Mechatronic Servo System Control - M. Nakamura S. Goto and N. Kyura Part 12 pps

1587 Master-Slave Sync hronous Po sitioning Con trol lo cus tra jectory tra jectory error 02 0 4 06 0 0 20 4 0 60 p x ( t ) [ mm] p y ( t ) [ mm] 0 1 234 0 20 4 0 60 T ime[ s ] p x ( t ) , p y ( t ) [ mm] p x ( t ) , p y ( t ) 0 1 234 − 0 . 2 0 0 . 2 T ime[ s ] e ( t ) [ mm] (a) Master-slave synchronous positioning control method locus trajectory trajectory error 020 4 060 0 20 4 0 60 p x ( t ) [ mm] p y ( t ) [ mm] 0 1 234 0 20 4 0 60 T ime[ s ] p x ( t ) , p y ( t ) [ mm] p x ( t ) p y ( t ) 0 1 234 −5 0 5 T ime[ s ] e ( t ) [ mm] (b) Conventional method locus trajectory trajectory error 02 0 4 06 0 0 20 4 0 60 p x ( t ) [ mm] p y ( t ) [ mm] 0 1 234 0 20 4 0 60 T ime[ s ] p x ( t ) , p y ( t ) [ mm] p x ( t ) , p y ( t ) 0 1 234 − 0 . 2 0 0 . 2 T ime[ s ] e ( t ) [ mm] (c) Tr ac king con trol metho db et we en two serv os ystems Fig. 7.7. Experimental results of master-slave synchronous positioning control metho dfor the step disturbancebased on XY table exp erimental resultswith the saw-tooth-shapecycle disturbance under the same conditions. The whole simulation results and experimental results are consistent. How- ever, from the trajectory error with the saw-tooth-shapecycle disturbance, the sa w-to oth-shap ec ycle disturbance can be sho wn in thes im ulation. But it can- not be shown in the experiment. The reasonisthatthe impact of quantization errorisalmost unchanged in order to connectthe A/D, D/Aconverterinto the con troller be twe en the XY table and the pe rsonalc omputer. (ii) Disturbanceinput in the actual equipment In order to approach the motionconditions adoptedinthe actualequipment, an experiment, in whichdisturbance wasdirectly added to the experiment equipment, wascarriedout. In the XY table,the y axisiscompletely moved alongthe directionofthe x axisinorder to make the y axismoving based on the x axis. Therefore, it is only possible to putdisturbance into the x 7.1T he Master-Sla ve Sync hronous Po sitioning Con trol Metho d1 59 0 1 234 − 0 . 2 0 0 . 2 T ime[ s ] e ( t ) [ mm] 0 1 234 −5 0 5 T ime[ s ] e ( t ) [ mm] (a) Master-slave synchronous positioning control method (b) Con ve nt ional metho d 0 1 234 − 0 . 2 0 0 . 2 T ime[ s ] e ( t ) [ mm] (c) Tracking control methodbetween two servosystems Fig. 7.8. Simulation results for the sawtooth state cycledisturbancewave-like based on XY table axis because the force opposite to the movementofthe x axiscan be added in the y axis. When theXYtable is moving,astep disturbance is gener- ated duetothe externalforce added on the y axisof700∼ 800[N]inthe XY axis. In theXYtab le,anexperimentwas carriedout basedonthe tracking control methodbetween two servosystems and the master-slave synchronous positioning control method. The input command is u x ( t )=28. 2(0 ≤ t ≤ 5). When addingdisturbance to the actual equipment, the experimental re- sults based on the master-slave synchronous positioningcontrol method and tracking controlmethodbetween two servosystems are illustrated in the Fig. 7.9,respectively.The left-hands side illustratesthe resultsofthe XY table locus,trajectory error e ( t )=p x ( t ) − p y ( t )ofthe y axiscorresponding to the x axis. From thelocus of theXYtable,position synchronization of the y axis output in the x axisoutput canberealized based on both methods. However, fromthe trajectory errorgraph, there appeared large errors about0.6[mm] after the beginning of theexperimentthe both methods. The reason is that the feedbacksignal is not input duringthe initial step in order to discretely approximate the differential of feedbacksignal of the x axisposition output forboth methods. In Fig. 7.9(c), the err or reduction is very slow afterthe be- ginning of theexperimentand the maximalerroramplitude is 0.15[mm] after droppingofthe constant. Butthe amplitude of theconstanterrorinFig. (a) is ve ry small at 0.07[mm]. Therefore, thee ffectiv eness of the master-sla ve syn- 1607 Master-Slave Sync hronous Po sitioning Con trol lo cus tra jectory error 0 1 00 0 1 00 p x ( t ) [ mm] p y ( t ) [ mm] 0 1 2345 0 0 . 2 0 .4 0 . 6 T ime[ s ] e ( t ) [ mm] (a) Master-slave synchronous positioning control method lo cus tra jectory error 0 1 00 0 1 00 p x ( t ) [ mm] p y ( t ) [ mm] 0 1 2345 0 0 . 2 0 .4 0 . 6 T ime[ s ] e ( t ) [ mm] (c) Tracking control methodbetween two servosystems Fig. 7.9. Experimental results with actual disturbancebased on XY table chronous positioning control methodwhen adding disturbance to the actual equipmentwas verified. 7.2C on tourC on trol with Master-Sla ve Sync hronous Positioning In the proposed master-slave position synchronization, thereexists alarge de- viation in thefollowing locus of objectivelocus at thecorner of thetrajectory whenusing contour control for anyobjectivetrajectory. In master-slave position synchronization, the response locus of the objec- tivetrajectory hasadeviation because of the response delaytothe objective trajectory at the corner of the trajectory.Therefore, the following two control methods added to the master-slave synchronous positioningcontrol method areproposed, namely,the metho dwith command of extending the linear in- terval before following alocus approximating the objective trajectory after putting into objectivetrajectory of theposition,and the metho dofhigh- precisioncontourcontrol with alittle time synchronizationatthe corner. The servocontrollersofindustrialmechatronic systems almost all control eachaxis independently.Inthe proposed method,itisnot necessary to change theexisting hardware andsoftware, which provide commands, it is only nec- essary to revise with simpledefinition of the master axis and the slave axis. 7.2C on tour Con trol with Master-Sla ve Sync hronous Po sitioning 161 7.2.1Derivation of the Contour Control Method with Master-Slave Synchronous Positioning (1)D efinition of the Problem Thec on trol ob jective is ak ind of mec hatronic systems whic hh as as tructure with two axesl ik ea nX Yt able.T he cont rolp urp ose is to realize high-precision contourcontr ol of amechatronic servosystem tracing an objectivetrajectory ev en without strictp rope rt y( K p ). Moreo ve r, fort he mech atronic serv os ystem with adefined master axis and slave axis, if theproblemfor themaximal two axescould be solved, it can be expanded to the multiple axes if the mecha- tronic servosystem with multiple axes also contains the same relationship between the master axis and the slave axi s. In thetypical processing as tap, since the impact of disturbance mixed into the control system on the slave axiscan be neglected comparing with thatonthe master axisinthe direction of rotation,therefore, adisturbance can be only mixedintothe master axis. In thecontourcontrol of themechatronic servosystem, the objectivelo- cusisapproximatedbydifferentlines (refer to 1.1.2 item 8). In the n th linear interval, the velocity v is putintomove fromthe objective point(x n ,y n )as originalpointtothe n +1th objectivepoint(x n +1 ,y n +1 ). If thecontrol objec- tive reaches the objectivepoint(x n +1 ,y n +1 ), thesame movementwill begin fromthis new originalpoint. Suchakind of movementwill stop until when reachingthe finalobjectivepoint. However, there exists adelayinthe servo system andthe finalpartofthe line trajectory is lost because the position outputcannotreachthe objective pointevenwhen the position input of the objectivelocus reaching the objectivepoint. Therefore, contourcontrol for the trajectory composedofthe line trajectorieswill be separately consid ered into alinear interval and acorner part. (2) Control Method withaLinear Interval In order to realize the correct contour control, it is necessary to makethe proportional relationship between the master-axis position output P x ( s )and the slave-axis position output P y ( s ). The control system with this relation adapts the master-sla ve sync hronous po sitioningc on trol method in tro duced in 7.1.Inthe master axis, thevelocityinput U x ( s )asstandard is the input and in the slave axis, themaster-axis position output is regarded as theinverse dynamics mo dification element F s ( s )ofthe slave axis. Theline between the n th objectivepoint(x n ,y n )and the n +1 objective point(x n +1 ,y n +1 )isgiven after multiplyingthe coefficient k c (regionAin Fig. 7.10). This master-axis position P x ( s )and slave-axis position P y ( s )isexpressed according to the 1st order model of the servosystem as P x ( s )= K px s + K px 1 s U x ( s )+ 1 s + K px D x ( s )(7.9a ) P y ( s )= K py s + K py k c F s ( s ) P x ( s )(7.9b ) 1627 Master-Slave Sync hronous Po sitioning Con trol where D x ( s )denotes velocitydisturbance. K px and K py have the meanings of K p 1 in equation (2.20) in the lowspeed 1st order model of 2.2.3 about the master axis and the slave axis, respectively.Mor eover, F s ( s )isasthe following equation when usinginverse dynamics basedonthe slave-axis feature. F s ( s )= s + K py K py . (7.10) Therefore, in the contour control in this region, the master-axis position input command is u x ( t )=L − 1 { U x ( s ) /s} andthe slave-axis revised input command is u y ( t )=L − 1 { k c F s ( s ) P x ( s ) } .They aregiven to the servosystem as the command of eachsampling time interval ∆t p . (3) Control Method withthe CornerPart Sincethere is aresponse delaycorresponding to the objective trajectory,the response locus will be missed to the objectivelocus in the corner part. In order to preventlocus deterioration due to suchamiss, after the position input reachingthe objective pointand at themoment of thefollowing locus reachingintothe distance withinthe time of v∆t p fromthe objective point (region BinFig. 7.10), the control method(Fig. 7.1)onthe linearinterval will be continued without achangeof k c andthe commandtime will be also lasted. The input scale of extended commandiswithin the radiu sof v∆t p fromobjectivepointand untilrealizing position output. The reason is that the adv ancing distancew ithin one sampling time in terv al ∆t p with the given objectivetangentvelocity v is v∆t p .When the following locus reaches within v∆t p ,i no rder to ch ange thep osition input commandi nt ot he nextl inear interval position input, v is as the velocityintroducedin7.2.1(1) generally, ev en generating lo cus deviation deterioration as Fig. 7.10. Since ∆t p is also very small,the errorisactually very slight.Moreover, proper v and ∆t p can x a x i s y a x i s ( x n , y n ) r v ∆ t M o v ing dir e c t ion R egion A R egion B Fig. 7.10. Contour control at the corner part 7.2C on tour Con trol with Master-Sla ve Sync hronous Po sitioning 163 be previouslyworkedout according to the allowable error. After the following locus reaches the radius v∆t p of theobjectivepointand basedonFig. 7.1, the next control of the linear interval will be carried out with acommand forsynchronization. The above proposal is thecontourcontrol method of master-slave synchronous positioning. 7.2.2P rop ert yA nalysis and Ev aluationo ft he Con tour Con trol Metho dw ith Master-Sla ve Sync hronous Po sitioning (1)P rop ert yA nalysis of the Con tour Con trolM etho dw ith Master-Sla ve Sync hronous Po sitioning The effectiveness of the contour control metho dofmaster-slave synchronous positioning is evaluated according to the analyticalsolution using amathemat- ical model of equation (7.9 a ) ∼ (7.10). In the equation (7.9a ), (7.9b ), ainverse Laplace transform (refer to appendix A.1) is conducted with U x ( s )=v x /s, D x ( s )=d x /s as p x ( t )=x (0)e − K px t +  e − K px t − 1 K px + t  v x + 1 − e − K px t K px d x (7.11 a ) p y ( t )=y (0)e − K py t + k c x (0)e − K px t + k c  e − K px t − 1 K px + t  v x + k c 1 − e − K px t K px d x (7.11 b ) where v x denotest he ve lo cit yo ft he x directioni ft he ob jective tangen tv e- locityis v . d x denotesdisturbance. Using equation (7.11 a ), (7.11 b ), locus error in the verticaldirection of the following locus correspondingtothe objective locus in the contour control methodofthe master-slave synchronous positioningcan be calculated. Locus error can be calculated according to | p x ( t )sin ϕ − p y ( t )cos ϕ | if angel between the x axisand the objective locus is calculated with ϕ .Then it is put into equation (7.11a ), (7.11 b ). And here, proportional constant k c is ch anged as tan ϕ by using ϕ ,a nd basedo nt he handling in the corner part with the contourcontrol method of master-slave synchronous positioning, thesmall va lues of x (0), y (0) as initial va lues for the next in terv al can be appro ximated to x (0) = y (0) =0because the following locus is made to approximate the objectivepoint. According to above procedure, e ( t )=| α (sin ϕ − tan ϕ cos ϕ ) | =0 (7.12) in theory,the locus errorwill be 0when mixedwith anykindsofdisturbances. But thereexist as α =  e − K px t − 1 K px + t  v x + 1 − e − K px t K px d x . (7.13) 1647 Master-Slave Sync hronous Po sitioning Con trol Namely,ifthere arenomodeling errors in the contour control methodof master-slave synchronous positioning and the slave axisistracing correctly the master axisatany time, it shows that the locus error of contour control is 0. (2)PropertyAnalysis of the Modeling Error In the contour control methodofmaster-slave synchronous positioning, the serv os ystem is expressed by the 1st order mo del. The po sition sync hronization is carriedout using itsinverse dynamics equation (7.10). In fact,itisvery diffi- cult to makethe propertyvalues K px , K py of eachaxis consistence completely because of the variation of momentofinertialaccordingtothe mechanical movementstates and the variation of thespring constant. Forexample, K px , K py are notconsistenteventhought that disturbance is not mixed. Therefore, the deteriorationoccurred in the contour control performance of thecontour control method of themaster-slave synchr onous positioningbecause thereex- ist modelingerrors in the K py of th emathematical model.The deterioration degree,i.e., robu stness of this contour control methodcorresponding to the modeling error K py ,isdiscussed.When K py of themodification element F s ( s ) is differentfrom ∆K py of theactual control objective, the relationship between the proposed methodand modeling errorcan be distinguished by investigating the control performance of themast er-slave synchronous positioningcontrol method.And here,the propertyisinvestigated when K py and ∆K py are dif- ferent fromthe previous assumptionofgain of themodification element F s ( s ) anddisturbance and the initial value is 0. the modificationelement ˆ F s ( s )is expressed if existing modelingerror K py as ˆ F s ( s )= s + K py + ∆K py K py + ∆K py . (7.14) The stationary term of lo cus error e ( t )u sing equation (7.14) is expressed as belowwhen F s ( s )i ne quation (7.11 a ), (7.9 b )a re ch anged in to ˆ F s ( s )a nd deviatedanalyticalsolution p y ( t )isused. e =     ∆K py K 2 py + K py ∆K py v x sin ϕ     (7.15) wherelocus error e ( t )isas e whichisnot changed depends on time t .This equation (7.15) expressesthe locus errorofcontourcontrol when theadopted contourcontrol method of themaster-slave synchronous positioningwith mo d- eling error ∆K py . If existing modelingerror, the significance of using the contourcontrol method of themaster-slave synchronous positioningisevaluated according to the comparison with contour control performance without completeposi tion synchronization.Inthe conventional methodwithout position synchroniza- tion, the locus error is expressed as belowif F s ( s ) P x ( s )inequation (7.11 a ) 7.2C on tour Con trol with Master-Sla ve Sync hronous Po sitioning 165 and (7.9b )ischanged into U x ( s ) /s andthe stationaryitem can be calculated if putintothe analytical solution p y ( t ) e =     K px − K py K px K py v x sin ϕ     . (7.16) This equation (7.16) is also about e whichisnot dep endent on time t .If ∆K py in equation (7.15) is changed,the locus errorwill be ad justed in the small scale comparing with the locus errorinequation (7.16) of conventional method. And here, K py = n xy K px when carryingout analysis. Thesolution of inequality of theconventional methodand the contourcontrol method of master-slave synchronous positioning are as (1 − n xy ) K px ≤ ∆K py < ∞ , (2 ≤ n xy )(7.17a ) (1 − n xy ) K px ≤ ∆K py < n xy (1 − n xy ) n xy − 2 K px , (1 <n xy < 2) (7.17b ) n xy (1 − n xy ) n xy − 2 K px <∆K py < (1 − n xy ) K px , (0 <n xy < 1). (7.17 c ) When the x axisisthe master axisand the y axisisthe slave axis, generally, in order to define the response propertyofthe slave axisfaster than that of themaster axisand K px ≤ K py ,the resultsinthe scale of n xy ≥ 1inequation (7.17 a ), (7.17 b )i sv ery imp ortant . In order to evaluate the appropriation,when these condition equations are regarded as ev aluationc riteriao ft he cont ourc on trol method of master- slave sync hronous po sitioning, as im ulation of con tour con trol is conducted. With conditions of ∆t p =10[ms], v =10[mm/s], K px =5[1/s], ther esults with fiv et yp es of K py ,7,10, 20, 30, 50[1/s], areillustrated in Fig. 7.11. The objec- tivelocus is performed(0 , 0) → (20, 20) as theobjectivepoint. Thehoriz ontal axisisthe ratio R y =(K py + ∆K py ) /K py of modeling errorcorresponding 0 .1 110 1 00 0 1 R y =(K p y + ∆ K p y ) / K p y E rro r e [ mm] n xy = K p y / K p x 1.0 1.4 2 . 0 4.0 6 . 0 1 0 . 0 4.0 6 . 0 1 0 . 0 Fig. 7.11. The relationship between the locus error e and the modeling error rate R y 1667 Master-Slave Sync hronous Po sitioning Con trol to theactual value K py .The left verticalaxis is the lo cus error e .The right verticalaxis is the ratio of n xy = K py /K px .Inthesefigures,the dotted line in the graphisthe locus errorofthe contourcontrol if thereisnoposition synchronization of theconventional metho d. If the lo cus error of the contour control methodofthe master-slave synchronous positioningisunder the dot- tedline, the significantofposition synchronization can be judged even there have been modelingerrors. Furthermore, the effective scale of theproposed methodcan be shown in the scale of the horizontal axis R y describedbythe unbrokenline in the graph. From theseresults, the scale, when the locus error of contour control methodofmaster-slave synchronous positioningwith R y < 1isbigger than that of the conventional method, exists. Therefore, the proposed methodis effectivewithin the limit scale in which R y > 1, n xy ≥ 2, andthe locus error is smaller than theconventional methodwith any ∆K py ,and 1 <n xy < 2. However, in fact,ifthe propertyofthe servosystem of the equipmentisknown clearly,the modeling errorof K py ,ingeneral, will be several percentstotimes of percents. From theresults of Fig. 7.11, the locus error is very slight when R y =1isequivalenttothe actualmodelingerror. Basedonthe above analyt- ical resultsofthe modeling errorand considering thatthe current state of the actual modeling error, in fact, it is possible to adapt effectively thecontour control with position synchronizationofthe contourcontrol method of the master-slave synchronous positioning. 7.2.3E xp eriment al Te st of the Con tour Con trolM etho do f Master-Slave Synchronous Positioning The experimentwith the contour control methodofmaster-slave synchronous positioning wascarriedout using an actualXYtable (refer to E.4 aboutexper- imentequipment). The experimental conditions are, ∆t p =10[ms], v =3(about 1/23 ratedspeed)[mm/s], the position loop gains of the 1st order model are K px =5[1/s] of master axis and K py =10[1/s] of theslave axis. Theobjective trajectory,illustrated in Fig. 7.12, is movedfrom1at (0, 0) to 2at → to 3at → to 4at → to 1. The disturbances are putintoas − 5[mm/s]step between 3 ∼ 7second at the region of 2 → 3and as − 1[mm/s]step between 3 ∼ 7second at the region of 3 → 4. Fig. 7.12(a) is aboutthe conventional methodbywhich each axisiscontrolled independently.Fig. (b) is about the contour control methodofthe master-slave synchronous positioning. Fig. (c) illustrates the results of the contour control meth od of master-slave synchronous position- ing when addingthe modeling error ∆K py =1[1/s]. The left graph shows the locus if the horizontal axis is the position output of the x axisand the verticalaxis is the position outputofthe y axis. Therightgraph shows the time change of thelocus error. In Fig. (a),the erroroccurred with an aver- age 0.1[mm]accordingtothe difference of servofeatures among axis ( K px =5, K py =10)a nd alsoo ccurred duet od isturbance influence. Th us, the follo wing locus cannot reachthe finalobjectivepoint. In Fig. (b), in all regions, the 7.2C on tour Con trol with Master-Sla ve Sync hronous Po sitioning 167 errorisabout 0.02[mm]ifcontourcontrol is correctlyperformed because of good position synchronizationeveninthe part added with disturbance.In addition, in the Fig. (c), thelocus erroris0.03[mm] when the precision of contour control is very high. In order to use the differential of the modificationelementofthe contour control meth od of master-slave synchronous positioning, thenoisemixed into the input signal of the slave axisshould be considered. If making differential (discrete) on asignal with much noise, the problemofamplitude increment will be generated. If using pulseoutput of theencoder which is often applied in the mec hatronic serv os ystem of the industrial field,t here will be no problem on thed iscrete of mas ter-axis outputv alu es.E ve nu singt he tach ogenerator output as position outputand these values includingthe integral value of the tachogenerator output, thereare alsonoproblems on theirdiscrete. In the currentexperiment, good resultswere obtained when usingthe tachogenerator output. Basedonthe proposed position synchronizationcontourcontrol method, good control performance canbeverified not only by theoretical analysis but also with experimentresults. [...]... digital controller, and necessary computing servo amplifier: power amplifier part providing output rated with input signal (current control part) servo controller: from position input of one axis of mechatronic servo system to current input of motor, including (position control part, ) velocity control part, current control part and power amplifier part servo parameters: parameters including in servo controller,... (command of movement, emergence stop, etc according to operation procedure) mechanism: construct the mechanic movement part of mechatronic servo system mechatronic servo system, mechatronic system: The overall title of servo part of mechatronic machine composed by (management part) reference input generator, position control part, velocity control part, current control part, power amplifier, motor and. .. and change torque of rotational movement according to input current objective point: angle part when approximating linearly the given objective locus in contour control position control system, servo, servo system: from position command of one axis of mechatronic servo system, to position output of motor, including position control part, velocity control part, current control part, power amplifier part, ... axis, etc., including control part of each motor in each articulated part of robot arm current control part: control current flowing in motor by current command of motor and current sensor command part: generation part of position and velocity (trajectory) for the mechatronic servo system movement driven by numerical input or taught data (management part) reference input generation part reference input... mechatronic servo system, to velocity output of motor, including velocity control part, current control part, power amplifier and motor part B Definition of control method contouring control: control to realize trajectory of position output of mechatronic servo system In industrial application, the curve of objective locus is approximated by line and the tangential velocity is always constant full-closed... current command calculated in current control part is transformed by D/A transformation, and feedback into current control part is determined by A/D transformation resolution to be driving force of motor velocity control part: control velocity of motor by velocity command of motor and observer value, generally used for ratio integral control velocity control part: from velocity input of one axis of mechatronic. .. order model: model expressed by 2nd order system of the overall mechatronic servo system In general, it can be adopted with the velocity of 1/20∼1/5 motor rated velocity Glossary 171 4th order model: model with the combination that motor rotation part and mechanism part of mechatronic servo system are expressed by two mass model, and electric part of servo controller is expressed by 2nd order model... the control system structured by feedback system with the information of each moveable tip or motion tip modified taught data method: method for improving the contour control performance through the delay dynamics compensation of mechatronic servo system position control: for realizing position of mechatronic servo system without problems in the part of locus It is called PTP(point to point) semi-closed... motor and mechanism part, and output rotation times reference input generator: compute something in each designated time interval for position control of each link based on objective trajectory and put into position control part sampling time: time interval till exporting torque (velocity) command, by observing state based on the (updated) data input when velocity control part (position control part) ... modeling control object with the actual control object reduced order model: General title of 1st and 2nd order model of mechatronic servo system two mass model: model constructed by connecting the system model, structured by the mechanism driven using motor, with motor inertia moment and load inertia moment with spring D Definition of performance locus: final results of position of mechatronic servo system . objectivelocus in contour control. position control system, servo, servosystem:from position command of one axis of mechatronic servosystem, to position output of motor, including position control part, . etc.according to op eration proc edure). mechanism:constructthe mechanic movementpart of mechatronic servosystem. mechatronic servosystem, mechatronic system: The overall title of servo part of mechatronic machine composed. contour control perfor- mance through the delaydynamics compensation of mechatronic servosystem. position control: for realizing position of mechatronic servosystem without prob- lems in the part of lo cus. It is called PTP(p oin tt op oin t). semi-closed