A novel symmetrical 3-DOF PKM and its performance comparison with 3-PRS PKM①

Liu Yuzhe (劉宇哲), Wu Jun, Wang Liping, Zhao Jingshan, Wang Jinsong, Yu Guang

(*State Key Laboratory of Tribology and Institute of Manufacturing Engineering, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, P.R.China) (**Beijing Key Laboratory of Precision/Ultra-precision Manufacturing Equipments and Control, Beijing 100084, P.R.China)

A novel symmetrical 3-DOF PKM and its performance comparison with 3-PRS PKM①

Liu Yuzhe (劉宇哲)***, Wu Jun②***, Wang Liping***, Zhao Jingshan*, Wang Jinsong***, Yu Guang***

(*State Key Laboratory of Tribology and Institute of Manufacturing Engineering, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, P.R.China) (**Beijing Key Laboratory of Precision/Ultra-precision Manufacturing Equipments and Control, Beijing 100084, P.R.China)

A novel symmetrical 3-degree-of-freedom (DOF) parallel kinematic manipulator (PKM) is firstly presented, which is named 3-P (Qu) RU. According to the structure feature, a double closed loop vector method is proposed to investigate this PKM.Based on this method, kinematic, velocity and error models of this manipulator are established respectively. Since3-PRS PKM has been applied successfully in practice and its structure is similar to the 3-P (Qu) RU PKM, corresponding models of a 3-PRS PKM are given and a performance comparison study between them is investigated on workspace, manipulator dexterity, position error and error sensitivity. The comparison results reveal that the 3-P (Qu) RU PKM has the advantage on velocity performance and the disadvantage on accuracy performance. This novel 3-P (Qu) RU PKM is an available selection for a tool head of a hybrid machine tool and the analysis is greatly helpful for the further applications of this manipulator.

comparison study, parallel kinematic manipulator(PKM), kinematics, velocity, error

0 Introduction

It is well known that the parallel kinematic manipulator (PKM)has potential advantages in high speed,heavy load handling and high response rate. Its drawback is on the limited workspace. On the contrary, the serial kinematic manipulator (SKM) has the advantage of large workspace and the disadvantage of what PKM owns[1,2]. With an increasing need for manufacturing structural aircraft parts, hybrid machine tools, combining the advantages of PKMs and SKMs, are becoming a research hotspot and attracting more and more attentions from both academia and industry.The successful applications include ECOSPEED series machine centers developed by DS-Technologie[3], the Tricept series machine developed by PkmtriceptSL[4]and the TriVarianthybrid PKM proposed by Huang[5]. Usually as tool heads of hybrid machine tools, PKMs with different structures are investigated widely.Among a mass of PKMs, the 3-degree-of-freedom (DOF) PKMs are used most frequently and are most suitable due to their simple kinematics[6].

The structure of 3-DOF PKMs can be mainly divided into asymmetrical and symmetrical structures. Asymmetrical structures have been proposed and analyzed by a lot of researchers. A hybrid machine tool with a 3-DOF parallel module, TripteorX7, was proposed and its static and dynamic models were set up[7]. The proposed models predicted the behavior of the structure and avoided costly trials. A 2-UPS-PRP PKM was investigated and its error analysis revealed that some identified workspace either amplified or reduced the input errors[8]. In addition, future work about accuracy evaluation, static forces and kinematics were mentioned. A novel asymmetrical 3-DOF PKM with two translational and one rotational motions was designed[9]. This structure, which could rotate 360 degrees around theY-axis, owned a higher rotational ability than general PKMs. A 2-PRU-PRRU PKM was created and studied on the aspect of kinematics[10]. Asymmetrical 3-DOF PKMs with 2-PRU-1PRS and 2-PRU-1PUR were respectively proposed by Xie[11]. Both of them aimed at a better orientation and eliminating parasitic motions. Another novel asymmetrical 3-DOF PKM, named TAM, with purely translational motions was presented and compared with the SKM400PKM on the conditional number of the Jacobin matrix[12]. An asymmetrical 3-DOF spherical PKM was studied in a mobility analysis[13]. The work provided plenty of theoretical foundations for different asymmetrical structures. Here, R, P, U and S represent revolute, prismatic, universal and spherical joint respectively.

However, asymmetrical 3-DOF structures may lead to an isotropic motions that are not suitable as a tool head in practice. In the successful applications in industry, symmetrical 3-DOF structures are more suitably used as a tool head like the Z3 tool head with a symmetrical 3-PRS structure in ECOSPEED and the symmetrical 3-UPS structure in Tricept.However, it is found that Tricept with an A/C-axis (rotations about theX- andZ-axes) tool head may scratch the finished surface during rotating the cutter in a high speed. Z3 with an A/B-axis (rotations about theX- andY-axes)tool head avoids this problem. One can see that the symmetrical 3-PRS structure owns more advantages in the machining of aircraft structural parts.As far as the authors can be aware, vast literatures about kinematics and dynamics analysis[14], architecture optimization[15], dimension synthesis[16], and sensitivity analysis[17]were presented for the 3-PRS structure. However, few structure innovations based on the 3-PRS structure are presented. Although there are some structure innovations proposed such as in Ref.[18], these structures do not have a better kinematic performance than the 3-PRS structure and the structure innovations just rely on combinations of existing limbs or joints, which limits the further improvement on the comprehensive performances.In this paper, a novel 3-DOF symmetrical PKM,named 3-P (Qu) RU,derived from a 3-PRS structure is put forward. Qu joint is a kind of novel joint which is a planar quadrilateral construction with four revolute joints. This 3-P (Qu) RU PKM is analyzed in the aspects of workspace, velocity, position error and error sensitivity. So does the 3-PRS PKM. In order to show the performance of the 3-P (Qu) RUPKM clearly, a comparison study between the two PKMs is presented.

The remainder of the paper is organized as follows. The next section presents the descriptions and system modeling of a 3-P (Qu) RU PKM. The double closed loop vectors method is proposed to establish the system model. In Section 2, the 3-PRS PKM is introduced.In Section 3, in order to guarantee a fair comparison, a comparison rule is set up. Moreover, the evaluation indices are presented. In Section 4, numerical examples are utilized to show the comparison study.The conclusions are organized in Section 5.

1 Description and system modeling of 3-P (Qu) RU PKM

1.1 Description of configuration and calculation of DOF

A virtual prototype of the 3-P (Qu) RU PKM is shown in Fig.1. It is composed of a mobile platform, a base platform, and three supporting limbs with an identical kinematic structure. Each supporting limb is composed of onePjoint, one Qu joint, oneRjoint and oneUjoint. Qu joint is a novel joint, the target of which is to increase the stiffness of the manipulator. This joint is composed of fourRjoints and is a planar scissor like quadrilateral construction. The fourRjoints are installed at four vertices of the quadrilateral construction.

Fig.1 The virtual prototype of 3-P (Qu) RU PKM

In order to determine the output motion of this PKM, its DOF number should be obtained. The formula for calculating the DOF of a manipulator is expressed as

NDOF=6(n-g-1)+f

(1)

whereNDOFrepresents the number of DOF in the manipulator,nrepresents the number of links,grepresents the number of joints andfrepresents the total number of joint DOF.

As each limb is the same, Eq.(1) can be rewritten as

NDOF=6(3n′+2-3g′-1)+3k

(2)

wheren’ andg’ represent the corresponding link and joint number of each limb, number 2 represents the link number for the mobile platform and the base platform andkrepresents the total number of joint DOF of each limb.

It is easy to know thatn’is always 1 smaller thang’. For each limb,Pjoint provides one DOF, Qu joint provides one DOF.Rjoint provides one DOF andUjoint provides two DOFs. Hence,kis equal to 5. Finally, it is known thatNDOF=3. Hence, this PKM has 3 output motions. It should be pointed out that the 3-P (Qu) RU PKM has the 3 same main motions and 3 parasitism motions with 3-PRS PKM.

1.2 System modeling based on the double closed loop vectors method

The traditional single closed loop vector for one limb is not suited to this 3-P (Qu) RU PKM due to the complexity of its limb structure. More equations and variables should be added in its corresponding models. Hence, the double closed loop vectors method is proposed and applied in the following kinematic model, velocity model and error model.

1.2.1 Kinematic model of 3-P (Qu) RU PKM

The architecture of a 3-P (Qu) RU PKM is shown in Fig.2. A fixed Cartesian reference coordinate systemO-XYZis fixed at the center of the base platformB1B2B3. A moving coordinate systemN-UVWis fixed at the center of the mobile platformA1A2A3. For simplicity and without losing the generality, let theX-axis point along the direction of vectorOB1and theU-axis point along the vectorNA1.Z-axis andW-axis are respectively perpendicular to the base platform and the mobile platform. Both axisYand axisVsatisfy the right-hand rule.BiCion the base platform fori=1, 2 and 3 represents the guide rail of each limb. Each P joint moves alongBiCi.Giis the midpoint ofEiFi. Furthermore, all the joints attached to the base and mobile platform are symmetrically distributed at vertices of the equilateral triangles. It means that the angles on the mobile platform between vectorsNA1,NA2andNA3keep the same 2/3 π rad and the angles on the base platform between vectorsOB1,OB2andOB3keep the same 2/3 π rad no matter in 3-P (Qu) RU PKM or in 3-PRS PKM.

Fig.2 Architecture of 3-P (Qu) RU PKM

In order to simplify the orientation expression, T-T rotation angle is introduced to describe the orientation of the mobile platform in both PKMs. The rotational matrix is expressed as

Angleψrotates around the normal direction of the moving platform and it is zero in an ideal situation. Furthermore, the accuracy of angleψis not necessary to guarantee when the machine tool machines a part. Hence,ψcan be ignored and only two angles are kept in the rotational matrix. More information about the meaning of the rotational matrix can be found in Ref.[6].

According to the above analysis, each limb has the same kinematic characteristic. The same vector loop of each limb is drown in Fig.3. If only one closed loop vector of each limb is employed, the position ofEiFicannot be determined. Hence, a double closed loop vectors method (OBiCiFiGiAiNandOBiCiDiEiGiAiN) is introduced.

Fig.3 The double closed loop vectors of 3-P (Qu) RU PKM

A kinematic function according to the vector loopOBiCiFiGiAiNis written as

(3)

Another kinematic function according to the vector loopOBiCiDiEiGiAiNis written as

=Ril3i

(4)

where Rirepresents the rotation matrix of each limb.ai,bi, l1i, l2i, l3i, l4iand l5irepresent the position vectors ofNiAi,OiBi,CiDi,CiFi,DiEi,EiFiandGiAiin the coordinate systemOi-XiYiZi, respectively.H represents the position vector of the mobile platform.pirepresents the input vector of eachPjoint.

The kinematic model of 3-P (Qu) RU PKM is derived based on Eqs(3) and (4). The following analyses are all based on the double closed loop vectors.

According to Eqs(3)and (4), the kinematic model is rewritten as

(5)

=Ril3i

(6)

The kinematic constraint equations with the norm of both sides of Eqs(5) and (6) are expressed as

(7)

=l3

(8)

EiFiare fixed atGiAiand it is assumed thatGiAiare perpendicular toEiFi. The vectors ofEiFiandGiAiare expressed by one parameterωias:l4i=l4[cosωi0 sinωi]Tandl5i=l5[cos(ωi+0.5π) 0 sin(ωi+0.5π)]T.ωirepresents the orientations ofEiFiin theXiZiplane.

Consideringωiis unknown, more expressions are needed.Multiplying both sides of Eq.(3) by (Riey)Tleads to:

(Riey)T(H+RT-TRiai)=0

(9)

where ey=[0 1 0]T.

Based on Eqs(7)～(9), the kinematic model is expressed as:

fj(X, P, P*)=0

(10)

where X=[xyzφθψ]T, P=[p1p2p3]T, P*=[ω1ω2ω3]T,j=1～9.

The double closed loop vectors method is presented above and its kinematic model is given in Eq.(10). Compared with 9 variables in the kinematic model of 3-PRS PKM, there are 12 variables in this model due to its special structure feature. The traditional single closed loop vector method for solving 9 variables is not suited to this 3-P (Qu) RU PKM and the double closed loop vectors method proposed in this paper is suitable.In this method, the 12 variables can be divided into the output and input variables. The output variable is X containing 3 main output variables and 3 parasitism output variables. The input variable is P and P*. P can be seen as the 3 main input variables and P*can be seen as the 3 parasitism input variables. In the solving process, 3 variables are known and 9 variables are unknown. The 3 known variables correspond to the 3 main output in invers kinematics or the 3 main input in forward kinematics. Hence, no matter in invers kinematics or forward kinematics, 9 equations based on Eqs(7)～(9) are needed. Taking the forward kinematics as an example, it is to solve X while P is known and P*is unknown. In order to obtain the solutions, the classical Newton iterative method is used to solve Eq.(10) as follows:

Sk+1=Sk-[J(Sk)]-1F(Sk)

(11)

where Sk+1represents X and P*in the next iterative process; and Skis solved in the previous iterative process. J(Sk) is the derivatives function for forward kinematics and F(Sk) is the forward kinematics.

Starting with an initial estimate S0=[X0P*0]T, the iterative process will end once the following expression is satisfied:

Sk+1-Sk≤ε

(12)

where ε is a specified tolerance.

After that, X and P*with reference to the corresponding P are obtained.

1.2.2 Velocity model of 3-P (Qu) RU PKM

In order to obtain a velocity model, taking the derivative of Eqs(3) and (4) with respect to time, it leads to:

(13)

(14)

Multiplying both sides of Eq.(13) by (Rin2i)Tand multiplying both sides of Eq.(14) by (Rin3i)T, they can be simplified as

(15)

Eq.(15) is rewritten as a matrix expression:

(16)

In Eq.(16), the number of rows in J is not equal to the number of columns in J. J cannot directly get a matrix inversion of itself. In order to give a further velocity analysis, the matrix of velocity transfer J should be able to be inversed. As mentioned above, this PKM has 3 main motions and 3 parasitism motions. Hence, the velocity parameters of 3 parasitism output are needed to be eliminated based on Eq.(9). By taking the derivative of Eq.(9) with respect to time, the relation between the velocity vectors of 3 main motions and 3 parasitism motions is expressed as:

(17)

Eq.(17) is solved as:

(18)

Furthermore, the expressions can be obtained as

(19)

(20)

wheret1,t2,g1,g2,g3andg4are the corresponding coefficients.

Taking Eqs(18)～(20) into Eq.(15), it leads to:

(21)

Eq.(21)isrewrittenasasquarematrixexpression:

(22)

1.2.3Errormodelof3-P(Qu)RUPKM

Atfirst,Eqs(3)and(4)istakenwiththefirstorderperturbation:

H+ΔH+(E+Δα×)RT-T·Ri·(ai+Δai)

=Ri·(bi+Δbi)+Ri·(E+ΔηR1i×)[ΔR1i

+(l4i+Δl4i)]+Ri·(pi+Δpi)+Ri·(l5i+Δl5i)

+Ri·(E+ΔηUi×)ΔUi

(23)

H+ΔH+(E+Δα×)RT-T·Ri·(ai+Δai)

=Ri·(bi+Δbi)+Ri·(pi+Δpi)+Ri·(l1i+Δl1i)

+Ri·(E+ΔηR2i×)[ΔR2i+(l3i+Δl3i)]

+Ri·(l5i+Δl5i)+Ri·(E+ΔηUi×)ΔUi

(24)

where ΔH represents the position error vector of the output; Δα represents the orientation error vector of the output; Δairepresents the error vector of ai. Δbirepresents the error vector of bi. Δpirepresents the error vector of pi.Δlmirepresents the error vector of lmiform=1 to 5; ΔηR1i, ΔηR2i, ΔηR3iand ΔηR4irepresent the orientation error vector of fourRjoints of Qu joint. ΔR1i, ΔR2i, ΔR3iand ΔR4irepresent the position error vector of fourRjoints of Qu joint; ΔηUirepresents the orientation error vector ofUjoint; ΔUirepresents the position error vector ofUjoint. Furthermore, it should be pointed out that theRjoint connectingEiFiwithGiAican be ignored because the main error of theRjoint is the rotational motion around its normal direction and it cannot influence the error of the mobile platform.

With ignoring the second and higher order terms, Eq.(23) is subtracted from Eq.(3) and Eq.(24) is subtracted from Eq.(4):

ΔH+Δα×RT-TRiai

=-RT-TRiΔai+RiΔbi+RiΔpi+RiΔR1i+RiΔl2i

×l4i)+RiΔUi+RiΔl5i

(25)

ΔH+Δα×RT-TRiai

×l4i)+RiΔUi+RiΔl5i

(26)

Multiplying both sides of Eq.(25) by (Riey)Tand multiplying both sides of Eq.(26) by (Ril3i)T, they lead to:

(Riey)T·ΔH+[(RT-TRiai)×(Riey)]T·Δα=(Riey)T·(-RT-TRiΔai+RiΔbi+RiΔpi+RiΔR1i

+(Riey)T·(RiΔUi+RiΔl5i)

(27)

(Ril3i)TΔH+[(RT-TRiai)×(Ril3i)]TΔα=(Ril3i)T(-RT-TRiΔai+Ri(Δbi+Δpi+Δl1i+ΔR2i))

+RiΔl3i)

(28)

Then the expression of the error model based on Eqs(27) and (28) can be written as:

(29)

where ΔP represents a vector with all error parameters of 3-P (Qu) RU PKM.

2 System modeling of 3-PRS PKM

Since the 3-PRS PKM has been analyzed by several researches, its kinematic model, velocity model and error model are briefly introduced.Its architecture is presented in Fig.4. The 3-PRS PKM is also composed of a mobile platform, a base platform and 3 supporting limbs with the identical kinematics. The base platform is connected with the mobile platform by each limb containing one P joint, one R joint and one S joint. In addition, P joint on each limb is actuated. The 3-PRS PKM has the 3 same main motions and 3 parasitism motions with the 3-P (Qu) RU PKM.

Fig.4 Architecture of 3-PRS PKM

In order to compare the two PKMs in convenience, a same naming rule is employed to describe the 3-PRS PKM in Fig.4. A fixed Cartesian reference coordinate systemO-XYZis fixed at the center of the base platformB1B2B3. A moving Cartesian reference coordinate systemN-UVWis fixed at the center of the mobile platformA1A2A3. For simplicity and without losing the generality, let theX-axis point along the direction of vectorOB1and theU-axis point along vectorNA1.Z-axis andW-axis are respectively perpendicular to the base platform and the mobile platform. BothY-axis andV-axis satisfy the right-hand rule.BiCion the base platform fori=1, 2 and 3 represents the guide rail of each limb. The guide rail of each limb is perpendicular to the base platform. Each P joint moves alongBiCi.

Some expressions and introductions mentioned in the model of 3-P (Qu) RU PKM are not presented again in this section.

A kinematic model according to vector loopOBiCiAiNis written as:

H+RT-TRiai=Ribi+Ripi+Rili

(30)

By taking the derivative of Eq.(30) with respect to time, the velocity model is written as:

(31)

whereξirepresents the angle velocity vector ofliandnlirepresents the unit direction vector ofli.

Multiplying both sides of Eq.(31) by (nli)T, it leads to:

(32)

Eq.(32) can be rewritten in the same matrixformat with Eq.(22):

(33)

The error model of 3-PRS PKM is given directly:

(Rili)TΔH+[(RT-TRiai)×(Rili)]TΔα

=(Rili)T(-RT-TRiΔai+RiΔbi+RiΔpi)

+(Rili)T(RiΔRi+RiΔli+RiΔSi)

+(li×li)TΔηRi

(34)

(Riey)TΔH+[(RT-TRiai)×(Riey)]TΔα

=(Riey)T(-RT-TRiΔai+RiΔbi+RiΔpi)

+(Riey)T(RiΔRi+RiΔli+RiΔSi)

+(li×ey)TΔηRi

(35)

where Δlirepresents the error vector of li;ΔηRirepresents the orientation error vector ofRjoints; ΔRirepresents the position error vector ofRjoint; ΔSirepresents the position error vector ofSjoint.

Furthermore, the error model of 3-PRS PKM is also obtained in the same format with Eq.(29) as:

(36)

where ΔP represents a vector with all error parameters of 3-PRS PKM.

3 Performance evaluation indices

To be fair, the structure parameters of 3-P (Qu) RU PKM are given based on the structure parameters of 3-PRS PKM. Two rules are set up:

1) Keep aiand biof both PKMs in the samelengths;

2) Atφ=0 andθ=0, the structure parameters of both the two PKMs should satisfy the following expressions:

(37)

(38)

According to the above rules, all the structure parameters of 3-P (Qu) RU PKM can be obtained with the structure parameters of 3-PRS PKM and the given lengths of l1i, l4iand l5i. After calculating Eqs(37)and (38), the structure parameters of both PKMs are given in the following. In 3-PRS PKM,a=200mm,b=260mm andl=460mm. In 3-P (Qu) RU PKM,a=200mm,b=260mm,l1=240mm,l2=533mm,l3=293mm,l4=100mm andl5=50mm.

3.1 Workspace

In order to evaluate the workspace performance, an index is introduced.

λ(X)=|qxmax-qxmin|

(39)

where qxmaxrepresents an output point containing the maximum value along X direction. qxminrepresents an output point containing the minimum value along X direction. To put this conveniently,λ(X) is represented byDAXD(distance alongXdirection).

The same operations are performed alongYdirection andZdirection.DAYDandDAZDrespectively represent the distances alongYdirection andZdirection. The biggerDAXD,DAYDandDAZDare, the bigger workspace the PKM has.

3.2 Manipulator dexterity

The character of the velocity transfer matrix for a PKM is seen as the ability to change the positions and orientations of the mobile platform. One of the frequently used indices based on the velocity transfer matrix is called dexterity. A dexterity index for a given output point is 1/κ.κrepresents the condition number of the velocity transfer matrix for 3-P (Qu) RU PKM in Eq.(22) or for 3-PRS PKM in Eq.(33). The condition numberκranges from 1 to ∞. For a given output point whichκis equal to 1, it means that the dexterity of the PKM is the best and the PKM is isotropic. Furthermore,GDI(global dexterity index)is given to evaluate both PKMs in a given output range:

(40)

whereμrepresents a value ofGDI,Vdenotes a given output rang. The biggerμis, the better the velocity performance the PKM has.

3.3 Position error

Although PKMs own many merits such as high speed and heavy load handling, the accuracy performance should be paid attentions to if a PKM is used as a tool head. Position error of a PKM is an important evaluation index of the accuracy performance. A determination of the position errors for a given output point is put forward in the following:

ΩH=|ΔH|

(41)

whereΩHrepresents the position error. ΔH for 3-P (Qu) RU PKM is calculated by Eq.(29) and one for 3-PRS PKM is calculated by Eq.(36). The bigger the value ofΩHis, the worse accuracy the PKM has.

3.4 Error sensitivity

The position error is influenced by the values of the given structural errors, which cannot comprehensively reflect the accuracy performance of a PKM. Error sensitivity is employed to estimate the influence of all error parameters on the output error. A determination of a global position error sensitivity for a given output range is put forward in the following:

(42)

whereVdenotes a given output range.ΔHx,ΔHyandΔHzrespectively represent elements of the position output error vector.ΔPHrepresents the unit position error parameter of a joint.σHpresents the influence of a unit error of a joint position error parameter on the position error of the output error.

Furthermore, the position error sensitivity and orientation error sensitivity should be discussed respectively. The similar determination of a global orientation error sensitivity for a given output range is also proposed:

(43)

whereΔαx,ΔαyandΔαzrespectively represent elements of the orientation output error vector.ΔPαrepresents the unit orientation error parameter of a joint.σαpresents the influence of a unit error of a joint orientation error parameter on the orientation error of the output error.

The error sensitivity reflects the influence of a unit error of a joint on the output error. This index estimates the influence of each single joint error on the output error rather than the influence of all joints error together on the output error. The bigger the error sensitivity is, the worse accuracy the PKM has.

4 Numerical examples

Based on the above evaluation indices, numerical examples are employed to show specific performances between the 3-P (Qu) RU PKM and 3-PRS PKM on workspace, manipulator dexterity, position error and error sensitivity.

4.1 Workspace comparison

In this section, kinematic workspaces of both PKMs are obtained with the same input range. Both of the PKMs own the same motion boundary on workspace. Moreover, it is seen that workspaces of both PKMs are totally the same since the values ofDAXD,DAYDandDAZDare the same as 46mm, 37mm and 100mm, respectively. Collections of every dispersed output point for both PKMs are plotted in Fig.5. The comparison result reveals that this 3-P (Qu) RU PKM owns the same kinematic workspace with a 3-PRS PKM, which ensures a basic machining trajectory requirement for a tool head.

Fig.5 The workspace of both PKMs

4.2 Dexterity comparison

The dexterity index of each PKM is calculated at each output points in a given output range withZ=700mm,φ=0～2π andθ=0～π/4. In order to present the dexterity of each PKM comprehensively, the dexterity distributions respectively along three DOF directions are given. Their specific situations areZDOF varying in 600mm to 800mm withφ=0 andθ=π/6,φDOF varying in 0 to 2π withZ=700mm andθ=π/6,θDOF varying in 0 to π/4 withZ=700mm andφ=0. Distributions of dexterity index for 3-P (Qu) RU PKM and 3-PRS PKM are respectively plotted in Fig.6.

It shows that both of the two PKMs present a similar dexterity distribution. The big values of dexterity index are both in the center and some specific edges of the given output range. Moreover, the biggestGDIvalue of the 3-PRS PKM is 0.0068 and smaller than that of the 3-P (Qu) RU PKM, being 0.0126, which means that the manipulator dexterity performance of the 3-P (Qu) RU PKM is better. According to the meaning of manipulator dexterity, the 3-P (Qu) RU PKM owns a better isotropic motion and performs well with regard to its force and motion transmission capability. Hence, it can be concluded that the 3-P (Qu) RU PKM has more advantages in the aspect of manipulator dexterity.

(a)Z=700mm,φ=0～2π andθ=0～π/4; (b)ZDOF varying in 600mm to 800mm withφ=0 andθ=π/6; (c)φDOF varying in 0 to 2π withZ=700mm andθ=π/6; (d)θDOF varying in 0 to π/4 withZ=700mm andφ=0

Fig.6 Distributions of dexterity for 3-P (Qu) RU PKM and

3-PRS PKM

4.3 Position error comparison

In order to obtain a fair error comparison between the two PKMs, errors of the common structural parameters are kept in the same.Structural errors of the 3-P (Qu) RU PKM and the 3-PRS PKM are given in random within a given range. The position error simulations of the two PKMs are performed in the same range with Section 4.2.

Fig.7 and Fig.8 show that both of the two PKMs present a similar position error distribution. The big values of position error are both at some specific edges of the given output range. Furthermore, the maximum values of position errors of the 3-P (Qu) RU PKM and the 3-PRS PKM are 1.919mm and 1.002mm respectively. The data show that the position error of the 3-P (Qu) RU PKM is bigger than that of the 3-PRS PKM. It reveals that the 3-PRS PKM is better than the 3-P (Qu) RU PKM in the aspect of position errors.

(a)Z=700mm,φ=0～2π andθ=0～π/4; (b)ZDOF varying in 600mm to 800mm withφ=0 andθ=π/6; (c)φDOF varying in 0 to 2π withZ=700mm andθ=π/6; (d)θDOF varying in 0 to π/4 withZ=700mm andφ=0

Fig.7 Position errors of 3-P (Qu) RU PKM

(a)Z=700mm,φ=0～2π andθ=0～π/4; (b)ZDOF varying in 600mm to 800mm withφ=0 andθ=π/6; (c)φDOF varying in 0 to 2π withZ=700mm andθ=π/6; (d)θDOF varying in 0 to π/4 withZ=700mm andφ=0

Fig.8 Position errors of 3-PRS PKM

4.4 Error sensitivity comparison

The difference between the two PKMs is the joints. Qu joint (including four R joints), R joint and U joint of the 3-P (Qu) RU PKM are different from R joint and S joint of the 3-PRS PKM. P joints of both PKMs are totally the same. In the following discussions, just the different joints are analyzed. As mentioned above in Section 1.2.3, the R joint of the 3-P (Qu) RU PKM is ignored.Hence, the error sensitivity of Qu joint (four R joints) and U joint of the 3-P (Qu) RU PKM are compared with R joint and S joint of the 3-PRS PKM.The simulation results of error sensitivities based on the global range and 4 poses are given in Tables 1 and 2. The global range isZ=700mm,φ=0～2π andθ=0～π/4. The 4 poses areZ=700mm,φ=0 andθ=π/6,Z=700mm,φ=π/2andθ=π/6,Z=700mm,φ=π andθ=π/6 andZ=700mm,φ=3π/2 andθ=π/6, respectively.

It is found that the biggest global position error sensitivity values of the two PKMs are both about 0.7 and the biggest global orientation error sensitivity values of the two PKMs are both about 0.35. The global error sensitivity of both the two PKMs keep in a similar level. It reveals that the unit error of each joint of both PKMs has a similar influence on the output error. However, the 3-P (Qu) RU PKM is worse than the 3-PRS PKM in the aspect of position errors based on the same structural errors. It can be thought that bigger output error value of the 3-P (Qu) RU PKM is caused by a plenty of joints. Hence, the 3-P (Qu) RU PKM containing more joints is worse than the 3-PRS PKM in the aspect of position errors. From the aspect of error sensitivity, it can be seen that accuracy performance of these two PKMs are basically on a similar level.

Table 1 Error sensitivity of 3-P (Qu) RU PKM with different poses

Table 2 Error sensitivity of 3-PRS PKM with different poses

5 Conclusion

In this paper, a novel symmetrical 3-DOF PKM, named 3-P (Qu) RU, is firstly presented and investigated. According to its structure feature, a double closed loop vectors method is proposed to establish its kinematic model, velocity transfer model and error model.In order to show the performance of the 3-P (Qu) RU PKM, a comparison study of numerical simulations with a 3-PRS PKM is presented. The results reveal that this 3-P (Qu) RU PKM owns the same kinematic workspace with the 3-PRS PKM. And it owns a better isotropic motion and performs well with regard to its force and motion transmission capability. However, it is found that the 3-P (Qu) RU PKM has a bigger output error value but keeps the similar level in the aspect of error sensitivity with the 3-PRS PKM. In general, the accuracy performance of 3-P (Qu) RU PKM is worse than that of 3-PRS PKM.

Compare with the 3-PRS PKM which is applied in the 5-axis hybrid machine tool successfully, it shows that the 3-P (Qu) RU PKM has potential applications in the field of hybrid machine tools. In order to develop the 3-P (Qu) RU PKM in practice, its accuracy performance should be improved by other further studies, including the accuracy design and kinematic calibration method. This paper gives an available selection for a tool head of a hybrid machine tool and the analysis in this paper is greatly helpful for further applications of the 3-P (Qu) RU PKM.

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Liu Yuzhe, born in 1990. He has been a candidate of doctoral research since 2012 in Mechanical Engineering Department of Tsinghua University. He received his B.S. degree in Mechanical Science and Engineering School of Huazhong University of Science and Technology in 2012. His current research interests include parallel mechanisms and accuracy assurance.

10.3772/j.issn.1006-6748.2017.02.003

②To whom correspondence should be addressed. E-mail: jhwu@mail.tsinghua.edu.cn

on Apr. 5， 2016

①Supported by the National Natural Science Foundation of China (No. 51575307, 51225503), the Science and Technology Major Project-Advanced NC Machine Tools & Basic Manufacturing Equipments (No. 2013ZX04004021, 2014ZX04002051) and Top-Notch Young Talents Program of China.