Sensitivity-Orthogonal Residual Learning for Adaptive Sideslip and Cornering Stiffness Estimation

Kyoungtae Ji; Dongyeol Won; Kyoungseok Han

cs.ROOct 20, 2026

Sensitivity-Orthogonal Residual Learning for Adaptive Sideslip and Cornering Stiffness Estimation

Authors: Kyoungtae Ji, Dongyeol Won, Kyoungseok Han

Organizations: Department of Automotive Engineering, Hanyang University, Seoul, South Korea · Department of Automotive Engineering (Automotive-Computer Convergence), Hanyang University, Seoul, South Korea

Abstract

Online adaptation of the cornering stiffness keeps a model-based sideslip estimator valid as tires change, but competes with learned model correction for the same filter innovation. A corrector with enough parameters can absorb the innovation pattern that a stiffness change produces, and adaptation is suppressed while the fit improves. We propagate the stiffness sensitivities of the predicted measurements through the vehicle states and construct a corrector orthogonal to them. A null-space parameterization enforces the constraint without a penalty coefficient and preserves the static least-squares stiffness estimate on each identification segment. In full-vehicle simulation, the stiffness response of the unconstrained corrector declines as its parameter count grows, whereas the orthogonal corrector responds at least as fully as the uncorrected filter at all sizes and lowers sideslip error by 12.7% to 22.0%. In a simulated drive with a stiffness step, the orthogonal corrector adapts as quickly as the uncorrected filter and lowers sideslip error during and after the transient. On public real-vehicle data, the 430-parameter unconstrained corrector loses most of its response to deliberate offsets of the filter's nominal stiffness, whereas the orthogonal corrector of the same size retains most of it, with a sideslip error slightly above the uncorrected filter's.

Figures & tables

Figure 1: One measurement–model mismatch, two explanations, schematic. (a) Measured output and model prediction at the nominal stiffness. (b) The mismatch removed by adapting the stiffness, which moves to the actual value. (c) The same mismatch removed by a learned correction added to the prediction, with the adapted stiffness left near nominal.

Figure 2: Overview of the method. (a) Offline identification of the corrector on segments of constant stiffness. (b) Online joint estimation with the corrector fixed. Blue: the elements added by the proposed method. The symbols are defined in Sections 3 and 4 .

Figure 3: Single-track model, top view ( $x$ forward, $y$ to the left). Blue: velocities and slip angles; orange: lateral axle forces.

Figure 4: Identification run with sinusoidal steering at stiffness scale $0.8$ , lateral-acceleration component, $R^{-1/2}$ -weighted. (a) Sensitivities $\boldsymbol{\psi}_{f}$ (solid) and $\boldsymbol{\psi}_{r}$ (dashed). (b) Fixed-stiffness filter innovation and the outputs of the unconstrained and the orthogonal corrector. Both correctors have 30 parameters.

Figure 5: Geometric view of the orthogonality constraint, schematic, in the plane spanned by

\operatorname{col}(\boldsymbol{\Psi})

and its orthogonal complement. (a) Unconstrained correction. (b) Sensitivity-orthogonal correction.

Role	Runs	Maneuver	$\lambda_{Ky}$
Identification	9	sinusoidal steering, slalom, double lane change	0.8, 1.0, 1.1
Identification	2	step steering at 80 and 50 km/h	1.0
Evaluation, intermediate scale	3	the three maneuvers above	0.9
Evaluation, one axle	8	sinusoidal steering, slalom	0.85 or 1.15, one axle
Evaluation, change during drive	1	step steering at 80 km/h, 360 s	1.0, 0.8 in 120–240 s

Table 1: Simulation runs. The stiffness factor

\lambda_{Ky}

applies to all tires unless stated.

Figure 6: Steering angle (top) and speed (bottom) of the identification maneuvers in simulation at the nominal stiffness. (a) Sinusoidal steering. (b) Slalom. (c) Double lane change.

Figure 7: Stiffness response on the scaled runs, basic correctors (30 parameters). One marker per run. Dashed line: full response. Thin lines: slopes through the origin, reported in Table 2 . (a) Front axle. (b) Rear axle.

Figure 8: Sideslip estimate on a double lane change at scale $0.9$ , a run not used for identification. Basic correctors (30 parameters).

Parameters	Estimator	$\beta$ RMSE [deg]	stiffness response
			front	rear
–	uncorrected	0.245	0.80	0.73
30	unconstrained	0.273	0.79	0.79
30	orthogonal (proposed)	0.214	0.92	0.84
130	unconstrained	0.247	0.65	0.64
130	orthogonal (proposed)	0.199	1.10	1.03

Table 2: Simulation results for all corrector sizes. Sideslip RMSE over the constant-stiffness runs, nine of which are identification runs, and stiffness response over the scaled runs. Rows with random Fourier features are means over the feature draws.

Figure 9: Simulation results against the number of corrector parameters. (a) Stiffness response, front/rear mean over the scaled runs. (b) Sideslip RMSE over the constant-stiffness runs. Markers: mean over the feature draws. Bars: minimum to maximum over the draws. Dashed: uncorrected filter. Dotted: full response.

Figure 10: One-axle stiffness changes at scales $0.85$ and $1.15$ , basic correctors (30 parameters). One marker per run. Rows: front axle scaled (a, b) and rear axle scaled (c, d). Columns: estimated change of the front axle (a, c) and of the rear axle (b, d), against the reference change of the scaled axle. Dashed lines: full response in (a, d), zero in (b, c). Thin lines: slopes through the origin.

	time to $90\%$ of effective change [s]		settled / effective [%]	$\beta$ RMSE [deg]
Estimator	front	rear	front / rear	$120$ – $140$ s / $140$ – $240$ s
uncorrected	6.5	7.6	101 / 96	0.234 / 0.255
unconstrained	11.0	11.8	98 / 95	0.325 / 0.256
orthogonal (proposed)	6.5	7.1	108 / 103	0.183 / 0.179

Table 3: Transient after the stiffness change at

120

s, basic correctors (30 parameters), unsmoothed estimates. Settled / effective: settled change in

200

–

240

s as a percentage of the effective stiffness change.

Figure 11: Stiffness change during the drive at $80$ km/h, all tires to $0.8$ of nominal from $120$ s to $240$ s (dashed lines). (a), (b) Stiffness estimate relative to the unchanged drive, $4$ s moving average, with the effective stiffness change. (c) Sideslip error, $8$ s moving root-mean-square value. Basic correctors (30 parameters).

Figure 12: Real-vehicle drives, steering angle (top) and speed (bottom). (a), (b) Identification drives. (c) One evaluation drive.

Estimator	$\beta$ RMSE [deg]	response, front	response, rear
uncorrected	0.79	0.99	1.00
unconstrained	0.79 [0.69, 0.92]	0.35 [0.28, 0.45]	0.24 [0.14, 0.37]
orthogonal (proposed)	0.83 [0.77, 0.91]	0.88 [0.73, 1.05]	0.70 [0.44, 0.98]

Table 4: Real-vehicle results with

430

parameters. For the correctors, entries report the mean over the feature draws, with the minimum and maximum in brackets.

Figure 13: Sideslip estimate on one evaluation drive against the optical reference. 430-parameter correctors, one feature draw.

Figure 14: Stiffness response on the vehicle, 430-parameter correctors. One marker per evaluation drive and offset, mean over the feature draws, with thin bars from the minimum to the maximum over the draws. Dashed line: full compensation. Solid lines: slopes through the origin, reported in Table 4 . (a) Front axle. (b) Rear axle.

Figure 15: Real-vehicle results against the number of corrector parameters. (a) Stiffness response, front/rear mean. (b) Sideslip RMSE at $\kappa=1$ . Markers: mean over the feature draws. Bars: minimum to maximum over the draws. Dashed: uncorrected filter. Dotted: full compensation.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Quantity	Simulation (CarMaker)	Real vehicle (IONIQ 5)
$l_{f}$ , $l_{r}$ [m]	$0.97$ , $1.57$	$1.563$ , $1.437$
$m$ [kg], $I_{z}$ [kg m 2 ]	$1598$ , $1800$	$2254$ , $2193$
$C_{f0}$ , $C_{r0}$ [kN/rad]	$89.9$ , $91.8$	$157.3$ , $276.4$
steering input [rad]	Map from the handwheel angle $\delta_{\mathrm{hw}}$ [rad] to the road-wheel angle: $\delta=0.0723\,\delta_{\mathrm{hw}}+8.4\times 10^{-5}\,\delta_{\mathrm{hw}}^{3}$	Dataset single-track angle
signals	CarMaker outputs at $100$ Hz	Optical velocity sensor (Kistler Correvit S-Motion) and rear-axle inertial unit, recorded at $1$ kHz and averaged over blocks of 10 samples to $100$ Hz
no stiffness update below	$v<5$ m/s	$v<3$ m/s

Appendix

Table 5: Vehicle parameters, estimator and identification settings, and the 15 regressors of the basic corrector. Vehicle parameters are from the CarMaker datasheet (simulation) and the dataset description (real vehicle). Process-noise entries are per sample at

\Delta t=0.01

s. Superscript m denotes a measured signal.

Vehicle Dynamics

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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.

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