IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 10, NO. 4, AUGUST 2005 391

Fault-Tolerant Control System of Flexible Arm forSensor Fault by Using Reaction Force Observer

Yu Izumikawa, Kazuhiro Yubai, Member, IEEE, and Junji Hirai, Member, IEEE

Abstract—In recent years, control system reliability has re-ceived much attention with increase of situations where computer-controlled systems such as robot control systems are used. In orderto improve reliability, control systems need to have abilities todetect a fault (fault detection) and to maintain the stability andthe control performance (fault tolerance). In this paper, we ad-dress the vibration suppression control of a one-link flexible armrobot. Vibration suppression is realized by an additional feedbackof a strain gauge sensor attached to the arm besides motor posi-tion. However, a sensor fault (e.g., disconnection) may degrade thecontrol performance and make the control system unstable at itsworst. In this paper, we propose a fault-tolerant control system forstrain gauge sensor fault. The proposed control system estimatesa strain gauge sensor signal based on the reaction force observerand detects the fault by monitoring the estimation error. Afterfault detection, the proposed control system exchanges the faultysensor signal for the estimated one and switches to a fault-modecontroller so as to maintain the stability and the control perfor-mance. We apply the proposed control system to the vibrationsuppression control system of a one-link flexible arm robot andconfirm the effectiveness of the proposed control system by someexperiments.

Index Terms—Fault-tolerance, flexible arm, sensor fault.

I. INTRODUCTION

IN RECENT years, control system reliability has receivedmuch attention with increase of situations where robots are

used. In order to improve reliability, control systems need tohave abilities to detect a fault (fault detection) and to maintainthe stability and the control performance (fault tolerance). Inmechanical systems, actuator faults and sensor faults to whichthis paper is dedicated are often concerned. Many studies havebeen done on fault-tolerant control system for sensor fault, andthey are mainly classified into two approaches:

1) design a controller which stabilizes both a normal and afaulty system by solving a robust stability or a simultane-ous stabilization problem [1], [2];

2) estimate a faulty sensor signal from normal sensor signals,and use the estimated signal instead of a faulty one [3].

The first approach tends to be conservative since a (single) con-troller is designed for large model uncertainty to guarantee astability of a normal and a faulty system. On the other hand, inthe second approach, which exploits sensor redundancy, a stabil-ity and a control performance can be maintained by exchanging

Manuscript received February 20, 2005; revised March 27, 2005. Recom-mended by Guest Editors K. Ohnishi, R. Oboe, and Y. Hori.

The authors are with the Department of Electrical and Electronic Engineering,Mie University, Tsu-shi, Mie-ken 514-8507, Japan (e-mail: izumi@ems.elec.mie-u.ac.jp; yubai@elec.mie-u.ac.jp; hirai@elec.mie-u.ac.jp).

Digital Object Identifier 10.1109/TMECH.2005.852442

a faulty sensor signal for an estimated one reconstructed frominput signals and normal sensor signals by utilizing interrela-tions of each sensor signal. However, from a practical point ofview, the second approach lacks estimation robustness againstplant deviations and disturbances. In this paper, we extend thesecond approach by implementing a sensor signal observer toovercome this problem.

This paper is concerned with the vibration suppression con-trol of a one-link flexible arm robot. In our control system,vibration suppression is realized by an additional feedback ofa strain gauge sensor attached to the arm besides motor angle.However, a sensor fault may degrade a control performance andmake the control system unstable at its worst. In our controlsystem, a strain gauge sensor is prone to fail in comparison withan encoder because it is attached directly to the arm. Then, wepropose a fault-tolerant control system for a disconnection faultof a strain gauge sensor. We focus on a reaction force observerand show that a strain gauge sensor signal can be estimatedby multiplying an appropriate gain to reaction force. Moreover,adaptation algorithm is applied to update of this gain to im-prove estimation accuracy and estimation robustness. A straingauge sensor fault is detected by monitoring an estimation errorbetween a sensor signal and an estimated value, and the con-trol system works with the estimated one after fault detection.However, the control system may be unstable after exchang-ing the sensor signal for the estimated one since the estimatedvalue includes estimation error and estimation delay, and theseeffects are not considered at the controller design. Therefore, afault-mode controller taking into account the estimation errorand the estimation delay should be designed and the controlsystems should switch from the normal-mode controller to thefault-mode controller after fault detection to maintain the stabil-ity and the control performance. Loop shaping design procedure(LSDP) is adapted as a controller synthesis to design both thenormal-mode and the fault-mode controller. Both controllers aredesigned according to the same controlled plant with differentweighting functions in consideration of the estimation error andthe estimation delay. As a result, it is possible to design the fault-mode controller easily in comparison with constructing anothersensor-less control system.

The rest of this paper is organized as follows. Section II de-scribes a mathematical model of a flexible arm robot and thevibration suppression control system. In Section III, a new fault-tolerant control system consisting of a sensor signal observer, afault detector, and switching controllers is proposed. The effec-tiveness of the proposed fault-tolerant control system is exam-ined through experiments in Section IV. Section V concludesthis paper.

1083-4435/$20.00 © 2005 IEEE

392 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 10, NO. 4, AUGUST 2005

Fig. 1. Configuration of the one-link flexible arm.

II. MATHEMATICAL MODEL OF FLEXIBLE ARM AND

VIBRATION SUPPRESSION CONTROL SYSTEM CONFIGURATION

Bernoulli-Euler beam theory and some simplifications givethe mathematical model from an input torque τ ref to motorangle θ and to distortion of the arm Sd written as (1) and (2),respectively. In these equations, higher order vibration modesmore than the second one are neglected for simplicity of thecontroller design.

Pθ =θ

τ ref=

1Jt

{1s2

+(φ

′(0))2

s2 + ω2

}(1)

PSd=

Sd

τ ref=

1Jt

{φ

′(0)φ

′′(x)

s2 + ω2

}(2)

s Laplace operator;x distance from the root of arm;Jt total moment of inertia;ω resonant frequency of arm;φ mode function;The strain gauge sensor is placed at the node of the second

order vibration mode (x = xs) so as to mainly detect the firstorder vibration mode. The configuration of the one-link flexiblearm is depicted in Fig. 1. Vibration suppression control andposition control are realized by feedbacking distortion of thearm Sd and motor angle θ through an appropriate stabilizingcontroller CN [4].

III. PROPOSED FAULT-TOLERANT CONTROL SYSTEM

Since the control system of the flexible arm has no alternatefor an encoder, the control system has to stop the operationimmediately after fault detection in an encoder. In the case ofa strain gauge sensor fault, it is possible to maintain a stabilityof the control system by interruption of a faulty sensor signalbecause a strain gauge sensor is an additional one to improve avibration suppression performance. However, this interruptionresults in poor vibration suppression. In the control system ofthe flexible arm, it is desirable to maintain not only a stability ofthe control system but also a vibration suppression performance

Fig. 2. Proposed fault-tolerant control system.

Fig. 3. Model of 2-mass resonant system.

after a strain gauge sensor fault. For this purpose, we propose anew fault-tolerant control system to

• estimate a strain gauge sensor signal;• detect a fault and exchange a faulty sensor signal for the

estimated one;• maintain a stability and a control performance after

switching from a faulty sensor signal to the estimatedone.

Fig. 2 illustrates a configuration of the proposed fault-tolerantcontrol system. The proposed system is constructed by addinga strain gauge sensor signal observer, a fault detector, and afault-mode controller CF to the (normal) control system of theflexible arm. The rest of this section describes the details aboutthese components.

A. Sensor Signal Observer

Since a sensor signal observer estimates a strain gauge sensorsignal from an input torque and a reaction force, we give a briefreview of a reaction force observer proposed in [5] first. Sec-ondly, we introduce a relationship between a reaction force anda strain gauge sensor signal, and show that a strain gauge sensorsignal can be constructed from a reaction force. Furthermore,an adaptive gain is incorporated into the sensor signal observerto improve estimation accuracy and robustfy estimation againstdisturbances and parameter variations.

1) Reaction Force Observer: A reaction force observer isproposed to estimate a reaction force τ reac acting on a motorin a 2-mass resonant system as shown in Fig. 3, and somesuccessful applications are reported (e.g., vibration suppressioncontrol of 2-mass resonant system [5] and sensor-less forcecontrol [6]). A flexible arm model (1) and (2) can be modeled as

IZUMIKAWA et al.: FAULT-TOLERANT CONTROL SYSTEM OF FLEXIBLE ARM FOR SENSOR FAULT BY USING REACTION FORCE OBSERVER 393

a 2-mass resonant system. In Fig. 3, the transfer function frominput torque τ ref to motor position θ is written as

θ

τ ref=

1Jm s2

{s2 + Ks/Jl

s2 + (Jm + Jl)Ks/Jm Jl

}. (3)

By comparing (3) with (1), a flexible arm can be modeled as a2-mass resonant system with an equivalent motor inertia Jm , anequivalent load inertia Jl , and an equivalent spring coefficientKs as defined from (4) to (6), respectively [7].

Jm =Jt

1 + φ′(0)2(4)

Jl =Jtφ

′(0)2

1 + φ′(0)2(5)

Ks =Jtφ

′(0)2ω2

(1 + φ′(0)2)2. (6)

Assume that a disturbance torque τdis entering at a motor inputis composed as (7), where D and τ fric are motor viscosity andfriction torque, respectively.

τdis = Dsθ + τ fric + τ reac. (7)

Another expression of τdis is obtained from Fig. 3 as follows:

τdis = τ ref − Jm s2θ. (8)

From (7) and (8), a reaction force τ reac is estimated by

τ reac = F (τ ref − Jm s2θ − Dsθ − τ fric) (9)

where F is the second order low-pass filter in order to avoid purederivation of θ and suppress the effect of the neglected higherorder vibration modes. τ fric is defined as follows:

τ fric =

τ c sgn(θ), if |θ| �= 0τ c sgn(τ ref), if θ = 0 and |τ ref | > τc

τ ref , if θ = 0 and |τ ref | ≤ τ c

(10)

where τ c denotes the identified coulomb friction torque. The cut-off frequency of F is determined as 200 [rad/s] in considerationof the fact that the first and the second order vibration modesexists around 25 [rad/s] and 300 [rad/s], respectively.

2) Adaptive Gain K: From Fig. 3, the transfer function fromτ ref to τ reac are written as

τ reac

τ ref=

Ks/Jm

s2 + (Jm + Jl)Ks/Jm Jl

=φ

′(0)2ω2/(1 + φ

′(0)2)

s2 + ω2. (11)

Comparing (2) with (11) yields the following relationship be-tween τ reac and Sd :

Sd = Kτ reac (12)

K =(1 + φ

′(0)2)φ

′′(x)

Jtφ′(0)ω2

. (13)

Equation (12) implies that Sd can be estimated from only avail-able signals, θ and τ ref . However, plant deviations, disturbances,and inevitable identified errors on plant parameters such as Jt

and φ may cause to poor estimation of Sd . In order to improve

Fig. 4. Sensor signal observer.

TABLE IPLANT PARAMETERS

estimation accuracy and robustfy estimation of Sd against plantdeviations, etc., an adaptive K is introduced in the sensor sig-nal observer. We adopt least squares estimation (see [8].) as theadaptation algorithm of K shown in (14) and (15) by regardingSd as a teaching signal.

K = Γτ reac(SdF − Sd) (14)

Γ = δΓ − (Γτ reac)2 (15)

where SdF is defined as FSd and δ is a forgetting factor. Ofcourse, gain adaptation is activated while there is no faultsince wrong information is obtained after a sensor fault oc-curs. A configuration of the sensor signal observer is shown inFig. 4.

3) Sensor Signal Estimation Results: We have performedthe strain gauge sensor signal estimation by the sensor signalobserver. The flexible arm is controlled by a stabilizing con-troller and plant parameters are listed in Table I. Superscripts+ and − represent clockwise and counter-clockwise, respec-tively. Fig. 5 depicts the sensor signal estimation result afterconvergence of K. Fig. 5 shows that the sensor signal observerestimates Sd well after convergence of K. Next, we performedthe sensor signal estimation with load change to confirm therobustness against the model deviation. The estimation result

394 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 10, NO. 4, AUGUST 2005

Fig. 5. Strain gauge sensor estimation result.

Fig. 6. Estimation result with load change.

is depicted in Fig. 6 together with the result with the fixed Kfor comparison. The fixed K is set to the value obtained in theprevious experiment after convergence. The sensor signal esti-mation with the fixed K results in poor estimation, especiallyafter 25.4 s, while the adaptive K works well to some extent.A comparison between the estimation results with the fixed Kand the adaptive K indicates the effectiveness of the adaptiveK. However, these estimation results include estimation errorand estimation delay caused by modeling error and the filter Fattached to the sensor signal observer. The control system maybe unstable after exchanging the sensor signal for the estimatedone. This is the reason why a fault-mode controller activatedafter fault detection should be designed to maintain the stabilityand the control performance.

B. Fault Detector

In order to switch the controller, the control system has todetect when the fault occurs. The residual signal, which is theindex of fault detection, should be nearly zero during fault-free,and take nonzero under faulty situation [9]. In this paper, weassume the faulty sensor signal deviates largely from the normalone at disconnection fault as stated in [10]. Since a complexfault detection scheme is not required under this assumption,

Fig. 7. Power spectrum density of Sd and Sd .

we adopt an estimation error between the sensor signal Sd andthe estimated value Sd as the residual signal r as shown in

r = Sd − Sd . (16)

The fault detector calculates r at each control period, and detectsthe fault as soon as the residual signal exceeds the prespecifiedthreshold rth.

C. Controller Design

Loop shaping design procedure (LSDP) is adopted as a con-troller synthesis to design the normal-mode controller CN andthe fault-mode controller CF . In this procedure, the open-loop characteristics of the controlled plant P = [Pθ , PSd

]T areshaped by a precompensator V and a postcompensator W . Astabilizing controller C∞ which guarantees robust stability mar-gin and the specified open-loop characteristics is designed forthe shaped plant G = WPV by solving two Riccati equations,see [11] for details. The resulting feedback controller for thecontrolled plant P is constructed as

C = V C∞W . (17)

We adopt LSDP from the following reasons.1) Loop-shaping by V and W needs no consideration on

phase characteristics.2) Since both input and output sensitivities can be considered

simultaneously, it is suitable for the controlled plant witha bad condition number such as the flexible arm.

3) Weighting functions are determined easily in comparisonwith H∞ control because LSDP is based on loop-shapingin open-loop as well as classical loop-shaping.

Especially, the third feature allows an intuitive selectionof weighting functions at designing CN and CF . While theweighting functions V and W = diag(W1,W2) for CN arespecified so as to satisfy a given control requirement withoutconsideration of any faults, CF must be designed in consider-ation of the estimation error Sd − Sd and the estimation delaycaused by F .

In order to specify V and W for CF , power spectrum densi-ties of Sd and Sd are examined. Fig. 7 illustrates power spectrum

IZUMIKAWA et al.: FAULT-TOLERANT CONTROL SYSTEM OF FLEXIBLE ARM FOR SENSOR FAULT BY USING REACTION FORCE OBSERVER 395

TABLE IIWEIGHTING FUNCTIONS V CW1 C AND W2

Fig. 8. Bode diagrams of CN and CF .

densities of Sd and Sd whose waveform is presented in Fig. 5.Power spectrum density of Sd is larger than that of Sd espe-cially in higher than 50 rad/s. This implies that estimation errorSd − Sd is significant in higher than 50 rad/s. Thus, only W2 ismodified so that the open loop gain of W2PSd

V has lower gainthan the fault-free scenario in higher than 50 [rad/s] at designingCF to maintain the stability of the control system, while V andW1 are identical to these for CN . Table II shows the weightingfunctions V and W for CN and CF determined based on theabove observation. Moreover, the low-pass filter F is added toW2 for CF in consideration of the estimation delay of Sd , andhence P is shaped by V and W

′= diag(W1, FW2) in design-

ing CF . However, the resulting feedback controller CF is con-structed according to (17) by using W = diag(W1,W2) sincethe sensor signal observer includes F . Bode diagrams of CN

and CF are shown in Fig. 8. CF has lower gain, especially fromSd to τ ref , than CN in high frequencies, while the bode diagramfrom θ to τ ref changes little as expected. Then the stability andthe tracking performance can be maintained by switching fromCN to CF after fault occurrence. Moreover, in order to avoidcontrol performance degradation caused by discontinuous τ ref

at controller switching from CN to CF , an initial state of CF

should be set appropriately so that an output of CF coincideswith that of CN at the controller switching [12].

Fig. 9. Residual signal r.

Fig. 10. Motor angle θ.

IV. EXPERIMENTAL RESULTS

In order to verify the effectiveness of the proposed fault-tolerant control system, we have simulated disconnection faultof the strain gauge sensor during control. In this experiment,disconnection fault occurs virtually at 10.35 [s], and the faultysensor indicates 1.0 [1/m], which is much larger than the fault-free scenario, after the disconnection fault occurrence [10]. Thethreshold rth is set to ±0.2 [1/m] for the disconnection fault.An initial value of adaptive gain K is set to the value obtainedin the experiment of sensor signal estimation in Section III.

The residual signal r depicted in Fig. 9 is nearly equal to zerowhen there is no fault. Once fault occurs, r exceeds rth, and afault detector indicates the occurrence of the strain gauge sensorfault immediately.

Figs. 10 and 11 show motor angle θ and the distortion of armSd , respectively together with no-switching case for compari-son. No-switching case means that only interruption of a faultysensor signal is performed. The tracking performance associated

396 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 10, NO. 4, AUGUST 2005

Fig. 11. Distortion of arm Sd .

with motor angle θ has little degradation. While the vibrationsuppression performance much degrades in no-switching case,the proposed case shows enough vibration suppression closeto the fault-free scenario. These results show that the proposedfault-tolerant control system maintains the stability and the over-all performance after the strain gauge sensor fault.

V. CONCLUSION

In this paper, we proposed a fault-tolerant control systemwith a sensor signal observer for a strain gauge sensor fault. Inthe proposed system, a strain gauge sensor fault is detected bymonitoring an estimation error between a sensor signal and anestimated signal, and the control system works with the esti-mated signal after fault detection. To construct the sensor signalobserver, we showed the relationship between the reaction forceand the strain gauge sensor signal and constructed the sensor sig-nal observer consisting of the reaction force observer and thegain K. Moreover, by introducing adaptive gain K, estimationaccuracy and estimation robustness against parameter variationsare improved. The stability and the control performance aremaintained by switching from the normal-mode controller tothe fault-mode controller that is designed taking into accountthe estimation error and the estimation delay. LSDP is adoptedto design the normal-mode and fault-mode controllers, and bothcontrollers are designed according to the same procedure, exceptfor specifying the different W2 in consideration of the estima-tion error and the estimation delay. As a result, it is possibleto design the fault-mode controller easily in comparison withconstructing another sensor-less control system instead of thefault-mode controller. The effectiveness of the proposed fault-tolerant control system was confirmed by some experiments.

REFERENCES

[1] H. Komatsu, T. Suzuki, S. Okuma, and Y. Yamaguchi, “Realizationof fault tolerant design control for the re-entry space vehicle using µ-synthesis,” in Proc. IEEJ Tech. Meeting Industrial Instrumentation andControl, vol. IIC-02-28, 2002, pp. 77–81, (in Japanese).

[2] Y. Hamada, S. Shin, and N. Sebe, “A design method for fault-tolerantmultivariable control systems—A condition for l-partial integrity,” SICETrans., vol. 34, no. 9, pp. 1184–1190, Sep. 1998.

[3] R. Isermann, R. Schwarz, and S. Stolzl, “Fault-tolerant drive-by-wiresystems,” IEEE Control Syst. Mag., vol. 22, no. 5, pp. 64–81, Oct. 2002.

[4] A. Inoue, S. Komada, and T. Hori, “Control of flexible arms by bendingmoment considering second mode vibration,” in Proc. Convention Record.IEEJ Industry Applications Soci., 1999, pp. 403–404, (in Japanese).

[5] K. Yuki, T. Murakami, and K. Ohnishi, “Vibration control of a 2mass resonant system by the resonance ratio control,” IEEJ Trans. Ind.Applicat., vol. 113-D, no. 10, pp. 1162–1169, Oct. 1993.

[6] K. Ohishi, M. Miyazaki, and M. Fujita, “Hybrid position and forcecontrol without force sensor,” RSJ J., vol. 11, no. 3, pp. 468–476, Apr.1993.

[7] S. Komada, K. Ohta, M. Ishida, and T. Hori, “Trajectory control offlexible arms using vibration suppression control by first order lag strainsignal feedback,” in Proc. 2nd Asian Control Conf., 1997, pp. 185–188.

[8] T. Suzuki, Adaptive Control. Tokyo, Japan: Corona, 2001, pp. 87–90. (InJapanese).

[9] X. D. Paul and M. Frank, “Fault detection via factorization approach,”Syst. Control Lett., vol. 14, no. 5, pp. 431–436, Jun. 1990.

[10] Y. Izumikawa, K. Yubai, and T. Hori, “Vibration suppression control offlexible arm robot by PD gain switching considering sensor failure,” inProc. 2002 IEEE Int. Conf. Industrial Technology, 2002, pp. 684–689.

[11] D. McFarlane and K. Glover, “A loop shaping design procedure usingH∞ synthesis,” IEEE Trans. Autom. Control, vol. 37, no. 6, pp. 759–769,Jun. 1992.

[12] H. Komatsu, T. Zanma, T. Suzuki, and S. Okuma, “Switching of H∞controllers under constraints and decision of controller state at instanceof the controller switching,” IEEJ Trans. Electron., Inform. Syst. Soci.,vol. 121-C, no. 12, pp. 1897–1904, Dec. 2001.

Yu Izumikawa received the B.E. and M.E. degreesfrom the Department of Electrical and ElectronicEngineering, Mie University, Mie, Japan, in 2000,and 2002, respectively. He is currently working to-ward the Ph.D. degree in the same department. Hisresearch interests include fault detection, and fault-tolerant control.

Kazuhiro Yubai (M’00) received the B.E., M.E., andPh.D. degrees in electrical engineering from NagoyaUniversity, Nagoya, Japan, in 1996, 1998, and 2000,respectively.

He is currently a Research Associate at Mie Uni-versity, Mie, Japan. His research interests include ro-bust control, system identification, and motion con-trol.

Junji Hirai (M’96) was born in Sizuoka, Japan, onOctober 27, 1951. He received the B.S. degree inelectrical engineering from the University of Tokyo,Tokyo, Japan, in 1976. He received the Ph.D. de-gree from Yokohama National University, Yoko-hama, Japan in 1999.

He joined the Yaskawa Electric Corporation,Japan, in 1982, and moved to Mie University in 2002,where he is presently a Professor in the Electrical andElectronic Engineering Department. His fields of in-terest include power electronics, motion control, and

robotics.Dr. Hirai is a member of the Institute of Electrical Engineers of Japan.