Cai-dong WANG，Guang-zhen CUI，Xin-jie WANG，Chao-hui ZHANG，Liang-wen WANG

（1 College of Mechanical and Electrical Engineering，Zhengzhou University of Light Industry，Zhengzhou 450002，China）

（2 Henan Key Laboratory of Intelligent Manufacturing of Mechanical Equipment，Zhengzhou University of Light Industry，Zhengzhou 450002，China）

Abstract：A novel parallel robotmechanism.3-R2H2S，which contains preloaded Hooke joints is proposed in this paper.To study its dynamic performance，the structure of the 3-R2H2S robot is simplified.The global index of the robot’s dynamic performance is derived from the first and second order influence coefficientmatrix.By varying different structural parameters，the velocity and inertial force performance indexes of the 3-R2H2S parallel mechanism are analyzed.The effect of the robot’smember size on the dynamic performance is discussed when the structural size is changed within the reachableworkspace.Finally，dynamic performance atlases are drawn using a computer simulation and the optimized parameter group is found.This study provides a theoretical basis to optimize both design and dynamic performance of parallel robots.

Key words：Parallel robot，Configuration analysis，Dynamic performance，Influence coefficientmatrix，Velocity performance，Performance atlas

1 Introduction

The payload of parallel robots is determined by their two（ormore）closed-loop motion chains［1］.Parallel robots have many advantages such as high stiffness，high payload capacity，high motion accuracy，small ratio of deadweight to load，and excellent dynamic properties compared with serial robots.They arewidely used in many areas，such as formachine tools，astronomy，medical devices，and physical sensors［2］.Parallel robots have attracted the attention ofmany researchers in recent decades.Studies of the kinematics and dynamic performance are of great significance for the development of high-precision robots［3-6］.Thus，the study of their performance has become an important topic in the field of robotics.Several novelmethods to investigate their kinematic performance have been proposed.For example，Gosselin defined the global condition index during an optimization study of robotic movement in 1991［7］.Zhao analyzed the methodology of the dimensional synthesis of a parallel robot with three translational degrees of freedom，while taking into accountanisotropic kinematic proper-ties［8］.Furthermore，Shi has proposed a global movement performance fluctuation index［9］.The screw theorywas investigated in Refs［10-11］via a kinematics and singularity analysis of parallel robots.U-sing screw theory，Jaime and José［12］performed a kinematics and singularity analyses of a 4-DOF parallelmanipulator.Joshi et al.［13］developed a Jacobian matrix for limited-DOF parallel kinematic machines，and Gallardo［14］analyzed the kinematics of a hybrid manipulator.Hao Qi［15］introduced the inverse dynamicmodel and the acceleration-performance function using the principle of a virtual planar parallel mechanism.By taking into account acceleration factor，velocity factor，gravity factor，and external force factor for the dynamic performance evaluation，the method is suitable to evaluate the dynamic performancemore accurately.

Because there is a close general connection between the properties and the dimensions of a mechanism，a performance atlas can be obtained，which uses the dimensions of the mechanism on one axis to represent the properties ofmechanism.Gao put forward a workspace-area-property atlas［16］，while Liu proposed both the global conditioning index and the global stiffness index atlases［17］.The performance atlas can help reveal the performance of differentmechanisms，and it can provide the basis for the designer to choose a suitable mechanism.In this paper we use the performance atlas to illustrate the relationship between motion and dynamic performance and the dimensions of a 3-R2H2S robot. When the conventionalmethod to analyze kinematics and dynamic of a parallelmechanism is used，the first and second order derivatives are difficult to solve.Huang et al.［18］proposed an influence coefficientmethod and studied the first/second-order kinematic influence coefficient matrices.These were later found to be Jacobian/Hessian matrices.Based on the Hessian matrix，a simplermethod to evaluate the accuracy of a planar parallel robotwas proposed by Briot［19］.By analyzing the numerical value of the acceleration deviation，Liu found that the Hessian matrix has amore important effect on the acceleration performance［20-21］.The influence coefficient method does not require any derivatization process，which depends on the kinematics size，the configuration，and the position of the parallelmechanism.Themethod is independent from motion parameters such as velocity and acceleration.Thus，the calculation process for the influence coefficientmethod is easy and simple，and it has many potential applications formechanism analysis.

A novel3-R2H2Sparallel robot is described and investigated in this paper.Themotion and dynamic performance of the 3-R2H2S robotwith differentmember sizeswas studied using the first and second order influence coefficientmatrix.The results provide an important basis for the performance analysis of parallel robots.

2 3-R2H 2S parallelm echanism and configuration analysis

Recently，many researchers focused on Delta parallel robots for high-speed applications to replace serial manipulators［22-24］.The 3-R2H2S parallelmechanism is described by improving the Delta robot structure in this paper，and theworkspace of the 3-R2H2S parallelmechanism is analyzed［25］.As shown in Fig.1，the base platform andmoving platform are connected by three similar kinematic chains，which are distributed uniformly.Each kinematic chain consists of an R joint（rotational joint），two H joints（Hooke joints）and two S joint（spherical joints），which were connected in sequence.Compared to the Delta robot，the connection joints of the active and passive arms have changed to two Hooke joints，which can be preloaded.The Hooke joint uses an open structure，which expands the swing angle range of the active arm effectively.The two parallel bars of the passive arm were fixed flexibly by a tension spring.This keeps two sections of the Hooke joint close to each other in the connection area，which eliminates the clearance for the Hooke joint during themovement.Because of the replace of two spherical jointswith two Hooke joints in each kinematic chain，the partial freedom of the passive arm was eliminated.Through the motion simulation，we found that the 3-R2H2S parallelmechanism overcomes the drawback of the conventional structure that the connection bar，which is connected by two spherical joints，spins about its own central axis［26］.The novel 3-R2H2S parallel mechanism moves smoothly and has larger workspace with excellentmotion performance.

According to the Kutzbach-Grubler formula for the degree of freedom of a spatial mechanism，the DOF can be expressed as：

Here，nis the number ofmembers of themechanism；mis the number of motion pairs in the mechanism；is the general degree of freedom of each pair.

According to Eq.（1），the number ofmembers isn=11，and the quantity of motion pairs ism=15，while the general degree of freedom of each pairs isThus，the degree of freedom of the 3-R2H2S robot is 3.Therefore，the kinematic performance of the novelmechanism is equivalent to the conventional Deltamechanism.

Fig.1 Three-dim ensiona Im ode I I of the 3-R2H2S robot

An analytic solution to an inverse kinematics problem can be obtained for the driving motors’rotating angle，when the pose（position and orientation）of the end-effector is known.Specifically，the rotation angle of three active arms，relative to the base platform，need to be calculated.The inverse kinematic solution of a robot plays a key role for parallel robot trajectory planning.The end-effector of the robot can reach the required pose using inverse kinematics.

The geometrymethod was used to solve the inverse kinematics problem for the 3-R2H2S mechanism in this paper.As shown in Fig.2，the circumcircle radius of the base platform isR，the circumcircle radius of themoving platform isr.The linkage length of an active or passive arm is denoted asAi Bi=la，BiCi=lb.The fixed coordinate frameO-XYZ，and the moving coordinate frameO′-X′Y′Z′are attached to the geometric center of the base and themoving platform，respectively.

Fig.2 Structure diagram for the 3-R2H2S.

The geometrymethod was used to solve the inverse kinematics problem for the 3-R2H2S mechanism in this paper.As shown in Fig.2，the circumcircle radius of the base platform is denoted byR，while the circumcircle radius of themoving platform is denoted byr.The linkage lengths of the active and passive arm are denoted byAi Bi=la，BiCi=lb.The fixed coordinate frameO-XYZ，and themoving coordinate frameO′-X′Y′Z′are attached to the geometric center of the base and themoving platform，respectively.

On the basis of the geometrical-restriction relation，the position vector ofAi（the rotational joint’s center）in the fixed coordinate frameO-XYZis：

Here，αiis the included angle betweenOAiand the positive direction of theXaxis.

Similarly，the position vector ofCi（the equivalent center on themoving platform）in moving coordinatesO′-X′Y′Z′can be obtained as follow：

Here，α′iis the included angle betweenO′C′and the direction of theX′axis.

The structure of the base and moving platform were designed analogously，thusαi=α′i.

Itwas assumed that the rotation angle betweenOAiandAi Biisθi.Thus，according to the geometrical-restriction relation，the position vector ofBi，with respect to a fixed coordinate system，can be obtained：

It was assumed that the coordinate of the moving platform center is（x，y，z）in the fixed reference frameO-XYZ.Because the poses of the moving platform and the base platform are kept parallel，the vectorOCiin the fixed coordinate frameO-XYZcan be expressed as：

Eq.（6）represents the solution to the inverse kinematics problem of the parallel robot.Eq.（6）can be simplified as follow：

According to Eq.（7），tican be expressed as：

The angleθican be obtained by：

Using the inverse kinematics solution，there are eight groups joint solutions for the 3-R2H2S mechanism，when the pose ofmoving platform is known.

In the previous eight groups’inverse solutions，some groups causemechanism interference during the actualmotion.Therefore，the key problem of the parallel robot’s inverse solution is tofind the optimal solution for the eight groups.

3 In fluence coefficien tm atrix of 3-R2H 2S robot

The influence coefficient matrix represent the essence of the robots mechanics，it is very helpful for the analysis of velocity，acceleration，and force of the parallel mechanism.To study the dynamic performance，it is necessary to simplify the model of the 3-R2H2Smechanism.The simplifications for the robot’smechanism were：（1）Maintaining the globalmovement performance.（2）The ability constraints for the moving platform of each kinematic chain cannot be changed.（3）The total freedoms of the mechanism cannot be changed.According to the principles of simplification，each kinematic chain of the parallel robot can be simplified to an RHH form.The simplified model is shown in Fig.3.Its kinematics and dynamic performance is equivalent to R2H2S.Because the three chains have a similar structure form，the first chain was used as an example to illustrate.The analysismethod of the other two kinematic chains is similar to the first chain.

Fig.3 Sim p Iified mode I for the first chain

Hooke joints can be decomposed equivalently into two revolute pairs，whose axes are orthogonal.The kinematic chain can be decomposed intofive basic pairs via simplification.Hence，in order to make each kinematic chain contain six basic pairs，a basic pairwas added on each kinematic chain using the virtualmechanism method［16］.The virtual pair should ensure that the influence coefficient matrix is not singular.Then，the first and second order influence coefficient matrices of the 3-R2H2S parallel robot can be derived using the virtualmechanism method.

3.1 First order influence coefficient matrix for the 3-R2H2S robot

3.1.1 First order influence coefficientmatrix of the kinematic chain

In the simplified model for the 3-RHH mechanism，the unit vector in the axial direction of the kinematic pairs in each kinematic chain can be expressed as（i=1，2，3，j=1，2，3，4，5，6）.It was assumed thatPis the position vector for the centerPof the moving platform in the fixed frameO-XYZ，Rjis the position vector of the kinematic pairjin the fixed frameOXYZ，the radius vector of the pairRjto pointPisP-Rj.Using vectorandP-Rj，Sj×（P-Rj）can be obtained.

In the first chain，the virtual revolute pair was added in front of.While the unit vector of the virtual revolute pair and radius vector can be set as arbitrary values，it is necessary to guarantee that the newly formed coefficientmatrix is not singular.Setting the unit vectoryields=（1，0，0）T.According to Fig.3，the unit vector is：

The first order influence coefficientmatrix for the first chain can be expressed as follows：

The solutionmethod for the first order influence coefficientmatrix of the other two kinematic chains is similar to G（1）.It needs to replace the axial unit vectorof the kinematic pair and radius vectorP-Rj，respectively.The first order influence coefficientmatrix G（i）of each kinematic chain is6×6 form.For the 3-R2H2S robotmechanism which has three kinematic chains，the robot has three 6×6 first order influence coefficientmatrixes.

3.1.2 First order integrated influence coefficientmatrix of the3-R2H2S robot

From G（1），G（2），G（3），the first order integrated influence coefficientmatrix［］of 3-R2H2S is obtained as follows：

Here，the superscriptHdenotes themoving platform，and the subscriptqdenotes the initial angular position of active parts.The activemembers in each kinematic chain are the second pairs after setting a pair using the virtual mechanism method.Thus， （G（1））-11：and（G（1））-12：，respectively，denote the elements of the first and second row in the inverse matrix of G（1），which is the first order influence coefficientmatrix of the first kinematic chain.The meanings of other elements in［］-1are similar.

3.2 Second order influence coefficientmatrix of the 3-R2H2SRobot

3.2.1 Second order influence coefficientmatrix of the kinematic chain

The kinematic pairs of a 3-R2H2S robot are all revolute pairs，so the second order influence coefficient matrix of each kinematic chain can be derived using the following equations［16］：

Here，［H］m：ndenotes the element in themrow andncolumn of the second order influence coefficientmatrix［H］.Thisway，the second order influence coefficient matrix of each kinematic chain can be obtained.The matrixes are 6×6 form，and each element in thematrix is a 6×1 vector.According toand P-Rj，the first and second row of the second order influence coefficientmatrix of the first kinematic chain can be obtained as follows：

Similarly，if the elements from the third row to the sixth row of the first kinematic chain can be obtained，and subsequently the second order influence coefficientmatrix of the first chain can be derived.The solvingmethod for the second order influence coefficient matrix of the other two kinematic chains is the same as the first chain.

3.2.2 Second order integrated influence coefficientmatrix of the3-R2H2SRobot

According to the first order integrated influence coefficientmatrix and the second order influence coefficientmatrix of each kinematic chain，the second order integrated influence coefficientmatrix can be obtained by the following formula［27］.

First，themiddle transitionmatrix is derived as follows：

The second order integrated influence coefficient matrix of 3-R2H2S can be expressed as：

4 Perform ance analysis of the 3-R2H 2S robot

The velocity and acceleration of the 3-R2H2Sparallel robotwere analyzed using global performance indexes based on the workspace in this paper：

Here，J∈｛G，H｝；ηJis the global performance index of robot，andWis the workspace of the robot.

Eq.（16）describes themean of the reciprocal condition numberkJwithin the reachableworkspace of the 3-R2H2S robot.ηJdenotes the dexterity and control precision of the robot.For 1≤kJ≤∞，0≤ηJ≤1，whenηJis closer to 1，the kinematics and dynamic performance of the robot improves.Thus，ifηJ=1，the isotropy of the robot’smechanism is optimal.

4.1 Velocity performance analysisof the3-R2H2S robot

The velocity of the 3-R2H2S parallel robot can be expressed as follows：

Here，Gis the first order influence coefficientmatrix，andis the generalized velocity vector of the active arm.Let

It was assumed thatis a condition number［25］when velocity performance index is calculated.The global velocity performance indexηGfor the different mechanism dimensions can be obtained based on Eq.（16）.The effect of robotmechanism size on the velocity performance is investigated in this section.According to the 3-R2H2S mechanism，the velocity performance can be studied by changing the size values forlaandlb，and keeping the radius of the base and moving platform unchanged.We set the initial value as follows：R=105 mm，r=50 mm，la=200～250mm（the step size is5mm），lb=450～500 mm（the step is 5 mm）.Then，121 mechanism groups can be obtained，and 1 000 pointswere evenly chosen within the workspace of each mechanism.Based on the above calculation，the program for the velocity performance index was developed using MATLAB.

The velocity performance atlas can be obtained by calculating using the method proposed in this papersee Fig.4.

Fig.4 Ve Iocity performance atIas for the 3-R2H2S robot

The atlas in Fig.4 shows thatηGis bigger forla=244～250 mm，lb=450～459 mm.According to Eq.（16），the biggerηGis，the smaller are the errors for input and output，and the performance of themechanism is improved too.As a result，the mechanism’s velocity performance is better when the member parameterswere selected within this range.

4.2 Inertial force performance analysis for the3-R2H2S robot

The inertia force performance determines the dexterity and the control precision of the parallel mechanism.The larger inertial force performance index is，the better is the inertial force performance of themechanism.The performance of the inertia force of the parallel robot depends onThere fore，were chosen as the evaluation index for the inertial force performance.Assuming tha，the biggerkG+His，the smaller is the inertia force，and the higher is the control precision.

Based on the above calculation，the computing program for the inertial force performance index was developed using MATLAB.By setting the initial parameters identical to the velocity analysis process，the inertia force performance atlas of the parallel robot can be obtained-see Fig.5.The results show that the inertia force performance index improves with increasing active arm size and the decreases with the passive arm size.The inertia force performance index is larger，when the sizes arewithin or near the following ranges：la=242～250 mm，lb=450～457 mm.The inertial force performance is better，the inertia force is smaller，and the dexterity of3-R2H2Sparallel robot is improved.

Fig.5 Inertia Iforce perform ance atIas for the 3-R2H2S robot

According to the above calculation and analysis，when both the active arm and passive arm parameters were selected within the rangesla=244～250 mm，lb=450～457 mm，the velocity and inertia force performances improved，including the entire kinematic characteristics of the 3-R2H2S parallel robot.Therefore，the member parameter group of the robot with good performance according to the above solving process can be obtained，which enables the theoretical basis and an effective method for structural optimization design.

5 Conclusions

The kinematics and dynamic performance indexes for the 3-R2H2S parallel robot were analyzed，based on the first and second order influence coefficientmatrices.These were derived using the influence coefficientmethod in this paper.The performances of a parallel robotwith various structure parameterswere analyzed using the velocity and inertia force performance evaluation index.The relationships between the velocity and inertia force performance，（the structure size）were studied，and the abstract performance indexes were transformed into the intuitive performance atlas with curves in the plane.Finally，the structuremember parameters of the 3-R2H2S parallel robotwith the bestmovement performance have been obtained.This method avoids blindness during the design process of the parallel robot，and it helps obtain themember parameters ofmechanism with the best dynamic performance.The results，which were obtained in this study，may be useful for kinematics and dynamic performance analysis，and structural optimization of the parallel robot before a prototype is built.

6 Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

7 Con flicts of Interest

The authors declare that they have no conflicts of interest.

Appendix

Notation

nthe number ofmembers of themechanism

mthe number ofmotion pairs in themechanismis the sum of the degrees of freedom of all pairs

Rthe circumscribed circle radius of the base platform

rthe circumscribed circle radius of themoving platform

lalength of active arm（mm）

lblength of passive arm（mm）

αithe included angle betweenOAiand theXaxis

α′ithe included angle betweenO′C′and theX′axis

θithe rotate angle of the active arm

ηJthe global performance index of the robot

Wthe workspace of the robot

vvelocity of the 3-R2H2S parallel robot（mm/s）

Gthe first order influence coefficientmatrix

˙qthe generalized velocity vector of the active arm（degree/s）