1426 The Mo dified Ta ugh tD ata Metho d 6.2.2 ExperimentalVerification for Modified TaughtData Method Using aGaussian Network (1)Conditions of the Experiment In order to verify the effectivenessofaGaussiannetwork basedonthe 2nd order model shown in 6.2.1, theexperimentofcontourcontrol using an XY table wasmade(refertothe experimentinstrument E.4). The controlofthe XY table is constructed by twoGaussian networks in equation (6.46) for independentaxes in order to conduct the independentmovementofthe x axis andthe y axis, respectively.The experimentalresults will be shown when the objectivetrajectory of theXYtable is as u x ( t )= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 4 . 8(0 ≤ t<0 . 5) 4cos  π ( t − 0 . 5) 2  + 4 5 cos  5 π ( t − 0 . 5) 2  (0. 5 ≤ t<4 . 5) 4 . 8(4 . 5 ≤ t ≤ 5) u y ( t )= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 0(0 ≤ t<0 . 5) 4sin  π ( t − 0 . 5) 2  + 4 5 sin  5 π ( t − 0 . 5) 2  (0. 5 ≤ t<4 . 5) 0(4 . 5 ≤ t ≤ 5). (2) Generation of the TeachingSignal In thedeterminationofthe initial parameters of theGaussian network, the definedv alue K p =5 [1/s] of the po sition lo op gain of the equipmen ti nt he equation (6.52) wa su sed, and the critical condition from K v =4K p to K v = 20[1/s]ofthe velocityloopgain usedinthe industrial field, which cannot be defineddirectly,was used. This K v whichwas notthe value measured by the actual device wasconsideredtocontainlarge errors. But the high-precision contourcontrol can be realizedbecause in the proposed methodthe Gaussian network forthe modification elementwas usedand the inverse dynamics can be constructed based on the learningfromthe actualequipment. Besides, the linearizable region condition of the equipmentwas consi dered as 15[cm] in themov able regionofthe table.The output scale of thetwo Gaussianunitsabout the position were setas − 7 . 5 ≤ φ ( r ) ≤ 7 . 5[cm] when x p max =10[cm].The maximalvelocityofthe equipmentwas consideredas 9.3[cm/s].The outp ut scale of thetwo Gaussianunitsabout velocitywere set as − 11. 325 ≤ φ ( dr/dt) ≤ 11. 325[cm/s]when x v max =15[cm/s]. Concerning the safetyofthe equipment, the output scale of thetwo Gaussianunitsabout acceleration were setas − 60. 4 ≤ φ ( d 2 r/dt 2 ) ≤ 60. 4[cm/s 2 ]whichwas notover the maximala ccelerationo f8 4.7[cm/s 2 ]. The teachingsignal of learningfor theabove Gaussiannetwork with initial parameters came fromthe output 6.2M od ified Ta ugh tD ata Metho dU sing aG aussian Net wo rk 143 −5 0 5 0 1 2345 −5 0 5 T ime[ s ] y a x i s [cm ] Objec t i v e tra jec t o ry Gaussi a n C onv ent iona l x a x i s [cm ] −5 0 5 −5 0 5 x [cm ] y [cm ] Gaussi a n o b jec t i v elo c us S t a rt point C onv ent iona l (a) Following trajectory (b) Following locus Fig. 6.9. Experimental results by using XY Table data obtained by the computer when the properties of the servosystem ex- pressed in the movementcan be given arbitrary andthe XY table wasmoved with the original objectivetrajectory in the experiment. The sampling time interval is ∆t p =10[ms] when makingthe teaching signal wasthe same as that of thecontourcontrol experiment. Therefore, the teaching signals were obtained as ( u l , x l )=(u ( l∆t p ), y ( l∆t p ), y ( l∆t p ), dy( l∆t p ) /dt, dy( l∆t p ) /dt, d 2 y ( l∆ t p ) /dt 2 , d 2 y ( l∆ t p ) /dt 2 ), l =0 , ···,5 00. Ho we ve r, the data obtained by the computer from the actual XY table we re only thev elo cit yo utput dy /dt of thetechogenerator obtained from the servomotor.The position output y wa st he nu merical in tegralo ft he ve lo cit yo utput andt he acceleration output ¨y wasthe numerical differential of the velocityoutput.Additionally, the ve- locityoutput dy/dt of thetechogeneratorwere the results whose noise have be en deleted by the band pass filter of 0 ∼ 10[Hz].W ith the learningr ate of η =0. 001 during the Gaussiannetwork learning,the learning process will stop when thecommon threshold of the x axisand the y axiswas belowthe 0.35[mm]. Therew ere 182l earningt imes when the data set of the teac hing signal ( u l , x l ), l =0, ···, 500 wasregarded as one time learning. (3)Experimental Results of the Contour Control By using the Gaussiannetwork shown in the Fig. 6.7 afterlearning, the ex- perimentalresults of contour control with the input of the XY table usingthe revised taught data revised by the Gaussian network were shown. Fig. 6.9(a) showsthe following trajectory of the experimental results in the Gaussian network afterlearning. Fig. 6.9(b) shows the following locus in the XY plate. Here, the objectivetrajectory without anyrevision wasused in theconven- tional method.Comparin gwith the conventional metho dwithout anyrevi- 1446 The Mo dified Ta ugh tD ata Metho d sion, the following trajectory wasregarded as the following locus wasclearly approaching the objectivewhen using the Gaussiannetwork to realize the revision. Therefore, the high-precision control can be realized. 6.3A Mo difiedT augh tD ata Method for aF lexible Mec hanism When the mo ve men to ft he robo ta rm be comes faster, the flexible mec hanism of the robot arm is necessary for the flexibilityofthe manipulator andflexible connection of the link. If neglecting the characteristics of flexibility,oscillation or overshoot in themovementofthe robotarm will occur.The contourcontrol performance will deteriorate andthe determination time of theposition will increase. According to the flexiblemechanism, the mathematicalmodel is made. Based on this equation, the taughtdata mo dification elementofthe former sectionisconstructed. The high-precision contourcontrol can be realizedin the robot manipulatorofthe flexiblemechanism. Then,the requirement of ahigh-speed, high-precision movementofama- nipulator in ind ustr y, the proposed technique as the control methodwhich canbring the current system into maximal effect is very important without huge change of hardwareinthe current system. 6.3.1Derivation of Contour Control with Oscillation Restraint Using the Modified TaughtData Method In order to realize contour control with oscillation restraintinthe movementof the flexible arm, the block diagram of thecontrol system in theone axisflexible arm shown in 6.10isconsidered. In the Fig. 6.10, R ( s )denotes the objective trajectory, Z ( s )denotes the position of the arm fulcrum, Y ( s )denotes the output(tip position of the arm), K p denotesthe position loop gain. The modified taughtdata method (refer to 6.1.1) is adopted with the modification elemen t F 3 ( s )for constructing the taught data revised fromthe objective trajectory of arm. In this section, although only one axis is considered, the realizationofcontrol with oscillation restraintfor oneaxis can also be adapted forthe multi-axis mechatronic servosystem. The dynamics of the servosystem whichcausesthe movementofthe arm is expressedbythe 1st order model (refer to the 2.2.3). Theflexible arm of the elasticitybodyisexpressed by the 2nd order system, where ζ L denotesthe damping factor and ω L denotesthe naturalangularfrequency. Therefore, the whole transfer function of the control system of this flexible arm is expressed as 6.3A Mo difiedT augh tD ata Metho df or aF lexible Mec hanism 145 G 3 ( s )= a 0 s 3 + a 2 s 2 + a 1 s + a 0 (6.56) a 0 = K p ω 2 L a 1 = ω 2 L +2ζ L ω L K p a 2 = K p +2ζ L ω L . In the modified taughtdata method,the modification element F 3 ( s )is derive du sing the po le assignmen tr egulatora nd the minim um order observ er fort he cont rols ystem to solv et he ch aracteristics of thec losed-lo op system and transfer it to the op en-lo op system whose relationship of the input and outputi se quiv alen tt ot he transferf unctiono ft he closed-loo ps ystem. Fo r the control system of equation (6.57), the modificationelementisas F 3 ( s )= b 5 s 5 + b 4 s 4 + b 3 s 3 + b 2 s 2 + b 1 s + b 0 ( s − γ 1 )(s − γ 2 )(s − γ 3 )(s − µ 1 )(s − µ 2 ) (6.57) b 0 = a 0 ( h 0 − g 0 ) b 1 = a 0 ( h 1 − g 1 )+a 1 ( h 0 − g 0 ) b 2 = a 0 (1 − g 2 )+a 1 ( h 1 − g 1 )+a 0 ( h 0 − g 0 ) b 3 = a 1 (1 − g 2 )+a 2 ( h 1 − g 1 )+h 0 − g 0 b 4 = a 2 (1 − g 2 )+h 1 − g 1 b 5 =1− g 2 g 0 = l 2 f 1 +(l 1 l 2 + k 2 ) f 2 +(l 2 2 + l 1 k 2 − l 2 k 1 ) f 3 g 1 = l 1 f 1 +(l 2 1 + k 1 ) f 2 +(l 1 l 2 + k 2 ) f 3 g 2 = f 1 + l 1 f 2 + l 2 f 3 h 0 = l 2 − a 0 f 2 − a 0 l 1 f 3 h 1 = l 1 − a 0 f 3 l 1 = − ( µ 1 + µ 2 ) l 2 = µ 1 µ 2 k 1 = − l 2 1 + l 2 − a 1 + a 2 l 1 k 2 = − l 1 l 2 − a 0 + a 2 l 2 - K p + 1 - s F ( s ) L ω L ω L ωs + s + 2 22 2ζ Objec t i v e tra jec t o ry R ( s ) M o t o r o utp ut Z ( s ) F ollow ing tra jec t o ry Y ( s ) Tau ght d a t a U ( s ) 3 S e rvo c ontroller a nd mot o r F lex i b le a r m Fig. 6.10. Blo ck diagram of mo dified taugh td ata metho df or flexible arm 1466 The Mo dified Ta ugh tD ata Metho d f 1 = − ( d 1 − a 2 d 2 +(a 2 2 − a 1 ) d 3 − a 0 − a 3 2 +2a 1 a 2 ) /a 0 f 2 = − ( d 2 − a 2 d 3 − a 1 + a 2 2 ) /a 0 f 3 = − ( d 3 − a 2 ) /a 0 d 1 = − γ 1 γ 2 γ 3 d 2 = γ 1 γ 2 + γ 2 γ 3 + γ 3 γ 1 d 3 = − ( γ 1 + γ 2 + γ 3 ) . In the equation (6.57), the modificationelementexpressed by the 1st order transfer function for the rigid body system shown in 6.1.1isexpanded into the fifth-order modificationelementincluding the observer. γ 1 , γ 2 , γ 3 arethe poles of the regulator and µ 1 , µ 2 arethe poles of the minimal order observer. From thetaughtdata u ( t )generated through the modificationelement F 3 ( s ), tracing correctly the objectivetrajectory without oscillation in theflexible arm can be realized. 6.3.2 ExperimentalVerification of Oscillation RestraintControl Using the Modified TaughtData Method Through the experimental device of the flexible arm whichemphasizes the arm elasticitycharacteristic of oneaxis of the mechatronic servosystem, the effectiv eness of the prop osed metho dc an be ve rified. With the metalp late in the flexible arm, the bo ttom edge of this flexible arm is installed in the base seat of the drivedevice whichconsistsofcombinationswith aDCservomotor andt he ball screw.T he cont rolp urp ose is to mak et he flexiblea rm correspo nd to the ob jective tra jectory without the oscillation from the static state of the base seat to another static state after moving to the objectiveposition.The size of them etal bo ardi sa sf ollow s, the length is 0.83[m], width is 0.028[m] and heightis0.002[m]. The mass is 351[g], the elasticitycoefficientis K = 73785. 2[g/s 2 ], the viscous frictional co efficien ti s D L =3. 626[g/s], then atural angular frequencyi s ω L =14 . 5[Hz],the damping factor is ζ L =3. 56 × 10 − 4 , andthe position loop gain is K p =15[1/s]. Theobjectivetrajectory is the moving trajectory with the velocityof0.03[m/s]. The design parameters in the equation (6.57) are the poles of the regulator γ = − 10 (three-fold root) andthe poles of the observer γ = − 20 (two-fold root). Fig. 6.11shows the experimentalresults of the proposed methodwith the equiv alen tv elo cit ym ove men tw ith 0.03[m/s] of the base seat. The horizon tal axis of the graph is time and the verticalaxis is the oscillation in the center of gravityofflexible arm. From theresults of the oscillation in the Fig. (a) with the modified taught data methodofthe proposed method,the maximal amplitude is 0.45[mm]. Themaxi mal value of the oscillation in the results of the equivalentvelocitymovementinFig. (b) is 2.0[mm]. Comparing with one another, the amplitudeofoscillationinthe center of gravityofthe arm is reduced to the 1/4. Theleft oscillation is from the modelingerrorwhich cannotbegenerated in the ideal simulation results. 6.3A Mo difiedT augh tD ata Metho df or aF lexible Mec hanism 147 0 510 15 − 0 . 3 − 0 . 2 − 0 .1 0 0 .1 0 . 2 0 . 3 T ime[ s ] C ent e r of g r a v i ty[cm ] (a) Modifiedtaughtdata method 0 51 0 15 − 0 . 3 − 0 . 2 − 0 .1 0 0 .1 0 . 2 0 . 3 T ime[ s ] C ent e r of g r a v i ty[cm ] (b) Uniform velocitymovement Fig. 6.11. Experimental result Theadaptivenesspossibilityofthe modeling errorofthe modifiedtaught data methodwas investigated. With the simulation, the scale of the oscillation arm when the design error is putinthe damping factor ζ L or thenatural angular frequency ω L wascalculated. Whenthe size of theoscillationofthe armwith the putdesign err or waswithin the allowance of modeling errorin order to letitbelow10[%]ofthe maximaloscillationwithout design error, and the natural angular frequency ω L is − 4 . 1 ∼ 2 . 8[%], then thesize of the oscillation became − 100 ∼ 3549[%]inthe damping factor ζ L . 7 Master-Slave SynchronousPositioningControl When onerobot manipulatorhas manylinks and eachofthem corresponds to on eaxis of the motor, it is very important to realize the synchronous po- sitioning of eachaxis in the high-precision contour control. In this chapter, we propose anew high-precision contour control not subject to the restriction of the currentconditions. It is adaptedfor themaster-slave synchronous po- sitioning control, whichsupp osesone axisasthe master-axisand another as theslave-axis without alar ge characteristicvalue K p of theservosystem. 7.1 The Master-Sla ve Sync hronousP ositioningC on trol Method The typical applicationswhichrequires synchronous movementbasedonthe relationship between the master axis and the slave axisare tapping pro cess work, installing tapping tools in therotated masteraxis and processing screw by an up anddownmovementofmaster axis(sending) with rotation, and so on. Since the process specification of the screw pitchofthe product is regular, if the rotationsofthe master axisand sending position are not synchronous, the screw pitch will be changed,ortools will be brokenanthe extreme case. The master-slave synchronous positioningmethodistogenerate modifica- tion term of inverse dynamics forthe servosystem and with this modification term, the position outputofthe master axisistaken as theinput signal of theslave axis. If theremixed with disturbance in the master axis, from the prop osed method, the slave-axis synchronous positioningmethodcan be im- plemented properly. The command of the servosystem of eachindustrialrobot axisisindepen- dentlygiven. The command of the slave axisisrevised by software. Therefore, since it is expected that the existing hardware is notchanged andthe desirable synchronous positioning can be realized, the value of anyindustrialapplica- tion of this method is very high. M. Nakamura et al.: Mechatronic Servo System Control, LNCIS 300, pp. 149–168, 2004. Springer-Verlag Berlin Heidelberg 2004 1507 Master-Slave Sync hronous Po sitioning Con trol 7.1.1NecessityofMaster-Slave Synchronous PositioningControl (1) Mathematical Mo del of the Ob jectiv eo ft he Master-Sla ve Sync hronous Po sitioning Con trol Concerning the control objectivewith the requirementofposition synchro- nization,the overall controlsystem with the control equip mentand the servo system arealmost all controlling master axes and slave axesindependently. Forthe actuator, many servomotorshave been used.Inorder to use high- performance deviceinthe servomotorsand their controlequipments, the prop ertyofvelocitycontrol of theservomotor is consideredasafixedcon- stantwhen the processing speedisnot very highand the propertyofthe position control is only considered (refer to 2.2.3). Therefore, the transfer function of the servosystem is expressed as P x ( s )= K px s ( s + K px ) U x ( s )+ 1 s + K px D x ( s )(7.1a ) P y ( s )= K py s ( s + K py ) U y ( s )(7.1b ) where, the x axisisthe master axis, the y axisisthe slave axis, P x ( s ), P y ( s )are the positions of the x axisand the y axis, U x ( s ), U y ( s )are the velocityinput referenceo ft he x axisa nd the y axis, K px , K py have the meanings of K p 1 in the equation (2.20) for the 1st order model written in the item 2.2.3 about the x axisand the y axis. Thedisturbance, expressed as D x ( s ), is only added in the master axis, supposed in the tap processing. The first item of equation (7.1a ) describesthe relationsh ip between the velocityinput U x ( s )a nd the po sition outputofthe x axis. Thesecond item describes the relationship between the disturbance D x ( s )inputing into the x axisand position outputofthe x axis. Thepropertyofcontrol system is describedby K px , K py .T heirv alues aredetermined by the structureofthe hardware. In addition, 1 /s before the servosystem denotes the integral from the velocityinput to the position input. The control purpose of themaster-slave synchronous positioningcontrol is to makethe position outpu tofthe x axisand the y axisare synchronous, that is,tomakethe following equation successfully P y ( s )=k c P x ( s )(7.2) where k c is the proportional constant. If the position output of the x axis andthe y axissatisfies equation (7.2), theposition synchronization can be realized. (2) Issues without Expectation of PositionSynchronization If the dynamics of the x axisand the y axi sare notconsideredand the velocity input U y ( s )o f y axisi s k c times of ve lo cit yi nput of the x axis, thep osition output of the y axisisas 7.1T he Master-Sla ve Sync hronous Po sitioning Con trol Metho d1 51 P y ( s )= k c K py s ( s + K py ) U x ( s ) . (7.3) The position outputerrorofthe y axistothe x axis, fromequation (7.1a ) and(7.3),isas k c P x ( s ) − P y ( s )= k c ( K px − K py ) ( s + K px )(s + K py ) U x ( s )+ k c s + K px D x ( s ) . (7.4) From equation (7.4), if thereisnopositi on synchronization,the position out- putofthe x axisand the position outputofthe y axisare notsynchronous because the position output error is not 0. Since the position loop gains of the x axisand the y axisare difference,there exists adeviationofposition output. From this case, if we use velocityinput referenceofthe x axiswithout change, thesynchronous actioncannotberealized because the position lo op gains of the x axisand the y axisare notthe same.Inaddition, without setting the compensation of the y axisfor thedisturbance D x ( s )ofthe x axisisanother reasonfor synchronization. 7.1.2 Derivation and PropertyAnalysis of the Master-Slave Synchronous Positioning ControlMethod (1) Derivation of the Master-Slave Synchronous Positioning ControlMethod In the former part, the problem that the k c times of ve lo cit yi nput reference of the x axisissimply used as the velocityinput referenceofthe y axiswas in- trod uced. In order to make the po sition of the y axiss ync hronizationw ith the po sition of axis x ,t he ve lo cit yi nput referenceo fa xis x is revised for comp en- sating the differentdynamics between axis x andaxis y .Ifthe velocityinput referenceo fa xis y is pe rformedl ik et his,t he po sition sync hronizationc an be realized. Ho we ve r, if pe rforming ar evision in the ve lo cit yi nput referenceo f axis x is only for the velocityinput referenceofaxis y ,the compensation for disturbance in axis x cannot be implemen ted and the high-precision po sition sync hronizationc annotb er ealized. But if the po sition output of axis x is feedbackasthe position input of y ,the impact of adisturbance in the axis x can be ove rcome by the feedbac ko ft he po sition output of axis x .I ft he only feedback in the position outputofaxis x without anychange, the syn- chronization of axis x with the movementdelaycausedbythe dynamics of axis y cannot be realized. Therefore, by using the inverse dynamics of axis y andrevising the feedbacksignal of the position output of axis x ,the position synchronizationcan be realized. Namely,inorder to change thedynamics of axis y into 1, feedforward compensation is performedaccordingtothe inverse dynamics of axis y . In order to realize the above properties,the inverse dynamics of the1st order system of axis yF s ( s )can be constructed as 1527 Master-Slave Sync hronous Po sitioning Con trol F s ( s )= s + K py K py . (7.5) The master-slave synchronous positioning control method, with the position outputofaxis x as theposition input of axis y ,can be given according to F s ( s ), is shown. This master-slave synchronous positioningcontrol method is based on the prerequisite of differentdynamics between axis x andaxis y .It can be alsoused for compensation for anyfatal effects of disturbance D x ( s ) mixed in to axis x .W hen feedback the po sition outputo fa xis x ,i ti sa ssumed that therea re no observ ational noises (Int he mech atronic serv os ystem, there are no observ ational noise be cause of the po sition test by pulsem easuremen t in the enco der). Moreo ve r, discussion is carriedo ut with the assumptiono f correctly modelingthe dynamics of axis y in the following part. When a modelingerrorexists, it is necessary to adjust correctly the value of K py in equation(7.5) to minimizethe modeling error. The block diagram of the master-slave synchronous positioning control methodisillustrated in Fig. 7.1. (2) PropertyAnalysis of the Master-Slave Synchronous Positioning ControlMethod The position outputofaxis y in the master-slave synchronous positioning control methodisas P y ( s )= k c K px s ( s + K px ) U x ( s )+ k c s + K px D x ( s ) . (7.6) U ( s ) F ( s ) D ( s ) - K p x + - K p y + + + k 1 - s s 1 - s 1 - s x x P ( s ) x P ( s ) y c Xax i s se rvo system S e rvo c ontroller M o t o r a nd mec h a nis m p a rt P o s i t ion loop M odificat ion element Yax i s se rvo system S e rvo c ontroller M o t o r a nd mec h a nis m p a rt P o s i t ion loop Fig. 7.1. Blo ck diagram of master-sla ve sync hronous po sitioning con trol metho d [...]... Master-Slave Synchronous Positioning Control There are two kinds of supposed disturbances in the required equipment when performing position synchronization Concerning these disturbances, simulation is made with (a) master-slave synchronous positioning control method, (b) without expectation of position synchronization and (c) a tracking control method between two servo systems [35] The tracking control. .. comparisons, the effectiveness of the master-slave synchronous positioning control method is verified The impact of a disturbance in master-slave synchronous positioning control method is due to the different operation in the computer for the controller for the differential of inverse dynamics Fs (s) expressed in equation (7.6) (ii) Saw-tooth-shape cycle disturbance The saw-tooth-shape disturbance refers to the... Fig 7.3 In order to compare the master-slave synchronous positioning control method with step disturbance, the simulation results of the tracking control method between the servo system without position synchronization is shown in Fig 7.4 From the left side, the locus of the XY table, time change of axis x and y and trajectory error e(t) = px (t) − py (t) of axis x and axis y are illustrated Fig 7.4(b)... axis x when exhibiting the saw-tooth-shape cycle disturbance However, from the graph of trajectory error, there are two times of trajectory error 0.3[mm] in Fig (c) comparing with 0.15[mm] in Fig (a) when considering the impact of the saw-tooth-shape cycle disturbance input between 7.1 The Master-Slave Synchronous Positioning Control Method 157 0∼2[s] If there are no saw-tooth-shape cycle disturbances... former part, an experiment is carried out with the same conditions Fig 7.7 illustrates the experimental results under the step disturbance with the master-slave synchronous positioning control method and simulation results of the tracking control method between two servo system without position synchronization Fig 7.8 illustrates the e(t) [mm] e(t) [mm] 0.2 0 5 0 −5 −0.2 0 1 2 3 Time[s] 0 4 (a) Master-slave... of axis x and disturbance Dx (s) in axis x is for avoiding the divergence of the position input reference of axis y Therefore, there is no problem when using (7.6) as the modification element Fs (s) and the effectiveness of the master-slave synchronous positioning control method can be verified 7.1.3 Experimental Test of the Master-Slave Synchronous Positioning Control Method By using the master-slave synchronous... two servo systems [35] The tracking control method between 154 7 Master-Slave Synchronous Positioning Control two servo systems in (c) is the method used to compensate for the velocity input of axis x by the position output feedback of axis x The velocity input contains the features of the ramp and the step After cutting the screw and returning to a trapezoidal wave as in Fig 7.2, it is function as ⎧... results are consistent with the situation of step disturbance Therefore, the effectiveness of the master-slave synchronous positioning control method was verified (2) Experiment of Master-Slave Synchronous Positioning Control In the former part, a simulation was made with a disturbance generated in the computer and good results were obtained Next, an experiment will be made with the actual XY table The experiment... saw-tooth-shape cycle disturbance, the simulation results of the tracking control method between the servo system without position synchronization is shown in Fig 7.6 The trajectory error e(t) = px (t) − py (t) of axis y to axis x is only shown, which is different from the simulation results with step disturbance Fig (b) has almost the same results when existing step disturbance From the Fig (c) and. .. Conventional method trajectory locus 0 0 4 py(t) 40 0 0 (a) Master-slave synchronous positioning control method trajectory trajectory error 60 0 px(t), py(t) e(t) [mm] 0 locus 0 0.2 e(t) [mm] 40 0 trajectory error 60 px(t), py(t) [mm] py(t) [mm] 60 px(t), py(t) 0 −0.2 0 1 2 3 Time[s] 4 0 1 2 3 Time[s] 4 (c) Tracking control method between two servo system Fig 7.4 Simulation results on step disturbance the . requirementofposition synchro- nization,the overall controlsystem with the control equip mentand the servo system arealmost all controlling master axes and slave axesindependently. Forthe actuator, many. position synchronization. Concerning these disturbances, simula- tion is made with (a) master-slave synchronous positioningcontrol method, (b) without exp ectationofposition synchronization and( c). master-slave synchronous positioningcontrol method was verified. (2) ExperimentofMaster-Slave Synchronous PositioningControl In theformerpart, asimulation wasmadewith adisturbance generated in the