development and verification of a series car modelica ... and veriﬁcation of a series car...

The Modelica Association Modelica 2006, September 4th-5th,2006

Development and Verification of a Series Car Modelica/DymolaMulti-Body Model to Investigate Vehicle Dynamics Systems

Christian Knobel∗ Gabriel Janin‡ Andrew Woodruff§

∗BMW Group Research and Technology, Munich, Germany, [email protected]‡Ecole Nationale Superieure de Techniques Avancees, Paris, France, [email protected]

§Modelon AB, Lund, Sweden, [email protected]

Abstract

The development and the verification of a Multi-body model of a series production vehicle in Mod-elica/Dymola is presented. The model is used to in-vestigate and to compare any possible configuration ofactuators to control vehicle dynamics with a generalcontrol approach based on model inversion and a non-linear online optimization.Keywords: Multi-body Vehicle Model, Vehicle Dynam-ics, Model Verification, Model Validation, Active Vehi-cle Dynamics Systems, Tire Model, Suspension Kine-matics, Suspension Compliance

1 Introduction

Systems for control of Vehicle Dynamics went to se-ries production for the first time in 1978 with the limi-tation of brake pressure to avoid locking the wheels toensure cornering under all braking conditions. Brak-ing individual wheels (independently from the driverscommands) to stabilize the vehicle at the driving limitwent to series production in 1995. In the last fewyears, control systems for vehicle dynamics with ad-ditional actuators to control steering, drive torque dis-tribution and wheel load distribution have entered themarket.All of these systems acting on the force allocationfrom the center of gravity (CG) to the four tire con-tact patches (TCP) and on the force transfer at theTCPs. This strong interdependence between thesesystems1 is the reason why independent operationof more than one of them is only possible witha loss of potential to prevent critical interferences.In [2] (cf. also [3], [4], [5], [6] and [7]) a gen-eral approach was introduced to investigate the ref-

1cf. [1] for an overview and a detailed classification of systemsfor vehicle motion control.

erence behavior (best possible allocation and transferof forces acting on the vehicle) for any configurationof actuators controlling vehicle dynamics including allsteering angles δ =

[δ1 δ2 δ3 δ4

]T , brake/drive

torques M =[M1 M2 M3 M4

]T , wheel loads

Fz =[Fz1 Fz2 Fz3 Fz4

]T and even camber angles

γ =[γ1 γ2 γ3 γ4

]T . The comparison of the ref-erence operation of different configurations may sup-port decisions in the future development of vehicle dy-namics. The investigation of the reference behavioralso supports controller development and the dynamicspecification of the actuators, or makes it possible toinvestigate the potential loss if the applied actuatorshave less dynamics compared to the ideal required dy-namics. The reconfigurable behavior of the generalapproach allows further investigation of the impact ofactuator failures on vehicle dynamics for reliability in-vestigations.

δ3

γ3M3

Fz3

β ψ&v

CG

1

2

4

3

Figure 1: Planar vehicle model with influencing vari-ables and vehicle motion y

Because the approach presented in [2] is based onmodel inversion of a plane vehicle model with theplane motion y =

[ψ β v

]T described by the yawrate ψ, the body slip angle β and the velocity v (cf.Figure 1), a verified multi-body model is needed as ve-

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Development and Verification of a Series Car Modelica/Dymola Multi-body Model to Investigate Vehicle DynamicsSystems


hicle model to ensure that the control approach worksalso with all effects neglected during the controllerdesign. Using the verified Modelica/Dymola Multi-body vehicle model presented in this paper as a vehi-cle model for the control approach allows investigationand comparison of all possible configurations of avail-able actuators quickly and easily.The possibility to model multi-body suspension as-semblies, controllers, hydraulic and mechatronic ac-tuators in one and the same environment was the rea-son to choose Dymola as the modeling and simulationtool. Development and verification of the model wasdone bottom-up. Multi-body front and rear suspen-sion, tires, steering system, power train and the bodywere modeled and then verified separately with testrig results as shown in Section 2. Creating the multi-body vehicle model by connecting those subsystemstogether is presented in Section 3 as well as the verifi-cation process of the full vehicle through objective testmaneuvers with a series car equipped with additionalmeasurement technology. Finally, an application ex-ample is presented in Section 4.

2 Development and Verification ofSubsystems

Using the Multi-Body Library [8] and the Vehicle Dy-namics Library [9], tire, suspension, body and envi-ronment for the multi-body model were constructed.Simple models for the steering and the power trainsystem are developed only for the verification of themulti-body vehicle model with a conventional seriesproduction vehicle.

2.1 Tire

Pacejka’s Magic Formula [10] is used to model theplane transfer behavior of the tire, which calculatesfirst the pure forces

F ′xoi = Dsin(C arctan(Bκ−E(Bκ− arctanBκ)))

F ′yoi = Dsin(C arctan(Bα−E(Bα− arctanBα)))

(1)

i ∈ {1 . . .4} designated with ′ to indicate the represen-tation with the wheel coordinate system out of the in-puts longitudinal slip κ, tire side slip angle α, wheelload Fz and camber angle γ. Secondly,

F ′xi = F ′

xoi cos(C arctan(Bα−E(Bα− arctanBα)))F ′

yi = F ′yoi cos(C arctan(Bκ−E(Bκ− arctanBκ)))

(2)

the interdependence between the longitudinal and lat-eral tire forces is considered where the peak parame-ter D(Fz,γ), the shape parameter C, the stiffness pa-rameter B(Fz,γ) and the curvature parameter E(Fz,γ)are different for (1), (2) and for longitudinal and lat-eral directions, respectively. For the identification ofthese parameters, an error minimization is used to fitthe model result to the test rig results of the tire used(cf. Figure 2 and Figure 3). Because braking (negativelongitudinal tire forces) is more important for vehicledynamics control than accelerating (positive longitu-dinal tire forces), different weighting factors are usedto get a better correlation of the negative longitudinaltire forces.

−30 −20 −10 0 10 20 30−8000

−6000

−4000

−2000

0

2000

4000

6000

8000

κ [%]

Fx [N

]

Tire Test RigModel Simulation

Figure 2: Longitudinal force Fx over longitudinal slipκ for different wheel loads Fz

−8000 −7000 −6000 −5000 −4000 −3000 −2000 −1000 00

1000

2000

3000

4000

5000

6000

7000

Fx [N]

Fy

[N]

Fz = 6340N α = −1°

Fz = 6340N α = −9°

Fz = 4565N α = −9°

Fz = 4896N α = −1°

Figure 3: Lateral force Fy over longitudinal force Fx

(longitudinal slip κ sweeps at different tire side slip an-gles α and wheel loads Fz) known as a Krempel Graph

168

C. Knobel, G. Janin, A. Woodruff


2.2 Suspension

For the front and rear suspension, ADAMS modelscould be used as source to model the McPhersonfront suspension (cf. Figure 4) and the semi-trailing

Rear Link

Front Link

Steering Link

ARB link

Spring

Strut (damper)

ARB

Upright

X Y

Z

Figure 4: McPherson front suspension design

arm rear suspension design (cf. Figure 5) in Model-ica/Dymola.

Upper Link

Lower Link

ARB link

Spring

Damper

ARB

Trailing Arm

X Y

Z

Figure 5: Semi-trailing arm rear suspension design

The default models for those suspension designs in theVehicle Dynamics Library [9] could not be used with-out customizing and modifying the design, the jointlocation and the kinematic relationships to match thebehavior of the source models in ADAMS.The McPherson front suspension used independentlower rods instead of the conventional control arm in[9]. The rear suspension used a trailing arm designwith two guiding links, and the body spring and anti-roll subsystems were attached to the upper guiding link(cf. [11]). Non-linear bump stops were added on thedamper’s tube of the front and rear suspension.The hard points for all joint locations of the modelneed to be adjusted to agree with the ADAMS sourcemodel.

The kinematics of the suspension models are verifiedusing a vertical travel sweep test rig which needs to bemodeled in Modelica/Dymola as well. Camber and toechanges are plotted to verify the kinematic behavior ofthe suspension models with the source model and thereal suspension of the used series vehicle (cf. Figure6).

-5 -4 -3 -2 -1 0 1 2 3 -100

-80

-60

-40

-20

0

20

40

60

80

100

γ (°)

Sus

pens

ion

Tra

vel (

mm

)

Dymola ModelSource ModelTest Rig LeftTest Rig Right

-5 -4 -3 -2 -1 0 1 2 3

γ (°)

-40 -30 -20 -10 0 10 20 30 -100

-80

-60

-40

-20

0

20

40

60

80

100

δt (min)

Sus

pens

ion

Tra

vel (

mm

)

-40 -30 -20 -10 0 10 20 30 δ

t (min)

Figure 6: Kinematic analysis: camber in degrees andtoe in angular minutes for front (left) and rear suspen-sion (right)

Simulation of the rigid suspension model (without anybushings) was impossible and caused singularity er-rors. After the implementation of bushings, simula-tion of the multi-body front and rear suspensions waspossible. However, a pure investigation of the rigidkinematics was only possible using very stiff bush-ings, which is the reason for the differences betweenthe source model and the Modelica/Dymola model inFigure 6.The kinematics of the real suspension could only beverified with bushings, which is again the main reasonfor the differences between the model results and thereal test rig results shown in Figure 6.

2.3 Further Subsystems of the Vehicle

After modeling the tires and suspensions, adding body,power train and steering system models, as well as thevehicle’s environment, is necessary to the completemulti-body vehicle model.The vehicle’s body is considered to be rigid and itsmass is distributed as follows: one summarized sprung

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mass, including the driver, one passenger and fuel isin the body whereas the unsprung mass of the wheelsincluding brake caliper and rotor and their links is dis-tributed to the four wheels. The vehicle’s inertias atthe CG are identified on a pendulum test rig.The complexity and accuracy of the power trainand steering models are rather low because they areonly used to get reasonable connections between thedriver’s inputs and the brake/drive torques M and thewheel steer angles δ. The former uses a speed con-troller and a differential gear to distribute the torquesto the four wheels similar to the series production vehi-cle. The latter consists a rack-and-pinion steering sys-tem including a rotational spring in the steering shaft.The study of a full-vehicle model requires the model-ing of its environment. The equations of motion canonly be solved by having a complete description ofphysical system. Therefore, the interaction betweenthe vehicle and the world must be taken into account.It consists the interactions between vehicle and driver,vehicle and air, and tire and road. The road is modeledby a flat surface with with a road friction coefficientµ. The aerodynamic drag force Faero

x =−12 Aρcxv2 ap-

plied at the center of gravity simplifies the interactionbetween air and vehicle. These environments are fromthe Vehicle Dynamics Library [9]. Only driver modelsneed to be built up to be able to simulate the objectivetest maneuvers for the verification in Section 3.The active control actuators are modeled by ideal rev-olute joints inserted at the rigid connection betweenthe suspension and wheel subsystems. This meant thewheels could be manipulated directly and without al-tering the suspension geometry. Passive systems arerepresented by constant values as inputs for the idealactuators.

3 Development and Verification ofthe Multi-body Vehicle Model

Connecting the subsystems from Section 2 creates themulti-body vehicle model.Important steps are the definition, design and imple-mentation of the model. A more important step isto check if the model matches real vehicle behav-ior. Therefore, the vehicle and the model behaviorare compared with objective test maneuvers such assteady state cornering (steady state behavior), steeringsteps (dynamic behavior in the time domain) and sine-sweeps (dynamic behavior in the frequency domain).Body side slip angle β and velocity v are measuredusing an optical Correvit sensor, roll ϕ and pitch θ

Figure 7: Chassis representation of the multi-body ve-hicle model

0 1 2 3 4 5 6 7 8 940

60

80

100

120

140

160

180

ay [m/s2]

δ sw [°

]

Model SimulationTest Maneuver

0 1 2 3 4 5 6 7 8 90

5

10

15

20

25

30

ay [m/s2]

dψ/d

t [°/

s]

0 1 2 3 4 5 6 7 8 9−3

−2

−1

0

1

2

3

ay [m/s2]

β [°

]

0 1 2 3 4 5 6 7 8 9−5

−4

−3

−2

−1

0

ay [m/s2]

φ [°

]

Figure 8: Steady state cornering, r=40m, dry road µ=1

are measured indirectly by suspension travel sensorsat all four wheels, and all translational accelerationsax,ay,az as well as the rotational rates ψ, ϕ, θ are mea-sured by a sensor cluster located at the CG. The steer-ing wheel angle δsw and the steering wheel torque Msw

are measured using an instrumented steering wheel.

The results of the verification without any fitting ofuncertain parameters like stiffness of the bushings, orfriction coefficient µ of the road are presented in Figure8 and Figure 9.

The main reason for the higher yaw rate generated bythe model in both maneuvers is the uncertain road fric-tion coefficient µ. The higher roll in both maneuversis caused by different stiffnesses of the body springsused in the model and the real vehicle.

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−5

0

5

10

15

20

25

30

35

40

45

t [s]

δ sw [°

]

Model SimulationTest Maneuver

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−2

0

2

4

6

8

10

12

14

16

18

t [s]dψ

/dt [

°/s]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−1

−0.5

0

0.5

t [s]

β [°

]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

t [s]

φ [°

]

Figure 9: Step steer, v=70 km/h, dry road µ=1

4 Application of the Model

Exchanging the conventional steering and power trainsystem of the verified vehicle model and using theideal actuators as described in 2.3 leads to a genericconfiguration for vehicle dynamics control. All steer-ing angles δ, drive/brake torques M , wheel loads Fz

and camber angles γ (cf. Figure 1) could be used pas-sively or actively controlled by the general allocationapproach presented in [2]. A non-linear online op-timization calculates the arbitrary parameters for theunder determined inverses of the over-actuated2 planemotion vehicle model (cf. 1). The number of the arbi-trary parameters depends on the available actuators forthe influencing variables. These arbitrary parametersare always used by the non-linear online optimizationto minimize the maximum adhesion potential utiliza-tion

η2i =

(Fxi

Fxi max

)2

+(

Fyi

Fyi max

)2

(3)

(0 ≤ ηi ≤ 1) of the four tires i ∈ {1 . . .4} is approx-imated by an elliptic relation. The forces Fxi max andFyi max depend on the wheel load Fzi and the camber an-gles γi. The control commands out of these optimiza-tion are used as inputs for the multi-body vehicle (cf.Figure 10). Inputs for the optimization are the torqueand forces u =

[MzCG FxCG FyCG

]T acting on thecenter of gravity. The allocation of these forces to theTCPs and the force transfer in the TCPs are optimized

2cf. [12] for definition and examples

Dynamic Inversion Static Inversion

y

Fz

Controlleryd Force

Allocation

udHMI-controller

Msw

sw

yyy xb xg

Figure 10: Control loop with the multi-body vehiclemodel

with the optimization objective

min max ηi (4)

This setup facilitates investigation into the referencebehavior (best possible allocation and transfer offorces acting on the vehicle) of the vehicle dynamicsfor every configuration of available actuators influenc-ing vehicle dynamics for the verified multi-body vehi-cle model.Changing the number of available actuation during adriving maneuver allows investigation into the impactof actuator failures on vehicle dynamics, which maysupport reliability investigations of active vehicle dy-namics systems.Such an investigation is presented as an exemplary ap-plication of the presented multi-body vehicle model(cf. Figure 11, Figure 12 and Figure 13). The frontright steering actuator of a vehicle (equipped with foursteering actuators, four drive torque actuators and fouractuators to control the wheel load distribution) fails.The failure occurs after one second of driving an openloop single lane change at a constant speed v=70 km/h(cf. Figure 13). All actuators and actuator dynam-ics are limited for the optimization (4) to values of ac-tual available actuators. The actuator fails in the worstcase situation at maximum steering angle of δ = 1.8degrees and is assumed to be self-locked after the fail-ure occurred. To compensate this fixed steering er-ror, the vehicle exhibits the same amount of body slipangle as can be seen in Figure 13. Steering back tostraight driving again the failure wheel becomes theouter wheel (with respect to the center of the corner)with more wheel load. Therefore, the optimization re-duces the wheel load at this wheel as much as possi-ble. However, the steering angles and drive torquesat the other wheels are much higher compared to theusual driving condition without failure (right side ofFigures 11, 12 and 13). The maneuver presented isclose to the physical driving limit since two tires havealready reached their maximum possible adhesion po-tential utilization (cf. Figure 12). The lateral acceler-

171



ation ay is reaching 4 m/s2 at the minimum and max-imum of the yaw rate ψ. This performance meets theactual specification of flat run-flat tires.

0 1 2 3−4

−3

−2

−1

0

1

2

3

4

δ [d

eg]

Wheel 1Wheel 2Wheel 3Wheel 4

0 1 2 3

0 1 2 3−500

0

500

M [N

m]

0 1 2 3

0 1 2 3

2000

4000

6000

8000

t [s]

Fz [N

]

0 1 2 3t [s]

Figure 11: Influencing variables of vehicle with steer-ing actuator failure (left) and usual working vehicle(right)

5 Conclusion and Outlook

The development and the verification of a multi-bodymodel of a series production vehicle is presented. Thisvehicle model was used to investigate and compareany possible configuration of actuators to control ve-hicle dynamics. In this context, Modelica/Dymola hasproven to be a practical environment for future devel-opment of vehicle models including mechatronic andhydraulic actuators, multi-body suspensions and con-trollers. To develop, verify and use Modelica/Dymolamodels in an efficient way, however, interfaces toCAD systems to import CAD model data and inter-faces to real time environments are desirable. Es-pecially for this project, a library for non-linear op-timization was missed, which was the reason why

0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

t [s]

η [−

]

Wheel 1Wheel 2Wheel 3Wheel 4

0 0.5 1 1.5 2 2.5 3 3.5t [s]

Figure 12: Adhesion potential utilization of failure(left) and usual vehicle (right)

0 1 2 3−15

−10

−5

0

5

10

15

t [s]

dψ/d

t [de

g/s]

FailureUsual

0 1 2 3−2

−1

0

t [s]

β [d

eg]

0 1 2 350

80

t [s]

v [k

m/h

]

Figure 13: Plane vehicle motion of the vehicle withsteering actuator failure (-) and usual working vehicle(:)

this part of the presented control approach was real-ized in MATLAB/Simulink. The presented Model-ica/Dymola multi-body model was implemented intoMATLAB/Simulink using the Simulink Interface ofDymola.The outlook of the presented project is to improve thematching of the model by using an error minimization.Parameters for the error minimization are probably theuncertain stiffness of the bushings and the road frictioncoefficient µ.The presented example of steering actuator failurecould be improved by adding a strategy of activelycontrolled camber (assuming the availability of sucha system) for steering actuator failures which counter-acts the failure force generated by the tire side slip an-gle αi of the failure wheel.

References

[1] J. Andreasson, C. Knobel, and T. Bunte. OnRoad Vehicle Motion Control - striving towardssynergy. In Proc. of 8th International Symposiumon Advanced Vehicle Control AVEC, 2006.

[2] C. Knobel, A. Pruckner, and T. Bunte. Opti-mized Control Allocation - A General Approachto Control and to Investigate the Motion of Over-

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actuated Vehicles. Submitted paper for 4th IFACSymposium on Mechatronic Systems, 09 2006.

[3] Y. Hattori, K. Koibuchi, and T. Yokoyama. Forceand Moment Control with Nonlinear OptimumDistribution for Vehicle Dynamics. In Proc. of6th International Symposium on Advanced Vehi-cle Control AVEC, 2002.

[4] E. Ono, Y. Hattori, Y. Muragishi, andK. Koibuchi. Vehicle Dynamics ControlBased on Tire Grip Margin. In Proc. of 7thInternational Symposium on Advanced VehicleControl AVEC, 2004.

[5] R. Orend. Steuerung der Fahrzeugbewegung mitminimaler Kraftschlussausnutzung an allen vierRadern. In Proc. of Steuerung und Regelung vonFahrzeugen und Motoren - Autoreg. VDI, 2004.

[6] J. Andreasson and T. Bunte. Global Chassis Con-trol Based on Inverse Vehicle Dynamics Mod-els. Presented at XIX IAVSD World Congress,10 2005.

[7] T. Bunte and J. Andreasson. Integrierte Fahrw-erkregelung mit minimierter Kraftschlussaus-nutzung auf der Basis dynamischer Inversion. InProc. of Steuerung und Regelung von Fahrzeu-gen und Motoren - Autoreg, Wiesloch, 2006.

[8] M. Otter, H. Elmqvist, and S. E. Mattsson. TheNew Modelica MultiBody Library. In Proc. ofthe 3. International Modelica Conference, pages311–330, 2003.

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