Experimental verification of nonlinear sliding mode observer for accurate estimation of vehicle sideslip angle and lateral tire forces

Abstract

This paper investigates the estimation of the sideslip angle and lateral tire-road forces for a class of nonlinear road vehicles. A Second-Order Sliding Mode Observer (SOSMO) is developed as an extended-state observer to simultaneously estimate lateral tire forces along with other measurable variables. The vehicle’s sideslip angle is then estimated independently using two distinct methods: dynamical equations and an inverse model-based estimation approach. The latter method introduces an innovative tire model that incorporates nonlinear tire-road friction characteristics, effectively simulating lateral force behavior. Comparative simulations and experimental analyses across two practical scenarios demonstrate that the proposed strategies, which do not require detailed tire-road interaction modeling, outperform conventional estimators such as the Extended Kalman Filter (EKF) and the State Dependent Riccati Equation Filter (SDREF). The experimental results particularly highlight the superior accuracy and efficiency of the developed SOSMO-based estimation strategies in providing estimates of the sideslip angle and lateral forces, offering reliable and cost-effective solutions compared to traditional methods.

Keywords

Estimation methods experimental study sideslip angle single-track model sliding mode observer vehicle dynamics

Introduction

Road vehicles are influenced by various factors including speed, lateral acceleration, yaw rate, and tire-road friction coefficients, all of which significantly impact their behavior and safety. Ensuring stability is paramount in automotive research, as it directly contributes to safety and efficiency. Numerous elements such as speed, lateral acceleration, yaw rate, and tire-road friction coefficients affect road vehicles, greatly influencing their performance and safety. However, real-time monitoring of these factors is challenging due to the constraints of sensor technology. While some factors can be directly measured, others, like lateral tire-road forces and sideslip angle, require costly equipment. An alternative approach uses sensors to capture basic outputs and applies observation algorithms to estimate all system states. Although estimating all variables is advantageous for controlling vehicle dynamics, focusing on the estimation of sideslip angle and lateral forces is especially vital in road vehicle applications.

Precisely determining the vehicle’s sideslip angle, which represents the angle between the direction of the vehicle’s center of mass velocity and its longitudinal axis, is crucial for maintaining lateral stability. This topic has been a focal point of extensive research since the 1950s. Researchers have investigated various methods, including both dynamic and kinematic models, each with unique benefits and drawbacks. Traditionally, estimating the sideslip angle without expensive optical sensors has involved using dynamic model-based techniques. These include deterministic state estimators like the Luenberger Observer (LO),^1,2 Sliding Mode Observer (SMO),^3,4 and High-Gain Observer (HGO),^5–7 as well as stochastic state estimators such as the traditional Kalman Filter (KF),^8,9 Extended Kalman Filter (EKF),¹⁰ and Unscented Kalman Filter (UKF).¹¹ While simulating vehicle dynamics is feasible, integrating a vehicle model with a physical or empirical tire model presents complexities due to factors like tire characteristics, vehicle specifications, and road conditions. Previous studies have employed specific tire models, such as the linear tire model,¹² and nonlinear models including the Pacejka tire model (Pacejka’s Magic Formula),¹³ the Dugoff tire model,¹⁴ the LuGre tire model,¹⁵ and the Brush tire model,¹⁶ to describe tire friction. However, the numerous parameters involved require calibration through extensive testing under various tire specifications and pressures, making the understanding of tire forces and tire-road friction quite complex. Therefore, it is crucial to develop practical methods that do not rely on prior knowledge of tire-road properties.

Over the past decade, there has been a significant increase in the use of nonlinear estimation methodologies in vehicular dynamics, building on earlier academic contributions. For instance, a nonlinear single-track vehicle model was explored using a simplified Magic tire model and employed an EKF to estimate sideslip angles without additional sensors.¹⁷ Their study demonstrated the effectiveness of this method through simulations and real-world experiments. Similarly, a basic nonlinear single-track vehicle model was used combined with a random-walk model for road friction, developing an extended LO to accurately estimate sideslip angles and road friction coefficients with high precision and robustness.¹⁸ In a later study, the State-Dependent Riccati Equation (SDRE) technique was utilized alongside the Random Walk Model to estimate the sideslip angle of road vehicles, showing superior performance compared to the conventional EKF.¹⁹ An innovative method was also introduced integrated with an EKF scheme for estimating vehicle sideslip angles in a double-track vehicle model, which provided more accurate estimates.²⁰ A novel approach was also proposed using the constrained UKF tailored to a single-track vehicle model for estimating sideslip angles, eliminating the need for specific tire interaction force modeling and achieving promising results in accurately estimating complex dynamic systems.²¹ Moreover, a unified observation algorithm was investigated to concurrently estimate tire side-slip angles and lateral tire forces for a double-track vehicle model. Although their adaptive SMO initially showed promise, the computational complexity of their compensation algorithm for estimating vehicle sideslip angles requires further investigation.²² The EKF was applied to estimate the states of a dual-track vehicle, validating accuracy through simulations on various road surfaces and driving maneuvers.²³ Furthermore, practical experiments were conducted using the EKF method to estimate lateral tire-road forces and friction, along with sideslip angles.²⁴ In 2022, a robust sideslip angle estimation method for commercial vehicles was introduced, combining a Levenberg–Marquardt backpropagation-based neural network to estimate time-varying tire cornering stiffness with a time-varying Kalman filter.²⁵ This approach eliminates the need for detailed tire models and demonstrates superior accuracy and robustness under diverse road conditions. In 2023, a robust sideslip angle estimation algorithm was developed using a hybrid kinematic-dynamic observer with a friction classifier to adapt to sudden changes in road grip. Experimental tests on various surfaces showed high accuracy, with errors under 1.5°.²⁶ An approach was introduced for estimating sideslip angles and yaw rates in a single-track vehicle model, utilizing the Information Fusion-based UKF and testing its feasibility with various techniques.²⁷ In 2024, two robust sideslip angle estimation methods were proposed. The first employed a robust unscented particle filter for distributed drive electric vehicles, combining UKF-based importance density with systematic resampling, and showed superior accuracy and robustness in simulations under non-Gaussian noise.²⁸ The second introduced an interacting multiple model-based approach using EKF and UKF variants with a Dugoff tire model, relying only on low-cost sensors without requiring friction coefficient knowledge.²⁹ Both methods demonstrated strong performance in simulations, though the absence of experimental validation remains a noted limitation.

Despite substantial progress in observer design for estimating vehicle sideslip angle and lateral tire forces, many existing methodologies continue to face critical limitations, particularly due to the inherent nonlinearities in tire-road interactions and the difficulty of accurately identifying tire model parameters across varying operating conditions. Conventional estimation techniques often rely on complex tire models, such as the Pacejka Magic Formula or the Dugoff model, which demand extensive experimental calibration and are typically sensitive to parameter uncertainties and road surface variations. These constraints can significantly undermine the reliability and generalizability of such approaches in practical, real-time automotive applications. To overcome these challenges, the present study introduces a novel observer framework that integrates advanced nonlinear estimation strategies, offering a robust and computationally efficient alternative to traditional estimation methods.

The key contributions of this work can be summarized as follows. A central achievement of this study is the design and implementation of a SOSMO framework specifically developed for single-track vehicle models. This second-order structure surpasses conventional first-order sliding mode observers by significantly improving estimation accuracy while preserving robustness to model uncertainties and external disturbances. The SOSMO effectively captures the rapid dynamics of lateral tire forces, achieving robust convergence without inducing the chattering effects typically associated with discontinuous control strategies. To further enhance the robustness and flexibility of sideslip angle estimation, two complementary estimation approaches are developed. The first approach employs a dynamic model-based formulation that combines measured vehicle states with observer-estimated tire forces to estimate the sideslip angle. The second approach adopts an inverse modeling strategy, utilizing an advanced nonlinear tire model to reconstruct the sideslip angle from observed vehicle responses. This model accurately reflects nonlinear tire-road friction characteristics, thereby improving estimation accuracy under varying road conditions. Furthermore, the observer framework integrates the nonlinear tire model to account for saturation and hysteresis effects in tire-road interactions, enabling high-fidelity estimation even during aggressive maneuvers. The proposed methods are thoroughly validated through both simulation studies and experimental testing under various steady-state and transient driving scenarios. Quantitative performance metrics, such as root mean square error (RMSE), confirm that the SOSMO-based framework delivers superior accuracy and robustness compared to widely used estimation methods like the EKF and SDRE approaches.

The structure of the paper is as follows: Section “Vehicle dynamics” introduces the reference single-track vehicle model, emphasizing the use of the four-parameter Pacejka Magic Formula for tire-road interaction modeling. Section “Construction of the sliding mode observer” details the design of the extended-state SOSMO alongside auxiliary observation strategies. Section “Simulation and experimental verification” presents numerical validations through simulations and experimental data, demonstrating the observer’s effectiveness. Finally, Section “Conclusion” concludes the study and highlights future directions, emphasizing the framework’s high estimation accuracy, computational efficiency, scalability, and suitability for real-time automotive applications.

Vehicle dynamics

In this study, the single-track vehicle model, depicted in Figure 1, serves as the foundational framework for designing state observers. Despite its simplified structure, the model is extensively recognized in the literature as a standard benchmark for evaluating state estimation techniques, owing to its advantageous balance between analytical tractability and essential dynamic representation. The vehicle is modeled with a front steering axle and is referenced with respect to both an inertial frame, denoted as Oxy, and a body-fixed frame, labeled O′xy.

Figure 1.

Reference frames of a two-degree-of-freedom vehicle model.³⁰

In order to accommodate real-time computations and streamline observer design, equation (1) incorporates the single-track vehicle model to illustrate vehicle lateral dynamics, focusing notably on critical states such as yaw rate and sideslip angle.¹⁹

m a_{y} = F_{y 1} \cos δ + F_{y 2}

(1a)

J_{z} \overset{\cdot}{r} = L_{1} F_{y 1} \cos δ - L_{2} F_{y 2}

(1b)

where $m$ denotes the total mass of the vehicle, $r$ represents the yaw rate, $δ$ indicates the steering wheel angle of the front tires, $J_{z}$ stands for the vehicle’s moment of inertia about the yaw axis, and $L_{1}$ and $L_{2}$ represent the distances from the front and rear axles to the center of gravity, respectively. The vehicle’s sideslip angle $β$ is defined as the angle between the longitudinal axis of the vehicle and the direction of its velocity vector. Additionally, the lateral velocity and acceleration are denoted as $V_{y}$ and $a_{y}$ , respectively, as follows:

V_{y} = V_{x} \tan β

(2a)

a_{y} = V_{x} r + {\overset{\cdot}{V}}_{y}

(2b)

where $V_{x}$ represents the longitudinal velocity. Additionally, $F_{y 1}$ and $F_{y 2}$ , defined in equation (3) following the established tire-road interaction model specific to the single-track vehicle, denote the lateral forces acting on the front and rear axles, respectively.²⁴

F_{yi} (α_{i}) = D_{i} \sin {Catan [B α_{i} - E (B α_{i} - atan (B α_{i}))]}

(3)

where $B$ denotes the stiffness factor, $C$ represents the form factor, and $E$ stands for the curvature factor. Additionally, for the tire-road friction coefficient $μ$ , $D_{i} = μ F_{zi}$ is defined as the peak value, where $F_{z 1} = \frac{mg L_{2}}{L_{1} + L_{2}}$ and $F_{z 2} = \frac{mg L_{1}}{L_{1} + L_{2}}$ . The wheel slip angles $α_{i}$ associated with the front and rear axles, respectively, are given by $α_{1} = atan (\frac{V_{y} + r L_{1}}{V_{x}}) - δ$ and $α_{2} = atan (\frac{V_{y} - r L_{2}}{V_{x}})$ .

Construction of the sliding mode observer

This section focuses on estimating both the lateral tire forces and the vehicle’s sideslip angle. To facilitate the observer design, the following widely accepted simplifying assumptions are adopted:

Assumption 1: The longitudinal velocity is considered constant, that is, ${\overset{\cdot}{V}}_{x} = 0$ .

Assumption 2: The sideslip angle is assumed to be small, such that $1 + \tan^{2} β ≅ 1$ . This implies the sideslip angle does not exceed 15°, even under demanding driving conditions. This assumption is justified by the fact that during normal driving, where traction is maintained, the sideslip angle typically remains below 2°.¹⁸ Larger sideslip angles generally occur only during vehicle instability, which is beyond the scope of this study.

Assumption 3: The steering angle, longitudinal velocity, yaw rate, and lateral acceleration are assumed to be measurable. These quantities are typically available through standard onboard sensors in automotive systems.

Remark 1: In steady-state conditions such as highway driving, and in scenarios where the vehicle speed remains relatively constant over time, the assumption of constant longitudinal velocity is a common modeling simplification. This approach reduces the complexity of the dynamic equations and facilitates analytical development. However, it is acknowledged that this assumption may lead to reduced accuracy during dynamic maneuvers involving rapid acceleration or braking, where longitudinal forces become more influential.

Given the first two assumptions, it follows that $a_{y} = V_{x} (r + \overset{\cdot}{β})$ . Consequently, for $i = 1$ and $2$ , the state-space model for the vehicle is described accordingly.

\overset{\cdot}{r} = {(- 1)}^{i} \frac{L_{3 - i} m}{J_{z}} a_{y} + Δ_{i}

(4)

thus defined as either $\overset{\cdot}{r} = - \frac{L_{2} m}{J_{z}} a_{y} + Δ_{1}$ or $\overset{\cdot}{r} = \frac{L_{1} m}{J_{z}} a_{y} + Δ_{2}$ , where $Δ_{1} = \cos δ \frac{L_{1} + L_{2}}{J_{z}} F_{y 1}$ and $Δ_{2} = - \frac{L_{1} + L_{2}}{J_{z}} F_{y 2}$ .

Assumption 1: It is assumed that the lateral interaction forces $F_{yi}$ satisfy the Lipschitz condition.³¹ This condition, supported by known Lipschitz constants $L_{Δ i}$ , ensures that $| {\overset{\cdot}{Δ}}_{i} | \leq L_{Δ i}$ .^32,33 In driving conditions with small slip angles, lateral forces vary smoothly, making the Lipschitz continuity assumption reasonable and commonly accepted.

The extended-state SOSMO framework is now formulated, inspired by the concept of an extended state observer, to estimate both the yaw rate and unknown lateral interaction forces concurrently. As depicted in the schematic diagram in Figure 2, the observation algorithm utilizes real-time data obtained directly through online measurements or indirectly derived variables. These include readily available parameters such as the steering angle of the front wheels, vehicle speed, lateral acceleration, and yaw rate. To enable the estimation of these two state variables, the following extended-state SOSMO is devised.

\begin{array}{l} \dot{\hat{r}} = {(- 1)}^{i} \frac{L_{3 - i} m}{J_{z}} a_{y} + {\hat{Δ}}_{i} - k_{0 i} L_{Δ i}^{\frac{1}{2}} | \hat{r} - r |^{\frac{1}{2}} s i g n (\hat{r} - r) \\ {\hat{\overset{\cdot}{Δ}}}_{i} = - k_{1 i} L_{Δ i} s i g n (k_{0 i} L_{Δ i}^{\frac{1}{2}} | \hat{r} - r |^{\frac{1}{2}} s i g n (\hat{r} - r) \end{array}

(5)

where $\hat{r}$ denotes the estimated yaw rate, and for $i = 1$ and $2$ , ${\hat{Δ}}_{i}$ represents the estimates of the augmented lateral interaction forces $Δ_{1}$ and $Δ_{2}$ , respectively. Additionally, $k_{0 i}$ and $k_{1 i}$ are positive constants, with $sign (.)$ denoting the sign function. The rational-based design of the observer (5) has been made based on a sliding variable defined as the yaw rate estimation error. This choice enables the observer to directly correct errors. The reaching law has also been constructed using the Super-Twisting Algorithm, which is well known for ensuring finite-time convergence of both the sliding variable and its derivative under bounded uncertainties. The observer updates the yaw rate estimate based on the known system model and includes a nonlinear correction term of the form $| \hat{r} - r |^{\frac{1}{2}} sign (\hat{r} - r)$ . This term ensures convergence while significantly reducing the chattering effects typically encountered in classical first-order sliding mode observers. In addition, a second state is introduced to estimate the augmented lateral interaction forces. The dynamics of ${\hat{Δ}}_{i}$ follow a sliding-mode-inspired law to ensure robust and accurate estimation.

Figure 2.

Schematic diagram of the proposed estimation framework.

Although equations (4) and (5) provide two mathematically equivalent expressions for $\overset{\cdot}{r}$ and $\dot{\hat{r}}$ , respectively, it is important to recognize that, as physical quantities, they cannot be accepted simultaneously. In a real system, such quantities must be uniquely defined. To ensure consistency and preserve physical interpretability, only one expression is retained in the final observer structure. For example, by considering the formulation involving $Δ_{1}$ , the estimation of $Δ_{2}$ is accordingly reformulated to depend directly on ${\hat{Δ}}_{1}$ , eliminating the need for a separate dynamic equation. Finally, to evaluate the convergence of estimations and thereby the stability of estimation errors, the closed-loop systems of observation errors are derived using equations (4) and (5), along with the definitions of observation errors denoted as $e_{r} = \hat{r} - r$ and $e_{Δ_{1}} = {\hat{Δ}}_{1} - Δ_{1}$ .

\begin{array}{l} {\overset{\cdot}{e}}_{r} = e_{Δ_{1}} - k_{01} L_{Δ 1}^{\frac{1}{2}} | e_{r} |^{\frac{1}{2}} s i g n (e_{r}) \\ {\overset{\cdot}{e}}_{Δ_{1}} = - {\overset{\cdot}{Δ}}_{1} - k_{11} L_{Δ 1} s i g n (k_{01} L_{Δ 1}^{\frac{1}{2}} | e_{r} |^{\frac{1}{2}} s i g n (e_{r})) \end{array}

(6)

To demonstrate the convergence of state estimates toward the actual state, while considering the Lipschitz condition and ensuring that all differential inclusions adhere to the upper semi-continuity property in the Filippov sense, the error dynamics can be compactly reformulated as follows:

\begin{array}{l} {\dot{e}}_{r} = e_{Δ_{1}} - k_{01} L_{Δ 1}^{\frac{1}{2}} | e_{r} |^{\frac{1}{2}} sign (e_{r}) \\ {\dot{e}}_{Δ_{1}} \in [- L_{Δ 1}, L_{Δ 1}] - k_{11} L_{Δ 1} sign (k_{01} L_{Δ 1}^{\frac{1}{2}} | e_{r} |^{\frac{1}{2}} sign (e_{r}) \end{array}

(8)

where the evolution of ${\overset{\cdot}{e}}_{Δ_{1}}$ and ${\overset{\cdot}{e}}_{Δ_{2}}$ transforms

{\overset{\cdot}{e}}_{Δ_{1}} \in [- (1 + k_{11}) L_{Δ 1}, (1 - k_{11}) L_{Δ 1}]

(9)

and the stability of the error dynamics is assured through the following lemma, which confirms that the observation error converges to a small neighborhood around zero, even in the presence of $F_{y 1}$ .

Lemma 1: The following dynamics associated with a system of relative degree m demonstrate finite-time stability.³⁴

\begin{matrix} {\overset{\cdot}{z}}_{0} = z_{1} - λ_{0} L^{m / m + 1} {| z_{0} |}^{m / m + 1} sign (z_{0}) \\ {\overset{\cdot}{z}}_{1} = z_{2} - λ_{1} L^{m - 1 / m} {| z_{1} - {\overset{\cdot}{z}}_{0} |}^{m - 1 / m} sign (z_{1} - {\overset{\cdot}{z}}_{0}) \\ \begin{matrix} ⋮ \\ {\overset{\cdot}{z}}_{m - 1} = z_{m} - λ_{m - 1} L^{1 / 2} {| z_{m - 1} - {\overset{\cdot}{z}}_{m - 2} |}^{1 / 2} sign (z_{m - 1} - {\overset{\cdot}{z}}_{m - 2}) \\ z_{m} \in - λ_{m} Lsign (z_{m} - {\overset{\cdot}{z}}_{m - 1}) + [- L, L] \end{matrix} \end{matrix}

(10)

Further details on the stability condition can be found in Davila et al.³⁴ and Razmjooei et al.³⁵ Applying Lemma 1 with $m = 1$ and suitably tuned design parameters, it becomes evident that the error dynamics (8) converge to small neighborhoods around zero.

After obtaining the estimation of ${\hat{Δ}}_{1}$ , it can be directly concluded that ${\hat{Δ}}_{2} = {\hat{Δ}}_{1} - \frac{L_{1} + L_{2}}{J_{z}} m a_{y}$ .

Remark 2: The estimates of the lateral tire forces can now be directly computed as

{\hat{F}}_{yi} = \frac{{(- 1)}^{i - 1} {\hat{Δ}}_{i} J_{z}}{(L_{1} + L_{2}) {(\cos δ)}^{2 - i}}

(11)

To estimate the vehicle’s sideslip angle, two methodologies are explored. The first approach employs dynamic formulations, where substituting equation (1a) into equation (2b) results in:

{\hat{V}}_{y}^{.} = \frac{{\hat{F}}_{y 1} \cos δ + {\hat{F}}_{y 2}}{m} - V_{x} r

(12)

where ${\hat{F}}_{y 1}$ and ${\hat{F}}_{y 2}$ have already been obtained using the proposed extended-state SOSMO. Subsequently, an estimation of the sideslip angle is approximated based on equation (2a) as follows:

\hat{β} = \tan^{- 1} (\frac{{\hat{V}}_{y}}{V_{x}})

(13)

where $\hat{β}$ is obtained without explicit knowledge of the tire model.

Proposition 1: An alternative approach is optionally proposed based on an innovative high-precision nonlinear tire model described in equation (14). This model represents a paradigm shift in automotive engineering’s conventional tire modeling approaches, eliminating reliance on linear models, avoiding the simplified yet less accurate Dugoff tire model, and moving beyond complex nonlinear models such as the Magic Formula.

{\bar{F}}_{yi} = a_{1 i} \tanh (a_{2 i} (- β + δ^{2 - i} + {(- 1)}^{2 - i} \frac{L_{i}}{V_{x}} r))

(14)

where $a_{ji}$ , for $j, i = 1$ and $2$ , represent model parameters. Since parameter identification of the tire model is crucial for obtaining accurate sideslip estimation, a wide range of operating conditions was considered, utilizing $21, 000$ random sets of experimental data for each tire. To ensure the reliability of the identification tests, experiments were conducted using a real sports vehicle on dry asphalt under two distinct practical scenarios. The vehicle was equipped with a rotary potentiometer to measure the steering angle and an inertial measurement unit integrated with a differential global positioning system to measure longitudinal velocity, yaw rate, sideslip angle, and lateral acceleration. All acquired data underwent preprocessing, including second-order low-pass Butterworth filtering to remove high-frequency noise, and normalization based on maximum signal magnitudes to ensure consistent comparisons. These steps enhanced the accuracy and reliability of the practical test data for analysis. It is important to note that not all of this data was used as output variables; for instance, the sideslip angle signal served as reference data in validation studies and for identification purposes.

Following data collection with the instrumented vehicle and computation of the lateral interaction forces on the front and rear axles (denoted as $F_{y 1}$ and $F_{y 2}$ ), the coefficients were determined using the least squares method. The orders were selected as the smallest values that produced an acceptable fitting error, maintaining model parsimony to prevent overfitting. The specified values chosen were $a_{11} = 1.4 \times 10^{5}$ , $a_{12} = 1.1 \times 10^{5}$ , $a_{21} = 55$ , and $a_{22} = 105$ . The resulting fit yielded Root Mean Square Error (RMSE) values of and $2.704 %$ for $F_{y 1}$ and $F_{y 2}$ , respectively. To demonstrate the accuracy of the proposed model, a numerical comparative analysis is conducted, encompassing both the Arctangent-based Dugoff tire model and the well-known Magic tire model used in equation (3).⁵ Figure 3 illustrates a comparison of these three distinct tire models, specifically focusing on the front tire’s performance under conditions of maximum lateral acceleration, reaching $8 m / s^{2}$ . As depicted in Figure 3, when the front tire’s slip angle remains below $3^{°}$ , the Arctangent model closely approximates the behavior of the Magic Formula curve, showing a striking resemblance. However, as the tire’s slip angle exceeds $3^{°}$ , the accuracy of the Arctangent model diminishes. In contrast, the proposed Hyperbolic Tangent (H-T) tire model exhibits an exceptionally broad accuracy spectrum, closely mirroring the characteristics of the Magic Formula curve. Even at tire slip angles exceeding $5^{°}$ , where the Magic Formula curve tends to flatten, the proposed model maintains close alignment with its values.

Figure 3.

Comparative analysis of front tire performance across three tire models at varying slip angles.

While the Arctangent model offers an acceptable representation for sideslip angles below $3^{°}$ , the proposed H-T model exhibits significantly superior performance, closely matching the highly efficient Magic Formula model. This heightened accuracy has the potential to enhance efficiency and precision in estimations, thereby improving overall automotive performance and safety through streamlined and accurate processes.

Remark 3: It is noteworthy that the single-track vehicle model under study has been equipped with the well-known Magic tire model (3), whereas the proposed innovative model (14) is designed specifically to offer precise and effective estimations of tire behavior.

Considering the proposed model (14), once the estimation of the uncertain terms $Δ_{i}$ and subsequently the lateral forces ${\hat{F}}_{yi}$ are ensured, the vehicle’s sideslip angle can be estimated. This is achieved by employing the inverse model of (14) and assuming ${\bar{F}}_{yi} = {\hat{F}}_{yi}$ , as follows:

\hat{β} = \frac{L_{i}}{V_{x}} r + (i - 2) δ - \frac{1}{a_{2 i}} \tanh^{- 1} (\frac{{\hat{F}}_{yi}}{a_{1 i}})

(15)

This study has successfully achieved its objectives by developing a methodology to indirectly estimate the sideslip angle, thereby reducing uncertainty without relying on costly measurement devices. The effectiveness of this approach and the innovative tire model, which streamline and maintain accurate estimations, will be demonstrated in the following evaluation section.

Simulation and experimental verification

This section presents the outcomes of the proposed frameworks in comparison to other high-performance references through computer simulations and real-world experiments. Initially, simulation results are briefly discussed for a scenario involving specific lane change maneuver parameters, which include abrupt steering inputs for quick lane changes. This is done to validate the approaches before prototyping in a cost-effective, safe, and flexible manner, providing detailed insights. Subsequently, extensive experiments are conducted for two scenarios, with their corresponding lateral accelerations, $a_{y}$ , plotted in Figure 4.

Figure 4.

Comparison of lateral accelerations for two simulation scenarios.

The illustration compares vehicle steering responses: one tests increasing steering angle at constant speed for circular driving assessment, while another examines lateral acceleration and yaw rate during dynamic maneuvers. These scenarios evaluate prior studies’ accuracy in estimating sideslip angle and lateral forces in varied real-world conditions. It is also worth noting that, in this study, the chattering effects commonly associated with sliding mode observers are mitigated by employing a continuous approximation of the sign function using the hyperbolic tangent. The parameters of this approximation, along with the observer gains, are optimized using the particleswarm function in MATLAB R2018a to achieve smooth and robust state estimation.

Computer simulation results

In this subsection, simulation results are showcased using Matlab/Simulink software for a scenario featuring a transient maneuver: a lane change at a constant longitudinal speed of $120 km / h$ , with a steering angle amplitude of $50^{°}$ , under high adhesion conditions. The goal is to examine the vehicle’s dynamic response during lateral movements, employing specific lane change maneuver parameters and abrupt steering inputs for rapid lane changes. To ensure a fair comparison, the evaluations are conducted using the nominal parameters $L_{1} = L_{2} = 1.15 m$ . Here, $L_{1}$ represents the distance from the front axle to the center of gravity, while $L_{2}$ represents the distance from the rear axle to the center of gravity. The mass of the vehicle is $m = 1600 kg$ , and the yaw movement inertia is $J_{z} = 1800 kg m^{2}$ .

Figure 5 presents yaw rate estimation results for the developed SOSMO and two leading strategies, SDRE¹⁹ and EKF.^10,36 While the estimation results and errors demonstrate close alignment between the estimated and actual yaw rates across all methods, the upper bounds of estimation errors initially highlight significant performance distinctions among these three approaches. Specifically, the extended-state SOSMO and SDRE exhibit error bounds approximately 10 times lower than that of EKF. Moreover, the extended-state SOSMO method shows superior performance with markedly smaller yaw rate estimation errors compared to SDRE. Quantitatively, the overall performance is now assessed by calculating the norm of yaw rate estimation errors, revealing an RMSE of $6.49 \times 10^{- 30}$ for extended-state SOSMO, outperforming SDRE with an RMSE of $1.35 \times 10^{- 28}$ , while EKF exhibits an RMSE of $7.29 \times 10^{- 28}$ .

Figure 5.

Comparative analysis of yaw rate estimation in the lane change scenario.

In relation to the auxiliary variable introduced in the extended-state SOSMO, which is specifically designed to estimate lateral tire forces with high accuracy, Figure 6 illustrates the effectiveness of the proposed method. The results show a significant reduction in estimation error, with the error bounds being approximately $176$ times smaller than the full variation range of the lateral tire forces, which spans up to $10000 N$ . More specifically, for the lateral forces $F_{y 1}$ and $F_{y 2}$ , which both reach peak values of approximately $8000 N$ during operation, the maximum absolute estimation errors were $40 N$ and $10 N$ , respectively. These correspond to relative errors of just $0.5 %$ for $F_{y 1}$ and $0.125 %$ for $F_{y 2}$ , indicating that the estimation remains highly accurate even under dynamic driving conditions. These values are well within the commonly accepted tolerance range of $5 - 10 %$ for lateral force estimation in practical applications. This result underscores the extended-state SOSMO’s capability to provide precise force estimation, even in the absence of direct comparative benchmarks in the cited literature. Such accurate force estimation significantly contributes to the reliable approximation of the sideslip angle, which is crucial for vehicle dynamics analysis.

Figure 6.

Lateral tire force estimation by the extended-state SOSMO in the lane change scenario.

Figure 7 presents the sideslip angle estimation performance for all the methods evaluated in this study. Focusing on the two proposed strategies introduced in Section “Construction of the sliding mode observer,” along with the SDRE approach, the results demonstrate that both proposed methods exhibit significantly lower estimation errors. Specifically, the first proposed method (direct approach) achieved an RMSE of $2.95 \times 10^{- 8}$ , while the second method (based on the H-T transformation) achieved an RMSE of $2.73 \times 10^{- 8}$ . In contrast, the SDRE method yielded an RMSE of $6.03 \times 10^{- 6}$ , and the Extended Kalman Filter (EKF) showed a substantially higher RMSE of $2.64 \times 10^{- 3}$ .

Figure 7.

Comparative analysis of sideslip angle estimation in the lane change scenario.

In addition to the RMSE values, the maximum absolute estimation errors for the sideslip angle, whose true value reaches up to approximately $0.04 rad$ , further highlight the differences in accuracy. The first and second proposed strategies resulted in maximum errors of $0.0001 rad$ ( $0.25 %$ ) and $0.000008 rad$ ( $0.02 %$ ), respectively. The SDRE method produced a maximum error of $0.00012 rad$ ( $0.3 %$ ), while the EKF exhibited the largest maximum error of $0.0015 rad$ ( $3.75 %$ ). All methods maintained errors below the commonly accepted threshold of $5 %$ ; however, the proposed observers achieved markedly higher precision. While all methods ultimately maintained estimation errors within the predefined $5 %$ confidence interval, the proposed observers and the SDRE approach demonstrated notably faster convergence to this bound. In contrast, the EKF required a significantly longer time to reach the same level of accuracy, reflecting its slower asymptotic convergence behavior.

These results highlight the considerable precision and accuracy achieved by the proposed estimation strategies, clearly demonstrating their significant performance superiority over the well-established SDRE method and the widely used EKF approach, particularly under demanding vehicle dynamics conditions. Furthermore, considering the lateral acceleration formula, its estimation can be derived using the estimated lateral forces ${\hat{F}}_{yi}$ as ${\hat{a}}_{y} = \frac{{\hat{F}}_{y 1} \cos δ + {\hat{F}}_{y 2}}{m}$ . In this context, Figure 8 demonstrates the significantly better performance of the proposed strategy compared to SDRE and EKF. It is also noteworthy that, although all methods satisfy the accuracy criterion defined by the confidence interval, the estimation error of the developed SOSMO converges to this bound in a shorter and well-defined time frame, underscoring its superior performance in terms of convergence speed.

Figure 8.

Comparative analysis of lateral acceleration estimation in the lane change scenario.

It is important to highlight that, although minor estimation error spikes were observed, the proposed methods consistently maintained very low magnitudes. For example, in Figure 6, the peak spike in estimation error is approximately 50 N, which is negligible compared to the overall variation in lateral tire forces (~8000 N). Similarly, Figure 7 demonstrates that the proposed observers exhibit minimal transient errors relative to the SDRE method and show a clear advantage over the EKF. These findings confirm the precision and robustness of the proposed observers, particularly in responding to abrupt dynamic changes. This evidences their strong potential for real-world applications, particularly in the accurate estimation of both measurable variables (e.g. yaw rate and lateral acceleration) and unmeasurable states (e.g. sideslip angle and lateral tire forces), which will be experimentally validated in the following sections.

Experimental results

In this section, experimental results from a real sports vehicle under steering pad and lane change scenarios on dry asphalt are presented, utilizing a rotary potentiometer for steering angle measurement and an inertial measurement unit with differential Global Positioning System for longitudinal velocity, yaw rate, sideslip angle, and lateral acceleration. All acquired data underwent rigorous pre-processing to eliminate errors and noise, ensuring accuracy and reliability for comparative analysis. It is important to note that not all of this data is used as output variables; for instance, the sideslip angle is utilized only as reference data in this study.

Open loop steering pad scenario

This maneuver, which involves maintaining a constant longitudinal speed of $120 km / h$ while the steering angle gradually and linearly increases over time under high-adhesion conditions, aims to evaluate the vehicle’s steady-state circular driving behavior.

Figure 9 presents the experimental results evaluating the performance of the three methods in estimating yaw rates. All three methods exhibit acceptable behavior, but the proposed method demonstrates superior precision, with an RMSE value of $1.98 \times 10^{- 7}$ , indicating high precision and reliability. In contrast, the EKF and SDRE, experiencing almost similar RMSE values of $2.29 \times 10^{- 3}$ , show lower accuracy. These findings substantiate the exceptional capabilities of the proposed method in practical applications of yaw rate estimation. Building on the proven efficacy of the extended-state SOSMO in estimating measurable yaw rate, Figure 10 presents the estimates of lateral tire forces, which affirm the practical capability of the developed extended-state observer to estimate unknown terms. This figure also illustrates that the lateral tire forces exhibit smooth and bounded variations, supporting the validity of the Lipschitz continuity assumption stated in Assumption 1. This consistent performance, observed across both previous simulation studies and current experiments, underscores the method’s robustness and reliability in vehicular applications, establishing it as a dependable alternative.

Figure 9.

Experimental comparative analysis of yaw rate estimation and errors in the first scenario.

Figure 10.

Experimental analysis of lateral tire force estimation by the extended-state SOSMO in the first scenario.

Figure 11 presents side-slip angle estimations for two proposed frameworks, as well as comparisons with SDRE and EKF methods. The results clearly demonstrate superior performance of both proposed strategies over SDRE, and notably, they significantly outperform the EKF method. Specifically, the EKF method shows a tendency toward divergence in estimation error. Quantitatively, the first strategy achieves an RMSE of $4.08 \times 10^{- 6}$ , while the H-T alternative achieves $3.27 \times 10^{- 6}$ . In contrast, SDRE and EKF methods exhibit RMSEs of $8.93 \times 10^{- 4}$ and $6.96 \times 10^{- 2}$ respectively, highlighting the substantial performance disparity and underscoring the clear superiority of the proposed strategies. These findings provide compelling evidence for integrating these innovative techniques into automotive research.

Figure 11.

Experimental comparative analysis of sideslip angle estimation and errors in the first scenario.

After validating the feasibility of the suggested strategies in a basic practical setting against two established methods, the next crucial step involves subjecting all approaches to rigorous evaluation under more demanding real-world conditions akin to those simulated in our initial studies. This subsection serves as a practical extension of the computer simulation results, facilitating an in-depth analysis of how each method performs under practical constraints. Such an examination will provide valuable insights into the robustness and adaptability of these strategies when applied in scenarios mirroring real-world complexities.

Lane change scenario

The vehicle’s dynamic response is currently being experimentally tested during a rapid lane change at $120 km / h$ with a 50° steering angle, mirroring conditions previously examined through simulations. Figure 12 illustrates the yaw rate estimation results, where the extended-state SOSMO method exhibits superior performance with a significantly smaller yaw rate estimation error, as indicated by an RMSE of $6.94 \times 10^{- 8}$ . In contrast, the SDRE and EKF methods display considerably larger estimation errors, with RMSE values of $7.91 \times 10^{- 4}$ . Comparing these numerical results with the simulation outcomes in Figure 5, obtained under ideal conditions with precise parameters and minimal errors, highlights how real-world experiments are affected by unpredictable factors like measurement inaccuracies and equipment limitations. Despite increased estimation errors, the key finding underscores the superiority of the proposed framework over the two well-known estimators. Figure 13 demonstrates the extended-state SOSMO’s effectiveness in accurately estimating previously unknown lateral tire forces through experimental validation, showcasing its capability to provide precise yaw rate estimations and successfully estimate unknown variables. The reliability of these results, consistent across both simulated scenarios (Figure 6) and practical experiments (Figure 13), positions the extended-state SOSMO as a reliable alternative method for real-world applications.

Figure 12.

Experimental comparative analysis of yaw rate estimation and errors in the second scenario.

Figure 13.

Experimental analysis of lateral tire force estimation by the extended-state SOSMO in the second scenario.

Finally, Figure 14 illustrates the sideslip estimation performance for both innovative methods and the SDRE and EKF frameworks. The findings indicate that both proposed techniques experience smaller error bounds, with an RMSE of $2.42 \times 10^{- 6}$ for the first one and $2.013 \times 10^{- 6}$ for the H-T based method, compared to the SDRE method with an RMSE of $3.12 \times 10^{- 4}$ , and EKF with an RMSE of $9.07 \times 10^{- 3}$ . These numerical values highlight a significant performance gap favoring the innovative approaches over the compared methods, emphasizing their precision and accuracy in estimating sideslip angles. Despite an increase in errors compared to simulation counterparts shown in Figure 7, which illustrate the influence of human factors and inherent variability in physical systems, the proposed methods demonstrate a lower sensitivity to these factors, reinforcing their effectiveness.

Figure 14.

Experimental comparative analysis of sideslip angle estimation and errors in the second scenario.

Quantitative performance comparison

To highlight the effectiveness of the proposed estimation strategies across different scenarios, Table 1 presents a comparative summary of RMSE values for yaw rate and sideslip angle estimation under both steering pad and lane change conditions.

Table 1.

RMSE comparison of yaw rate and sideslip angle estimation across different scenarios.

Scenario	Method	Yaw rate	Side slip angle
Lane change/simulation	SOSMO	$6.49 \times 10^{- 30}$	$2.95 \times 10^{- 8}$
	H-T transformation	–	$2.73 \times 10^{- 8}$
	SDRE	$1.35 \times 10^{- 28}$	$6.03 \times 10^{- 6}$
	EKF	$7.29 \times 10^{- 28}$	$2.64 \times 10^{- 3}$
Lane change/experiment	SOSMO	$6.94 \times 10^{- 8}$	$2.42 \times 10^{- 6}$
	H-T transformation	–	$2.013 \times 10^{- 6}$
	SDRE	$7.91 \times 10^{- 4}$	$3.12 \times 10^{- 4}$
	EKF	$7.91 \times 10^{- 4}$	$9.07 \times 10^{- 3}$
Steering pad/experiment	SOSMO	$1.98 \times 10^{- 7}$	$4.08 \times 10^{- 6}$
	H-T transformation	–	$3.27 \times 10^{- 6}$
	SDRE	$2.29 \times 10^{- 3}$	$8.93 \times 10^{- 4}$
	EKF	$2.29 \times 10^{- 3}$	$6.96 \times 10^{- 2}$

This table confirms that the proposed estimation strategies not only outperform classical methods such as SDRE and EKF in simulation but also maintain their superiority under real-world experimental conditions. Notably, the proposed SOSMO exhibits extremely low estimation error in yaw rate prediction, while both proposed methods for sideslip estimation achieve significantly higher accuracy and faster convergence compared to benchmark approaches. For instance, in the lane change simulation scenario, the H-T transformation method yields the most accurate sideslip angle estimation, surpassing all other approaches. The SOSMO also provides a sideslip estimate nearly comparable to that of the H-T transformation, whereas SDRE and EKF exhibit substantially higher errors. Moreover, while simulation results represent ideal conditions, experimental scenarios introduce additional uncertainties such as sensor noise and unmodeled disturbances. Under experimental lane change conditions, for example, SOSMO continues to perform strongly, with yaw rate and sideslip angle errors of $6.94 \times 10^{- 8}$ and $2.42 \times 10^{- 6}$ , respectively. Leveraging its high accuracy in yaw rate estimation, the H-T transformation method also maintains solid performance in estimating the sideslip angle. In contrast, SDRE and EKF show significant performance degradation, with yaw rate errors reaching $7.91 \times 10^{- 4}$ and a substantial sideslip angle error of $9.07 \times 10^{- 3}$ for EKF compared to the proposed methods. A similar trend is observed in the steering pad experiment, where SOSMO consistently delivers high estimation accuracy, while EKF again exhibits the highest sideslip angle error, suggesting poor generalization across different maneuvering scenarios. Despite these challenges, the proposed frameworks maintain estimation errors well within accepted limits, thereby demonstrating their robustness and practical applicability. These findings are poised to advance vehicle dynamics and control systems, enhancing safety and efficiency in transportation.

Remark 4: While the fractional power term $| \hat{r} - r |^{\frac{1}{2}} sign (\hat{r} - r)$ in observer (5) improves convergence for large errors, it may slow down near the sliding surface. Adaptive gain strategies, as proposed in Razmjooei et al.,^37,38 can address this limitation and will be further developed in future studies.

Conclusion

This research tackled the intricate challenge of precisely estimating sideslip angles and lateral tire-road forces in single-track road vehicles through experimental approaches. An extended-state SOSMO framework was introduced to achieve rapid convergence in estimating yaw rate, lateral tire forces, and sideslip angles. The study presented two distinct methodologies for sideslip angle estimation: the first method employed dynamical equation manipulations to yield acceptable estimations, while the second method utilized an inverse model-based approach integrating a novel tire model with nonlinear tire-road friction characteristics to achieve highly accurate estimations. Comparative analysis through simulations and practical experiments across various scenarios convincingly demonstrated the superior efficiency and accuracy of these methodologies over conventional alternatives such as the SDRE filter and EKF. For instance, during an open-loop steering pad maneuver designed to evaluate steady-state circular driving behavior, the proposed extended-state SOSMO achieved an impressive RMSE of $1.98 \times 10^{- 7}$ in yaw rate estimation. In contrast, the SDRE and EKF methods yielded RMSE values of $2.29 \times 10^{- 3}$ , highlighting the substantial performance advantage of the SOSMO framework. Moreover, the extended-state observer proved adept at accurately estimating lateral tire forces, a crucial factor for precise sideslip angle estimation. This capability was consistently demonstrated across both simulated scenarios and practical experiments, underscoring its superior performance compared to traditional estimation methods. These findings established the SOSMO-based methodologies as robust and reliable choices for advancing vehicle state estimation capabilities, with significant implications for future research and development in automotive engineering. However, a key limitation of these techniques is the reduced convergence speed near the sliding surface, which can lead to increased estimation error. Future research may focus on enhancing convergence uniformly across all regions, particularly near equilibrium points, while also extending the applicability of these methods to a wider range of vehicle types. Additionally, validating performance using double-track models and simplifying estimation strategies to minimize parameter dependency could further advance SOSMO-based approaches in vehicle dynamics and control.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Ethical approval

This article does not contain any studies with human or animal participants.

ORCID iDs

Hamid Razmjooei

Mohammad Hossein Shafiei

Data availability

The data supporting this study are available from the corresponding author upon request.

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