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冗余自由度机器人运动分析与仿真设计含CAD图

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冗余自由度机器人运动分析与仿真设计含CAD图,冗余,自由度,机器人,运动,分析,仿真,设计,CAD
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Kinematic analysis and simulation of a new-type robotwith special structureShuai GuoHua-Wei LiJian-Cheng JiZhi-Fa MingReceived: 12 March 2014/Accepted: 6 November 2014/Published online: 7 December 2014? Shanghai University and Springer-Verlag Berlin Heidelberg 2014AbstractCommon methods, such as Denavit-Hartenberg(D-H) method, cannot be simply used in kinematic analysisof special robots with hybrid hinge as it is difficult to obtainthe main parameters of this method. Hence, a homogeneoustransformation theory is presented to solve this problem.Firstly, the kinematics characteristic of this special structureis analyzed on the basis of the closed-chain theory. In such atheory, closed chains can be transformed to open chains,which makes it easier to analyze this structure. Thus, it willbecome much easier to establish kinematics equations andget the solutions. Then, the robot model can be built in theSimmechanics(atoolboxofMATLAB)withtheseequationsolutions. It is necessary to design a graphical user interface(GUI) for robot simulation. After that, the model robot andreal robot will respectively move to some spatial pointsunder the same circumstances. At last, all data of kinematicanalysis will be verified based on comparison between datagot from simulation and real robot.KeywordsMixed hinge structure ? Homogeneoustransformation ? Kinematic analysis ? Kinematic simulation1 IntroductionUsually,weuseDenavit-Hartenberg(D-H)methodtoanalyzethekinematicsofrobot.Thekeyofthismethodistoestablishaset of D-H parameters which indicate the relationship of thecoordinatesystemsofalljoints1.Therearealotofstudiesonthe serial chain industry robots, however few studies are onsuch industry robots with complex structures. For example, akind of large universal industry robot with hybrid hingestructure allows its arm to move backwards in low energyconsumption.Themotionlawofthisrobotdiffersfromthatofother robots and its D-H parameters cannot be determinedeasily. Two solutions are proposed to deal with this specialstructure. One is to transform the structure into equivalentserial joints (revolute or prismatic joints). Then we candetermine the D-H parameters one by one based on D-Htheory 2. The other is to transform closed chains to openchains so that we can calculate the relationship between theactive and passive joints. Then, the complex structure istransformed into simple serial structure 3. The first methodneedstoanalyzethecharacteristicsofeveryjointinthishybridhinge, thus it is sophisticated and error-prone 4. Meantime,inthesecondmethod,althoughageneraltheorytoanalyzeallclosed chains is introduced, the breakdown structure is notpresented and this theory is not verified by experiments.Considering these two methods, it will be easier for us todetermine the relationship between coordinate systems ofjoints if we know the kinematics characteristic of the specialstructure. Hence the motion characteristic of the specialstructureisanalyzedbasedontheclosed-to-openchaintheoryin the paper. Then the coordinate systems are directly estab-lishedateachactivejoint.Theadvantageofthemethodisthatit can avoid judging the motion of each kinematic pair in thestructure 5. In addition, it is convenient to build the simu-lation model mentioned below.2 Kinematics characteristic of new-type robotFigure 1 shows the large universal robotZX165U ofKawasaki Company. Such robots can keep the translationS. Guo (&) ? H.-W. Li ? J.-C. Ji ? Z.-F. MingKey Laboratory of Intelligent Manufacturing and Robotics,School of Mechatronics Engineering and Automation, ShanghaiUniversity, Shanghai 200444, Peoples Republic of Chinae-mail: guoshuai123Adv. Manuf. (2014) 2:295302DOI 10.1007/s40436-014-0094-xmotion of its end-effector when it is palletizing. During thisprocess, fewer motors work, thus less energy is consumed.Furthermore, the damper settled on the base is an importantcomponent to keep the object stable and safe in the wholeprocess 6. All joints except the joint 2 (JT2) are the sameas those in usual robots. In this paper, we will only discusshow the spinning of joint 2 affects the kinematics charac-teristic of the whole robot 7, 8.Firstly, we theoretically analyze the kinematics charac-teristic of the special structure. Figure 2a demonstrates amixed system consisting of closed chain structure. Thereare n links and Liis the name of a certain mid-link in thesystem. Figures 2b, c are two kinds of logic open-chainsystems generated from Fig. 2a. The producing mode ofthese two open-chain systems is shown as following 3, 9.All links with which the closed chain intersects with out-side world are determined (such as Lmand Lm?4in Fig. 2a).And then all parallel links connected to these links (LmandLm?4) are figured out. Thus we can easily choose twodifferent routes (from Lmto Ln), where no parallel linkexists. Finally, two logic open-chain systems are generatedaccording to corresponding selected routes. As a result, allnon-redundant links exist in the logic open-chain systems,as shown in Figs. 2a, b.Considering the characteristic of the robot we arestudying, there are totally 8 links. So n will be equal to 8 ifm is equal to 1 (see Fig. 2). The degree of freedom (DOF)(N) is equal to 6 (the number of active joints), and thenumber of passive joints (P) is equal to 3. We assume thevariables of the active joints as qA;i(i = 1, 2, _, 6) and thevariables of the passive joints as qP;j(j = 1, 2, 3). In thetheoretical basis and research scope of rigid body dynam-ics, rigid displacement is only determined by the six activejoints exclusively 10. The relative equation isqP;j qP;jqA;1;qA;2?;qA;6;j 1;2;3:1Equation (1) shows the displacement dependency betweenpassive and active joints, which can be written asqP;j qP;jq1;q2;?;q9;j 1;2;3:2There are other two geometric constraints. One is thatlink Lm?1is fixed on the base. The other is that the closedchain is a parallelogram structure. Consequently, from Eq.(2) we can obtain?q4 ?q5 q3 q2:3Namely, the Lm?2moves the same as the Lm?3does. So themanipulator fixed on the link Lm?5keeps executing trans-lation consistently.Figure 3 shows the simulation model of the robot. Onlythe joint 2 is rotating in the simulation process. Figure 3ashows the initial state of the robot meaning that all theangles of joints are 0. Figures 3b, c demonstrate the cor-responding status of the robot when the joint 2 moves p/4clockwise and anticlockwise respectively. The manipulatorkeeps translation motion all the time, which is consistentwith the actual situation (see Fig. 3).Now, we can make a conclusion that the end-effector ofthe robot executes translation motion while all joints arefixed except the rotating joint 2.Fig. 1 Industry robot consisting of hybrid hinge structuresFig. 2 Closed-chain system and open-chain systems296S. Guo et al.1233 Kinematics analysisThe aim of kinematics analysis is to describe the kinematicrelationship between joints and end-effector. The analysissolutions can demonstrate how to control robot motion, andestablish the dynamic equation and the model for erroranalysis 11, 12.3.1 Forward kinematics and homogeneoustransformationThe position of joints and orientations of rigid bodies canbe described by coordinate systems fixed on joints 10. Sowe build coordinate system on every active joint 12.In Fig. 4, the coordinate system of ZX165U robot isestablished. First of all, the coordinate system 1 is estab-lished and it coincides with the base coordinate system 0.For convenience, the Z axis is parallel to the axis of eachrotating joint and the direction of Z axis is determined bythe rotation direction and fixed conventionally by righthand rule. The second axis is exactly consistent with one ofthe certain axes in previous coordinate system. Finally, thelast axis is determined by the right hand rule again.The dimension parameters in the coordinate system areshowninFig. 4,inwhichtheaxesZ1Z6areconsistentwiththerevolute joints JT1JT6, L1= 670 mm, L3= 1 100 mm,L4= 270mm,L5= 288mm,L6= 1100mm,L7= 228mm.According to homogeneous transformation formula4wecanobtainallmatrices:01T;12T;22tT;2t3tT;3t3T;34T;45T;56T;6eT. The character e represents end-effectorcoordinate;1iiTrefers to the transformation relationshipfrom the coordinate i-1 to i;1iiRrefers to the revolutionrelationship from the coordinate i-1 to i; D refers to thetranslation vector from the coordinate i-1 to i. In Fig. 4, forexample, coordinate system 1 shifts L2distance along the?Y, and shifts L1distance along the ?Z, then anticlockwiserotates p/2 around the ?Y. Then it can be transformed tocoordinate 2. The transformation matrix is12T TZL1TYL2RYp=2;5where TZ(L1) and TY(L2) refer to translation matrices; L1and L2are translation lengths along the corresponding axes;RY(h) refers to rotation matrix; h is anticlockwise rotationangle around the ?Y.When other matrices are computed, the transformationmatrix from end-effector coordinate to base coordinate isobtained 13, 14. The solution to the forward kinematics isas follows0eT 01T12T22tT2t3tT3t3T34T45T56T6eT nxnynz0oxoyoz0axayaz0pxpypz126643775;6wherenxc6c1c4? s1s3s4 ? s6c5c1s4 c4s1s3 c3s1s5;nyc6c4s1 c1s3s4 ? s6c5s1s4? c1c4s3 ? c1c3s5;nzs6s3s5? c3c4c5 ? c3c6s4;ox ? c6c5c1s4 c4s1s3 c3s1s5 ? s6c1c4? s1s3s4;oy ? c6c5s1s4? c1c4s3 ? c1c3s5 ? s6c4s1 c1s3s4;ozc6s3s5? c3c4c5 c3s4s6;Fig. 3 Motion status of robot manipulator when only JT2 moves to 0, -p/4 and p/4Fig. 4 Homogeneous coordinate system of Kawasaki ZX165UKinematic analysis and simulation297123axs5c1s4 c4s1s3 ? c3c5s1;ays5s1s4? c1c4s3 c1c3c5;azc5s3 c3c4s5;pxL3s1s2? L6c3s1? L5s1? L7c3c5s1 L7c1s4s5 L7c4s1s3s5;pyL5c1 L6c1c3? L3c1s2 L7c1c3c5 L7s1s4s5? L7c1c4s3s5;pzL1 L4 L3c2 L6s3 L7c5s3 L7c3c4s5;7where si= sin hi, ci= cos hi, sij= sin (hi?hj), cij= cos(hi?hj), i, j = 1, 2, _, 6 .3.2 Inverse kinematicsThe issue of robot inverse kinematics is to calculate all thecorresponding joint angles when the pose and location ofrobot are given 13. In other words, the joint variables hi(i = 1, 2, _, 6) can be determined based on kinematicsequation if the values of n, o, a, p and the needed geometricparameters are known 14. Equation (7) is used to get thesolution to the inverse kinematics equation. This paperdetermines the first three joint variables hi(i = 1, 2, 3) bymeans of algebraic method, and the common inversetransformation method 15 is used to determine the lastthree joint variables hi(i = 4, 5, 6).First of all, there is only h1left and other variables areeliminatedbyeliminationmethod16,17forEq.(7),suchaspx? L7ax ?s1L5 L6c3? L3s2;8py? L7ay c1L5 L6c3? L3s2;9pz? L1? L4? L7az L3c2 L6s3:10AssumeA L5 L6c3? L3s2:11Equation (11) is substituted back into Eqs. (8) and (9).Thus Eq. (12) is obtained.A2 px? L7ax2 py? L7ay2;sin h1 ?px? L7ax=A;cos h1 py? L7ay=A;tan h1 ?px? L7ax=py? L7ay;8:12therefore,h1 arctan ?px? L7ax=py? L7ay;h12 ?2p;2p?:13In Eqs. (10) and (11), the quadratic components of bothsides are added together, and we getAsin h2? Bcos h2 C;14whereB pz? L1? L4? L7az;C L26? L23 L25?2L5A A2 B2=2L3:Substituting sin u=A and cos u=B into Eq. (14) leadstoDcos h2 u C;D ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2 B2p:(15Thus,h2 arccos C=D ? u;h22 ?p=3;5p=12?:16We can substitute h2into Eqs. (10) and (11) and obtainh3 arctan B ? L3? cos h2=A L3sin h2? L5;h32 ?2p=3;25p=18?:17Also h4, h5, h6are determined as follows.Equation (18) is deducted by multiplying both sides ofEq. (6) with all inverse transformation matrices from joint1 to joint 3,3t3T?12t3tT?122tT?112T?101T?10eT 34T45T56T6eT:18So we get the equation below:n3xo3xa3xp3xn3yo3ya3yp3yn3zo3za3zp3z00012666437775c6s4c4c5s6c4c5c6?s4s6?c4s5?L7c4c5s5s6c6s5c5L6L7c5c4c6?c5s4s6?c4s6?c5c6s4s4s5L7s4s500012666437775:19The left part of Eq. (19) contains known joint variablesi.e., h1, h2and h3, while the other side contains the jointvariables h4, h5and h6to be solved. As a result,c5 azs3 ayc1c3? axs1s3;h5 arccos azs3 ayc1c3? axs1c3;h52 ?13p=18;13p=18?:8:20Get rid of the singular solution, h5=0,? c4s5 s3c1ay? s3s1ax? c3az;h4 arccos c3az? s3c1ay s3s1ax=s5;h42 ?2p;2p?;h6 arccos s3oz c3c1oy? c3s1ox=s5;h62 ?2p;2p?:8:214 Analysis of simulationModel simulation can verify the correctness of the kine-matic analysis, and provide intuitive understanding andrigorous data 1, 18. The robot model discussed in thispaper is built relying on the powerful function of theMATLAB platform.298S. Guo et al.1234.1 The toolbox and the modelThere are two qualified requirements 19, 20. The first oneis to build an accurate robot model. The second one is tocontrol the model for communication. With module data-base, we can build a complex mechanical model to realizethe simulation of the mechanical structure. Meantime,driver and the sensor modules are effective ways to connectSimmechanics with Simulink 21, 22.As shown in Fig. 5, the model system can simulate thesystem controlled by the designed GUI. As shown inFig. 5, there are six active joints, three passive joints andthree fixed joints.Figure 6 shows the module chart of the robot (KawasakiZX165U).The following functions are included in the designedGUI (see Fig. 7):(i)controlling the process of simulation;(ii)forwarding and reviewing the information of jointangle;(iii)obtaining the information of manipulator pose andthen saving the data.4.2 Verification of the kinematics analysisFigure 8 shows the picture of the real robot ZX165U. Inexperiment, the end-effector of the robot goes through twodifferent ways, among which one looks like the letter Wwhile the other looks like an S (see Figs. 9 and 10).Fig. 5 Generated model in SimmechanicsFig. 7 GUI designed for controlling the robot modelFig. 6 Module chart of Kawasaki ZX165U robotKinematic analysis and simulation299123The ZX165U robot can be driven to any arbitrary pointsin a reachable space as long as a set of appropriate values(joint angles or poses) are settled. In this experiment, fivegroups of data about points which indicate joint angles areput into the controller. Then the controller controls therobot to pass through these five points by linear interpo-lation (see Fig. 9) and curve interpolation (see Fig. 10).During this process, the pose and position of manipulatorcan be obtained at each corresponding point from teachingpendant. Then we let the robot model follow the sameprocess by using the same data of 5 points mentionedabove, but this time we use GUI to input data. Intuitively,the paths are drawn in a 3D coordinate system which onlyinvolves position values of robot but does not involve posedata. Figure 10 indicates the reverse.From Figs. 9 and 10, it can be apparently seen that theactual path matches the simulation path within a certainrange of minor errors. These minor errors cannot be noticedunless the paths are magnified by many times. As a matterof fact, the accuracy is already close to 0.02 mm, as shownin Tables 1 and 2.We selected five arbitrary sets of data from processabove and put all information in Table 1. Table 2 showsthe data indicating inverse kinematics that are obtained indifferent ways. When the real and simulation robots moveto the same designated pose and position, the data of theirjoint angles will be recorded. In Tables 1 and 2, r- ands- stand for words real and simulation respectively.In Table 1, the mean error between simulation result andactual result is 0.0206 mm with the maximum error of 0.037mm. Specifically, the position errors are 0.0258 mm, 0.0218mm and 0.0142 mm corresponding to X, Y and Z. Such pre-cisioncansatisfyanyindustrialrequirement.O,AandTstandfor X, Y, Z Euler angles, respectively. Furthermore, the poseerrorisalmostnegligible.Actually,wecanneverseetherobotmoving when it is shifted by a tiny distance. In Table 2, themean error is 0.0011? by rough calculation with the wrongdataNo.5 without JT4. It means that the model robot canreach a certain point within an acceptable error range (tinyerror is incognizable). Just as mentioned in the previous sec-tion, the kinematics analysis is absolutely correct and can beproved by the comparison analysis of the data.5 ConclusionsThis paper analyzes and verifies the kinematics character-istic (translation motion) of a new-type robot with a mixedhinge structure. On the basis of that, the kinematic analysiscan be easily carried out. Then the kinematic equation isestablished and solved with homogeneous transformationtheory. The robot model built with kinematic equationsolution can be used for simulation and acquire the simu-lation data via a designed GUI. Five groups of these dataare selected, analyzed and compared. The error betweenreal data and simulation data is calculated through theanalysis. As shown in Tables 1 and 2, the errors are slight,which means that the simulation of the model can abso-lutely meet the actual requirement. Furthermore, the cor-rectness of kinematics analysis is proved on the basis of theappropriate data analysis. This article hopes to providesome inspiration for those who are studying or going tostudy this kind of robot with a hybrid hinge structure.Fig. 8 Physical picture of the robot ZX165UFig. 9 Real and simulation W pathsFig. 10 Real and simulation S paths300S. Guo et al.123AcknowledgementsThis research was supported by the Key Sci-entific and Technological Project of Shanghai Science and Technol-ogy Commission (Grant No. 12111101004).References1. Bi LY, Liu LS (2012) 6-axis robot design and simulation basedon simulationX. In: Yang DH (ed) Informatics in control, auto-mation and robotics, vol 133. Springer, Heidelberg, pp 5375442. Song T, He YY, Wang P et al (2013) Kinematic analysis of seriesindustrial robot with hybrid hinge structure. Mach Des Res 5:893. Fathi HG, Olivier C, Ruvinda G et al (2000) Modeling and setpoint control of closed-chain mechanisms: theory and experi-ment. IEEE Trans Control Syst Technol 8(8):8018154. Wang KS, Terje KL (1988) Structure design and kinematics of arobot manipulator. Robotica 6(4):2993095. Selig JM (2011) On the geometry of point-plane constraints onrigid-body displacements. Acta Appl Math 116:1331556. Suguru A, Masahiro S (2006) Human-like movements of roboticarms with redundant DOFs: virtual spring-damper hypothesis totackle the Bernstein problem. In: Proceedings of IEEE interna-tional conference on robotics and automation, Orlando, May 20067. Choi HB, Konno A, Uchiyama M (2009) Closed-form forwardkinematics solutions of a 4-DOF parallel robot. Int J ControlAutom Syst 7(5):8588648. Pisla D, Gherman B, Vaida C et al (2012) Kinematic modeling ofa 5-DOF hybrid parallel robot for laparoscopic surgery. Robotica30(7):109511079. Gallardo-Alvarado J (2005) Kinematics of a hybrid manipulatorby means of screw theory. Multibody Sys Dyn 14(3/4):345366Table 1 Values of the end-effector pose calculated from six angles (forward kinematics)No.JT1 XJT2 YJT3 ZJT4 OJT5 AJT6 T1Angles/(?)-109.48522.272-14.039-237.41917.240170.801r-pose/mm-2 063.292-669.6811 554.426-176.225112.738-161.616s-pose/mm-2 063.300-669.6461 554.450-176.226112.738-161.6172Angles/(?)-107.58721.784-28.428-291.43713.683225.816r-pose/mm-1 963.292-569.6811 254.426-176.224112.738-161.616s-pose/mm-1 963.310-569.6441 254.450-176.225112.738-161.6163Angles/(?)-102.88517.2107.830-196.06731.841125.679r-pose/mm-2 063.292-437.8412 079.731-176.224112.737-161.616s-pose/mm-2 063.300-437.8382 079.730-176.224112.737-161.6154Angles/(?)-88.64917.809-17.936-158.89926.68084.877r-pose/mm-2 025.97610.8851 433.336-168.714132.354-157.510s-pose/mm-2 025.97010.9141 433.340-168.716132.345-157.5115Angles/(?)-62.99527.106-1.952-144.98452.93961.589r-pose/mm-2 025.990915.3271 721.282-168.716132.353-157.514s-pose/mm-2 025.970915.3331 721.300-168.718132.353-157.515Table 2 Angles of six joints calculated from the end-effector pose (inverse kinematics)No.JT1 XJT2 YJT3 ZJT4 OJT5 AJT6 T1Pose/mm2 025.976477.1061 244.825-168.715132.355-157.511r-angle/(?)-74.67024.682-24.595-135.56228.21953.006s- angle/(?)-74.66824.682-24.595-135.56228.20853.0082Pose/mm-2 063.293-835.5241 689.418-176.225112.737-161.616r-angle/(?)-113.90824.153-7.343-229.34024.738163.047s- angle/(?)-113.90924.153-7.343-229.34224.738163.0493Pose/mm-1 963.292-369.6811 254.426-176.225112.738-161.615r-angle/(?)-101.47119.467-29.217-311.3729.473243.137s- angle/(?)-101.47119.468-29.217-311.3739.474243.1384Pose/mm-2 243.50037.7661 647.109-176.225112.738-161.616r-angle/(?)-88.54624.849-8.980-160.58214.64387.526s- angle/(?)-88.54624.8498-8.980-160.58114.64387.5255Pose/mm-2 256.349-170.7711 897.625-175.823101.698-161.497r-angle/(?)-94.34424.2350.035-180.81111.734109.330s- angle/(?)-94.34524.2360.035-179.19311.735109.328Kinematic analysis and simulation30112310. Kuo SR, Yang YB (2013) A rigid-body-qualified plate theory forthe nonlinear analysis of structures involving torsional actions.En
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