Bond Graph Based Fault Diagnosis of 4W-Vehicles Suspension Systems I: Passive Suspensions Luis Silva1,*, Diego Delarmelina2,*, Sergio Junco1, Nacer K. M’Sirdi3, Hassan Noura3 1

Departamento de Electrónica, 2 Escuela de Ingeniería Mecánica Facultad de Ciencias Exactas, Ingeniería y Agrimensura Universidad Nacional de Rosario Ríobamba 245 Bis – S2000EKE Rosario – Argentina

3

Laboratoire des Sciences de l’Information et des Systèmes (LSIS) Domaine Scientifique de St Jérôme Avenue Escadrille Normandie Niémen, 13397 Marseille Cedex 20, France [email protected], [email protected], [email protected] *

On stage at LSIS during the execution of this research.

Keywords: Vehicle dynamics, fault detection and isolation, analytical redundancy, diagnostic bond graphs. Abstract: This paper applies the model-based ARR technique, implemented on the so-called Diagnostic Bond Graph, to the problem of detecting and isolating faults in vehicle suspensions. A fourteen degrees of freedom (DoF), fourwheeled vehicle Bond Graph model adapted from models available in the literature is used as starting point. The main contribution of the paper is the proposition of a simplified Diagnostic Bond Graph that, on the ground of the chosen measurements, allows solving the FDI problem on a reduced subsystem decoupled from the wheel dynamics. This renders unnecessary using the complex and uncertain ground-tire interaction model. The simulation results presented illustrate the method ability of monitoring and isolating all the possible suspension faults considered.

1. INTRODUCTION The great complexity of current technical systems and processes calls for the development of methods for fault detection and isolation (FDI), aimed at helping in the assurance of the systems safety of operation. This is important because of economic, human safety, and environmental reasons, among others. Model-based FDI (MB-FDI), one of the answers to this challenge, has evolved to a discipline with extensive theoretical studies and many applications. A main reason for this is the fact that MB-FDI not only provides techniques for direct online application, but also allows the off-line study of a plenty of faulty scenarios without any risk for the real system. One of the most popular MB-FDI methods is the Analytical Redundant Relationships or ARR-approach. It generates residuals from constraints (usually equations) among known (measured) system variables, which are parameterized by coefficients whose nominal values in normal system operation are known to a good level of approximation. If the evaluation of the residuals

as function of the measured variables yields values under certain thresholds around zero (a tolerance for model uncertainties and disturbances), then the system operates in a non-faulty condition. In contrast, a residual above its threshold is an indication of a faulty operation condition, which is interpreted as deviations of the parameters from their nominal values, or as a structural modification of the constraint itself. As bond graphs (BG) are most suited to the efficient modeling and simulation of physical systems [1], [2], significant research efforts have been oriented towards the development of BG-based FDI techniques. The diagnostic bond graph method (DBG), an implementation on the BG domain of the ARR-approach to FDI, counts among the most recent succesful contributions in the FDI field. It has been chosen here as a tool to solve a FDI problem on a relatively complex fourwheel vehicle suspension model. This is a problem of undoubtedly utmost importance when it comes to guarantee the secure and comfortable operation of a ground vehicle. Several BG models [3], [4], [5], have been taken into consideration in order to formulate the BG of the vehicle underlying the DBG developed to solve the FDI problem. The rest of the paper is organized as follows. Section 2 describes both the ARR and the DBG techniques in an ad-hoc manner on a quarter-car vehicle model, which is useful for the sequel. Section 3 presents the BG of the full 4-wheel vehicle. Section 4 addresses the construction of the DBG, the residuals expressions, and the fault signature matrix. Finally, Section 5 concludes providing some simulation results that illustrate the performance of the proposed fault diagnosis method.

2. ARR-BASED FDI ON BG The scheme of Figure 1 outlines a typical MB-FDI system. There, the block MM stands for some mathematical model producing an output whose evaluation and further processing allows to diagnose the fault. How to do it using the ARR technique on Bond Graphs is explained next.

Fig. 1. General framework for model based FDI.

2.1 Analytical Redundant Relationships An ARR is a (partial) dynamic model written in terms of only known input and measured system variables; when equated to zero it expresses the condition of normal operation. A residual is the ARR’s deviation from zero due to a fault. Consider for instance the quarter-car model in Fig. 2, with two speeds and an effort being measured as depicted. The effort equation of the junction 1:V2 provides the ARR1 (1) as well as its associated residual Res1 (2). As part of the MM in Fig. 1, Res1 would be fed by the three measurements V1, V2 and FKs, and parameterized by the nominal values of its coefficients m2 and Bs. Thus, Res1 is able to detect a fault in the damper, for instance. Indeed, in normal condition, Res1 would yield a zero output value (only approximately in practice, due to model uncertainty, and measurement and signal processing errors), while in the presence of a faulty damper Res1 would deviate from zero. Remark 1. For further reference, it should be kept in mind that the terms of Res1 are generalized efforts, i.e., power covariables of the generalized flow V2, the measured variable.

m2 V&2 − FKs − Bs ⋅ (V1 − V2 ) = 0 Res1 = m2 V&2 − FKs − Bs ⋅ (V1 − V2 )

To illustrate all of this, consider again the quarter car model. The DBG given in Fig. 3 uses only a submodel of the BG in Fig. 2, because its purpose is diagnosing faults only in the suspension (composed by damper and spring) and, as the wheel speed V1 is measured, the wheel submodel is unnecessary (and so the road profile related input V0). Throughpower 1- and 0-junctions have been added next to the MSf:V2 and MSe:FKs respectively, in order to capture the residuals Res1 (2) and Res2 (3) with the usual convention on BGs (cf. Remark 1). The presence of only Ks in Res2 indicates its ability to detect a fault in the spring but not in the damper, while the opposite happens with Res1. Thus, the conditions for the Monitorability (Mb) and Isolability (Ib) of both faults are given. This is shown by the Fault Signature Matrix in Table 1.

Fig. 3. DBG of quarter car BG.

(1)

Res2 = (1 / K s ) ⋅ F&Ks + V2 − V1

(3)

(2)

Table 1. Fault Signature Matrix for Quarter-car Model Component Ks Bs

Res1 1 0

Res2 0 1

Mb 1 1

Ib 1 1

— Res1

— Res2 — m2 position Fig. 2. Quarter-car models. a) Mechanical sketch; b) Bond graph.

2.2 The Diagnostic Bond Graph and the FSM The ARR-based method reviewed in this section uses the DBG as the MM producing the residuals. A DBG [6]-[8] is a BG of the plant to which the measured signals as well as the original input sources are injected, the formers via modulated sources. Feeding the measurements enforces causality inversion in all storages (in most cases), thus yielding a DBG in derivative causality, which is a beneficial fact because in this way the initial conditions of the states are not needed [6], [10]. On the other hand, and most important, the power covariable associated to the measured (power) variable being injected by each modulated source directly yields a residual (cf. Remark 1).

— road profile

normal operation

5% decrease in Ks

normal operation

5% decrease normal in Bs operation

Fig. 4. FDI of suspension on quarter-car DBG.

The injection of a sinusoidal road profile on the scheme of Fig. 3 has been simulated with 20sim™ [12] as shown in Fig. 4. Instead of the (real and instrumented) Vehicle-and-road-system its BG model as in Fig. 2 has been used. The correspondence between the entries of Table 1 and the residuals is clearly seen.

•

Remark 2. A full FDI scheme would need complements and refinements such as observers to obtain non measurable variables, adaptive thresholds for residuals to cope with model uncertainties and measurement noise, etc. None of these issues is considered here, but the derivation of the basic scheme for FDI of suspension faults on a full 4-wheeled vehicle BG model.

Fx = m v x + mω y v z − mω z v y

3. BG OF 4-WHEELED VEHICLE

Tx = I xx ω x + ω y I zz ω z − ω z I yy ω y

3.1 Model Assumptions

T y = I yy ω y + ω z I xxω x − ω x I zz ω z

Fig. 5 shows a vehicle model with body, suspensions and tires, which has 14 DoF: the translational displacements (longitudinal, lateral and vertical) and the rotations (roll, pitch and yaw) of the (rigid) body, the rotations of the 4 wheels, and the vertical displacements of the 4 suspensions. The steering angle (δ) is the same for both frontal wheels. Two different frames are considered, the local frame (x,y,z) attached to the body’s center of mass and aligned with its principal directions, and the global frame (X,Y,Z) fixed to the earth. The body geometric features and local forces are referred to the former system, while all components are referred to the later. The body has mass m and moments of inertia Ixx (roll), Iyy (pitch) and Izz (yaw) about the x-, y- and z-axis, respectively. The suspension is represented by a spring-damper system (Ks, Bs) as shown in Fig. 5. The wheel parameters are: mass mu, moment of inertia Iω , and tire elastic coefficient Kt. Pacejka’s models are used for both the longitudinal and the lateral dynamics of the tire-road interaction [9].

•

Fy = m v y + mω z v x − mω x v z

(4)

•

Fz = m v z + mω x v y − mω y v x •

•

(5)

•

Tz = I zz ω z + ω x I yy ω y − ω y I xxω x

Fig. 6. BG model of the body Euler’s equations.

Vertical dynamics The tire-suspension vertical dynamics shown in Fig. 7 replicates the BG of Fig. 2 at each vehicle corner, where the suspension is attached to the complex body model instead of being just to a simple I-storage.

Fig. 5. Full car suspension model.

3.2 Bond Graph Model of 4W-Vehicle In the sequel the BG models of the body, suspension and wheel-road subsystems are discussed, as well as the submodels necessary to get the overall vehicle BG via subsystem coupling.

Euler’s equations for 3-D body motion The non-linear differential equations (4) and (5) -known as Euler’s equations- represent the body motion in the 3D local frame. Fig. 6 shows its BG representation [2]. The forces Fx,y,z and the momenta Tx,y,z result from the effect of the tire and suspension subsystems. Each of them is next modeled decomposed in the x-, y-, and z-axes, which correspond to the longitudinal, lateral and vertical dynamics, respectively.

Fig. 7. BG of one tire-suspension subsystem.

The force Fz exerted on the body center of mass is computed in (6), where Fzi is the i-th suspension force.

Fz =

∑

i=4 i =1

Fzi

(6)

Lateral dynamics The total component Fy over the body results from the contribution of each lateral force, each of them being determined by Pacejka’s lateral model. In the following equations FzNi is the i-th tire normal force, and sai is the i-th slip angle.

Fy = F lat1. cos δ + F lat 2. cos δ + F lat 3+ F lat 4 Flat i = f NL ( Fz N i ; sai )

(7)

According to [9], the equations for the slip angles (frontal and rear) are the followings: vy vy b.ω z a.ω z ; sa3, 4 = − + sa1, 2 = δ − − 2 2 2 2 vx

vx + v y

vx

vx + v y

Longitudinal dynamics The total force Fx in (8) is composed by all the longitudinal forces of the tires and the projection of the two lateral forces (Flat1 and Flat2) over the x-axis. The model also considers the aerodynamic force, whose value depends on the longitudinal speed Vx.

Fx = F long1. cos δ + F long 2. cos δ + F long 3+ F long 4 Flong i

− F lat1.sin δ − F lat 2.sin δ − Faero = f NL ( Fz N i ; slipi ) Pacejka’s longitudinal model

slipi =

R .ωWheel i − Vx R .ωWheel i

, Faero =

(8)

1 C x .ρ .Vx2 . A 2

;

⎡v x ⎤ ⎡v X ⎤ ⎢ ⎥ ⎢ ⎥ = ΨΘΦ v ⎢v y ⎥ ⎢ Y ⎥ ⎢v ⎥ ⎢⎣v Z ⎥⎦ ⎣ z⎦

The inverse chain of transformations is necessary to apply the total vehicle weight (WeightZ, aligned with the Z-axis) to the body (in its local frame). In this way, Weightx Weighty and Weightz are added to the total forces over each axis. The overall inverse transformation can be easily obtained knowing that inverse and transposed matrices can be exchanged because dealing with orthogonal matrices. ⎡Weight x ⎤ ⎡ 0 ⎤ ⎢ ⎥ ⎥ t t t⎢ = Φ Θ Ψ Weight 0 y⎥ ⎢ ⎢ ⎥ ⎢Weight ⎥ ⎢Weight Z ⎦⎥ ⎣ z⎦ ⎣

Vehicle full model The transmission of every force (as force as well as torque) to the sprung mass (represented by Euler’s equations (4), (5) modeled in Fig. 6) can be represented by three sets of transformers that capture the set of the previously presented equations: TF-lateral, TF-vertical and TF-longitudinal. This is indicated in Fig. 8 (placed at the end of this paper), which depicts the vehicle full BG model. These multiport transformers return the set of velocities to the submodels of the vertical, lateral and longitudinal dynamics.

4. DIAGNOSTIC BOND GRAPH

The coefficient Cx stands for the aerodynamic resistance; ρ for the air density; and A for the vehicle’s total frontal area.

Moments over the body The set of equations (9) stands for the moments over the body produced by the above mentioned forces. TM is the sum of the moments that the engine produces over the four wheels.

Tx = ( Fy1 + Fy 2 + Fy 3 + Fy 4 ).h + ( Fz1 + Fz 3 ).(−c) + ( Fz 2 + Fz 4 ).d Ty = −( Fx1 + Fx 2 + Fx3 + Fx 4 ).(h − R ) + ( Fz1 + Fz 2 ).(−a ) + ( Fz 3 + Fz 4 ).b + TM

⎡ω x ⎤ ⎡ω X ⎤ ⎢ ⎥ ⎢ ⎥ = ΨΘΦ ω ⎢ω y ⎥ ⎢ Y ⎥ ⎢ω ⎥ ⎢⎣ω Z ⎥⎦ ⎣ z⎦

The decision on the physical quantities to be measured is crucial to the construction of the DBG. In the sequel, the following measurements are considered: the efforts produced by all four suspension springs, the vertical speeds of all four wheels centers of mass, the speed of the body center of mass, and the body rotational speed. The knowledge of the vertical speeds of the wheels allows disregarding the tire-road submodel when constructing the DBG, whose only purpose is to diagnose faults in the suspensions. The setup in Fig. 9 illustrates this idea. This is a very important issue from a practical point of view, since the tire-road interaction is not only very complex, but also very uncertain.

(9)

Tz = ( Fx1 + Fx 3 ).c + ( Fx 2 + Fx 4 ).(−d ) + ( Fy1 + Fy 2 ).a + ( Fy 3 + Fy 4 ).(−b)

Coordinate transformation In order to evaluate the evolution of the vehicle respect to the ground it is necessary to convert the variables (x,y,z) of the local frame into the variables (X,Y,Z) of the global frame. This renders possible the calculation of the angular and linear speeds in the global frame. The overall transformation can be thought as a chain of the three rotations Φ , Θ and Ψ , which rotate the amount of the roll, the pitch and the yaw angle, respectively [2]. These transformations operate on any vector via succesive multiplication of the three component vector by the corresponding rotation matrices Φ , Θ and Ψ .

Fig. 9. Overview of diagnostic scheme used in this study.

DBG generation The BG-based FDI-scheme is shown in Fig. 10. The bottom part is the vehicle model (or the vehicle itself, in the case of an implementation), and the upper part in the box gives an overall view of the DBG. The latter receives the corner speed (Zi_speed) (each corner speed being calculated from the three sensed body variables and the body dimensions), the

wheel’s center of mass speed (mui_speed) and the spring effort (Ksi_effort) of each suspension, with i = 1-4. The DBG of the suspension, the only shown in detail in Fig. 10, is completed with the oval box labelled DBG BODY, whose details are brought in by Figs. 11 (the DBG of the Euler’s equations) and 12 (the DBG of the transformers). Looking into the details it is seen that all storages are in derivative causality, which makes the DBG insensitive to the system initial conditions.

and Fz’4 (where “prime” is there to distinguish these forces from the “real” ones in the faulty system model). Fig. 11 shows the sub-BG of Euler’s equations in derivative causality. It outputs the forces Fx,y,z and the torques Tx,y,z, which are used to compute the efforts F’1,2,3,4 over the DBG of the coupling sub-BG corresponding to the transformer set TF of Fig. 8. This is done by the DBG of Fig. 12, which deserves some explanation.

Fig. 11. DBG of Euler’s equations in derivative causality.

Fig. 10. Total DBG and total vehicle bond graph model.

As suggested in Fig. 10, two residuals are obtained for each suspension, as follows: 1 d Res1 = ⋅ ( Ks1 _ effort ) − ( Z1 _ speed − mu1 _ speed ) Ks1 dt 1 d Res 2 = ⋅ ( Ks 2 _ effort ) − ( Z 2 _ speed − mu2 _ speed ) Ks 2 dt 1 d Res3 = ⋅ ( Ks3 _ effort ) − ( Z 3 _ speed − mu3 _ speed ) Ks3 dt Res 4 =

1 d ⋅ ( Ks 4 _ effort ) − ( Z 4 _ speed − mu4 _ speed ) Ks 4 dt

ResA= F1′ − (Ks1 _ effort) − Bs1 ⋅ (Z1 _ speed− mu1 _ speed) ResB= F2′ − (Ks2 _ effort) − Bs2 ⋅ (Z2 _ speed− mu2 _ speed) ResC = F3′ − (Ks3 _ effort) − Bs3 ⋅ (Z3 _ speed− mu3 _ speed) ResD= F4′ − (Ks4 _ effort) − Bs4 ⋅ (Z 4 _ speed− mu4 _ speed) The symbols F’1,2,3,4 in the expressions of ResA,B,C,D stand for the efforts calculated by the block DBG BODY after processing the four corner speeds, ωz , Vx, Vy, and TM. For a reason that will become apparent next, these efforts are not, as a-priori expected, the four corner vertical forces Fz’1, Fz’2, Fz’3

The efforts in (11) -defined from the set of effort equations (10) (cf. (9)et(6))-, are used to calculate the desired set of corner_speed efforts with the TF DBG of Fig. 12. There, the assumption of small angular displacements has been done, which results in a linear BG parameterized by the body dimensions [11]. The sub-BG implementing (11), necessary to couple the DBGs of Figs. 11 and 12, is not shown.

Tx = Fy .h + ( Fz1 + Fz 3 ).(−c) + ( Fz 2 + Fz 4 ).d T y = − Fx .(h − R ) + ( Fz1 + Fz 2 ).(−a) + ( Fz 3 + Fz 4 ).b − TM

(10)

Fz = Fz1 + Fz 2 + Fz 3 + Fz 4 + Weight z

Tx' = Tx − Fy .h T y' = Ty + Fx .(h − R ) + TM

(11)

Fz' = Fz − Weight z The sub-model TF DBG calculates four efforts from the system of three independent equations (11). It means that the set F’1,2,3,4 returned by the DBG does not equal the wanted set Fz’1,2,3,4. In fact, they are equal up to an undetermined and arbitrary constant K, as illustrated in Fig. 13. A quartet of forces F = [F1 F2 F3 F4]T produces a unique combination of accelerations in the corners. Clearly, the particular set of forces in Fig. 13 does not produce any acceleration. Thus, we can infer that, inversely, the information on the corner acceleration is not enough for knowing F. The force that the DBG calculates is F = [F1+K F2–K F3–K F4+K]T, where K is unknown.

body, the components where a fault may occur are those corresponding to the suspensions. According to [6], if the component fault is shown at least in a residual, then it can be monitored (Mb=1). If the residuals combination is unique, the fault is also isolable (Ib=1).

Table 2. Fault Signature Matrix for 4-W Vehicle Model Component Res1 Res2 Res3 Res4 Res5 Res6 Res7 0 0 0 0 0 0 Ks1 1 Bs1 0 0 0 0 1 1 1 Ks2 0 0 0 0 0 0 1 Bs2 0 0 0 0 0 0 1 Ks3 0 0 0 0 0 0 1 Bs3 0 0 0 0 0 0 1 Ks4 0 0 0 0 0 0 1 Bs4 0 0 0 0 0 0 1

Fig. 12. TF DBG.

This means that the residuals ResA,B,C,D will remain different from zero even under non-faulty condition. But this can be easily solved if the residuals are generated as the linear combination (12) of the former residuals. Indeed, the new residuals are zero under non-faulty conditions.

Fig. 13. A particular load configuration on the body.

Res5 = ResA+ ResB = F1 − (Ks1 _ effort) − Bs1 ⋅ (Z1 _ speed − mu1 _ speed) + + F2 − (Ks2 _ effort) − Bs2 ⋅ (Z 2 _ speed − mu2 _ speed)

The final part of any FDI method consists in processing the residuals. The processing algorithm depends on how the full FDI system is implemented (chosen system model, kind of sensors, sensing method, noise presence, etc). This paper addresses only the core FDI-algorithm, and does not present any further processing except for the calculation of the sensitivities respect to the parameters that are likely to suffer a fault. It is concluded from the equations below that no fault can be detected while the suspensions are in their steady state. S1, Ks1 =

1 ∂Res1 d =− ⋅ ( Ks1 _ effort ) ∂Ks1 Ks1 2 dt

S 2, Ks 2 =

1 ∂Res 2 d =− ⋅ ( Ks 2 _ effort ) 2 dt ∂Ks 2 Ks 2

S 3, Ks 3 =

1 ∂Res3 d =− ⋅ ( Ks3 _ effort ) ∂Ks 3 Ks 3 2 dt

S 4, Ks 4 =

1 ∂Res 4 d =− ⋅ ( Ks 4 _ effort ) 2 dt ∂Ks 4 Ks 4

∂Res5 = −( Z1 _ speed − mu1 _ speed ) ∂Bs1

S 5, Bs 2 =

∂Res5 = −( Z 2 _ speed − mu 2 _ speed ) ∂Bs 2

S 6, Bs1 =

∂Res6 = −( Z1 _ speed − mu1 _ speed ) ∂Bs1

S 6, Bs 3 =

∂Res6 = −( Z 3 _ speed − mu 3 _ speed ) ∂Bs3

− F1 + (Ks1 _ effort) + Bs1 ⋅ (Z1 _ speed − mu1 _ speed) + +

S 7, Bs1 =

∂Res7 = ( Z1 _ speed − mu1 _ speed ) ∂Bs1

F4 − (Ks4 _ effort) − Bs4 (Z 4 _ speed − mu4 _ speed)

S 7, Bs 4 =

∂Res7 = −( Z 4 _ speed − mu 4 _ speed ) ∂Bs 4

F1 − (Ks1 _ effort) − Bs1 ⋅ (Z1 _ speed − mu1 _ speed) + + F3 − ( Ks3 _ effort) − Bs3 ⋅ (Z3 _ speed − mu3 _ speed)

(12)

Res7 = −ResA+ ResD =

Ib 1 1 1 1 1 1 1 1

Sensitivity analysis

S 5, Bs1 =

Res6 = ResA+ ResC =

Mb 1 1 1 1 1 1 1 1

Fault Signature Matrix The fault signature matrix in Table 2 is built from the residuals (12). It is important to mention that the total suspension forces (F1, F2, F3 and F4) are calculated using the body dynamic evolution and the body parameters (m, Ixx, Iyy, Izz, a, b, c, d and h). Under the hypothesis of non-faulty sensors and

5. SIMULATION RESULTS In order to verify its performance, the FDI scheme presented above has been intensively studied via simulation under very different drive conditions in both normal and faulty

situations. For all possible faults the residual evolutions were as predicted by the fault signature matrix. All the simulations have been performed using the software 20-sim™ [12], that allowed for a specification of the simulation problem directly in the BG language for both of its main components, the complex system BG and the DBG. The results of three different simulation experiments out of the whole set are presented next. All of them were generated with the vehicle being initially at rest and receiving the same inputs: Motor torque on each of wheels 1 and 2 = 95.0 Nm; motor torque on each of wheels 3 and 4 = 0.0 Nm; road vertical speed 1 = sin(20.t) m/s; road vertical speed 2 = 0.25 sin(30.t) m/s; road vertical speed 3 = 0.3 sin(25.t) m/s; road vertical speed 4 = 0 m/s; steering angle command: the last curve plotted in figure 14.

second maneuver (cf. the third term on the right-hand side of the force Fx = m. V&x + m.ωy.Vz - m.ωz.Vy , knowing that during the curve Vy≠0 and at high speed ωz is high) and the rather irregular shape of Vy (the suspensions are strongly loaded during this second interval). Fig. 15 plots the X-Y trajectory described by the vehicle in the global frame. The next two figures illustrate the residuals evolution in faulty conditions. Fig. 16 shows the effects of the following faults in the springs of the first and third suspensions: 5% reduction in Ks1 during the time interval (5s, 11.5s), and 5% increase in Ks3 from 15 to 22 seconds. Fig. 17 shows the effects of faults in the dampers of the second and first suspensions: 15% increase in Bs2 from 5 to 10 seconds, and 5% reduction in Bs1 from 15 to 21 seconds.

Fig. 16. Residuals under faults in two suspension springs.

Fig. 14. Center of mass vertical position, Vx ,Vy and steering angle.

Fig. 17. Residuals under faults in two suspension dampers.

The simulation results confirm that the residuals evolve as predicted by the fault signature matrix.

6. CONCLUSIONS Fig. 15. X-Y trajectory.

The normal behavior of the vehicle under the action of the referred inputs is shown in Figs. 14 and 15. The center of mass vertical Z_position is shown on the top of Fig. 14, followed by Vx , Vy (as referred to the body’s local frame), and the steering angle. The transient in the Z_position during the first few seconds, which is due to the vehicle’s weight, is followed by the vertical oscillations induced by the ground profile (road vertical speeds). The first steering command happens at low speed (Vx ≈15 km/h), while the second occurs at higher speed (Vx ≈60 km/h). This explains the plateau in Vx during the

A model based system for fault detection and isolation in vehicle suspensions has been presented. Bond graph modeling and simulation of the four-wheel vehicle dynamics addressed in this paper not only confirmed the capability of this formalism to deal whith complex physical systems, but also allowed to solve the FDI-problem completely in the BG domain. To this aim, the diagnostic bond graph method has been used, a technique that generates residuals sensitive to the faults on the base of analytical redundant relationships. Including the vertical speed of the wheels among the measured variables enabled formulating a simpler DBG (with regards to the original vehicle full BG model) that avoids using

the complex and uncertain tire-ground interaction model and thus preserves the FDI system from uncertainties that would affect its accuracy. As confirmed by simulation results, the system designed is able to detect and isolate all the faults considered in the suspensions components (passive dampers and springs). Further work will address observer design, FDI on active suspension models as well as fault tolerant suspension control. Acknowledgments: The authors wish to thanks Dr. Norberto Nigro from the School of Mechanical Engineering, Faculty of Engineering, National University of Rosario, for fruitful discussions on automobile dynamics.

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[5] C. Niesner, G. Dauphin-Tanguy, D. Margolis, F Guillemard, M. Pengov. “A 4 wheel vehicle bond graph model including uncertainties on the car mass and the centre of mass position”, Proc. ICBGM’05, 2005, pp.179184. [6] K. Medjaher, A. K. Samantaray, B. Ould Bouamama, “Diagnostic Bond Graphs for Direct Residual Evaluation”, Proc. ICBGM’05, 2005, pp.307-312. [7] K. Sia, A. Naamane, “Bond Graph: a suitable tool for component faults diagnosis” Proc. ICBGM’03, 2003, pp. 89-102. [8] B. Ould Bouamama, A. K. Samantaray, M. Staroswiecki, G. Dauphin-Tanguy, “Derivation of Constraint Relations from Bond Graph Models for Fault Detection and Isolation”, Proc. ICBGM’03, Simulation Series, Vol.35, No.2, 2003, pp.104-109. [9] F. Aparicio Izquierdo, C. Vera Alvarez, V. Díaz López, “Teoría de los vehículos automóviles”, Ed. Madrid: Escuela Técnica Superior de Ingenieros Industriales de Madrid, 1995. [10] S. K. Ghoshal, A. K. Samantaray, A. Mukherjee, “Improvements to Single Fault Isolation using Estimated Parameters”, Proc. ICBGM’05, 2005, pp. 301-306. [11] D. José Bel Cacho, “Formulación e implementación de una metodología de elementos finitos para el análisis de modelos bond graph de sistemas discretos-continuos”, Tesis doctoral, Dept. Mech. Eng., Zaragoza Univ., Spain. [12] Controllab Products B.V., 20-sim™, www.20sim.com/.

Fig. 8. Overall BG-model of vehicle.