# Multiobjective Optimization of 6-DOF Parallel Manipulator for Desired Total Orientation Workspace

Authors: Hongbin Qiang; Lihang Wang; Jisong Ding; Lijie Zhang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

A desired total orientation workspace for a parallel manipulator is usually an essential requirement in a practical application. At present, for the multiobjective optimization method of 6-DOF parallel manipulator for desired total orientation workspace, it is needed to predefine maximal and minimal lengths of the legs to serve as the constraint, and then the numerical method is used to solve the length of the legs and judge whether the solved maximal and minimal leg lengths meet the constraint. Predefining maximal and minimal length of the legs limits of the optimal range, the numerical method has heavy calculation burden and low calculating accuracy. In this paper, a hybrid method for solving the maximal and minimal lengths of the legs of 6-DOF parallel manipulator with desired total orientation workspace is proposed, and the actuator stroke length is calculated according to the maximal and minimal leg lengths. By judging whether the actuator stroke length can be solved to serve as the constraint, without the predefined maximal and minimal leg lengths to serve as the constraint, the optimal range is enlarged. Aiming at the physical size of the parallel manipulator and the proposed desired workspace condition index (DWCI), the optimization of the geometric parameters of the parallel manipulator is conducted based on the multiobjective optimization algorithm (NSGA-II), which is subject to the actuator stroke length. Stewart platform is set as the example; the geometric parameters of the platform whose workspace contains the desired total orientation workspace are optimized and the hybrid method is proved to be more accurate and efficient compared to the traditional numerical method. This method provides the optimization guidance for engineering designers to design the parallel manipulator for desired total orientation workspace.


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## Body

## 1. Introduction

Parallel manipulator has the advantages of compact structure, strong bearing capacity, high precision of motion, and low inertia and therefore is widely used in the flight simulator, ship motion simulation, ship stabilized platform, space docking manipulator, parallel machine tools, and robot wrist [1, 2]. However, the limited workspace and dexterity are the major defects of the parallel manipulator [3–6]. As a result, it is of great importance to design a set of geometric parameters of the parallel manipulator with ideal workspace and good performance.At present, in the design of the parallel manipulator for the workspace requirements, there are two types [6]. One is to get the geometry parameters of the parallel manipulator that maximizes the workspace. Lou [6] defined the effective regular workspace and aimed at the maximal side length of the effective regular workspace for the constant orientation, and the optimization of the geometric parameters is conducted by the controlled random search (CRS) method, which is subject to the dexterity and the leg lengths. Toz [7] defined the dexterous workspace by the condition index and minimum singular value (MSV) and used the particle swarm optimization (PSO) to optimize the dexterous workspace of asymmetric parallel manipulator, which is subject to kinematics and geometric constraints. Aiming at the condition number, stiffness, and the reachable workspace solved by geometric method, Shirazi [8] optimized the geometric parameters of the 6-DOF parallel manipulator.However, maximizing the workspace blindly does not meet the engineering requirements, since a desired regular shape workspace is usually an essential requirement in practice. The other type is to obtain the geometric parameters of a parallel manipulator whose workspace contains a desired workspace. Liu [9] used analytic method to design geometric parameters of a DELTA manipulator to get a desired total orientation workspace. In [10], a genetic algorithm was applied to optimize geometric parameters of a DELTA manipulator whose workspace contains a desired three-dimensional workspace. As for a 6-DOF parallel manipulator whose workspace is six- dimensional, it is necessary to optimize geometric parameters of parallel manipulator for six- dimensional workspace. At present, some scholars have optimized the geometric parameters of the 6-DOF parallel manipulator for a desired lower-mobility workspace. Liu [11] optimized the geometric parameters of the 6-DOF parallel manipulator for a desired single DOF workspace. Cirillo [12] used genetic algorithm to optimize the geometric parameters of the 6-DOF parallel manipulator in aeronautics applications for the known trajectory. In addition, some scholars have designed geometric parameters of parallel manipulator for a desired total orientation workspace [13]. Zhao [14] exploited an algebraic algorithm to optimize the length of the legs of a parallel manipulator to get a desired total orientation workspace. Fu [15] proposed a novel 3-leg 6-DOF parallel manipulator whose three translational and three rotational motions are decoupled, and an algorithm of optimal designing is proposed to find the smallest dimensional parameters of the proposed robot for the desired workspace based on analytical solutions of the forward kinematics. Yang [16] gave the size and position of the desired total orientation workspace, and the leg lengths were obtained by the six-dimensional discretization of the workspace based on the numerical method, and then aiming at the linear combination of the physical size and dexterity index, the geometric parameters of the 6-DOF platform were optimized by the genetic algorithm, which is subject to the given leg lengths. Xie [17] solved the global transmission indexes for the desired total orientation workspace with different sets of geometric parameters; the optimal geometric parameters were obtained based on the changing characteristics of indexes with structural parameters.At present, there are few studies on multiobjective optimization of 6-DOF parallel manipulator for desired total orientation workspace. In general, numerical method is used to discretize the six- dimensional desired total orientation workspace to solve the length of the legs; however, it consumes a lot of time. Furthermore, the solved length of the legs should be subject to the predefined maximal and minimal length, which limits the optimization design of the geometric parameters. Therefore, a hybrid method for solving the maximal and minimal length of the legs of 6-DOF parallel manipulator with desired total orientation workspace is proposed in this paper, and it is more accurate and efficient compared to the numerical method. And the actuator stroke length is calculated according to the maximal and minimal leg lengths, by judging whether the actuator stroke length can be solved to serve as the constraint; without the predefined maximal and minimal leg lengths to serve as the constraint, the optimal range is enlarged. And then aiming at the physical size of the parallel manipulator and the proposed DWCI, the optimization of the geometric parameters of the parallel manipulator is conducted based on the multiobjective optimization algorithm (NSGA-II), which is subject to the actuator stroke length. Stewart platform is set as the example; the geometric parameters of the platform whose workspace contains the desired total orientation workspace are optimized and the hybrid method is proved to be more accurate and efficient compared to the traditional numerical method. This method provides the optimization guidance for engineering designers to design the parallel manipulator for desired total orientation workspace.


## 2. 6-DOF Parallel Manipulator

The structural sketch of 6-DOF parallel manipulator is shown in Figure1. The manipulator is composed of the base platform, the moving platform, the legs, and the universal (spherical) joints. The basic platform is fixed and the moving platform is at the end of the manipulator. Two platforms are connected by the legs and the universal (spherical) joints.Figure 1
The structure sketch of 6-DOF parallel manipulator.LettingAi (i=1,…,6) denote the center points of the universal (spherical) joints which connect the legs and base platform andBi (i=1,…,6) denote the center points of the universal (spherical) joints which connect the legs and moving platform, Ai and Bi are at both ends of the same leg. A symmetrical hexagon is constructed by the point Ai, and Oa is the center of the symmetrical hexagon, the pointOa is the origin of the reference coordinate frame Oa, and the direction ofz axes perpendicular to the base platform points upwards, the direction ofx axes parallel to the base platform points to the midpoint ofA1 andA6, and the direction ofy axes is determined by the right-hand rule. A symmetrical hexagon is constructed by the pointBi, andOb is the center of the symmetrical hexagon, the pointOb is the origin of the reference coordinate frame Ob, and the direction ofz axes perpendicular to the moving platform points upwards, the direction ofx axes parallel to the moving platform points to the midpoint ofB1 andB6, and the direction ofy axes is determined by the right-hand rule. The reference coordinate frame Oa and the moving coordinate frame Ob are assigned to the two platforms, respectively. The pointOa is the origin of Oa located at the base platform and the pointOb is the origin of Ob located at the moving platform. The vectors ai=[xai,yai,0] and bi=[xbi,ybi,0] represent the vectors of the pointsAi andBi in Oa and Ob, respectively. In the initial state, thezaxis of Oa is recombined with thez axis of Ob, thexaxis andyaxis of Oa are parallel to thexaxis andyaxis of Ob, and the vector of the point Ob in Oa is h=[0,0,H]. The geometric parameters of the parallel manipulator can be determined by the vectors ai, bi, and h.


### 2.1. Inverse Kinematics

The total orientation workspace of the 6-DOF parallel manipulator can be denoted asΩ=x,y,z,α,β,γ, where the position in the workspace of the manipulator translation is defined as P=x,y,z and the posture in the workspace of the manipulator orientation is defined as Q=α,β,γ. In this paper, the Roll-Pitch-Yaw (RPY) angles are used to describe the orientation transformation. First rotate coordinate frame Ob around thex axis of an angle γ (Yaw), then rotate the resulting coordinate frame around they axis of an angle β (Pitch), and finally rotate coordinate frame around thez axis of an angle α (Roll). The transformation matrix from moving coordinate frame to the reference coordinate frame can be expressed as follows:(1)RBA=Rz,αRy,βRx,γ=cαcβcαsβsγ-sαcγcαsβcγ+sαsγsαcβsαsβsγ+cαcγsαsβcγ-cαsγ-sβcβsγcβcγwhere cα≡cos⁡α, sα≡sin⁡α, cβ≡cos⁡β, sβ≡sin⁡β, cγ≡cos⁡γ, and sγ≡sin⁡γ.According to the position P and posture Q in the workspace and the transformation matrixRBA, the leg length of parallel manipulator can be solved. Theith leg vector can be obtained by the difference between the vectors bi and ai in Oa. Theith actuator can be expressed by (2). The vector diagram for theith actuator is given in Figure 2. Ob represented by the dotted line is the initial state, and represented by the solid line is the arbitrary position state in the total orientation workspace.(2)li=RBA·bi+pBA+h-ai=lixliylizT=cαcβxbi+cαsβsγ-sαcγybi+x-xaisαcβxbi+sαsβsγ+cαcγybi+y-yai-sβxbi+cβsγybi+z+Hwhere PBA denotes the position P.Figure 2
The vector diagram for theith actuator.Then the length of theith leg can be obtained by (3) and the square of the length of theith leg can be got by (4).(3)eix,y,z,α,β,γ=lix2+liy2+liz2(4)fix,y,z,α,β,γ=lix2+liy2+liz2


## 2.1. Inverse Kinematics

The total orientation workspace of the 6-DOF parallel manipulator can be denoted asΩ=x,y,z,α,β,γ, where the position in the workspace of the manipulator translation is defined as P=x,y,z and the posture in the workspace of the manipulator orientation is defined as Q=α,β,γ. In this paper, the Roll-Pitch-Yaw (RPY) angles are used to describe the orientation transformation. First rotate coordinate frame Ob around thex axis of an angle γ (Yaw), then rotate the resulting coordinate frame around they axis of an angle β (Pitch), and finally rotate coordinate frame around thez axis of an angle α (Roll). The transformation matrix from moving coordinate frame to the reference coordinate frame can be expressed as follows:(1)RBA=Rz,αRy,βRx,γ=cαcβcαsβsγ-sαcγcαsβcγ+sαsγsαcβsαsβsγ+cαcγsαsβcγ-cαsγ-sβcβsγcβcγwhere cα≡cos⁡α, sα≡sin⁡α, cβ≡cos⁡β, sβ≡sin⁡β, cγ≡cos⁡γ, and sγ≡sin⁡γ.According to the position P and posture Q in the workspace and the transformation matrixRBA, the leg length of parallel manipulator can be solved. Theith leg vector can be obtained by the difference between the vectors bi and ai in Oa. Theith actuator can be expressed by (2). The vector diagram for theith actuator is given in Figure 2. Ob represented by the dotted line is the initial state, and represented by the solid line is the arbitrary position state in the total orientation workspace.(2)li=RBA·bi+pBA+h-ai=lixliylizT=cαcβxbi+cαsβsγ-sαcγybi+x-xaisαcβxbi+sαsβsγ+cαcγybi+y-yai-sβxbi+cβsγybi+z+Hwhere PBA denotes the position P.Figure 2
The vector diagram for theith actuator.Then the length of theith leg can be obtained by (3) and the square of the length of theith leg can be got by (4).(3)eix,y,z,α,β,γ=lix2+liy2+liz2(4)fix,y,z,α,β,γ=lix2+liy2+liz2


## 3. The Maximal and Minimal Leg Lengths and the Actuator Stroke Length

The most usual types of workspace of 6-DOF parallel manipulator are reachable workspace, constant orientation workspace, orientation workspace, total orientation workspace, etc. Total orientation workspace is all the locations that may be reached with all the orientations among a set defined by ranges on the orientation angles [13]. The desired total orientation workspace is set by engineering designers according to engineering requirements. The desired total orientation workspace is a subset of the total orientation workspace, and it is usually geometry in a regular shape, such as a cube, a ball, and a cylinder. In this paper, a cube is set as the desired total orientation workspace Ωt, as follows:(5)Ωt=x,y,z,α,β,γ∣xmin≤x≤xmax,ymin≤y≤ymaxzmin≤z≤zmax,αmin≤α≤αmaxβmin≤β≤βmax,γmin≤γ≤γmax


### 3.1. Numerical Method for Calculating the Maximal and Minimal Leg Lengths

Numerical method [18] involves discretizing the six-dimensional desired total orientation workspace, calculating the leg lengths for each pose (discretized positions and postures), and getting the maximal and minimal leg lengths. Specific steps are as follows.( 1 ) Set a set of geometric parameters of the 6-DOF parallel manipulator, which includes ai, bi, and h.( 2 ) Divide the desired total orientation workspace Ωt into the position workspace Ωp and the posture workspace Ωq. The position workspace and the posture workspace can be expressed by (5) and (6), respectively, as follows:(6)Ωp=x,y,z∣xmin≤x≤xmax,ymin≤y≤ymaxzmin≤z≤zmax(7)Ωq=α,β,γ∣αmin≤α≤αmax,βmin≤β≤βmaxγmin≤γ≤γmax(3) Discretize the position and posture workspace by the grid method. The translation step lengths alongx axis,y axis, andz axis are Sx, Sy, and Sz. The orientation step lengths aroundx axis,y axis, andz axis are Sα, Sβ, and Sγ. The number of discrete points in each dimension can be obtained by (8) as follows:(8)Nv=floorvmax-vminSv+1where floor(▪) denotes a function for rounding down and the subscript v denotes each dimension (v=x,y,z,α,β,γ). Then, the total number of discrete points in all dimensions can be obtained by (9). The discretized position and posture can be expressed by (10) and (11).(9)NT=NxNyNzNαNβNγ(10)xm1,ym2,zm3=xmin+m1Sx,ymin+m2Sy,zmin+m3Sz(11)αn1,βn2,γn3=αmin+n1Sα,βmin+n2Sβ,γmin+n3Sγwhere m1=0,1,…,Nx; m2=0,1,…,Ny; m3=0,1,…,Nz; n1=0,1,…,Nα; n2=0,1,…,Nβ; n3=0,1,…,Nγ.( 4 ) According to (3), calculate the six leg lengths for each pose.( 5 ) Get the six maximal and minimal leg lengths by comparing the leg lengths in each position.


### 3.2. Algebraic Method for Calculating the Maximal and Minimal Leg Lengths

Equation (3) used to calculate the leg lengths is a six-dimensional function that involves the variables of x,y,z,α,β,γ. Therefore, the maximal and minimal leg lengths are the maximum and minimum of a six-dimensional function. The functional maximum and minimum exist in the functional extremum or the boundary value. The extremal points of (3) are identical to (4) [19]. Therefore, for the convenience of solving the extremal points, we can obtain the extremal points of (3) by solving the extremal points of (4). Equation (12) is the partial derived function of (4), and the extremal points of (4) are obtained by (12) [20].(12)∂f∂x=2cαcβxbi+cαsβsγ-sβcγybi+x-xai∂f∂y=2sαcβxbi+sαsβsγ+cαcγybi+y-yai∂f∂z=2-sβxbi+cβsγybi+z+H∂f∂α=2(xai-xsαcβxbi+sαsβsγybi+cαcγybi+y-yaicαcβxbi+cαsβsγybi-sαcγybi∂f∂β=2xai-xcα+y-yaisαcβsγybi-sβxbi-cβxbi+sβsγybiz+H∂f∂γ=2x-xaicαsβcγ+sαsγ+y-ybisαsβcγ-cαsγ+z+HcβcγybiThe maximum and minimum of six-dimensional function exist in the extremum of six-dimensional function or the maximum and minimum of five-dimensional function (one of the six variables is the boundary point); the maximum and minimum of a five-dimensional function exist in the extremum of five-dimensional function or the maximum and minimum of four-dimensional function (two of the six variables are the boundary points). By this analogy, the rest can be done as shown in Figure 3. Particularly, the maximum and minimum of a one-dimensional function exist in the extremum of one-dimensional function or the maximum and minimum of a zero-dimensional function (all of the six variables equal to the boundary points). Since each variable has two boundary values and arbitrary combinations with other variables, the number of the functions of k variables can be obtained by (13)Nm=26-m×C66-mm=1,2,⋯,6wherem is the number of variables of the function.Figure 3
Solving process of function of m(m=1,…,6) variables.The solving process of function of k variables is shown in Figure3. The algebraic method contains solving the function ofm(m=1,…,6) variables, getting the extremal points ofm(m=1,…,6) variables and 64 boundary points, excluding the extremal points that are not in the desired total orientation workspace, and calculating the extremum of (3) by the obtained extremal points and boundary points. Specific steps are shown as follows.( 1 ) Set a set of geometric parameters of the 6-DOF parallel manipulator, which includes ai, bi, and h.( 2 ) According to (12) and Figure 3, the function ofm(m=1,…,6) variables is solved.( 3 ) Get the extremal points ofm(m=1,…,6) variables and 64 boundary points.( 4 ) Exclude the extremal points that are not in the desired total orientation workspace.( 5 ) Get the extremum and the boundary value of (3).( 6 ) Get the six maximal and minimal leg lengths by comparing the leg lengths obtained in step (5).


### 3.3. Hybrid Method for Calculating the Maximal and Minimal Leg Lengths

The numerical method takes much time to calculate the maximal and minimal leg lengths, due to the large calculation quantity on the leg length for each discretized pose. And the precision is affected by the step lengths; usually a smaller step length leads to the higher precision and more time. The algebraic method takes less time to calculate the explicit maximal and minimal leg lengths. However, it is difficult to obtain the extremal points of the six-element function, and the extremal points of the function can be only obtained with the specified orientation variables (α,β,γ) which are equal to the boundary points.Therefore, a hybrid method including the numerical method and algebraic method is proposed, where the numerical method is used for orientation variables and the algebraic method is used for translation variables. Firstly, the three-dimensional posture workspace is discretized and the discretized postures are obtained. Then the functions of 3 variables (x,y,z) in every discretized posture are solved. Finally, the maximal and minimal leg lengths are obtained. The extremal points can be obtained by the hybrid method based on the three-element function without solving six-element function. And compared with the numerical method, it is more accurate and efficient.Equation (14) is the partial derived function of the functions of 3 variables (x,y,z) and (15) is the extremal points.(14)∂f∂x=cαcβxbi+cαsβsγ-sβcγybi+x-xai∂f∂y=sαcβxbi+sαsβsγ+cαcγybi+y-yai∂f∂z=-sβxbi+cβsγybi+z+H(15)x=xai-cαcβxbi-cαsβsγ-sβcγybiy=yai-sαcβxbi-sαsβsγ+cαcγybiz=sβxbi-cβsγybi-HThe solving process of functions ofn(n=1,2,3) variables is shown in Figure 4. Specific steps of the hybrid method for solving the maximal and minimal leg lengths of 6-DOF parallel manipulator for desired total orientation workspace are shown as follows.Figure 4
Solving process of function of n(n=1,2,3) variables.( 1 ) Set a set of geometric parameters of the 6-DOF parallel manipulator, which includes ai, bi, and h.( 2 ) Discretize the posture workspace by the grid method. The orientation step lengths around x axis, y axis, and z axis are Sα,Sβ,Sγ. The discretized postures can be got by (11). The number of discrete points in each dimension can be got by (8). The total number of discretized postures can be got by (16).(16)NTP=NαNβNγ( 3 ) Select one posture in NTP discretized postures.( 4 ) According to (14) and Figure 4, solve the function ofn(n=1,2,3) variables.( 5 ) Get the extremal points ofn(n=1,2,3) variables and 8 boundary points.( 6 ) Exclude the extremal points that are not in the desired total orientation workspace.( 7 ) Get the extremum and the boundary value of (3).( 8 ) Get the maximal and minimal leg lengths of each discretized posture.( 9 ) Select the next discretized posture in step (3) until all of theNTP discretized postures have been calculated.( 10 ) Get the six maximal and minimal leg lengths by comparing the leg lengths in theNTP discretized postures.


### 3.4. Actuator Stroke Length

The maximal and minimal leg length solved for the desired total orientation workspace should be between the maximum and minimum of the actuator length. The maximal and minimal values of the actuator can be determined by dead length and stroke length of the actuator. The dead length is the part which is not in the extension part of the cylinder, and the stroke length is the part in the extension part of the cylinder [21], as shown in Figure 5. According to Figure 5, the actuator stroke length can be obtained, as shown in (17).(17)lsmin≥lmax-lminlsmin≥lmax-ld2lsmax≤lmin-ldwhere lmin denotes the minimum of leg length, lmax denotes the maximum of leg length,ld denotes the dead length of actuator, andls denotes the actuator stroke length. Hydraulic cylinders, pneumatic cylinders, or electric cylinders are generally used as actuators, and the dead lengthld usually can be determined according to the actual condition.Figure 5
The schematic diagram of actuator.According to (17), the minimum lsmin and the maximum lsmax of the actuator stroke length can be got by (18) and (19), respectively. Equation (20) can be defined based on (18) and (19), which is expressed as follows:(18)lsmin=maxlmax-lmin,lmax-ld2(19)lsmax=lmin-ld(20)g=lsmax-lsminIf the value of g in (20) is nonnegative, it means there is the solution to the actuator stroke length of the 6-DOF parallel manipulator under this set of geometric parameters; thus an actuator can be selected to achieve the maximum and minimum leg lengths for the desired total orientation workspace. Otherwise, if the value of g is negative, there is no solution to the actuator stroke length.


## 3.1. Numerical Method for Calculating the Maximal and Minimal Leg Lengths

Numerical method [18] involves discretizing the six-dimensional desired total orientation workspace, calculating the leg lengths for each pose (discretized positions and postures), and getting the maximal and minimal leg lengths. Specific steps are as follows.( 1 ) Set a set of geometric parameters of the 6-DOF parallel manipulator, which includes ai, bi, and h.( 2 ) Divide the desired total orientation workspace Ωt into the position workspace Ωp and the posture workspace Ωq. The position workspace and the posture workspace can be expressed by (5) and (6), respectively, as follows:(6)Ωp=x,y,z∣xmin≤x≤xmax,ymin≤y≤ymaxzmin≤z≤zmax(7)Ωq=α,β,γ∣αmin≤α≤αmax,βmin≤β≤βmaxγmin≤γ≤γmax(3) Discretize the position and posture workspace by the grid method. The translation step lengths alongx axis,y axis, andz axis are Sx, Sy, and Sz. The orientation step lengths aroundx axis,y axis, andz axis are Sα, Sβ, and Sγ. The number of discrete points in each dimension can be obtained by (8) as follows:(8)Nv=floorvmax-vminSv+1where floor(▪) denotes a function for rounding down and the subscript v denotes each dimension (v=x,y,z,α,β,γ). Then, the total number of discrete points in all dimensions can be obtained by (9). The discretized position and posture can be expressed by (10) and (11).(9)NT=NxNyNzNαNβNγ(10)xm1,ym2,zm3=xmin+m1Sx,ymin+m2Sy,zmin+m3Sz(11)αn1,βn2,γn3=αmin+n1Sα,βmin+n2Sβ,γmin+n3Sγwhere m1=0,1,…,Nx; m2=0,1,…,Ny; m3=0,1,…,Nz; n1=0,1,…,Nα; n2=0,1,…,Nβ; n3=0,1,…,Nγ.( 4 ) According to (3), calculate the six leg lengths for each pose.( 5 ) Get the six maximal and minimal leg lengths by comparing the leg lengths in each position.


## 3.2. Algebraic Method for Calculating the Maximal and Minimal Leg Lengths

Equation (3) used to calculate the leg lengths is a six-dimensional function that involves the variables of x,y,z,α,β,γ. Therefore, the maximal and minimal leg lengths are the maximum and minimum of a six-dimensional function. The functional maximum and minimum exist in the functional extremum or the boundary value. The extremal points of (3) are identical to (4) [19]. Therefore, for the convenience of solving the extremal points, we can obtain the extremal points of (3) by solving the extremal points of (4). Equation (12) is the partial derived function of (4), and the extremal points of (4) are obtained by (12) [20].(12)∂f∂x=2cαcβxbi+cαsβsγ-sβcγybi+x-xai∂f∂y=2sαcβxbi+sαsβsγ+cαcγybi+y-yai∂f∂z=2-sβxbi+cβsγybi+z+H∂f∂α=2(xai-xsαcβxbi+sαsβsγybi+cαcγybi+y-yaicαcβxbi+cαsβsγybi-sαcγybi∂f∂β=2xai-xcα+y-yaisαcβsγybi-sβxbi-cβxbi+sβsγybiz+H∂f∂γ=2x-xaicαsβcγ+sαsγ+y-ybisαsβcγ-cαsγ+z+HcβcγybiThe maximum and minimum of six-dimensional function exist in the extremum of six-dimensional function or the maximum and minimum of five-dimensional function (one of the six variables is the boundary point); the maximum and minimum of a five-dimensional function exist in the extremum of five-dimensional function or the maximum and minimum of four-dimensional function (two of the six variables are the boundary points). By this analogy, the rest can be done as shown in Figure 3. Particularly, the maximum and minimum of a one-dimensional function exist in the extremum of one-dimensional function or the maximum and minimum of a zero-dimensional function (all of the six variables equal to the boundary points). Since each variable has two boundary values and arbitrary combinations with other variables, the number of the functions of k variables can be obtained by (13)Nm=26-m×C66-mm=1,2,⋯,6wherem is the number of variables of the function.Figure 3
Solving process of function of m(m=1,…,6) variables.The solving process of function of k variables is shown in Figure3. The algebraic method contains solving the function ofm(m=1,…,6) variables, getting the extremal points ofm(m=1,…,6) variables and 64 boundary points, excluding the extremal points that are not in the desired total orientation workspace, and calculating the extremum of (3) by the obtained extremal points and boundary points. Specific steps are shown as follows.( 1 ) Set a set of geometric parameters of the 6-DOF parallel manipulator, which includes ai, bi, and h.( 2 ) According to (12) and Figure 3, the function ofm(m=1,…,6) variables is solved.( 3 ) Get the extremal points ofm(m=1,…,6) variables and 64 boundary points.( 4 ) Exclude the extremal points that are not in the desired total orientation workspace.( 5 ) Get the extremum and the boundary value of (3).( 6 ) Get the six maximal and minimal leg lengths by comparing the leg lengths obtained in step (5).


## 3.3. Hybrid Method for Calculating the Maximal and Minimal Leg Lengths

The numerical method takes much time to calculate the maximal and minimal leg lengths, due to the large calculation quantity on the leg length for each discretized pose. And the precision is affected by the step lengths; usually a smaller step length leads to the higher precision and more time. The algebraic method takes less time to calculate the explicit maximal and minimal leg lengths. However, it is difficult to obtain the extremal points of the six-element function, and the extremal points of the function can be only obtained with the specified orientation variables (α,β,γ) which are equal to the boundary points.Therefore, a hybrid method including the numerical method and algebraic method is proposed, where the numerical method is used for orientation variables and the algebraic method is used for translation variables. Firstly, the three-dimensional posture workspace is discretized and the discretized postures are obtained. Then the functions of 3 variables (x,y,z) in every discretized posture are solved. Finally, the maximal and minimal leg lengths are obtained. The extremal points can be obtained by the hybrid method based on the three-element function without solving six-element function. And compared with the numerical method, it is more accurate and efficient.Equation (14) is the partial derived function of the functions of 3 variables (x,y,z) and (15) is the extremal points.(14)∂f∂x=cαcβxbi+cαsβsγ-sβcγybi+x-xai∂f∂y=sαcβxbi+sαsβsγ+cαcγybi+y-yai∂f∂z=-sβxbi+cβsγybi+z+H(15)x=xai-cαcβxbi-cαsβsγ-sβcγybiy=yai-sαcβxbi-sαsβsγ+cαcγybiz=sβxbi-cβsγybi-HThe solving process of functions ofn(n=1,2,3) variables is shown in Figure 4. Specific steps of the hybrid method for solving the maximal and minimal leg lengths of 6-DOF parallel manipulator for desired total orientation workspace are shown as follows.Figure 4
Solving process of function of n(n=1,2,3) variables.( 1 ) Set a set of geometric parameters of the 6-DOF parallel manipulator, which includes ai, bi, and h.( 2 ) Discretize the posture workspace by the grid method. The orientation step lengths around x axis, y axis, and z axis are Sα,Sβ,Sγ. The discretized postures can be got by (11). The number of discrete points in each dimension can be got by (8). The total number of discretized postures can be got by (16).(16)NTP=NαNβNγ( 3 ) Select one posture in NTP discretized postures.( 4 ) According to (14) and Figure 4, solve the function ofn(n=1,2,3) variables.( 5 ) Get the extremal points ofn(n=1,2,3) variables and 8 boundary points.( 6 ) Exclude the extremal points that are not in the desired total orientation workspace.( 7 ) Get the extremum and the boundary value of (3).( 8 ) Get the maximal and minimal leg lengths of each discretized posture.( 9 ) Select the next discretized posture in step (3) until all of theNTP discretized postures have been calculated.( 10 ) Get the six maximal and minimal leg lengths by comparing the leg lengths in theNTP discretized postures.


## 3.4. Actuator Stroke Length

The maximal and minimal leg length solved for the desired total orientation workspace should be between the maximum and minimum of the actuator length. The maximal and minimal values of the actuator can be determined by dead length and stroke length of the actuator. The dead length is the part which is not in the extension part of the cylinder, and the stroke length is the part in the extension part of the cylinder [21], as shown in Figure 5. According to Figure 5, the actuator stroke length can be obtained, as shown in (17).(17)lsmin≥lmax-lminlsmin≥lmax-ld2lsmax≤lmin-ldwhere lmin denotes the minimum of leg length, lmax denotes the maximum of leg length,ld denotes the dead length of actuator, andls denotes the actuator stroke length. Hydraulic cylinders, pneumatic cylinders, or electric cylinders are generally used as actuators, and the dead lengthld usually can be determined according to the actual condition.Figure 5
The schematic diagram of actuator.According to (17), the minimum lsmin and the maximum lsmax of the actuator stroke length can be got by (18) and (19), respectively. Equation (20) can be defined based on (18) and (19), which is expressed as follows:(18)lsmin=maxlmax-lmin,lmax-ld2(19)lsmax=lmin-ld(20)g=lsmax-lsminIf the value of g in (20) is nonnegative, it means there is the solution to the actuator stroke length of the 6-DOF parallel manipulator under this set of geometric parameters; thus an actuator can be selected to achieve the maximum and minimum leg lengths for the desired total orientation workspace. Otherwise, if the value of g is negative, there is no solution to the actuator stroke length.


## 4. Multiobjective Optimization Algorithm

The Elitist Nondominated Sorting Genetic Algorithm version II (NSGA-II) is one of the most classical and popular multiobjective evolutionary algorithms, which is proposed by Deb [22] et al. In this paper, NSGA-II is employed to solve the multiobjective optimal design problems of 6-DOF parallel manipulator for desired total orientation workspace; the specific process is shown in Figure 6, and the simulated binary crossover (SBX) operator and polynomial mutation operator are applied to the algorithm [11].Figure 6
The detailed process of NSGA II.


### 4.1. Targets

By the determined volume of the desired total orientation workspace, the larger physical size of the parallel manipulator leads will meet the workspace requirements better. So the physical size of the parallel manipulator needs to be optimized to achieve higher economy and practicability. In general, the maximal leg length [14] or the volume of the manipulator [16] is the target of the physical size of the parallel manipulator.One of the kinematics and dynamics indexes should be selected as the target to ensure the good performance of the manipulator. The global performance indexes such as the global condition index (GCI) [23], global manipulability index, and global transmission index [17] are calculated over the whole workspace. However, manipulator always works in the desired workspace rather than the whole workspace. Therefore, a performance index is proposed in this paper, which is used for the parallel manipulator within the desired workspace. For example, the desired workspace condition index (DWCI) can be got by (21)k=∫WηdW∫WdWwhere W denotes the desired workspace and η denotes the local conditional index (LCI).When the Monte Carlo method is used to randomly generate m poses for the desired workspace, the DWCI can be obtained by (22) [24],(22)k=∑i=1mηimwhere m denotes the number of random poses and ηi denotes the local conditional index of the ith (i=1,…,m) random poses.The range ofk is (0,1], where 0 means the poor dexterity and 1 means the best dexterity. The larger the k is, the more flexible the mechanism is [23].


### 4.2. Constraint

The actuator stroke length of parallel manipulator for desired total orientation workspace is taken as the constraint. If (20) is nonnegative, the constraint is satisfied. If (20) is negative, the constraint is not satisfied.Calculate the maximal and minimal leg lengths by the hybrid method proposed in this paper, and then calculate the stroke length. The specific process is shown in Figure6.


## 4.1. Targets

By the determined volume of the desired total orientation workspace, the larger physical size of the parallel manipulator leads will meet the workspace requirements better. So the physical size of the parallel manipulator needs to be optimized to achieve higher economy and practicability. In general, the maximal leg length [14] or the volume of the manipulator [16] is the target of the physical size of the parallel manipulator.One of the kinematics and dynamics indexes should be selected as the target to ensure the good performance of the manipulator. The global performance indexes such as the global condition index (GCI) [23], global manipulability index, and global transmission index [17] are calculated over the whole workspace. However, manipulator always works in the desired workspace rather than the whole workspace. Therefore, a performance index is proposed in this paper, which is used for the parallel manipulator within the desired workspace. For example, the desired workspace condition index (DWCI) can be got by (21)k=∫WηdW∫WdWwhere W denotes the desired workspace and η denotes the local conditional index (LCI).When the Monte Carlo method is used to randomly generate m poses for the desired workspace, the DWCI can be obtained by (22) [24],(22)k=∑i=1mηimwhere m denotes the number of random poses and ηi denotes the local conditional index of the ith (i=1,…,m) random poses.The range ofk is (0,1], where 0 means the poor dexterity and 1 means the best dexterity. The larger the k is, the more flexible the mechanism is [23].


## 4.2. Constraint

The actuator stroke length of parallel manipulator for desired total orientation workspace is taken as the constraint. If (20) is nonnegative, the constraint is satisfied. If (20) is negative, the constraint is not satisfied.Calculate the maximal and minimal leg lengths by the hybrid method proposed in this paper, and then calculate the stroke length. The specific process is shown in Figure6.


## 5. The Example Application

### 5.1. The Desired Total Orientation Workspace and Mechanism Configuration

The proposed optimization method is used to optimize the ship motion simulation platform. The ship moves in six DOF, which includes heave (alongz axis), sway (along y axis), surge (along x axis), roll (around x axis), pitch (around y axis), and yaw (around z axis). The amplitudes of roll, pitch, and heave are larger compared to yaw, sway, and surge and have great influence on ship navigation. Therefore, the desired total orientation workspace of ship movement is set as shown in Table 1.Table 1
The desired total orientation workspace.
x axis y axis z axis Translation Amplitude(mm) ±35 ±35 ±50 Rotation Amplitude(°) ±10 ±8 ±5The Stewart parallel manipulator is chosen as the ship motion simulation platform, whose joint position is shown in Figure7,.where a symmetrical hexagon is constructed by the pointAi, which is inscribed in a circle with the radius of Ra around the pointOa.la denotes the length of the shorter edge of the symmetrical hexagon andθa denotes the central angle corresponding to the short sidela. Also, a symmetrical hexagon is constructed by the pointBi, which is inscribed in a circle with the radius ofRb around pointOb. lb denotes the length of the shorter edge of the symmetrical hexagon andθb denotes the central angle corresponding to the short sideslb.Figure 7
Positions of the joints on the base and moving platform.The central anglesθa andθb can be obtained by (23) and (24) as follows:(23)θa=2arcsin⁡0.5laRa(24)θb=2arcsin⁡0.5lbRbThe vectors ai and bi can be obtained by (25) and (26) as follows:(25)ai=Racos⁡ηiRasin⁡ηi0(26)bi=Rbcos⁡φiRbsin⁡φi0where(27)φi=iπ3-θb2,i=1,3,5φi=i-1π3+θb2,i=2,4,6ηi=i-1π3+θa2,i=1,3,5ηi=iπ3-θa2,i=2,4,6Therefore, the geometric parameters of Stewart parallel manipulator can be determined byRa,Rb, andH.


### 5.2. The Cost Function

R a,Rb, andH are chosen as the design variables of the cost function.The volume of the Stewart parallel manipulator and the DWCI are chosen as targets of the optimization algorithm. The volume of the Stewart parallel manipulator can be got by (28) as follows:(28)V=Sh=πRa2hThe small length of short side leads to the good dexterity of the Stewart parallel manipulator [25]. To ensure the manipulator performs better without the interference of universal (spherical) joints installation, the short side lengths are set as 150mm. Hydraulic cylinders are chosen as the actuator, and the dead length ld of the hydraulic cylinder is 500mm. The orientation step length is set as Sα=Sβ=Sγ=1°.The cost function can be expressed as(29)MinimizeVRaRbH,-kRaRbHSubjectto300≤Ra≤700mm;Subjectto250≤Rb≤700mm;Subjectto600≤H≤900mm;SubjecttoRa≥Rb;Subjecttola=lb=170mm;SubjecttoSα=Sβ=Sγ=1°;Subjecttold=500mm;SubjecttoΩt=x,y,z,α,β,γ∣xmin≤x≤xmax,ymin≤y≤ymaxzmin≤z≤zmax,αmin≤α≤αmaxβmin≤β≤βmax,γmin≤γ≤γmaxSubjectto∣xmin≤x≤xmax,ymin≤y≤ymaxzmin≤z≤zmax,αmin≤α≤αmaxβmin≤β≤βmax,γmin≤γ≤γmax;Subjecttog=lsmax-lsmin≥0.


### 5.3. Optimization Results

All the parameters of NSGA-II are presented in Table2.Table 2
The parameters of NSGA-II.
parameters values Population size 50 Number of iterations 300 Crossover probability 0.9 Mutation probability 1/3 Distribution index for SBX 20 Distribution index for polynomial mutation 100The Pareto front, which is the optimization solutions including 50 sets of geometric parameters, is shown in Figure8.V denotes the volume of the Stewart parallel manipulator, andk denotes the DWCI. The total orientation workspace of the Stewart parallel manipulator of each set of geometric parameters contains the desired total orientation workspace represented by Table 1. Known from Figure 8, the large volume of manipulator V leads to the high value of the DWCI k. The optimal set of geometric parameters can be selected among the 50 sets of geometric parameters according to the requirements of manipulability, stiffness, or power.Figure 8
The optimization solutions.A set of geometric parameters is selected to verify the hybrid method proposed in this paper, as shown in Table3.Table 3
The set of geometric parameters for verifying the hybrid method.
Design variables Targets R a R b H V -k 522.71mm 322.91mm 853.95mm 7.33×108mm3 -0.25917The numerical method and the hybrid method are, respectively, used to obtain the maximal and minimal lengths of 3 legs, and the results are shown in Table4. It can be known that the small step length leads to the high precision and long time. The computer runs Windows 7 operating system and has the Intel(R)Core(TM) i7-4770CPU@3.4GHz processor and a 8.00GB memory. Compared with the numerical method, the hybrid method is more accurate and efficient, which is suitable for the optimal calculation.Table 4
The maximal and minimal length of 3 legs.
Step length Numerical method Hybrid method 5mm, 1° 2mm, 1° 1° Maximum(mm) 1041.0121 1041.0121 1041.0121 pose (35,-35,50,5,8,10) (35,-35,50,5,8,10) (35,-35,50,5,8,10) Minimum(mm) 796.3186 796.3167 796.3160 pose (-20,-35,-50,-5,-8,-10) (-23,-35,-50,-5,-8,-10) (-22.01,-35,-50,-5,-8,-10) time/s 457.47 8425.23 0.33The total orientation workspace defined by the set of geometric parameters of the Stewart parallel manipulator and the desired total orientation workspace shown in Table1 are shown in Figure 9 [26]. It can be observed that the total orientation workspace contains the desired total orientation workspace.Figure 9
The desired total orientation workspace and total orientation workspace.


## 5.1. The Desired Total Orientation Workspace and Mechanism Configuration

The proposed optimization method is used to optimize the ship motion simulation platform. The ship moves in six DOF, which includes heave (alongz axis), sway (along y axis), surge (along x axis), roll (around x axis), pitch (around y axis), and yaw (around z axis). The amplitudes of roll, pitch, and heave are larger compared to yaw, sway, and surge and have great influence on ship navigation. Therefore, the desired total orientation workspace of ship movement is set as shown in Table 1.Table 1
The desired total orientation workspace.
x axis y axis z axis Translation Amplitude(mm) ±35 ±35 ±50 Rotation Amplitude(°) ±10 ±8 ±5The Stewart parallel manipulator is chosen as the ship motion simulation platform, whose joint position is shown in Figure7,.where a symmetrical hexagon is constructed by the pointAi, which is inscribed in a circle with the radius of Ra around the pointOa.la denotes the length of the shorter edge of the symmetrical hexagon andθa denotes the central angle corresponding to the short sidela. Also, a symmetrical hexagon is constructed by the pointBi, which is inscribed in a circle with the radius ofRb around pointOb. lb denotes the length of the shorter edge of the symmetrical hexagon andθb denotes the central angle corresponding to the short sideslb.Figure 7
Positions of the joints on the base and moving platform.The central anglesθa andθb can be obtained by (23) and (24) as follows:(23)θa=2arcsin⁡0.5laRa(24)θb=2arcsin⁡0.5lbRbThe vectors ai and bi can be obtained by (25) and (26) as follows:(25)ai=Racos⁡ηiRasin⁡ηi0(26)bi=Rbcos⁡φiRbsin⁡φi0where(27)φi=iπ3-θb2,i=1,3,5φi=i-1π3+θb2,i=2,4,6ηi=i-1π3+θa2,i=1,3,5ηi=iπ3-θa2,i=2,4,6Therefore, the geometric parameters of Stewart parallel manipulator can be determined byRa,Rb, andH.


## 5.2. The Cost Function

R a,Rb, andH are chosen as the design variables of the cost function.The volume of the Stewart parallel manipulator and the DWCI are chosen as targets of the optimization algorithm. The volume of the Stewart parallel manipulator can be got by (28) as follows:(28)V=Sh=πRa2hThe small length of short side leads to the good dexterity of the Stewart parallel manipulator [25]. To ensure the manipulator performs better without the interference of universal (spherical) joints installation, the short side lengths are set as 150mm. Hydraulic cylinders are chosen as the actuator, and the dead length ld of the hydraulic cylinder is 500mm. The orientation step length is set as Sα=Sβ=Sγ=1°.The cost function can be expressed as(29)MinimizeVRaRbH,-kRaRbHSubjectto300≤Ra≤700mm;Subjectto250≤Rb≤700mm;Subjectto600≤H≤900mm;SubjecttoRa≥Rb;Subjecttola=lb=170mm;SubjecttoSα=Sβ=Sγ=1°;Subjecttold=500mm;SubjecttoΩt=x,y,z,α,β,γ∣xmin≤x≤xmax,ymin≤y≤ymaxzmin≤z≤zmax,αmin≤α≤αmaxβmin≤β≤βmax,γmin≤γ≤γmaxSubjectto∣xmin≤x≤xmax,ymin≤y≤ymaxzmin≤z≤zmax,αmin≤α≤αmaxβmin≤β≤βmax,γmin≤γ≤γmax;Subjecttog=lsmax-lsmin≥0.


## 5.3. Optimization Results

All the parameters of NSGA-II are presented in Table2.Table 2
The parameters of NSGA-II.
parameters values Population size 50 Number of iterations 300 Crossover probability 0.9 Mutation probability 1/3 Distribution index for SBX 20 Distribution index for polynomial mutation 100The Pareto front, which is the optimization solutions including 50 sets of geometric parameters, is shown in Figure8.V denotes the volume of the Stewart parallel manipulator, andk denotes the DWCI. The total orientation workspace of the Stewart parallel manipulator of each set of geometric parameters contains the desired total orientation workspace represented by Table 1. Known from Figure 8, the large volume of manipulator V leads to the high value of the DWCI k. The optimal set of geometric parameters can be selected among the 50 sets of geometric parameters according to the requirements of manipulability, stiffness, or power.Figure 8
The optimization solutions.A set of geometric parameters is selected to verify the hybrid method proposed in this paper, as shown in Table3.Table 3
The set of geometric parameters for verifying the hybrid method.
Design variables Targets R a R b H V -k 522.71mm 322.91mm 853.95mm 7.33×108mm3 -0.25917The numerical method and the hybrid method are, respectively, used to obtain the maximal and minimal lengths of 3 legs, and the results are shown in Table4. It can be known that the small step length leads to the high precision and long time. The computer runs Windows 7 operating system and has the Intel(R)Core(TM) i7-4770CPU@3.4GHz processor and a 8.00GB memory. Compared with the numerical method, the hybrid method is more accurate and efficient, which is suitable for the optimal calculation.Table 4
The maximal and minimal length of 3 legs.
Step length Numerical method Hybrid method 5mm, 1° 2mm, 1° 1° Maximum(mm) 1041.0121 1041.0121 1041.0121 pose (35,-35,50,5,8,10) (35,-35,50,5,8,10) (35,-35,50,5,8,10) Minimum(mm) 796.3186 796.3167 796.3160 pose (-20,-35,-50,-5,-8,-10) (-23,-35,-50,-5,-8,-10) (-22.01,-35,-50,-5,-8,-10) time/s 457.47 8425.23 0.33The total orientation workspace defined by the set of geometric parameters of the Stewart parallel manipulator and the desired total orientation workspace shown in Table1 are shown in Figure 9 [26]. It can be observed that the total orientation workspace contains the desired total orientation workspace.Figure 9
The desired total orientation workspace and total orientation workspace.


## 6. Conclusions

A hybrid method for solving the maximal and minimal leg lengths of 6-DOF parallel manipulator with desired total orientation workspace is proposed. The maximal and minimal leg lengths are obtained by the hybrid method, which consists of the numerical method used for orientation variables and algebraic method used for translation variables. The extremal points can be obtained by the hybrid method based on the three-element function without solving six-element function. And compared with the numerical method, it is more accurate and efficient.The actuator stroke length is obtained to be the constraints, which is calculated by the maximal and minimal length of the legs, which enlarges the optimal range without the predefined maximal and minimal length of the legs.Aiming at the physical size of the parallel manipulator and the proposed DWCI, the optimization of the geometric parameters of the parallel manipulator is conducted based on the multiobjective optimization algorithm (NSGA-II), which is subject to the actuator stroke length. Stewart platform is set as the example; the geometric parameters of the platform whose workspace contains the desired total orientation workspace are optimized and the hybrid method is proved to be more accurate and efficient compared to the traditional numerical method. This method provides the optimization guidance for engineering designers to design the parallel manipulator for desired total orientation workspace.



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