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Sensorless Torque Estimation using Adaptive Kalman Filter andDisturbance Estimator

Sang-Chul Lee, Student Member, IEEE, Hyo-Sung Ahn, Member, IEEE

Abstract— This paper presents a stochastic estimationmethod and a signal processing based method for estimatingdisturbance torques without using any force sensors. The firstmethod will address a robustness against measurement noisesby estimating noise covariance. The second method will showseveral practical merits. By containing system models insideof the estimator, the total disturbance torque injected into theplant is estimated. The experimental results conducted usinga master-slave manipulator show the validity of two proposedmethods.

I. INTRODUCTION

NOWADAYS as science advances, the works that arehard to be executed by human hands and need a

minute control are increasing. Teleoperation and telesurgeryare representative examples of such works. In the field oftelesurgery, improving transparency and haptic feedback isa very important issue. To achieve them and to be able toimprove in actual situations, proper force control and forcecoordination between the human operator and the reactionforce from the environment are necessary. To realize a preciseposition control and force feedback, Kodak Tadano et alsuggested a master-slave system that uses force sensor andpneumatic cylinders [1], and A. Wroblewska summarizeddemands of force feedback systems and illustrated surgicaltools [2]. In these ways, reflected force can be easily mea-sured, because they are based on force sensors. However,there are several drawbacks related to space, cost, frequencybandwidth and infection. To overcome disadvantages ofmethods using force sensors, several methods have beenproposed for force estimation. One of the popular solutionsis the state feedback observer [3]. By including disturbanceas a state variable, the state feedback observer is designedbased on error dynamics. Another solution is the disturbanceobserver [4]–[8]. By using a part of inverse transfer function,the disturbance torque is estimated. Another famous solutionis the Kalman filter. We can find many reports on states orparameters estimation using the Kalman filter [9]–[13].

The estimation methods mentioned above can be usedto estimate internal system parameters and exogenous dis-turbance. However, there exist some remarkable followingissues. (i) many estimation researches assumed that mea-surement data are accurate at every processing time. i.e.,there is no measurement noise. In practice, disturbancecompensation under the noisy measurement is necessary for

Authors are with Distributed Control and Autonomous Systems Lab-oratory, Department of mechatronics, Gwangju Institute of Science andTechnology (GIST), Korea, E-mail: [email protected]

more accurate control. (ii) The state feedback observer needsanalytic feedback gain calculation. (iii) Disturbance observeris designed based on the assumption that the disturbance isinjected into the rotor as a torque. Thus if there exist othertypes of disturbances, the estimated torque may be differentfrom the actual torque that we want to estimate. (iv) Theestimation based on Kalman filter takes a long operationtime; so reducing operation time is desirable.

In order to overcome the problems mentioned above , thispaper proposes two solutions. The first estimation methodis a stochastic approach which uses adaptive Kalman Filter(AKF). The discrete Kalman filter (DKF) needs a covariancematrix of the measurement vector properly selected in ad-vance. However, the AKF based estimation method updatesthe measurement covariance matrix at each processing time.The second estimation method is the signal processing baseddisturbance estimator. The purposes of the disturbance esti-mator are to estimate the torques asymptotically stably andto ensure a fast response. This paper is organized as follows.In section II, basic estimation approaches are described.In section III the adaptive Kalman filter based estimationmethod is proposed. The disturbance estimator is proposedIn section IV. Experimental results are shown in section V,and finally, in section VI of this paper the conclusion is made.

II. BASIC CONCEPT OF DISTURBANCE ESTIMATION

In the bilateral teleoperation, the transparency is one ofthe desired goals. When the transparency has achieved, anoperator feels the massless and infinitely stiff operation, andexperiences the immediate sensation of manipulating theremote environment. To achieve the transparency, many re-searchers proposed varied types of teleoperation architectures[15]. Fig.1 depicts the structure of the experimental systemwhich used in experimental test, and it is one of the bilateralteleoperation systems. The structure has bilateral symmetrywith an opposite device. In order to achieve the transparencyand feel the sensation from the remote site, teleoperationarchitectures need properly measured force information. Upto date, however, many teleoperation systems have beenusing force sensors for haptic(force) feedback and control. Toovercome the drawbacks of the force sensors, two methodsproposed in this paper estimate the disturbance torque byusing an input and an output signal of the motor installedon the each joint of the manipulator. As a result of theestimation, the estimated disturbance torque information, theangular position, and(or) the angular velocity of each jointwill be transmitted to the opposite side without any force

Fig. 1. Structure of the teleoperation system

sensors (see Fig.1). Thus, control designers will be able toapply many teleoperation architectures and their own controlstrategies with the haptic(force) feedback.

III. ADAPTIVE KALMAN FILTER BASEDDISTURBANCE ESTIMATION

A. System model

In this section, we treat the disturbance estimation usingnoisy measurement. A linear stochastic model of a DC motorused for the stochastic approach is represented as follows:

x(t) = Fx(t) +Bu(t) (1)y(t) = Hx(t) + v(t) (2)

where the matrix F ∈ Rn×n represents the system matrixand the matrix B ∈ Rp×p represents the control input matrix.The matrix H ∈ Rm×n is the output matrix, and v ∈ Rm isthe measurement noise vector. By using an equivalent blockdiagram of the DC motor as depicted in Fig.2, the linearstochastic model is obtained as follows:

x1 = x2

x2 = −B

Jx2 +

K

JI − 1

JDin (3)

y = x1 + v

where B,K, J, I, and Din denote viscous coefficient, torqueconstant, moment of inertia, control signal(armature current),and the disturbance torque, respectively. Noting that x =[θ ω

]T , we obtain the vector matrix form of the linearstochastic model.

[θω

]=

[0 10 −B

J

] [θω

]+

[0KJ

]I +

[0

− 1J

]Din

(4)

1

JK

inD

I

motor

w

1

s

B

J

1

s q

Fig. 2. Block diagram of a DC motor

where θ, and ω represent angular position(x1), and angularvelocity(x2), respectively. Equation (4) shows the disturbancetorque(Din) is acting as a second input. We assume thedisturbance torque(Din) is independent to the state variablesand unbounded. Based on the additional assumptions thatthe characteristic of the disturbance is nearly constant andsampling speed is fast enough, the disturbance is includedin the system model as a state variable. By integrating thedisturbance into the system model(4), and discretization, theextended model obtained as follows : θ(k + 1)

ω(k + 1)Din(k + 1)

=

I + Ts

0 1 00 −B

J − 1J

0 0 0

θ(k)ω(k)Din(k)

+ TS

0KJ0

I(k)

(5)

y(k) =[1 0 0

] θ(k)ω(k)Din(k)

+ v(k) (6)

where TS represent the sampling time. We assumethe measurement noise(v(k)) is the zero mean gaussiannoise(v(k)˜N(0, Rk)). The following AKF based distur-bance observer estimates Din by the extended model (5-6).

B. AKF based disturbance observer

In the extended system model (5-6), as the pair of thesystem matrix and the output matrix is observable, fullorder estimation is guaranteed by the DKF. Even the DKFguarantees the full order estimation, in order to yield the bestresult, the DKF needs an accurately selected measurementcovariance matrix in advance. However, obtaining theaccurate covariance matrix is not easy. Moreover, themeasurement covariance matrix can be changed with time.To overcome the problems mentioned above, this papersuggests the AKF based disturbance observer. For an LTIsystem, the recursive algorithm of the AKF is described asfollows [14], [16]:

Time update1 ) Project the state ahead

xk|k+1 = Fk−1xk−1|k−1 +Bkuk (7)

2 ) Project the error covariance ahead

Pk|k+1 = KkPk−1|k−1FkT +Q∗

k (8)

Measurement update1) Kalman gain

Kk = Pk|k−1HTk (HkPk|k−1H

Tk + R∗

k)−1 (9)

2) Update estimate with measurement Zk

xk|k = xk|k−1 +Kk(zk −Hkxk|k−1) (10)

3) Update the error covariance

Pk|k = (I −KkHk)Pk|k−1 (11)

where F ∈ Rn×n, B ∈ Rn×n, K ∈ Rn×m, and P ∈ Rn×n

represent the system matrix, the input matrix,the kalmangain, and the covariance state matrix, respectively. H ∈Rm×n, Q ∈ Rn, and zk ∈ Rm denote the output vector,the covariance vector of system noise, and measurement,respectively. The AKF based disturbance observer is differentto the DKF in terms of an adaptive covariance matrixR∗

k ∈ Rm×m in (9). The covariance matrix is updated bya covariance uncertainties online estimator represented asfollows [16]:

R∗(ki) =1

N

i∑j=i−N+1

{[zj −H(kj)x(kj)][zj −H(kj)x(kj)]T

+H(kj)P (kj)HT (kj)}

(12)

The first term in the bracket represents a squared errormatrix between the measurements and the estimated states,and diagonal elements of the second term are the covariancesof state variables. In the first term, the noisy(clear) measure-ment makes the big(small) squared error term. Thus, R∗

k isincreased(decreased), and it decreases(increases) the Kalmangain Kk. Consequently, the dependence of the estimation

result between previously estimated states and current mea-surement will be affected by the quality of measurement. Aswe can see in (12), the covariance uncertainties estimatorneeds N number of recent measurements. By an average ofthe recent N number of terms in the bracket, the AKF baseddisturbance observer updates the measurement covariancematrix R∗

k at each processing time. In addition, large numberof the measurement will offer a more accurate measurementcovariance matrix(R∗

k) estimation.

IV. DISTURBANCE ESTIMATOR

A. System model

This section proposes a signal processing based estimationmethod. For the estimation, the DC motor model is used (seeFig.2). In Fig.3, the upper part describes the system model,and the middle and lower part represent the disturbanceestimator. In the system model, an input disturbance Dinput

and an output disturbance Doutput are described. The inputdisturbance Dinput can be any kind of disturbance, e.g., loadforce(torque), gravitational torque, parametric fluctuation orany combinations of them. In order to validate the totaldisturbance estimation, Doutput is injected. We assume thedisturbances are independent to the system and unbounded.Subscript n denotes the nominal value of parameters.

B. Disturbance estimator

The disturbance estimator consists of an extraction partand an estimation part. The two parts are cascade-connected.As a first step, the extraction part calculates the systemoutput generated by the input and output disturbances. Afterthat, the estimation part generates the final estimation result.Each process is described as follows:

1) Extraction part:The composition of the extraction part is same to the

system model, but it does not have any disturbances. Thecomputation of the extraction part is described as follows:

θD =

(− K

s(Js+B)I +

1

s(Js+B)Dinput +

1

sDoutput

)+

(Kn

s(Jns+Bn)I

)(13)

The term in the upper bracket represents the actualposition result(θI,D) generated by the control input(I)and both the input disturbance(Dinput) and the outputdisturbance(Doutput). The term in the lower bracketshows the position result((θI ) generated by the controlinput(I) only. Due to the subtraction, where the parametersK,J, andB are same with Kn, Jn, andBn, the output ofthe extraction part(θD) is the angular position generatedby the disturbances only. Otherwise, the output containscontain the angular position generated by the parametricfluctuation(uncertainty) also. It implies all the dynamicsnot considered in extraction part are considered as the

1

JK

inputD

I

motor

,I Dw

1

s

B

J

1

s

,I Dq

outputD

1

nJnK

Iw

1

s

n

n

B

J

1

sIq

1

nJPID

*w

1

s

n

n

B

J

1

s

Dq

ˆtotalu D=

*

Dq

Extraction

Estimation

Disturbance Estimator

T

u

Fig. 3. Block diagram of motor with disturbance

disturbance.

2) Estimation part:The estimation part calculates the final estimation result

by using a control loop. The reference signal of the controlloop is output of the extraction part(θD). Except the torquecoefficient(Kn) of the extraction part and a PID controllerof the estimation part, the open-loop transfer functions areexactly same. If the estimation part is stable, the feedbacksignal(θ∗D) follows the reference(θD), and the output ofthe PID controller approaches the total disturbance. Conse-quently, designing the disturbance estimator is concluded bymaking the estimation part asymptotically stable. In order toobtain a stable region of the PID gains, state space equationsis used. Except the PID controller, the open-loop dynamicsof the estimation part is described as follows:

x1 = x2

x2 = −Bn

Jnx2 +

1

Jnu (14)

y = x1

where x =[θ∗D ω∗ ]T

. In order to determine errorequations, let e = θD − θ∗D = θD − x1 = e1. Then errorequations obtained as follows:

e1 = −x1 = e2

e2 =Bn

Jnx2 −

1

Jnu (15)

To determine the control signal(u) in (14), the PID control

law is applied:

u = KP e+KI

∫edτ +KD

de

dt(16)

The PID control law(15) together with e0 = e1 and errorequations (14), error dynamics described as follows:

e0 = e1

e1 = e2 (17)

e2 = −(KI

Jn

)e0 −

(KP

Jn

)e1 −

(Bn

Jn+KD

)e2

If the characteristic matrix of the error dynamics is Hurwitz,error approaches zero asymptotically. It indicates θ∗D and ualso asymptotically approach θD and the total disturbanceDtotal, respectively. In order to calculate stable region ofthe PID gains, Routh−Hurwitz theorem is used. Calculatedregion of the PID gains are written as follows :

KI

Jn> 0 ,

KP

Jn> 0 ,

Bn

Jn+KD > 0 (18)

As we mentioned above, the estimated total disturbancecontains not only exogenous disturbances(Dinput andDoutput) but also all the discordance between the realsystem model and the model of the extraction part. Inaddition, simply removing all the viscous friction loops inFig.3, the viscous friction also estimated as the disturbance.It shows the flexibility of the disturbance estimator.

V. EXPERIMENTAL RESULTS

To validate proposed two estimation methods, experimen-tal tests were conducted using a master-slave system. Theoverall setup is shown in Fig.4, and the specifications ofmotor drivers and motors are shown in Tables I and II. Forthe tests, the slave manipulator was fixed, and the operatormanipulated the master to give an order to the slave device.

A. AKF based disturbance observer

The experimental test of the AKF based estimationmethod is conducted by two ways. At the first, we confirmedthe improvement of the AKF based method in comparisonwith the DKF. In order to obtain noisy measurement, samezero mean gaussian noise having 0.5 of the standard deviationis added to the measurements of both the AFK and DKF.In addition, wrong measurement covariance (Rk = 0.22)is given to the DKF. Due to the manipulator is fixed, themean of the measurement is ideally zero. Fig.5 shows themeasurements, i.e., control signal(I) of the slave motor,and the position discordance between the master and slavemanipulators. The AKF based observer also tested under thesame condition. We set the wrong measurement covarianceas (0.22), and we set the accumulation number(N ) in (12)as 5. The estimation results of the DKF and AKF basedobserver are shown in Fig.6. Due to the DKF dose nothave the measurement covariance adaptation algorithm, the

Fig. 4. Experimental setup : master-slave manipulator

TABLE ISPECIFICATION OF MOTOR DRIVERS

Maxon - ADS 50/5Output voltage VCC min. 12 VDC; max. 50VDCOutput current Depending on load, continuous 5A

TABLE IISPECIFICATION OF MOTORS

Maxon - RE30Assigned power rating(W) 60

Max. continuous current(A) 4Torque constant(mNm / A) 25.9

Max. continuous torque(mNm) 86.2

estimation is based on the given wrong measurement covari-ance matrix. As the Fig.6(a) demonstrates, the estimationresult of the DKF containing a lot of noise. If it used forthe haptic feedback and the position control, it will makehuge vibration on the manipulator. On the other hand, eventhe given initial measurement covariance matrix of the AKFwas mismatched to the actual noise, the AKF calculatedrelatively clear estimation result(see Fig.6(b)) based on thecovariance uncertainties online estimator(12). Such resultdemonstrates the AKF based disturbance observer correctsthe measurement covariance matrix at each processing time.Moreover, such result indicates possibility of the reliableestimation under the noisy circumstance. As the second test,we applied huge number(Rk = 12) of the standard deviationto the measurement noise. Fig.7(a) shows the estimationresult when the number of accumulation(N ) is 5. On theother hand, when we used 30 number of measurement data,as shown in Fig.7(b) significantly improved estimation resultis obtained. As depicted above, the disturbance estimationusing noisy measurement is achieved.

B. Disturbance Estimator

This subsection validate the disturbance estimator. Forthe test, same bilateral teleoperation system is used. Two

Fig. 5. Measured armature current and Position difference(DKF)

Fig. 6. Disturbance estimation. (a) DKF, (b) AKF (N=5).

Fig. 7. Disturbance estimation using different number of measurementaccumulation (AKF), standard deviation = 1

measurements of the disturbance estimator, the controlsignal and the angular position of the slave manipulatorare depicted in Fig.8. From the two measurements,extraction part calculates the position(θD) by using (13),and then estimation part calculates the total disturbance. Inthe experimental test, PID gains KP = 0.135, KI = 0.01,KD = 0.008 are used to make estimation part asymptoticallystable. We can find that a set of PID gains satisfies condition(17) in the continuous time domain. Of course PID gainscan be selected by trial and error also. In Fig.9(a), theoutput of the extraction part(θD : reference of the estimationpart) is denoted as reference, and the feedback signal ofthe estimation part(θ∗D) is represented as tracking. Trackingperformance can be used as an estimation performance indexin real time. The total estimation result is shown in Fig.9(b).In the test, the angular position is used for the measurementof the disturbance estimator, however, even the designeruses the angular velocity as the measurement but notposition, disturbance estimator will precisely estimate thetotal disturbance just by eliminating the last one integrator ateach extraction and estimation part. It shows the flexibility

Fig. 8. Measurement (Disturbance Estimator)

Fig. 9. Estimation result of the disturbance estimator

of the disturbance estimator again.

VI. CONCLUSIONS

This paper proposed a stochastic estimation method anda signal processing based method for the purpose of dis-turbance torque estimation without force sensors. The AKFbased method presented robustness against the measurementnoise. When the measurements have a lot of noise, and itscharacteristic is unknown, this method can be one of thereliable methods.

Through the section IV and V, we proposed several meritsof the disturbance estimator, and they can be summarizedas follows: (i) Wherever the disturbance injected betweentwo measurements(I and θI,D), the disturbance estimatorestimates the total disturbance as asymptotically stable. (ii)The disturbance estimator inherits the structure of the systemmodel. Thus it provide the instinctive and direct application.(iii) Unrestricted selection of the measurement type is possi-ble. (iv) We can determine the types of disturbance includedin the total disturbance by considering sub-model of thedisturbance estimator. In addition to the merits mentionedabove, when the disturbance estimator used for feedbackcompensation, the disturbance estimator will try to makethe system act as same with the sub-model. Because allthe effects not considered in the sub-model is estimatedas the disturbance. In consequence, direct and fast loopshaping could be achieved. Feedback control, loop shapingand extending to the nonlinear disturbance estimator are ourfuture works. In this paper, a DC motor model is usedfor total disturbance estimation. However, the disturbanceestimator can be used in countless system simply having the

plant model as its sub-model. In addition, we used a PIDcontroller for the disturbance estimation. However, any kindof controller and control technique can be used which canmake the estimator stable.

ACKNOWLEDGEMENT

This research was supported by the institute of Medi-cal System Engineering(iMSE) in the GIST, and by theMKE(The Ministry of Knowledge Economy), Korea, underthe ITRC(Information Technology Research Center) supportprogram supervised by the NIPA(National IT Industry Pro-motion Agency) (NIPA-2010-C1090-1031-0006)

REFERENCES

[1] Tadano K and Kawashima K, Development of a Master Slave Systemwith Force Sensing Using Pneumatic Servo System for LaparoscopicSurgery, IEEE Int. Con. on Robotics and Automation, pp. 947 - 952,2007.

[2] A. Wroblewska, Methods of the force measurement for robotic surgicaltools, Proc. the Fifth International Workshop on, Robot Motion andControl, pp. 51 - 54, 2005.

[3] Giuseppe S. Buja, Disturbance torque estimation in a sensorless DCdrive, Proc. IECON. Int. Conference on Industrial Electronics, Control,and Instrumentation., vol.2, pp. 977 - 982, 1993.

[4] Satoshi Komada, Muneaki Ishidar, Kouhei Ohnishi, and TakamasaHori, Disturbance Observer Based Motion Control of Direct DriveMotors, IEEE Transaction on Energy Conversion, vol.6, Issue.3, pp.553 - 559, 1991.

[5] Kenji Kaneko, Kiyoshi Komoria, Kouhei Ohnishi and Kazuo Tanie,Manipulator Control based on a Disturbance Observer in the Opera-tional Space, Proc. IEEE Int. Conf. on Robotics and Automation, 1994,vol.2, pp. 902 - 909, 1994.

[6] Liu C.S. and Peng H, Disturbance observer based tracking control,Journal of Dynamic Systems, Measurement, and Control, vol.122,Issue.2, pp. 332 - 335, 2000.

[7] Satoshi Komadar, Muneaki Ishidar, Kouhei Ohnishiir and TakamasaHorii, Motion control of linear synchronous motors based on distur-bance observer, 16th Annual Conference of IEEE Industrial ElectronicsSociety, 1990. IECON ’90., vol.1, pp. 154 - 159 , 1990.

[8] Edi Leksono, Toshiyuki Murakami and Kouhei Ohnishi, ObserverBased Robust Force Control in cooperative Motion Systems, Proc. the24th Annual Conf. of the IEEE Industrial Electronics Society, IECON’98., vol.3, pp. 1801 - 1806, 1998.

[9] P.P.AcarnIey, J.K.AI-Tayie, Estimation of speed and armature temper-ature in a brushed DC drive using the extended Kalman filter, Proc.IEE Electric Power Applications, vol.144, Issue.1, pp. 13 - 20, 1997.

[10] Zhengyun Ran, Huade Li, and Shujin Chen, Application of OptimizedEKF in Direct Torque Control System of induction motor, Int. Con. onInnovative Computing, Information and Control, ICICIC ’06., vol.1,pp. 331 - 335, 2006.

[11] Murat Barut, Seta Bogosyan and Metin Gokasan, EKF Based Sensor-less Direct Torque Control of IMs in the Low Speed Range, Proc. theIEEE International Symposium on Industrial Electronics, ISIE 2005.,vol.3, pp. 969 - 974, 2005.

[12] Yong Liu, Zi Qiang Zhu and David Howe, Instantaneous Torque Esti-mation in Sensorless Direct-Torque-Controlled Brushless DC Motors,IEEE Transactions on Industry Applications., vol.42, Issue.5, pp. 1275- 1283, 2006.

[13] Jing-Ping Jiang, Shen Chen and Pradip K.Sinha, Optimal FeedbackControl of Direct-Current Motors, IEEE Transactions on IndustrialElectronics., vol.37, Issue.4, pp. 269 - 274, 1990.

[14] Jung-Han Kim and Jun-Ho Oh, Disturbance Estimation using SlidingMode for Discrete Kalman Filter, Proc. the 37th IEEE Conference onDecision and Control., vol.2, pp. 1918 - 1919. 1998.

[15] Edvard Naerum and Blake Hannaford, Global Transparency Analysisof the Lawrence Teleoperator Architecture,

[16] Richard Bellman, Stochastic models, estimation. and control Vol.2,1994.