Incorporating a Model of Vehicle Dynamics in a Diagnostic ... · rately modeling the vehicle dynamics and estimating the state of the vehicle, these disturbance torques can be included

AVEC ’04

Incorporating a Model of Vehicle Dynamics in a

Diagnostic System for Steer-by-Wire Vehicles

Christopher D. GaddaStanford University

Paul YihStanford University

J. Christian GerdesStanford University

Dept. of Mechanical EngineeringStanford, California 94305-4021, USA

Phone: 650-906-6417Fax: 650-723-3521

Email: [email protected]

This paper examines the benefits of incorporating vehicle dynamics modeling into the designof a diagnostic system for a steer-by-wire vehicle. The use of a model of vehicle dynamicsimproves the speed and accuracy of the diagnoses, by eliminating a significant source ofdisturbance input to the steer-by-wire system model. A method for reducing the effectsof modeling uncertainty on diagnostic system performance based on spectral fault charac-teristics is also presented. The techniques discussed are demonstrated on an experimentalsteer-by-wire vehicle.

Topics / Vehicle Diagnostics, Active Safety

1. INTRODUCTION

The numerous benefits of steer-by-wire technol-ogy have yet to be realized in a production vehicle.The potentially catastrophic nature of a steeringsystem failure mandates that any practical steer-by-wire system be extremely reliable. One approach forachieving the necessary level of reliability incorpo-rates a fast and accurate diagnostic system that canisolate a fault and respond by switching to a redun-dant component or a modified control law that canaccommodate the fault.

The area of drive-by-wire research draws fromthe years of work done on fly-by-wire technologies,diagnostic systems included, and many of the resultsapply directly. A probabilistic analysis of the failurerates of steer-by-wire systems using various forms ofredundancy coupled with diagnostic techniques de-scribed in [1], shows that steer-by-wire systems canbe designed to have an overall reliability rate of 10−9

failures/hour, the same as imposed on the aviationindustry. But diagnostic systems for aircraft havecertain design freedoms that are not available tothose for ground vehicles. The expense of triply re-dundant sensors, actuators, and controllers, all com-mon practices in fly-by-wire designs, are prohibitivein production automobiles. Furthermore, aircraftare typically tens of seconds or more from any pos-sible source of collision, so diagnostic systems foraircraft have ample time to correctly identify thesource of a fault and choose appropriate action. Thisis clearly not the case in an automobile where thedecision must be made much more quickly to pre-

vent a collision.

Previous work in this area includes a baysiannetwork approach to diagnostic systems [7], [8].In [5], a technique for diagnosing actuator faults inthe presence of unknown disturbances and modellingerrors is developed. Isermann presents a discussionof many of these diagnostic techniques in [2], [3],and their use in fault-tolerant drive-by-wire systemsin [4].

In any practical diagnostic system, noise andmodeling uncertainty ultimately limit the accuracyand speed of fault detection and isolation. In thecase of a steer-by-wire system, the lateral forces onthe road wheels create disturbance torques on thesteering motor. If the diagnostic system is designedwithout knowledge of the vehicle dynamics, thesedisturbance torques can only be regarded as a noisesource. This reduces the accuracy of the diagnosesand/or increases the fault detection time. By accu-rately modeling the vehicle dynamics and estimatingthe state of the vehicle, these disturbance torquescan be included in the model and need not adverselyaffect performance.

We use simple linear models of the steering sys-tem and vehicle dynamics in the design of an ex-ample diagnostic system for a steer-by-wire vehi-cle. The imperfections in the models used resultin a diagnostic system that produces residuals thatrespond to the driver command signal, even whenthere is no fault present. This effect is addressedthrough the use of residual filters designed to exploitthe differences in spectral content between the resid-

AVEC ’04

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er-

by-w

ire s

yste

m Current-trusting

observerr

!

iYaw rate-trusting

observer

+

+

H(s)

H(s)

Thr e

sho

ld logic-

-

"1

"2

"1’

"2’

DC Motor

Motor

Controller

i

vVehicle

Dynamics#a

! $

r

Steering

Controller id

!d

Fig. 1: Steer-by-wire system block diagram

uals produced by an actual fault and those whichresult from modeling inaccuracies. Experimentalresults collected on a steer-by-wire vehicle demon-strate the effectiveness of the techniques presented.

2. DIAGNOSTIC SYSTEM

2.1 Fault TypesWhile there exist many ways to classify differ-

ent failure modes for a steer-by-wire system, oneparticularly useful classification approach is the sim-ple dichotomy of time scale: gradual faults and sud-den faults. Gradual faults occur as components wearout or overheat, but do not require immediate detec-tion, since by definition they are not changing veryquickly. Fault detection techniques which are suit-able for detecting gradual faults can operate overa longer time scale. A sudden fault, however, suchas a wiring harness failure may require immediatecorrective action to maintain vehicle stability anddriver control. Thus, detection time is of criticalimportance for sudden faults.

Techniques which focus on parameter estima-tion or adaptive filtering such as recursive least-squares, extended Kalman filtering, or instrumentalvariables are well suited for the detection of a grad-ual fault, but may result in detection times whichare unacceptably long in response to a sudden fault.The focus of this paper is on detection and isolationtechniques for sudden faults, which when used inparallel with parameter estimation techniques willprovide a system capable of diagnosing faults thatdevelop over any time scale.

2.2 System DescriptionWe develop an example diagnostic system for

a 1997 Corvette which has been adapted to steer-by-wire. A block diagram of the system is shown infigure 1. The steer-by-wire system uses a DC motorwith a gearbox to drive the pinion of the originalpower steering system, so the following linear modelof the motor dynamics is used:

˙xm = Amxm + Bm

[i τa

]T (1)

where

xm =[

δ δ]T

Am =[

0 10 − bm

Jm

]

Bm =[

0 0rsrprgηkm

Jm− 1

Jm

]where δ is the steering angle, Jm is the effectivemoment of inertia of the steering system, bm is theeffective damping of the steering system, rs is thesteering ratio, rp is the torque amplification factorof the power steering system, rg is the gear ratio ofthe gearbox connecting the DC motor to the pinion,η is the combined efficiency of the motor and gear-box, and km is the motor constant relating torqueto current. The inputs to this model are the currentto the motor, i, and the aligning torque, τa.

The aligning torque is a function of the vehi-cle state and represents a significant disturbance tothe steer-by-wire system. In the absence of a suit-able model of vehicle dynamics, the aligning torquecould be modeled as a stochastic process or a norm-bounded unknown signal, but due to the magnitudeof the aligning torque, the resulting diagnostic sys-tem would be very insensitive to faults. For thisreason, we incorporate a model of vehicle dynamicsinto the design of our diagnostic system.

The planar dynamics of the vehicle are mod-eled using the bicycle model, where left and righttire forces are considered in aggregate. Small angleapproximations are used and lateral tire force is as-sumed to be proportional to the tire slip angle, sothat a linear model of the planar vehicle dynamicsis developed, given by the following:

xv = Avxv + Bvδ (2)

where

xv =[

β r]T

Av =[− C0

mV − 1 + C1mV 2

C1Iz

− C2IzV

]Bv =

[Cf

mVCf aIz

]and

C0 = Cαf + Cαr

C1 = Cαrb− Cαfa

C2 = Cαfa2 + Cαrb2

where β is the sideslip angle, r is the yaw rate, Iz isthe polar moment of inertia of the vehicle, Cαf andCαr are the front and rear cornering stiffnesses, re-spectively, a and b are the distances from the center

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ld logi c-

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"1

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Motor

Controller

i

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Fig. 3: Diagnostic system block diagram

δαf�

r

αr

uy,CG

ux,CG

Fy,r

Fy,f

βCG

γCG

ψUCG

Fig. 2: Bicycle model

of gravity to the front and rear axles, respectively,m is the mass of the vehicle, and V is the vehiclevelocity.

The aligning torque is related to the vehiclestate by the following equation:

τa = −Cαf (tp + tm)(β +a

Vr − δ) (3)

where tp and tm are the pneumatic and mechanicaltrail of the tire, respectively. In order to arrive at alinear model, tm and tp are assumed to be constantand known, as are Cαf and Cαr. Combining (1),(2), and (3) yields the following state-space modelof the vehicle and steering system dynamics:

x = Ax + Bi (4)

where

x =[

β r δ δ]T

A =

− C0

mV − 1 + C1mV 2

Cαf

mV 0C1Iz

− C2IzV

aCαf

Iz0

0 0 0 1C3Jw

aC3JwV − C3

Jw− bw

Jw

B =

000

rsrpkM rgηJw

and

C3 = (tp + tm)Cαf

A block diagram presented in figure 3 shows thebasic structure of the diagnostic system. Measure-ments of yaw rate, motor current, and steering an-gle are provided to a pair of state observers, each ofwhich computes separate estimates of the state vari-ables. From each of these state estimates, the steerangle estimates are compared against the measuredsteer angle to produce two residuals. These resid-uals are then filtered (as described in section 2.4)and then compared against thresholds to detect andisolate a fault.

Each of the observers have the following struc-ture:

˙x = Ax + Bi + L(r − r) (5)

The gain vector L is chosen such that the observerdynamics will be stable and “trust” either the mea-sured current or the measured yaw rate, while “dis-trusting” the other. This is accomplished by solvingthe steady-state continuous-time Kalman filter de-sign problem with process and sensor noise weight-ing factors adjusted appropriately. To create the ob-server which depends heavily on the measured cur-rent, the process noise variance is set to be 1/10,000the value of the sensor noise variance. To createthe observer which depends heavily on the mea-sured yaw rate, the process noise variance is set tobe 10,000 times the value of the sensor noise vari-ance. This technique obviates the difficult processof choosing pole locations for a pole-placement ap-proach or manually adjusting gains. Furthermore,it always results in a stable observer. It should benoted that the process and sensor noise variance val-ues are not chosen based on actual noise models foreither sensor, and the resulting observer is not anoptimal state estimator. The noise variances are

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simply a convenient way of tuning the resulting ob-server’s sensitivity to a particular input.

By using two separate observers, one which de-pends primarily on the yaw rate measurement andone that depends primarily on the motor currentmeasurement, a single fault can be both detectedand, to an extent, isolated. When the steer-by-wiresystem is working correctly, the observers will con-verge to accurate state estimates and both residualswill be small. If a fault occurs which affects the yawrate sensor the residual which relies on yaw rate,ρ2, will have much larger response than the resid-ual which relies on the motor current, ρ1. When afault occurs which affects the motor current sensoror some aspect of the motor model, such as a changein friction or motor constant, ρ1 will have a muchlarger response than ρ2. If a fault occurs which af-fects the steering angle sensor, both ρ1 and ρ2 willhave a significant response.

With this simple example diagnostic systemthere are only two residuals, hence only four possiblediagnoses using this technique. In a practical appli-cation of this approach, additional residuals wouldbe formed from other measurements, increasing thevariety of faults that can be detected and the de-gree to which a fault condition can be isolated. Thetechnique presented here focuses on the diagnosis ofthe motor, as an accurate diagnosis of the motorfacilitates the diagnosis of other components of thesteer-by-wire system.

2.3 Modeling UncertaintyIncluding a model of the vehicle dynamics in the

models used by the diagnostic system significantlyreduces the uncertainty in the aligning torque τa,but also creates a dependence on the accuracy of thevehicle dynamics model. Several factors limit the ac-curacy of this model. The dynamics are linearizedand thus cannot accurately represent coulomb fric-tion in the steering system, nor tire saturation nearthe limits of handling. Also, the model depends on anumber of parameters such as vehicle mass, vehiclecenter of gravity, or tire cornering stiffness, whichmay not be known very precisely, or may changeover time.

System identification techniques, such as recur-sive least-squares or instrumental variables can beused to estimate the various uncertain parameters inthe model. These parameter estimates may alreadybe available as the result of a portion of the diagnos-tic system looking for gradual faults. This approachwill reduce modeling errors, but even with perfectparameter estimates, nonlinear and other unmod-eled dynamics still limit how accurately the mod-els will be able to predict the behavior of the ac-tual system. This, in combination with sensor noise,will limit how small the residuals will be for an un-faulted system, which in turn limits the sensitivityof the diagnostic system to an actual fault.

2.4 Residual FilteringFiltering the residuals before comparing them

to a set of fixed thresholds is another approach forhandling modeling errors. If the residuals for theun-faulted system are too large, linear filters can beused to remove energy in select frequency ranges toreduce the overall amplitude of the residual. Whenthe frequency content of an un-faulted residual dif-fers significantly from that of the residual when afault is occurring, residual filtering can improve de-tection performance significantly.

The input to the actual system is primarily thedriver command, which generally does not have sig-nificant frequency content above ∼ 2 Hz. The ac-tual system and the observers are driven by thissame signal, so the component of the residualsformed due to model inaccuracies is expected toalso have little frequency content above ∼ 2 Hz.A suddenly occurring fault, however, produces abroad range of frequency content in the residuals.This spectral difference between faulted and un-faulted residuals makes it possible to design linearfilters which reject the frequency content of a nomi-nal residual while passing the frequencies associatedwith a suddenly occurring fault, thereby reducingthe effects of model inaccuracy and increasing thesensitivity of the fault detection. While by no meansoptimal, a simple second-order high-pass filter witha cutoff frequency of 10 rad/s, given by the followingtransfer function, significantly improves the perfor-mance of the diagnostic system:

H(s) =s2

s2 + 20s + 100(6)

This filter technique has some disadvantages aswell. The use of a filter which eliminates the DCcomponent of a signal means that all residuals willreturn to zero a short time after a fault occurs, evenif the fault is still present. Also, the use of high-passfiltering renders the diagnostic system insensitive tofaults which develop on a time-scale below the cut-off frequency of the filters. As mentioned previously,the technique presented here is only one part of com-plete system and must be used in combination withother techniques to detect more gradually occurringfaults.

3. SIMULATION

This technique is demonstrated in simulationfor a vehicle traveling at a constant speed of 6.5 m/sfor two different types of fault. Figure 4 shows theresiduals for a simulated bias shift of 1◦ on the steer-ing angle sensor. The horizontal dotted lines repre-sent the detection threshold level. The simulatedfault occurs at 5 seconds into the simulation, and asexpected, both residuals respond immediately. Fig-ure 5 shows the residuals for a simulated bias shift of2.8◦/s on the yaw rate sensor. Again, the simulated

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4 4.5 5 5.5 6−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

Time (s)

Res

idua

ls (

rad)

ρ’1 (Based on current)

ρ’2 (Based on yaw rate)

Fig. 4: Simulated residuals for 1◦ steer angle biasshift

4 4.5 5 5.5 6−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Time (s)

Res

idua

ls (

rad)

ρ’1 (Based on current)

ρ’2 (Based on yaw rate)

Fig. 5: Simulated residuals for 2.8◦/s yaw rate biasshift

fault occurs at t = 5 seconds, but in this case onlythe yaw rate-based residual, ρ′2 responds. In bothcases, the affected residuals return to zero withinroughly five time constants of the high-pass filter,even though the simulated faults persist.

4. EXPERIMENT

Using the steer-by-wire vehicle and softwaresimulation of fault conditions, we collected experi-mental data demonstrating the techniques discussedhere. All of the tests were performed with the vehi-cle moving at a roughly constant speed of approxi-mately 6.5 m/s, while being piloted through a slalomcourse. The peak lateral acceleration during thesetests was kept under 5 m/s2, so the linear tire modelapproximation is valid. Two types of faults weresimulated by modifying the software controlling thesteer-by-wire system: a 1◦ steering angle sensor biasshift and a 2.8◦/s yaw rate sensor bias shift.

0 5 10 15 20 25 30−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time (s)

Res

idua

ls (

rad)

ρ1 (Based on current)

ρ2 (Based on yaw rate)

Fig. 6: Residuals for un-faulted system

10 12 14 16 18 20−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

Time (s)

Res

idua

ls (

rad)

ρ’1 (Based on current)

ρ’2 (Based on yaw rate)

Fig. 7: Filtered residuals for un-faulted system

Figure 6 shows the unfiltered residuals collectedduring an experiment where no simulated faults oc-curred. In both residuals, the driver command sig-nal can be seen; it is particularly noticeable in thecurrent-based residual, ρ1. The horizontal dottedlines represent threshold levels for each residual,chosen such that no false positives occurred duringany of our experiments. In figure 7 the filtered resid-uals for the same experiment are plotted, showinghow effective the high-pass filtering is at eliminatingthe effects of model inaccuracies. In this plot a newthreshold level is shown, also chosen such that nofalse positives were recorded. This threshold is anorder of magnitude lower than the thresholds usedin the unfiltered case.

Figure 8 shows the residuals during a simulated1◦ steering angle bias shift. Just as in simulationboth residuals have an immediate response, thenquickly return to zero. In figure 9 a simulated 2.8◦/syaw rate bias shift is shown only affecting ρ′2, whichalso returns quickly to zero.

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7 7.5 8 8.5 9−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

Time (s)

Res

idua

ls (

rad)

ρ’1 (Based on current)

ρ’2 (Based on yaw rate)

Fig. 8: Filtered residuals for 1◦ steer angle bias shift

4 4.5 5 5.5 6−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Time (s)

Res

idua

ls (

rad)

ρ’1 (Based on current)

ρ’2 (Based on yaw rate)

Fig. 9: Filtered residuals for 2.8◦/s yaw rate biasshift

5. CONCLUSION

These results illustrate the effectiveness of a di-agnostic system based on vehicle dynamics modelsand residual filtering in detecting suddenly occur-ring faults. The experiments show how modelinginaccuracies can be readily accommodated when thespectral content of a fault differs significantly fromthat of normal system responses. This representsonly a component of a complete diagnostic system,as it is designed to work in parallel with a sys-tem which can diagnose a gradually worsening faultcondition. The successful combination of diagnos-tic techniques which can accurately isolate suddenfaults quickly and gradual faults before they becomeserious enough to affect vehicle handling is an av-enue for further research.

6. ACKNOWLEDGMENTS

The authors would like to acknowledge Nis-san Motor Corporation for sponsoring research indiagnostic systems for steer-by-wire vehicles, withspecial thanks to Toshimi Abo, Kazutaka Adachi,Takeshi Mitamura, Dr. Kimio Kanai, and Masa-haru Asano for their support of this project.

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