00asme maas schulte froehleke

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 5, NO. 2, JUNE 2000 165

Model-Based Control for Ultrasonic MotorsJürgen Maas, Thomas Schulte, and Norbert Fröhleke

Abstract—A complete model-based control for traveling-wave-type ultrasonic motors is presented in this paper. The controlscheme consists of inner control loops with respect to the oscil-lation systems, offering all meaningful degrees of freedom foradjusting the traveling bending wave, and outer control loops fortorque and speed. After a brief review on modeling the actuatorand presentation of a parameter identification method, thecontrol scheme is developed and verified by measurements on aprototype drive system. Several measures for the compensation ofnonlinearities and temperature effects are developed and achievedimprovements are discussed with respect to the special propertiesof this novel actuator. Finally the developed drive is applied to an“active control stick.”

Index Terms—Active control stick, motion control, nonlinear os-cillation systems, piezoelectric ultrasonic motors, vibration con-trol.

I. INTRODUCTION

PIEZOELECTRIC ultrasonic motors (USM’s) have at-tracted considerable attention as a new type of actuator

for servo drive applications, e.g., as actuators for robotics. Inparticular, rotary traveling-wave-type USM’s combine featuressuch as high driving torque at low rotational speed, highholding torque, low electromagnetic interference, and are alsomore compact than conventional geared motors.

Fig. 1(a) shows an exploded view of a rotary traveling-wave-type USM with its basic components. The driving principle [1]is based on a two-stage energy conversion process.

In the first energy conversion stage, mechanical vibrationsof the stator are excited to its eigenfrequency by a ring-shapedpiezoelectric ceramic actuator which is bonded to the “lowersurface” of the stator. The piezoelectric ceramic ring is dividedinto two excitation systems (two phases) whose sectors are al-ternately positively and negatively polarized. Each system is fedby a sinusoidal voltage in the ultrasonic range (40–45 kHz) gen-erating a standing wave on the stator [see Fig. 1(b)]. Since a spa-tial phase shift of a quarter wave length is introduced betweenthe excitation systems, a traveling bending wave is generatedby a correct superposition of both standing waves achieved withproper amplitudes and a well-defined temporal phase shift.

The rotor is pressed against the stator by means of a discspring. In most traveling-wave motors, the rotor is coated witha special lining material, realized by an elastic contact layer,

Manuscript received December 15, 1999; revised February 1, 2000. Thiswork was supported by the Deutsche Forschungs Gemeinschaft (DFG).

J. Maas was with the Institute for Power Electronics and Electrical Drives,University of Paderborn, D-33098 Paderborn, Germany. He is now withDaimlerChrysler AG Research and Technology, D-60528 Frankfurt, Germany(e-mail: [email protected]).

T. Schulte and N. Fröhleke are with the Institute for Power Electronics andElectrical Drives, University of Paderborn, D-33098 Paderborn, Germany(e-mail: [email protected]).

Publisher Item Identifier S 1083-4435(00)05159-0.

(a)

(b)

Fig. 1. (a) Basic components of rotary traveling-wave-type USM. (b) Drivingprinciple of traveling-wave-type USM.

in order to obtain good force transmission [see Fig. 1(b)]. Dueto the traveling bending wave, the surface points of the statorperform elliptic motions containing a vertical and a horizontalcomponent. While a normal pressure distribution is producedat each wave crest by the vertical movement, frictional force isgenerated inside the elastic contact layer driving the rotor bythe horizontal movement. This second energy conversion stageis characterized by extreme nonlinear contact mechanism be-tween stator and rotor.

Due to the complex and nonlinear characteristics of USM’s,intensive research efforts are required to optimize the perfor-mance of USM’s. Besides several research activities with re-spect to the motor design and materials, an appropriate drivecontrol must be applied for sufficient operation, since the motorparameters are influenced heavily by temperature (e.g., mechan-ical eigenfrequency and speed characteristic; see, also, [2]). Thebehavior of the motor is characterized by a nonlinear mechan-ical oscillation system and nonlinear torque generation (see Sec-tions II-D and V-B) [3].

Several control strategies have been proposed, which arebased on black-box modeling and fuzzy or neuro methods [4],[5]. Since these control schemes are developed without any

1083–4435/00$10.00 © 2000 IEEE

166 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 5, NO. 2, JUNE 2000

Fig. 2. Overall scheme of the developed USM-drive system.

analytical model approach, their potential of optimization islimited. They neglect decisive nonlinearities and motor asym-metries and make use of frequency modulation to adjust thespeed. Consequently, the well-known pullout phenomenon (seeSection IV-C) of USM’s occurs, by which a large advantageousrange of the drive’s performance cannot be utilized.

In the following, a complete model-based control scheme ispresented which avoids the latter disadvantages. Fig. 2 showsthe overall scheme of the developed drive system, with detailsbeing explained in [7]. The electrical excitation of motor vi-brations is applied by a specially designed two-phase voltage-source converter. Due to dielectric properties of the piezoelec-tric ceramic, characterized by a capacitance in each phase, in-ductors and have to be added in series. Thus, the USMbecomes an integral part of a resonant converter. Two small sec-tors of the piezoelectric ceramic layer (see Fig. 1) are utilized fordetecting both standing waves denoted by sensor signals(as-signed to phase 1) and (assigned to phase 2). In order to dealwith the motor asymmetries and cross couplings caused by thepiezoelectric ceramic layer and the vibrating stator, the feedingfull-bridge inverter stage is driven by adequate control quantitieswhich guarantee all necessary degrees of freedom: switchingfrequency , phase angles and , as well as dutycycles and of both phases (see Fig. 3). These control quan-tities enable an independent adjustment of voltage amplitudesand phases to obtain optimized motor conditions. The measure-ments in Fig. 3 illustrate that the USM must be fed by differentvoltage amplitudes and an electrical phase shift which differsfrom 90 in order to compensate motor asymmetries for opti-mized conditions. The corresponding driving signals, ,

, of the transistors are generated by a digital modulatorcontrolled by a digital signal processor (DSP).

Fig. 3. Measured voltagesv and currentsi of USM and measuredoutput voltagev of inverter stage resulting from modulator signalsQ ,Q ,Q , andQ set by control values! , ' , ' , � , and� .

As shown by measured electrical state variables in Fig. 3,the quantities of the electrical and mechanical resonant circuitsare characterized by sinusoidal waveforms in the ultrasonic fre-quency range. Only the fundamental components of motor volt-ages and sensor signals are used as feedback informa-

MAAS et al.: MODEL-BASED CONTROL FOR USM’S 167

Fig. 4. Simulation model of converter-fed ultrasonic motor divided into its functional modules.

tion for control which are determined by analog phase-sensitivedemodulation circuits. Additionally, the position of the rotor isdetected by an incremental encoder.

The block named “DSP-based control system” is focused onin this paper. The main goal is a speed-controlled USM driveoffering high performance for servo applications as requiredfor an “active control stick,” presented in Section VI. Since thebending wave is decisive for the torque generation (comparablewith the current of conventional electrical motors), it seems tobe meaningful to apply inner controls for the electrical and me-chanical oscillation systems. Their design is described in Sec-tion IV. The inner controls consider compensation measures forasymmetries and cross couplings and the pullout phenomenonis avoided by using amplitude modulation instead of frequencymodulation. Thus, the motor can be operated at the mechanicalresonance optimizing the drives performance remarkably. Bymeans of an inverse contact model, the nonlinear torque gen-eration is compensated. Therefore, an open-loop torque controlis applicable such that proven speed control schemes can be im-plemented as discussed in Section V. First, a brief review onmodeling is given in Section II and a parameter identificationmethod is proposed in Section III.

II. M ODEL OF INVERTER-FED USM

USM’s are characterized by a two-stage energy conversionprocess as previously explained. By the conversion process, theenergy is transferred from electrical input to mechanical outputterminals using different types of functional modules summa-rized in Fig. 4. Since the modules consist of electrical as wellas mechanical components, the new actuator represents a typicalexample of a mechatronic system completed by a control for en-visaged operation mode and performance. While the electricaland mechanical oscillation systems can be described by lineartransfer properties, the piezoelectric energy conversion and themicromechanical contact between stator and rotor are charac-terized by nonlinear behavior. Appropriate models are essentialfor optimization of the overall drive performance and designingthe control. The simulation model derived in [8] describes thetotal drive system by combining the electrical and mechanicalpartial models as indicated in Fig. 4. On one hand, the simula-tion model reflects the ultrasonic oscillations (see Fig. 3) of theelectrical and mechanical resonant circuits and is, therefore, in-dispensable for detailed studies of the motor performance. Ho-ever, on the other hand, this so-calledoriginal modelcannot beused straightforward when designing the control, because am-

Fig. 5. Comparison between original model and averaged model for motorvoltagev .

plitudes and phases of the ultrasonic oscillations can only beinfluenced by the control signals , , , , and .

In order to obtain an appropriate model for the control de-sign, an averaged model is derived in [10] based on the originalmodel in [8]. Since this paper deals with the design of the drivecontrol, essential details of the averaged model are selected to bedescribed in the following. According to the resonant structures,the ultrasonic oscillations contain a major fundamental compo-nent shown by measurements in Fig. 3.The sinusoidal waveformof each ultrasonic oscillation can be well approximated by

(1)

The averaged model in [10] reflects the slow dynamic be-havior of the fast-changing ultrasonic oscillations usingtheir time-varying fundamental Fourier coefficients and

(in Cartesian coordinates) as new state quantities and is,thus, predestined for designing the control.

In Fig. 5, a comparison between simulation results obtainedby the original and averaged model are illustrated for motorvoltage . Since the envelope of the ultrasonic oscillation

is well approximated by the amplitude , calcu-lated by the state variables of the averaged model through

(2)

the results verify the selected approach based on the averagingmethod.

A. Model of Electrical Subsystem

The electrical subsystem comprises the inverter stage and theelectrical resonant tanks represented by the series inductances


, of the converter and the capacitances, of thepiezoelectric ceramic (see Fig. 2), taking losses of the compo-nents additionally into account. Inputs of the resonant circuitsare the fundamental voltage components of the inverter stage

(3)

resulting from control inputs , , , , and of the two-phase resonant converter. The output quantities are given by thevector of fundamental model voltages

(4)

The dynamic behavior between input voltageand outputvoltage is given by the transfer matrix

(5)

using indexes 1 and 2 for the two-phase model of the electricalsubsystem

(6)

describes the feedback of the vibrating stator on the feedingconverter by the piezoelectric effect (see Section II-B). Eachphase is characterized by a multivariable plant denoted bytransfer function for direct dynamical couplings betweenthe time-varying Fourier coefficients and denoted by transferfunctions for dynamical cross couplings. For the controldesign these transfer functions can be reduced to second-ordermodels, containing the eigenvalues

, as outlined in [11] given by

(7)

(8)

B. Piezoelectric Energy Conversion Process

The motor voltages excite the mechanical subsystem of thestator via the inverse piezoelectric effect. Usually, a scalar andlinear approach is used for describing the piezoelectric ceramicactuator. In [6], a model for a piezoelectric stack actuator wasderived and it was proven successfully by measurements. AMaxwell resistive capacitance (MRC) was introduced in theelectrical subsystem in series [Fig. 6(a)]. Its behavior is given bya rate-independent hysteresis between stored chargeand ap-plied voltage [Fig. 6(b)], while the behavior of the block

(a)

(b) (c)

Fig. 6. MRC. (a) Model of piezoelectric actuator with MRC element. (b)Rate independent hysteresis of MRC element. (c) Equivalent circuit of thepiezoelectric actuator with MCR.

linear piezo actuator is described by linear properties. Since nospecific boundary conditions are given, this model is used as anappropriate model approach for piezoelectric motors. For a de-tailed elaboration of the model refinement, see [15].

With respect to the averaged model, the describing functionof the MRC contains a real and an imaginary part because of itshysteresis. Within a small operation range, it can be describedby series-connected linear capacitance and resistance

. Since it is not possible to distinguish betweenand the total piezoelectric capacitance, is consideredonly as illustrated by the equivalent electric circuit in Fig. 6(c).

The voltage drop across in Fig. 6 lowers the effectivevoltage in comparison to measurable motor voltage. Inequation (4), model voltages are introduced in order to ob-tain a simple model structure describing the electrical oscillationsystem without impact of the mechanical oscillation system.Therefore, piezoelectric feedback voltages must be intro-duced on one hand in (6) as additional inputs (see Fig. 4). Onthe other hand, feedback voltages and must be takeninto account when calculating from measurable motor volt-ages , too. is given by (9), shown at the bottom of the nextpage. Voltage vector excites the mechanical subsystem by ac-celeration vector

(10)

represents the modal mass of vibrating stator, containsthe electromechanical coupling by coefficients, , and term

models small direct cross couplings between the mechanicalmodes (for details refer to [11]). Due to inaccuracies when man-ufactured and inhomogenities when bonding, the piezoelectricactuator cross excitation of the two-phase piezoelectric actuatormust be taken into account considered by coefficients, inmatrix .


Reflecting the general principle of the electromechanical en-ergy conversion, piezoelectric feedback voltagesare calcu-lated by

(11)

with matrix containing transposed and capacitances, of the piezoelectric actuator. Vector represents the

fundamental components of the traveling bending wave in theaveraged model.

C. Model of Vibrating Stator

The dynamics of the vibrating stator are well modeled by atwo-mass approximation describing both standing waves underconsideration of asymmetries by a small difference with respectto their eigenfrequencies. The dynamic transfer behavior of theaveraged model between input quantityand output quantityis given by

(12)

Using indexes 1 and 2 for both standing waves, the transfer ma-trix

(13)

models the multivariable plant properties of one mechanical res-onant tank. By simplification of the transfer functions above,second-order models are obtained containing the eigenvalues

(14)

(15)

D. Nonlinear Stator–Rotor Contact Mechanism and RotorModel

Modeling the friction mechanism, an elastic model for thecontact layer is applied in [8], describing the nonlinear interac-tion between the models of stator and rotor. The contact modelis derived for the general case of a nonperfect traveling wave,which means in comparison to a perfect traveling wave that,in this case, the amplitudes of both standing waves are dissim-ilar and/or the temporal phase shift between them differs from90 . Feedback impacts of contact forces on the stator are mod-eled by a modal forcing vector . Contact forces acting on therotor are described by motor torque opposing the appliedload torque . Modeling the rotor dynamics the drive’s speed

(s/r/min) is taken into account by the modelgiven in Fig. 4. For the control design, a one-mass system isassumed represented by inertia. Under consideration of av-eraging method for the vibrating stator, the fundamental de-scribing functions of are of interest only, while for the rotordynamic, the dc value of is important.

The eigenvalues of the mechanical vibrator are affected by thenonlinear impacts of contact forces . The viscoelastic prop-erties of the contact layer and the feedback of frictional forcescause, on one hand, an increase of total stiffness of the vibratingsystem by which a shift of the vibrator’s eigenfrequencyof about kHz results. On the other hand the vibration’sdamping increases, yielding variations of coefficient.

Under consideration of contact forces, the vibrating systemreflects a nonlinear large signal model, whose eigenvalues de-pend strongly on the operating point (in particular, on the am-plitude of the traveling bending wave). However, an onlinecalculation of for control measures is not feasible due tothe complex nonlinear characteristic, which requires high com-putation demands. Moreover, the practicability failed, too, be-cause of inaccuracy of the contact model and uncertainties ofparameters caused by temperature effects. Therefore, an alterna-tive solution is proposed to obtain optimized drive performanceunder consideration of the nonlinear feedback. As is commonlyknown, such systems can be treated by linearization at the oper-ating point and an adaptive control with respect to the operatingpoint.

Besides the variations of eigenvalues by contact forces, theinfluence of switching frequency must be considered, too.This has an impact on the dynamics of the oscillation systems.As indicated by the signal line for in Fig. 4, the switching

(9)


frequency is varied by the control during operation. Thus, thesmall-signal behavior of the stator model is derived by lineariza-tion in case of small disturbances of , , and . Aftersuitable simplifications, [11], the linearized stator model at theoperating point is obtained by

(16)

As indicated by (16), impact on the bending wavecausedby the discussed disturbances are described by the same transfermatrix as derived in (13). Thus, a small frequency modu-lation can be approximated by an amplitude modulation. More-over, effects of disturbance can be eliminated by adequatefrequency modulation of (see Section IV-B).

Besides the nonlinear impacts analyzed above, slow-varyingparameter variation occurs additionally, caused by temperatureeffects, which influence the electrical as well as mechanicalsubsystem due to parameter dependencies of the piezoceramiclayer, stator material, and contact layer.

III. I DENTIFICATION OF MODEL PARAMETERS

The control presented in Sections IV and V is based on theaveraged model and parameterized by suitable identificationmethods. While parameters of the electrical and mechanicalresonant tanks are estimated online (Section III-B), the pa-rameters of the piezoelectric coupling matrix and thevalues of and must be known in advance. In thissection, a parameter identification method is presented basedon investigations in the frequency domain.

A. Offline Identification of Piezoelectric Actuator Model

For parameter identification of the piezoelectric actuatormodel, its input and output quantities must be known. Sincethe electrical input signals (motor voltages and currents

) can only be measured, but not the output quantities ofthe piezoelectric actuator [acceleration(see Fig. 4)], theoutput quantities of the mechanical resonant system mustbe utilized. Therefore, the parameters of the piezoelectricactuator model can only be identified, when the mechanicalsubsystem is included [15]. Thus, the resonant frequency andthe damping of the mechanical subsystem must be consideredwithin the parameter identification, although these values arenot of interest here. Since these parameters depend on the op-eration point (due to feedback forces; see Section II-D), theidentification task is impossible without additional measures.In order to deal with these nonlinearities, the rotor might beremoved before performing the measurements. However, inthis case, the mechanical vibration system and, therefore, thepiezoelectric actuator is not loaded as compared to practicaloperation, because the inherent damping of the stator withoutthe rotor is very low.

This disadvantage can be avoided when performing measure-ments with a constant bending wave in order to keep the param-eters of the mechanical oscillation system constant. A constant

TABLE IRESULTS OF THEPARAMETER IDENTIFICATION RELATED TO PIEZOELECTRIC

COUPLING AND CROSSCOUPLING

bending wave is achieved using the control in Section IV, whichis adjusted by provisional parameters in a first step.

Measured results are evaluated by a Simplex search method,varying arbitrary initialized parameters of the model in such away that an appropriate loss functionis minimized. This lossfunction is calculated from measured output signalsandoutputs calculated by the model using the measured electricalinput quantities

(17)

During the experiment, the switching frequencyis variedwithin a range of 400 Hz around the resonant frequency,while motor voltages , currents , and modal amplitudes

are measured every 50 Hz at frequencies( measuringpoints). The parameter identification is performed for differentamplitudes of the bending wave chosen to be m, m,

m, and m and a temporal phase shift of 90 (per-fect traveling wave; see Section V). Table I shows the results ofparameter identification with respect to the piezoelectric cou-pling coefficients , and cross couplings , , while inFig. 7, results for and are illustrated. In accordancewith theory given in [15], variations of the amplituderesult insmall deviations of parametersand only. However, in con-trast to the results above, identification results ofandshow an increase of both parameters when the amplitudeand,therefore, the amplitudes of currents are increased. This isaffected by the hysteresis behavior of the piezoceramic, sincethe approximation by the linear equivalent electric circuit is onlyvalid in a small operation range, as discussed in Section II. Thismotivates the need to perform the parameter identification in theoperation range of interest, as mentioned above.

B. Online Identification of Resonance Parameters

With known parameters of the piezoelectric actuator, theinput quantities of the electrical and of the mechan-ical subsystem can be calculated online. Since the USM driveis excited by sinusoidal signals at any time, it is possible todetermine the parametersand of the resonant tanks by


(a)

(b)

Fig. 7. Parameter identification of (a)C and (b)R and atw. 1: 0.5�m.2: 0.6�m. 3: 0.7�m. 4: 0.8�m.

evaluation of its frequency responses at the operating point. Thefrequency response of resonant tanks is described by the staticgain of transfer matrices and containing the dampingcoefficients and the frequency deviations ofcorresponding eigenvalues. Under consideration of steady-stateoperation mode and by applying appropriate simplifications,the following identification equations were derived in [11] forthe electrical and mechanical subsystems:

(18)

(19)

(eigenfrequency at operating point) can be assumedas constant within the operating range. Since the impact offast-changing parameter variations (caused by nonlinearities)will be compensated by measures explained in Section IV,only slow-varying characteristics (e.g., caused by temperatureeffects) must be identified. Thus, a pseudostationary evaluationof transfer functions can be performed by using low-passfiltering of the inputs and outputs of resonant tanks. Then, thecomplex identification task is reduced to simple algorithms.

IV. V OLTAGE AND BENDING WAVE CONTROL

In order to design a high-performance speed control, an ap-propriate torque adjustment is required. For this task, an innerbending wave control is indispensable, since the torque of theUSM depends on the bending wave, as mentioned above. Theproposed control scheme is depicted in Fig. 8. It is based on theaveraged model outlined in Section II. Since the ultrasonic os-cillations of the motor are characterized by cascaded electrical

and mechanical resonant systems, it is obvious to control themotor by means of an inner voltage and an outer bending wavecontrol loop. Controlled quantities are the sine and cosine com-ponents of model voltages , and modal amplitudes ,which are measured by phase-sensitive demodulation circuits inFig. 2. In the following, the two-component controllers are des-ignated as vector controllers. Since the USM is characterized byasymmetries between phases, nonideal motor properties can becompensated only, when an asymmetrical feeding by the motorvoltages is applied (see Fig. 3). Thus, both phases must be con-trolled independently. Between the cascaded control structureand averaged model in Fig. 4(a), close coherence exists opti-mizing the drive control by the following systematic measures[11].

• The reference values of the inverter stage are passed tothe vector modulation algorithm in order to calculate theregulating quantities , , , and . A compensationof mechanical feedback on the converter is introduced.The voltage control is denoted by the vector controllers

.• A compensation of piezoelectric couplings between the

vibration modes is achieved by introducing the inversemodel of the piezoelectric actuator

(20)

• By this measure, both phases can be controlled indepen-dently and, therefore, an identical vector control schemecan be applied for the bending wave controlas used for the voltage control. The command inputs ofbending wave control are the amplitude of travelingbending wave and the phase between both standingwaves. These reference values are obtained from thespeed control and are transformed from polar to Cartesiancoordinates, as explained in Section V.

• Remaining regulating quantity of the converter istracked in such a way that the vibration modes of thestator are excited at resonance by means of(see Sections IV-B and IV-D).

• Utilizing the power performance of the drive controlfor wide-range operations with respect to high dynamicresponses, an online adaptation of control parametersand a disturbance compensation are recommended dueto nonlinear contact impacts and temperature influences(see Section IV-B).

A. Vector Control Scheme for Resonant Structures

The state variables and of resonant tanks are controlledby vector controllers , which are identical in structure forthe electrical and mechanical oscillation systems. Designingtransfer matrices of electrical plant (5) and of mechan-ical plant (13) must be considered, which are characterized bytransfer functions describing dynamical cross couplings be-tween the Cartesian input and output quantities. In order to re-duce the control design of one resonant system to a single-loopcontrol for each Cartesian quantity, a decoupling structure is re-quired. The decoupling of resonant structure(describing


Fig. 8. Voltage and bending wave vector control scheme.

Fig. 9. Vector control scheme for resonant structure.

and ) is achieved when decoupling matrix is introducedbetween plant and control by

(21)

This decoupling measure guarantees an operation at mechan-ical resonance, which is one important goal of the control. Asinvestigated in [11], the decoupled plant canbe well approximated by a first-order lag containing dampingcoefficient as time constant and frequency deviation asdominant static gain. Besides these dynamics, the behavior ofthe underlaid systems (denoted by ) and the measuring sys-tems must be taken into account for the design of the vectorcontrol in Fig. 9. While underlaid systems of the voltage con-trol are given by deadtimes of the digital control and modulator,the underlaid system of the bending wave control is given bythe closed-loop voltage control. For the single-loop controllers

, proportional–integral (PI) controllers are designed in [11],applying an analytical design method called “digital amount op-timum.” An analytical synthesis method is privileged due to theaimed application of an online adaptation of vector controllers.

In order to verify the vector control scheme for resonantstructures, the converter is loaded by a capacitanceandresistance , which are equivalent to motor parametersfor typical operation. Fig. 10 illustrates a comparison betweenresults obtained by simulation and measurement in case ofa step-like command and disturbance change. As is seen,first, a good agreement between simulation and experimentis obtained, second, the decoupling structure works fine, andthird, the design method offers acceptable dynamic responses.

B. Compensation of Nonlinearities and Temperature Effects

Results so far consider only linear time-invariant motor char-acteristics. When applying the cascaded vector control schemefor the drive control, appropriate measures for compensation ofnonlinear feedback forces and temperature effects must be de-veloped.

In Section II, a linearized model (16) of the vibrating systemis derived. In order to meet the assumption for linearization,first, the controllers have to be adapted, and second, the distur-bances by , and must be eliminated. Tempera-ture effects are compensated by adaptation of controllers usingparameters obtained by the online identification in Section III.Additionally, the fast-changing disturbances can be suppressedwhen using the remaining, regulating quantity of the con-verter. By introducing a fast frequency precontrol , thefrequency is modulated in such a way that the dominant distur-bance of the fast-varying eigenfrequency shift in (16) [describedby ] is compensated through

with

(22)

By this measure, the disturbance of the linearizedmodel (16) is eliminated together with achieving a fast adapta-tion of the control, since the frequency deviation , whichhas an impact on the dynamics of transfer matrix, is alwaysconstant. By investigation, it turned out that the characteristic

can be well approximated by a linear relationwithin the operating range of . Due to smaller influences of


(a)

(b)

Fig. 10. (a) Command response of voltage control byv = 300 V at t = 0.(b) Disturbance response caused byz = 15 V at t = 2 ms.

, its impact is neglected and must be compensated by therobustness of the control.

As introduced above, the mechanical subsystem is excitedat resonance for optimized operation mode (Section IV-D).This requirement is satisfied when applying a slow frequencytracking in addition to the fast frequency precontrol. A correctstationary tracking of resonance frequency is achieved whenestablishing a control unit by

(23)

using parameters identified by (19).The total scheme of online identification, adaptation, and fre-

quency modulation is depicted in Fig. 11. By implementing anI-controller for the slow frequency tracking, the requirement in(23) is guaranteed ( initial value). Under common consid-eration of the combined structure of the I-controller and onlineidentification, the frequency tracking represents a first-order lagand not a closed-loop frequency control.

Fig. 11. Online identification, adaptation, and frequency modulation.

(a)

(b)

Fig. 12. (a) Nonlinear frequency characteristics of mechanical subsystem. (b)Instability caused by turn-off of bending wave control att = 0.

C. Pull-Out Phenomenon and Hysteresis Effect

Nonlinear contact impact causes stability problems with re-spect to the frequency response, since the well-known pull-outphenomenon of traveling-wave USM’s occurs, as observedin [2], [4]. As illustrated in Fig. 12(a), this breakdown ( )appears when the feeding frequency is decreased belowthe resonance frequency . By increasing until the actualresonance frequency is attained again (), the rotor starts


to revolve abruptly. Since a hysteresis loop is obtained whenchanging in this way, the effect is designated as hysteresiseffect.

The reason is related to a degressive stiffness behavior ofthe contact mechanism, since variations of the amplitudein-fluence the resonance frequency , as mentioned above. Forinvestigation of the nonlinear frequency characteristics of me-chanical resonant tanks, the gain is calculated by

(24)

using the contact model in [8]. In Fig. 12(a), the calculated re-sults are compared with measurements. While outside the hys-teresis bandwidth single-valued curves are obtained, inside thebandwidth three solutions are calculated. Two of them are stableoperating points, indicated also by the measured curve. It can beshown that operating points marked by the dashed line charac-terize instabilities in the case of frequency control, which mustbe taken into account when designing the drive control.

Previously published control schemes always use frequencymodulation for controlling the amplitude. It is obvious thatthe pull-out phenomenon can be avoided only when a certaindistance between resonance frequencyand switching fre-quency is introduced (e.g., see [4]), consequently losing awide range of the performance of the USM, which is discussedin Section IV-D. Since amplitude modulation is applied in Fig. 8instead of frequency modulation for controlling, operatingpoints indicated as instabilities in Fig. 12(a) can be stabilized.Thus, the pull-out phenomenon can be suppressed in general of-fering an additional degree of freedom for frequency tracking.

In Fig. 12(b), the effect of stabilization is examined for oper-ating points which are unstable in the case of frequency control.The reference value of the controlled amplitude is chosen to be

m and the difference is about Hz,which characterizes the operating pointin Fig. 12(a). At time

ms, the bending wave control is turned off. Due to smalldisturbances, in practice, the operating point of the voltage con-trolled drive shifts from unstable point to stable point or

(high speed or standstill). Operating pointsor belongto the hysteresis curve in Fig. 12(a). As shown by these measure-ments, operation of USM drives at the mechanical resonanceand below this critical point is possible when using a suitabledrive control.

D. Experimental Results of Bending Wave Control

When operating the drive at the mechanical resonance fre-quency, the optimal operating point of the vibrator is found min-imizing the voltage demand for excitation, reducing losses ofthe converter and piezoelectric ceramic (see Fig. 14). In orderto prove the voltage decrease, measurements for different de-viations between and eigenfrequencies within anappropriate range of the feeding frequency are performed anddepicted in Fig. 13. Since the bending wave is kept constant bythe control, eigenfrequencies can be assumed as constant,too, and the voltage curves characterize the inverse behavior oflinear mechanical resonant circuits. A remarkable reduction ofthe voltage demand of the USM is achieved in contrast to pre-viously published control schemes, which keep a certain fre-

Fig. 13. Voltage requirement versus�! for constant modal amplitudes.

Fig. 14. Power distribution of USM drive.

Fig. 15. Large signal response of bending wave control for step change ofw.

quency distance to of several hundred hertz to avoidthe pull-out phenomenon.

In Fig. 14, the power losses within the USM drive are shown.In particular, losses affected by (see Section II-B) can beminimized when operating the USM at resonance, as mentionedabove.

In Fig. 15, measured results of a transient response are de-picted. A large step-like command of the amplitude from

m to m [ nm to nm;


Fig. 16. Speed and torque control scheme.

Fig. 17. Set-point adjustment for optimized force transmission.

see (25)] at ms is performed. The phase and frequencydeviations are kept constant to and .

As indicated by the measurements, the rise time of the modalamplitudes is only about 1 ms, and note that the decouplingworks sufficiently. By the fast frequency precontrol, a large shiftof of about Hz is modulated, tracking the eigenfre-quencies during the transient reaction of the amplitude. Incomparison with the results in Fig. 10, deviations occur due tothe neglected impacts of disturbance . However, a step-likecommand of such a large amount is not applied by the speedcontrol. Thus, the measured response satisfies requirements ofthe outer control loops.

V. TORQUE AND SPEEDCONTROL

While the torque of conventional electromagnetic motorsis dependent almost linearly on the current of the motor, thetorque of USM’s is dependent nonlinearly on the ampli-tudes of both standing waves, the temporal phase shift

, and the rotor’s speed , (e.g., see [9]). For realizing ahigh-performance speed control, the concept shown in Fig. 16is proposed, based on the compensation of the nonlinear torquegeneration of the USM by its inverse characteristic, called theinverse contact model.

This concept yields two important advantages.

• On the one hand, open-loop torque control is enabled.• On the other hand, a speed control loop can be realized by

applying a linear speed controller, which is well knownfrom conventional electrical drives with linear loads.

A. Set-Point Adjustment for Torque Control

Since there is more than one degree of freedom for adjustingthe speed–torque characteristic by the bending wave, an appro-

priate set-point adjustment must by chosen. In [12], it is pointedout that an operation with a perfect traveling wave goes alongwith minimized losses caused by the friction mechanism. Theseconditions are obtained when amplitudes are equal to asingle value and phase is kept to 90 . Thus, variations ofspeed–torque curves should be performed by adjustingfirst.Second, it is considered that amplitudecannot be decreasedunder a certain threshold due to stick effects at low ampli-tude which cause a sudden stop of the rotor rotation. Operatingthe drive in this inherent dead zone, a nonperfect traveling waveis required by adjusting phase and controlling the ampli-tude above the threshold. Therefore, a set-point adjustment ischosen which guarantees minimized losses within a wide oper-ation range, but continuous behavior within the whole operationrange, as illustrated in Fig. 17.

• In any operation mode, the amplitudes of both standingwaves are equal, established by .

• In the high-speed/high-torque region, the USM is oper-ated with a perfect traveling wave, obtained by adjusting

and .• In the low-speed/low-torque region, the phase shift

is used for the control and the amplitude is kept to itsthreshold .

The Cartesian reference values of the bending wave controlscheme are calculated by

(25)

Under consideration of the statements given above, measuredcurves of the USM under study are depicted in Fig. 18. While inthe high-speed/high-torque range the amplitude is varied from

m to m in steps of m(solid lines), in the low-speed/low-torque region, the phaseis adjusted to (dashedlines) controlling the amplitude above the threshold for contin-uous operation. Torque and speed of the USM are limited to

N m and r/min to prevent destruction.

B. Inverse Contact Model by Neural Network

Since analytical contact models describing the nonlineartorque generation are not sufficiently accurate, and are com-plicated, a numerical approach based on measurements (seeFig. 18) is predestined for realizing the inverse contact model.


Fig. 18. Measured speed–torque curves.

For this task, a basis function network (BFN) is proposed fordescribing the inverse speed–torque characteristic.

In order to generate the reference values of the bending wavecontrol from desired torque under consideration of actualmotor speed , a two-dimensional BFN with linear splinesis implemented for the inverse contact model which is trainedoffline by the sample points of Fig. 18. It serves for interpolationbetween the samples of a lookup table.

Since regulating quantities , are not varied indepen-dently (by this approach, only one quantity is varied at the sametime), it is possible to represent the set-point adjustment by onescalar and normalized regulating quantity when introducingthe transformation defined in

for

for

for

(26)

Thus, one BFN is sufficient for modeling the inverse charac-teristics and output quantity represents the actual regulatingquantity either for operations in the high-speed/high-torque orlow-speed/low-torque regions. Using (26), the net should be de-signed by:

1) nonlinear transformation , satisfying an even dis-tribution within the low-speed/low-torque range, since theno-load speed is roughly proportional to (seeFig. 18);

2) transparent parameter , which represents theboundary value between the two different operation

modes and is introduced for scaling them in order toachieve an almost even slope of approximated inversemodel within the common neural net for ;

3) deviation between training (reference) data andnet output denoted by andthe size of neural layer, requiring a compromise betweenaccuracy and the amount of neurons.

The BFN designed for the inverse contact model consists of41 13 neurons for the speed and torque input. Parameteris set to 0.4. After learning, a final fault of less than 3%within the whole input ranges is obtained. This accuracy is re-quired because of the large dependence on torque versus speed(see Fig. 18). An appropriate training algorithm is explained in[12].

However, the BFN represents the inverse speed–torque char-acteristic with output quantity depicted in Fig. 19. The lim-itation of the bending wave’s amplitude by is consideredby . For generating the regulating quantities and

, the inverse transformation of (26) is performed.

C. Speed Control Scheme

Since the reference values of the bending wave control arecalculated from the desired torque value by the BFN underconsideration of actual motor speed, the remaining nonlin-earity is compensated and an almost linear relation between ref-erence torque and motor torque is obtained. Thus, thecommand behavior of the new actuator approaches that of con-ventional electrical drives and is, thus, attractive for applicationsin the field of servo systems. Proven schemes can be appliedwhen designing the speed control depending on the application,as illustrated in Fig. 16. Since the mechanical load of the testsetup can be described by a one-mass system, a PI speed con-troller is the proper choice for the application.


Fig. 19. Inverse speed–torque characteristic withw as output quantity.

D. Experimental Results of Speed Control

In Fig. 20, the measured transient responses of the speed con-trol are shown, while in Fig. 21, the operation is illustratedby the corresponding speed–torque trajectories within the op-erating range of Fig. 18.

In case(a), a step-like command from the low-speed/low-torque region (operating point with r/min and) into the high-speed/high-torque region (operating point

with r/min and ) is performed. During the firsttime interval ms, the reference value of phaseis increased by the BFN due to reference torque, while thereference value of the amplitude is kept to m.When attains 90 at , the speed is regu-lated by varying the reference value of the wave’s amplitude,operating the drive with a perfect traveling wave. After a shortrise time of about ms, the desired speed is attained and thedrive’s speed settles smoothly on r/min. Since theUSM drive is accelerated during this response, the speed–torquetrajectory for case(a) passes almost the first (motory) quadrantin Fig. 21.

Case(b) depicts the inverse operation: . Due to thesufficient compensation of nonlinear torque generation by theBFN, the response time, overshoot, and shape of the curve isapproximately identical when compared to case(a). Since thedrive is decelerated in case(b), the corresponding speed–torquetrajectory passes almost the second (nonmotory) quadrant inFig. 21.

Case(c) shows a speed reversal of the drive. During intervalms, the reference torque is increased to its

boundary value of N m. After its boundary value is attained,a linear deceleration of the drive’s speed toward appearsdue to the constant motor torque. At time ms, the refer-ence torque is decreased below its boundary and the speed con-trol works finally in a linear operation mode, accelerating thedrive toward r/min. In the intermediate interval, the

controlled phase is varied smoothly from its negative to its pos-itive maximum.

In case(d), the disturbance behavior is investigated per-forming a step-like load change from zero to N m:

. Due to the relative small inertia of the drivesystem, the speed is affected largely by the load disturbance,but corrected within ms to its reference value ofr/min.

As indicated by elaborated measurements, first, the designedtorque and speed control scheme ensures the optimized condi-tions as outlined by the speed–torque characteristics in Fig. 18for high-speed/high-torque and low-speed/low-torque opera-tion. Second, the inverse contact model by the BFN, appliedfor compensation of nonlinear torque generation, operatessatisfactory as indicated by the expected transient responses ofthe speed control.

VI. USM DRIVE APPLIED TO AN ACTIVE

CONTROL STICK (ACS)

Finally, an ACS realized by means of the torque- and speed-controlled USM drive is noted as an interesting kind of appli-cation. One main disadvantage of the conventional side stick,realizing the “fly-by-wire” concept in modern aircrafts, is theabsence of any feedback from the rudders. In order to repro-duce feedback forces artificially (called active force feedbackand outlined in Fig. 22), an actuator has to be introduced in themechanical assembly of the stick. By the ACS, it is also pos-sible to generate an adjustable reset force, to couple the pilot’sand copilot’s sticks by wire and to produce stick vibrations as amechanical warning signal for the pilot. Due to their high powerdensity and high torque at low rotational speed, piezoelectricmotors seem to be advantageous for this application. In cooper-ation with DaimlerChrysler AG and SFIM-Industries, a uniaxialprototype ACS (see Fig. 23) [13] was successfully realized by


(a) (b) (c) (d)

Fig. 20. Experimental results of speed control.

Fig. 21. Speed–torque trajectories of Fig. 20 within speed–torque region ofFig. 18.

the USM drive using controls developed in the previous sec-tions.

The control scheme for the ACS reproduces the reset forcerepresented by a nonlinear spring characteristic and superim-poses feedback torque . Since no torque measurement

system is applied for the prototype ACS, it is realized byopen-loop torque control using the inverse contact model.

For this task, the speed control in Fig. 16 is extended by anadditional outer position control loop and the reference value(desired position of the stick) is generated by the inverse re-lation of the nonlinear reset characteristic (see Fig. 22. As theinput signal for the inverse relation, the reference torqueis used (operation without torque measurement system) regu-lating the ACS in such a way that the nonlinear reset charac-teristic is reproduced as desired when torqueresulting frompilot’s force is applied to the stick. Feedback can be super-imposed easily by , as illustrated in Fig. 22. The controlsystem reacts stably in any way and necessary limitations ofspeed and torque can be realized easily by limitation of theirreference values.

Fig. 24 shows measured torque curvesversus stick po-sition of the realized uniaxial ACS in comparison with thedesired torque given by the nonlinear reset characteristic.The operation range of interest is passed through by constantspeed ( ; steady-state operation) in both directions (in-crease of stick position and decrease of stick position). The de-viations are about 15% of the maximum torque value, whichrepresents inaccuracies of the inverse contact model (open-looptorque control) mainly affected by the neglected temperature ef-fects on the torque generation. However, these deviations cannot


Fig. 22. Control scheme for ACS.

Fig. 23. Prototype ACS and test setup.

be sensed by the pilot. In practice, the accuracy can be increasedby a torque measurement system, as illustrated by the dashedblock and signal line in Fig. 22. In this case, open-loop torquecontrol means an alternative operation mode or, rather, a pos-sibility for monitoring the system and, therefore, an increasein safety. The designed ACS control turns out to be a suitablemeasure for this kind of application. The proposed concept ofthe ACS and active force feedback is also interesting for otherkinds of “by-wire” application.

Fig. 24. Nonlinear reset characteristic reproduced by the ACS.

VII. CONCLUSION

This paper has proposed a novel control scheme for traveling-wave-type USM’s. In order to obtain high performance, a sys-tematic design approach by a motor model is applied using theinverse model structure for the cascaded control. Since all domi-nant nonlinearities are taken into account, an attractive dynamicbehavior and a remarkable reduction of losses are achieved. Byouter control loops for torque and speed, compensating the non-linear speed–torque characteristic, first, the command behaviorapproaches that of conventional electrical drives, and second,losses caused by the friction mechanism are minimized. Theouter controls are based on inner control loops for the elec-trical and mechanical oscillation systems. The nonlinear andtime-variable behavior of the oscillation systems is almost com-pensated by online identification of parameters, adaptation ofcontrollers, and frequency tracking. By these measures, oper-ation at the mechanical resonance is enabled, minimizing thelosses of the electrical subsystem remarkably. The potential of


optimization with respect to dynamics and efficiency is demon-strated by experimental results, and the applicability of the drivecontrol to servo systems is pointed out by an ACS realized by aUSM.

REFERENCES

[1] T. Sashida and T. Kenjo,An Introduction to Ultrasonic Mo-tors. Oxford, U.K.: Clarendon, 1993.

[2] S. Furuya, T. Maruhashi, Y. Izuno, and M. Nakaoka, “Load-adaptive fre-quency tracking control implementation of two-phase resonant inverterfor ultrasonic motor,”IEEE Trans. Power Electron., vol. 7, pp. 542–550,July 1992.

[3] N. Hagood and J. McFarland, “Modeling of a piezoelectric rotary ultra-sonic motor,”IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 42,pp. 210–224, Mar. 1995.

[4] T. Senjyu, H. Miyazato, S. Yokoda, and K. Uezato, “Speed control ofultrasonic motors using neural network,”IEEE Trans. Power Electron.,vol. 13, pp. 381–387, May 1998.

[5] F.-J. Lin, “Fuzzy adaptive model-following position control for ultra-sonic motor,”IEEE Trans. Power Electron., vol. 12, pp. 261–268, Mar.1997.

[6] M. Goldfarb and N. Celanovic, “Modeling piezoelectric stack actuatorsfor control of micromanipulation,”IEEE Contr. Syst. Mag., vol. 17, pp.69–79, June 1997.

[7] J. Maas, P. Krafka, N. Fröhleke, and H. Grotstollen, “Prototype driveand modulation concepts for DSP-controlled ultrasonic-motors poweredby resonant converters,” inProc. 6th European Conf. Power Electronicsand Applications, vol. 1, Seville, Spain, Sept. 1995, pp. 1777–1782.

[8] J. Maas, P. Ide, N. Fröhleke, and H. Grotstollen, “Simulation modelfor ultrasonic motors powered by resonant converters,” inConf. Rec.IEEE-IAS Annu. Meeting, vol. 1, Lake Buena Vista, FL, Oct. 1995, pp.111–120.

[9] J. Maas, P. Ide, and H. Grotstollen, “Characteristics of inverter-fed ultra-sonic motors—Optimization of stator/rotor-interface,” inProc. 5th Int.Conf. New Actuators, Bremen, Germany, June 1996, pp. 241–244.

[10] J. Maas and H. Grotstollen, “Averaging model for inverter-fed ultrasonicmotors,” inProc. IEEE PESC’97, vol. 1, St. Louis, MO, June 1997, pp.740–746.

[11] J. Maas, T. Schulte, and H. Grotstollen, “Optimized drive control forinverter-fed ultrasonic motors,” inConf. Rec. IEEE-IAS Annu. Meeting,vol. 1, New Orleans, LA, Oct. 1997, pp. 690–698.

[12] , “Controlled ultrasonic motor for servo-drive applications,” inProc. 4th Europ. Conf. Smart Structures and Materials—2nd Int. Conf.Micromechnanics, Intelligent Materials and Robotics, Harrogate, U.K.,July 1998, pp. 701–708.

[13] T. Schulte, H. Grotstollen, H.-P. Schöner, and J.-T. Audren, “Active con-trol stick driven by a piezoelectric motor,” inProc. 3rd Int. Symp. Ad-vanced Electromechanical Motion Sys., vol. 1, Patras, Greece, July 1999,pp. 583–588.

[14] J. Maas and T. Schulte, “High performance speed control for ultrasonicmotors,” inProc. IEEE/ASME Int. Conf. Advanced Intelligent Mecha-tronics, vol. 1, Atlanta, GA, Sept. 1999, pp. 91–96.

[15] T. Schulte and N. Fröhleke, “Parameter identification of ultrasonicmotors,” inProc. IEEE/ASME Int. Conf. Advanced Intelligent Mecha-tronics, vol. 1, Atlanta, GA, Sept. 1999, pp. 97–102.

Jürgen Maas was born in Nieheim, Germany, in1965. He received the Dipl.-Ing. and Ph. D. degreesin electrical engineering from the University ofPaderborn, Paderborn, Germany, in 1993 and 1998,respectively.

From 1993 to 1998, he was a Scientific ResearchAssistant at the Institute for Power Electronics andElectrical Drives, University of Paderborn. Hisresearch interests were focused on power suppliesand control strategies for piezoelectric drives. Since1998, he has been with the Laboratory for Power

Electronics and Actuators, DaimlerChrysler AG Research and Technology,Frankfurt, Germany, developing novel mechatronic actuator systems to improvethe ride comfort of automobiles.

Thomas Schultewas born in Detmold, Germany,in 1970. He received the Dipl.-Ing. degree inelectrical engineering in 1998 from the Universityof Paderborn, Paderborn, Germany, where he iscurrently working toward the Ph. D. degree in theDepartment of Electrical Engineering and Informa-tion Technology, Institute for Power Electronics andElectrical Drives.

He is also currently a Research Assistant at theUniversity of Paderborn, working on controllers andpower supplies for piezoelectric motors. His interests

are control systems, digital control, piezoelectric drives, and new actuators.

Norbert Fröhleke was born in Paderborn, Germany,in 1951. He received the B. Sc. degree in electricalengineering from the University of Paderborn,Paderborn. Germany, the M.S. degree from theTechnical University of Berlin, Berlin, Germany, andthe Ph.D. degree from the University of Paderbornin 1975, 1984, and 1991, respectively.

He has worked in the fields of numerical controlof tooling machines, electronics maintenance of elec-trical power stations, and teaching from 1976 to 1978and in the field of power electronics from 1983 to

1984 at the former AEG. Since 1989, he has been coordinating the researchactivity in power electronics, supervising a large number of B.Sc./M.Sc. stu-dents, and performing the aquisition of projects in the Department of ElectricalEngineering and Information Technology, Institute for Power Electronics andElectrical Drives, University of Paderborn, where he has also lectured. His re-search interests include switched-mode power converter analysis, modeling andcontrol for various applications, high-frequency magnetics, and computer-aideddevelopment of power electronic circuits and magnetic components, piezoelec-tric drives, and new actuators.

00asme maas schulte froehleke

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