Volume 16 Number 6
December 2019
Article Contents
Song Lu, Yang-Min Li and Bing-Xiao Ding. Multi-objective Dimensional Optimization of a 3-DOF Translational PKM Considering Transmission Properties. International Journal of Automation and Computing, vol. 16, no. 6, pp. 748-760, 2019. doi: 10.1007/s11633-019-1184-9
Cite as: Song Lu, Yang-Min Li and Bing-Xiao Ding. Multi-objective Dimensional Optimization of a 3-DOF Translational PKM Considering Transmission Properties. International Journal of Automation and Computing, vol. 16, no. 6, pp. 748-760, 2019. doi: 10.1007/s11633-019-1184-9

Multi-objective Dimensional Optimization of a 3-DOF Translational PKM Considering Transmission Properties

Author Biography:
  • Song Lu received the M. Sc. degree in mechanical design and theory from Shantou University, China in 2013. He is currently a Ph. D. degree candidate in electromechanical engineering at Faculty of Science and Technology, University of Macau, China. His research interests include robotics, parallel manipulators, and mechanism design. E-mail: robot.slu@gmail.com (Corresponding author) ORCID iD: 0000-0003-3414-0362

    Yang-Min Li received the B. Sc. and M. Sc. degrees from Jilin University, China, in 1985 and 1988, respectively, and the Ph. D. degree from Tianjin University, China in 1994, all in mechanical engineering. He is currently a full professor with the Department of Industrial and Systems Engineering, Hong Kong. He is a Member of the American Society of Mechanical Engineers and the Canadian Society of Mechanical Engineers. He is an Associate Editor of the IEEE Transactions on Automation Science and Engineering, Mechatronics, and International Journal of Control, Automation, and Systems.His research interests include robotics, parallel manipulators, micromanipulators multibody dynamics and control. E-mail: yangmin.li@polyu.edu.hk ORCID iD: 0000-0002-4448-3310

    Bing-Xiao Ding received the Ph. D. degree in electromechanical engineering from University of Macau, China in 2018. He is currently an assistant professor with the Department of Mechanical, Jishou University, China. His research interests include robotics, micro-positioning, smart actuators, hysteresis compensation and nonlinear system control. E-mail: bingxding@hotmail.com ORCID iD: 0000-0002-1655-5225

  • Received: 2018-09-07
  • Accepted: 2019-04-09
  • Published Online: 2019-07-19
通讯作者: 陈斌, bchen63@163.com
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Multi-objective Dimensional Optimization of a 3-DOF Translational PKM Considering Transmission Properties

Abstract: Multi-objective dimensional optimization of parallel kinematic manipulators (PKMs) remains a challenging and worthwhile research endeavor. This paper presents a straightforward and systematic methodology for implementing the structure optimization analysis of a 3-prismatic-universal-universal (PUU) PKM when simultaneously considering motion transmission, velocity transmission and acceleration transmission. Firstly, inspired by a planar four-bar linkage mechanism, the motion transmission index of the spatial parallel manipulator is based on transmission angle which is defined as the pressure angle amongst limbs. Then, the velocity transmission index and acceleration transmission index are derived through the corresponding kinematics model. The multi-objective dimensional optimization under specific constraints is carried out by the improved non-dominated sorting genetic algorithm (NSGA II), resulting in a set of Pareto optimal solutions. The final chosen solution shows that the manipulator with the optimized structure parameters can provide excellent motion, velocity and acceleration transmission properties.

Song Lu, Yang-Min Li and Bing-Xiao Ding. Multi-objective Dimensional Optimization of a 3-DOF Translational PKM Considering Transmission Properties. International Journal of Automation and Computing, vol. 16, no. 6, pp. 748-760, 2019. doi: 10.1007/s11633-019-1184-9
Citation: Song Lu, Yang-Min Li and Bing-Xiao Ding. Multi-objective Dimensional Optimization of a 3-DOF Translational PKM Considering Transmission Properties. International Journal of Automation and Computing, vol. 16, no. 6, pp. 748-760, 2019. doi: 10.1007/s11633-019-1184-9
    • Owing to low moving inertia, high stiffness, high dexterity, and high payload-to-weight ratio, the low-mobility parallel kinematic manipulator (PKM) is typically regarded as a very promising candidate for practical engineering applications[1, 2]. In particular, the outer actuator-driving parallel manipulators (OADPM) enable the drive motors to be installed on stationary frames and choose light thin rods as motion arms, resulting in effectively reducing the overall quality of manipulator and making the end effector achieve high speed and acceleration[3]. Therefore, OADPMs are being widely used in material handling, packaging and other fields. Along with efficiency enhancement of advanced industrial automation equipment, higher performances in terms of motion precision and acceleration operations are all required for configuration design and dimension optimization of PKMs[4].

      Dimension synthesis is an important link in the design of the high-speed parallel manipulator[5]. The main methods used currently are choosing algebraic features of the Jacobian matrix[6] (such as condition number, singular value, determinant) as the local performance evaluation index[7]. For instance, Mazare et al.[8] employed the condition number of the Jacobian matrix as the dexterity index, which is very common and reasonable in the kinematics performance evaluation. In order to achieve desired excellent performance, the motion transmission[9], velocity transmission[10, 11] and acceleration transmission[12] characteristic should be taken into account for the dimensional synthesis when the parallel manipulator executes a high-efficiency practical operation. A large amount of effort has contributed to the research of motion/force transmission of mechanisms. Some specific indices including transmission angle, pressure angle, and transmission factor, were accordingly proposed to evaluate its quality. Ball[13] defined the virtual coefficient between the transmission wrench screw and out twist screws as a transmission factor for evaluating the transmission performance of spatial mechanisms. Wang et al.[14] proposed a general transmission index to evaluate the motion/force transmissibility of fully parallel manipulators based on the virtual coefficient. Tsai and Lee[15] introduced the concept of a generalized transmission wrench screw to characterize the transmission properties of mechanisms. Based on this concept, they defined the measurements of transmissibility and manipulability, and investigated the transmission performance of the variable lead screw mechanism. Takeda et al.[16] presented a transmission index based on the minimum value of the pressure angle′s cosine at the connection point of the limb with the moving platform of a single-DOF (degree-of-freedom) parallel manipulator. Zhao[17] chose the tetrahedron volume comprised of the unit vectors of the distal links as the performance index to measure the transmission behavior of the distal links: the larger the volume, the better the transmission performance. To deal with the dimensional synthesis of the Delta PKM under transmission requirements, Zhang et al.[18] defined the pressure/transmission angles of spatial limbs of the Delta parallel manipulator. By employing the reciprocal product of a wrench and a twist as a linear functional, Huang et al.[19] analyzed the force/motion transmissibility of the Stewart PKM. By virtue of screw theory, Zhang et al.[20, 21] presented the motion-force transmission performance evaluation index, of an overconstrained 2PRR-PRPS (the underlined letter means the joint is an actuator) parallel mechanism, resulting in the local transmission index and global transmission. Since the objective function, the constraint function and optimization algorithm used for the dimensional synthesis of the parallel robot may be varied, the design result of the dimensional synthesis is usually not unique[22]. Moreover, little work has been focused on the dimensional synthesis of the parallel mechanism while simultaneously considering the motion transmission, velocity transmission and acceleration transmission. Thus, the motivation of the paper is to investigate the methodology of the dimensional synthesis of the three translational degrees of freedom parallel robot while considering the motion transmission, velocity transmission and acceleration transmission.

      It is important to note that the dimensional optimization of PKM is a typical multi-objective optimization problem[23]. The traditional weight-sum optimization method[24, 25] that combines all different performance indices in a weighted sum function, is not suit for such a complicated optimization mission because of some drawbacks[26], 1) the weighting factors are selected arbitrarily and 2) only obtain a single Pareto solution but not the complete optimum solutions. To overcome these shortcomings of the traditional method, some advanced intelligent algorithms inspired by natural biological evolution processes are proposed, such as a genetic algorithm[27], artificial intelligence approach[28], particle swarm method[29], etc. Among them, the improved non-dominated sorting genetic algorithm (NSGA II)[30, 31] introduced the elite strategy, crowing distance distribution and non-dominated quick sort optimization mechanism, which makes the algorithm not only have good convergence and distribution at same time, but also possess faster convergence speed, thus this algorithm has been widely used.

      The requirements of high positioning precision, low inertia and outstanding structure compactness for three-dimensional additive manufacturing production has motivated the development of a symmetrical 3-PUU PKM[32] which is taken as our research object. The focus of this article is largely on multi-objective dimensional optimization analysis of the symmetrical 3PUU PKM. The paper is organized as follows: Section 2 gives the system description and inverse kinematics analysis. Section 3 is devoted to motion transmission index definitions based on the pressure angle. Section 4 is devoted to the performance index derivation of velocity transmissibility. Section 5 is devoted to the performance index derivation of acceleration transmissibility. Section 6 gives the process framework of dimensional optimization of the 3-prismatic-universal-universal (PUU) PKM. Section 7 describes the non-dominated sorting genetic algorithm-II (NSGA-II). Conclusions are drawn in the final section.

    • The configuration of the simulated 3-PUU parallel kinematic manipulator shown in Fig. 1 is composed of a moving platform, a fixed base and three identical kinematic chains. To improve the whole rigidity of the manipulator, we introduce an ideal virtual constraint in each kinematic chain, as shown in Fig. 2. Generally, virtual constraints play repeated constraining roles in movement characteristics of the mechanism, and do not change the final number of degrees of freedoms of the manipulator. By virtue of employing the desired virtual constraints, the force status between struts and their contiguous moving components is dramatically improved, and the reliability of the transmission is further maintained. Hence, the stiffness and stability of the mechanism system can be effectively guaranteed and enhanced. The four-bar crank mechanism with a redundant constraint to be analyzed is illustrated in the following diagram. Each limb connects the fixed base to the moving platform by a prismatic joint followed by two universal joints in sequence, where the P joint is implemented by a lead screw actuation system driven by a servomotor. The constant-length limbs are driven by their corresponding sliders along the vertical guideways which are perpendicular to the ground and arranged into an axial symmetrical structure. The axes of the two intermediate revolute pairs of universal joints on the sliders are parallel to these of universal joints on the moving platform, such configuration enables the moving platform with 3-DOF purely translational motion in Cartesian space. Its schematic diagram is annotated in Fig. 3. The reference frame O-xyz, attached to a stationary pedestal, and the moving frame O′-x′y′z′, attached to a mobile platform, are similarly placed at the center of the equilateral triangle $\Delta {B_1}{B_2}{B_3}$ and $\Delta {A_1}{A_2}{A_3}$. The x-axis is set to direct along the segment $\overrightarrow {O{B_1}} $ and parallel to the x′ axis, and the z and z′ axes are perpendicular to the triangle $\Delta {B_1}{B_2}{B_3}$ and $\Delta {A_1}{A_2}{A_3}$, respectively. Define ${{r}}=[x \quad y\quad z]^{\rm T}$ as the position vector of moving platform with respect to the fixed frame O-xyz.

      Figure 1.  CAD model of the 3PUU PKM

      Figure 2.  Schematic representation of the 3PUU PKM

      Figure 3.  Schematic representation of the 3PUU PKM

    • As depicted in Fig. 3, the closed loop position vector equation of point O′ can be expressed as

      ${{r}} = {{{b}}_i} + {q_i}{{{e}}_i} + {{{d}}_i} + l{{{w}}_i} - {{{a}}_i},\;(i = 1,2,3)$

      (1)

      where ${{{e}}_i} =[0 \quad 0 \quad 1]^{\rm T}$, ${{{a}}_i} =[ {s\cos {\beta _i}} \quad { - s\sin {\beta _i}} \quad 0]^{\rm T}$, ${{{b}}_i} =[{S\cos {\beta _i}} \;\; { - S\sin {\beta _i}} \;\; 0]^{\rm T}$, and ${{{d}}_i}\!=[{- d\cos {\beta _i}}\;\; {d\sin {\beta _i}}$$ 0]^{\rm T}$. The symbolic variables s, S, qi, ${\beta _i}$, l, ei, wi, di, ai and bi stand for the radius of the moving platform, the radius of the stationary pedestal, the displacement of kinetic slider i, the angle of point Bi in the fixed coordinate system O-xyz, the length of strut, the unit vector along the lead screw, the vector along the strut i, the vector from the central shaft of lead screw i to the center point of universal joint Ci, the position vector of point Ai in the moving coordinate O′-x′y′z′ and the position vector of point Bi in the fixed coordinate system O-xyz, respectively.

      Taking the Euclidean norm on both sides of (1), yields

      ${q_i} = {{{r}}^{\rm T}}{{{e}}_i} - \sqrt {{{({{{r}}^{\rm T}}{{{e}}_i})}^2} - {{\left\| {{{r}} + {{{a}}_i} - {{{b}}_i} - {{{d}}_i}} \right\|}^2} + {l^2}} $

      (2)

      ${{w}}{}_i = \frac{{{{r}} + {{{a}}_i} - {{{b}}_i} - {{{d}}_i} - {q_i}{{{e}}_i}}}{l}.$

      (3)
    • The transmissibility performance of the manipulator has a close relationship in correspondence with the transmission angle of acute links. The pressure angle is defined as the acute angle between the directions of the force and the velocity of the point receiving the force on the follower. As an important parameter of the transmission mechanism, the transmission angle is defined as the angle between the output link and the coupler (Fig. 4), which can be used to measure the force transmission performance and mechanism transmission efficiency. The greater the transmission angle is, the better transmission performance becomes and the higher the transmission efficiency achieved. On the other hand, if the transmission angle changes are small, the transmission performance and transmission efficiency will decrease accordingly. For the planar mechanism, when the transmission angle is $90^\circ $, the motion/force transmissibility performance is best. The high-speed 3PUU parallel manipulator is a typical space mechanism with three translational degrees-of-freedom. As shown in Fig. 5, its kinematic chain connecting the support slider and moving platform still possesses certain features of four-bar linkage mechanisms. Besides, three lines, each of which represents the connecting link, will intersect at a point. For smooth operation of any mechanism without jerky movement, the maximum value of the pressure angle should be less than or equal to the allowable pressure angle, and generally the value of the allowable pressure is less than $50^\circ $. Referring to four-bar linkage mechanisms, the pressure angle ${\gamma _i}$ amongst three spatial limbs is defined

      Figure 4.  Transmission angle of the four-bar mechanism

      Figure 5.  Transmission angle of the 3PUU PKM

      ${\gamma _1} = {\rm arc}\cos \left( {\frac{{{{w}}_1^{\rm T}({{{w}}_3} \times {{{w}}_2})}}{{\left| {{{{w}}_3} \times {{{w}}_2}} \right|}}} \right)$

      (4)

      ${\gamma _2} ={\rm arc}\cos \left( {\frac{{{{w}}_2^{\rm T}({{{w}}_1} \times {{{w}}_3})}}{{\left| {{{{w}}_1} \times {{{w}}_3}} \right|}}} \right)$

      (5)

      ${\gamma _3} ={\rm arc}\cos \left( {\frac{{{{w}}_3^{\rm T}({{{w}}_2} \times {{{w}}_1})}}{{\left| {{{{w}}_2} \times {{{w}}_1}} \right|}}} \right).$

      (6)

      To vividly explain the geometric meaning of ${\gamma _i}$, take ${\gamma _3}$ as example. We suppose the neighboring Limbs 1 and 2 are stationary. The distal Limbs 1 and 2 becomes a fictitious link. Thus, ${\gamma _3}$ represents the angle between the driving force (along w3) imposed by the distal Links 3 to the fictitious link at point, and the velocity (along {${{{w}}_2} \times {{{w}}_1}$}) of the same point.

    • The pressure angle is a parameter with a specific physical unit and is inappropriate as the direct evaluation index. Therefore, by dimensional normalization processing, its sine is selected as the evaluation index, namely

      ${\chi _i} = \sin ({\gamma _i}),\;({{i}} = 1,2,3).$

      (7)

      Taking the transmissibility of a single kinematic chain as the sole evaluation index of monolithic manipulators is unsuitable. The aim of transmission property analysis of the space 3PUU parallel manipulator is to maximize the combination effect of transmission performance of the three kinematic branches. Hence, the norm of transmission vector ${{\chi }} = ({\chi _1},{\chi _2},{\chi _3})$ is chosen as the evaluation index of the monolithic manipulator.

      ${\eta _1} = \left\| {{\chi }} \right\| = \sqrt {\chi _1^2 + \chi _2^2 + \chi _3^2}.$

      (8)

      We know ${\kappa _1}$ is a local transmission index which is dependent on the pose of the manipulator, and it only can estimate the motion/force transmissibility of the manipulator under specific poses during the entire workspace. Further, to evaluate the total transmission within the whole workspace, we define the global motion/force transmission index (GMTI) as

      ${\kappa _1} = \dfrac{{\displaystyle\int_W {{\eta _1}{\rm d}W} }}{{\displaystyle\int_W {{\rm d}W} }}$

      (9)

      where W represents the whole workspace. Hence, the mathematical description of the parallel mechanism dimension optimization in regard of force/motion transmissibility performance is constructed as

      $\begin{gathered} {f_1} = \mathop {\max }\limits_{{{x}} \subset {{X}}} ({\kappa _1}) = \mathop {\min }\limits_{{{x}} \subset {{X}}} ( - {\kappa _1}) \\ {\rm s.t.}\begin{array}{*{20}{c}} {}&{{W_0} \subset W}. \end{array} \end{gathered} $

      (10)

      As an example, the architectural parameters of 3-PUU PKM are designed as s = 0.070 m, S = 0.370 m, l = 0.380 m, and d = 0.070 m. The local pressure angle distribution amongst three spatial limbs is given in Fig. 6. As shown in Fig. 7, the task workspace of 3PUU PKM is a cylinder with radius is R = 0.09 m and height is h = 0.2 m. The global motion/force transmission is shown in Fig. 8 (a). Obviously, we can find that the biggest pressure angle value occurs at the center of the task workspace, the worst GMTI occurs around the boundary of the workspace. Moreover, the local pressure angle distribution amongst three spatial limbs is $120^\circ $ centrosymmetric. The global motion/force transmission along with different radius of moving platform (S) is shown in Fig. 9.

      Figure 7.  Task workspace of the 3PUU PKM

      Figure 6.  Pressure angle distribution amongst three spatial limbs

      Figure 8.  Local distribution of transmission performance indices

      Figure 9.  Global motion transmissibility versus the variation of l

    • Taking the derivative of (1) with respect to time, the velocity equations of the moving platform are described as

      ${{v}} = {\dot q_i}{{e}} + l{{{\omega }}_i} \times {{{w}}_i},\;(i = 1,2,3)$

      (11)

      where v, ${\dot q_i}$ and ${{{\omega }}_i}$ denote the velocity of moving platform, velocity of i-th slider and angular velocity of i-th strut, respectively.

      Taking dot multiplying on both sides of (11) expression with the vector ${{{w}}_i}$, the velocity of the i-th slider can be readily obtained as

      ${\dot q_i} = \frac{{{{w}}_i^{\rm T}}}{{{{w}}_i^{\rm T}{{e}}}}{{v}}.$

      (12)

      Writing (12) for i = 1, 2, 3, and rearranging in a matrix form gives

      ${{{J}}_q}{\dot{ q}} = {{{J}}_x}{\dot{ X}},\;{{J}} = {{J}}_q^{ - 1}{{{J}}_x}$

      (13)

      ${{{J}}_q} = {\rm diag}\{ {{w}}_1^{\rm T}{{e}},\;{{w}}_2^{\rm T}{{e}},\;{{w}}_3^{\rm T}{{e}}\},\;{{{J}}_x} = {[{{{w}}_1},\;{{{w}}_2},\;{{{w}}_3}]^{\rm T}}$

      (14)

      where ${\dot{ q}} = {[{\dot q_1},\;{\dot q_2},\;{\dot q_3}]^{\rm T}}$ and ${\dot{ X}} = {[\dot x,\;\dot y,\;\dot z]^{\rm T}}$ denote the vector that contains the slider velocities and the velocity vector of the moving platform, respectively. The Jacobian matrix J is a 3-rows and 3-columns matrix which represents the mapping relationship from the end-effector Cartesian velocities to the active joint velocities.

      Take the norm of the driving joint velocity vector in kinematic performance,

      ${\left\| {{\dot{ q}}} \right\|^2} = {{\dot{ q}}^{\rm T}}{\dot{ q}} = {{\dot{ X}}^{\rm T}}{{{J}}^{\rm T}}{{J\dot X}}.$

      (15)

      Since ${{{J}}^{\rm T}}{{J}}$ is a real symmetric matrix, there exists an orthogonal matrix Q that satisfies the following diagonalization processing of ${{{J}}^{\rm T}}{{J}}$,

      ${{{Q}}^{\rm T}}{{{J}}^{\rm T}}{{JQ}} = {\rm diag}\left\{ {\lambda _1},\;{\lambda _2},\;{\lambda _3}\right\}$

      (16)

      where ${\lambda _1}$, ${\lambda _2}$ and ${\lambda _3}$ are the eigenvalues of matrix ${{{J}}^{\rm T}}{{J}}$, we define ${{Qu}} = {\dot{ X}}$, (15) can be expanded as the following expression:

      ${\left\| {{\dot{ q}}} \right\|^2} = {{{u}}^{\rm T}}{{{Q}}^{\rm T}}{{{J}}^{\rm T}}{{JQu}} = \sum\limits_{i = 1}^3 {u_i^2{\lambda _i}}. $

      (17)

      According to the above (17), we can get

      ${\left\| {{\dot{ q}}} \right\|^2} \le {\lambda _{\max }}\sum\limits_{i = 1}^3 {u_i^2} $

      (18)

      where ${\lambda _{\max }}$ is the maximum eigenvalue of matrix ${{{J}}^{\rm T}}{{J}}$. On the basis of the orthogonal matrix Q, i.e., ${{{Q}}^{\rm T}} = {{{Q}}^{ - 1}}$, we can deduce

      $\sum\limits_{i = 1}^3 {u_i^2} = {({{{Q}}^{ - 1}}{\dot{ X}})^{\rm T}}{{{Q}}^{ - 1}}{\dot{ X}} = {{\dot{ X}}^{\rm T}}{{{Q}}^{\rm - T}}{{{Q}}^{ - 1}}{\dot{ X}} = {\left\| {{\dot{ X}}} \right\|^2}.$

      (19)

      Furthermore, (18) can be rewritten as

      $\left\| {{\dot{ q}}} \right\| \le \sqrt {{\lambda _{\max }}} \left\| {{\dot{ X}}} \right\|.$

      (20)

      The physical meaning of ${\eta _2} = \sqrt {{\lambda _{\max }}({{{J}}^{\rm T}}{{J}})} $ represents the transmission relationship between the velocity of the moving platform under the given poses and the velocity boundary of the actuated joints. When the movement of the moving platform in Cartesian space is determined, if reduce the maximum eigenvalue of matrix ${{{J}}^{\rm T}}{{J}}$, the maximum value of the actuated joints′ velocity will be decreased. Hence, the maximum eigenvalue of the matrix ${{{J}}^{\rm T}}{{J}}$ can be chosen as a measure index to describe the velocity transmissibility from operation space to joint space. To thoroughly display the velocity transmissibility during the whole task workspace, global velocity transmissibility index (GVTI) is introduced as

      ${\kappa _2} = \frac{{\displaystyle\int_W {{\eta _2}{\rm d}W} }}{{\displaystyle\int_W {{\rm d}W} }}$

      (21)

      where GVTI measures the average value of the velocity transmissibility index value over the prescribed cylindrical workspace. Hence, the mathematical description of the parallel mechanism dimension optimization in regard of velocity transmissibility performance is constructed as

      $\begin{gathered} {f_2} = = \mathop {\min }\limits_{{{x}} \subset {{X}}} ({\kappa _2}) \\ \;\;{\rm s.t.}\begin{array}{*{20}{c}} {}&{{W_0} \subset W}. \end{array} \end{gathered} $

      (22)

      Like the above motion transmission simulation, the global velocity transmission distribution is shown in Fig. 8 (b). Obviously, we can find that the best velocity transmission value occurs at the center of the task workspace, the worst GVTI occurs around the boundary of the workspace. Moreover, the global velocity transmission distribution is $120^\circ $ centrosymmetric. The global velocity transmission value along with a different radius of moving platform (S) is shown in Fig. 10.

      Figure 10.  Global velocity transmissibility versus the variation of l

    • Taking the derivative of (21) with respect to time yields

      ${{a}} = {\ddot q_i}{{e}} + l[{{\dot{ \omega }}_i} \times {{{w}}_i} + {{{\omega }}_i} \times ({{{\omega }}_i} \times {{{w}}_i})],\;(i = 1,2,3)$

      (23)

      where ${{a}}$, ${\ddot q_i}$ and ${{\dot{ \omega }}_i}$ denote the acceleration of moving platform, acceleration of slider i and angular acceleration of strut i, respectively.

      Taking dot multiply by ${{{w}}_i}$ on both sides of (23) and rewriting in matrix form finally results in

      ${\ddot{ q}} = {{J\ddot X}} + {{U}}$

      (24)

      ${{U}} = {[{u_1},\;{u_2},\;{u_3}]^{\rm T}} = {\left[\frac{l}{{{{w}}_1^{\rm T}{{e}}}},\;\frac{l}{{{{w}}_2^{\rm T}{{e}}}},\;\frac{l}{{{{w}}_3^{\rm T}{{e}}}}\right]^{\rm T}}$

      (25)

      where ${\ddot{ q}} = {[{\ddot q_1},\;{\ddot q_2},\;{\ddot q_3}]^{\rm T}}$ and ${\ddot{ X}} = {[\ddot x,\;\ddot y,\;\ddot z]^{\rm T}}$ denote the acceleration of sliders and moving platform, respectively.

      The acceleration characteristic of the parallel mechanism describes the transmission relationship between the generalized acceleration of the moving platform and the acceleration of actuated joints, which rest with the Jacobian matrix J and U. By means of (24), we can get

      $\begin{split} & {\left\| {{\ddot{ q}}} \right\|_2} = {\left\| {{{J\ddot X}} + {{U}}} \right\|_2}\le\\ &\quad {\left\| {{{J\ddot X}}} \right\|_2} + {\left\| {{U}} \right\|_2}\le \\ &\quad {\left\| {{J}} \right\|_2}{\left\| {{\ddot{ X}}} \right\|_2} + {\left\| {{U}} \right\|_2}\le \\ & \quad \sqrt {{\lambda _{\max }}({{{J}}^{\rm T}}{{J}})} {\left\| {{\ddot{ X}}} \right\|_2} + \sqrt {\sum\limits_{i = 1}^3 {u_i^2} }.\end{split} $

      (26)

      According to (26), the acceleration bound of the driving joints are determined simultaneously by the norm of Jacobian matrix J and norm of matrix U. Under a given velocity and acceleration of moving platforms, the smaller the maximum eigenvalue of the matrix J and the norm of the matrix U are, the lower the acceleration bound of the actuated joints suffer, which coincides with the dimension optimization problem solved for PKM acceleration transmissibility characteristics. Hence, the objective function of PKM dimension optimization based on acceleration transmissibility characteristics ${\eta _3}$ is defined as

      ${\eta _3} = \left( {{\sigma _{\max }}({{J}}) + \sqrt {\sum\limits_{i = 1}^3 {u_i^2} } } \right)$

      (27)

      where ${\eta _3}$ represents the generalized acceleration transmissibility factor between the moving platform and actuated joints. Furthermore, to thoroughly display the velocity transmissibility during the whole task workspace, global acceleration transmissibility index (GATI) is introduced as

      ${\kappa _3} = \frac{{\displaystyle\int_W {{\eta _3}{\rm d}W} }}{{\displaystyle\int_W {{\rm d}W} }}.$

      (28)

      Hence, the mathematical description of parallel mechanism dimension optimization in regard of acceleration transmissibility performance is constructed as

      $\begin{gathered} {f_3} = = \mathop {\min }\limits_{{{x}} \subset {{X}}} ({\kappa _3}) \\ \;\;{\rm s.t.}\begin{array}{*{20}{c}} {}&{{W_0} \subset W}. \end{array} \end{gathered} $

      (29)

      The equation mentioned above is, on the premise of satisfying the constraints of basic workspace, finding the structural parameter vector ${{\beta }}$ in parameter space X to make the objective function ${f_{ac}}({{\beta }})$ achieve its minimum, and can result in the desired dimension with the lowest acceleration of actuated joints under the given motion of the moving platform. Like the above motion transmission simulation, the global acceleration transmission distribution is shown in Fig. 8 (c). Obviously, we can find that the best acceleration transmission value occurs at the center of the task workspace, the worst GATI occurs around the boundary of the workspace. Moreover, we can also readily discover that the global velocity transmission distribution is likewise $120^\circ $ centrosymmetric. The global acceleration transmission value along with a different radius of moving platform (S) is shown in Fig. 11.

      Figure 11.  Global acceleration transmissibility versus the variation of l

    • According to the real structure configuration, the radius of the moving platform (s), the radius of the base platform (S), and the length of strut (l) are chosen as the reasonable design variables of the multi-objective dimensional optimization of the 3PUU PKM.

    • 1) As shown in Fig. 3, the main dimensional parameters of the 3-DoF parallel manipulator should be met with certain geometric constraints. Taking into account that the fixed base should be designed as large as possible to equip the whole manipulator, the following constraint condition should be established as

      $\left\{ \begin{aligned} & {s_{\min }} \le s \le {s_{\max }} \\ & {S_{\min }} \le S \le {S_{\max }} \\ & {l_{\min }} \le l \le {l_{\max }}. \end{aligned} \right.$

      (30)

      From the topview of the manipulator, as shown in Fig. 12, the geometrical constraints which allow for the moving platform to have an available space within the structure limitations must be considered, namely

      Figure 12.  Topview of the 3PUU PKM

      $d + l + 2s \le 2S.$

      (31)

      2) The second constraint on the 3PUU is the connecting rod must be inward oblique, so (Sd) is relatively larger than s, we design

      $\frac{{S - d}}{s} \ge 1.2.$

      (32)

      3) Moreover, the moving platform can reach the boundary of task workspace, thus

      $s + d + l - s \ge R({W_t}).$

      (33)

      4) In addition, as shown in Fig. 13, the angle between the projection vector of ${{w}}_i^*$ on plane ${B_1}{B_2}{B_3}$ and its corresponding fixed base radius vector $( - {{{b}}_i})$ should be less than or equal to $90^\circ $

      Figure 13.  Angle between the strut and linear screw

      ${\theta _i} = \arccos \left(\frac{{{{w}}_i^* \cdot ( - {{{b}}_i})}}{{\left\| {{{w}}_i^*} \right\| \cdot \left\| { - {{{b}}_i}} \right\|}}\right) \le 90^\circ .$

      (34)
    • The multi-objective optimization problem of the 3-DOF 3PUU PKM can be described as

      $F({{\beta }}) = [{f_1}({{\beta }}),\;{f_2}({{\beta }}),\;{f_3}({{\beta }})]$

      (35)

      where ${{\beta }} = {(s,S,l)^{\rm T}}$ is a vector consisting of three decision variables. Every decision variable ${\beta _i}$ must exist between its low boundary $\beta _i^{(L)}$ and its up boundary $\beta _i^{(U)}$, by which the corresponding decision space is formulated. ${g_i}({{\beta }})$ and ${h_k}({{\beta }})$ are called constraint functions. If a solution x satisfies all constraints and variable boundaries, it is called a feasible solution. The set of all feasible solution is called the feasible region. The objective function $F({{\beta }})$ forms a multidimensional space called a target space. There exists a mapping relationship between the N-dimension decision vector and the M-dimension objective vector.

      $\begin{aligned} & \min F({{\beta }}) = [{f_1}({{\beta }}),\;{f_2}({{\beta }}),\;{f_3}({{\beta }})] \\ & \quad\quad {\rm s.t.}\quad{\left\{ {\begin{aligned} & {g_j}({{\beta }}) \ge 0, \quad {(j = 1,2, \cdots ,J)} \\ & {h_k}({{\beta }}) = 0, \quad {(k = 1,2, \cdots ,k)} \\ & \beta _i^{(L)} \le {\beta _i} \le \beta _i^{(U)}, \quad {(i = 1,2, \cdots ,n)}. \end{aligned}} \right.} \end{aligned} $

      (36)
    • For a nontrivial multi-objective optimization problem, there does not exist a single solution that simultaneously optimizes each objective but a set of solutions. It is not possible to compare the advantages and disadvantages among these solutions in terms of the objective function. The main task of solving the multi-objective optimization problem is: Firstly, find many representative Pareto optimal solutions that meet the specific requirements without preference; then, according to design requirements and engineering practical experience, select the most satisfactory optimization result from the Pareto optimal solutions.

    • The NSGA-II is currently one of the most popular multi-objective evolutionary algorithms, it reduces the complexity of non-dominated genetic algorithms and possesses the advantages of fast run speed and rapid convergence. The NAGA-II is improved on the basis of the first generation of non-dominated sorting genetic algorithm, the improvements are mainly concerned with three aspects: 1) propose the rapid non-dominated sorting algorithm which reduces the calculation complexity, make the next generation of population select from the double space which merges the parent populations and offspring populations, so as to retain the most outstanding individuals; 2) the introduction of elite strategy ensures some excellent individual species will not be discarded in the process of evolution, so as to improve the accuracy of the optimized results; 3) the employment of crowded degree and crowded degree comparison operators not only overcomes the defect of setting shared parameters by humans in NSGA, but also regards as the comparison standard between population individuals, resulting in letting the quasi Pareto individual domain be evenly extended to the entire Pareto domain and ensuring the population diversity. The multi-objective optimization problem in this paper is solved based on NSGA-II, of which algorithm evolutionary processes are as follows:

      1) Randomly generate the parent population ${P_0}$ of initial population size N, then produce offspring population ${Q_0}$ of population size N through the genetic operators (crossover and mutation).

      2) As shown in Fig. 14, merge the parent population ${P_n}$ and progeny population ${Q_n}$ into the combined population of population scale 2N; according to the non-dominated serial number (level), all of the individuals are non-dominated sorting quickly and classified into level ${F_1}$, ${F_2}$, ${F_3}\cdots$. Calculate the individual local congestion distance of each non-dominated layer and re-sort.

      Figure 14.  Algorithm procedure of NAGA-II

      3) Based on the sorting result, choose the N individuals as the new parent population ${P_{n + 1}}$.

      4) Create new offspring population ${Q_{n + 1}}$ through performing the selection, crossover and mutation operations.

      5) Repeat Step 2 to Step 4 until the set maximum iteration number of the algorithm is reached.

      The function gamultiobj embedded in the GADST toolbox in Matlab software is based on the NSGA-II algorithm and can solve the multi-objective optimization problem with fine convenience, thus, this paper uses the gamultiobj function to optimize the structural parameters of the 3PUU parallel manipulator which is a typical multi-objective optimization problem. The dimensional parameter optimization process of the 3PUU parallel mechanism based on NSGA-II is shown in Fig. 15. The objectives and design variables for all the Pareto optimal solutions are shown in Table 1. Fig. 16 shows the distribution of the Pareto front which represents the set of optimal Objective 1 and Objective 2. It is obvious to note that the distribution of Pareto front possesses excellent diversity and uniformity. Fig. 17 shows the distribution of objectives in the objective space. Fig. 18 shows the distribution of solutions in the solution space. As listed in Table 2, the computation presented 21 solutions in the Pareto front. We can find that the motion transmissibility index and the velocity transmissibility index are fundamentally conflicting objectives. If GMTI increases, GVTI decreases and vice versa. The designers may choose an appropriate solution according to practical demands, herein, we chose the twelfth solution s = 0.075 7, S = 471 7, l = 0.479 4. Based on the twelfth solution, the comparative analysis of motion transmissibility, velocity transmissibility and acceleration transmissibility distribution according to initial structure and optimized structure parameters are shown in Fig. 19. It can be observed that the transmission quality under the optimized structure parameters is improved. In other words, the 3PUU PKM with the optimized parameters possesses excellent transmissibility performance.

      Numbers (m)S (m)l (m)f1f2f3
      10.071 50.500 00.490 0–1.1490.2330.148
      20.050 00.465 30.497 9–2.2810.3910.232
      30.097 60.493 70.500 0–0.9640.2150.139
      40.063 10.461 30.455 1–1.1790.2430.159
      50.051 40.432 90.446 8–1.8780.3320.210
      60.051 90.436 10.451 2–1.8900.3340.211
      70.066 50.468 60.470 9–1.2830.2560.164
      80.092 60.496 60.487 1–0.8300.2100.137
      90.053 30.462 20.495 0–2.1310.3830.229
      100.069 30.474 20.480 8–1.3160.2610.167
      110.053 20.457 80.464 6–1.6810.2990.189
      120.075 70.471 70.479 4–1.2300.2520.161
      130.080 10.481 10.484 2–1.1030.2310.149
      140.090 60.493 60.484 5–0.8490.2120.138
      150.077 20.470 00.477 5–1.2030.2490.161
      160.074 40.465 10.459 6–1.0410.2280.150
      170.053 50.454 80.444 0–1.2340.2510.165
      180.060 00.467 50.469 0–1.3880.2670.171
      190.091 20.487 30.488 9–0.9670.2170.141
      200.095 40.466 60.477 8–1.0270.2270.148
      210.069 80.489 70.496 6–1.3150.2580.162

      Table 2.  Pareto optimal solutions of the 3PUU PKM dimensional optimization

      Figure 16.  Pareto front of Objective 1 and Objective 2

      Figure 17.  Distribution of Pareto solution in objective space

      Figure 18.  Local distribution of Pareto solution in solution space

      Figure 15.  Dimensional optimization process based on NSGA-II

      smin (m)smax (m)Smin (m)Smax (m)lmin (m)lmax (m)
      0.0500.1000.3000.5000.3000.500

      Table 1.  Constraint parameters of the 3PUU PKM

      Figure 19.  Comparison of transmission performance based on the initial parameters and optimized parameters

    • In this paper, a general and systematic framework for the multi-objective dimensional optimization of a 3-DOF PKM considering motion transmissibility, velocity transmissibility and acceleration transmissibility is presented. By employing a pair of desired virtual constraints, a stiffness-enhanced 3PUU parallel manipulator with three translational DOFs is proposed. A motion transmissibility performance index based on transmission angles with clear physical meaning is presented. The performance of velocity transmissibility and acceleration transmissibility are derived by virtue of the kinematics model of the parallel manipulator. Because of the introduction of the elite strategy, crowing distance distribution and non-dominated quick sort optimization mechanisms, the improved non-dominated sorting genetic algorithm (NSGA II) not only has good convergence and distribution at same time, but also possesses faster convergence speed. Thanks to NSGA II, the best appropriate solution of the set of Pareto optimal solutions is selected. Detailed numerical examples and comparisons demonstrate the application and show the correctness of the proposed approach for implementing multi-objective dimensional optimization.

    • This work was supported by National Natural Science Foundation of China (Nos. 51575544 and 51275353), the Macao Science and Technology Development Fund (No. 110/2013/A3) and Research Committee of University of Macau (Nos. MYRG2015-00194-FST and MYRG203 (Y1-L4)-FST11-LYM).

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