Active damping applied to HSM-drivenmechanical loads with elasticity ?

Riccardo Antonello, Angelo Cenedese and Roberto Oboe

University of Padova, Department of Management and Engineering,Vicenza, Italy (Tel: +39-0444-998844; (e-mail:

riccardo.antonello@gest.unipd.it; angelo.cenedese@unipd.it;roberto.oboe@unipd.it)

Abstract: Hybrid Stepper Motors (HSM), together with the microstepping driving technique,are widely used in many motion control applications, given their low cost and high reliability.On the other hand, being controlled in an open loop fashion, they cannot achieve high levelsof performance, this mainly due to the absence of a load-side position sensor. In this paper,we address the problem of controlling the motion of a mechanical load, driven by a HSM,in presence of a flexible mechanical transmission between motor and load. This is a typicalindustrial scenario, in which the problem of the oscillations arising from the excitation of themechanical resonance by various disturbances (including torque ripple) is usually addressed byseverely limiting the overall dynamic performance. In this paper, we propose the use of anactive damping strategy, which allows for the improvement of the dynamic response and anexcellent rejection of the oscillations caused by the torque ripple. The proposed technique doesnot require the re-design of the existing equipments, since it is based on an enhancement of thestandard microstepping, in which the angle of the stator flux is properly modulated, to producea compensating torque and, in turn, damp the oscillatory modes. Such modulation is basedon the proper processing of the measurements obtained from a load-side MEMS accelerometer,which can be easily fitted into existing setups. Experimental results confirm the effectiveness ofthe proposed solution.

Keywords: Hybrid stepper motor, active damping, MEMS accelerometers

1. INTRODUCTION

Hybrid Stepper Motors (HSM) are widely used in manymotion control applications, where there are tight costconstraints and it is acceptable to have a limited level ofperformance. Their structure is composed of a cylindricalpermanent magnet, polarized axially, covered by toothedsoft steel cores, as shown in Fig. 1. This type of motorworks as a stepping motor by the combined principlesof the permanent-magnet motor and the variable reluc-tance motor (Kenjo [1984]). The two cores are angularlydisplaced by half of a teeth and this, combined with theparticular toothed structure of the stator (with the samepitch of the rotor), allows steps with the width of a quarterof a teeth, when the two phases of the stator windingsare driven in an on/off fashion (see Fig. 2). HSMs drivenin such a way share with all stepper motor the problemof a non-smooth torque generation, together with an os-cillatory behavior in step-to-step motion. A widely ap-plied driving technique that partially alleviates the aboveproblems, is the so-called microstepping (Rahman andGrantham [1990]), in which the phase currents are twosinusoids in quadrature (see Fig. 3). With this driving

? This work has been partially supported by the ProgrammaOperativo F.S.E. 2007-2013 Regione Veneto, Codice Progetto2105/101/17/2214/2009 (”Modellizzazione Orientata al Controllo diSistemi Meccatronici”).

technique, the stator flux rotates smoothly, thus limitingtorque ripple and rotor oscillations.

We will discuss later some details on torque generation inHSM, but it is well known (Ahn et al. [2005]) (Bodsonet al. [1993]) (Blauch et al. [1993]) that, even with the useof microstepping, torque ripple is not easy to be avoided,especially due to the presence of a large detent torque. Thisfact has often limited the application of HSM in all thosesystems in which the presence of mechanical resonancescould amplify the effects of torque ripple, resulting in anunacceptable load oscillation. Some authors have proposedthe use of complex ripple minimization techniques, but allof them require a rotor position sensor, thus reducing thecost benefit of having an open loop position actuator (Ahnet al. [2005]) (Kos et al. [2006]) (McInroy et al. [1995]).

Vibration suppression and control in mechanical loadswith resonant modes is a widely studied subject and sev-eral techniques have been proposed, like input shaping andactive damping (Miu [1993]). As for input shaping, thisrefers to a set of techniques aiming at reducing the overall(or residual) energy around the resonance frequency and,in turn, limit the oscillatory behavior. Minimum jerk andreference filtering are usually the simplest methods toavoid the excitation of the resonant modes, while minimumenergy control aims at the design of reference trajectoriesthat will end with null energy stored in the resonantelement of the controlled system. All the input shaping

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Fig. 1. Hybrid stepper motor.

Fig. 2. Square wave driving of 2–phase stepper motor.

Fig. 3. Microstepping: sinusoidal driving of 2–phase step-per motor.

techniques, however, are best suited for point-to-pointmotions, and they do not provide any remedy against theload oscillations due to disturbances, like torque ripple,always present in HSMs. Active damping has found a largeapplication in the control of flexible structures, and it hasbeen implemented either by distributing damping actua-tors along the structure to be controlled (e.g. piezoelectricactuators on the modal nodes of an elastic beam) or, incase a single actuator is available, by properly controllingthe generated torque. In principle, the idea is simple:inject a force/torque, equivalent to that produced by aviscous damper, to reduce the amplitude of the vibrations,caused either by disturbances or transients in point–to–point motions. This idea, of course, has a straightforwardapplication in all those systems provided by a torque-controllable actuator (e.g. DC motor, AC brushless motorsetc.), but it has never been applied in position controlledactuators, like HSMs.

In this paper we propose the application of an active damp-ing technique to a system with mechanical resonances anddriven by a HSM. We will show in the following how it ispossible to generate a torque for the compensation of theoscillatory behavior by using a phase modulation of thecurrents applied to the stator windings. The modulation,in turn, is driven by the velocity of the load, obtained byprocessing the output of a low-cost MEMS accelerometer.The paper is organized as follows. Section 2 describesthe positioning unit for surveillance camera that has beenused as a case study for the application of the proposedtechnique. Section 3 describes the mechanical model ofthe system to be controlled, with some details on thefactors that may lead to plant changes and, in turn to

robustness problems. Section 4 describes the control lawdesign, together with the solution adopted to generatethe damping torque and the limits on performance posedby the actuator used. The experimental results obtainedare reported in Section 5, while conclusions and futureresearch directions are presented in Section 6.

2. CASE STUDY - POSITIONING UNIT FORSURVEILLANCE CAMERA

In order to test the feasibility of an active vibrationsuppression scheme, when the load is driven by a HSM,we used a positioning unit for surveillance cameras, shownin Fig. 4. This is a pan–tilt unit, with adjustable focusand zoom. In pan and tilt, the camera is moved by twoHSMs, connected to the load through a reduction unit,composed of two gears and a toothed belt (Fig. 5). Thepositioning unit can be controlled manually, via a userconsole, or automatically, with a video processing unit thatcommands the pan and tilt angle in order to track a movingtarget.

This unit operates outdoor, often facing harsh environ-mental conditions and it can be mounted on a wide varietyof supports, ranging from concrete walls to long (andflexible) supporting poles. Additionally, it is subjected tovarious disturbances (ranging from wind gusts to birdshanging on the unit) and some of the operating parameters(e.g. friction of the sealing rings, stiffness of the belts, etc.)may widely change with the temperature (the unit must befully operative from -40 to +60 centigrades). Finally, usingan elastic transmission belt, the system presents mechani-

Fig. 4. Positioning unit for surveillance camera.

Fig. 5. Reduction unit with gears and toothed belt.

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Rotor acceleration FFT

Acceleration[rad

/s2]

Harmonic order

0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

Fig. 6. Torque harmonics of the HSM.

cal resonances, which may amplify the effects of the torqueripple generated by the HSM and other disturbances.

The unit comes without a load position sensor, so thetracking of a moving target is performed simply by actingon the HSM position, without any feedback. Rotations ofthe unit at a constant angular rate ωt (e.g. when followinga target moving at a constant speed) are obtained with amicrostepping drive, i.e. by applying to the stator windingstwo quadrature sinusoidal currents with a fixed frequencyωu = (Nr/NG)ωt, where Nr is the number of pole pairsof the HSM and NG is the gear ratio. In such operatingconditions, the torque ripple generated by the HSM iscomposed of several harmonics, with a base frequencyequal to that of the stator currents. Such harmonicsare generated by both the driving amplifiers (which mayintroduce differences in offsets and amplitudes of the phasecurrents) and motor imperfections (Bascetta et al. [2010]).As an example, the HSM used in moving the pan angleof the unit has a measured torque ripple spectrum as inFig. 6.

3. MECHANICAL SYSTEM MODELING ANDIDENTIFICATION

In order to perform the design of the active damping ofthe vibrational modes for the system described in theprevious section, it is vital to derive an analytical model,by means of which it is also possible to evaluate the effectsof the variation of a single mechanical parameter, like thecamera case inertia (e.g. due to a bird or to a change inthe zooming level) or the support stiffness.

In Fig. 7 we report the multibody model of the camerapositioning unit, w.r.t. the pan angle θ2 motion. The left

Fig. 7. Multibody model of the camera positioner.

part of the model is the fixed part of the unit (Sun),anchored to the ground through an elastic connectionKbase; the right part represents the HSM, with inertia Jsh.The two parts are connected via the springs Kb1 and Kb2,accounting for the transmission belt elasticity.

Also, we indicate with θ1 the relative angular displacementof the base w.r.t. the fixed structure, with θ2 the hubrotation angle (output), with θsh the rotor field angle w.r.t.θ2, and with θu the stator field angle w.r.t. θ2 (input).

In order to get the full model of the system, we need todefine the input torque, generated by the HSM, and driv-ing the hub rotation. It is worth noticing that, using themicrostepping technique, the stator field can be orientedwith an arbitrary angle θu, by driving the stator windingswith two current in quadrature, namely i1 = I cos(θu) andi2 = I sin(θu). In absence of a load torque, the rotor alignsits magnetic axis with the stator field (i.e. θu = θsh). Whenan external load is applied, this causes a displacementbetween rotor and stator flux, according to

τm = KT I sin (θu − θsh) (1)

This, in turn, can be linearized for small displacement,around the equilibrium, yielding

τm ≈ KT I (θu − θsh) (2)

Clearly, the above result shows that the HSM behaves likea spring, with an equivalent stiffness depending on themotor torque constant KT and the level of the drivingcurrent I.

The next step is to derive the transfer function of thesystem by considering the stator flux angular position θuas the system input and the hub (camera) angle θ2 as thesystem output. It can be proved that such transfer functionhas three pairs of complex conjugate poles and two pairsof complex conjugate zeros:

Θ2(s)

Θu(s)=

(s2 + 2ξz1wz1s+ w2z1)

(s2 + 2ξp1wp1s+ w2p1)

× · · ·

· · · × (s2 + 2ξz2wz2s+ w2z2)

(s2 + 2ξp2wp2s+ w2p2)(s2 + 2ξp3wp3s+ w2

p3), (3)

Θ2(s) (Θu(s)) being the Laplace transform of θ2 (θu).

It is worth noticing that in all the above derivation, forthe sake of simplicity, no difference between electrical andmechanical angle of the HSM motor has been explicitlyshown, even if they have been considered in the derivationof (3). In practice, the motor used in the positioning unithas Nr = 50 teeth, corresponding to 1.8◦ of step angle.As for the ratio between motor and load position, it isNG = θ2/θu = 1/7.

We identified experimentally the frequency response ofthe actual system, by using both a sweep signal and afiltered white noise as a driving signal θu. Notably, theload position cannot be sensed directly, since this wouldrequire major (unwanted) modification to the hardwaredesign and sensible component cost increase. Instead, weused a low–cost MEMS accelerometer, that can be easilyplaced on top of the camera case (see Fig. 8). Clearly,the measured tangential acceleration is proportional to theload angular acceleration θ̈2, from which it is possible torecover the angular position.

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Fig. 8. MEMS accelerometer placed on top of the cameracase.

Magnitude[dB]

Bode diagram - G(s) = Θ2(s)/Θu(s)

Phase[deg]

Frequency [Hz]

GroundNeoprene

Free

101 102

101 102

-200

-100

0

100

200

-80

-60

-40

-20

0

Fig. 9. Experimental frequency response Θ2(s)/Θu(s) withdifferent support stiffnesses.

We report in Fig. 9 the frequency response obtained withspectral analysis methods and limited-bandwidth whitenoise excitation. Three different types of connection withfloor are considered, namely with high, medium and zerostiffness Kbase (the latter obtained by using a ball–bearingsupport). As an expected result, the pole-zero pair atlow frequency disappears in case of zero stiffness, with aconsequent order reduction in the plant to be controlled.Excluding the latter condition, being not realistic, wecan observe that the system shows two evident resonancepeaks, which can possibly amplify the torque ripple. Witha concrete base, the resonant frequencies are 17.55 Hz and25.95 Hz, respectively.

4. CONTROL LAW DESIGN

In order to design the active damping control, the ex-perimental frequency response has been fitted into theanalytical model whose structure is defined in (3). TheBode plot for nominal plant (concrete base and nominalcamera inertia) is shown in Fig. 10. Note that the staticgain corresponds to the gear ratio NG.

The rationale behind the active damping strategy is toset up an inner loop, capable of reducing the height of

Bode diagram - identified G(s) = Θ2(s)/Θu(s)

Magnitude[dB]

Phase[deg]

Frequency [Hz]100 101 102 103

100 101 102 103

-180

-135

-90

-45

0

-80

-60

-40

-20

0

Fig. 10. Fitted Bode plot (by using a frequency-domainleast-square fitting procedure) of Θ2(s)/Θu(s) innominal conditions.

the resonant peaks as well as the oscillations that maybe caused by various disturbances. In approaching thedesign of such inner control, we must take into accountseveral constraints, due to the use of HSM and to thespecific application under study. As for the latter, wemust consider that there is no precise knowledge aboutthe transfer function, since this may widely change duringoperation (e.g. when a bird lands on the camera, the loadinertia increases and, in turn, all resonance frequenciesdecrease). Of course, an on-line identification procedurecould be implemented (e.g. Wang and Tomizuka [2008]),but this is not compatible with the uninterruptible 24-7 operating scenario of the surveillance camera and thepossible sudden and large variations in plant parameters.Then, a typical active damping scheme, with a feedbacktuned on the plant resonances (see, for instance, Han et al.[2008]) cannot be applied to the camera positioning unitof interest. Given the above limitation, we resorted to asimple and robust solution, making use of a load velocityfeedback, equivalent to inserting a viscous damper betweenload and base. This, in turn, is equivalent to commandingthe actuator to generate a torque proportional to the loadspeed.

HSMs, however, are position actuators, so the question is:how to command HSM to generate a damping torque? Theanswer is handed by (2), which says that if we modulatethe stator flux angle θu, we modulate the generated torqueas well (this under the hypothesis of small differencesbetween stator and rotor angles). Using this simple con-sideration, a vibration damping inner loop can be realizedin the form shown in Fig. 11, where the load side velocityis obtained by integrating the load acceleration, measuredusing the MEMS accelerometer. Remarkably, the proposedmethod to generate a compensating torque in HSM–drivensystem can be applied also in systems with fixed plant pa-rameters, for which more sophisticated damping schemescan be applied (Han et al. [2008]). In practice, if the targetis going to be followed with a constant camera angularspeed ωt, this is achieved by setting the phase currents inthe HSM as in (4)-(5), where θur represents the desired

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load position:

i1 = I cos(θur) = I cos(ωurt) = I cos(Nr/NG ωt t) (4)

i2 = I sin(θur) = I sin(ωurt) = I sin(Nr/NG ωt t) (5)

When the active damping inner loop is active, the actualangular position for the stator flux is obtained as follows

θu = θur −K θ̇2 (6)

i.e. by subtracting a signal proportional to the load angularvelocity to the reference angle θur. This, as shown above,is equivalent to ask to the HSM to generate a torqueproportional to the load angular velocity, as required bythe active damping strategy. It is worth noticing that theactual implementation of the inner loop must take intoaccount that the MEMS accelerometer may have some biasand high frequency noise in its output, so a band–pass filtermust be placed in the feedback path of the control schemein Fig. 11.

θ̈2Accelerometer

BPFK1

s

Θ2(s)

Θu(s)

MEMSθur θu

θ̇2

Fig. 11. Active damping inner loop.

The design of the inner loop clearly reduces to the choiceof the feedback gain K. Actually, if the damping of theresonant modes is the control objective, a high value of Kwould lead to better results, as shown in Fig. 12.

Bode diagram - Closed loop with different active damping gains

Magnitude[dB]

Phase[deg]

Frequency [Hz]

K = 0

K = 0.1

K = 1

K = 5

100 101 102 103

100 101 102 103

-180

-135

-90

-45

0

-80

-60

-40

-20

0

Fig. 12. Input/output frequency response with differentcompensation gains.

On the other hand, this choice leads to two problems.The first is that the amplitude of the compensation signalmay become larger than a quarter of the HSM step width,and this may cause the rotor of the HSM to fail to followthe reference profile. The second problem is related to thepromptness of the system. Typical reference positions havea ramp-like profile, and the steady-state error increases

as the bandwidth of the system decreases. Fig. 13 showsthe simulated time plot of the error (in [deg]) betweenreference and actual position in case of K = 1 andK = 100, respectively. It can be seen that with the secondchoice for the gain, the reference following error is ratherlarge, so this choice would lead the actual camera positionto lag behind the desired one, and, in turn, to loose theview of the moving target within the image plane. Then,the best choice for the compensation gain K has beendone by considering the maximum slope for the motioncommands (about 180◦/s) for the device considered.

Trackingerror[deg]

Position tracking error et = NG θur − θ2 (case K = 1)

Trackingerror[deg]

Time [s]

Position tracking error et = NG θur − θ2 (case K = 100)

0 50 100 150 200

0 1 2 3 4 5

0

50

100

150

0

0.5

1

1.5

Fig. 13. Ramp reference following error (top plot: caseK = 1; bottom plot: case K = 100)

5. EXPERIMENTAL RESULTS

The active damping control presented in the previoussection has been applied to the camera positioning unitand its effectiveness proven in different operating condi-tions. Fig. 14 shows the comparison of the spectra of theaccelerations measured by the MEMS accelerometer, withand without the active damping. The rotating speed ischosen so that the first harmonic of the torque ripple fallsin the resonant peak of the frequency response. As a result,a reduction of 95% of the vibration amplitude is achieved.

Frequency [Hz]

Acceleration[m

/s2] Non-compensated

Compensated

10 15 20 25 30

0

1

2

3

Fig. 14. Vibration attenuation at constant speed.

The same test at constant speed has been performedwith different supports (with different rigidity), in orderto test the insensitivity of the solution to the mountingconditions. Fig. 15 shows the results obtained with teflonand neoprene supports, confirming the robustness of themethod.

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Acceleration FFT

Acceleration[rad

/s2]

Frequency [Hz]

Non-compensated

Compensated

Acceleration FFT

Acceleration[rad

/s2]

Frequency [Hz]

Non-compensated

Compensated

15 20 25 30 3515 20 25 300

5

10

15

0

5

10

15

Fig. 15. Vibration attenuation with different supports(teflon on the right, neoprene on the left) – constantspeed reference.

The robustness against support variations has been testedalso in transient conditions (constant acceleration) and theresults reported in Fig. 16 confirm the ability of the systemto provide an effective damping of all vibratory modes.

Acceleration FFT

Acceleration[rad

/s2]

Frequency [Hz]

Non-compensated

Compensated

Acceleration FFT

Acceleration[rad

/s2]

Frequency [Hz]

Non-compensated

Compensated

0 50 100 1500 50 100 1500

1

2

3

4

0

1

2

3

4

Fig. 16. Vibration attenuation with different supports(teflon on the right, neoprene on the left) – constantacceleration reference.

Finally, the robustness against load inertia variations hasbeen verified, by placing ad additional weight on top ofthe camera. The results shown in Fig. 17 confirm againthe insensitivity of the proposed method to wide changesin operating conditions.

Acceleration FFT

Acceleration[rad

/s2]

Frequency [Hz]

Non-compensated

Compensated

Acceleration FFT

Acceleration[rad

/s2]

Frequency [Hz]

Non-compensated

Compensated

0 50 100 1500 50 100 1500

1

2

3

4

0

1

2

3

4

Fig. 17. Vibration attenuation with different load inertia(nominal on the left, doubled on the right) – constantacceleration reference.

6. CONCLUSIONS

Active damping control is widely adopted for reducing thevibrations in motion control systems. This technique, how-ever, has been used so far only in those systems providedwith a torque-controlled actuator. The reported applica-tion shows how it is possible to extend the applicationof this method even to motion control systems drivenby HSM. The proposed solution can be easily retrofittedon existing hardware, being based on the use of load-side MEMS accelerometers and a simple modification of

the microstepping technique. Experimental results showthe robustness and effectiveness of a simple load velocityfeedback in suppressing vibrations. The method developedfor driving the HSM in order to produce a compensationtorque can be applied to other mechatronic systems andits use in conjunction with more complex control schemes,possibly adaptive, will be the subject of future research.

ACKNOWLEDGEMENTS

Authors will like to thank Videotec (Schio, Vicenza, Italy)for providing (and allowing to hack) the camera posi-tioning unit. Thanks also to Andrea Parisotto, DanielePellizzer, Marco Peruzzo and Mattia Schiesaro for theirvaluable contribution to the experimental activity.

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