research article estimating friction parameters in...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 249674 8 pageshttpdxdoiorg1011552013249674

Research ArticleEstimating Friction Parameters in Reaction Wheels forAttitude Control

Valdemir Carrara and Heacutelio Koiti Kuga

INPE Avenida dos Astronautas 1758 12227-010 Sao Jose dos Campos SP Brazil

Correspondence should be addressed to Valdemir Carrara valcarraragmailcom

Received 11 March 2013 Accepted 12 May 2013

Academic Editor Antonio F Bertachini A Prado

Copyright copy 2013 V Carrara and H K Kuga This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The ever-increasing use of artificial satellites in both the study of terrestrial and space phenomena demands a search for increasinglyaccurate and reliable pointing systems It is common nowadays to employ reaction wheels for attitude control that provide widerange of torque magnitude high reliability and little power consumption However the bearing friction causes the response ofwheel to be nonlinear which may compromise the stability and precision of the control system as a whole This work presents acharacterization of a typical reaction wheel of 065Nms maximum angular momentum storage in order to estimate their frictionparameters It used a friction model that takes into account the Coulomb friction viscous friction and static friction accordingto the Stribeck formulation The parameters were estimated by means of a nonlinear batch least squares procedure from dataraised experimentally The results have shown wide agreement with the experimental data and were also close to a deterministicmodel previously obtained for this wheelThismodel was then employed in aDynamicModel Compensator (DMC) control whichsuccessfully reduced the attitude steady state error of an instrumented one-axis air-bearing table

1 IntroductionThis paper presents a Dynamic Model Compensator (DMC)control of a reactionwheel in current controlmodeThe erroris compensated by means of a mathematical model of thewheel dynamics and the bearing friction Reaction wheels areactuators largely employed in attitude control subsystems inorder to provide attitude pointing and stability of artificialsatellites They consist of a Brushless DC (BLDC) motorcoupled to a high inertia flywheel The torque applied tothe wheel is sensed by the satellite in the opposite directionallowing the attitude control based on information of inertialsensors like gyroscopes sun sensors magnetometers andstar sensors Reaction wheels are devices that must operatecontinuously for several years in vacuum conditions sub-jected to wide variations in temperature and high radiationdoses So its reliability and quality are essential to the satellitehealthThese requirements pose great challenge to a reactionwheel design which makes such equipment highly complexand expensive Reaction wheels are classified according totheir capability of storing angularmomentum from the smallones employed in microsatellites to large ones appropriated

for orbital stations and communication satellites Normallyreactionwheels are operated either in current (or equivalentlytorque) mode or in speed mode In current mode theelectronics delivers the necessary current to the motor inorder to achieve the commanded torque In speed mode asecondary outer control loop regulates the current to elimi-nate the error between the commanded angular speed and theflywheel speed which ismeasured by some sort of rate sensor(usually Hall effect sensor or optical incremental encoder)The speed mode control avoids the bearing friction effectswhich causes a nonlinear behavior in the current controlmode However speed control introduces more complexityin the electronics and also causes some delay in the wheelresponse In order to assure linearity in the current modeand eventually disregard the speed control mode this worksuggests mitigating the effects of friction by adopting a DMCcontroller in current control loop This compensator wasapplied to an off-the-shelf reaction wheel that operates inboth current and speed mode The friction model includesCoulomb viscous and static or breakaway torques Withthe aim of evaluating the control performance the static

2 Mathematical Problems in Engineering

friction was replaced by the Stribeck friction which unlikethe previous one does not present discontinuities when themotor reverses its rotation sense All friction parameters andthe motor coefficient were obtained by a least squares fitof data collected from several experiments performed withthe wheel in current mode The experiments consisted of acontinuously varying current command in order to stimulatethe wheel through various speeds and sense inversions so asto assure correct parameters identification andmodel fidelityThe DMCwas then introduced in the attitude control loop ofan air-bearing table that emulates the frictionless conditionsfound in space The table has a fiber optic gyroscope formeasuring angular rate (that provides the reference for theattitude after integration) the reaction wheel a system ofradio-modem for reaction wheel telemetry and commandand a power supply battery A small computer fan was placedin the air-bearing table so as to yield a small torque whichwill be dully balanced by the attitude control procedure Byproper selection of the initial conditions plus the fan torquethe wheel will be forced by the attitude control to reverseits direction of rotation The results show that there is asignificant gain when the DMC is implemented in the controlloop when compared with the simple current mode controlwith control performance comparable to the speed modeA comparison of the statistical method for determiningfriction and motor parameters with a deterministic methodin which each parameter has been obtained from a dedicatedexperiment to highlight its influence is also presented in thisstudy

In an interesting work Robertson and Stoneking [1] statethat the Guidance navigation and control subsystem (orsometimes attitude and orbit control subsystem) presents ahigh number of critical failures in satellites when comparedwith other subsystems From these failures almost 30 canbe assigned to the reaction wheel which confirms that agood wheel design or selection makes difference In Carraraand Milani [2] the friction parameters of a reaction wheelcommanded in current are raised from experimental meansIn that work the wheel is subject to specific commands inorder to highlight a particular parameter These are thencalculated by manual curve adjustment based on minimumquadratic variation The model used took into account theCoulomb and viscous frictions In Carrara [3] the samemodel was used in an attitude controller of an air-bearingtableThe controller used command in current with dynamiccompensation based only on Coulomb and viscous frictionsWith this method it was possible to reduce the error duringwheel rotation inversion by an order of magnitude Later acomparison between the two forms of control in Carrara et alwas made [4] which showed that the dynamic compensatorintroduces an error comparable but slightly higher to controlmode in the angular velocity of the wheel

The wheel friction parameters plus the Stribeck friction(in fact a continuous and differentiable model of staticfriction or departure friction) were estimated by a Kalmanfilter in Fernandes et al [5] but with nonconclusive resultsbecause of the scarcity of accurate experimental data Fewworks in the literature relate friction models with reactionwheels bearings [6 7] On the other hand several papers have

Figure 1 Experiment mounted on the air-bearing table

friction models and estimation of parameters in rotors as inOlsson et al [8] and Canudas de Wit and Ge [9] includinga dynamic model for friction by Canudas de Wit et al [10]and Canudas de Wit and Lischinsky [11] both based on theLuGre (Lund-Grenoble) friction model [12] This dynamicmodel was later employed in Carrara et al [13] to estimate theparameters of a reaction wheel by Kalman filtering A modelof the slip-stick phenomenon was simulated by Karnopp[14] while Al-Bender et al [15] presented a dynamic modelbased on the generalized Maxwell-slip friction HirschornandMiller [16] proposed a dynamic controller for systemwitha bristle model for nonlinear friction effects Amin et al [17]suggest using a nonlinear observer to estimate and to com-pensate for the friction which presented good concordancewith experimental data Hensen et al [18] experimentallyobtained the friction parameters of the dynamic LuGremodelby means of an apparatus and Ramasubramanian and Ray[19] employed the extended Kalman-Bucy filter (EKBF) toestimate the Dahl model friction coefficients Armstrong-Helouvry et al [20] compiled the existing frictionmodels andcompensation methods in a comprehensive work

This work proposes to estimate by means of a nonlinearleast squares procedure the parameters of the friction ofa reaction wheel shown in the photo of Figure 1 consid-ering not only the Coulomb and viscous frictions but alsothe Stribeck friction The so-estimated parameters are thencompared to those obtained in Carrara and Milani [2] andCarrara [3] In the following sections the formulation offriction model and of the estimation of parameters will bepresented The experimental results appear next togetherwith the comparison between both methods statistical anddeterministic The conclusions are presented in sequence Atypical reaction wheel [21] acquired by the Space Mechanicsand Control Division of National Institute for Space Research(INPE) was employed for data acquisition and analysis

2 Mathematical Model

For gathering the necessary data for this work a setup madeby Carrara andMilani [2] was used In a bearing table systemof one degree of freedom in rotation (Figure 1) a reactionwheel with maximum capacity of 065Nms commanded by

Mathematical Problems in Engineering 3

120596

c

TTs

Coulomb

Viscous

Stribeck

Figure 2 Friction torque model used in the parameter estimation

current via serial interface a fiber optic gyroscope of oneaxis (not used in this work) a command and telemetryelectronics a radio modem for communication with theequipment and a battery for power supply were installedThe programs needed to command the wheel and make thecurrent readings and angular velocity were written in C++and run on a computer that is external to the table

Themathematical model of a reaction wheel is analogousto themodel of aDCmotor in which inertia includes besidesthe rotor inertia the inertia of the flywheel attached to theaxis of the wheel In the model considered here the viscousfriction Coulomb friction and the friction of Stribeck wereincluded The differential equation describing the motion is

119879119908= 119869119908 + 119887120596 + sgn (120596) [119888 + 119889119890minus120596

21205962

119904 ] (1)

where 119879119908is the motor torque the wheelrsquos and rotor inertia is

119869119908 119887 is the coefficient of viscous friction 119888 is the Coulomb

friction torque 119889 is the starting torque 120596 is the angularvelocity of the wheel and 120596

119904is known as Stribeck speed

[8 9] The torque model is displayed graphically in Figure 2The starting torque 119889 can be decomposed on the differencebetween the static torque 119879

119904and the Coulomb torque 119888 that

is 119889 = 119879119904minus 119888 Neglecting nonlinear effects present in current

to torque conversion one can consider that the torque appliedto the motor is proportional to the current in the stator 119868 inthe form

119879119908= 119896119898119868 (2)

In current control mode one commands the current 119868 onthe wheel and gets telemetry readings of angular velocity120596 and current itself which may be slightly different fromthat commanded due to the presence of an internal currentcontrol loop to the wheel For the estimation of parametersby means of a least squares procedure the state to solve for iscomposed of the angular velocity themotor constant viscousfriction coefficient Coulomb torque and static torque Sincethe inertia of the wheel cannot be estimated independently ofother parameters the inertia value supplied by the manufac-turer of 119869

119908= 15 times 10

minus3 kgm2 was adopted The state to beestimated is then

x = (120596119896119898

119869119908

119887

119869119908

119888

119869119908

119879119904

119869119908

)

119879

= (11990911199092119909311990941199095)119879

(3)

Stribeck speed 120596119904could also be estimated but preliminary

tests showed that the noise present in the measurements atlow speed where this parameter is important does not allowa good estimate of its value In addition the estimated valuesof the remaining parameters are barely affected by 120596

119904 As a

result one adopted to this speed the 4 rpm value obtainedindirectly through a mapping of the average current as afunction of the angular velocity of the wheel at low speedsusing the speed control mode

From (1) the dynamical model for the estimation processis drawn

1= 1199092119868 minus 11990931199091minus sgn (119909

1) [1199094+ (1199095minus 1199094) 119890minus1199092

11205962

119904 ] (4)

Once the dynamical part is represented by only one (time)variable (rotation 119909

1) and the remaining states are param-

eters the nonnull elements of the corresponding Jacobianmatrix of partial derivatives are

1205971

1205971199091

= minus1199093+ 2

1199091

1205962119904

sgn (1199091) (1199095minus 1199094) 119890minus1199092

11205962

119904

1205971

1205971199092

= 119868

1205971

1205971199093

= minus1199091

1205971

1205971199094

= minus sgn (1199091) (1 minus 119890

minus1199092

11205962

119904 )

1205971

1205971199095

= minus sgn (1199091) 119890minus1199092

11205962

119904

(5)

The data were generated with the wheel subjected to twocommand profiles both with small amplitude so as to keepit at low speeds and with periodic reversals in the directionof rotation as shown in Figure 3 The first profile consistedof multiple sinusoidal cycles in which each period had theamplitude and the period chosen randomly within certainlimits The second profile had random amplitude constantcurrent in each actuation and reverse direction every 30seconds similar to a square wave

The bearing temperature and atmospheric pressure insidethe reaction wheel were monitored during the entire run ofthe profiles with duration of 300 seconds each Althoughit is plausible that the temperature affects the friction andas a consequence the behavior of the wheel this influencehas not been taken into account in this model since thevariation of both during the experiment was small less than1∘C in temperature Note that particularly in Figure 3(b)the Coulomb torque causes changes of inflection in thecurve of the angular velocity when it reverses its directionof rotation This is an indication that these experiments areable to provide information for this and other estimationparameters which will be presented in the following section

3 Estimation Procedure

Theprocedure of parameter estimation from (1) was based onthe batch least squares methodThe weighted loss function 119869


0 50 100 150 200 250 300minus400

minus300

minus200

minus100

0

100

200

300

400

500

Time (s)

Commanded current (mA)Angular velocity (rpm)

(a)

0 50 100 150 200 250 300minus150

minus100

minus50

0

50

100

150

200

250

Time (s)


(b)

Figure 3 Sinusoidal (a) and stepwise (b) profiles commanded to the reaction wheel

considering a priori information in norm notation is givenby

119869 =1003817100381710038171003817y minusHx1003817100381710038171003817

2

Rminus1 +1003817100381710038171003817x119900 minus x1003817100381710038171003817

2

Pminus1119900

(6)

where sdot represents the norm of a matrix or vector y is thevector containing 119898 measurements H is the 119898 times 119899 matrixthat relates the measurements to the state x of 119899 elements x

119900

is the a priori state value R is the 119898 times 119898 covariance matrixof measurement errors and P

119900is the covariancematrix of the

errors on the a priori state Initially the loss function is in theform

119869 =

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[

[

(Pminus1119900)12

x119900

(Rminus1)12

y]

]

minus [

[

(Pminus1119900)12

(Rminus1)12

H]

]

x100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

(7)

where (sdot)12 represents a square rootmatrix of (sdot) Utilizing anorthogonal transformation T of for example Householderthat does not change the norm one triangularizes the systemso that

119869 =

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

T[[

(Pminus1119900)12

x119900

(Rminus1)12

y]

]

minus T[[

(Pminus1119900)12

(Rminus1)12

H]

]

x100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

[y1

y2

] minus [H1

0 ] x10038171003817100381710038171003817100381710038171003817

2

=1003817100381710038171003817y1 minusH

1x10038171003817100381710038172

+1003817100381710038171003817y21003817100381710038171003817

2

(8)

Note that after the orthogonal transformationH1is a 119899times119899

triangular matrix 0 is a119898 times 119899matrix of zeros and y1and y2

are vectors of sizes 119899 and 119898 respectively resulting from theapplication of the orthogonal transformationTTherefore theminimum of the loss function is simply

119869min =1003817100381710038171003817y21003817100381710038171003817

2 if 1003817100381710038171003817y1 minusH1x1003817100381710038171003817 = 0 (9)

which is the least squares solution according to Lawson andHanson [22] Once the matrixH

1is triangular the resolution

of

y1= H1x (10)

is trivial (back substitution) and x is the estimated statevector This approach was coded in Fortran and adapted[23] to solve the nonlinear problem of estimation of frictionparameters In this caseH is the matrix of partial derivativeswith components given by (5) and yx and x

119900are now

deviations from the nominal Because of the nonlinear natureof the problem from an initial condition x

119900(a priori) the

solution is obtained iteratively and converges quickly in fewiterations

4 Estimation of Friction Parameters

Some of the friction parameters of this wheel were estimatedby Carrara and Milani [2] and Carrara [3] in previous worksBecause very specific methods for individual computation ofthe friction parameters were used in those works they werenamed deterministic methods in contrast with the statisticalmethod employed in this study By the deterministic meth-ods the viscous friction coefficient 119887 = 516 times 10

minus6Nmsthe Coulomb friction 119888 = 08795 times 10minus3Nm and the motorconstant 119896

119898= 00270NmA were obtained

In the parameter estimation procedure the departure statevector was set to

x119900= (0 18 000344 05863 0)

119879 (11)

which corresponds to the values of the deterministic meth-ods as defined by (3) The profile 2 was used to estimatethe values of the parameters 119909

2 1199093 1199094 and 119909

5 The profile


Table 1 Friction parameters of the reaction wheel

Parameter Deterministic StatisticalMotor constant 119896

11989800270 00228

Viscous coefficient 119887 516 times 10minus6

483 times 10minus6

Coulomb torque 119888 08795 times 10minus3

08795 times 10minus3

Static torque 119879119904

mdash 09055 times 10minus3

0 50 100 150 200 250 300minus200

minus100

0

100

200

300

400

500

Time (s)

Ang

ular

velo

city

(rpm

)

DeterministicEstimatedMeasured

Figure 4 Results of deterministic and statistical methods in theprofile 1

1 (sinusoidal) was used to validate the estimated parametersIn the least squares procedure one assumed that rotationmeasurements had a standard deviation of about 5 rpm Thestate vector after convergence of the procedure was

x119900= (0 15205 000322 05863 06037)

119879

(12)

Considering the inertia value 119869119908= 15 times 10

minus3 kgm2 thefriction parameters result in 119887 = 483times10minus6Nms 119888 = 08795times10minus3Nm and 119896

119898= 00228NmA Table 1 shows the results

obtained herein It is realized that the highest difference wasencountered in the motor constant which was 15 below thedeterministic method The Coulomb torque did not presentmeaningful difference within the accuracy tolerance adoptedin its computation

Figure 4 shows to profile 1 a comparison of the measured(Figure 3) and estimated speeds by the deterministic andstatistical methods Note that both methods present similarresults however the error with respect to actual measure-ments is still relatively high

Figure 5 shows the same results for the stepwise profile 2(Figure 4) Note clearly the better closeness of the statisticaladjustment (compared to the deterministic one) to theexperimental rotation measurements

Figure 6 shows the measurements (in red) and theresiduals between the measured rotations and the values

0 50 100 150 200 250 300

300

Time (s)

Ang

ular

velo

city

(rpm

)


minus200

minus150

minus100

minus50

0

50

100

150

200

250


estimated by bothmethods (deterministic and statistical) forthe sinusoidal profile 1 Increasing residuals near the zerocrossings is verified where the friction models have lowerperformance

Figure 7 shows the residuals between measured and esti-mated rotations for the stepwise profile 2 The best per-formance of the statistical adjustment at low speeds (50ndash100 rpm) is also quite pronounced

The results indicate problems in the wheel response at lowangular rates mainly in transitions crossing the zero levelNevertheless themodel obtained by the statistical estimationof parameters behaves better in this range In practical termsthis model when used in a control system provides a smoothtransition through zero and can eliminate the need to definea dead zone facilitating the design and implementation of thecontrol system On the other hand it should be noted that themathematical model used in both methods is symmetricalwith respect to the direction of rotation However there isevidence [24] that bearings may be asymmetric although thedegree of asymmetry is in general small

41 Torque Mode Control In order to emphasize the nonlin-ear friction effect in the controller performance a cooler fanwas attached to the air-bearing table and oriented in sucha way that it introduces a small but constant torque Theinitial velocity of the wheel was adjusted so that a zero-speedcrossover occurs during the control action A PID controllerwas used to control the attitude of the air-bearing table basedon the integrated signal of the FOG gyroThe PID gains wereadjusted to minimize or to avoid the overshoot response inattitude andwere kept constant during thewhole experimentThe air-bearing table dynamics can be modeled as a one-axisrigid body with inertia 119869 and the fan disturbance torque 119879

119889

119869Ω = 119879119908+ 119879119889 (13)


0 50 100 150 200 250 300minus400

minus300

minus200

minus100

0

100

200

300

400

Time (s)

Resid

uals

(rpm

)


(a)

0 50 100 150 200 250 300minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Time (s)

Resid

uals

(rpm

)


(b)

Figure 6 Estimation residuals of profiles 1 (a) and 2 (b)

0 100 200 300 400 500Time (s)

minus1000

minus500

0

500

1000

1500

2000

Ang

le er

ror (

mill

ideg

rees

)

Figure 7 Attitude error during zero-speed crossing and withexternal disturbance (cooler fan)

whereΩ is the table angular velocity asmeasured by the FOGgyro and 119879

119908is the wheelrsquos reaction torque

Figure 7 shows the attitude error with a null referencesignal while Figure 8 shows the commanded current equalto the PID signal that is 119868 = 119906 where 119906 is the PID outputThemaximum attitude error occurs during wheelrsquos reversionat elapsed time of 230 seconds approximately The torquegenerated by the fan could be estimated based on the angularmomentum variation resulting in 046 times 10minus3Nm and it ispractically constant The attitude error reaches 15 degreesafter zero-speed crossing followed by an error of 02 degreein steady state From controller viewpoint this means thatthe pointing requirement is no longer accomplished during

0 100 200 300 400 500Time (s)

minus60

minus40

minus20

0

20

40

60

80

100

Curr

ent (

mA

)

Figure 8 Control signal (motor current) under external distur-bance

zero-speed crossing Most of the error is due to the long timethe integral controller takes to compensate the fast changingin the friction torque during wheel reversion As it will beshown the DMC controller changes the control signal asquick as the friction torque allowing the PID to respond onlyto the external disturbance torque

42 DynamicModel Compensator Control Using a nonlinearcontroller to handle the zero-speed problem of the reactionwheel is a natural consequence of the fact that the mathemat-ical model represents the behavior of the wheel reasonablywell It is therefore straightforward to use this model as anonlinear compensator for the controller and to make the


r+ +

+e u

120596

120596PID Reaction

wheel

1

s

120579

1

Js+

+

TdDMC

TwI

minus

Figure 9 PID controller with RW dynamic model compensation

0 50 100 150 200 250 300 350Time (s)

minus200

minus100

0

100

200

300

400

500

600

Ang

le er

ror (

mill

ideg

rees

)

Figure 10 Attitude error of the air-bearing table with DMC control

wheel action directly proportional to the PID signal [11] Sincethe table responds to only an acceleration of the wheel thecontrol command will be in the form

119868 = 119906 +119887

119896119898

120596 +sgn (120596)119896119898

[119888 + 119889119890minus12059621205962

119904 ] (14)

where 119906 is the PID control signal For a null wheel angularvelocity the compensator takes the form

119868 = 119906 +(119888 + 119889) sgn (119906)

119896119898

(15)

Figure 9 shows a simplified block diagram of theDynamic Model Compensator (DMC) control The newcontroller was tested under the same condition as the torquemode control but incorporating the dynamic compensationAs can be seen in Figures 10 and 11 (analogous to Figures7 and 8) the error was almost negligible with a maximumdeviation of only 01 degree during wheel reversion and ittook about 20 s to reach the steady state The large error ofalmost 06 degree is due to the initial step response of thecontrol at the beginning of the experiment and will not beconsidered as a steady error The control signal is shown inblack in Figure 11 and separated in its two components thefriction dynamic compensator (in red) and the PID signal(blue curve) It is clear in this graph that the PID control isapproximately constant as it would be expected due to thedisturbing torque of the fan The PID controller gains werekept identical to those used previously although they could

Time (s)

minus60

minus40

minus20

0

20

40

60

80

Curr

ent (

mA

)

0 50 100 150 200 250 300 350

Friction model

PID u

Current I


be adjusted in order to achieve a better performance since thedynamics is now almost linear due to themodel compensator

The effect of the Stribeck friction is barely seen inFigure 11 which indicates that this friction is not so signif-icant for the wheelrsquos behavior In fact the same experimentwas carried out without the Stribeck friction model in theDMC (not shown in this paper) which showed similarresults However it is not recommended to simply neglect theStribeck factor since it introduces some sort of hysteresis thatshould be important during motor starting and reversing

5 Conclusions

This paper presented a computational and mathematicalmodel for an off-the-shelf reaction wheel [21] obtainedfrom nonlinear models of Coulomb viscous and Stribeckfrictions based on testing and experimental measurementsof the behavior of the wheel Previous work did not includeStribeck friction and the values of the parameters of friction(Coulomb and viscous) were obtained deterministically [23] Estimation of Stribeck friction via extended Kalmanfiltering was attempted [5] but resulted nonconclusive due tothe limited data used Another work used the LuGre Model[13] with extended Kalman filter but only at simulation level(no actual data)

Based on the more complete proposed model a nonlin-ear estimation of states and parameters by the method ofleast squares using data from two experiments one withsinusoidal profile and another with positive and negativelevels where the transitions by zero were exercised numeroustimes (24 times in the sinusoidal profile 1 and 9 timesin the stepwise profile 2) were accomplished As expecteddegraded performance of the models in the crossings byzero was noted but with better fit of statistical methodA nonlinear Dynamic Model Compensator (DMC) for thereaction wheel control was then implemented in order tomake the wheel behavior linear The controller showedimproved performance in this new condition and reached themaximum error of only 01 degrees at zero-speed crossing


The DMC presented also smooth responses near zero asexpected with errors smaller than the ones presented withthe deterministic parameter estimation method [4] Due tothis the compensator significantly reduced the nonlineareffects that occur in the response of the wheel during thereversals of direction avoiding the model discretization anddecreasing the complexity of the control synthesis in thistype of actuator Future works suggest the use of this modelin a control system of position (angle) or angular velocityand corresponding performance comparisons in terms ofresponse time performance and accuracy

Disclosure

The authors are solely responsible for the printed materialincluded in this paper

Acknowledgment

The authors thank the project FUNDEP-FINEP-SIA-11382lowast3which provided the support for acquiring equipment used inthe experiments

References

[1] B Robertson and E Stoneking ldquoSatellite GNampC AnomalyTrendsrdquo in Proceedings of the Annual AAS Rocky MountainGuidance and Control Conference vol 113 of Advances in theAstronautical Sciences pp 531ndash542 2003

[2] V Carrara and P G Milani ldquoControl of one-axis air bearingtable equipped with gyro and reaction wheelrdquo in Proceedingsof the 5th Brazilian Symposium on Inertial Engineering (SBEINrsquo07) Rio de Janeiro Brazil November 2007

[3] V Carrara ldquoComparison between means of attitude controlwith reaction wheelsrdquo in Proceedings of the 6th BrazilianSymposium on Inertial Engineering (SBEIN rsquo10) Rio de JaneiroBrazil November 2010

[4] V Carrara R Siqueira and D Oliveira ldquoSpeed and currentmode strategy comparison in satellite attitude control with reac-tion wheelsrdquo in Proceedings of the 21st International Congressof Mechanical Engineering (COBEM rsquo11) Natal Brazil October2011

[5] D C Fernandes H K Kuga V Carrara and R A Romano ldquoAmodel of a reaction wheel with friction parameters estimationby Kalman filterrdquo in Proceedings of the 7th Brazilian Congress ofMechanical Engineering (CONEM rsquo12) Sao Luis Brazil August2012

[6] M L B Moreira R V F Lopes and H K Kuga ldquoEstimationof torque in a reaction wheel using a Bristle model for frictionrdquoin Proceedings of the 18th International Congress of MechanicalEngineering Ouro Preto Brazil November 2005

[7] G Shengmin and H Cheng ldquoA comparative design of satelliteattitude control system with reaction wheelrdquo in Proceedingsof the 1st NASAESA Conference on Adaptive Hardware andSystems (AHS rsquo06) pp 359ndash362 June 2006

[8] H Olsson K J Astrom C Canudas de Wit M Gafvert andP Lischinsky ldquoFriction models and friction compensationrdquoEuropean Journal of Control vol 4 no 3 pp 176ndash195 1998

[9] C Canudas de Wit and S S Ge ldquoAdaptive friction compen-sation for systems with generalized velocityposition friction

dependencyrdquo in Proceedings of the 36th IEEE Conference onDecision and Control pp 2465ndash2470 San Diego Calif USADecember 1997

[10] C Canudas de Wit H Olsson K J Astrom and P LischinskyldquoNew model for control of systems with frictionrdquo IEEE Trans-actions on Automatic Control vol 40 no 3 pp 419ndash425 1995

[11] C Canudas de Wit and P Lischinsky ldquoAdaptive frictioncompensation with partially known dynamic friction modelrdquoInternational Journal of Adaptive Control and Signal Processingvol 11 no 1 pp 65ndash80 1997

[12] K J Astrom and C Canudas-de-Wit ldquoRevisiting the LuGrefriction modelrdquo IEEE Control Systems Magazine vol 28 no 6pp 101ndash114 2008

[13] V Carrara A G Silva and H K Kuga ldquoA dynamic frictionmodel for reaction wheelsrdquo in Proceedings of the 1st IAAConference on Dynamics and Control of Space Systems (DyCoSSrsquo12) vol 145 of Advances in the Astronautical Sciences pp 352ndash343 Porto Portugal March 2012

[14] D Karnopp ldquoComputer simulation of slip-stick friction inmechanical dynamic systemsrdquo Transactions of the ASME Jour-nal of Dynamic Systems Measurement and Control vol 107 no1 pp 100ndash103 1985

[15] F Al-Bender V Lampaert and J Swevers ldquoThe generalizedMaxwell-slip model a novel model for friction simulation andcompensationrdquo IEEE Transactions on Automatic Control vol50 no 11 pp 1883ndash1887 2005

[16] R M Hirschorn and G Miller ldquoControl of nonlinear systemswith frictionrdquo IEEETransactions onControl Systems Technologyvol 7 no 5 pp 588ndash595 1999

[17] J Amin B Friedland and A Harnoy ldquoImplementation of afriction estimation and compensation techniquerdquo IEEE ControlSystems Magazine vol 17 no 4 pp 71ndash76 1997

[18] R H A Hensen M J G van de Molengraft andM SteinbuchldquoFrequency domain identification of dynamic friction modelparametersrdquo IEEE Transactions on Control Systems Technologyvol 10 no 2 pp 191ndash196 2002

[19] A Ramasubramanian and L R Ray ldquoComparison of EKBF-based and classical friction compensationrdquo Transactions of theASME Journal of Dynamic Systems Measurement and Controlvol 129 no 2 pp 236ndash242 2007

[20] B Armstrong-Helouvry P Dupont and C C deWit ldquoA surveyof models analysis tools and compensation methods for thecontrol of machines with frictionrdquo Automatica vol 30 no 7pp 1083ndash1138 1994

[21] J A A Engelbrecht Userrsquos Manual for the SunSpace ReactionWheel and Gyroscope Subsystem SunSpace Matieland SouthAfrica 2005 (SS01-106000)

[22] C L Lawson and R J Hanson Solving Least Squares ProblemsPrentice-Hall Englewood Cliffs NJ USA 1974 Prentice-HallSeries in Automatic Computation

[23] H K Kuga Orbit determination of artificial satellites byestimation techniques and state smoothing techniques [PhDdissertation] INPE Sao Jose dos Campos Brazil 1989 (INPE-4959-TDL388)

[24] C Canudas de Wit and K J Astrom ldquoAdaptive friction com-pensation in DC-motor driversrdquo IEEE Journal of Robotics andAutomation vol 3 no 6 pp 681ndash685 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


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Journal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


friction was replaced by the Stribeck friction which unlikethe previous one does not present discontinuities when themotor reverses its rotation sense All friction parameters andthe motor coefficient were obtained by a least squares fitof data collected from several experiments performed withthe wheel in current mode The experiments consisted of acontinuously varying current command in order to stimulatethe wheel through various speeds and sense inversions so asto assure correct parameters identification andmodel fidelityThe DMCwas then introduced in the attitude control loop ofan air-bearing table that emulates the frictionless conditionsfound in space The table has a fiber optic gyroscope formeasuring angular rate (that provides the reference for theattitude after integration) the reaction wheel a system ofradio-modem for reaction wheel telemetry and commandand a power supply battery A small computer fan was placedin the air-bearing table so as to yield a small torque whichwill be dully balanced by the attitude control procedure Byproper selection of the initial conditions plus the fan torquethe wheel will be forced by the attitude control to reverseits direction of rotation The results show that there is asignificant gain when the DMC is implemented in the controlloop when compared with the simple current mode controlwith control performance comparable to the speed modeA comparison of the statistical method for determiningfriction and motor parameters with a deterministic methodin which each parameter has been obtained from a dedicatedexperiment to highlight its influence is also presented in thisstudy

In an interesting work Robertson and Stoneking [1] statethat the Guidance navigation and control subsystem (orsometimes attitude and orbit control subsystem) presents ahigh number of critical failures in satellites when comparedwith other subsystems From these failures almost 30 canbe assigned to the reaction wheel which confirms that agood wheel design or selection makes difference In Carraraand Milani [2] the friction parameters of a reaction wheelcommanded in current are raised from experimental meansIn that work the wheel is subject to specific commands inorder to highlight a particular parameter These are thencalculated by manual curve adjustment based on minimumquadratic variation The model used took into account theCoulomb and viscous frictions In Carrara [3] the samemodel was used in an attitude controller of an air-bearingtableThe controller used command in current with dynamiccompensation based only on Coulomb and viscous frictionsWith this method it was possible to reduce the error duringwheel rotation inversion by an order of magnitude Later acomparison between the two forms of control in Carrara et alwas made [4] which showed that the dynamic compensatorintroduces an error comparable but slightly higher to controlmode in the angular velocity of the wheel

The wheel friction parameters plus the Stribeck friction(in fact a continuous and differentiable model of staticfriction or departure friction) were estimated by a Kalmanfilter in Fernandes et al [5] but with nonconclusive resultsbecause of the scarcity of accurate experimental data Fewworks in the literature relate friction models with reactionwheels bearings [6 7] On the other hand several papers have

Figure 1 Experiment mounted on the air-bearing table

friction models and estimation of parameters in rotors as inOlsson et al [8] and Canudas de Wit and Ge [9] includinga dynamic model for friction by Canudas de Wit et al [10]and Canudas de Wit and Lischinsky [11] both based on theLuGre (Lund-Grenoble) friction model [12] This dynamicmodel was later employed in Carrara et al [13] to estimate theparameters of a reaction wheel by Kalman filtering A modelof the slip-stick phenomenon was simulated by Karnopp[14] while Al-Bender et al [15] presented a dynamic modelbased on the generalized Maxwell-slip friction HirschornandMiller [16] proposed a dynamic controller for systemwitha bristle model for nonlinear friction effects Amin et al [17]suggest using a nonlinear observer to estimate and to com-pensate for the friction which presented good concordancewith experimental data Hensen et al [18] experimentallyobtained the friction parameters of the dynamic LuGremodelby means of an apparatus and Ramasubramanian and Ray[19] employed the extended Kalman-Bucy filter (EKBF) toestimate the Dahl model friction coefficients Armstrong-Helouvry et al [20] compiled the existing frictionmodels andcompensation methods in a comprehensive work

This work proposes to estimate by means of a nonlinearleast squares procedure the parameters of the friction ofa reaction wheel shown in the photo of Figure 1 consid-ering not only the Coulomb and viscous frictions but alsothe Stribeck friction The so-estimated parameters are thencompared to those obtained in Carrara and Milani [2] andCarrara [3] In the following sections the formulation offriction model and of the estimation of parameters will bepresented The experimental results appear next togetherwith the comparison between both methods statistical anddeterministic The conclusions are presented in sequence Atypical reaction wheel [21] acquired by the Space Mechanicsand Control Division of National Institute for Space Research(INPE) was employed for data acquisition and analysis

2 Mathematical Model

For gathering the necessary data for this work a setup madeby Carrara andMilani [2] was used In a bearing table systemof one degree of freedom in rotation (Figure 1) a reactionwheel with maximum capacity of 065Nms commanded by


120596

c

TTs

Coulomb

Viscous

Stribeck




119879119908= 119869119908 + 119887120596 + sgn (120596) [119888 + 119889119890minus120596

21205962

119904 ] (1)









119879119908= 119896119898119868 (2)


119908= 15 times 10


x = (120596119896119898

119869119908

119887

119869119908

119888

119869119908

119879119904

119869119908

)

119879

= (11990911199092119909311990941199095)119879

(3)



119904 As a



1= 1199092119868 minus 11990931199091minus sgn (119909

1) [1199094+ (1199095minus 1199094) 119890minus1199092

11205962

119904 ] (4)




1205971

1205971199091

= minus1199093+ 2

1199091

1205962119904

sgn (1199091) (1199095minus 1199094) 119890minus1199092

11205962

119904

1205971

1205971199092

= 119868

1205971

1205971199093

= minus1199091

1205971

1205971199094

= minus sgn (1199091) (1 minus 119890

minus1199092

11205962

119904 )

1205971

1205971199095

= minus sgn (1199091) 119890minus1199092

11205962

119904

(5)






0 50 100 150 200 250 300minus400

minus300

minus200

minus100

0

100

200

300

400

500

Time (s)


(a)

0 50 100 150 200 250 300minus150

minus100

minus50

0

50

100

150

200

250

Time (s)


(b)



119869 =1003817100381710038171003817y minusHx1003817100381710038171003817

2

Rminus1 +1003817100381710038171003817x119900 minus x1003817100381710038171003817

2

Pminus1119900

(6)


119900




119869 =

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[

[

(Pminus1119900)12

x119900

(Rminus1)12

y]

]

minus [

[

(Pminus1119900)12

(Rminus1)12

H]

]

x100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

(7)


119869 =

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

T[[

(Pminus1119900)12

x119900

(Rminus1)12

y]

]

minus T[[

(Pminus1119900)12

(Rminus1)12

H]

]

x100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

[y1

y2

] minus [H1

0 ] x10038171003817100381710038171003817100381710038171003817

2

=1003817100381710038171003817y1 minusH

1x10038171003817100381710038172

+1003817100381710038171003817y21003817100381710038171003817

2

(8)




119869min =1003817100381710038171003817y21003817100381710038171003817

2 if 1003817100381710038171003817y1 minusH1x1003817100381710038171003817 = 0 (9)



of

y1= H1x (10)


119900are now









x119900= (0 18 000344 05863 0)

119879 (11)


2 1199093 1199094 and 119909

5 The profile




11989800270 00228


483 times 10minus6





0 50 100 150 200 250 300minus200

minus100

0

100

200

300

400

500

Time (s)

Ang

ular

velo

city

(rpm

)




x119900= (0 15205 000322 05863 06037)

119879

(12)








0 50 100 150 200 250 300

300

Time (s)

Ang

ular

velo

city

(rpm

)


minus200

minus150

minus100

minus50

0

50

100

150

200

250






119889

119869Ω = 119879119908+ 119879119889 (13)


0 50 100 150 200 250 300minus400

minus300

minus200

minus100

0

100

200

300

400

Time (s)

Resid

uals

(rpm

)


(a)

0 50 100 150 200 250 300minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Time (s)

Resid

uals

(rpm

)


(b)


0 100 200 300 400 500Time (s)

minus1000

minus500

0

500

1000

1500

2000

Ang

le er

ror (

mill

ideg

rees

)





0 100 200 300 400 500Time (s)

minus60

minus40

minus20

0

20

40

60

80

100

Curr

ent (

mA

)





r+ +

+e u

120596

120596PID Reaction

wheel

1

s

120579

1

Js+

+

TdDMC

TwI

minus


0 50 100 150 200 250 300 350Time (s)

minus200

minus100

0

100

200

300

400

500

600

Ang

le er

ror (

mill

ideg

rees

)



119868 = 119906 +119887

119896119898

120596 +sgn (120596)119896119898

[119888 + 119889119890minus12059621205962

119904 ] (14)


119868 = 119906 +(119888 + 119889) sgn (119906)

119896119898

(15)


Time (s)

minus60

minus40

minus20

0

20

40

60

80

Curr

ent (

mA

)

0 50 100 150 200 250 300 350

Friction model

PID u

Current I




5 Conclusions





Disclosure


Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120596

c

TTs

Coulomb

Viscous

Stribeck




119879119908= 119869119908 + 119887120596 + sgn (120596) [119888 + 119889119890minus120596

21205962

119904 ] (1)









119879119908= 119896119898119868 (2)


119908= 15 times 10


x = (120596119896119898

119869119908

119887

119869119908

119888

119869119908

119879119904

119869119908

)

119879

= (11990911199092119909311990941199095)119879

(3)



119904 As a



1= 1199092119868 minus 11990931199091minus sgn (119909

1) [1199094+ (1199095minus 1199094) 119890minus1199092

11205962

119904 ] (4)




1205971

1205971199091

= minus1199093+ 2

1199091

1205962119904

sgn (1199091) (1199095minus 1199094) 119890minus1199092

11205962

119904

1205971

1205971199092

= 119868

1205971

1205971199093

= minus1199091

1205971

1205971199094

= minus sgn (1199091) (1 minus 119890

minus1199092

11205962

119904 )

1205971

1205971199095

= minus sgn (1199091) 119890minus1199092

11205962

119904

(5)






0 50 100 150 200 250 300minus400

minus300

minus200

minus100

0

100

200

300

400

500

Time (s)


(a)

0 50 100 150 200 250 300minus150

minus100

minus50

0

50

100

150

200

250

Time (s)


(b)



119869 =1003817100381710038171003817y minusHx1003817100381710038171003817

2

Rminus1 +1003817100381710038171003817x119900 minus x1003817100381710038171003817

2

Pminus1119900

(6)


119900




119869 =

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[

[

(Pminus1119900)12

x119900

(Rminus1)12

y]

]

minus [

[

(Pminus1119900)12

(Rminus1)12

H]

]

x100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

(7)


119869 =

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

T[[

(Pminus1119900)12

x119900

(Rminus1)12

y]

]

minus T[[

(Pminus1119900)12

(Rminus1)12

H]

]

x100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

[y1

y2

] minus [H1

0 ] x10038171003817100381710038171003817100381710038171003817

2

=1003817100381710038171003817y1 minusH

1x10038171003817100381710038172

+1003817100381710038171003817y21003817100381710038171003817

2

(8)




119869min =1003817100381710038171003817y21003817100381710038171003817

2 if 1003817100381710038171003817y1 minusH1x1003817100381710038171003817 = 0 (9)



of

y1= H1x (10)


119900are now









x119900= (0 18 000344 05863 0)

119879 (11)


2 1199093 1199094 and 119909

5 The profile




11989800270 00228


483 times 10minus6





0 50 100 150 200 250 300minus200

minus100

0

100

200

300

400

500

Time (s)

Ang

ular

velo

city

(rpm

)




x119900= (0 15205 000322 05863 06037)

119879

(12)








0 50 100 150 200 250 300

300

Time (s)

Ang

ular

velo

city

(rpm

)


minus200

minus150

minus100

minus50

0

50

100

150

200

250






119889

119869Ω = 119879119908+ 119879119889 (13)


0 50 100 150 200 250 300minus400

minus300

minus200

minus100

0

100

200

300

400

Time (s)

Resid

uals

(rpm

)


(a)

0 50 100 150 200 250 300minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Time (s)

Resid

uals

(rpm

)


(b)


0 100 200 300 400 500Time (s)

minus1000

minus500

0

500

1000

1500

2000

Ang

le er

ror (

mill

ideg

rees

)





0 100 200 300 400 500Time (s)

minus60

minus40

minus20

0

20

40

60

80

100

Curr

ent (

mA

)





r+ +

+e u

120596

120596PID Reaction

wheel

1

s

120579

1

Js+

+

TdDMC

TwI

minus


0 50 100 150 200 250 300 350Time (s)

minus200

minus100

0

100

200

300

400

500

600

Ang

le er

ror (

mill

ideg

rees

)



119868 = 119906 +119887

119896119898

120596 +sgn (120596)119896119898

[119888 + 119889119890minus12059621205962

119904 ] (14)


119868 = 119906 +(119888 + 119889) sgn (119906)

119896119898

(15)


Time (s)

minus60

minus40

minus20

0

20

40

60

80

Curr

ent (

mA

)

0 50 100 150 200 250 300 350

Friction model

PID u

Current I




5 Conclusions





Disclosure


Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 300minus400

minus300

minus200

minus100

0

100

200

300

400

500

Time (s)


(a)

0 50 100 150 200 250 300minus150

minus100

minus50

0

50

100

150

200

250

Time (s)


(b)



119869 =1003817100381710038171003817y minusHx1003817100381710038171003817

2

Rminus1 +1003817100381710038171003817x119900 minus x1003817100381710038171003817

2

Pminus1119900

(6)


119900




119869 =

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[

[

(Pminus1119900)12

x119900

(Rminus1)12

y]

]

minus [

[

(Pminus1119900)12

(Rminus1)12

H]

]

x100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

(7)


119869 =

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

T[[

(Pminus1119900)12

x119900

(Rminus1)12

y]

]

minus T[[

(Pminus1119900)12

(Rminus1)12

H]

]

x100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

[y1

y2

] minus [H1

0 ] x10038171003817100381710038171003817100381710038171003817

2

=1003817100381710038171003817y1 minusH

1x10038171003817100381710038172

+1003817100381710038171003817y21003817100381710038171003817

2

(8)




119869min =1003817100381710038171003817y21003817100381710038171003817

2 if 1003817100381710038171003817y1 minusH1x1003817100381710038171003817 = 0 (9)



of

y1= H1x (10)


119900are now









x119900= (0 18 000344 05863 0)

119879 (11)


2 1199093 1199094 and 119909

5 The profile




11989800270 00228


483 times 10minus6





0 50 100 150 200 250 300minus200

minus100

0

100

200

300

400

500

Time (s)

Ang

ular

velo

city

(rpm

)




x119900= (0 15205 000322 05863 06037)

119879

(12)








0 50 100 150 200 250 300

300

Time (s)

Ang

ular

velo

city

(rpm

)


minus200

minus150

minus100

minus50

0

50

100

150

200

250






119889

119869Ω = 119879119908+ 119879119889 (13)


0 50 100 150 200 250 300minus400

minus300

minus200

minus100

0

100

200

300

400

Time (s)

Resid

uals

(rpm

)


(a)

0 50 100 150 200 250 300minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Time (s)

Resid

uals

(rpm

)


(b)


0 100 200 300 400 500Time (s)

minus1000

minus500

0

500

1000

1500

2000

Ang

le er

ror (

mill

ideg

rees

)





0 100 200 300 400 500Time (s)

minus60

minus40

minus20

0

20

40

60

80

100

Curr

ent (

mA

)





r+ +

+e u

120596

120596PID Reaction

wheel

1

s

120579

1

Js+

+

TdDMC

TwI

minus


0 50 100 150 200 250 300 350Time (s)

minus200

minus100

0

100

200

300

400

500

600

Ang

le er

ror (

mill

ideg

rees

)



119868 = 119906 +119887

119896119898

120596 +sgn (120596)119896119898

[119888 + 119889119890minus12059621205962

119904 ] (14)


119868 = 119906 +(119888 + 119889) sgn (119906)

119896119898

(15)


Time (s)

minus60

minus40

minus20

0

20

40

60

80

Curr

ent (

mA

)

0 50 100 150 200 250 300 350

Friction model

PID u

Current I




5 Conclusions





Disclosure


Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












11989800270 00228


483 times 10minus6





0 50 100 150 200 250 300minus200

minus100

0

100

200

300

400

500

Time (s)

Ang

ular

velo

city

(rpm

)




x119900= (0 15205 000322 05863 06037)

119879

(12)








0 50 100 150 200 250 300

300

Time (s)

Ang

ular

velo

city

(rpm

)


minus200

minus150

minus100

minus50

0

50

100

150

200

250






119889

119869Ω = 119879119908+ 119879119889 (13)


0 50 100 150 200 250 300minus400

minus300

minus200

minus100

0

100

200

300

400

Time (s)

Resid

uals

(rpm

)


(a)

0 50 100 150 200 250 300minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Time (s)

Resid

uals

(rpm

)


(b)


0 100 200 300 400 500Time (s)

minus1000

minus500

0

500

1000

1500

2000

Ang

le er

ror (

mill

ideg

rees

)





0 100 200 300 400 500Time (s)

minus60

minus40

minus20

0

20

40

60

80

100

Curr

ent (

mA

)





r+ +

+e u

120596

120596PID Reaction

wheel

1

s

120579

1

Js+

+

TdDMC

TwI

minus


0 50 100 150 200 250 300 350Time (s)

minus200

minus100

0

100

200

300

400

500

600

Ang

le er

ror (

mill

ideg

rees

)



119868 = 119906 +119887

119896119898

120596 +sgn (120596)119896119898

[119888 + 119889119890minus12059621205962

119904 ] (14)


119868 = 119906 +(119888 + 119889) sgn (119906)

119896119898

(15)


Time (s)

minus60

minus40

minus20

0

20

40

60

80

Curr

ent (

mA

)

0 50 100 150 200 250 300 350

Friction model

PID u

Current I




5 Conclusions





Disclosure


Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 300minus400

minus300

minus200

minus100

0

100

200

300

400

Time (s)

Resid

uals

(rpm

)


(a)

0 50 100 150 200 250 300minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Time (s)

Resid

uals

(rpm

)


(b)


0 100 200 300 400 500Time (s)

minus1000

minus500

0

500

1000

1500

2000

Ang

le er

ror (

mill

ideg

rees

)





0 100 200 300 400 500Time (s)

minus60

minus40

minus20

0

20

40

60

80

100

Curr

ent (

mA

)





r+ +

+e u

120596

120596PID Reaction

wheel

1

s

120579

1

Js+

+

TdDMC

TwI

minus


0 50 100 150 200 250 300 350Time (s)

minus200

minus100

0

100

200

300

400

500

600

Ang

le er

ror (

mill

ideg

rees

)



119868 = 119906 +119887

119896119898

120596 +sgn (120596)119896119898

[119888 + 119889119890minus12059621205962

119904 ] (14)


119868 = 119906 +(119888 + 119889) sgn (119906)

119896119898

(15)


Time (s)

minus60

minus40

minus20

0

20

40

60

80

Curr

ent (

mA

)

0 50 100 150 200 250 300 350

Friction model

PID u

Current I




5 Conclusions





Disclosure


Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










r+ +

+e u

120596

120596PID Reaction

wheel

1

s

120579

1

Js+

+

TdDMC

TwI

minus


0 50 100 150 200 250 300 350Time (s)

minus200

minus100

0

100

200

300

400

500

600

Ang

le er

ror (

mill

ideg

rees

)



119868 = 119906 +119887

119896119898

120596 +sgn (120596)119896119898

[119888 + 119889119890minus12059621205962

119904 ] (14)


119868 = 119906 +(119888 + 119889) sgn (119906)

119896119898

(15)


Time (s)

minus60

minus40

minus20

0

20

40

60

80

Curr

ent (

mA

)

0 50 100 150 200 250 300 350

Friction model

PID u

Current I




5 Conclusions





Disclosure


Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Disclosure


Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article estimating friction parameters in...

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