# Kinematics Comparative Study of Two Overconstrained Parallel Manipulators

Authors: Qiang Yan; Bin Li; Yangmin Li; Xinhua Zhao
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

A comparison study of kinematics characteristics of two overconstrained 2-RPU&SPR parallel manipulators (PMs) is introduced in this paper. The two 2-RPU&SPR PMs have the same kinematics properties in terms of one translational degree of freedom (DOF) and two rotational DOFs kinematics outputs. But there are some differences between the two PMs as far as joints distribution is concerned, leading to the differences in respect of workspace and dexterity of the two PMs. Firstly, based on screw theory, the structural characteristics and DOFs of the two PMs are analyzed. Secondly, the inverse and forward displacements problems for the two PMs are formulated by analytic formulae. Some numerical examples are simulated by software. Thirdly, based on algorithm for the direct displacement solution, the workspace characteristics of the two PMs are analyzed and compared. Then, the Jacobian matrices of the mechanisms are formulated. Based on the Jacobian matrices, the dexterities of the two PMs are established and compared. Finally, according to the comparisons of the properties between the two PMs, some useful conclusions are provided.


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## Body

## 1. Introduction

In recent years, lower-mobility PMs have played a more important role in both services and industrial applications. Compared to 6-DOF parallel manipulators, the lower-mobility structures have simpler mechanical design, lower manufacturing cost, and larger workspace. In terms of characteristics of DOFs, the lower-mobility PMs can be classified into three categories, namely, pure translational PMs, pure rotational PMs, and mixed PMs. Compared to the top two types PMs, the mixed PMs have both translational DOFs and rotational DOFs, and this type of PMs displays good potential for future applications in the field of motion simulator or a mixed orientating/positioning tool. Many scholars have done some research on this type of PMs and developed their own design.In 1983, Hunt [1] developed a 3-RPS PM, which was a classical mixed PM, and the related research work about the 3-RPS PM could be found in many literatures. The kinematic analyses concerning positon analysis, synthesis, dynamic modeling, workspace analysis, instantaneous motions analysis, and so forth can be found in the literature [2–8]. According to changeable layout angle of actuators Li and Xu [9] developed a parallel manipulator, which is called 3-PRS PM, and the kinematic performances of the mechanism are analyzed. Liu et al. [10–12] presented a type of new 3-DOF PM which has two translational freedoms and one rotational freedom with three nonidentical chains. The inverse and forward kinematics problems and singularity problems for this type PMs are presented. Recently, Li et al. [13] proposed a novel overconstrained 2-RPU&SPR PM, and the kinematic performance of the PM is analyzed. Yang [14] developed a family of 1T2R PMs with three limbs in his monograph. Qi et al. [15] discussed the kinematic optimum design of the 3-PUS/PU with 1T2R DOFs PM. Apart from the mixed 3-DOF PMs mentioned above, some other similar 3-DOF structures can be seen in the literature [16–19].The comparison study of two overconstrained 2-RPU&SPR parallel manipulators, which have owned one translational DOF and two rotational DOFs, is analyzed in this paper. The content is disposed in the following structure. Firstly, based on screw theory, the structural characteristics and DOFs of the two PMs are described in Sections 2 and 3, respectively. Some numerical examples are depicted in Section 4. In Section 5, the workspace concerning the two PMs is acquired by using the algorithm for the direct displacement solution, and some conclusions are concluded. The general Jacobian matrix for the two PMs is obtained, and the dexterity comparison is analyzed in Section 6. According to the analysis of the platform motion performances and based on the two structures, some conclusions and future studies are drawn in Section 7.


## 2. Structure Characteristics of the Two Overconstrained 2-RP_U&SP_R Manipulators

Type one 2-RPU&SPR PM involves a static plate and a mobile platform which is connected by two same RPU limbs and one SPR limb as shown in Figure 1. Here, R, U, and S are the abbreviation of the revolute, universal, and spherical joints. The driving prismatic joints are P which are underlined. Placing the reference frame B-xyz on the base, similarly, the mobile frame A-uvw is set on the mobile plate. Points B and A are middle points of lines B1B2 and A1A3, respectively. The axes of x and u are separately parallel to B1B2 and A1A2, the axes of z and w are perpendicular to the base platform and the moving platform, respectively. Bi (i=1,2) and A3 are the intersection of the axes of the revolute joints and actuated prismatic joints, B3 is the middle point of the spherical joint, and the centroid of the universal joints is A1 and A2 separately. It should be extraordinarily obviously noted that, for type one 2-RPU&SPR PM, the universal joints of the RPU limbs which are connected on the moving platform are shared.Figure 1
Type one 2-RPU&SPR PM.Here, type two 2-RPU&SPR PM has two different universal joints for RPU limbs relative to type one 2-RPU&SPR PM. The value of the distance concerning the two universal joints is A1A2 in Figure 2.Figure 2
Type two 2-RPU&SPR PM.The direction matrix of theA-uvw relative to the B-xyz frame can be found by using three Euler angles ψ, ϕ, and θ, and the three parameters are satisfying x-z-y conventions conditions:(1)R=cθcϕsψsθ-cψcθsϕcψsθ+sψcθsϕsϕcψcϕ-sψcϕ-sθcϕsψcθ+cψsθsϕcψcθ-sψsθsϕ=uxvxwxuyvywyuzvzwz=uvw.Here, the unit vectors of the frameA-uvw relative to the frame B-xyz are u, v, and w; “s” and “c” are the abbreviations of sin and cos.


## 3. Mobility Analysis of the Two Mechanisms via Screw Theory

Reciprocal screw theory is used for analyzing constraint forces, which is exerted on the mobile plate, in this paper. As for a parallel manipulator, the reciprocal screws associated with a serial limb stand for the constraints of wrench. These wrench constraints exerted on the mobile platform are realized through serial limbs. Using the properties of the reciprocal screw theory, the outcomes of mobility analysis concerning the two type 2-RPU&SPR PMs will not depend on the change of coordinate system.Taking into account the versatility, each limb is set for a local coordinate systemAi-xiyizi, i=1, 2, 3 in Figure 3. The limb twist system of the RPU limb in Figure 3(a) is given by(2)$1=100000,$2=010000,$3=0000-sin⁡α-cos⁡α,$4=1000-q12cos⁡αq12sin⁡α,where qi=AiBi, i=1,2,3.Figure 3
Twist systems of RPU limb and SPR limb.
(a)
RPU limb (b)
SPR limbConsidering the reciprocity between twist and wrench, the constraint system for the RPU limb can be described as(3)$1r=100000,$2r=000001.Here,$1r denotes a constraint force, it is passing through the centroid of universal joint and paralleling to the axis of joint screw $1, and $2r is the abbreviation of constraint couple. The direction of the constraint couple is norm to the axes of joint screws $1 and $2 which are passing through the centroid of the joints.In Figure3(b), the limb twist system concerning the SPR limb can be described as follows: (4)$1=100000,$2=0000sin⁡β-cos⁡β,$3=0sin⁡β-cos⁡β000,$4=1000-q3cos⁡β-q3sin⁡β,$5=0-cos⁡β-sin⁡β-q300.The constraint screw concerning the limb of SPR can be described: (5)$r=1000-q3cos⁡β-q3sin⁡β.The system in (5) demonstrates a constraint force passing through the spherical joint center and it is parallel to the axis of $1 in each configuration.Using (3) and (5), we can know that two RPU limbs constrain a translational DOF of the mobile plate. The direction is the same as the constraint force. Meanwhile, it restricts a rotational DOF and the direction is the same as the constraint couple. Similarly, the SPR restricts a translational DOF of the mobile platform along the direction of the constraint force. So, the two types 2-RPU&SPR PM have the same DOFs, which include one translational DOF and two rotational DOFs.


## 4. Inverse and Forward Position Analyses

### 4.1. Inverse Position Analysis

Inverse position analysis of the two PMs involves the determination of the limb lengths given the pose of the mobile platform.The vectorr=xyzT which represents point A in the B-xyz can be described by using the loop closure equation for both type 2-RPU&SPR PMs.(6)r=bi+qiwi+AiA0→-aii=1,2,r=bi+qiwi-aii=3.Here, ai=Rai0, the length and unit vector of limb i are qi, and wi respectively, and the position vectors of Ai are a10=a1-k00T, a20=a2k00T, and a30=a30-10T, respectively, which is measuring inA-uvw. The position vectors of Bi are b1=b1-100T, b2=b2100T, and b3=b3010T which are measured inB-xyz.ai and bi are the values of distance concerning AAi and BBi,  a1=a2=a3=a, b1=b2=b. For type one 2-RPU&SPR PM, k=0. For type two 2-RPU&SPR PM, k=1.The constraint concerning the RPU limb is imposed by the revolute joint; it restricts the vectors of wi and bi; meanwhile, it is normal to the unit vector c which represents the axis of the revolute joint. So, on both sides of  (6), making the dot product withc yields(7)r+aiTc=0,i=1,2.Here, c=010T.In the same way, the constraint of the SPR limb which is exerted on the revolute joint restricts the vectors of w3 and a3; meanwhile, it is perpendicular to the vector c3 which is the axis of revolute joint. So, on both sides of (6), making the dot product with c3 yields(8)r-biTc3=0,i=3.Here, c3=Rc30, c30=100T.Based on the geometric constraint of the two PMs, as shown in Figure2, the axis of the jointR2 can just only move in the plane Bxz, we can get another equation, and it can be listed as follows: (9)uy=0.Through (7), (8), and (9), the equation which concerns the two PMs can be listed as follows: (10)ϕ=0,(11)y=acos⁡ψ,(12)x=ztan⁡θ.Here,ϕ, y, and x stand for the parasitic motions which is relative to the mobile plate. The parasitic motions concerning the two PMs can contribute to perplex kinematics, require real-time compensation, increase difficulty in calibration, and even lead to damage. Given a set of unconstrained variables ψθz, the parasitic motion of the two PMs can be computed by using (10), (11), and (12). Then, the value of length concerning the limb can be listed as follows:(13)qi=r-bi+ai,i=1,2,3.


### 4.2. Forward Position Analysis of the Two Mechanisms

The forward position analysis concerning the two PMs involves the determination of the pose of the mobile plate. Firstly, on both sides of (13), making Euclidean norm and taking some necessary addition or subtraction, the simplified equations can be presented as follows:(14)q12=ztan⁡θ-tcos⁡θ-asin⁡ψsin⁡θ+b2+z+tsin⁡θ-asin⁡ψcos⁡θ2,(15)q22=ztan⁡θ+tcos⁡θ-asin⁡ψsin⁡θ-b2+z-tsin⁡θ-asin⁡ψcos⁡θ2,(16)q32=ztan⁡θ+asin⁡ψsin⁡θ2+2acos⁡ψ-b32+z+asin⁡ψcos⁡θ2.( 1 ) When q1≠q2, through subtraction and addition with respect to (14) and (15), simplifying the equations, we can acquire(17)z=f1ctan⁡θ+asin⁡ψcos⁡θ,(18)q12+q22=2z2cos⁡θ2-4azsin⁡ψcos⁡θ+2a2sin⁡ψ2+2b2+2t2-4btcos⁡θ.Here,f1=q12-q22/4b.Then, substituting (17) into (18) and (16) gives(19)4btcos3⁡θ-mcos2⁡θ+4btcos⁡θ=2f12-m,(20)q32=f12sin⁡θ2+4af1sin⁡ψsin⁡θ-4ab3cos⁡ψ+4a2+b32,where m=q12+q22-2t2-2b2.( 2 ) When q1=q2, doing some subtractions with respect to (14) and (15), the equations can be simplified as follows:(21)θ=0.Meanwhile, (14) and (16) can be rearranged as follows:(22)z=q12-b-t2+asin⁡ψ,(23)q32=2acos⁡ψ-b32+z+asin⁡ψ2.Substituting (22) into (23) yields(24)q32=4a2+q12-b-t2+b32-4ab3cos⁡ψ+4asin⁡ψq12-b-t2.Accordingly, given a set of valuesq1q2q3, when q1≠q2, through using (19) and (22), the values of θ and ψ can be computed. So, the direction matrix of R can be determined; through (12), (11), and (17), the values of x,y, and z can be gotten. When q1=q2, through (21) and (24), the values of θ and ψ can be computed. Similarly, the direction matrix of R can be acquired; in the same way, through (12), (11), and (22), the values ofx, y, and z can be obtained.So the analyses of inverse and the forward position concerning the two PMs can be computed directly from Sections4.1 and 4.2; meanwhile, the parasitic motions concerning the mobile plate can be described in a simplified way, which is enormously significant for mechanisms.


### 4.3. Numerical Examples

The concerned parameters of the two PMs are defined as follows:a=100 mm, b=300 mm,and b3=500 mm. Based on the inverse position analysis relative to type one, given four different groups of inputs ψθz, through computing, the corresponding parameters are listed in Table 1; meanwhile, the corresponding configurations relative to the computed solutions are described in Figure 4. In the same way, as for the forward position analysis, given two different groups of inputs q1q2q3, output parameters are computed and listed in Table 2; the corresponding configurations relative to the solutions are described in Figure 5. Similarly, given four different groups of inputs ψθz to type two PMs, the corresponding results are computed and described in Table 3, and the corresponding poses relative to results are described in Figure 6. Given two different groups of inputs q1q2q3 to forward position analysis, the output parameters concerning type two PMs are listed in Table 4, and the corresponding configurations concerning the computed solutions are depicted in Figure 7.Table 1
Inverse position analysis for type one mechanism.
Inputs Outputs ψ   d e g . θ   d e g . z   m m q 1   m m q 2   m m q 3   m m x   m m y   m m Case (a) 25 35 700 1014.5651 685.7525 951.7624 490.1453 90.6308 Case (b) −25 35 700 1096.7629 765.2621 872.5787 490.1453 90.6308 Case (c) 25 −35 700 685.7525 1014.5651 951.7624 −490.1453 90.6308 Case (d) −25 −35 700 765.2621 1096.7629 872.5787 −490.1453 90.6308Table 2
Forward position analysis for type one mechanism.
Inputs Outputs q 1   m m q 2   m m q 3   m m ψ   d e g . θ   d e g . x   m m y   m m z   m m Case (a) 653 816 941 61.0059 −17.1845 −225.3812 48.4719 728.7883 Case (b) 816 653 941 61.0059 17.1845 225.3812 48.4719 728.7883Table 3
Inverse position analysis for type one mechanism.
Inputs Outputs ψ   d e g . θ   d e g . z   m m q 1   m m q 2   m m q 3   m m x   m m y   m m Case (a) 25 35 700 995.0845 727.6026 957.0014 490.1453 90.6308 Case (b) - 25 35 700 1078.8 802.9790 878.2902 490.1453 90.6308 Case (c) 25 - 35 700 656.5877 1043.3 957.0014 - 490.1453 90.6308 Case (d) - 25 - 35 700 739.2408 1123.4 872.5787 - 490.1453 90.6308Table 4
Forward position analysis for type one mechanism.
Inputs Outputs q 1   m m q 2   m m q 3   m m ψ   d e g . θ   d e g . x   m m y   m m z   m m Case (a) 653 816 941 50.4875 16.3288 −212.3484 97.5455 −296.9190 Case (b) 816 653 941 50.4875 16.3288 186.7299 97.5455 261.0975Figure 4
Different configuration solutions for inverse position analysis of type one 2-RPU&SPR PM.
(a) (b) (c) (d)Figure 5
Different configuration solutions for inverse position analysis of type one 2-RPU&SPR PM.
(a) (b)Figure 6
Different configuration solutions for inverse position analysis of type two 2-RPU&SPR PM.
(a) (b) (c) (d)Figure 7
Different configuration solutions for forward position analysis of type two 2-RPU&SPR PM.
(a) (b)


## 4.1. Inverse Position Analysis

Inverse position analysis of the two PMs involves the determination of the limb lengths given the pose of the mobile platform.The vectorr=xyzT which represents point A in the B-xyz can be described by using the loop closure equation for both type 2-RPU&SPR PMs.(6)r=bi+qiwi+AiA0→-aii=1,2,r=bi+qiwi-aii=3.Here, ai=Rai0, the length and unit vector of limb i are qi, and wi respectively, and the position vectors of Ai are a10=a1-k00T, a20=a2k00T, and a30=a30-10T, respectively, which is measuring inA-uvw. The position vectors of Bi are b1=b1-100T, b2=b2100T, and b3=b3010T which are measured inB-xyz.ai and bi are the values of distance concerning AAi and BBi,  a1=a2=a3=a, b1=b2=b. For type one 2-RPU&SPR PM, k=0. For type two 2-RPU&SPR PM, k=1.The constraint concerning the RPU limb is imposed by the revolute joint; it restricts the vectors of wi and bi; meanwhile, it is normal to the unit vector c which represents the axis of the revolute joint. So, on both sides of  (6), making the dot product withc yields(7)r+aiTc=0,i=1,2.Here, c=010T.In the same way, the constraint of the SPR limb which is exerted on the revolute joint restricts the vectors of w3 and a3; meanwhile, it is perpendicular to the vector c3 which is the axis of revolute joint. So, on both sides of (6), making the dot product with c3 yields(8)r-biTc3=0,i=3.Here, c3=Rc30, c30=100T.Based on the geometric constraint of the two PMs, as shown in Figure2, the axis of the jointR2 can just only move in the plane Bxz, we can get another equation, and it can be listed as follows: (9)uy=0.Through (7), (8), and (9), the equation which concerns the two PMs can be listed as follows: (10)ϕ=0,(11)y=acos⁡ψ,(12)x=ztan⁡θ.Here,ϕ, y, and x stand for the parasitic motions which is relative to the mobile plate. The parasitic motions concerning the two PMs can contribute to perplex kinematics, require real-time compensation, increase difficulty in calibration, and even lead to damage. Given a set of unconstrained variables ψθz, the parasitic motion of the two PMs can be computed by using (10), (11), and (12). Then, the value of length concerning the limb can be listed as follows:(13)qi=r-bi+ai,i=1,2,3.


## 4.2. Forward Position Analysis of the Two Mechanisms

The forward position analysis concerning the two PMs involves the determination of the pose of the mobile plate. Firstly, on both sides of (13), making Euclidean norm and taking some necessary addition or subtraction, the simplified equations can be presented as follows:(14)q12=ztan⁡θ-tcos⁡θ-asin⁡ψsin⁡θ+b2+z+tsin⁡θ-asin⁡ψcos⁡θ2,(15)q22=ztan⁡θ+tcos⁡θ-asin⁡ψsin⁡θ-b2+z-tsin⁡θ-asin⁡ψcos⁡θ2,(16)q32=ztan⁡θ+asin⁡ψsin⁡θ2+2acos⁡ψ-b32+z+asin⁡ψcos⁡θ2.( 1 ) When q1≠q2, through subtraction and addition with respect to (14) and (15), simplifying the equations, we can acquire(17)z=f1ctan⁡θ+asin⁡ψcos⁡θ,(18)q12+q22=2z2cos⁡θ2-4azsin⁡ψcos⁡θ+2a2sin⁡ψ2+2b2+2t2-4btcos⁡θ.Here,f1=q12-q22/4b.Then, substituting (17) into (18) and (16) gives(19)4btcos3⁡θ-mcos2⁡θ+4btcos⁡θ=2f12-m,(20)q32=f12sin⁡θ2+4af1sin⁡ψsin⁡θ-4ab3cos⁡ψ+4a2+b32,where m=q12+q22-2t2-2b2.( 2 ) When q1=q2, doing some subtractions with respect to (14) and (15), the equations can be simplified as follows:(21)θ=0.Meanwhile, (14) and (16) can be rearranged as follows:(22)z=q12-b-t2+asin⁡ψ,(23)q32=2acos⁡ψ-b32+z+asin⁡ψ2.Substituting (22) into (23) yields(24)q32=4a2+q12-b-t2+b32-4ab3cos⁡ψ+4asin⁡ψq12-b-t2.Accordingly, given a set of valuesq1q2q3, when q1≠q2, through using (19) and (22), the values of θ and ψ can be computed. So, the direction matrix of R can be determined; through (12), (11), and (17), the values of x,y, and z can be gotten. When q1=q2, through (21) and (24), the values of θ and ψ can be computed. Similarly, the direction matrix of R can be acquired; in the same way, through (12), (11), and (22), the values ofx, y, and z can be obtained.So the analyses of inverse and the forward position concerning the two PMs can be computed directly from Sections4.1 and 4.2; meanwhile, the parasitic motions concerning the mobile plate can be described in a simplified way, which is enormously significant for mechanisms.


## 4.3. Numerical Examples

The concerned parameters of the two PMs are defined as follows:a=100 mm, b=300 mm,and b3=500 mm. Based on the inverse position analysis relative to type one, given four different groups of inputs ψθz, through computing, the corresponding parameters are listed in Table 1; meanwhile, the corresponding configurations relative to the computed solutions are described in Figure 4. In the same way, as for the forward position analysis, given two different groups of inputs q1q2q3, output parameters are computed and listed in Table 2; the corresponding configurations relative to the solutions are described in Figure 5. Similarly, given four different groups of inputs ψθz to type two PMs, the corresponding results are computed and described in Table 3, and the corresponding poses relative to results are described in Figure 6. Given two different groups of inputs q1q2q3 to forward position analysis, the output parameters concerning type two PMs are listed in Table 4, and the corresponding configurations concerning the computed solutions are depicted in Figure 7.Table 1
Inverse position analysis for type one mechanism.
Inputs Outputs ψ   d e g . θ   d e g . z   m m q 1   m m q 2   m m q 3   m m x   m m y   m m Case (a) 25 35 700 1014.5651 685.7525 951.7624 490.1453 90.6308 Case (b) −25 35 700 1096.7629 765.2621 872.5787 490.1453 90.6308 Case (c) 25 −35 700 685.7525 1014.5651 951.7624 −490.1453 90.6308 Case (d) −25 −35 700 765.2621 1096.7629 872.5787 −490.1453 90.6308Table 2
Forward position analysis for type one mechanism.
Inputs Outputs q 1   m m q 2   m m q 3   m m ψ   d e g . θ   d e g . x   m m y   m m z   m m Case (a) 653 816 941 61.0059 −17.1845 −225.3812 48.4719 728.7883 Case (b) 816 653 941 61.0059 17.1845 225.3812 48.4719 728.7883Table 3
Inverse position analysis for type one mechanism.
Inputs Outputs ψ   d e g . θ   d e g . z   m m q 1   m m q 2   m m q 3   m m x   m m y   m m Case (a) 25 35 700 995.0845 727.6026 957.0014 490.1453 90.6308 Case (b) - 25 35 700 1078.8 802.9790 878.2902 490.1453 90.6308 Case (c) 25 - 35 700 656.5877 1043.3 957.0014 - 490.1453 90.6308 Case (d) - 25 - 35 700 739.2408 1123.4 872.5787 - 490.1453 90.6308Table 4
Forward position analysis for type one mechanism.
Inputs Outputs q 1   m m q 2   m m q 3   m m ψ   d e g . θ   d e g . x   m m y   m m z   m m Case (a) 653 816 941 50.4875 16.3288 −212.3484 97.5455 −296.9190 Case (b) 816 653 941 50.4875 16.3288 186.7299 97.5455 261.0975Figure 4
Different configuration solutions for inverse position analysis of type one 2-RPU&SPR PM.
(a) (b) (c) (d)Figure 5
Different configuration solutions for inverse position analysis of type one 2-RPU&SPR PM.
(a) (b)Figure 6
Different configuration solutions for inverse position analysis of type two 2-RPU&SPR PM.
(a) (b) (c) (d)Figure 7
Different configuration solutions for forward position analysis of type two 2-RPU&SPR PM.
(a) (b)


## 5. The Workspace Analysis and Comparison of the Two Mechanisms

Given a group of limb lengthsq1q2q3, through corresponding equations in Section 4, the other parameters involving parasitic motions are computed directly. So when the restrictions to the limb lengths are set up, the parasitic motion workspace and the reachable workspace of the two mechanisms can be obtained directly. Accordingly, some comparisons concerning the two PMs are obtained.


### 5.1. Case Studies and Comparison of the Two Mechanisms

The concerning parameters of two PMs are defined as follows:a=100 mm, b=300 mm, b3=500 mm, t=100 mm, 600 mm ⩽q1⩽900 mm, 600 mm ⩽q2⩽900 mm, and 600 mm ⩽q3⩽900 mm. The workspace concerning two PMs is analyzed by software. In Figures 8 and 9, the workspace of parasitic motion concerning the two PMs is described. The reachable workspace along the directions of x-,y-, z-axes are depicted in Figures 10, 11, and 12. The positional workspace of the mobile plate is described in Figure 13.Figure 8
The distribution ofθ.
(a)
Type one 2-RPU&SPR PM (b)
Type two 2-RPU&SPR PMFigure 9
The distribution ofψ.
(a)
Type one 2-RPU&SPR PM (b)
Type two 2-RPU&SPR PMFigure 10
The reachable workspace inx-axis.
(a)
Type one 2-RPU&SPR PM (b)
Type two 2-RPU&SPR PMFigure 11
The reachable workspace iny-axis.
(a)
Type one 2-RPU&SPR PM (b)
Type two 2-RPU&SPR PMFigure 12
The reachable workspace inz-axis.
(a)
Type one 2-RPU&SPR PM (b)
Type two 2-RPU&SPR PMFigure 13
The reachable workspace of the PMs.
(a)
Type one 2-RPU&SPR PM (b)
Type two 2-RPU&SPR PMFrom Figures8 to 13, we can see that the reachable spaces of the two types of the 2-RPU&SPR PMs are similar in shape, but there exists some difference in size. For type one 2-RPU&SPR PM, the reachable workspace of the moving platform in the direction of x, y,andz is bigger than that of type one PM, and the volume of the reachable space of type one PM is bigger than that of type two.


## 5.1. Case Studies and Comparison of the Two Mechanisms

The concerning parameters of two PMs are defined as follows:a=100 mm, b=300 mm, b3=500 mm, t=100 mm, 600 mm ⩽q1⩽900 mm, 600 mm ⩽q2⩽900 mm, and 600 mm ⩽q3⩽900 mm. The workspace concerning two PMs is analyzed by software. In Figures 8 and 9, the workspace of parasitic motion concerning the two PMs is described. The reachable workspace along the directions of x-,y-, z-axes are depicted in Figures 10, 11, and 12. The positional workspace of the mobile plate is described in Figure 13.Figure 8
The distribution ofθ.
(a)
Type one 2-RPU&SPR PM (b)
Type two 2-RPU&SPR PMFigure 9
The distribution ofψ.
(a)
Type one 2-RPU&SPR PM (b)
Type two 2-RPU&SPR PMFigure 10
The reachable workspace inx-axis.
(a)
Type one 2-RPU&SPR PM (b)
Type two 2-RPU&SPR PMFigure 11
The reachable workspace iny-axis.
(a)
Type one 2-RPU&SPR PM (b)
Type two 2-RPU&SPR PMFigure 12
The reachable workspace inz-axis.
(a)
Type one 2-RPU&SPR PM (b)
Type two 2-RPU&SPR PMFigure 13
The reachable workspace of the PMs.
(a)
Type one 2-RPU&SPR PM (b)
Type two 2-RPU&SPR PMFrom Figures8 to 13, we can see that the reachable spaces of the two types of the 2-RPU&SPR PMs are similar in shape, but there exists some difference in size. For type one 2-RPU&SPR PM, the reachable workspace of the moving platform in the direction of x, y,andz is bigger than that of type one PM, and the volume of the reachable space of type one PM is bigger than that of type two.


## 6. The Dexterity Analysis and Comparison of the Two Mechanisms

The velocity of each limb can be gotten by differentiating (6), so the equations can be described as follows:(25)r˙=q˙iwi+ωi×qiwi-ω×aii=1,2,3.Here, the velocity of ith linear actuator is q˙i, the angular velocity of link AiBi is ωi, “×” represent the cross product between vectors, and r˙=x˙y˙z˙T and ω=ωxωyωzT stand for the linear and angular velocity of the mobile plate, respectively.Through dot multiplying on both sides of (25) with wi, the passive variables ωi are eliminated; this yields(26)q˙i=wir˙+ai×wi·ωi=1,2,3.If the two PMs are far from singularities, we can rewrite (26) in the matrix form:(27)q˙a=JaX˙.Here, q˙a=q˙1q˙2q˙3, X˙=r˙ω,  Ja=w1Ta1×w1Tw2Ta2×w2Tw3Ta3×w3T.The values of inverse velocity concerning the PMs can be computed by (27). Because the two PMs described in the paper possess only 3-DOF, we can obviously pay attention to the fact that the linear and angular velocity components of the mobile plate are not all independent.Through differentiating (10), (11), and (12) with respect to time, respectively, we can get the parasitic motions equations: (28)ϕ˙=0,(29)y˙=-asin⁡ψψ˙,(30)x˙=z˙tan⁡θ+zsec2⁡θθ˙.Through differentiating the direction matrix R concerning time, the equations can be gotten:(31)R˙=SωR,where Sω=0-ωzωyωz0-ωx-ωyωx0.Then, through (32) and using ψ˙ and θ˙, ω=ωxωyωzT are described: (32)ω=cos⁡θ001-sin⁡θ0ψ˙θ˙.Let x˙=ψ˙θ˙z˙T; the implied relations are described by using (29), (30), (31), and (32):(33)X˙=Jrx˙.Here Jr=0zsec2θtan⁡θ-asin⁡ψ00001cos⁡θ00010-sin⁡θ00.Through (27) and (33), the equations of relation can be obtained: (34)q˙=Jx˙,where J=JaJr is the Jacobian matrix of the mechanism, which includes the effect of the mechanical constrains on the mechanism. J is called the constrained Jacobian matrix of 2-RPU&SPR SPM.From (34), the Jacobian matrix of type one 2-RPU&SPR PM can be obtained as follows:(35)JT1=A11A12A13A21A23A24A31A32A33,where (36)A11=abcos⁡ψsin⁡θ-a2sin⁡ψcos⁡ψJ1,A12=zsec2⁡θasin⁡ψsin⁡θ-b+absin⁡ψcos⁡θJ1,A13=-b+asin⁡ψsin⁡θtan⁡θ+asin⁡φcos⁡θJ1,A21=-absin⁡θcos⁡ψ-a2sin⁡ψcos⁡ψJ2,A22=zsec2⁡θb+asin⁡ψsin⁡θ-absin⁡ψcos⁡θJ2,A23=tan⁡θb+asin⁡ψsin⁡θ+asin⁡ψcos⁡θJ2,A31=-2ab3sin⁡ψ+a2sin⁡ψcos⁡ψJ3,A32=-azsin⁡ψsin⁡θsec2⁡θJ3,A33=-asin⁡ψsin⁡θtan⁡θ-asin⁡ψcos⁡θJ3;J1=a2+b2-2absin⁡ψsin⁡θ,J2=a2+b2+2absin⁡ψsin⁡θ,J3=a2+b32-2ab3cos⁡ψ.The Jacobian matrix of type two 2-RPU&SPR PM can be calculated as follows:(37)JT2=M11M12M13M21M22M23M31M32M33,where(38)M11=-a2sin⁡ψcos⁡ψ+absin⁡θcos⁡ψD1,M12=z-b-acos⁡θ+asin⁡ψsin⁡θsec2⁡θ+absin⁡ψcos⁡θ+absin⁡θD1,M13=tan⁡θ-b-acos⁡θ+asin⁡ψsin⁡θ+asin⁡θ+acos⁡θsin⁡ψD1,M21=-a2sin⁡ψcos⁡ψ-absin⁡θcos⁡ψD2,M22=zsec2⁡θb+acos⁡θ+asin⁡ψsin⁡θ+absin⁡θ-abcos⁡θsin⁡ψD2,M23=tan⁡θb+acos⁡θ+asin⁡ψsin⁡θ-asin⁡θ+asin⁡ψcos⁡θD2,M31=a2sin⁡ψcos⁡ψD3,M32=-azsin⁡θsec2⁡θsin⁡ψD3,M33=-asin⁡θtan⁡θsin⁡ψ-asin⁡ψcos⁡θD3,D1=2a2+b2+2abcos⁡θ-2absin⁡θsin⁡ψ,D2=2a2+b2+2abcos⁡θ+2absin⁡θsin⁡ψ,D3=a2+b32-2ab3cos⁡ψ.From Figure14, we can see that the distributions of dexterity of the two types PMs are similar in shape, but there exists some difference. The dexterity of type one 2-RPU&SPR PM is better than that of type two PM.Figure 14
The dexterity of the PMs.
(a)
Type one 2-RPU&SPR PM (b)
Type two 2-RPU&SPR PM


## 7. Conclusions

This paper investigates the comparison study of two overconstrained 2-RPU&SPR PMs. The two 2-RPU&SPR PMs have identical kinematics limbs but different in joints distribution. The structure characteristics and DOFs of the two PMs are analyzed based on screw theory. Both of the inverse and forward displacements of the two PMs are derived by analytic formulae and some numerical examples are depicted. Unlike most parallel robots, these two overconstrained 2-RPU&SPR PMs have explicit solutions for inverse and forward kinematics issues. The reachable workspaces of the two mechanisms are developed based on the forward position analysis and comparison analysis is carried out. The Jacobian matrix of the two PMs is derived, and dexterity comparison studies are provided. The stiffness and dynamics comparison of the two overconstrained 2-RPU&SPR PMs will be studied in the future.



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*Source: 5091405-2016-10-30.xml*