5.2C on tour Con trol Metho dw ith Av oidance of To rque Saturation 111 (3)Contour Control ConsideringTorque Saturation In thec on tourc on trol of an industrialm ec hatronic serv os ystem, motioni s pe rformedi nt he regionw ithout generating torque saturation. In order to implemen ti t, the tra jectory of mec hatronic serv os ystem should be determined without torque saturation.F ig. 5.8 illustrates the con tour con trol structureo f am ec hatronic serv os ystem. The con tour con trol considering torque saturation is divided in to two big parts.O ne is the generation part of thet ra jectory in wo rking co ordinatesw ithout torque saturation. Another is the comp ensation part of dynamics of the mechatronic servosystem. Forgenerationoftrajectory ( w x ( t ) ,w y ( t )), alocus is generatedbysat- isfying the working precision  between the objectivelocus ( r x ,r y )and the generatedlocus ( w x ,w y )without torque saturationinamechatronic servo system as shown in Fig. 5.8 firstly.The velocitygiven in locus ( w x ,w y )gen- eration is approximated with the objectivevelo city v with alimitationinthe regionwithout torque saturation. If directly usingthe generatedtrajectory ( w x ( t ) ,w y ( t )) as an input trajec- tory ( u x ( t ) ,u y ( t )), following the locus ( x, y )generated from the locus ( w x ,w y ) will be degraded because of the dynamics of the mechatronic servosystem. If usingthe inverse dynamics of themechatronic servosystem in equation (5.11) without torque saturation, the input trajectory ( u x ( t ) ,u y ( t )) can be adoptedwith revised generated trajectory ( w x ( t ) ,w y ( t )). Then,any delay of them ec hatronic serv os ystem is comp ensated, and the follo wing tra jectory ( p x ( t ) ,p y ( t )) is consistentwith the generated trajectory ( w x ( t ) ,w y ( t )). More- over, the following locus ( x, y )issatisfied with working precisionof  . (4) Trajectory Generation Considering Torque Saturation Fo ra no bj ectiv el oc us ( r x ,r y )generated from two lines for approximating the trajectory shown in Fig. 5.7,the trajectory generation method,ifgener ating atrajectory alongthe time shift under the limitation of thetorqueofthe mechatr onic servosystem, is explained below. 1. When thereexists an angle in the objectivelocus ( r x ,r y ), the angle will be approximatedbyacircle satisfying working precision  . 2. Radius r of thec ircle included in the lo cus ( w x ,w y )i sc alculatedb ya tangentvelocitybetween the minimal radius r min (= v 2 /A max )satisfying torque constraints andthe maximalacceleration A max . ( r , r ) xy xy xy ( w ( t ) , w ( t )) ( u ( t ) , u ( t )) ( x ( t ) , y ( t )) v Objec t i v elo c us Objec t i v e veloc i ty G ener a t ed tra jec t o ry I nput tra jec t o ry F ollow ing tra jec t o ry T r a jec t o ry gener a t o r I n v e rse d y n a mic s M e c h a tronic s e rvo system Fig. 5.8. Contour control structure of mechatronic servosystem including torque saturation 1125 To rque Saturation of aM ec hatronic Serv oS ystem a) If r ≥ r min :generated trajectory ( w x ( t ) ,w y ( t )) is calculated for chang- ingthe objective tangentvelocityintotangentvelocity. b) If r<r min :trajectory is generatedaccordingtothe following proce- dure. i. In the region from t 1 to t 2 ,the tangentvelocityisdecelerated with maximaldecelerationof − A max from v to v min (the tangent velocityis v min = √ A max r if the acceleration of radius r circle is A max ). ii.Inthe regionfrom t 2 to t 3 ,the locus is describedbycircle. iii. In the region from t 3 to t 4 ,the tangentvelocityisaccelerated with amaximal acceleration of A max from v min to v . 3. In th ebeginningpointand end pointofthe objective locus, acceleration and decelerationare performedwith amaximal acceleration A max . Based on theabove introduced procedure, atrajectory ( w x ( t ) ,w y ( t )) can be generatedwithout torque saturation and the generated locus ( w x ,w y )can be made consistentwith the objectivelocus ( r x ,r y )within the working precision  . In the case of 2b, trajectory generation can be derivedby w x ( t )= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ vt cos θ c 1 ( t ≤ t 1 ) w x ( t 1 )+  v ( t − t 1 ) − A max ( t − t 1 ) 2 2  cos θ c 1 ( t 1 <t≤ t 2 ) w x ( t 2 )+r  sin  θ c 1 + v min ( t − t 2 ) r  − sin θ c 1  ( t 2 <t≤ t 3 ) w x ( t 3 )+  v ( t − t 3 )+ A max ( t − t 3 ) 2 2  cos θ c 2 ( t 3 <t≤ t 4 ) w x ( t 4 )+vtcos θ c 2 ( t 4 <t) (5.17 a ) w y ( t )= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ vt sin θ c 1 ( t ≤ t 1 ) w y ( t 1 )+  v ( t − t 1 ) − A max ( t − t 1 ) 2 2  sin θ c 1 ( t 1 <t≤ t 2 ) w y ( t 2 )+r  cos  θ c 1 + v min ( t − t 2 ) r  − cos θ c 1  ( t 2 <t≤ t 3 ) w y ( t 3 )+  v ( t − t 3 )+ A max ( t − t 3 ) 2 2  sin θ c 2 ( t 3 <t≤ t 4 ) w y ( t 4 )+vtsin θ c 2 ( t 4 <t) (5.17 b ) wherethe time interval of deceleration andaccelerationis t 4 − t 3 = t 2 − t 1 = ( v − v min ) /A max ,d escribing the time of the circle is t 3 − t 2 = r ( θ c 2 − θ c 1 ) /v min . This meth od is performedunder conditionof2b r<r min andwith the lowest 5.2C on tour Con trol Metho dw ith Av oidance of To rque Saturation 113 limitationofvelocityfor preventing arapid change in velocity. Besides, the control time becomes longer in order to describeacircle. The high-precision contour control will be performedunder the conditions of that following the locus ( x, y )atthe angle part als oshould be satisfied torque constraints, and the generated locus ( w x ,w y )should be in agreementwith the objectivelocus ( r x ,r y )within the working precision  . (5) DelayCompensation Based on Inverse Dynamics In order to compensate forthe dynamics of themechatronic servosystem, the trajectory should be revised by using inverse dynamics.Although the inverse dynamics of equation (5.11) contains asecond-order differential, the trajec- tory ( w x ( t ) ,w y ( t )) is possible to obtain a2nd order differential, compensation based on inverse dynamics canberealized to design acceleration without torque saturation. The inverse dynamics of amechatronic servosystem as in equation (5.11) withouttorquesaturation is expressedas F ( s )= s 2 + K v s + K p K v K p K v . (5.18) The input trajectory ( u x ( t ) ,u y ( t )) is derived according to arevised trajectory ( w x ( t ) ,w y ( t )) based on inverse dynamics (5.18) as u x ( t )=w x ( t )+ 1 K p dw x ( t ) dt + 1 K p K v d 2 w x ( t ) dt 2 (5.19 a ) u y ( t )=w y ( t )+ 1 K p dw y ( t ) dt + 1 K p K v d 2 w y ( t ) dt 2 . (5.19 b ) When input trajectory ( u x ( t ) ,u y ( t )) are adopted as the command of the mec hatronic serv os ystem, the follo wing tra jectory ( p x ( t ) ,p y ( t )) can be in good agreement with the generated trajectory ( w x ( t ) ,w y ( t )). (6)Contour Control AlgorithmConsidering Torque Saturation The procedure of contour control considering torque saturation is illustrated as below. 1. Atrajectory is generatedbasedonequation (5.17 a ), (5.17 b )accordingto the procedure of 5.2.1(4)fromthe objective trajectory ( r xi ( t ) ,r yi ( t )). 2. An input trajectory is calculated for comp ensating delayofdynamics by using inverse dynamics of equation (5.19) 3. Input commandofobjectivetrajectory,whichcan compensate forthe dynamics delay of themechatronic servosystem, is given. 1145 To rque Saturation of aM ec hatronic Serv oS ystem 0 51 0 0 5 x a x i s pos i t ion [ r e v ] y a x i s pos i t ion [ r e v ] C onv ent iona lme t hod C ons ider only w o r king p r e c i s ion C ons ider inv e rsedy n a mic s Objec t i v elo c us L o c us 0 51 0 0 5 x a x i s pos i t ion [ r e v ] y a x i s pos i t ion [ r e v ] C onv ent iona lme t hod C ons ider only w o r king p r e c i s ion C ons ider inv e rsedy n a mic s Objec t i v elo c us L o c us 0 2 4 y a x i s pos i t ion [ r e v ] Objec t i v e tra jec t o ry C onv ent iona lme t hod W i t hout inv e rsedy n a mic s W i t h inv e rsedy n a mic s y a x i s tra jec t o ry 0 2 4 y a x i s pos i t ion [ r e v ] Objec t i v e tra jec t o ry C onv ent iona lme t hod W i t hout inv e rsedy n a mic s W i t h inv e rsedy n a mic s y a x i s tra jec t o ry −10 0 1 0 y a x i s veloc i ty [ r e v/s] −10 0 1 0 y a x i s veloc i ty [ r e v/s] 0 1 2 −50 0 5 0 T ime [ s ] y a x i s acceler a t ion [ r e v/s 2 ] 0 1 2 − 2 0 2 T ime [ s ] y a x i s to r q u emonit o r [V] (a) Simulation (b) Experiment Fig. 5.9. Experimental results and simulation results corresponding to the objective tra jectory of two lines 5.2.2E xp eriment al Ve rificationo fC on tour Con trol Considering Torque Saturation (1)E xp eriment Using DEC-1 In order to verify the effectivenessofthe contourcontrol method avoiding torque saturation, acomputersimulation and experimentusing theDEC-1 (experimentequipmentreferringE.1) were carriedout. As contourcontrol approaches,three methods arecompared,i.e., conventional methodwith orig- inal objectivetrajectory usuallyused in the industrial field,considering only working precisionwithout performing accel eration anddeceleration, and con- tour control avoidingtorquesaturation.The conditions of computersimula- tion and experimentare as below: position loop gain K p =10[1/s], velo city 5.2C on tour Con trol Metho dw ith Av oidance of To rque Saturation 115 loop gain K v =56[1/s], maximal acceleration A max =80[rev/s 2 ,sampling time interval 10[ms],working precision  =0. 1[rev], objective tangentvelocity v =13 . 1[rev/s].The objective trajectory is given as dr x ( t ) dt =9. 26 (0 ≤ t ≤ 1 . 08[s]) (5.20 a ) dr y ( t ) dt =  9 . 26 (0 ≤ t ≤ 0 . 54[s]) − 9 . 26 (0. 54 <t≤ 1 . 08[s]). (5.20 b ) Input trajectory ( u x ( t ) ,u y ( t )) is derived according to the procedure of 5.2.1(4). In Fig. 5.9,the computer simulationresults and experimental results are illus- trated. The acceleration outputinthe experimentalresults is measured by a torque monitor. As shown in Fig. 5.9,the following locus generated overshoot is basedonthe conventional method. This overshoot is notpermitted to oc- cur in contour control in industry(referto1.1.2 item 3). However, overshoot does notoccur in the proposed methodwhichconsiders working precision. In addition, the following locus has alarge errorcompared with objective locus in the conventional method, butinthe proposed method,the following locus is almost the same as the objectivelocus whenconsidering working precision. In the experimentalresults, the locus error is 0.17[rev]. From theacceleration outputinthe experimentalresults shown in the figure, torque saturationis generated. The torque saturation is 3[V]response of the torque monitor. Con- cerning the bad impact of the conventional method, the tangentvelocityby conventional methodwill become larger than the objectivetangentvelocity v = − 9 . 26[rev/s].A tt he pe ek po in t, the ve lo cit yi s − 11. 5[rev/s] in thes im u- lationand − 11. 0[rev/s] in theexperimental results. However, in the contour control method avoidingtorquesaturation,the tangentvelocityisalso con- sisten tw ith the ob jectiv et angen tv elo cit y. Fr om theser esults, the prop osed methodiseffectiveincomparingother two methods. (2)Experiment Using an ArticulatedRobot Arm (Performer MK3S) The proposed contour control methodconsidering torque saturation was adoptedf or an articulated robo ta rm (P erformerM K3S; exp erimen td evice refers to E.3). Thereare nonlineartransforms between working coordinates andjointcoordinatesadoptedinthe articulated robotarm. As introduced ab ove,the contourcontrol method avoidingtorquesaturation cannotbe adoptedwithout change. If generating trajectory considering torque satura- tion in working co ordinatesand compensating fordelayinjointcoordinates, the proposed methodcan be adopted. In the delay compensation in jointco- ordinates, modifiedtaughtdata method (refer to section 6.1)isused here. Besides, the relationship between maximalacceleration a max in jointcoor- dinatesand maximalacceleration A max in working coordinatesiscalculated according to coordinate transform by using Jacobianwith areference input time in terv al. 1165 To rque Saturation of aM ec hatronic Serv oS ystem Although PerformerMK3S uses 5axes for a5-freedom-degree articulated robot arm, only two axesare usedinthe experiment. The servomotor in eachaxis is connected with the servocontroller forcarrying outvelocityand current control. The servocontroller is connectedwith the computer when performing position control. In eachaxis, an AC servomotor (rate dspeed 3000[rpm])isused and driving arm through decelerationdevice. The con- ditions of the device are: position loop gain K p =25[1/ s],velocityloop gain K v =150[1/ s],maximumacceleration a max =11 . 0[rad/ s 2 ], sampling time interval ∆t =6[ms](refer to section 3.1), length of arm l 1 =0. 25[m], l 2 =0. 215[m], gear ratioofeachaxis n 1 =160, n 2 =161. In theexperiment, the value multiplyingposition loop gain K p in the error between position input and motorposition output areput into the motorasvelocityinput through aD/A converter. (i)Supposedtorque saturation generation ThePerformer MK3S used in the experimentcan output very largeamounts of torque. In order to verify the significance of theproposed method,the supposedtorquesaturation can be generatedbythis device. Thismethod focuses on velocityinput. If the actualmeasured angular acceleration output multiplyin gvelocityloopgain K v with the error between velocityinput v i andoutput v f satisfied | K v ( v i − v f ) | >a max (5.21) velocityinput v i is ch anged as v i =sign(v i − v f )  a max K v  + v f (5.22) angular accelerationisnot over a max .T orques aturation is ch angeable de- pended on thedevice type.Basedonthe proposed method,the experimentis realizedi nt he same devicec onsidering va rious torque prope rties. (ii) Simulationand experimental results Fig. 5.10illustrates the locus for four methods in 5.11, synthesized velocity and simulation resu lts and experimental results of the Baxis acceleration with saturation. (a) conventional metho d(objective trajectory is used as input of the robot arm without anychange),(b)conventional methodinthe state with supp ose dtorquesaturation generati on,(c) contour control method(consider- ing precision) considered torque saturation, (d) contour control method(con- sidering velocity) considered torque saturation are adopted. The conditions of the simulation are designated tangentvelocity v =0. 15[m/s],objectivelo- cus0. 05[m]length two lines of (0. 135, 0 . 365) ∼ (0. 185, 0 . 365) ∼ (0. 185, 0 . 415) whichisturned as avertical angle. As introduced in 5.2.1(4), maximalac- celeration a max in join tc oo rdinatesa nd maximala ccelerationi nw orking co- ordinatesgiven from the objectiveare calculatedas A max =1. 0[m/s 2 ]. In 5.2C on tour Con trol Metho dw ith Av oidance of To rque Saturation 117 0.14 0.16 0.18 0.36 0.38 0.4 0.42 x [m] y [m] 0.14 0.16 0.18 x [m] 0 0.1 0.2 Velocity[m/s] 0 0.2 0.4 0.6 -20 -10 0 10 20 Time[s] B axis acceleration[rad/s 2 ] 0 0.2 0.4 0.6 Time[s] (a) Without torquesaturation (b) With torque saturation 0.14 0.16 0.18 0.36 0.38 0.4 0.42 x [m] y [m] 0.14 0.16 0.18 x [m] 0 0.1 0.2 Velocity[m/s] 0 0.2 0.4 0.6 0.8 1 -20 -10 0 10 20 Time[s] B axis acceleration[rad/s 2 ] 0 0.2 0.4 0.6 0.8 Time[s] (c) Proposed method (considering precision) (d) Prop osed metho d (considering ve lo cit y) Fig. 5.10. Simulation results 1185 To rque Saturation of aM ec hatronic Serv oS ystem 0.14 0.16 0.18 0.36 0.38 0.4 0.42 x [m] y [m] 0.14 0.16 0.18 x [m] 0 0.1 0.2 Velocity[m/s] 0 0.2 0.4 0.6 -20 -10 0 10 20 Time[s] B axis acceleration[rad/s 2 ] 0 0.2 0.4 0.6 Time[s] (a) Without torquesaturation (b) With torque saturation 0.14 0.16 0.18 0.36 0.38 0.4 0.42 x [m] y [m] 0.14 0.16 0.18 x [m] 0 0.1 0.2 Velocity[m/s] 0 0.2 0.4 0.6 0.8 1 -20 -10 0 10 20 Time[s] B axis acceleration[rad/s 2 ] 0 0.2 0.4 0.6 0.8 Time[s] (c) Proposed method (considering precision) (d) Proposed method (considering ve lo cit y) Fig. 5.11. Experimental results 5.2C on tour Con trol Metho dw ith Av oidance of To rque Saturation 119 thecontourcontrol considering torque saturation forfocusing on precisionin Fig. (c), the working precisionis  =1. 0 × 10 − 4 [m] focusingonlocus, min- imal velocity v min =0. 0155[m/s] is given when velocityisdecreased to 10% of objectivevelocity. In the contour control considering torque saturationfor focusing on velocityinFig. (d), ther eexists adecrease of contour control pre- cision when the response cannot be fit forthe situation that velocityisover dropped at thecorner at theoperation of laser cutting, or input current of laser is overreduced, or increasing currentcost so much time at velocityincreasing. Forthesecase s, thedelayissue of input currentresponse will disappear when velocityisonly equalto70% of theobjectivevelocity. Then,the working pre- cision wascalculatedunder the conditionthatthe velocitywas decreasedtill 70%ofobjectivevelocity. If  =0. 005[m], minimal velocity v min =0. 1[m/s] is given when velocityisdecreased to 70%ofobjectivevelocity. The common pole of regulator in Fig. (c) and (d) wasgiven as γ = − 30. In thefollowing locus of Fig. (a), the deterioration of locus as roundn ess at the corner part of the simulation and experimentcan be found. The rea- sonfor deterioration is th edelaydynamics of the robot arm and it can be understood even from the results of acceleration to be notlinked to torque saturation. On the other hand,the marked part of Baxis acceleration exist 0.33 ∼ 0.44[s] saturation by observing the resultsofeachaxis acceleration in the simulation and experimental results in Fig. (b). In addition, the error in the experimentissmaller in the simulation results and experimental results. With same trendatthe marked part of thefollowing locus, in the simulation error is 1.35[mm], butinthe experimentis0.74[mm]. At the marked combined velocity, in the simulation the overshoot is 0.3[m/s], but in the experimentis 0.12[m/s]. Overshootmust be avoided as much as possible in order to improve precision(referto1.1.2 item 3). From thesimulation and experimental results in Fig. (c), thereare no overshoots in thefollowing locus results. From the combinedvelocity, spending more time than Fig. (a) and (b) at the marked corner part for usin gnecessary minimal velocity. Hence, the dynamics of the robot arm is compensated and thereisnotorquelimitation. In addition, the minimal velocityissatisfied as v min =0. 015[m/ s] .F romt he simu lationa nd experimentalresults in Fig. (d), thereisnoovershoot in thefollowing locus results, and the designated working precisionissatisfied as  =0. 005[m]. Sp ending time is notl onger than Fig. (a),( b), andt here is no torque limita- tion.Additionally, fromthe synthesis velocity, minimal velocityislarger than v min =0. 1[m/s] in order to reducethe velocityatthe marked corner part. From theabove simulation and experimental results, the contour control methodconsidering torque saturation satisfies working precisionand mini- malvelocitywithin the torque saturation, and it can be realizedwithin the limitationofcontourcontrol performance. 6 TheModified Taught Data Method In order to realize the movementofanindustrialrobot, thegiven objectivetra- jectory is alwaysused without anychangewhen their coordinate values which are the taughtdata obtained fromthe teaching.Therefore, in the movement resp onse of therobot at theplayback, the errors between the objectivelo- cusand the following locus of the robot appeared because of the time delay generated at eachaxis. In this chapter, the modifiedtaughtdata method is prop osed in order to impr ove the precision of the trajectory in the contour control. 6.1 Modified Taught Data Method Usinga Mathematical Mode l In the operation of the robot, the practician, who is performing the teach- ing of therobot in theindustrialfield, improvedthe precisionofthe contour control of therobot successfullythroughthe teaching points with alittle over movementfromthe actualobjectivepoints at th ecorner part of objective locus (modified taughtdata).However, thismetho dcan be only adapte dfor thelimited actionsituation. From theinvestigation of the adopted methodbythe practicianand the reasons of performance improvement, the deterioration of control performance owing to the dynamics delayofthe mechatronic servosystem and the real- izationmetho dofdynamic compensation (modified taughtdata) have been found. With the model of amechatronic servosystem in chapter1,the mod- ifiedtaughtdata method with pole assignmentregulatorfor thedynamic compensation wasproposed and the construction of the modificationelement wasintroduced. In order to use thismethodfor thesemi-closed pattern which is without asensor for measuringthe tipposition of therobot arminthe mechatr onic servosystem (refer to 1.1.2 item 5), the modificationelement wasrevised from the closed-loop form with the control lawtothe open-loop M. Nakamura et al.: Mechatronic Servo System Control, LNCIS 300, pp. 121–147, 2004. Springer-Verlag Berlin Heidelberg 2004 [...]... the mechatronic servo system When the modification element F1 (s) is adopted in the mechatronic servo system G1 (s), the control system of the robot arm after revision can be changed as −γ Y (s) = R(s) (6 .9) s−γ From the comparison between the original mechatronic servo system (6.3) and the revised mechatronic servo system (6 .9) , the modification element changes the pole of the mechatronic servo system. .. coordinates of a mechatronic servo system, the relationship between the input and output of the each independent coordinate axis can be expressed independently as Y (s) = G(s)U (s) (6.1) where U (s) denotes taught data, Y (s) the following trajectory of the mechatronic servo system and G(s) the dynamics of the mechatronic servo system The teaching playback robot refers to the semi-closed type control system. .. (6.10) and (6.11) is selected (3) Modified Taught Data Method Based on the 2nd Order Model (i) Mathematical model When the velocity of the motion of the mechatronic servo system becomes high and the velocity of the servo motor is between 1/5 ∼ 1/20 of the rated value, considering the characteristics of the velocity control of the servo motor and the control system of the whole mechatronic servo system. .. the control performance of the mechatronic servo system, it is necessary to satisfy the following equation so that the response of the control system of the mechatronic servo system after revision is faster than that before revision γ ≤ −Kp (6.10) 6.1 Modified Taught Data Method Using a Mathematical Model 125 Then, the velocity limitation of the servo motor, i.e., the actuator of the mechatronic servo. .. in (2. 29) of the middle speed 2nd order model in 2.2.4, respectively (ii) Modification element The mechatronic servo system is expressed by a state-space representation based on the 2nd order model (6.12) The modification element can be derived by the pole assignment regulator (refer to appendix A.3) and the minimum Modification element R(s) F2 (s) U(s) Mechatronic servo system Motor and Servo controller... R(s) F(s) U(s) Mechatronic servo system G(s) Y(s) Fig 6.1 Block diagram of the modified taught data method 6.1 Modified Taught Data Method Using a Mathematical Model 123 the mechatronic servo system will be expressed by the state-space representation and the modification element will be designed with the pole assignment regulator (refer to appendix A.3) in order to change the mechatronic servo system F (s)G(s)... closed-loop control system (2) A modified taught data method based on the 1st order model (i) Mathematical model Firstly, deriving the modification element easily, the 1st order model of the mechatronic servo system is derived by the modified taught data method When the actuator of the mechatronic servo system, i.e., the velocity of the servo motor, is moved under 1/20 of the rated value, the whole control. .. element R(s) F1 (s) U(s) (6.4) Mechatronic servo system Servo Motor and controller mechanism part + - Kp 1 s Y(s) Position loop Fig 6.2 Block diagram of the modified taught data method based on the 1st order model 124 6 The Modified Taught Data Method If the state equation (6.4) can be derived with the assumption dr(t)/dt 0, the pole assignment regulator can be adopted and the modification term of the... servo motor, i.e., the actuator of the mechatronic servo system, must be considered when using the modified taught data method in the actual mechatronic servo system When the maximum velocity of the servo motor is Vmax , the velocity limitation is shown as, |Kp {u(t) − y(t)}| ≤ Vmax (6.11) The left-hand side of (6.11) denotes the velocity input of the servo motor In fact, the computer simulation of the... expressed in the state space of the equation (6.3) of the mechatronic servo system, the modification element F1 (s) is derived by the pole assignment regulator (refer to the appendix A.3) For the objective trajectory r(t), assume dr(t)/dt 0 From the equation (6.3) and the assumption dr(t)/dt 0, the mechatronic servo system expressed by a state-space representation is changed as dx(t) = −Kp x(t) + Kp . servosystem, i.e., the velocityofthe servomotor,ismovedunder 1/20 of theratedvalue, thewhole controlsystem of the mechatronic servosystem includingcontrol equipment, servosystem and mechanism shown. ), the con trol system of the rob ot arm after revision can be changed as Y ( s )= − γ s − γ R ( s ) . (6 .9) From thecomparison between the original mechatronic servosystem (6.3 )and the revised mechatronic servosystem (6 .9) , the modificationelementchanges the. theratedvalue, considering the characteristics of thevelocitycontrol of theservomotor and thecontrol system of thewhole mechatronic servosystem shown in Fig. 6.3, it is necessary to express eachcoordinate independentlywith