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1.College of Mechanical and Vehicle Engineering，Taiyuan University of Technology，Taiyuan 030024，P.R.China；2.School of Electronic Information and Electrical Engineering，Shanghai Jiao Tong University，Shanghai 200240，P.R.China；3.State Key Laboratory of Robotics and System，Harbin Institute of Technology，Harbin 150001，P.R.China

Abstract: Heavy-load transfer robots are widely used in automobile production and machinery manufacturing to improve production efficiency.In order to meet the needs of large billet transfer，a 4-DOF transfer robot is designed in this paper，which consists of parallel four-bar mechanisms.The Jacobian matrix referring to the mapping matrix from the joint velocity to the operating space velocity of the transfer robot can be solved by the differential-vector method.The mean value of the Jacobian matrix condition number in the workspace is used as the global performance index of the robot velocity and the optimization goal.The constraint condition is established based on the actual working condition.Then the linkage length optimization is carried out to decrease the length of the linkage and to increase the global performance index of velocity.The total length of robot rods is reduced by 6.12%.The global performance index of velocity is improved by 45.15%.Taking the optimized rod length as the mechanism parameter，the distribution of the motion space of the transfer robot is obtained.Finally，the results show that the proposed method for establishing the Jacobian matrix of the lower-mobility robot and for the optimization of the rods based on the velocity global performance index is accurate and effective.The workspace distribution of the robot meets the design requirements.

Key words：parameter optimization；motion analysis；mechanism design；transfer robot；heavy-load

0 Introduction

With the development of automation technolo?gy，industrial robots have gradually replaced tradi?tional manpower in simple and repetitive tasks.In addition，because of the great application prospect of industrial robots，this is known as one of the three pillars of industrial automation［1］.As a kind of industrial robot，heavy-load transfer robots have been widely used in various industries，such as auto?mobile production，machinery manufacturing，pal?letizing logistics，especially in the harsh environ?ment not suitable for human work［2-3］.

In the 1950s，the American Devol Company designed and developed the earliest industrial robot.The robot needs to input a predetermined motion path in advance.It is also known as a teach-repeat industrial robot.The motion accuracy and response speed are relatively low due to adopting open-loop control.By the end of the 1970s，these characteris?tics of industrial robots represented by PUMA ro?bots had been improved.Since then，the countries around the world have carried out structural im?provements and performance optimization of trans?fer robots with the goal of high stability，high reli?ability，and low weight ratio.American Par Sys?tems Company introduced a Cartesian coordinate in?dustrial robot.Japan developed SCARA robotic arm that contains both slider joints and revolute joints.YASKAWA Company used the parallel fourbar mechanism for mechanical arm structure design，and developed a new type of robot with low-weight and high-load.Foreign industrial robots are mainly light-load robots，which are mainly used in fast-re?sponse working environments and small spaces.For example，the ABB-IRB6640 robot with a transfer speed of up to 3.3 m/s can be loaded with a payload of 200 kg.In China，the research on industrial ro?bots started late.After more than half a century of development，research institutions and robot compa?nies of China have made many achievements.For example，Harbin Boshi Robot Company has devel?oped a professional robot with a payload of 300 kg.Siasun Robotics developed an industrial robot with independent intellectual property rights and with a payload of 120 kg.Harbin Institute of Technology and Shanghai University independently developed transfer robots with independent intellectual proper?ty rights.At present，articulated robots，as the main robots，have the advantages of large work?space and high flexibility.However，the positional accuracy of the industrial robots under high-speed and heavy-load conditions has reduced because of the complexity of rotary joint structures.Therefore，it is of great significance to design and develop a new type of heavy-load transfer robot.

Yang et al.［4］proposed a detailed scheme de?sign for heavy-load transfer robots based on the par?allel equations and transfer constraint conditions.Fu?kui et al.［5］put forward a hoisting mobile robot Han?Grawler，which has high mobility and can freely choose and adjust the route.The robot can achieve linear translational motion at a speed of 0.1 m/s and rotational motion at a speed of 8.5 °/s by installing a mechanical restraint suspension mechanism.Zhu et al.［6］used the D-H method to establish the kinemat?ics model of the 150 kg heavy-load robot indepen?dently.The singular posture of the robot was devel?oped and analyzed based on the inverse kinematics analysis.Ma et al.［7］designed a simplified 3-DOF parallel robot based on the FANUC transfer manip?ulator.The size of the mechanism was optimized by using the velocity global performance index and the acceleration global performance index.Rioux et al.［8］aimed at the navigation problem of the transfer robot when moving large heavy objects.They proposed a system solution from the incremental construction of the environment map and the calculation of collisionfree trajectories to the execution of these trajecto?ries.The solution significantly increased the transfer payload of the robot when the main hardware of the robot was not changed.According to the structural characteristics of heavy-duty transport robots，Yang et al.［9］used ADAMS to obtain the relationship be?tween error fluctuations and the movement of key parts of the robot by conducting kinematics simula?tion.The peak torque characteristics of key parts were analyzed to guide the selection of the drive mo?tor.Wang et al.［10］analyzed the velocity global per?formance index of a 5-DOF parallel mechanism based on the Jacobian matrix to obtain the influence of the size of movable platform and fixed platform on mechanism performance，and the dimension of the mechanism with better velocity performance.

In order to improve the control accuracy of the palletizing robot operation，Xiong et al.［11］took ABB-IRB660 palletizing robot as the research ob?ject，and carried out workspace calculation and tra?jectory planning to achieve the optimal motion time under the premise of considering the boundary condi?tions.Hawley et al.［12］proposed a model-based mo?tion controller to realize stable motion control when pushing and transporting heavy objects.The effec?tiveness of this control strategy was verified by NAO transfer robot experiments.Liu et al.［13］pro?posed a geometric kinematics solution for the 4-DOF parallelogram palletizing robot.Compared with the D-H method，this algorithm has the charac?teristics of simple computation and convenient realtime control，but it is only suitable for specific forms of robots.

Aiming at an existing heavy-load palletizing ro?bot，Wei et al.［14］constructed a motion equation to analyze the kinematics characteristics.The mass of the structure was optimized by using the response surface method and the topology method.At pres?ent，a large number of domestic heavy-load transfer robots rely on imports［15-17］.The internationally re?nowned heavy-load robot manufacturers occupy more than 80% of China’s market share［3］.The de?sign schemes of the heavy-load robot were carried out by conducting kinematic analysis，robot design optimization and motion control according to the ac?tual requirements of the robot to realize general transferring or special scenarios application.

Therefore，to carry out the mechanism design and kinematics analysis of heavy-load transfer ro?bots，it is necessary to decrease the driving source or motion coupling constraint scheme by taking actu?al workspace as a constraint.According to the tech?nical parameter requirements of the robot，based on principles of high rigidity，high strength and simple control，this paper proposes a configuration scheme of a material transfer robot with multi-parallel fourbar mechanisms and carries out the overall mecha?nism design.The forward kinematic problem of the robot is calculated by using the D-H parameter method.The inverse kinematic problem is solved by using the inverse transform method and geometric analytic method.The Jacobian matrix of the robot is obtained by synthesizing the total differential equa?tion method and the vector product method.The op?timization of the length of the robot rod and the anal?ysis of workspace are carried out.

1 Mechanism Design of Heavy?Load Transfer Robot

The overall schematic diagram of the heavyload transfer robot is shown in Fig.1，including the upper arm，the elbow，the vertical arm，the longitu?dinal parallel linkage，the transverse parallel link?age，the auxiliary triangle plate，driving linkage of the elbow，supporting linkage of the elbow，the long rod，the short rod，the driving linkage and clamp connecting rod of the linear motion mecha?nism.The slewing support rotates around the fixed base through the hydraulic motor installed on the slewing support.Since the whole machine rotates around the bottom rotation axis，when the bottom rotation degree of freedom is removed，the mecha?nism can be converted to a plane mechanism.The degrees of freedom of the robot can be calculated by

Fig.1 Schematic diagram of the transfer robot

whereN，Pl，PhandP' are the numbers of compo?nents，low pairs，high pairs and redundant con?straints，respectively.

From Fig.1，it can be seen thatN=19，Pl=26，Ph=0 andP'=1.The degree of freedom of the plane mechanism is 3.Finally，the space rotation de?gree of freedom is added to the overall degree of freedom of the mechanism.The robot has 4 degrees of freedom.

To improve the capacity of the robot，four par?allel four-bar mechanisms are used in the robot.The first parallel four-bar mechanism is composed of the upper arm，the longitudinal parallel linkage，the slewing support and bottom edge of the auxiliary tri?angle plate.The bottom edge of the auxiliary trian?gle plate is always parallel to the ground during the movement.The auxiliary telescopic hydraulic cylin?der pushes the robot to achieve horizontal expan?sion.The second parallel four-bar mechanism is composed of the elbow，the transverse parallel link?age，the upper part of the vertical arm and the right side of the auxiliary triangle plate.The vertical arm always maintains upright during the movement.The third parallel four-bar mechanism is composed of the upper arm，the elbow，the driving link of the elbow and the supporting link of the elbow，which assists the lifting hydraulic cylinder to promote vertical lift?ing movement of the robot.The hinge point of the short rod and the long rod of the linear motion mech?anism is at the middle point of the long rod.The fourth parallel four-bar mechanism is composed of two long rods，the driving linkage and the clamp linkage.The clamp connecting rod keeps horizontal all the time.The driving linkage is equipped with a sliding block，which forms a sliding pair with the guide rail installed on the vertical arm.The linear motion mechanism is driven by the driving hydraulic cylinder installed on the vertical arm.

2 D?H Coordinate System and Ja?cobian Matrix

2.1 D?H coordinate system establishment and coordinate transformation

There are many parallelograms in the robot.The kinematic simplification can be conducted［18］.According to the characteristics of the parallelo?gram，AD//BCandEF//BC，the rodsAD，BC，andEF，which are shown in Fig.1，are in the same direction at any time.Therefore，rodBCcan be used as the equivalent rod for the upper arm.In the same way，in the parallelogram CGHK，the rodCKcan be used as the equivalent rod for the el?bow.According to the preliminary designed trans?fer mechanism，the simplified diagram of the trans?fer robot mechanism marked with the D-H coordi?nate system is shown in Fig.2.The upper arm lengthl2，the elbow lengthl3，the vertical rod lengthl4，the lengths of the clamping rods lengthsl5andl6are given in Fig.2.The rod parameters of the robot are shown in Table 1，whereθiis the an?gle of rotating around theZi-1?axis,fromXi-1?ax?is toXi?axis in the D?H coordinate system which is shown in Fig.2.

Fig.2 Simplified diagram of transfer robot mechanism

Table 1 Linkage parameters of transfer robot

As shown in Fig.2，θ4is the acute angle be?tween the elbow and the vertical rod.Since the verti?cal rod remains vertical at all time，θ4is mainly de?termined by the posture of the elbow.The lengths of the two clamping rods are equal，l5=l6.The hori?zontal angles of the two rods are equal.According to the geometric parameters of each member of the robot and the relationship of the positions between each other，it can be obtained that

According to the method of establishing the DH coordinate system and the simplified diagram of the transfer robot，a homogeneous transformation matrixi-1Tibetween adjacent coordinate systems∑Oi-1and∑Oiis established.

Therefore，the D-H transformation matrix be?tween each rod can be obtained as

whereci=cosθi，si=sinθi.From Eq.（2），cos(θ2+θ3)=-sinθ4，sin(θ2+θ3)=-cosθ4，cos(θ2+θ3+θ4)=0，sin(θ2+θ3+θ4)=-1，cos(θ2+θ3+θ4+θ5)=sinθ5，sin(θ2+θ3+θ4+θ5)=-cosθ5，cos(θ2+θ3+θ4+θ5+θ6)=sinθ5，sin(θ2+θ3+θ4+θ5+θ6)=cosθ5.

The transformation matrix0T6of the robot endeffector relative to the base coordinate system is ex?pressed as

wherenx=cosθ1sinθ5，ny=sinθ1sinθ5，nz=cosθ5，ox=-cosθ1cosθ5，oy=-sinθ1cosθ5，oz=sinθ5，ax=sinθ1，ay=-cosθ1，az=0，px=cosθ1(l2cosθ2-l3sinθ4+2l5sinθ5)，py=sinθ1(l2cosθ2-l3sinθ4+2l5sinθ5)，pz=l2sinθ2-l3cosθ4-l4+d1.

2.2 Comparison simulations between the origi?nal robot and the equivalent mechanism

In order to verify the equivalence of the mecha?nism，the three-dimensional model of the mecha?nism is imported into ADAMS for dynamic compar?ative analysis.The 3D model of mechanism in AD?AMS is shown in Fig.3.The step functions in the driving process are shown in Table 2.

Table 2 Step functions of prismatic pairs

Fig.3 3D model of mechanism in ADAMS

The relationship between the driving displace?ment and the rotation angle of the rod is shown in Fig.4.The relationship between the displacement of the first drive and angleABCof the upper arm is shown in Fig.4（a）.The relationship between the displacement of the third drive and angleBEFis shown in Fig.4（b）.Through the diagrams of the re?lationship between the driving rod and the connect?ing rod，the displacement of the endL-point in thexandydirections can be clearly observed.

Fig.4 Relationship between the driving displacement and the rotation angle

In Fig.5，theL-point motion trajectory curves at the end of the three-dimensional model and equiv?alent mechanism are compared.The comparison of changes of theL-point in thexdirection is shown in Fig.5（a）.The comparison of changes of theL-pointin theydirection is shown in Fig.5（b）.According to the analysis of the trajectory curve comparison di?agram，it can be obtained that the trajectory of the three-dimensional model is consistent with that of the equivalent mechanism.Thus，the correctness of the results is verified，which provides a theoretical basis for the kinematic analysis and size optimization of the mechanism.

Fig.5 Comparison of L-point trajectory curves of 3D model and equivalent mechanism

2.3 Verification of D?H parameters and kine?matics equations

In order to verify the correctness of the ob?tained D-H parameters and kinematics equations，the three-dimensional model of the transfer robot is generated by MATLAB Robotics Toolbox，and the robot control panel is generated at the same time.The control panel can adjust the angles of each joint to make the joint rotate.The robot simulation mod?els are shown in Fig.6.The angles of each joint in the initial state of the robot in Fig.6（a），F(xiàn)ig.6（b）and Fig.6（c）areθ1=0，θ2=0，θ3=0，θ4=3π/2，θ5=0，θ6=-π，θ1=π/2，θ2=37.5π/18，θ3=π/2，θ4=142.5π/18，θ5=-π/2，θ6=0 andθ1=82.5π/18，θ2=135π/18，θ3=π/2，θ4=π/4，θ5=-113π/18，θ6=46π/18，respectively.

Fig.6 Robot simulation models

These special values are brought into transfor?mation matrix0T6for verification.Letd1=179 mm，l2=172 mm，l3=658 mm，l4=170 mm，l5=85 mm，the position coordinates of the end effector are con?sistent with the values obtained from the 3D model generated in MATLAB.Thus，the correctness of the results is verified，which provides a theoretical basis for the workspace analysis of the mechanism and the solution of the Jacobian matrix.

3 Jacobian Matrix Solution

The Jacobian matrix refers to the mapping ma?trix from the joint velocity to the operating space ve?locity，which satisfies

whereVis the operating space velocity，the joint velocity andJthe Jacobian matrix of the robot.

For a robot withnjoints，its Jacobian matrix is a 6×n-order matrix.The first three rows represent the transfer ratio of the operator’s linear velocityv.The last three rows represent the transfer ratio of the operator’s angular velocity.Each column repre?sents the corresponding joint velocityto the trans?fer ratio of the operator’s linear velocity and angular velocity.The Jacobian matrix can be divided as

whereJlis the mapping matrix of joint linear veloci?ty to Cartesian space velocity andJathe mapping matrix of joint angular velocity to Cartesian space velocity.

In order to solve the Jacobian matrix，the total differential method and the vector method are com?bined，which can avoid a large number of matrix op?erations and matrix inversion operations when com?pared to the total differential method or the vector method，respectively［19］.Therefore，the mapping matrixJlis solved by using the total differentiation method.The mapping matrixJais solved by using the vector method.

3.1 Jacobian matrix Jl solution based on the to?tal differential method

The position of the robot end manipulator rela?tive to the robot base coordinate system is expressed as0P6，shown as

According to the definition of the Jacobian ma?trix，the mapping from the velocity of the end opera?tor to the joint velocity can be got by solving total differential0P6，shown as

3.2 Jacobian matrix Ja solution based on the vector method

The movement of each joint of the robot has an impact on the velocity of the robot’s end manipula?tor.Based on the relationship between the joint ve?locity and the end velocity，velocityωiat the end of the robot caused by the revolute joint satisfies

Then the angular velocity of the robot end satisfies

whereZiis theZaxis of the robot coordinate system∑Oi.The mapping from the end angular velocity of the robot to the joint angular velocity is expressed as

where0Zirepresents the coordinate of theZaxis of the coordinate system ∑Oi，which is relative to the base coordinate system.That is to say，the column vector is composed of the third column elements of0Ti.Therefore

Deduced from Eq.（6），the operator’s angular velocity can be given as

The differential of Eq.（2）can be expressed as

Then Eq.（12）can be simplified as

Therefore，Jacan be expressed as

Based on Eqs.（8）and（15），the Jacobian ma?trix of the transfer robot can be given as

Eq.（2）shows that the number of independent joint angles of the transfer robot is 4，which belongs to the robot with few degrees of freedom.The Jaco?bian matrix obtained by Eq.（16）is a rectangular ma?trix with 6 rows and 4 columns.This kind of Jacobi?an matrix cannot perform inverse matrix operation.This brings troubles to robot singularity analysis and robot control.For the robot proposed in this paper，the linearly independent velocity in Cartesian space is determined.By recombining the mapping coeffi?cients in the Jacobian matrix corresponding to the linearly independent velocity，the square Jacobian matrix of the robot with few degrees of freedom can be obtained.

Since the row vectors in the first，second，and sixth rows of Eq.（16）are linear correlation.The vectors in the fourth and fifth rows are linear correla?tions.There is a set of constants that are not all ze?ros.So

whereJ(i，：) is theith row vector of the Jacobian matrix of the robot.Choosing one velocity compo?nent from (vx，vy，ωz) and (ωx，ωy) as the subordi?nate component［16］，the remaining four components can be used as independent velocity components.There are 6 possible combinations.Taking (vy，ωy)as the subordinate velocity component，a 4×4 Jaco?bian matrix with few degrees of freedom can be ob?tained as

4 Size Optimization Based on Ve?locity Global Performance Index

4.1 Design variables and constraints

The simplified structure of the transfer robot is shown in Fig.7.

Fig.7 Schematic diagram of simplified robot mechanism and parameters

The workspace of the robot is related to the up?per arm lengthl2，the elbow lengthl3，the vertical rod lengthl4，the heightd1from the turning center to the ground，and the pitch angleθ2of the upper arm and the pitch angleθ3of the elbow.The maxi?mum workspace of the robot is also related to the rod lengthl5.In order to reduce the total length of the rods to save material，settingl4=2l5.Taking into account the interference between the rods and the limitation of the height of the base，60°≤θ2≤150°，and 30°≤θ3≤135° are determined.Thend1，l2，l3，andl5are determined to be the design variables，which can be expressed as

In order to make the robot meet the require?ments of the material transfer operation and avoid the interference and collision between the robot rods and between the rods and the ground，the restriction conditions of the rod size include rod length restric?tion and workspace restriction.

4.1.1 Constraint conditions of rod lengths

Considering the height of the furnace door of the heating furnace，the height of the die cavity of the die forging press and the height of the work?shop，the design height of the clamp center is be?tween 850—2 800 mm.Therefore，the constraints are as follows

After the clamp is close to the heating furnace，the material clamping is completed by the linear mo?tion mechanism.Considering the wall thickness of the heating furnace and the position of the material in the furnace，the rod size of the linear motion mechanism meets the requirements

4.1.2 Workspace constraints

When the pitch angle of the robot’s upper arm and elbow reaches the maximum and the linear mo?tion mechanism is fully deployed，the end of the ro?bot can reach the farthest horizontal point.Com?bined with the requirements of the horizontal travel parameters of the transfer robot，in order to make the maximum stretch distance of the clamp of the ro?bot not less than 4 200 mm，the constraints can be obtained by

Converting Eq.（20）to Eq.（22）into nonlinear constraint functions，we can obtain

According to the above constraints，the prima?ry rods sizes ared1=900 mm，l2=1 500 mm，l3=2 200 mm，l4=2l5=1 200 mm.

4.2 Objective function determination

The performance indexes used for kinematic size optimization are the condition number of the Ja?cobian matrix and operability［17］，isotropy，static stiffness，the smallest or the largest singular value，etc.The condition number of the Jacobian matrix di?rectly determines the mapping accuracy of joint space velocity to operating space velocity.The num?ber also reflects the degree of distortion of the trans?fer relationship between the joint input velocity and the end-effector output velocity.The minimum val?ue of the Jacobian matrix condition number is 1.When the Jacobian condition number of the robot is 1 in its working space，the corresponding configura?tion is kinematic isotropic，and the transmission per?formance of the mechanism is the best.The larger the Jacobian condition number is，the larger devia?tion of the end manipulator’s velocity caused by the deviation of the input angular velocity.At the same time，the distortion of the transfer relationship be?tween the motion input and the output is more seri?ous when solving the inverse solution of the joint an?gular velocity.Therefore，the Jacobian matrix con?dition numberkis used as an evaluation index of ro?bot motion performance.

From the definition of the matrix condition number，the relationshipkwith the Jacobian matrixJis

where ‖J‖ is the norm of the Jacobian matrixJ，and ‖J-1‖ the norm of the Jacobian inverse matrix.

Since 1 norm，∞ norm and 2 norm change with the joint angleθiand the rod lengthli，the size judgment must be carried out first in the calculation process.Therefore，the condition number of the Ja?cobian matrix is defined by the F norm

Because the Jacobian matrix of the robot is de?termined by its configuration，different joint angles correspond to different Jacobian matrices.There?fore，the Jacobian condition number is a local perfor?mance index.Gosselin proposed a velocity global performance indexηto represent the average value of the Jacobian condition number in the workspace，which can be expressed as

The range ofηis[1，∞).The smallerηis，the better global performance of the robot velocity is.Taking minimum ofηas the optimization objective，the objective function can be given as

5 Size Optimization Results and Workspace Analysis

From Eq.（18），the inverse matrix of the Jaco?bian matrix of the heavy-load robot is

From Eq.（25），the Jacobian condition numberkis

From Table 1 and Eq.（2），θ4∈[-15°，165°].According to Eq.（26），the global performance in?dex of velocityηis the function of lengthliof mem?ber，which satisfies

With the initial valueX0=[1 500，2 200，600]selected，F(xiàn)(x)=min(η) as the optimization objec?tive and Eq.（23）as the constraint condition，and the constrained minimization function fmincon in MATLAB optimization toolbox is called，the opti?mization result is obtained which isX=[1 614.23，1 807.64，501.58].From Eq.（23），d1=800.After optimization，the sizes of the manipulator rods ared1=800 mm，l2=1 600 mm，l3=1 800 mm，andl4=2l5=1 000 mm.Table 3 shows the change of the global performance index of velocityηand the total lengthL2+L3+L4of the rods before and after the size optimization.After op?timization，the total length of the rods is reduced by 6.12%.The global speed performance is improved by 45.15%，and the robot has less consumables and lighter weight.

Table 3 Comparison of optimization results

The robot workspace refers to the set of all space points which can be reached by the robot end manipulator under normal working conditions.From Eq.（4），the position coordinates of the end clamp of the robot are

According to the range of motion of each joint angle and the optimized rod lengthd1=800 mm，l2=1 800 mm，l3=1 800 mm，l4=2l5=1 000 mm in Table 1.As shown in Fig.8，the workspace of the robot is obtained by MATLAB simulation.The results show that the robot workspace meets the space requirements of the task.

Fig.8 Workspace distributions of transfer robot

6 Conclusions

A configuration design scheme of a 4-DOF transport robot with multi parallel four-bar mecha?nisms is proposed.In order to design a robot with better velocity transmission performance in the whole workspace，the optimization of the length of the rod and the spatial analysis of the movement of the transfer robot are carried out by taking the global performance index of velocity as the optimization goal.The main conclusions are as follows：

（1）Based on the technical parameters and task requirements of the heavy-load transfer robot，a 4-DOF material transport robot with multi parallel four-bar mechanisms is designed.The structure of the robot takes the parallelogram motion chain as the main structure to improve the strength and stiff?ness of the robot.The Jacobian matrix of the robot is obtained by using the D-H parameter method，to?tal differential equation method and vector product method.

（2）Taking the lengths of the main rods as de?sign variables，the global performance index of ve?locity as the objective function，and the actual work?ing condition of the robot as the constraint condi?tion，the length of the rod is optimized by MAT?LAB simulation.Based on the optimized length of the rod，the workspace of the end manipulator in the global coordinate system is solved by programming.The workspace of the robot fully meets the design requirements.