Inverse-dynamics observer design for a linear single-track vehicle model with distributed tire dynamics

Luigi Romano; Ole Morten Aamo; Jan Åslund; Erik Frisk

eess.SYMar 8, 2026

Inverse-dynamics observer design for a linear single-track vehicle model with distributed tire dynamics

Authors: Luigi Romano, Ole Morten Aamo, Jan Åslund, Erik Frisk

Organizations: Department of Electrical Engineering, Linköping University, Linköping, Sweden · Department of Engineering Cybernetics, NTNU, Trondheim, Norway · Department of Electrical Engineering, Linköping University, Linköping

Abstract

Accurate estimation of the vehicle's sideslip angle and tire forces is essential for enhancing safety and handling performances in unknown driving scenarios. To this end, the present paper proposes an innovative observer that combines a linear single-track model with a distributed representation of the tires and information collected from standard sensors. In particular, by adopting a comprehensive representation of the tires in terms of hyperbolic partial differential equations (PDEs), the proposed estimation strategy exploits dynamical inversion to reconstruct the lumped and distributed vehicle states solely from yaw rate and lateral acceleration measurements. Simulation results demonstrate the effectiveness of the observer in estimating the sideslip angle and tire forces even in the presence of noise and model uncertainties.

Figures & tables

Figure 1: Schematic of the single track model.

Figure 2: Schematic representation of the ODE-PDE interconnection ( 1 ).

Parameter	Description	Unit	Value
$v_{x}$	Longitudinal speed	$\textnormal{m}\,\textnormal{s}^{-1}$	$50$
$m$	Vehicle mass	kg	1300
$I_{z}$	Vertical moment of inertia	$\textnormal{kg}\,\textnormal{m}^{2}$	2000
$l_{1}$	Front axle length	m	1.4
$l_{2}$	Rear axle length	m	1
$F_{z1}$	Front vertical force	N	$2660$

Table I: Model parameters

Figure 3: Behavior of the lumped states and steering inputs, for

\eta=1

and

\gamma=500

: (a) kinematic variables; (b) axle forces; (c) steering inputs.

Figure 4: Dynamics of

\mathinner{\!\left\lVert(\bm{X}(t),\bm{z}(\cdot,t))\right\rVert}_{\mathcal{X}}

\mathinner{\!\left\lVert(\hat{\bm{X}}(t),\hat{\bm{z}}(\cdot,t))\right\rVert}_{\mathcal{X}}

, and

\mathinner{\!\left\lVert(\tilde{\bm{X}}(t),\tilde{\bm{z}}(\cdot,t))\right\rVert}_{\mathcal{X}}

for

\eta=1

and

\gamma=500

Figure 5: Evolution of the PDE state $\tilde{z}_{1}(\xi,t)$ , along with its IC (blue line) and BC (orange line).

Vehicle Dynamics

Vehicle dynamics studies how vehicles move and respond to forces from tires, terrain, aerodynamics, and control inputs.

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RAGE-XY: RADAR-Aided Longitudinal and Lateral Forces Estimation For Autonomous Race Cars

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In this work, we present RAGE-XY, an extended version of RAGE, a real-time estimation framework that simultaneously infers vehicle velocity, tire slip angles, and the forces acting on the vehicle using only standard onboard sensors such as IMUs and RADARs. Compared to the original formulation, the proposed method incorporates an online RADAR calibration module, improving the accuracy of lateral velocity estimation in the presence of sensor misalignment. Furthermore, we extend the underlying vehicle model from a single-track approximation to a tricycle model, enabling the estimation of rear longitudinal tire forces in addition to lateral dynamics. We validate the proposed approach through both high-fidelity simulations and real-world experiments conducted on the EAV-24 autonomous race car, demonstrating improved accuracy and robustness in estimating both lateral and longitudinal vehicle dynamics.

Abstract

Vehicle Dynamics

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