A Probabilistic Vehicle Diagnostic System Using Multiple ModelsMatthew L. Schwallmschwall@stanford.edu

J. Christian Gerdesgerdes@cdr.stanford.edu

Design DivisionDepartment of Mechanical Engineering

Stanford UniversityStanford, California 94305-4021

Bernard Bakerbernard.baeker@daimlerchrysler.com

Thomas Forchertthomas.forchert@daimlerchrysler.com

DaimlerChrysler AGResearch and Technology

Mechatronic and Diagnosis, REM/SDStuttgart, Germany

Abstract

In addition to being accurate, it is important that diag-nostic systems for use in automobiles also have low de-velopment and hardware costs. Model-based methodshave shown promise at reducing hardware costs sincethey use analytical redundancy to reduce physical re-dundancy. In addition to requiring no extra sensors, thediagnostic system presented in this paper also allows forhigh accuracy and low development costs by using in-formation from multiple simple models. This is madepossible by the use of a Bayesian network to processmodel residuals. A hybrid, dynamic Bayesian networkis used to model the temporal behavior of the faults anddetermine fault probabilities. A prototype of the systemhas been implemented and tested on a Mercedes-BenzE320 sedan. This paper describes the prototype systemand presents results demonstrating the system’s advan-tages over traditional residual threshold techniques.

Introduction

The goal of on-board vehicle fault detection and isolation(FDI) is to identify faults before they damage the vehicle orcreate a dangerous situation for the occupants. In order tobe practical to implement, such systems should require min-imal additional hardware, such as sensors and computationalpower, and also have low complexity and development cost.

Model-Based Techniques

Model-based methods are the preferred means of achiev-ing these goals. These methods use analytical redundancyin order to reduce the need for costly physical redundancy,such as extra sensors, and have been shown to be successfulin a wide variety of applications (Isermann & Balle 1996).As shown in Fig. 1, model-based techniques consist of twostages. In the first stage, measurements of system variablesand parameters are compared with values predicted by dy-namic models in order to generate a residual. This residualis then analyzed in the second decision-making stage to de-termine if a fault has occurred.

Copyright c© 2003, American Association for Artificial Intelli-gence (www.aaai.org). All rights reserved.

Residual

SystemModel

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DecisionMaker

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Figure 1: Generalized structure of model-based diagnostics

Multiple Model DiagnosticsFor complex systems such as a vehicle, it is often impracticalto represent the entire system with a single model. Instead,separate models are used for different subsystems and dis-similar observable modes such as temporal and frequency-based signals.

In theory, there are many benefits in using a multiplemodel approach. First, the many representations provide aneasy means for incorporating system information providedfrom a variety of sources. Another possible advantage is thatdesign models can be used directly and need not be adaptedto different ontologies or assumption sets. Finally, integrat-ing several forms of information yields a more complete di-agnostic system that can take advantage of relationships be-tween different modeling domains. The challenge is to de-velop a framework for integrating the models that exploitsall of these advantages.

A challenge common to both single and multiple modeltechniques is how to distinguish between a fault and noise onthe residuals. Many model-based diagnostic systems simplycompare the residuals against a threshold to determine if afault has occurred. This requires carefully designed, well-behaved residuals, which often requires extensive modelingand high computational effort. This can translate into unac-ceptable engineering and on-board computer costs. Further-more, most techniques lack a structure for setting thresholds.

The system described in this paper uses a Bayesian net-work (BN) for residual processing. Because of the use ofthis AI technique, the system is capable of using less ro-

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bust residuals. This lowers the costs associated with resid-ual generation and reduces the total effort required to createa diagnostic system. In addition to enabling the integrationof multiple models, this method permits the use of multi-ple modes of reasoning, and has therefore been referred toas Multi-Modal Diagnostics (MMD). The next section dis-cusses the probabilistic Bayesian network framework. Thefollowing two sections introduce the vehicle system and de-scribe the implementation of the diagnostic system. Experi-mental results are then presented and discussed.

Probabilistic Residual ProcessingInherent in all diagnostic methods is some uncertainty aboutthe true state of the system, due to sensor noise and unmod-eled dynamics. For this reason, all techniques must havesome sort of threshold that determines the boundary betweenaccepting a signal as the result of noise, and declaring afault. The methodology presented here differs from previ-ous techniques in that thresholds are not set on residuals, buton the probability that a fault has occurred. This threshold isan intuitive quantity that can be easily understood in termsof risks and costs. In addition, as shown in Schwall andGerdes (2001), optimal thresholds can be determined giventhe cost of a false alarm and the additional costs associatedwith missing a fault.

Static Bayesian Network RepresentationBayesian (belief) networks are often used for diagnostics,and the technique presented here is not the first to applythem to diagnosing a dynamic system. However, previousmethods, such as that presented by Lerner et al., model thesystem directly with a Bayesian network. This requires acomplicated transformation for each model and results ina highly complex network that poses challenging computa-tional problems.

Figure 2 shows how a BN can be used for processing mul-tiple residuals. In this paper, observable nodes, such as themodel residuals, are shown in shaded boxes. This BN re-lates the state of three faults to the probability distributionfor two residuals. For example, the probability distributionfor Residual 1 depends on the states of its parents, Fault 1and Fault 2. The probability of either an individual fault, orthe joint probability of the faults can be inferred given thestate of the residuals.

Residual 1

Fault 1

Fault 2

Fault 3

Residual 2

Figure 2: An example diagnostic Bayesian network

Faults can be assumed to have two states—fault and no-fault. Residuals, however, are continuous values, as theyrepresent the difference between a measured and an esti-mated scalar value. Therefore, the fault nodes in the networkare discrete, the observed residual nodes are continuous, and

the resulting network is a hybrid Bayesian network. For asingle residual R with a set of parent faults F , P (r | F )is represented by a continuous conditional probability distri-bution (CPD). It is convenient to represent P (r | F ) by sep-arate probability density functions P (r | f) for every possi-ble state f of the parent faults F . The Gaussian distributionis commonly used both because it is valid for a normally dis-tributed noise assumption and also simplifies calculations incertain types of networks. However, for the diagnostic net-work presented here, there is little computational incentiveto use one form of distribution over another, because residu-als are the only continuous nodes, and they are observed.

Representation of Time Dependence

The appearance and persistence of faults is a temporal pro-cess. In order to represent this dependence, a dynamicBayesian network (DBN) is used. Two significant assump-tions must be made in order to compactly represent the DBNby a 2-time-slice Bayesian network (2-TBN).

The first is that the system is time-invariant. For our sys-tem this implies that the probability of a fault given the pre-vious state of the fault, P (f(t+1) | f(t)), is constant for allt. For components that wear, this probability could slowlychange over the life of the vehicle. Fortunately, such a pro-cess is slow enough that it could be updated each time thecar is started, and does not impair our ability to use a DBN.

The second assumption is the Markov assumption, whichstates that the future is conditionally independent of the pastgiven the present. This is reasonable when considering faultsand implies that, given the status of the faults at time t, theprobability of faults at time t + 1 does not depend on faultsprior to t.

Time = t+1

Residual 1

Fault 1

Fault 2

Fault 3

Residual 2

Residual 1

Fault 1

Fault 2

Fault 3

Residual 2

Time = t

Figure 3: An example dynamic Bayesian network

The 2-TBN corresponding to adjacent time slices of theBN shown in Fig. 2 is shown in Fig. 3. The lack of edgesfrom the residuals at t to the residuals at t + 1 implies thatthe residuals at t + 1 are conditionally independent of theresiduals at t given the state of the faults at time t or t + 1.This means that the only correlation between residuals atdifferent time steps is through the mechanisms of the faults.

In order to assess whether or not this is a valid assump-tion, it is necessary to analyze the mechanisms that causeresiduals. As was shown in Fig. 1, residuals exist becausethe model predicts a different value than what a sensor in-dicates. Expanding on this representation with the use ofa Box-Jenkins model structure (Box, Jenkins, & Reinsel1994), Fig. 4, shows that there are two possible sources of a

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Residual

SystemModel

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Input

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MeasuredOutput

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Figure 4: Residual generating system structure

residual. One is due to differences between the system andthe system model, and the other is due to noise.

Errors in the system model can either be undesirable dy-namics and parameter errors, which are the faults, or dynam-ics that are a part of the functioning system but are not mod-eled due to their complexity. The components of a residualdue to undesirable errors are correlated through the mecha-nism of the fault and are therefore conditionally independentgiven the fault state. Residuals due to other unmodeled dy-namics and parametric errors in the system model will becorrelated with the system input. These correlations must beaccounted for in order to maintain the accuracy of the solu-tion. As will be seen below, they can be dealt with by usingan extra node called a status parameter provided the type ofinput that excites the unmodeled dynamics is known.

The other contribution to residuals is noise. Noise is in-troduced into the system as an independent random input.However, after this input has been shaped by the noise dy-namics, its contribution to the residual is autocorrelated.Thus, in order for our assumption to be valid, the BN mustsample the residuals slower than the noise dynamics.

Prototype SystemA prototype of the diagnostic system has been implementedon a Mercedes-Benz E320 sedan in order to diagnose thecar’s handling system. As the focus of this paper is theimplementation of the AI system, the models are onlybriefly described. For more detail regarding the models, see(Schwall & Gerdes 2001). The section after this will de-scribe the hardware and software implementation.

The input to the diagnostic system comes from six sensorslocated on the vehicle: four wheel speed sensors, a yaw rategyro and a steering angle sensor. For the purpose of thisexample, ten faults will be considered. The faults consist ofa failure of any of the four tires, as well as faults of any ofthe six sensors. In order to diagnose the system, three simplemodels are used; the bicycle model of vehicle handling, amodel of yaw rate given left and right wheel speeds, and amodel for predicting longitudinal slip of the driven wheels.

Vehicle ModelsThe classic bicycle model of vehicle cornering consists of apair of dynamic equations relating steering angle to yaw rate

and sideslip angle. The model assumes a constant longitudi-nal velocity.

By comparing the yaw rate predicted by the bicycle modelwith the value measured with the gyro, a residual can begenerated which will indicate a fault in one of the parametersused in the model.

R1 = rexpected(bicycle model) − rmeasured(gyro) (1)

When turning, the outside wheels of a car rotate fasterthan the inside wheels. The second model compares the leftand right wheel speeds in order to estimated the yaw rate ofthe vehicle (Frank, Palkovics, & Gianone 2000).

Estimates of the yaw rate can be computed using datafrom both the front and rear wheel pairs and compared withthe yaw rate measured by the yaw rate gyro. Thus:

R2 = rexpected(front wheels) − rmeasured(gyro) (2)

R3 = rexpected(rear wheels) − rmeasured(gyro) (3)

Wheel slip is defined by the relative difference in velocityof the vehicle and the wheel’s rotational rate. For a two-wheel drive vehicle that is not braking, the undriven wheelshave zero slip. Assuming that both wheels on the sameside of the vehicle have the same effective radius, the wheelspeeds of the driven and undriven wheels are sufficient tocalculate the actual wheel slip

Carlson and Gerdes (2002) demonstrated that a simplelongitudinal model with linear tire stiffness can reasonablyapproximate the slip arising from low levels of acceleration.Therefore, given the car’s mass and an experimentally deter-mined longitudinal tire stiffness, slip can be estimated fromthe vehicle’s acceleration, which is calculated by differenti-ating the undriven wheel speeds.

The actual and estimated slip for each of the driven wheelscan be compared to yield one residual for each of the rightand left wheel pairs.

R4 = Sexpected(acceleration) − Smeasured(right wheels) (4)

R5 = Sexpected(acceleration) − Smeasured(left wheels) (5)

Since all four wheels slip during braking, these residualsare only valid when the vehicle is not braking. In the future,the model could be extended to cover braking if desired.

Bayesian Network StructureFigure 5 shows a single time slice of the DAG that resultsfrom drawing the causal relationships implied by the mod-els. Here, the five residuals are shown in shaded solid boxes.In addition to the residuals and faults, the graph contains dis-crete “hidden” nodes shown in dashed boxes. Some of thesenodes, such as Cαf and Cαr representing the front and rearcornering sicknesses, are included in order to make the BNeasier to generate and understand. They are used when mul-tiple faults effect a residual through the same mechanism orparameter.

The observed (shaded) “hidden” nodes such as Not Brak-ing and Constant Velocity are status parameter nodes that arenecessary to represent the assumptions used when generat-ing the residuals. They function because of the structure of

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the CPDs of their child residuals. The CPDs indicate thatin the presence of a vehicle status parameter, there may be ahigh likelihood of an abnormal residual even in the absenceof any faults. This provides a means to “explain away” alarge residual by the presence of a status parameter.

The DBN is based on this single-slice BN with intercon-nections between each fault at time t and the same fault attime t + 1. Therefore, in order to solve the DBN exactly,we need to solve for and maintain the joint probabilities ofall possible fault combinations. For the 10 binary faults, thisdistribution contains 210 probabilities. This distribution wascalculated and maintained at each time step for the systemdescribed in this paper, and is acceptable for diagnostic sys-tems of similar scope given modern computing power. How-ever, for larger systems such as might eventually be used onproduction vehicles, approximate solution methods wouldbe valuable and are an area of active research.

Conditional Probability DistributionsAs seen in Schwall and Gerdes (2002) , the noise on resid-uals generated with automotive dynamic models does notperfectly fit a normal distribution, though the approximationis quite close. For the prototype system, the residual distri-butions were approximated by normal distributions, both inthe fault and no-fault cases.

Estimating the mean µ and variance σ2 of the residualdistributions in the no-fault case is straightforward, sincethis data is readily available. These parameters were esti-mated by taking data with the fully functioning vehicle andare shown in Table 1. The residuals’ variances are high dueto the simplicity of the models and demonstrate the potentialof the method when using low-cost models.

Table 1: No-fault residual distribution parametersResidual µ σBicycle Model (rad/s) 0.005 0.020Front Wheel Speed Yaw Rate (rad/s) -.014 0.015Rear Wheel Speed Yaw Rate (rad/s) -.008 0.018Right Wheel Slip 0.0 0.004Left Wheel Slip 0.0 0.004

Data was not taken in the presence of faults; rather, thefaults evaluated were simulated. The residual distributionsin the presence of faults were estimated as having the samemean as the no-fault distribution and larger variances. Inthis case, the standard deviations were estimated as 10 timesthat of the no fault cases. The variance for the residual withmultiple faults was made the same as that with one fault.This was done so that exceptionally large residuals wouldnot necessarily suggest multiple faults.

The a priori fault probabilities are based on very roughapproximations of vehicle component life. They correspondapproximately to one failure of each tire every 30,000 milesand one failure every 100,000 miles for the sensor faults.

System ImplementationThe research prototype system has been implemented in thevehicle in a manner driven by the desire for easy develop-

ment, frequent changes, and future upgrades. For this rea-son, the hardware and software are considerably differentthan what would be used in a production application, wherecost would be a driving factor and economies of scale wouldallow for custom hardware and optimized software.

System Hardware and Sensors

Figure 6: Mercedes-Benz E320 research vehicle

Single Board PC

Gyros

Figure 7: Trunk area showing single board computer systemand gyros

The diagnostic system consists of two computers. Oneis a Versalogic VSBC-6 single board PC, configured withan additional Versalogic VCM-DAS-1 data acquisition mod-ule and Softing CAN-AC2-104 Controller Area Network(CAN) board. As show in Fig. 7, this system is mountedin the trunk of the vehicle. It communicates over ethernetwith a laptop computer inside the vehicle.

The research vehicle is a 1999 Mercedes-Benz E320sedan donated to Stanford University by DaimlerChrysler.As standard features, the vehicle has Anti-lock Braking Sys-tem (ABS) and Electronic Stability Program (ESP). As partof the sensor package for these systems, the vehicle has low-resolution wheel speed sensors on all four wheels, a steeringangle sensor, and a yaw rate gyro.

The wheel speed signals are available on the vehicle’sCAN bus and are read using the Softing board. The steeringangle value is not available over CAN, but instead is read

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Steering Angle Sensor (F10)

Bicycle Model (R1)

RF Tire (F1)

LR Tire (F4)

RR Tire (F3)

LF Tire (F2)

RF Wheel Speed Signal

Front Wheel Speeds vs Yaw Rate (R2)

Rear Wheel Speeds vs Yaw Rate (R3)

LF Wheel Speed Signal

LR Wheel Speed Signal

RR Wheel Speed Signal

Yaw Gyro (F9)

Right Wheel Slip (R4)

Left Wheel Slip (R5)

Constant Velocity

RF Wheel Speed

Sensor (F5)

LF Wheel Speed Sensor

(F6)

LR Wheel Speed Sensor

(F8)

RR Wheel Speed sensor

(F7)

Not Braking

Cαf Cαr

Figure 5: Bayesian network structure for vehicle stability control system

directly and interpreted using a custom designed board. Theyaw rate signal is also not available over CAN, and there-fore a separate Bosch automotive-grade yaw rate gyro hasbeen installed in the trunk and its signal is read using thedata acquisition module. For the production implementationof such a diagnostic system, all of the sensor values couldbe made available over CAN and no extra sensor hardwarewould be required.

Matlab Simulink and xPC Environment

In order to facilitate rapid development and frequentchanges, the MathWorks’ Matlab Simulink and xPC Targetprototyping environments were used. The laptop inside thevehicle acts as the host computer and runs Simulink whilethe single board PC runs xPC Target.

The residual-generating system models are created inSimulink and downloaded to the single board computerwhere they process the sensor signals and send this infor-mation over the ethernet connection to the laptop. The mod-els run at 100Hz, but the laptop only samples the residualsat 1Hz. The high update rate on the models is done in or-der to maintain the accuracy of the dynamic models. Theresidual sampling rate reflects the update rate of the dynamicBayesian network running on the laptop. 1Hz was chosen inorder to sample slower than the noise dynamics and fulfillthe assumption that the residuals are not autocorrelated.

The DBN is programmed into Matlab using the Bayes NetToolbox written by Kevin Murphy (Murphy 2001). Neitherthe programming language nor the code are optimized forspeed, however, they allow for easy changes and can solvethe BN at the required 1Hz rate. The result is a real-timediagnostic output available on the laptop screen inside thecar.

Bike Model ResidualRear Yaw ResidualFront Yaw Residual

Time(s)

Nor

mal

ized

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idua

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ues

20 40 60 80-5

0

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Figure 8: Residuals with simulated wheel speed sensor fault

ResultsTesting is ongoing, and this section presents representativeresults showing the performance of the system. The faultsexamined have primarily been small magnitude sensor faultssimulated in software, as these pose a greater challenge tothe system than obvious physical faults such as a short cir-cuit. Tire failures have not been tested due to the difficultyand danger in creating such a fault.

In one test, data was taken for 100 seconds of driving atspeeds of 20-50 km/h. During the interval from 60 to 90seconds into the run, a 1% bias was added to the left frontwheel speed signal. This simulates a fault in the wheel speedsensor, such as the loss of a tooth on the sensor. Figure 8shows the output of the bicycle model and two wheel speedyaw rate estimation models. The residual values have been

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Left Slip ResidualRight Slip Residual

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Figure 9: Residuals with simulated wheel speed sensor fault

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Figure 10: Diagnostic system output for simulated wheelspeed sensor fault

normalized by their standard deviations. As is expected, thefront wheel speed yaw rate residual was affected by the fault,while the rear wheel speed yaw rate residual and bicyclemodel were not.

Figure 9 shows the two slip model residuals for the sametest. The left slip residual is influenced by this fault. It isinteresting to note a few areas, such as around 25 and 40s,where some of the slip residuals are rather large even thoughthere is not fault. This is where the brakes were applied,influencing the slip residuals.

Figure 10 shows the DBN’s assessment of fault likeli-hood. The probability of no fault is very close to 1 exceptduring the fault from 60 to 90 seconds. During this interval,the system accurately isolates the failure as left front wheelspeed sensor. The system was not fooled by the high slipresiduals during the periods of braking, because the brakingstatus parameter “explained away” the large residuals.

Both the Bayesian network and a trained eye can detect

that between 60 and 90 seconds, two of the five residuals arebehaving abnormally. However, the deviation is slight andattempts at placing residual thresholds would have failed.There is no value that the residuals consistently exceed dur-ing the fault that they do not exceed during regular driving.Therefore, a system that relied on thresholds could not havedetected this fault.

ConclusionThe prototype diagnostic system implemented demonstratesthe power of multiple model diagnostics when residuals areintegrated using AI. The use of a hybrid, dynamic, Bayesiannetwork enables the detection of faults that could not be de-tected using techniques based on residual thresholds. In ad-dition, the network provides an easy framework for includ-ing information such as modeling assumptions.

Due to the availability of models and sensor information,automotive systems are ideal applications for this technol-ogy. While the prototype presented examines only a smallnumber of faults, these faults effect safety critical systemsincluding ABS and ESP. If put into production, the addi-tional cost of such a system would be limited to the process-ing power necessary for the models and Bayesian networksince no extra sensors are required. Future advances in ap-proximate DBN solution methods would help enable the useof cheaper processors and mitigate these costs.

ReferencesBox, G.; Jenkins, G.; and Reinsel, G. 1994. Time SeriesAnalysis, Forecasting and Control. Englewood Cliffs, NewJersey: Prentice Hall.Carlson, C. R., and Gerdes, J. C. 2002. Identifying TirePressure Variation by Nonlinear Estimation of Longitudi-nal Stiffness and Effective Radius. In Proceedings of AVEC2002 6th International Symposium of Advanced VehicleControl.Frank, P.; Palkovics, L.; and Gianone, L. 2000. UsingWheel Speed and Wheel Slip Information for ControllingVehicle Chassis Systems. In 5th International Symposiumon Advanced Vehicle Control (AVEC).Isermann, R., and Balle, P. 1996. Trends in the Applicationof Model Based Fault Detection and Diagnosis of Techni-cal Processes. In Proc. of IFAC 13th Triennial World Con-ference.Lerner, U.; Parr, R.; Koller, D.; and Biswas, G. 2000.Bayesian Fault Detection and Diagnosis in Dynamic Sys-tems. In Proceedings of the Seventeenth National Confer-ence on Artificial Intelligence (AAAI), 531–537.Murphy, K. 2001. The Bayes Net Toolbox for Matlab. InComputing Science and Statistics, volume 33.Schwall, M. L., and Gerdes, J. C. 2001. Multi-Modal Di-agnostics for Vehicle Fault Detection. In Proceedings ofIMECE 2001, DSC–24600.Schwall, M. L., and Gerdes, J. C. 2002. A ProbabilisticApproach to Residual Processing for Vehicle Fault Detec-tion. In Proceedings of the American Controls Conference,2552–2557.

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