MAGNETIC BEARING REACTION WHEEL MICRO-VIBRATION …

MAGNETIC BEARING REACTION WHEEL MICRO-VIBRATION SIGNATUREPREDICTION

G. Borque Gallego1, L. Rossini1, E. Onillon1, T. Achtnich2, C. Zwyssig2, R. Seiler3, D. Martins Araujo4, and Y.Perriard5

1CSEM SA, Rue Jaquet-Droz 1, CH-2002 Neuchâtel, Switzerland. Email: [email protected],[email protected], [email protected]

2Celeroton AG, Industriestrasse 22, CH-8604 Volketswil, Switzerland. Email: [email protected],[email protected]

3ESA/ESTEC, Keplerlaan 1, NL-2200 AG Noordwijk, The Netherlands. Email: [email protected], Esplanade des Particules 1, CH-1211 Geneva, Switzerland. [email protected]

5Integrated Actuators Laboratory, EPFL, Rue de la Maladière 71, CH-2000 Neuchâtel, Switzerland. Email:[email protected]

ABSTRACT

This article presents an analytical micro-vibration modelfor magnetic bearing reaction wheels (MBRWs) based ona rotordynamics model combined with the active con-trol loop. This model is validated employing a fully-active Lorentz-type MBRW demonstrator and a multi-component dynamometer capable of measuring the ex-ported micro-vibrations. The results show an accuratedescription of the generated vibration by only consider-ing the unbalance forces as excitation.

Keywords: Magnetic bearing reaction wheel; vibration;modelling.

1. INTRODUCTION

The demand for cost reduction in space missions has ledto the development of small spacecraft with more capa-ble payloads, which often implies experiencing a highersensitivity to mechanical disturbances. In general, the re-quirement of high pointing accuracy for new telecom-munication, Earth observation and scientific telescopemissions has tightened the restrictions concerning toler-able levels of on-board disturbances [1], such as micro-vibration generated by different spacecraft actuators, withreaction wheels (RWs) being frequently identified as themost critical sources for small satellites. In order to com-ply with these requirements, magnetic bearing actuatorsof compact size and low mass have been identified as apromising solution for small spacecraft.

The main potential advantages of magnetic bearingsand the progress in some technological aspects, likethe development of rare-earth magnets (i.e. samarium-cobalt magnets), resulted in the first efforts of developingmagnetically-suspended reaction wheel systems. Thesemain advantages are, among others, a virtually infinitelifetime, due to friction-less and maintenance-free oper-

ation, and better control of bearing dynamic properties.Different magnetic bearing RW topologies, e.g. [2–4],have been presented in the past. They mainly focus onultra-low disturbance generation, and they usually relyon hybrid topologies, i.e. combining active and passivestabilisation of rotor’s degrees of freedom, and focusedon rather large spacecraft platforms, such as Aerospa-tiale’s families of MBRWs [5], employed in SPOT 1, 2or 3 missions among others. These satellites were of ap-proximately 2 tons of weight and needed high restrictionson allowed disturbances.

The growth in importance of small satellites – under500 kg of mass – in space missions has led to the study ofthe applicability of magnetic bearings for these reducedplatforms in the recent years [6, 7]. Nevertheless, thechallenges involved in the miniaturisation and develop-ment of a competitive compact magnetic bearing reactionwheel remain to be solved.

A new and more compact ultra-high-speed magneticbearing and motor topology has been proposed and de-veloped by the Swiss Federal Institute of Technology inZurich (ETHZ) and Celeroton AG [8–11], with promis-

MountingPlate

MBRWRotor

MBRWCasing

Figure 1. Miniature magnetic bearing reaction wheeldemonstrator by Celeroton AG. Rotor with inertia disk(left) and external casing (right).

_____________________________________________________________________________________________ Proc. 18. European Space Mechanisms and Tribology Symposium 2019, Munich, Germany, 18.-20. September 2019

ing capabilities for space applications, such as reactionwheels for attitude and orbit control systems (AOCS). Forstudying the capabilities of such a technology for spaceapplications, a magnetic bearing reaction wheel demon-strator has been developed [12,13] that is shown in figure1.

In order to assess if different reaction wheel systems com-ply with the allowed mechanical noise requirements inspace missions, several on-ground qualification tests areusually performed, like, for instance, at the European Re-action Wheel Characterisation Facility (RCF) [14]. Themeasured force and torque rely on the combination of dif-ferent multi-component piezoelectric dynamometric sen-sor measurements, rigidly mounted, on one side, to anisolation table (big seismic mass over pneumatic isola-tors), and on the other side, to the test specimen.

Extensive work has already been done for measuring andcharacterising the micro-vibration type of disturbancesgenerated by conventional ball-bearing reaction wheels(BBRWs) [15,16], but to the authors knowledge, no sim-ilar study has been performed for assessing the noisesignature of magnetic bearing reaction wheels. In thisarticle, a characterisation of the main micro-vibrationsources for these magnetic actuators is performed, and ananalytical model is presented and validated by compari-son with experimental micro-vibration data of the afore-mentioned miniature magnetic bearing reaction wheeldemonstrator developed by Celeroton.

2. MICRO-VIBRATION SOURCES

Due to the high impact of reaction wheels on satel-lite’s pointing accuracy, the noise signature and the mainsources of these disturbances for common BBRWs havebeen extensively addressed in the past [15,16]. These dis-turbances are commonly known as micro-vibration, dueto their low amplitude and periodic components, depen-dent on rotor’s speed, and are common to any rotatingmachinery.

The main sources of periodic reaction wheel micro-vibration can be classified, as performed in [15], accord-ing to their origin or cause in: unbalance-driven , bearing-driven, and motor-driven disturbances.

2.1. Unbalance-Driven Disturbances

This component of the micro-vibration is caused by un-even distribution of mass in the rotor around its geometri-cal axis, and it is acknowledged to be the most significantdisturbance in reaction wheels. It consists of two types ofunbalances: static unbalance, which appears as a periodicforce perpendicular to the rotation axis, and dynamic un-balance, measurable as a cross-torque, also perpendicularto the spin axis. Both types can be modelled by consid-ering an additional mass in the rotor, and expressed asfollows:

Fs = [Fx, Fy]> = msrsΩ2[sin(Ωt), cos(Ωt)]>,

Td = [Tx, Ty]> = mdrdddΩ2[sin(Ωt), cos(Ωt)]>,(1)

where Ω is the wheel rotation speed, ms and md are thestatic and dynamic unbalance point-masses, respectively,rs and rd the radius between the geometric axis of rota-tion and the static and dynamic masses, and dd the axialdistance between the dynamic unbalance masses. Theseunbalance models consider a centrifugal force and torquein radial direction (Fx, Fy , Tx, Ty), synchronous with ro-tation speed. Due to the interaction with other bodies ofthe reaction wheel assembly (RWA), such as the stator,other modulations of these unbalance can be measured,appearing as higher integer harmonic orders of the rota-tion frequency.

One of the main advantages of magnetic bearings, due tothe lack of physical contact between rotor and stator, isthe possibility of including active unbalance control, thatwould considerably reduce the exported vibrations of thereaction wheel.

2.2. Bearing-Driven Disturbances

For conventional ball bearing RWs, other non-integermodulations of the harmonic micro-vibrations appear asa result of imperfections in the bearing components, orig-inated from the manufacturing process, such as wavi-ness or geometrical irregularities in the surface of bear-ing parts. Ball bearings also generate random/transientdisturbances, mainly due to lubrication issues, suchas performance deterioration at close speeds close tozero, limited number of zero-crossings, momentary non-homogeneous oil distribution inside the bearings thatgenerate additional friction force or other disturbancesoriginated from ageing lubricant.

All these disturbances will not appear in a magneticallylevitated RW, due to the contactless, and thus, friction-less operation of these actuators. In practice, some per-manent magnet and winding imperfections, power elec-tronics and sensors will generate some parasitic forces tothe rotor and stator, appearing as integer or non-integerharmonic orders of the rotation frequency. The character-istics of such forces will greatly depend on the magneticbearing topology and control approach employed, as hap-pens for the motor-driven disturbances.

2.3. Motor-Driven Disturbances

Torque ripple and torque instabilities are the main com-ponents of motor-dependent disturbances that appearabout rotation axis as a distorting motor torque (Mz). Theformer is understood as any periodic/deterministic torquedisturbance derived from power electronics switching,variation in airgaps between permanent magnets, currentmeasurement errors and cogging torque [16]. The latteris referred to as any random or transient disturbance onthe motor torque, such as random torque noise.


Homopolar

Side

Hetero

polar

Side

Stator

MotorWinding

RadialBearingWindings

AxialBearingWinding

SensorPCB

PermanentMagnets

Rotor Rim

BackIron

Figure 2. Schematic cross-sectional view of the magneticbearing reaction wheel demonstrator topology.

3. MINIATURE MAGNETIC BEARING RE-ACTION WHEEL DEMONSTRATOR FORSMALL SATELLITES

In this section, the miniature magnetic bearing topologywill be shown [13]. As mentioned before, the actuator isa Lorentz-type AMB, i.e. all the 6 degrees of freedom(DoF) of the rotor are actively controlled and the forcegeneration is based on electrodynamic principle (Lorentzforce).

3.1. Magnetic Bearing and Motor Topology

As shown in Fig. 2, the system is composed of two mainparts for bearing and motor – heteropolar and homopolarsides – located at the two axial extremes of the rotor, andprovided with an inertial disk that increases its momen-tum storage and exchange capacity.

On the one hand, the heteropolar side is responsible forgenerating radial forces and motor torque to the rotor,by means of the interaction between the magnetic fluxcreated by a diametrically-magnetised permanent magnet(PM) and the current applied to a three-phased skewedbearing winding with two pole-pairs, for the force gen-eration, and with a three-phased skewed motor windingwith a single pole-pair for torque generation. The het-eropolar configuration of this part of the machine requiresa synchronous modulation of both motor and bearing cur-rents with rotor’s orientation, typical in any field-orientedcontrol of PMSMs. The magnetic circuit is shared byboth, motor and bearing actuators (self-bearing), and isclosed by a stator core, where the bearing and motorwindings are located in the machine’s air-gap (slotless de-sign).

On the other hand, the homopolar side has no stator core,and is responsible for generating axial and radial forces.These forces result from the Lorentz force between themagnetic field created by two axially magnetised perma-nent magnets pointing towards each other and two sepa-rated ring-wound coils for the axial forces, and with a ra-dial bearing winding which is essentially the same as theheteropolar’s radial bearing winding. As before, a com-bined magnetic circuit is shared between axial and radialbearings. Nevertheless, the homopolar configuration isindependent from rotor’s orientation, and thus, actuation

Figure 3. Rigid rotor on compliant bearings and asso-ciated reference frame for rotordynamics model (Source:[18]).

control loop is simplified, as no field-oriented control isrequired.

The information regarding rotor position and orientationis obtained via two printed-circuit-board (PCB)-basedsensors placed at the two axial extremes of the stator.Radial position measurements are based on eddy-currentsensors, and axial and angular positions on Hall-effectstray-field measurements [17].

4. ANALYTICAL MICRO-VIBRATION MODEL

In order to predict the generated vibrations during op-eration, an analytical model is presented, that combinesthe dynamics of the rotating system and the control looprequired for magnetically levitate the rotor. In order tostudy the generated vibrations in radial directions, the 4degrees-of-freedom model of radial dynamics presentedin [18] or [11] is used. This model is employed not onlyfor estimating the level of micro-vibrations generated bythe magnetic bearing reaction wheel, but also to studythe general behaviour and stability of the system underdifferent conditions.

As shown in expression (1), the unbalance forces are ex-clusively radial, and if we consider small radial displace-ments which holds for this rotordynamics study, the axialdynamics is completely decoupled from the radial one.However, in practice, measurements have shown that themaximum axial forces are at least two orders of magni-tude lower than the radial ones, which confirms the pre-vious assumption, and thus axial dynamics are not con-sidered in the following models.

The coordinates of the rigid body are illustrated in figure3, whose movement while rotating about axis z at speedΩ can be fully characterised by the radial position of itsgeometrical centre (xc, yc) and the radial tilting of themain axes of inertia (φx, φy) about axes x and y, respec-tively.

4.1. Plant Model

Considering the generalised coordinate vector q =[xc, φy, yc,−φx]>, the equations of motion of an un-


damped rigid rotor magnetically levitated and subject torotor unbalance can be expressed in matrix form as fol-lows

M q +G(Ω)q + Sq = fb + fub(Ω), (2)

being Ω the rotation speed of the rotor, fb the magneticbearing forces applied by the actuator’s currents, and fub

the unbalance forces. The matrices appearing in the ex-pression are: M , the mass matrix, G(Ω) the gyroscopicmatrix, and S the stiffness matrix. These matrices havethe following form:

M = diag(m,Jxy,m, Jxy);

G(Ω) = Ω

0 0 0 00 0 0 −Jz0 0 0 00 Jz 0 0

;

S =

S11 S12 0 0S12 S22 0 00 0 S11 S12

0 0 S12 S22

;

where m represents the rotor mass, Jxy and Jz the trans-verse and polar inertia of the rotor, respectively. Thesubindeces 11 denote a linear-to-linear property, e.g.force in X and displacement in X direction, 12 denote alinear-to-angular property, e.g. torque about X and dis-placement in Y direction, and 22 denote a angular-to-angular property, e.g. torque about X and tilting in Xdirection.

The generalised bearing forces fb are applied to the rotorby means of the electromagnetic force controlled via theinput currents u = [id,1, iq,1, id,2, iq,2]> to the magneticbearings. The three-phase currents of the radial bearingwindings are expressed here in a field-oriented manner,by employing a dq-transformation, as commonly done forrotational machinery [9]. Considering the ideal actuators,these forces can be modelled as follows

fb =

1 0 1 0lb1 0 −lb2 00 1 0 10 lb1 0 −lb2

·· diag(χb1, χb1, χb2, χb2)u =

= V u,

(3)

being lb1 and lb2 the axial distance between the bearingplanes and the centre of mass, and χb1 and χb2 the forceconstants of each magnetic bearing. In practice, the rela-tion between currents and forces can result in a nonlinearexpression including cross-couplings between axes, butfor the studied Lorentz-type the nonlinearities are small,as it will be shown in the experimental data.

The generalised unbalance forces fub that are applied tothe rotor are originated by the residual unbalance, whosemagnitude is characterised by ε, the eccentricity of thecentre of mass with respect to the geometrical centre of

the rotor, and χ the tilting of the main axis of inertia w.r.t.the symmetry axis, resulting in the following structure

fub(Ω) = Ω2

mε cos(Ωt+ α)χ(Jxy − Jz) cos(Ωt+ β)

mε sin(Ωt+ α)χ(Jxy − Jz) sin(Ωt+ β)

. (4)

Note that expressions (1) and (4) are equivalent and theunbalance expressed in one form can be easily convertedto the other. In order to include such information on themodel, the residual unbalance of the rotor can be mea-sured employing a balancing machine, which generallywill provide it in terms of msrs and mdrd of (1). Then,considering the rotor total massm, inertia, (Jxy, Jz), andthe distance between balancing planes dd, it can be trans-formed into (4) for the rotordynamics analysis:

ε =msrsm

; χ =mdrdddJxy − Jz

.

Finally, the displacement sensors are located at the twoaxial extremes of the rotor ls1 and ls2. Its measurementsysens can be related to the generalised coordinates, con-sidering small displacements and angles, with the follow-ing linear form

ysens =

xs1ys1xs2ys2

=

1 ls1 0 00 0 1 ls11 −ls2 0 00 0 1 −ls2

q. (5)

4.2. Controller

The controller architecture is based on a linear-quadratic-gaussian (LQG) observer and controller, and its detaileddesign is presented in [11].

As shown in figure 4, the control loop for the aforemen-tioned multi-input-multi-output (MIMO) is composed ofa controller gain K, a Kalman filter state-space observer,a current limiter and its associated anti-windup for theintegral state, and a gravity compensator. At high speeds(>60 000 rpm), a generalised notch filter is enabled to re-ject exported forces due to rotor unbalance, but due to thelower speeds targeted for a RW application, it is omittedhere.

The feed-forward component of the currents uff is com-puted by considering the gravity magnitude and directiong and injecting directly the current required to apply sucha force by means of the inverse matrix V −1.

In order to extract the state vector from the noisy sensormeasurements, a Kalman filter of the form[

q(k|k)ˆq(k|k)

]= Nk

[q(k − 1|k − 1)ˆq(k − 1|k − 1)

]+

+Ok[u(k − 1)− uff (k − 1)]+

+Mkym(k),

(6)


Figure 4. Control scheme of the magnetic bearing reaction wheel demonstrator by Celeroton (Source: [11]).

is used. The estimated state [q, ˆq]> at time k is updatedby using the estimated state at the previous sample timek − 1, controller output without gravitational component(u − uff ) and sensor measurements ym, multiplied bythe observer gain matrices Mk, Nk and Ok.

Considering a reference position qref = 0, to fix the ro-tor at the geometrical centre of the airgap, the full statefeedback LQG controller

upc(k) = −K

eint(k|k)q(k|k)ˆq(k|k)

(7)

is employed. The controller matrix gainK is obtained byminimising a quadratic cost function dependent on statesand currents, as typically done for LQG control.

In order to limit the overheating in the bearings, the max-imum Joule losses in windings are fixed by imposing aquadratic limit to the input currents u. Whenever thethreshold is reached the current is limited and integrationis stopped via ssat.

4.3. Simulation Environment

The previous plant and controller dynamic equationsare combined in a simulation model implemented us-ing MATLAB. This simulation environment is capable ofperforming the controller design and analyse the closed-loop behaviour of the system. Two types of studies areavailable:

1. Frequency-based analysis: study of the closed-loopstability and performance in the frequency domain,representative for the steady-state behaviour of thelinear/linearised system [8].

2. Time-based analysis: study of the closed-loop be-haviour in the time domain, capable of assessing thestability and performance of the system includingnon-linearities and transient responses.

The latter is the analysis type employed for the micro-vibration simulations due mainly to the possibility of in-

cluding non-ideal characteristics of the system such as ac-tuator’s saturation or non-linearities of the bearings. Fur-thermore, the same analysis tools employed for the ex-perimental data can be used in order to obtain a moreequitable comparison.

In order to do that, the continuous-time equations of mo-tion shown in (2) are considered to describe the dynam-ics of the system, and the connections of the control loopare performed as shown in figure 4. The evolution of theclosed-loop system between each sampling time is thenobtained by numerically integrating the dynamics usingthe fourth-order Runge-Kutta (RK4) method. This wayall magnitudes and variables, such as rotor displacements,bearing currents and exported forces, are obtained in thetime domain and analysis tools similar to the ones ex-plained in [15] are implemented to obtain the frequencydecomposition of the variables for each speed.

In order to analyse the test results, a waterfall plot, i.e.force/torque FFT amplitude vs. speed vs. frequency, al-lows identifying the noise signature of the wheel. Then,to facilitate the viewing and comparison, a Worst-CasePlot is employed, where the maximum amplitude at ev-ery frequency for all the measured speeds is chosen, i.e.maximum force amplitude vs. frequency, allowing theidentification of the maximum force or torque for all therange of studied frequencies.

5. EXPERIMENTAL VALIDATION

In order to validate the vibration estimation generatedby the proposed analytical method, the following com-parison is proposed: the forces measured by an exter-nal state-of-the-art multi-component dynamometric plat-form, FD, shown in figure 5, and the simulation esti-mations using the previously defined analytical model,FS , are compared. This measurement validation set-uphas been selected because this type is extensively usedfor space qualification of equipment in terms of micro-vibration charaterisation. The validation is performedby running the wheel for its whole speed range (from0 rpm to 30 000 rpm) in steps of 1000 rpm, measuringthe forces using the dynamometric platform and bearingcurrents, and simulating the same procedure in the previ-


IsolationTable

Multi-ComponentDynamometer

MBRWDemonstratorXX

YY

ZZ

Figure 5. Validation test set-up consisting of the MBRWdemonstrator fixed to a Kistler 9255A mounted over anisolation table with its associated X, Y and Z directions.

ously presented simulation model.

An experimental modal testing to identify the combineddynamics of the demonstrator and the test bench hasbeen performed, represented in a transfer function ma-trix (TFM) H(ω). This TFM allows to relate the forcesin the frequency domain generated by the demonstra-tor estimated in simulation, FS(ω), and the forces mea-sured by the dynamometric platform, FD(ω), in orderto perform the comparison with its measurements. It isworth mentioning that, as it will be shown in the ex-perimental measurements, an unexpected resonance at400 Hz in Y direction was measured, independent of thetested specimen. Even though, the real exported micro-vibrations cannot be measured with this equipment, asthe test equipment itself heavily influences the measuredforces, it can be used to validate the proposed analyticalmodel combined with the experimental TFM H(ω).

Firstly, the measurements and simulations in the time do-main measured at constant speed for the whole speedrange are transformed into the frequency domain via aFFT. The resulting force/torque amplitude vs frequencyvs rotor speed is arranged in the form of a Waterfall plot,as shown in figure 6 for the multi-component dynamome-ter measurements. Secondly, the simulation data is mul-tiplied by TFM H(ω) for each frequency ω to obtain thesimulated exported forces H(ω)FS(ω). Then, the max-imum peak for all speeds at each frequency is taken toform a worst-case plot, allowing to directly compare themeasured and simulated force and torque amplitudes.

The results of the performed comparison between theexported forces measured by the multi-component dy-namometer, FD(ω), and the simulated forces filtered bythe structure dynamics TFM, H(ω)FS(ω), is shown infigure 7. It can be seen that the simulations are able tocorrectly extract the main components and characteristicsof the measured micro-vibration level and show and accu-rate estimation of the exported forces and torques, whileslightly underestimating the amplitude of the forces andtorques.

Vibrations at frequencies higher than 500 Hz correspondto higher order harmonics of the rotation speed, which arenot considered in the model. Nevertheless, specially for

torques, these vibrations at higher frequencies are greatlyamplified by the structure of the test bench and, in prac-tice, much lower amplitudes are expected.

6. CONCLUSION

The proposed analytical micro-vibration model for amagnetic bearing reaction wheel has been proposed andvalidated by comparing its simulations to the measure-ments of a multi-component dynamometer. The resultsshow that, with the only consideration of unbalance in themodel, the simulated level of micro-vibrations accuratelymatches the real vibration level. This implies that for amagnetically-levitated RW, the main source of vibrationsis the unbalance of the rotor.

In the future, considering this validated micro-vibrationmodel, an appropriate unbalance control technique willbe implemented, which will profit from the magneticbearing opportunities for the whole speed range andshow the capabilities of such a technology. The micro-vibration levels reported in this article are obtained withno vibration suppression algorithm, and thus, an impor-tant improvement can be expected.

ACKNOWLEDGMENTS

The authors would like to thank the Swiss Space Cen-ter (SSC) and the European Space Research and Tech-nology Centre (ESA/ESTEC) for the technical adviceand the possibility of employing their facilities forthe validation process, their support and funding un-der ESA Contract No. 4000108465/13/NL/PA. Further-more, this activity has been performed in the frame ofa Networking/Partnering Initiative (NPI) project involv-ing the Swiss Center for Electronics and Microtechnol-ogy (CSEM), the company Celeroton AG, and the Eu-ropean Research and Technology Centre (ESA/ESTEC),under ESA Contract No. 4000119286/17/NL/MH/GM,where a study on active reduction of micro-vibration foractive magnetic bearings is being undertaken.

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