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Review and experimental evaluation of models for drivability simulation with focus on tire modeling

Übersicht und experimentelle Bewertung von Modellen für die Fahrbarkeitssimulation mit Fokus auf der Modellierung des Reifens


Recent research showed a significant role of the interaction between traction and torsional vibrations on control design in passenger cars. However, there is a large diversity in the proposed models for drivability control design and validation. This paper gives an overview of popular models in drivability simulation and addresses the quantitative evaluation of these models in a wide range of operating points. Experiments have been performed with diverse excitation signals. Based on these experiments, a number of popular models for control design and validation are identified and compared. A new model is proposed, which will be shown to be a good trade-off between model accuracy and complexity. The results give a guidance for control engineers during the model selection process for either controller concept design, parametrization or validation.


Aktuelle Forschungsergebnisse haben einen wichtigen Einfluss der Wechselwirkung zwischen Traktions- und Drehschwingungen auf das Regelungsdesign in Personenkraftwagen gezeigt. Es gibt jedoch eine große Vielfalt an vorgeschlagenen Modellen für den Entwurf und die Validierung der Fahrbarkeitssteuerung und -regelung. Dieser Beitrag gibt einen Überblick über gängige Modelle in der Fahrbarkeitssimulation und befasst sich mit der quantitativen Bewertung dieser Modelle in einem breiten Bereich von Betriebspunkten. Es wurden Experimente mit verschiedenen Anregungssignalen durchgeführt. Basierend auf diesen Experimenten werden eine Reihe von gängigen Modellen für den Reglerentwurf und die Validierung identifiziert und verglichen. Es wird ein neues Modell vorgeschlagen, das sich als guter Kompromiss zwischen Modellgenauigkeit und Komplexität erweisen wird. Die Ergebnisse geben eine Orientierungshilfe für Regelungstechnikingenieure während des Modellauswahlprozesses für die Konzeption, Parametrierung oder Validierung von Fahrbarkeitsfunktionen.

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  1. All models have been implemented by physical modeling techniques in Simscape™.

  2. The motor torque has been estimated from electrical quantities. The estimator cannot be presented, due to confidentiality reasons.

  3. These algorithms are implemented in Mathwork’s Simulink Optimization and global Optimization Toolbox™ and 100 sampling points have been used.

  4. In [49], the RMS value is given for continuous time. In this work, the trapezoidal rule has been applied for numerical integration.


\(\dot{*}\) :

Derivative with respect to time

\(\hat{*}\) :

Estimated value

\(\overline{*}\) :

Normalized value

\(\tilde{*}\) :

Physical parameter value

\(*_{0.5}\) :

Median of a value

\(\alpha\) :

Average road elevation

\(\alpha_{t}\) :

Tire model parameter

\(\beta_{t}\) :

Tire model parameter

\(\gamma_{t}\) :

Tire model parameter

\(\epsilon\) :


\(\varepsilon\) :


\(\zeta\) :

Axle load ratio

\(\theta\) :

Parameter set

\(\vartheta\) :

Torsion of drivetrain

\(\vartheta_{b}\) :

Relative angle of contact planes

\(\vartheta_{t}\) :

Torsion of tire

\(\mu\) :

Coefficient of traction

\(\rho_{\text{air}}\) :

Air density

\(\tau\) :

Approximate normalized position in backlash

\(\sigma\) :

Longitudinal relaxation length

\(\varphi_{m}\) :

Motor angle

\(\varphi_{w}\) :

Wheel angle

\(\omega_{0}\) :

Natural Frequency

\(\omega_{m}\) :

Rotational motor speed

\(\omega_{t}\) :

Rotational tire speed

\(\omega_{w}\) :

Rotational wheel speed

\(A_{\text{air}}\) :

Cross sectional area of vehicle

\(D\) :

Damping ratio

\(F_{\alpha}\) :

Climbing resistance

\(F_{\text{air}}\) :

Aerodynamic drag force

\(F_{m,r}\) :

Resistance acting on vehicle mass

\(F_{t}\) :

Traction force

\(F_{rr,f/r}\) :

Rolling resistance at front/rear wheel

\(F_{w,z}\) :

Vertical wheel force

\(F_{x}\) :

Longitudinal Force

\(J_{m}\) :

Motor-sided inertia

\(J_{r}\) :

Inertia of two rims

\(J_{t}\) :

Inertia of two tires

\(J_{ts}\) :

Tire-sided inertia

\(J_{w}\) :

Inertia of two wheels

\(J_{ws}\) :

Wheel-sided intertia

\(K_{i}\) :

Gain factors of LPV system

\(N\) :

Number of samples

\(N_{\text{chirp}}\) :

Length of chirp signal

\(N_{\text{PRMS}}\) :

Length of PRMS signal

\(N_{\text{sigs}}\) :

Number of signals

\(T_{m}\) :

Motor torque

\(T_{r}\) :

Wheel resistance torque

\(T_{w}\) :

Wheel torque

\(a_{\text{rms}}\) :

Root mean squared value of \(a_{wx}\)

\(a_{wx}\) :

Frequency weighted acceleration

\(a_{\tau x}\) :

Running root mean square value of \(a_{x}\)

\(a_{x}\) :

Longitudinal acceleration

\(b\) :

Backlash angle

\(b_{\text{fit}}\) :

Fit of simplified backlash model to ideal model at contact

\(c_{d}\) :

Drag coefficient

\(c_{rr}\) :

Coefficient of rolling resistance

\(c_{\text{slip}}\) :

Slip stiffness

\(d\) :

Viscous damping factor of drivetrain

\(d_{c}\) :

Viscous damping factor of chassis

\(d_{t}\) :

Viscous damping factor of tyres

\(f\) :

Cost function

\(g\) :

Gravitational acceleration

\(i_{\text{gears}}\) :

Overall gear ratio

\(k\) :

Stiffness coefficient of drivetrain

\(k_{c}\) :

Stiffness coefficient of chassis

\(k_{t}\) :

Stiffness coefficient of tires

\(l_{f/r}\) :

Distance of center of front/rear wheel to COG in stand-still

\(m\) :

Vehicle mass

\(n\) :

Sampling instant

\(n_{p}\) :

Sampling instant at peak

\(p\) :

Fitting parameter of tanh2 backlash model

\(r\) :

Wheel radius

\(s\) :

Laplace variable

\(s_{\text{rel}}\) :

Relative inertial position of the chassis to the car body

\(t\) :


u :

Input vector in state-space system

v :

Inertial vehicle speed

\(v_{c}\) :

Inertial wheel and chassis speed

\(w_{\text{chirp}}\) :

Weighting of chirp signal

\(\bf{x}\) :

State vector of state-space system

\(y\) :

System output

\(\bf{y}\) :

Output vector from state-space system

\(y_{c}\) :

Second order characteristics


Normalised root mean square fit


Normalised root mean square median fit


Relative root mean square median fit


Relative error of \(a_{\text{rms}}\)


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Thanks to Florian Brunner for support in ECU-Bypass Coding, Sven Dannenberg for support in transmission control and Andreas Daasch and Dieter Schwarzmann for the provision of a test vehicle.


This research has been partly funded by IAV GmbH, Gifhorn, Germany

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Correspondence to Korbinian J. Figel.




Table 5 Parameter values of models A, B.1–3
Table 6 Parameter values of the models C.1, C.2, D.1 and D.2

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Figel, K.J., Schultalbers, M. & Svaricek, F. Review and experimental evaluation of models for drivability simulation with focus on tire modeling. Forsch Ingenieurwes 83, 105–118 (2019).

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