4.1E nco der Resolution 81 avelocitydetector is generated as the detection noise of rotation velocity, ripple-type velocitycan be preventedbysmoothing this detection noise with alow pass filter. However, aripple-typevelocityfluctuation of asoftware servo system cannotbesmoothed by alow pass filter because variedfrequencyin- troducedlaterisrelated with the objectivevelocity. Therefore, it is necessary to determine the encoder resolution for forcingthe velocityfluctuation within theallowance region. (2) PresentCondition of Encoder ResolutionDetermination Thedeterminationofpresentencoder resolution in theindustrialmechatronic software servesystem is carried out according to the necessityofpositioning precisionofamechatronic servosystem [4] .When performing contourcontrol, theencoder resolution calculated frompositioningprecisionisused without change. When required, controlperformance cannot be obtained, the encoder resolution with test error will be regulated. The determination of encoder res- olution cannotberealized theoretically for the required control performance. Therefore, in thischapter, the theoretical determination method forencoder resolution forcontrol performance,especially about contour control issue, con- sidering the relationship between ripple-typevelocityfluctuation andencoder resolution,isproposed. 4.1.2A Mathematical Mo del and ResolutionJ udgement for Enco der Resolution (1) AM athematical Mo del of aS oft wa re Serv oS ystem An industrial mechatronic servosystem is alwaysunder the velocitycondition of motion of theoperatedmotor at 1 / 20 ∼ 1 / 5ofmaximumvelocity. Its dynamics is expressed by the 2nd order system as (refer to the 2.2.4) Y ( s )= K p K v s 2 + K v s + K p K v U ( s )(4.1) where Y ( s )isthe position outputofthe servosystem, U ( s )isthe position in- putofthe servosystem. K p , K v have the meaning of K p 2 , K v 2 in the equation (2.29) of the middle sp eed 2nd order mo del in the item 2.2.4, resp ectiv ely . The control system of the mechatronic servosystem expressed by (4.1) is pickedout fromthe software servosystem shown in figure4.1 for encoder resolution analysis. The model of software servosystem for simplifying the analysis is shown in figure4.2. From thestructureofthe software servosystem (Fig. 4.1), the velocityfeedbackcalculatedbasedondifference computation is easily obtained from the external input. However, thisexternal input, as asimple external input, is the same as the velocitysignal in Fig. 4.1.This externalinput is the continuous feedbackofthe velocityoutput in an analogue 82 4Q uan tization Error of aM ec hatronic Serv oS ystem - - K p K + + uy v d y/d t d y/d t 1 s - 1 s - 22 V eloc i ty feedback signa l D i s c r e t i z a t ion a nd q u a n t i z a t ion Fig. 4.2. Software servosystem model for encoder resolution analysis servo system. But in asoftware servosystem, it is adiscrete feedback. The basic unit of the po sition signal is 1[pulse]. The ve lo cit ys ignal is calculated with the difference computation of the position signal. The basic unit of the velocitysignal according to difference computation is 1 /∆t p [pulse/s], where ∆t p [s] is the sampling time. (2)Relationship between Control Performanceand Encoder Resolution The relativeequation between velocityfluctuation,occurred according to en- coder resolution, and servoparameters is derived. In thispart, thevelocity fluctuationisanalyzedwhen the motion of theservomotor is under the con- stan tv elo cit y, whic hi sa lw ay sa doptedi nt he industrial field( refert oi tem 8 of 1.1.2). The flowofsignal is as Fig. 1.1.2. 1. The difference divided according to ve lo cit yr esolution 1 /∆ t p ,d etermined by difference computation of theposition signal, is accumulated. When the accum ulatedv aluei so ve rt he ve lo cit yr esolution, the ve lo cit yf eedbac k signali sa dded with 1 /∆ t p .This added velocityfeedb acksignal is the reason for the velocityfluctuation. 2. According to the velocityloopgain K v added into the velocityfeedback signal, the input of the motori sv ariedw ith the step of K v /∆t p [pulse/s 2 ]. 3. The change of velocityoutput of themotor basedonthe added velocity feedbacksignal is as ( K v /∆t p ) × ∆t p = K v [pulse/s], according to the integral of the input of the motorbasedonthe sampling time interval ∆t p . That is to say, the size of velocityfluctuation,occurred by the signal added into velocityfeedbackaccordingtothe effectofvelocityresolution, is consistent with the value of velocityloopgain K v .This relation can be expressed, if considering the unit, as ∆N = 60K v R E (4.2) where ∆N[rev/min] denotesthe velocityfluctuation amplitude with the ripple-type shape, R E [pulse/rev] denotes the encoder resolution defined by the pulse number of the encoder when the motorrotates throughone cycle. 4.1E nco der Resolution 83 This derivedequation (4.2)isthe fundamental equation fordetermining the following encoder resolution. Next, the relationship between the velocityfluctuation periodwith the ripple-typeshape andvelocityofthe objective trajectory is der ived. If the ve- locityofthe objective trajectory is as V ref [pulse/s], the velocityfeedback, ob- tained from the difference computation, is changed as (  V ref ∆t p  ) /∆t p when the velocityresolutionis1/∆t p ,where  x  is the maximalinteger below x . From 1, this errorisaccumulated in eachsampling time interval. Sincethe velocityfluctuation with theripple-typeshape occurred when the error is over 1 /∆t p ,The sampling time n at themoment of over1/∆t p is as n  V ref −  V ref ∆t p  ∆t p  = 1 ∆t p . (4.3) From (4.3), thevelocityfluctuation frequency f r [Hz] is calculated by f r = 1 n∆t p = V ref ∆t p −V ref ∆t p  ∆t p . (4.4) From (4.4), thevelocityfluctuation frequency f r is depended on the velocity of objectivetrajectory V ref .Inorder thatthe velocityfluctuation frequency f r is not changed into amonotonic functionabout V ref ,alowpass filter cannot be adoptedfor smoothing. (3)Determination of Encoder Resolution By using(4.2),the relation equation between velocityfluctuation andencoder resolution derivedby4.1.2(2), the determination equation of the encoder res- olution can be obtained. Whenthe motorisrotated with aconstantvelocity, the ratio between the scale of the velocityfluctuation andthe maximalve- locity, called velocityfluctuation ratio R N ,isadoptedasaspecification of am ec hatronic serv os ystem, in order to express clearly the motionl ev el of ve lo cit yo ft he motor. Fr om this po in to fv iew, in the soft wa re serv os ystem, the velocityfluctuation ratio R N generatedi nt he encod er resolution can be expressed by R N = ∆N N max (4.5) where, N max denotesthe maximalvelocity[rev/min] of theservomotor.Ifwe put (4.5)into(4.2),basedonthe solutionofthe encoder resolution R E ,the encoder resolution can be determinedby R E = 60K v R N N max . (4.6) The equation (4.6)isthe finalderived result in this section. Accordingto this equation, pr oper encoder resolution R E can be decidedfor satisfying the velocityfluctuation ratio R N ,d etermined according to the application of the servomotor fromthe maximalvelocity N max andvelocityloopgain K v . 84 4Q uan tization Error of aM ec hatronic Serv oS ystem 4.1.3Experimental Verificationofthe Encoder Resolution Determination (1)Experimental Verificationofthe Relationship between the Encoder Resolution and Control Performance From theexperiment, the relationship between the enco der resolution and the velocityfluctuation is verified. In the experiment, DEC-1(refer to the exper- imentdeviceE.1) wasadopted. Actually, DEC-1 wasoriginallyconstructed with an analogueservosystem. However, in thisexperiment, asoftware servo system using acomputerwas used. Thatistosay,the pulseoutput of the servomotor is accumulatedbyacounter equippedinthe computer. The com- puterprogram implementsthe servocontroller.Its outputisput into servo (a) Experimentresults of software servosystem (b) Simulation results of software servosystem (c)Experimental results of analogue servosystem Fig. 4.3. Verification of velocityripple in software servosystem 4.1E nco der Resolution 85 amplifier by using aD/A converterfor constructing the software servosystem. The resolut ion of D/A conversion is adoptedwith reduction by a1/100 am- plifier fromthe D/Aconverter, which can permit ± 5[V]with aresolution of 12[bit]. Since 1[bit] is about 2.44 × 10 − 5 [V], the effect of resolution to control performance canbeneglected. In addition, the velocityofthe servomotor is measured using digital data storage providing velocitydetector (tachogener- ator) outputequippedwith aload generator.This tachogenerator output is 7[V]with arotational frequencyof1000[rev/min] of theservomotor.Since thereare many factorsofnoises in thetachogenerator,the 100[Hz] lowpass filter is adoptedtoeliminate these noise factors. The resolution of the encoder installed in the servomotor is 2000[pulse/rev]. But fromthe tested two in- crease and decrease signals of the encoder outputasputting themintothe pulsecounter, the original 1[pulse] is changed into 4[pulse].Throughthe 4 timescircuit, it can be obtained as R E =8000[pulse/rev]. The maximalve- locityis N max =1000[rev/min], the sampling time interval ∆t p =4[ms] (refer to 3.1). The position loop gain and velo cityloopgain areset as K p =12[1/s] and K v =68[1/s]sothatthere is no oscillation or overshoot in theanalogue servosystem (refer to 2.1.2). Since velocityfluctuation is oneofthe problems in the industrial field, forbig velocityfluctuation in lowspeed, ramp input for DEC 1is u ( t )=40t [pulse], i.e., rotation speed of motoris0.3[rev/min] for lowspeed. In the steady state, the experimental results and simulation results are illustrated in Fig. 4.3.FromFig.(a), (b), in the steady state, the amplitudeinexperimental results and in simulation results are both 0.004[V]. The frequency in both about is 40[Hz]. The shapeofthe wavesare both tri- angular. From theabove,itcan be verifiedthatthe experimentalresults and simulationresults are almost the same. In Fig.(a)ofexperimental results, the size of velocityfluctuation is about0.004[V], i.e., 0.57[rev/min]. This value is almost the same as the size ∆N =60 × 68/ 8000 =0. 51[rev/min] of velocity fluctuation calculated by equation (4.2). In addition, the velocityfluctuation frequencyisalso consistentwith the frequency 40[Hz] calculated by equation (4.4). To verify,the experimentalresults of an analogue servosystem with same conditionsare illustratedinFig.(c). In the analogue servosystem, the velocity fluctuation does not occur at all. The velocityfluctuation in Fig.(a) is verified thatitisthe cause of theresolution of software servosystem by the experiment of 4.1.2(2). (2)Application of Encoder Resolution Determination Usingequation (4.6)derived by 4.1.2(3),the example of determining the encoder resolution is illustrated. In DEC-1 adopted in the previous exper- iment, the necessary encoder resolution is R E =60 × 68 × 1000/ 1000 = 4080[pulse/rev]obtainedfromequation (4.6)ifthe velocityfluctuation ra- tion is given as R N =1× 10 − 3 .I nc on trast,i ft he installed encod er res- olution is actually R E =8000[pulse/rev], the velocityfluctuation ratiois 86 4Q uan tization Error of aM ec hatronic Serv oS ystem R N =60 × 68/ (1000 × 8000) =5. 1 × 10 − 4 .Fromthis pointofview, according to the encoder resolution determination equation (4.6), the encoder resolution can be easily determined from the required velocityfluctuation ratio. 4.2 To rqueR esolution In the soft wa re serv os ystem, the feedbac ko ft he motorc urren te quiv alen tt o the torque is carried out through am icro-computer.B et we en the po we ra mpli- fierfor driving the motorand the micro-computeristhe A/D, D/Aconversion. Thetheoretical relation between the A/D, D/A conversion quantization error andcontrol performance must be clarified. The appropriate mathematicalmodel for the relationship between the torque resolution of the software servosystem and control performance is derived. According to the solutionofthe mathematicalmodel, the positioningpreci- sion by equation (4.8)and the position flu ctuation of the ramp response by equation (4.15)∼ (4.17), with regard to the torque resolution, can be clarified. Accordingtothe bit number proposed in the A/D, D/A converter, the con- trolperformance of the servosystem can be clearly estimated. Additionally, the minimal necessary bit number of the D/A, A/D conversion fortesting out torque command and currentfeedback, in order to implementthe neces- sary controlperformance of the software servosystem, can be determinedby equation (4.25). 4.2.1Mathematical Model of the Mechatronic Servo Systemfor Torque Resolution The conceptual graph of the discussed software servosystem in this section is shown in Fig. 4.4.The software servosystem is shown in Fig. 4.4.Inorder to construct the controlcircuit of the servocontroller using micro-computer software, the torque (current) commandoutput fromthe controlcircuit is quantized. Therefore, the currentreference input to the poweramplifier actu- ally needsaD/A converter. The block diagram of the2nd order system of the servosystem includingtorquequantizationisillustrated by Fig. 4.5. K p [1/s], K v [1/s] ha ve the meanings of K p 2 , K v 2 in the middle sp eed 2nd order mo del equation (2.29) of item 2.2.4. In addition, the sampling time in terv al of the velocityloopis ∆t v [s]. The serv os ystem is usually constructed with po sition feedbac k, ve lo cit yf eedbac ka nd current feedback .T he po sition feedbac ka nd velocityfeedbackrefer to the feedbackofthe actualmotor output forthe servocontroller.The current feedback refers to the feedback of powerampli- fied.Itisnot changed into theactual torque. Forthe mathematicalmodel of the servosystem in the block diagram of Fig. 4.5, theposition feedback andvelocityfeedbackiswidely considered. The currentfeedbackissimply assumed as the outputo ft he po we ra mplifier. The con trol metho do ft he ve - locityloopisPcontrol or PI control. But the entire prop ertyofthe velo city 4.2T orque Resolution 87 D / A S e rvo c ontroller Q u a n t i z a t ion C ompute r P o w e r a mplifier I nput ( c urrent ) O utp ut M o t o r Fig. 4.4. Structure of software servosystem K p K 1 - ++ v U ( s ) Y ( s ) s 1 - s S e rvo c ontroller M o t o r V eloc i ty loop P o s i t ion loop Q u a n t i z a t ion Fig. 4.5. The2nd order model of software servosystem including torque quantiza- tion loop is expressedbythe 1st order system.The position control and velocity control are combined into the 2nd order system (refer to item 2.2.4). In this section, the torque quantizationwith A/D, D/A conversion,asa problem, is expressedaccordingtothe quantizationterm in Fig. 4.5.Bythe function f ( · )for quantization of torque, themathematical model of aservo system includingthe torque quantizationisas d 2 y ( t ) dt 2 = f  K p K v u ( t ) − K p K v y ( t ) − K v dy( t ) dt  . (4.7) Formeasuringthe rotation angle of theservomotor by apulse[pulse] ac- cording to the encoder, the rotation angle u of motor as aposition com- mandisexpressed by apulse. The angular velocityinput, as theveloc- itycommand, is K p { u ( t ) − y ( t ) } [pulse/s]. The angular acceleration inp ut, regarded as the torque command to torque quantization, is K v [ K p { u ( t ) − y ( t ) }−dy( t ) /dt][pulse/s 2 ]. In order to mak et he angular acceleration quan ti- zation function f ( x )a st he step-wise functiono fF ig. 4.6,t he input angular acceleration x [pulse/s 2 ]i sq uant ized by the angular acceleration resolution R A [pulse/s 2 ]. In addition, considering the effect of torque quantizationonthe control performance,itassumed that position and velocitywithout quantizationare feedback with continuous values. In the actual software servosystem, the en- coder resolution of the servomotor is infinite. Thatis, theposition andvelocity information is continuously obtained at the desired state. Compared with the actual software servosystem with an encoder,the controlperformance with this assumption is the maxim um po ssible. The condition of deriving torque resolution is considered as thep rerequisite condition. In the soft wa re serv os ys- 88 4Q uan tization Error of aM ec hatronic Serv oS ystem 0 x [ p u l s e /s 2 ] f ( x ) [ p u l s e /s 2 ] R A R A Fig. 4.6. Quan tization of angular acceleration tem, forrealizing the required control performance,the A/D, D/Aconversion is carriedout with torque resolution capable of satis fying the lowerlimitation. Moreover, since introducing thisassumption, the analysis of the effect on the control performance of torqueresolution becomes easy and it is possible to derivethe torque resolution conditionequation by 4.2.4(1),(2). Theappro- priation of this conditionequation in 4.2.4(4)iscompletely expressedbya computer simulation taking into accountthe encoder of the servomotor. 4.2.2 Deterioration of PositioningPrecision Due to Torque Quantization Error (1)P osition Determination of the Soft wa re Serv oS ystem Fo rd etermining the po sition of the soft wa re serv os ystem, the effect of the torque quantizationerrorisconsidered. The positioningerror E s p = P ref − y ( ∞ )[pulse], whichisthe errorofobjectiveposition P ref [pulse] and the steady-state value of the position output y ( ∞ )[pulse], is determined based on the servoparameter K p , K v andthe angular acceleration resolution.The relationship equation is derivedtheoretically. As illustrated in Fig. 4.7,the servomotor is rotatedwith aconstantvelocityinput according to the objec- tive position P ref .The position can be determined. If the angular acceleration R A is quan tized, the ve lo cit yo ft he serv om otor will be alsoq uant ized in eac h sampling time interval ∆t v of thevelocityloop. Thatis, in the servosys- tem with the angular acceleration quantization, the velocityoutput is only changed with theunit of R A ∆t v [pulse/s]. This quantized resolution is called the angular velocityresolution. From this case, forthe servosystem with an- gular accelerationquantization, the velocityfeedbackiscarriedout untilthat angular velocityoutput becomes 0[pulse/s]. When the angular velocityoutput becomes zero, the velocityfeedbackiscut off andthe steady state is continued until the position outputbecomes constant. 4.2T orque Resolution 89 (2)Relationship between PositioningError andAngular AccelerationResolution At the momentthatthe input is equaltothe objective position P ref ,the input to the quantizationterm of Fig. 4.5 is expressed by K v ( K p ( P re f − y ) − dy/dt). When this value largerthanthe angular acceleration resolution R A , the position and velocityisfeedback. If the angular acceleration resolution is not full, that is, dy/dt =0[pulse/s], the outputofthe quantizationterm is 0 and the position outputremains constant. In the steady state that the position outputisconstant, thesize of the input to the quantizationterm is expressed by | K p K v E s p | with the positioning error E s p ,asFig. 4.7.When thisvalueisnot full of resolution R A of theangular acceleration,the position error E s p can be expressed by K p , K v , R A as | E s p | < R A K p K v . (4.8) From (4.8), theupper limitofthe position error E s p is proportional with the angular accelerationresolution R A andinversely proportional to theposition, velocityloopgain K p , K v . 4.2.3 Deterioration of Ramp Response Due to Torque Quan tization Error (1)Ramp Response of the Software ServoSystem Next, with regard to the ramp input of the software servosystem, the effect of torque quantizationerrorisconsidered. The objective trajectory of the servo motorisgiven with the constantvelocity V re f [pulse/s]. When the angular accelerationisquantized in each R A ,i ft he ob jective angular ve lo cit yi st he integer timesofthe angular velocityresolution, the angular velocityoutput is notchanged formaking the objective angular velocityconsistentwith an- gular ve lo cit yo utput.H owe ve r, if the ob jective angular ve lo cit yi sn ot the 0 0 T ime P o s i t ion E p s P r ef Fig. 4.7. Deteriorationo fp osition con trol in soft wa re serv os ystem 90 4Q uan tization Error of aM ec hatronic Serv oS ystem 0 0 0 T ime V eloc i ty T d T u T f E v r = R A ∆ t v V r ef V u V d P o s i t ion E p r E d E u Fig. 4.8. Deterioration of ramp response in software servosystem integer times of theangularvelocityresolution, the angular velocityoutput is changed because of inconsistence between objectiveangularvelocityand angular velocityoutput. Fig. 4.8 illustratedthe variation of theangularvelocityoutput.The upper part of Fig. 4.8 showsthe position fluctuation and the bottom part shows the angular velocityfluctuation.FromFig. 4.8,the response is dividedintotwo states: one is that the angular velocityoutput is belowthe objective angu lar velocity(scale of T d [s]) and another is that the angular velocityoutput is over the objectiveangularvelocity(scale of T u [s]). (2)State of Angular VelocityOutput under ObjectiveAngular Velocity V re f At the state of that the angular velocityoutput is belowthe objective angu- lar velocity V ref ,f romt he angular ve lo cit yq uant ization, the outputa ngular velocityisas V d =  V re f / ( R A ∆t v )  R A ∆t v [pulse/s] (where  x  is expressed as the maximalinteger below x ). The error V re f − V d between objectiveangular velocityand angular velocityoutput is made integral as theposition output error. If the angular acceleration input is overhalf of theangularacceleration resolution R A / 2(refertoFig. 4.6), the positivepulseequivalenttothe angu- lar acceleration resolution is generated. Whengenerating the pulse and the position outputerroris E d [pulse], the angular acceleration input is expressed as K v ( K p E d − V d )with the loop of Fig. 4.5.When thisvalueishalf of the angular acceleration resolution R A / 2, thefollowing relationship equation is [...]... the demanded precision of the ramp response can be determined (4) Numerical Example of Torque Resolution Determination The effectiveness of using the relationship between the derived control performance of the software servo system and the bit number of the A/D, D/A conversion in the software servo system is verified here The designed position loop gain and the velocity loop gain of the servo controller... response can be obtained from the bit number of the torque resolution of the actual operated software servo system (5) Relationship Among Control Performance, Torque Resolution and Servo Parameter The relationship amongst the control performance, torque resolution and servo parameter of the software servo system is summarized as below ... (4. 17) RAv = r Elimv ∆tv (4.21) The angular acceleration resolution RA is needed from equation (4.20) and equation (4.21) (4.22) RA ≤ min(RAv , RAp ) That is, when the angular acceleration resolution RA can be determined for satisfying the equation (4.22), the restraint of the deterioration of ramp response within the demanded allowance can be realized 94 4 Quantization Error of a Mechatronic Servo System. .. (4.22), and the calculation of the bit number B of the torque resolution using equation (4.25) for B = 15[bit] 4.2 Torque Resolution 95 When existing quantization of the position information in the actual software servo system, in order to investigate the degree of the obtained the control performance based on the torque resolution, the computer simulation is made using torque resolution and considering... Ev expressed by equation (4. 17) , and the angular acceleration resolution RA can be converted into the bit number B of the torque resolution by equation (4.25) This relationship is shown in Fig 4.9 According to the use of Fig 4.9, although the bit number of the torque resolution cannot be worked out from the required control performance, the positioning precision and the control performance of the ramp... response exists (4) Amplitude and Cycle of Position Fluctuation From Fig 4.8, if the position has deviation Eu + Ed , when the error between the objective velocity and tracing velocity Vref − Vd is continued at the time Td , the time Td of that the angular velocity output is continuously below the objective angular velocity is as 92 4 Quantization Error of a Mechatronic Servo System Ed + Eu Vref − Vd RA... the bit number of the D/A, A/D conversion which is adopted for the current reference and feedback of the software servo system First of all, the angular acceleration resolution is converted into the torque resolution RT [Nm] using the pulse number P [pulse/rev] equivalent to the moment of inertia JM [kgm2 ] of the motor and the encoder of one time rotation RT = 2πRA JM P (4.23) Next, the bit number of... consistent between considering the torque quantization and not considering the torque quantization Moreover, in the desired state without considering the quantization of position, if comparing the design values and simulation results, the expected control performance can be obtained by using the derived torque resolution about the positioning precision and the position fluctuation of the ramp response The... combining the equation (4.10) and equation (4.12) The amplitude of the r velocity Ev [pulse/s] is as r Ev = RA ∆tv (4. 17) from the angular acceleration resolution From above derived equation (4.15), the equation (4. 17) is the relationship equation expressing the relation among fluctuation period Tf , amplitude of r r position fluctuation Ep , velocity fluctuation amplitude Ev and angular acceleration resolution... gain of the servo controller are Kp = 40[1/s] and Kv = 200[1/s], respectively The sampling time interval for the velocity loop is ∆tv = 50[µs] The rated values of the servo motor are JM = 0.13 × 10−4 [kgm2 ], Tmax = 1. 47[ Nm], P = 5000[pulse/rev] For making the positioning precision as (4.19), the deterioration of the ramp r r response as Elimp = 1[pulse] and Elimv = 1[pulse/s], the angular acceleration . control system of the mechatronic servosystem expressed by (4.1) is pickedout fromthe software servosystem shown in figure4.1 for encoder resolution analysis. The model of software servosystem. Experimentresults of software servosystem (b) Simulation results of software servosystem (c)Experimental results of analogue servosystem Fig. 4.3. Verification of velocityripple in software servosystem 4.1E nco der Resolution 85 amplifier. Servo Systemfor Torque Resolution The conceptual graph of the discussed software servosystem in this section is shown in Fig. 4.4.The software servosystem is shown in Fig. 4.4.Inorder to construct