arXiv:1012.2787v1 [cs.RO] 13 Dec 2010  Proceedings of the ASME 2010 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2010 August 15-18, 2010, Montreal, Quebec, Canada  DETC2010-28650  COMPARISON OF PLANAR PARALLEL MANIPULATOR ARCHITECTURES BASED ON A MULTI-OBJECTIVE DESIGN OPTIMIZATION APPROACH  Damien CHABLAT,   St  ́ ephane CARO,   Raza UR-REHMAN,   Philippe WENGER  Institut de Recherche en Communications et Cybern  ́ etique de Nantes UMR CNRS n ◦   6597 1 rue de la No  ̈ e, 44321 Nantes, France Email:   { chablat, caro, ur-rehman, wenger } @irccyn.ec-nantes.fr  ABSTRACT  This paper deals with the comparison of planar parallel ma- nipulator architectures based on a multi-objective design opti- mization 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 contri- butions of the paper are highlighted with the study of 3- P RR , 3- RP R   and 3- R RR   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 depen- dency of almost all the performance measures. This makes the problem computationally complex and yields the traditional so- lution approaches inefficient. As reported in [2], since the perfor- mance 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 com- putational efforts and capabilities for mapping design parame- ters into design criteria, and hence turning out with a multiobjec- tive design optimization problem. Indeed, the optimal geomet- ric parameters of a PKM can be determined by means of a the resolution of a multiobjective optimization problem. The solu- tions of such a problem are non-dominated solutions, also called Pareto-optimal solutions. Therefore, design optimization of par- allel mechanisms is a key issue for their development. Several researchers have focused on the optimization prob- lem of parallel mechanisms the last few years. They have come up either with mono- or multi-objective design optimization problems. For instance, Lou et al. [3, 4] presented a general ap- proach for the optimal design of parallel manipulators to maxi- mize 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-Degree- Of-Freedom (DOF) spatial parallel manipulators for given po- sition 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 satisfy- ing two compulsory requirements of the workspace and accuracy. Similarly, Ceccarelli et al. [8] dealt with the multi-criterion op- 1   Copyright   c ©   2010 by ASME
timum design of both parallel and serial manipulators with the focus on the workspace aspects, singularity and stiffness prop- erties.   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 trans- lational parallel manipulator. Stock and Miller [13] formulated a weighted sum multi-criterion optimization problem with ma- nipulability 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 dynam- ics 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 fre- quency with different configurations. Krefft [16] also formulated a multi-criterion elastodynamic optimization problem for paral- lel mechanisms while considering workspace, velocity transmis- sion, inertia, stiffness and the first natural frequency as optimiza- tion objectives. Chablat and Wenger [17] proposed an analytical approach for the architectural optimization of a 3-DOF transla- tional 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 vari- ous 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 specifica- tions. 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 mech- anism 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 ge- netic algorithm for the three architectures and finally a compari- son is made between them.  MANIPULATORS UNDER STUDY  Figure 1(a)–(c) illustrate the architectures of the planar par- allel manipulators (PPMs) under study, which are named 3- P RR , 3- RP R   and 3- R RR   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   A 1 ,   A 2   and   A 3 , ( C 1 ,   C 2   and   C 3 , respec- tively) lie at the corners of a triangle, of which point   O   (point   P , resp.) is the circumcenter. Each leg of the 3- P RR   PPM is com- posed of a   P , a   R   and a   R   joint in sequence.   Each leg of the 3- RP R   PPM is composed of a   R , a   P   and a   R   joint in sequence. Likewise, each leg of the 3- R RR   PPM is composed of three   R  joints in sequence. The three   P   joints of the 3- P RR   and the 3-  RP R   PPMs are actuated while the first   R   joint of each leg of the 3- R RR   PPM is actuated.  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 di- hedron   (   ~ PX , ~ PY   ) , point   C   being its center and   ~ PX   parallel to seg- ment   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 geometric parameters of the manipulators are defined as follows: (i)   R   is the circumradius of triangle   A 1 A 2 A 3   of cir- cumcenter   O , i.e.,   R   =   OA i ; (ii)   r   is the circumradius of triangle  C 1 C 2 C 3   of circumcenter   P , i.e.,   r   =   PC i ,   i   =   1 , . . . ,   3; (iii)   L b   is the length of the intermediate links, i.e.,   L b   =   B i C i   for the 3-  P RR   PPM.   L b   is also the maximum displacement of the pris- matic joints of the 3- RP R   PPM. Similarly,   L b   is the length of the two intermediate links of the 3- R RR   PPM, i.e.,   L b   =   A i B i   =   B i C i ; (iv)   r   j   is the cross-section radius of the intermediate links; (v)   r p : 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 de- scribed in [18]. Figures 2 to 4 illustrate the flexible models of the legs of the 3- P RR , 3- RP R   and 3- R RR   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  θ i .   Be- sides, 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 3-  P RR   PPM contains sequentially: (i) a rigid link between the manipulator base and the   i th   actuated joint (part of the base plat- form) described by the constant homogeneous transformation matrix   T i Base ; (ii) a 1-dof actuated joint, defined by the homo- 2   Copyright   c ©   2010 by ASME
PSfrag replacements  O P A 1  A 2  A 3  B 1  B 2  B 3  C 1  C 2  C 3  ρ 1  ρ 2  ρ 3  R r  φ  (a) 3- P RR PPM  PSfrag replacements  O P A 1  A 2  A 3  B 1  B 2  B 3  C 1  C 2  C 3  ρ 1  ρ 2  ρ 3  R r  φ  (b) 3-R P R PPM  PSfrag replacements  O P A 1  A 2  A 3  B 1  B 2  B 3  C 1  C 2  C 3  ρ 1  ρ 2  ρ 3  R r  φ  (c) 3-R R R PPM  FIGURE 1 .   THE THREE PLANAR PARALLEL MANIPULATORS UNDER STUDY  Base platform (Rigid)  A c  R  Rigid body Rigid body 1-dof   6-dof 6-dof spring   spring spring  L   r  A c  FIGURE 2 .   FLEXIBLE MODEL OF THE 3- P RR   PPM’S KINE- MATIC CHAINS  geneous matrix function   V a ( q i  0 )   where   q i  0   is the actuated coor- dinate; (iii) a 1-dof virtual spring describing the actuator me- chanical stiffness, which is defined by the homogeneous ma- trix function   V s 1  (  θ   i  0  )   where  θ   i  0   is the virtual spring coordinate  PSfrag replacements  Base platform (Rigid)  A c  R  Rigid body Rigid body 1-dof   6-dof 6-dof   spring spring spring  L   r  A c  FIGURE 3 .   FLEXIBLE MODEL OF THE 3-R P R   PPM’S KINE- MATIC CHAINS  corresponding to the translational spring; (iv) a 1-dof passive   R - joint at the beginning of the leg allowing one rotation angle   q i  2 , which is described by the homogeneous matrix function   V r 1 ( q i  2 ) ; (v) a rigid leg of length   L   linking the foot and the movable plat- 3   Copyright   c ©   2010 by ASME
Base platform (Rigid)  A c  R  Rigid body Rigid body Rigid body 1-dof   6-dof 6-dof 6-dof spring   spring spring spring  L   L   r A c  FIGURE 4 .   FLEXIBLE MODEL OF THE 3- R RR   PPM’S KINE- MATIC CHAINS  form, which is described by the constant homogeneous transfor- mation matrix   T i L ; (vi) a 6-dof virtual spring describing the leg stiffness, which is defined by the homogeneous matrix function  V s 2  ( θ   i  1   · · ·  θ   i  6  ) , with  θ   i  1 ,  θ   i  2 ,  θ   i  3   and  θ   i  4 ,  θ   i  5 ,  θ   i  6   being the virtual spring coordinates corresponding to the spring translational and rotational deflections; (vii) a 1-dof passive   R -joint between the leg and the platform, allowing one rotation angle   q i  3 , which is described by the homogeneous matrix function   V r 2 ( q i  3 ) ; (viii) a rigid link of length   r   from the manipulator leg to the geometric center of the mobile platform, which is described by the con- stant homogeneous transformation matrix   T i r ; (ix) a 6-dof vir- tual spring describing the stiffness of the moving platform, which is defined by the homogeneous matrix function   V s 3  (  θ   i  7   · · ·  θ   i  12  ) ,  θ   i  7 ,   θ   i  8 ,  θ   i  9   and  θ   i  10 ,  θ   i  11 ,  θ   i  12   being the virtual spring coordinates corresponding to translational and rotational deflections of link  C i P ; (x) a homogeneous transformation matrix   T i End   that char- acterizes the rotation from the 6-dof spring associated with link  C i P   and the manipulator base frame. As a result, the mathematical expression defining the end- effector location subject to variations in all above defined coor- dinates of a single kinematic chain   i   of the 3- P RR   PPM takes the form:  T i   =   T i Base V i a  ( q i  0  )   V s 1  (  θ   i  0  )   V r 1  ( q i  1  )   T i L V s 2  (  θ   i  1   · · ·  θ   i  6  )  V r 2 ( q i  2 ) T i r V s 3  (  θ   i  7   · · ·  θ   i  12  )   T i End   (1) Similarly, the mathematical expressions associated with the kinematic chains of the 3- RP R   and 3- R RR   PPMs are obtained. From [18], the kinetostatic model of the i th   leg of the   X - PPMs can be reduced to a system of two matrix equations, namely,  [   S i  θ   | X   J i q  J i q   0 2 × 2  ] [   f i  δ   q i  ]  =  [  δ   t i  0 2  ]  (2) where   X   stands for 3- P RR , 3- RP R   or 3- R RR . The sub-matrix  S i θ   | X   =   J i  θ   | X   K i  θ   | X  − 1 J i  θ   | X T   describes the spring compliance rela- tive to the geometric center of the moving platform, and the sub- matrix   J i q   takes into account the passive joint influence on the moving platform motions.   J i  θ   is the Jacobian matrix related to the virtual springs and   J i q   is the one related to the passive joints.  K i  θ   | X  − 1   describes the compliance of the virtual springs.  K i  θ   | 3 P RR  − 1   =      K i act  − 1   0 1 × 6   0 1 × 6  0 6 × 1   K i link  − 1   0 6 × 6  0 6 × 1   0 6 × 6   K ip f  − 1       (3a)  K i  θ   | 3 RP R  − 1   =      K i link  − 1   0 6 × 1   0 6 × 6  0 1 × 6   K i act  − 1   0 1 × 6  0 6 × 6   0 6 × 1   K ip f  − 1       (3b)  K i  θ   | 3 R RR  − 1   =         K i act  − 1   0 1 × 6   0 1 × 6   0 1 × 6  0 6 × 1   K i link 1  − 1   0 6 × 6   0 6 × 6  0 6 × 1   0 6 × 6   K i link 2  − 1   0 6 × 6  0 6 × 1   0 6 × 6   0 6 × 6   K ip f  − 1          (3c) where   K i act   is the 1   ×   1 stiffness matrix of the i th   actuator,   K i link   is the 6   ×   6 stiffness matrix of the intermediate link for the 3- P RR  and 3- RP R   PPMs while   K i link 1   and   K i link 2   are the 6   ×   6 stiffness matrices of the first and second intermediate links of the i th   leg of 3- R RR   PPM.   K ip f   is the 6   ×   6 stiffness matrix of the i th   link of the moving platform. The compliance matrix of each link is expressed by means of the stiffness model of a cantilever beam, namely,  K i L  − 1   =              L  EA   0   0   0   0   0 0   L 3  3 EI z   0   0   0   L 2  2 EI z  0   0   L 3  3 EI y   0   −   L 2  2 EI y   0 0   0   0   L  GI x   0   0 0   0   −   L 2  2 EI y   0   L  EI y   0 0   L 2  2 EI z   0   0   0   L  EI z              (4)  L   being the length of the corresponding link,   A   is its the cross- sectional area, i.e.,   A   =  π r 2  j   for the links of the manipulators legs and   A   =  π r 2  p   for the links of the moving platform.   I y   and   I z   are the polar moments of inertia about   y   and   z   axes, resp.   I y   =   I z   =  π r 4  j   / 4 for the links of the manipulators legs and   I y   =   I z   =  π r 4  p / 4 for the links of the moving platform.   I x   =   I z   +   I y   is the polar moment of inertia about the longitudinal axis of the link.   E   and   G   are the Young and shear moduli of the material. Accordingly, the Cartesian stiffness matrix   K i   of the i th   leg defining the motion-to-force mapping is obtained from Eq. (2).  f i   =   K i  δ   t i   (5) with   f i   being the wrench exerted on the i th   leg of the manipulator and at the geometric center of the moving platform while  δ   t i   is the small-displacement screw of the moving-platform. 4   Copyright   c ©   2010 by ASME
Finally, the Cartesian stiffness matrix   K   of the manipulator is found with a simple addition of the three   K i   matrices, namely,  K   =  3  ∑  i = 1  K i   (6)  MULTIOBJECTIVE OPTIMIZATION PROBLEM  A multiobjective optimization problem (MOOP) is formu- lated in this section in order to compare 3- P RR , 3- RP R   and 3- R RR   PPMs.   In scope of this study, the manipulators are compared with regard to their mass in motion and their regular workspace size, i.e., the two objective functions of the MOOP, defined below. Moreover, the MOOP is subject to constraints on the manipulator dexterity and stiffness. It means that for a given external wrench, the displacements of the moving platform have to be smaller than given values throughout the obtained maxi- mum regular dexterous workspace.  Objective Functions Mass in Motion of the Manipulators   The compo- nents in motion of the manipulators are mainly their moving plat- form and the links of their legs. As a consequence, the mass in motion for the three PPMs under study is expressed as follows:  m P RR   =   3   m link   +   m p f   (7a)  m RP R   =   3   m link   +   m p f   (7b)  m R RR   =   6   m link   +   m p f   (7c)  m link   is the mass of links of the legs and are supposed to be the same while   m p f   is the mass of the moving platform. The mass of the prismatic or revolute actuators does not appear in Eqs. (7a)- (c) as it is supposed to be fixed for the 3- P RR   PPM and close to the base for the 3- RP R   PPM.  m p f   =  π   r 2  p   r  ν   (8a)  m link   =  π   r 2  j   L  ν   (8b) where  ν   is the material density. Finally, the first objective function of the MOOP is ex- pressed as:  f 1   ( x ) =   m X   →   min   (9)  x   being the vector of design variables, i.e., the geometric parame- ters of the manipulator at hand, and   X   stands for 3- P RR , 3- RP R  or 3- R RR .  Regular workspace size   The quality of the manipula- tor workspace is of prime importance for the design of Parallel Kinematics Machines (PKMs). It is partly characterized by its size and shape. Moreover, the lower the amount of singularities throughout the workspace, the better the workspace for continu- ous trajectory planning. The workspace optimization of parallel manipulators can usually be solved by means of two different formulations. The first formulation aims to design a manipulator whose workspace contains a prescribed workspace and the sec- ond one aims to design a manipulator, of which the workspace is as large as possible.   However, maximizing the manipulator workspace may result in a poor design with regard to the manip- ulator dexterity and manipulability [12, 19].   This problem can be solved by properly defining the constraints of the optimiza- tion problem. Here, the multiobjective optimization problem of PPMs is based on the formulation of workspace maximization, i.e, the determination of the optimum geometric parameters in order to maximize a regular-shaped workspace. In the scope of the paper, the regular-shaped workspace is supposed to be a cylinder of radius   R w , for which at each point a rotation range   ∆  φ   =   20   ◦   of the moving-platform about the   Z - axis has to be reached. Figure 5 illustrates such a regular-shaped workspace, whose   x c ,   y c   and  φ c   are its center coordinates and the rotation angle of the moving-platform of the manipulator in the home posture.  PSfrag replacements  R w  ∆  φ  ( x c ,   y c ,  φ c )  FIGURE 5 .   A REGULAR-SHAPED WORKSPACE  Consequently,   in   order   to   maximize   the   manipulator workspace, the second objective of the optimization problem can be written as:  f 2   ( x ) =   R w   →   max   (10)  Constraints of the Optimization Problem  The constraints of the optimization problem deals with the geometric parameters, the dexterity and the accuracy of the ma- nipulators. Moreover, the constraints have to be defined in order to obtain a singularity-free regular-shaped workspace. 5   Copyright   c ©   2010 by ASME
Constraints on the Geometric Parameters   For the three PPMs under study, the kinematic constraints are handled with their inverse kinematics.   It means that the inverse kine- matics is solved in order for the postures of the PPM to belong to the same working mode throughout the manipulator regular- shaped workspace. Besides, for the 3- P RR   PPM, the lower and upper bounds of the prismatic lengths  ρ i   are defined such as 0   ≤   ρ i   ≤ √ 3 R   in order to avoid collisions. To obtain feasible dis- placements of the prismatic joints, the range of the 3- RP R   PPM is defined such that   L / 2   ≤  ρ i   ≤   L .  Constraint on the Manipulator Dexterity   The ma- nipulator dexterity is defined by the condition number of its kine- matic Jacobian matrix. The   condition number  κ F   ( M )   of a   m   ×   n  matrix   M , with   m   ≤   n , based on the Frobenius norm is defined as follows  κ F   ( M ) =   1  m  √  tr ( M T   M ) tr   [( M T   M ) − 1 ]   (11) Here, the condition number is computed based on the Frobenius norm as the latter produces a condition number that is analytic in terms of the posture parameters whereas the 2-norm does not. Besides, it is much costlier to compute singular values than to compute matrix inverses. The terms of the direct Jacobian matrix of the three PPMs under study are not homogeneous as they do not have same units. Accordingly, its condition number is meaningless.   Indeed, its singular values cannot be arranged in order as they are of dif- ferent nature. However, from [20] and [21], the Jacobian can be normalized by means of a   normalizing length . Later on, the con- cept of   characteristic length   was introduced in [22] in order to avoid the random choice of the normalizing length. For instance, the previous concept was used in [23] to analyze the kinetostatic performance of manipulators with multiple inverse kinematic so- lutions, and therefore to select their best   working mode . Accordingly, for the design optimization of the three PPMs, the minimum of the inverse condition number  κ − 1   ( J )   of the kinematic Jacobian matrix   J   is supposed to be higher than a pre- scribed value, say 0.1, throughout the regular-shaped workspace, for any rotation of its moving-platform, i.e.,  min   (  κ − 1   ( J ) )   ≥   0 . 1   (12)  Constraints on the moving-platform pose errors  The position and orientation errors on the moving-platform are evaluated by means of the stiffness models of the manipulators. Let   ( δ   x ,  δ   y ,  δ   z )   and   (  δ φ x ,  δ φ y ,  δ φ z )   be the position and orien- tation errors of the moving-platform subject to external forces  ( F x ,   F y ,   F z )   and torques   (  τ z ,  τ y ,  τ z ) . The constraints on the pose errors on the moving-platform are defined as follows:  δ   x   ≤  δ   x max  δ   y   ≤  δ   y max  δ   z   ≤  δ   z max  δ φ x   ≤  δ φ   max x  δ φ y   ≤  δ φ   max y  δ φ z   ≤  δ φ   max z  (13)  (  δ   x max ,  δ   y max ,  δ   z max )   being the maximum allowable position errors and   (  δ φ   max x   ,  δ φ   max y   ,  δ φ   max z  )   the maximum allowable ori- entation errors of the moving-platform.   These accuracy con- straints can be expressed in terms of the components of the mech- anism stiffness matrix and the wrench applied to the moving- platform. Let us assume that the accuracy requirements are:  √  δ   x 2   +  δ   y 2   ≤   0 . 0001 m   (14a)  δ   z   ≤   0 . 001 m   (14b)  δ φ z   ≤   1 deg   (14c) If the moving-platform is subject to a wrench whose components are   ∥ ∥ F x , y  ∥ ∥   =   F z   =   100 N and  τ z   =   100 Nm, then the accuracy con- straints can be expressed as:  k min xy   ≥   ∥ ∥ F x , y  ∥ ∥   / √  δ   x 2   +  δ   y 2   =   10 6   N . m -1   (15a)  k min z   ≥   F z /  δ   z   =   10 5   N . m -1   (15b)  k min  φ z   ≥  τ z /  δ φ z   =   10  π / 180 N . m . rad -1   (15c)  Design Variables of the Optimization Problem  Along with the above mentioned geometric parameters ( R ,   r ,   L b ) of the PPMs, the radius   r   j   of the circular-cross-section of the intermediate bars defined and the radius   r p   of the circular-cross- section of the platform bars are considered as design variables, also called decision variables.   As a remainder, the moving- platform is supposed to composed of three circular bars of length  r . As there are three PPMs under study, the PPM type is an- other design variable that has to be taken into account.   Let  d   denote the PPM type:   d   =   1 stands for the 3- P RR   PPM;  d   =   2 stands for the 3- RP R   PPM; and   d   =   3 stands for the 3-  R RR   PPM. As a result, the optimization problem contains one discrete variable, i.e.,   d , and five continuous design variables, i.e.,   R ,   r ,  L b ,   r   j   and   r p . Hence, the design variables vector   x   is given by:  x   =   [   d R r L b   r   j   r p  ] T   (16)  Formulation of the Optimization Problem  The Multiobjective Design Optimization Problem of PPMs can be stated as:   Find the optimum design variables   x   of PPMs  6   Copyright   c ©   2010 by ASME
in order to minimize the mass of the mechanism in motion and to maximize its regular shaped workspace subject to geometric, kinematic and accuracy constraints.  Mathematically, the problem can be written as: minimize   f 1 ( x ) =   m X   (17) maximize   f 2 ( x ) =   R w  over   x   =   [   d R r L b   r   j   r p  ] T  subject to :   g 1   :   L b   +   r   ≥   R  2  g 2   : 0   <  ρ i   <   √ 3 R g 3   :  κ − 1   ( J )   ≥   0 . 1  g 4   :   k min xy   ≥   F x , y  √  δ   x 2   +  δ   y 2   =   10 6  g 5   :   k min z   ≥   F z  δ   z   =   10 5  g 6   :   k min  φ z   ≥  τ z  δ φ z  =   10  π / 180  x lb   ≤   x   ≤   x ub  where   x lb   and   x ub   are the lower and upper bounds of   x , respec- tively.  RESULTS AND DISCUSSIONS  The multiobjective optimization problem (17) is solved by means of modeFRONTIER [24] and by using its built-in mul- tiobjective optimization algorithms.   MATLAB code is incor- porated in order to analyze the system and to get the numer- ical values for the objective functions and constraints that are analyzed in modeFRONTIER for their optimality and feasibil- ity.   The lower and upper bounds of the design variables are given in Tab. 1.   The components of the PPMs are supposed to be made up of steel, of material density   d   =   7850 kg/m 3  and Young modulus   E   =   210   ×   10 9   N/m 2 .   For each iteration,  TABLE 1 .   LOWER AND UPPER BOUNDS OF THE DESIGN VARIABLES  Design Variable   d   R   [m]   r   [m]   L b   [m]   r   j   [m]   r p   [m]  Lower Bound   1   0.5   0.5   0.5   0   0  Upper Bound   3   4   4   4   0.1   0.1  the regular-shaped workspace is evaluated for the corresponding design variables and a discretization of this workspace is per- formed.   The constraints of the optimization problem are also evaluated at each grid point of the regular-shaped workspace to check whether they are satisfied or not. A multiobjective genetic  TABLE 2 .   modeFRONTIER   ALGOTITHM PARAMETERS  Scheduler   MOGA-II  Number of iterations   200  Directional cross-over probability   0 . 5  Selection probability   0 . 05  Mutation probability   0 . 1  DNA (DeoxyriboNucleic Acid) string   0 . 05 mutation ratio  DOE algorithm   Sobol  DOE number of designs   30  Total number of iterations   30   ×   200   =   6000  algorithm (MOGA) is used to solve MOOP (17) and to obtain the Pareto frontier in the plane defined by the mechanism mass and the workspace radius.   modeFRONTIER   scheduler and Design Of Experiments (DOE) parameters are given in Tab. 2. MATLAB is used to evaluate each individual of the current population (gen- erated by the   modeFRONTIER   scheduler).   MATLAB   returns the output variables that are analyzed by   modeFRONTIER   for the feasible solutions according to the given constraints. At the end, the Pareto-optimal solutions are obtained from the generated fea- sible solutions.  PSfrag replacements  Mass   m t   [kg] Workspace Radius   R w   [m] ID-I ID-II ID-III  0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.9 0.9 1  0   200   400   600   800   1000   1200   1400   1600  0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1  0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8  FIGURE 6 .   PARETO FRONTIER OF MOOP (17)  7   Copyright   c ©   2010 by ASME
The Pareto frontier, solution of MOOP (17), is depicted in Fig. 6 whereas the design parameters and the corresponding ob- jective functions for two extreme and one intermediate Pareto optimal solutions, as shown in Fig. 6, are given in Tab. 3. The CAD designs illustrating the three foregoing solutions are also shown in Fig. 8. It appears that all Pareto-optimal solutions of MOOP (17) are 3- P RR   PPMs.   Accordingly, Fig. 7 illustrates the Pareto Frontiers associated with the three planar parallel manipulator architectures. It is noteworthy that the Pareto-optimal solutions associated with the 3- P RR   PPM architectures are better than the Pareto-optimal solutions associated with the 3- RP R   and 3-  R RR   PPM architectures.  3- P RR  3- RP R  3- R RR  Rw[m] Mass[kg]  2 . 5  2 . 0 1 . 5 1 . 0 0 . 5  − 0 . 5  0 0   1000   2000   3000   4000  5000  FIGURE 7 .   PARETO FRONTIERS ASSOCIATED WITH THE 3-  P RR , 3- RP R , AND 3- R RR   PLANAR PARALLEL MANIPULATOR ARCHITECTURES  Figures 9(a)–(c) and 10(a)–(c) show the evolution of the de- sign variables as a function of   R w   along the Pareto Frontier as- sociated with each PPM architecture. It is noteworthy that the higher   R w , the higher the design variables. It is apparent that the variations in variables   R ,   r ,   L b   and   r   j   with respect to (w.r.t.)   R w  are almost linear whereas the variations in   r p   w.r.t.   R w   is rather quadratic. This is due to the fact that the higher the size of the mechanism the higher the bending of the moving platform links whereas the intermediate links are mainly subjected to tension and compression.  CONCLUSIONS  In this paper, the problem of dimensional synthesis of par- allel kinematics machines was addressed. A multiobjective de- sign optimization problem was formulated in order to determine optimum structural and geometric parameters of any parallel kinematics machine.   The proposed approach is similar to that used in [25] but we took into account the mass and the regular workspace instead of considering the entire volume of the ma- nipulator. The proposed approach was applied to the optimum design of three planar parallel manipulators with the aim to min- imize the mass in motion of the mechanism and to maximize its regular shaped workspace. Other performance indices can be used as constraints.   However, they cannot necessarily be used as objective functions as the latter are usually formulated as a sum of an index over all the manipulator workspace. As another constraint, we could use the collisions between the legs of the manipulator.  REFERENCES  [1] A. M. Hay and J. A. Snyman. Methodologies for the opti- mal design of parallel manipulators.   International Journal for Numerical Methods in Engineering , 59(11):131–152, 2004. [2] J. P. Merlet.   Parallel Robots . Kluwer Academic Publishers, Norwell, MA, USA, 2006. [3] Y. Lou, G. Liu, N. Chen, and Z. Li. Optimal design of paral- lel manipulators for maximum effective regular workspace. In   Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems , pages 795–800, Alberta, 2005. [4] Y. Lou, G. Liu, and Z. Li. Randomized optimal design of parallel manipulators.   IEEE Transactions on Automation Science and Engineering , 5(2):223–233, 2008. [5] E. Ottaviano and M. relli. Workspace and optimal design of a pure translation parallel manipulator-tsai manipulator.  Meccanica , 35(3):203–214, May 2000. [6] E. Ottaviano and M. Ceccarelli. Optimal design of capaman (cassino parallel manipulator) with prescribed workspace. In   2 nd   Workshop on Computational Kinematics KC2001 , pages 35–43, Seoul, South Korea, 2001. [7] F. Hao and J.-P. Merlet.   Multi-criteria optimal design of parallel manipulators based on interval analysis.   Mecha- nism and Machine Theory , 40(2):157–171, 2005. [8] M. Ceccarelli, G. Carbone, and E. Ottaviano. Multi criteria optimum design of manipulators. In   Bulletin of the Polish Academy of Sciences Technical Sciences , volume 53, 2005. [9] C. M. Gosselin and J. Angeles.   The optimum kinematic design of a planar three-degree-of-freedom parallel manip- ulator.   ASME Journal of Mechanisms, Transmission and Automation in Design , 110:35–41, 1988. [10] C. M. Gosselin and J. Angeles.   The optimum kinematic design of a spherical three-degree-of-freedom parallel ma- nipulator.   Journal of Mechanisms, Transmissions and Au- tomation in Design , 111(2):202–207, 1989. [11] H.   H. Pham and I-M. Chen.   Optimal synthesis for workspace and manipulability of parallel flexure mecha- nism. In   Proceeding of the 11 th   World Congress in Mech-  8   Copyright   c ©   2010 by ASME
TABLE 3 .   THREE PARETO OPTIMAL SOLUTIONS  Design ID  Design Variables   Objectives  d   R   [m]   r   [m]   L b   [m]   r   j   [m]   r p   [m]   m t   [kg]   R w   [m]  I   1   1.412   0.319   0.620   0.026   0.023   44.5   0.110  II   1   3.066   1.283   1.896   0.036   0.056   484.8   1.207  III   1   3.872   1.947   1.977   0.039   0.096   1545.6   1.609  Regular Workspace  (a) ID–I  PSfrag replacements  Regular Workspace  (b) ID–II  PSfrag replacements  Regular Workspace  (c) ID–III  FIGURE 8 .   CAD DESIGNS OF THREE PARETO-OPTIMAL SOLUTIONS OF MOOP (17)  0   0 . 2   0 . 4   0 . 6   0 . 8   1   1 . 2   1 . 4   1 . 6  1 . 8  0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5  4 . 0 4 . 5  R R   r L  0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 8 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0  [m]  R w   [m]  (a) 3- P RR   PPM  PSfrag replacements  0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8  0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5  4 . 0 4 . 5  R R   r L  0   0 . 1   0 . 2   0 . 3   0 . 4   0 . 8   0 . 6   0 . 7   0 . 8   0 . 9  1 . 0  [m]  R w   [m]  (b) 3- RP R   PPM  PSfrag replacements  0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8  0 0   0 . 5 0 . 5 1 . 0 1 . 0 1 . 5 1 . 5 2 . 0 2 . 0 2 . 5  3 . 0 3 . 5 4 . 0 4 . 5  R R   r L  0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 8 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0  [m]  R w   [m]  (c) 3- R RR   PPM  FIGURE 9 .   DESIGN VARIABLES   R ,   r ,   L b   AS A FUNCTION OF   R w   ALONG THE PARETO FRONTIER ASSOCIATED WITH THE MANIPU- LATOR AT HAND  anism and Machine Science , pages 2069–2073, Tianjin, China, Apr. 1–4 2003. [12] R. E. Stamper, L.-W. Tsai, and G. C. Walsh. Optimization of a three-dof translational platform for well-conditioned workspace. In   Proceedings of the IEEE International Con- ference on Robotics and Automation , pages 3250–3255, New Mexico, 1997. [13] M. Stock and K. Miller. Optimal kinematic design of spa- tial parallel manipulators: Application of linear delta robot.  Transactions of the ASME, Journal of Mechanical Design , 125(2):292–301, 2003. [14] C. Menon, R. Vertechy, M.C. Markot, and V. Parenti- Castelli. Geometrical optimization of parallel mechanisms based on natural frequency evaluation:   application to a spherical mechanism for future space applications.   IEEE Transactions on Robotics , 25(1):12–24, Feb 2009. [15] H. Li, Z. Yang, and T. Huang.   Dynamics and elasto- dynamics optimization of a 2-dof planar parallel pick and place robot with flexible links.   Journal of Structural and Multidisciplinary Optimization , 38(2):195–204, 2009. [16] M. Krefft and J. Hesselbach. Elastodynamic optimization of parallel kinematics. In   Proceedings of the IEEE Interna-  9   Copyright   c ©   2010 by ASME
0   0 . 2   0 . 4   0 . 6   0 . 8   1   1 . 2   1 . 4   1 . 6  0 . 02 0 . 04 0 . 06 0 . 08  0 . 10 0 . 12  0  0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 0 . 01 0 . 02 0 . 03 0 . 04 0 . 05 0 . 06 0 . 07  r   j   r p  [m]  R w   [m]  (a) 3- P RR   PPM  0.00 0   0 . 1   0 . 2   0 . 3   0 . 4   0 . 8   0 . 6   0 . 7   0 . 8   0 . 9  0 . 10 0 . 12 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5  0 . 01 0 . 02 0 . 03 0 . 04 0 . 05  0 . 06 0 . 07  r   j   r p  [m]  R w   [m]  (b) 3- RP R   PPM  0 . 02 0 . 04 0 . 06 0 . 08  0 . 10 0 . 12  0 0   0 . 5   1 . 0   1 . 5   2 . 0  2 . 5 0 . 01 0 . 02 0 . 03 0 . 04 0 . 05 0 . 06 0 . 07  r   j   r p  [m]  R w   [m]  (c) 3- R RR   PPM  FIGURE 10 .   DESIGN VARIABLES   r   j ,   r p   AS A FUNCTION OF   R w   ALONG THE PARETO FRONTIER ASSOCIATED WITH THE MANIPU- LATOR AT HAND  tional Conference on Automation Science and Engineering , Edmonton, Canada, Aug 1-2 2005. [17] D. Chablat and P. Wenger. Architecture optimization of a 3-dof parallel mechanism for machining applications, the orthoglide.   IEEE Transactions On Robotics and Automa- tion , 19(3):403–410, 2003. [18] A. Pashkevich, D. Chablat, and P. Wenger. Stiffness anal- ysis of overconstrained parallel manipulators.   Mechanism and Machine Theory , 44(5):966 – 982, 2009. [19] P. Wenger and D. Chablat. Kinematic analysis of a new par- allel machine tool: The orthoglide. In   Proceedings of the 7 th   International Symposium on Advances in Robot Kine- matics , Portoroz, Slovenia, 2000. [20] Z. Li. Geometrical consideration of robot kinematics sin- gularities.   The International Journal of Robotics and Au- tomation , 5(3):139–145, 1990. [21] B. Paden and S. Sastry. Optimal kinematic design of 6r ma- nipulator.   The International Journal of Robotics Research , 7(2):43–61, 1988. [22] F. Ranjbaran, J. Angeles, M.A. Gonzalez-Palacios, and R. Patel. The mechanical design of a seven-axes manipu- lator with kinematic isotropy.   ASME Journal of Intelligent and Robotic Systems , 14(1):21–41, 1995. [23] D. Chablat, Ph. Wenger, S. Caro, and J. Angeles.   The isoconditioning loci of planar 3-dof parallel manipulator. In   Proceedings of DETC’2002, ASME Design Engineering Technical Conference , Montreal, Quebec, Canada, 29 Sep– 2 Oct 2002. [24] ESTECO. modefrontier, version 4.0.3, 2008. [25] O. Altuzarra, O. Salgado, A. Hernandez, and J. Ange- les. Multiobjective optimum design of a symmetric parallel schnflies-motion generator.   ASME Journal of Mechanical Design , 131(3):031002–1–031002–11, 2009. 10   Copyright   c ©   2010 by ASME
WS  P R R L F  B MP  P R R L F  P R R L F  A   R   R RL   RL 1DOF BP 6DOF   DOFp Sp   Sp   Sp L   r Rig  6DOF Sp 6DOF Sp RL   RL BP Rig
6DOF Sp 6DOF Sp BP Rig RL   RL
WS
Ci Oi Bi xb yb  kl  kt  Ai P betai  0.0 0.5 1.0 1.5 0   1000  Mass [kg]  Rw [m]
x22   x23   x24   x25   x26   x27   x28   x29   x30   x31  v17  v18  v19  v20  v21  s09 s10 s11  x12   x13   x14   x15   x16   x17   x18   x19   x20   x21  v12  v13  v14  v15  v16  s14 s15 s16  s20  s21  s22  s24  s25


x22   x23   x24   x25   x26   x27   x28   x29   x30   x31  v17  v18  v19  v20  v21  s09 s10 s11  x12   x13   x14   x15   x16   x17   x18   x19   x20   x21  v12  v13  v14  v15  v16  s14 s15 s16  s28  s29  s31  s32  s33
x12   x13   x14   x15   x16   x17   x18   x19   x20  v12  v13  v14  v15  v16  v17  v18  v19  v20  v21  s01 s02 s03 s05 s06 s07
x12   x13   x14   x15   x16   x17   x18   x19  v12  v13  v14  v15  v16  v17  v18  v19  v20  s01 s02 s03 s05 s10 s11