.00 ©2010 IEEE 2778