Analysis of a PM DC Motor Model for Application in Feedback Design for Electric Powered Mobility Vehicles

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1 Analysis of a PM DC Moor Model for Applicaion in Feedback Design for Elecric Powered Mobiliy Vehicles P. Wolm 1, X.Q. Chen 1, J.G. Chase 1, W. Peigrew 2, C.E. Hann 1 1 Deparmen of Mechanical Engineering, Universiy of Canerbury PO Box 4800, Chrischurch 8140, New Zealand pwo19@suden.canerbury.ac.nz, {xaoqi.chen, geoff.chase}@canerbury.ac.nz z 2 Dynamic Conrols Ld, 17 Prin Place, Middleon, Chrischurch 8015, New Zealand wpeigrew@dynamicconrols.com Absrac - Accurae modelling of permanen magne (PM) DC moors is a prerequisie for expedien feedback design of elecric powered mobiliy vehicles. This paper idenifies he parameers in he ideal equaions for PM DC moors and considers he mehods used o measure and calculae hem accuraely. However, he ideal PM DC moor equaions oucomes are rarely accurae compared wih acual resuls and measuremens. The model inaccuracy arising from he exising heoreical model, which is ofen used in lieraure and by researchers, has been observed. The work discusses he mehod o increase he model accuracy, and analyse he model for he applicaion in feedback design for a arge vehicle. Tess have been conduced on he acual moors wih a real-ime daa acquisiion insrumen. Resuls have been obained and analysed o validae he improved PM DC model. Index Terms PM DC moor model, elecric powered mobiliy vehicles, feedback design, fricional orque. I. INTRODUCTION In he design of elecric powered mobiliy vehicles used for medical purposes, high safey sandards mus be me. In order o design, verify and une sysems such as feedback conrols ha can be used for a variey of differen ypes of mobiliy vehicles, an accurae model of such vehicles would expedie he process and aid grealy in reaching or exceeding safey requiremens. Alhough such models would comprise boh physical dynamics and elecrical dimensions, he elecrical drivers which are normally PM DC moors would be an iniial saring poin for accurae model developmen. Quie ofen daa plae values are incomplee or missing enirely. Confirmaion of any values provided is a mus for any model or coniguous bach of PM DC moors being used paricularly as he drivers are usually geared and reaed as a single sysem which has an effec on parameers. Once he moor-sysem parameers are known each conrol sysem is closer o being made unique o each wheelchair, scooer, ec. Many openly available sources of relevan engineering exs [1-3] and academic papers [4-6] lis he basic parameers for DC moors bu leave i o he reader o inuiively ascerain ha he lised parameers apply only o an ideal model. Of paricular concern is he reamen of fricion wihin a DC moor sysem. Rizzoni [1] saes fricion losses in he load are represened by he viscous fricion coefficien, implying ha no oher fricion is considered or exiss. Meanwhile, Wesphal [2] goes farher saing The only appreciable fricion effec in operaion is viscous fricion and coulomb and saic fricions are negleced giving a hin ha here are oher possible parameers o be considered. Oher sources menion various oher parameers such as T f, fricional orque [7] which is no fully expanded upon. This work presens an accurae model of an acual geared PM DC moor and he difference in he respecive resuls beween he ideal heoreical equaions and a model wih an enhanced equaion o acual measuremens. Firsly he ideal equaions and parameers and he mehods of measuring and calculaing he parameers are discussed. Then he resuls of he ideal model are conrased agains acual values. This is followed by discussion of he enhanced equaion wih an addiional parameer including is mehod of calculaion. Acual resuls are hen compared o he oupus of a Simulink model of he geared PM DC moor wih he enhanced equaion. II. PM DC MOTOR MODEL A. Ideal Equaions The equivalen circui model for an ideal DC moor is depiced in Figure 1. Figure 1: Ideal DC moor model.

2 Using Kirchoff s volage law and summing he volages hroughou a simple circui of a DC moor drive sysem resuls in di V Rai + La + V d = (1) where R a, L a and V emf are he armaure (or roor) resisance, armaure inducance and he back emf. The equaion of moion for a DC moor drive, ignoring any load, gives orque as a second order ODE in roaion as T J θ + Bθ = (2) where J and B are he moor ineria (or equivalen sysem ineria in a geared moor sysem refleced on he moor shaf) and he viscous fricion coefficien, also known as viscous damping coefficien (or equivalen viscous fricion for a geared moor sysem). The elecrical and mechanical componens are coupled in wo ways. Firs, an approximae relaion generally describes moor orque as a linear funcion of curren in he moor: emf B. Measuremen and Calculaion of Parameers Torque Consan k : Rearranging (3) provides he orque consan in erms of orque versus curren. Hence, his requires concurren orque and curren measuremens. Figure 2 shows he es se-up for orque measuremen. I consiss of a geared PM DC moor wih wheel rim, wo Newon force meers, a digial muli-meer (DMM) o measure volage a he moor erminals, a DMM clamp meer o measure curren, and a rope o provide a load on he moor. A seady 24 V was inpu, and he moor was allowed o reach a consan speed. The load was seadily increased unil moor sall. Moor curren, he forces on boh ends of he rope, and moor erminal volage o ensure ha volage was seady a he moor erminals were measured and recorded a se inervals. The moor orque was calculaed by muliplying he difference beween he wo forces on each end of he rope wih he radial disance beween wheel hub and where he rope resed on he wheel rim, i.e. T = Fr. T = k i (3) where k is he moor orque consan. Secondly, he back emf in he moor is linearly relaed o he moor roaional velociy, V emf = k e θ (4) where k e is variously known as he speed, elecrical, moor or back emf consan. The combined elecrical and dynamic relaionships resul in a sysem of equaions ha govern a DC moor sysem s response. Equaions (3) and (4) are subsiued ino (2) and (1) respecively o resul in he final sysem equaions of an ideal PM DC moor model; a 1s and 2nd order ODE ha can provide wo resuls, i and θ: k i J θ Bθ + = (5) di V = R eθ ai + La + k (6) d Represening (5) and (6) in a model of sae space form provides B θ J di = ke d La k J R L a a 0 θ + 1 V i La (7) θ θ = [ 1 0] (8) i Figure 2: Tes rig o calculae orque consan. Figure 3 shows he resuls wih he slope of he line providing he orque consan; Nm/A. Torque (Nm) Torque Consan, k y = x R 2 = Curren (A) Figure 3: Resuls of orque versus curren es Speed Consan, k e : If only looking a seady sae values, rearranging (6) o give slope-inercep form resuls in

3 V i θ = ke Ra i + (9) where θ is he moor shaf roaional speed and no speed of he gear oupu shaf. From his only seady sae values of inpu volage, curren and moor shaf velociy need o be measured. The es se-up included he same geared PM DC moor wih wheel, a seady 24 V supply, DMM o measure erminal volage, DMM clamp meer o measure curren, a means o conrol moor velociy, i.e. volage a he moor erminals, which in his case was a variable resisance, and a digial rpm gauge ha measured he rpm of he wheel which was aken o equal he moor shaf velociy. Figure 4 provides he resuls of measuremens made a muliple erminal volage inpus under no load condiions wih k e equalling he slope of he resuling line; rad/s/v. I should be noed ha in an ideal DC moor, he speed consan is equal o he orque consan. The measuremens obained by experimen indicae his is no he realiy. V / i ke & Ra y = 1.685x R 2 = θ_do / i Figure 4: Seady sae resuls of (9). Armaure Resisance, R a : By uilising (9) and he resuls for he speed consan, armaure resisance can be aken from he equaion of he line. I is he y axis inercep value, ohms (Ω), as seen in Figure 4. Viscous Fricion Coefficien, B: Under seady sae condiions (5) becomes Rearranging (10) provides k i B θ = (10) B = k i / θ or B k i / ω = (11) A range of seady sae curren and corresponding roaional velociy values were measured wih he same se-up as used o calculae he speed consan. The range in values was used o calculae B such ha i/ω became i/ ω, wih k already known. The fricion coefficien also akes ino accoun he gearing and eddy curren losses in he moor iron which increase wih speed. Hence B ends o be a lumped erm. Armaure Inducance, L a : To measure he armaure inducance a digial meer capable of measuring and recording insananeous or real-ime changes was needed. The insrumen used was a Fluke 105B Scopemeer. The es se-up of he geared PM DC moor also required a seady dc volage supply. A fixed volage was supplied o he a-res moor sysem, wih he roor/wheel locked. The rae of curren rise wih ime was capured by he meer and inducance was calculaed using (12) following his paragraph. Back emf is zero since here is no roor roaion. Iniial curren is zero and subsequenly negligible when compared wih rae of change of curren wihin he firs insance of moor sar-up. As a resul (1) becomes, when rearranged di L a = V / (12) d Wihin he firs 3 ms he curren wen from 0 o 44 A wih a 12 V sep inpu resuling in L a = 0.82 mh. Moor/Sysem Ineria, J: Once again he real-ime measuring meer, he Fluke 105B Scopemeer was needed and used he same es se-up as for armaure inducance bu wih a means o conrol volage inpu and hence moor curren levels. In his case he rae of volage change needed o be measured. The moor roor was held locked unil a se curren level was reached, overcoming he iniial effecs of roor inducance, and hen released. Once again he roaional velociy is iniially zero and subsequenly negligible when compared wih he rae of roaional acceleraion wihin he firs insance of moor release. To calculae moor/sysem ineria (5) was rearranged aking ino accoun negligible iniial velociy o give J ( k i) / θ = (13) The roaional acceleraion was iniially in unis of V/s bu was hen muliplied by he speed consan o conver o unis of rad/s 2. Wih a seady release curren of 17.2 A he resuling sysem ineria was kg. m 2. III. GEARED PM DC MOTOR A. Measured Oupus The relevan measured oupus have been menioned under II.B for he speed consan where seady sae values of volage inpu, moor curren and oupu angular velociy of he moor shaf were recorded. In paricular, he velociy of he roaing wheel was aken o be equal o he roaional velociy of he of he geared PM DC moor shaf. This is a resul of lumping he momen of ineria of he moor roor wih he gearing and wheel rim including he ire. Figure 5 indicaes he acual relevan seady sae measuremens over an arbirary ime period, hence he areas of no daa beween seady sae values.

4 Figure 5: Measured seady sae values of PM DC geared moor. I can be seen ha he oupu roaional velociy increases as expeced as does he moor curren alhough very slighly, also as expeced. This is prediced in (1) where under higher roaional velociies he back emf becomes predominan. Hence volage is limied inernally in he moor and herefore also moor curren. Once a load is applied, back emf is decreased as he moor slows. This increases he volage difference inernally hereby increasing curren flow and he orque o mainain a given roaional velociy for a given volage inpu and loading. B. Measured Oupus versus Ideal Model As saed for he sae space model of an ideal PM DC moor, he oupu can be in erms of roaional velociy of he oupu shaf/roor or he moor curren. The roaional velociy oupu is imporan in ha i is used o calculae he driving forces provided by he wheels of a wheelchair or oher mobiliy vehicles. Inernal moor curren can be used if necessary in feedback design for wha is known as load compensaion. Thus, hese are wo oupus ha are mos relevan wih regard o modelling and feedback design and which are mos necessary o compare o acual values. Figure 6 shows he measured curren and he equivalen ideal model oupu for he same given volage inpus. Figure 6: Acual curren measured versus equivalen ideal model oupu. The Simulink model has inerpolaed beween he seady sae volage inpus o give he coninuous graph of volage. Furhermore, he Simulink model inerpreed he iniial volage enered, which was no zero, as a sep. Hence, here is an iniial curren spike which emulaes wha would acually happen in a PM DC moor when a sudden sep inpu of volage occurs and recorded coninuously. Noe ha he subsequen curren spikes are mued. They would be similar o he iniial curren spike. The muing is due o he relaively shallow slope of he volage rise beween inpus due o Simulink s solver inerpolaion. However he inen is accurae as are he seady sae values for he given parameers. Mos noable is he large difference beween he acual curren measured and he ideal heoreical model s equivalen oupu. Figure 7 provides conras beween he measured roaional velociy and he equivalen oupu of he moor model for given inpu volages. Figure 7: Acual roaional velociy measured versus equivalen ideal model oupu. I is clear ha he model oupu of roaional velociy is much closer o he acual measuremens. The iniial increase and gradual levelling of he model s roaional velociy (as seen o he far lef of he Figure 7) is also in keeping wih wha would be seen of an acual PM DC moor wih volage sep inpus. The subsequen changes are he same as for he subsequen changes given in he curren comparison of Figure 6. IV. ENHANCED PM DC MOTOR MODEL EQUATION A. Addiional Parameer As saed in he inroducion here have been hins of addiional sources of fricion. Jang and Lee [8] make menion of DC moor fricion modelling and in paricular he Coulomb fricion model and menion fricional orque as do Rahman and Hoque [7]. However neiher makes he connecion beween Coulomb fricion and fricional orque and how hey may be idenified. Kara and Eker [9] discussed Coulomb fricion using he same symbology as for fricional orque and Sepp [10] goes on o clearly lay ou he fricion model including viscous, Coulomb and sicion or saic fricion. Coulomb fricion is considered o be a consan rearding force bu is disconinuous over zero crossings. Tha is, when a

5 moor reverses direcion i mus come o a sop a which poin Coulomb fricion drops o zero and hen opposes he reversed direcion. In effec Coulomb fricion is consan when roaional velociy is no zero. Boh Kara and Eker and Sepp consider Coulomb fricion as a non-linear effec, alhough Kara and Eker sae he linear model is capable of supplying he required robusness propery when he mechanical sysem roaes in a single direcion. In he case of modelling for feedback conrol of wheelchairs or oher mobiliy vehicles, dynamic effecs such as over or under-seer are noiceable under moion and he higher he velociies, he more noiceable. In hese cases Coulomb fricion can be reaed as a linear consan. Any non-lineariy associaed, and sicion as well, would only be a facor wih sop-sar dynamics such as jerk moion. For he purposes of his paper Coulomb fricion is reaed as linear. The enhanced DC moor model is shown in Figure 8. Figure 9: Acual curren measured versus equivalen enhanced model oupu Figure 8: Enhanced DC moor model. Through he fricion model, now recognising Coulomb fricion, τ f, (2) becomes = J θ + Bθ + T (14) Under seady sae condiions, subsiuing in (3) and rearranging provides Coulomb fricion in unis of Nm. τ f k i Bθ τ f = (15) The same series of measuremens as were used o calculae he speed consan were he values of inpu ino (15), namely curren and roaional velociy. Boh he orque consan and viscous fricion coefficien had been calculaed previously. The average of he series of calculaions provided he Coulomb fricion value. B. Measured Oupus versus Enhanced Model Figure 9 and Figure 10 compare he acual measured values for curren and roaional velociy respecively wih he equivalen enhanced model oupus for given volage inpus. As can be seen in boh Figure 9 and Figure 10 he model oupus closely follow boh acual measured values for curren and roaional velociy. Figure 10: Acual roaional velociy measured versus equivalen enhanced model oupu Table 1 provides he resuls of mean square error (MSE) mehod as a means o validae he enhanced PM DC moor model using he seady sae values found in Figures 7 and 8. Table 1 measuremen resuls of curren and roaional velociy. Inpu Volage (V) Curren (A) Roaional Velociy (Rad/s) V. CONCLUSIONS The experimenal es bed comprising a fron wheel drive vehicle moorised by wo PM DC moors, acuaors/sensors, and real-ime daa acquisiion sysem has been repored in [11]. This paper addresses accurae modelling for a geared PM DC moor as a firs sep o creaing a full model for he dynamics of a wheelchair or mobiliy vehicle. Since he DC moors are he only source of direcly conrollable driving force in he above menioned vehicles, i is essenial ha hey be accurae. The comparison of ideal PM DC moor model oupu and experimenal resuls shows errors in applying he sandard model o he geared PM DC moor. This work inroduced Coulomb fricion orque as an addiional parameer. The

6 improved model is more accurae, and has been validaed by experimens. There is a close agreemen beween he model oupu and he measuremen daa. By creaing he accurae moor model a crucial sage in pracical conrol problems for wheelchairs and mobiliy vehicles has been se. Wih he above solid basis, an adequae dynamic sysem model can be developed o es he reliabiliy of designed conrol. ACKNOWLEDGEMENT This work is suppored by he research gran DCLX0601 from Foundaion of Research, Science and Technology (FRST). The auhors would like o hank he suppor and echnical inpus from Dynamic Conrols Limied. REFERENCES [1] Rizzoni, G. (2004) Principles and Applicaions of Elecrical Engineering, McGraw-Hill, New York, pp. 869 & 875. [2] Wesphal, L.C. (2001) Handbook on Conrol Sysems Engineering, Springer, pp [3] Ainyas, Y. (2000) Manufacuring Auomaion: Meal Cuing Mechanics, Machine Tool Vibraions, and CNC Design, Cambridge Universiy Press, pp [4] Chen JX, Guo YG, Zhou JG, Jin JX, Performance Analysis of a Surface Mouned Permanen Magne Brushless DC Moor using an Improved Phase Variable Model, Record of he 2007 IEEE Indusry Applicaion Conference, pp [5] Kim CG, Lee JH, Kim HW, Youn MJ, Sudy on Maximum Torque Generaion for Sensorless Conrolled Brushless DC Moor wih Trapezoidal Back EMF, IEE Proceedings Elecric Power Applicaions, Vol 152, No 2, March 2005, pp [6] Haji H, Toliya HA, Roor Eccenriciy Faul Deecion of a DC Moor, Proceedings of 27 h Annual Conference of IEE, Indusrial Elecronics Sociey, Vol 1, 2001, pp [7] Rahman MA, Hoque MA, On-Line Self-Tuning ANN-Based Speed Conrol of a PM DC Moor IEEE/ASME Transacions on Mecharonics, Vol 2, No 3, Sep 1997, pp [8] Jang JO, LEE PG, Neuro-fuzzy Conrol for DC Moor Fricion Compensaion, Proceedings of he 39h IEEE Conference on Decision and Conrol, Vol 4, 2000, pp [9] Kara T, Eker I, Nonlinear Modelling and Idenificaion of a DC Moor for Bidirecional Operaion wih Real Time Experimens, Journal of Energy Conversion and Managemen, Vol 45, 2004, pp [10] Sepp K, Fricion Modelling in Linear Track Car Pendulum Sysem, [ May 2008, unpublished. [11] Chen XQ, Wolm P, Ellio R, Chase JG, Peigrew W, A Wireless Tesbed of Fron Wheel Drive Wheelchair for Sabiliy Conrol Prooyping, Conference Proceedings Compuaional Inelligence Roboics and Auonomous Sysems, ISBN , pp , Palmerson Norh, New Zealand, 28h 30h November, 2007.