Comparison of Planar Parallel Manipulator Architectures based on a Multi-objective Design Optimization Approach

This paper deals with the comparison of planar parallel manipulator architectures based on a multi-objective design optimization approach. The manipulator architectures are compared with regard to their mass in motion and their regular workspace size, i.e., the objective functions. The optimization problem is subject to constraints on the manipulator dexterity and stiffness. For a given external wrench, the displacements of the moving platform have to be smaller than given values throughout the obtained maximum regular dexterous workspace. The contributions of the paper are highlighted with the study of 3-PRR, 3-RPR and 3-RRR planar parallel manipulator architectures, which are compared by means of their Pareto frontiers obtained with a genetic algorithm. INTRODUCTION The design of parallel kinematics machines is a complex subject. The fundamental problem is that their performance heavily depends on their geometry [1] and the mutual dependency of almost all the performance measures. This makes the problem computationally complex and yields the traditional solution approaches inefficient. As reported in [2], since the performance of a parallel manipulator depends on its dimensions, the latter depend on the manipulator application(s). Furthermore, numerous design aspects contribute to the Parallel Kinematics Machine (PKM) performance and an efficient design will be one that takes into account all or most of these design aspects. This is an iterative process and an efficient design requires a lot of computational efforts and capabilities for mapping design parameters into design criteria, and hence turning out with a multiobjective design optimization problem. Indeed, the optimal geometric parameters of a PKM can be determined by means of a the resolution of a multiobjective optimization problem. The solutions of such a problem are non-dominated solutions, also called Pareto-optimal solutions. Therefore, design optimization of parallel mechanisms is a key issue for their development. Several researchers have focused on the optimization problem of parallel mechanisms the last few years. They have come up either with monoor multi-objective design optimization problems. For instance, Lou et al. [3, 4] presented a general approach for the optimal design of parallel manipulators to maximize the volume of an effective regular-shaped workspace while subject to constraints on their dexterity. Hay and Snyman [1] considered the optimal design of parallel manipulators to obtain a prescribed workspace, whereas Ottaviano and Ceccarelli [5, 6] proposed a formulation for the optimum design of 3-DegreeOf-Freedom (DOF) spatial parallel manipulators for given position and orientation workspaces. They based their study on the static analysis and the singularity loci of a manipulator in order to optimize the geometric design of the Tsai manipulator for a given free-singularity workspace. Hao and Merlet [7] discussed a multi-criterion optimal design methodology based on interval analysis to determine the possible geometric parameters satisfying two compulsory requirements of the workspace and accuracy. Similarly, Ceccarelli et al. [8] dealt with the multi-criterion op1 Copyright c © 2010 by ASME timum design of both parallel and serial manipulators with the focus on the workspace aspects, singularity and stiffness properties. Gosselin and Angeles [9, 10] analyzed the design of a 3-DOF planar and a 3-DOF spherical parallel manipulators by maximizing their workspace volume while paying attention to their dexterity. Pham and Chen [11] suggested maximizing the workspace of a parallel flexible mechanism with the constraints on a global and uniformity measure of manipulability. Stamper et al. [12] used the global conditioning index based on the integral of the inverse condition number of the kinematic Jacobian matrix over the workspace in order to optimize a spatial 3-DOF translational parallel manipulator. Stock and Miller [13] formulated a weighted sum multi-criterion optimization problem with manipulability and workspace as two objective functions. Menon et al. [14] used the maximization of the first natural frequency as an objective function for the geometrical optimization of the parallel mechanisms. Similarly, Li et al. [15] proposed dynamics and elastodynamics optimization of a 2-DOF planar parallel robot to improve the dynamic accuracy of the mechanism. They proposed a dynamic index to identify the range of natural frequency with different configurations. Krefft [16] also formulated a multi-criterion elastodynamic optimization problem for parallel mechanisms while considering workspace, velocity transmission, inertia, stiffness and the first natural frequency as optimization objectives. Chablat and Wenger [17] proposed an analytical approach for the architectural optimization of a 3-DOF translational parallel mechanism, named Orthoglide 3-axis, based on prescribed kinetostatic performance to be satisfied in a given Cartesian workspace. Most of the foregoing research works aimed to improve the performance of a given manipulator and the comparison of various architectures for a given application or performance has not been considered. In this paper, the mechanisms performance are improved over a regular shaped workspace for given specifications. As a result, we propose a methodology to deal with the multiobjective design optimization of PKMs. The size of the regular shaped workspace and the mass in motion of the mechanism are the objective functions of the optimization problem. Its constraints are determined based on the mechanism accuracy, assembly and the conditioning number of its kinematic Jacobian matrix. The proposed approach is applied to the optimal design of Planar Parallel Manipulators (PPMs) with the same mobility and set of design parameters. The non-dominated solutions, also called Pareto-optimal solutions, are obtained by means of a genetic algorithm for the three architectures and finally a comparison is made between them. MANIPULATORS UNDER STUDY Figure 1(a)–(c) illustrate the architectures of the planar parallel manipulators (PPMs) under study, which are named 3-PRR, 3-RPR and 3-RRR PPMs, respectively. Other families of PPMs are described in [2]. Here and throughout this paper, R, P, R and P denote revolute, prismatic, actuated revolute and actuated prismatic joints, respectively. The manipulators under study are composed of a base and a moving platform (MP) connected by means of three legs. Points A1, A2 and A3, (C1, C2 and C3, respectively) lie at the corners of a triangle, of which point O (point P, resp.) is the circumcenter. Each leg of the 3-PRR PPM is composed of a P, a R and a R joint in sequence. Each leg of the 3-RPR PPM is composed of a R, a P and a R joint in sequence. Likewise, each leg of the 3-RRR PPM is composed of three R joints in sequence. The three P joints of the 3-PRR and the 3RPR PPMs are actuated while the first R joint of each leg of the 3-RRR PPM is actuated. Fb and Fp are the base and the moving platform frames of the manipulator. In the scope of this paper, Fb and Fp are supposed to be orthogonal. Fb is defined with the orthogonal dihedron ( ~ Ox, ~ Oy), point O being its center and ~ Ox parallel to segment A1A2. Likewise, Fp is defined with the orthogonal dihedron ( ~ PX , ~ PY ), point C being its center and ~ PX parallel to segment C1C2. The manipulator MP pose, i.e., its position and its orientation, is determined by means of the Cartesian coordinates vector p = [px, py] T of operation point P expressed in frame Fb and angle φ , namely, the angle between frames Fb and Fp. The geometric parameters of the manipulators are defined as follows: (i) R is the circumradius of triangle A1A2A3 of circumcenter O, i.e., R = OAi; (ii) r is the circumradius of triangle C1C2C3 of circumcenter P, i.e., r = PCi, i = 1, . . . ,3; (iii) Lb is the length of the intermediate links, i.e., Lb = BiCi for the 3PRR PPM. Lb is also the maximum displacement of the prismatic joints of the 3-RPR PPM. Similarly, Lb is the length of the two intermediate links of the 3-RRR PPM, i.e., Lb = AiBi = BiCi; (iv) r j is the cross-section radius of the intermediate links; (v) rp: the cross-section radius of links of the moving platform, the latter being composed of three links. Stiffness Modeling The stiffness models of the three manipulators under study are obtained by means of the refined lumped mass modeling described in [21]. Figures 2 to 4 illustrate the flexible models of the legs of the 3-PRR, 3-RPR and 3-RRR PPMs, respectively. The actuator control loop compliance is described with a 1-dof virtual spring and the mechanical compliance of each link with a 6-dof virtual spring in each flexible model denoted θi. Besides, the moving platform of the manipulators is supposed to be composed of three links of length r connected to its geometric center P. From Fig. 2, the flexible model of the legs of the 3PRR PPM contains sequentially: (i) a rigid link between the manipulator base and the ith actuated joint (part of the base platform) described by the constant homogeneous transformation matrix TBase; (ii) a 1-dof actuated joint, defined by the homo2 Copyright c © 2010 by ASME