Sage Journals: Discover world-class research

Abstract

This study presents a second-order sliding mode observer (SOSMO) framework developed to improve the estimation of lateral tire forces in vehicles. The framework incorporates two distinct approaches for approximating the sideslip angle: one based on dynamical equations and another employing an inverse model estimation technique with a new tire model. The suggested tire model captures the nonlinear characteristics of tire–road friction, enabling a more accurate representation of lateral force behavior. Comparative evaluations are conducted using a single-track vehicle model based on the Pacejka formula under two scenarios: the open-loop steering pad maneuver and the lane change maneuver. Simulation results demonstrate the superior performance of the proposed methods compared to two established observers, namely, the extended Kalman filter and the state-dependent Riccati equation (SDRE) filter, even in the absence of detailed tire–road interaction models. Notably, in a steady-state circular driving scenario, the second approach achieves a 99% smaller error compared to the first approach and a 99.38% smaller error relative to the SDRE filter. In a transient maneuver scenario, the second approach achieves a 10.71% smaller error than the first approach and a 99.63% smaller error compared to the SDRE filter. Robust studies under external disturbances further confirm the proposed methods’ precision and reliability in estimating sideslip angle and lateral tire forces, offering a cost-effective alternative to traditional tire–road interaction models.

Keywords

Finite-time estimation framework nonlinear tire model sideslip angle sliding mode observer tire–road interaction

Introduction

Road vehicles are influenced by various factors, such as speed, lateral acceleration, yaw rate, and tire–road friction coefficients, which are crucial for performance and safety. However, real-time monitoring of these factors is impeded by sensor limitations. While some factors can be measured directly, others, such as lateral tire–road forces and sideslip angles, necessitate expensive equipment. A more effective approach involves using sensors to obtain standard outputs and employing observation algorithms to estimate system states.

The prioritization of accurately estimating the sideslip angle and lateral forces in road vehicle applications, which has motivated this study, has been extensively explored by researchers since the 1950s. Various methodologies have been investigated, including dynamic and kinematic models, each offering unique advantages and limitations (Liu et al., 2018). Traditionally, dynamic model-based estimation techniques have included deterministic state estimators such as the sliding mode observer (SMO; Stéphant et al., 2007) and high-gain observer (Alai et al., 2024), as well as stochastic state estimators like the traditional Kalman filter (Park, 2022), extended Kalman filter (EKF; May et al., 2023), and unscented Kalman filter (UKF; Antonov et al., 2011), all of which have been applied without relying on costly optical sensors.

Over the past decade, there has been a significant increase in the use of nonlinear estimation techniques in the field of vehicle dynamics, building on prior academic work. For instance, Gadola et al. explored a nonlinear single-track vehicle model that integrated a simplified Magic tire model, using an EKF to estimate sideslip angles without the need for additional sensors. Their research demonstrated the effectiveness of this method through both simulations and practical experiments (Gadola et al., 2014). Similarly, Ding et al. employed a simple nonlinear model for a single-track vehicle, incorporating a random-walk model to account for road friction. They developed an extended Luenberger observer to accurately estimate sideslip angles and road friction coefficients, achieving notable precision and robustness in their estimates (Ding et al., 2014). In a subsequent investigation, the state-dependent Riccati equation (SDRE) method was applied to estimate the sideslip angle of road vehicles, in conjunction with the Random Walk Model. This approach showcased improved performance compared to the traditional EKF (Strano and Terzo, 2017). A novel approach was presented for estimating vehicle sideslip angles within a double-track vehicle model, considering variations in tire pneumatic trail (Xia et al., 2018). Their enhanced model, combined with the EKF framework, yielded more precise estimates. Then, an innovative method was introduced using a constrained UKF for a single-track vehicle model to estimate sideslip angles (Strano and Terzo, 2018). This approach allowed the estimator to operate independently of the specific modeling of tire interaction forces, demonstrating promising results in accurately estimating complex dynamic systems. In addition, Cheng et al. investigated a unified observation algorithm designed to simultaneously estimate tire sideslip angles and lateral tire forces for a double-track vehicle model. Although their adaptive SMO initially demonstrated potential, the computational complexity of the compensation algorithm used for estimating vehicle sideslip angles warrants further investigation (Cheng et al., 2019). In 2020, a method combining deep neural networks with nonlinear Kalman filters was proposed for vehicle sideslip angle estimation, improving accuracy through adaptive measurement updates (Kim et al., 2020). While effective in simulations and experiments, challenges include computational complexity and limited validation under diverse real-world conditions. An EKF was also employed to estimate the states of a dual-track vehicle, demonstrating its accuracy through simulations conducted across various road surfaces and driving maneuvers (Lenzo et al., 2020). In 2022, an interacting multiple-model Kalman filter was developed integrating two EKFs for vehicle lateral velocity estimation (Park, 2022). The method showed improved sideslip angle and tire stiffness estimation but was limited by its reliance on a single GPS sensor, affecting performance in weak GPS environments and extreme driving conditions. In addition, while it improved accuracy in the tire nonlinear region, it did not address performance under abrupt steering inputs during rapid lane changes. Wang et al. proposed a second-order fault-tolerant EKF designed to enhance the robust estimation of yaw rate, lateral acceleration, and sideslip angle for a single-track vehicle model, even under varying data loss distributions (Wang et al., 2023). By incorporating an arbitrary discrete distribution to model data loss within the second-order fault-tolerant EKF framework, their method achieved an improvement of over 23% in sideslip angle estimation accuracy compared to conventional EKF methods. In 2023, a technique was introduced for estimating sideslip angles and yaw rates in a single-track vehicle model, utilizing an information fusion-based UKF and assessing its effectiveness through various testing methods (Zhao et al., 2023). Recently, a hybrid estimator was developed for vehicle sideslip angles, integrating the principles of a UKF with a convolutional neural network based on a nonlinear single-track vehicle model (Bertipaglia et al., 2024). This integration enables the neural network to leverage physical laws while providing pseudo-measurements to the UKF, effectively combining model-based and data-driven approaches using standard inertial measurement units and tire force data. Experimental results from 216 maneuvers indicated that this hybrid methodology significantly improved estimation accuracy over existing techniques, achieving a 25% reduction in mean squared error compared to traditional dynamic EKF, particularly in critical areas relevant to active vehicle control systems.

From the reviewed strategies, it can be concluded that enhancing estimation accuracy can be achieved through two principal approaches: refining tire models and developing advanced estimation techniques. First, from the perspective of tire model development, it can be observed that various specific tire models have been employed to describe tire friction, including linear tire models (Venhovens and Naab, 1999) and nonlinear models such as the Pacejka tire model (Pacejka’s Magic Formula; Pacejka, 2005), the Dugoff tire model (Han and Huh, 2011), the LuGre tire model (Lian et al., 2015), and the Brush tire model (Jin et al., 2017). The numerous parameters associated with these models necessitate extensive calibration through testing across different tire specifications and pressures, which complicates the understanding of tire forces and tire–road friction, rendering their direct application for estimation impractical. Consequently, developing methodologies that do not rely on prior knowledge of tire–road properties is essential. While integrating a vehicle model with a physical or empirical tire model also introduces complexities due to factors such as tire properties, vehicle specifications, and road conditions, this study, drawing inspiration from the validated performance of the arctangent tire model in literature (Gao et al., 2010), proposes the hyperbolic tangent (H-T) tire model. The H-T model, both computationally efficient and conceptually aligned with the high-accuracy Magic Formula model, demonstrates substantial improvements over the arctangent model in terms of estimation accuracy and performance.

Second, from the perspective of estimation techniques, a comprehensive study was conducted employing various Kalman filtering-based estimators to assess the accuracy of unmeasured vehicle states (Leanza et al., 2024). Nonlinear estimation techniques, such as UKF and the Cubature Kalman filter (CKF), were compared with EKF using real-world experimental data. The results indicated that although more complex models and estimation methods exist, simpler models often provide a more favorable trade-off between accuracy and computational demands for typical applications. In contrast, specialized scenarios require more intricate models and advanced estimation methods. As this study forms part of an ongoing research project aimed at advancing deterministic estimation methods as alternatives to the widely adopted Kalman filters, it holds the potential to improve upon the results reported by highly promising methods. The study seeks to identify the most suitable model and estimation algorithm, balancing accuracy with computational efficiency. In addition, it highlights the advantages of tuning parameters optimally through optimization algorithms, which simplifies the process and enhances overall performance.

Considering these objectives and drawing inspiration from the comprehensive review in literature (Leanza et al., 2024; Zhu et al., 2024) on the current state of vehicle sideslip angle estimation, this paper aims to advance the discussion with contributions that diverge from existing approaches in two significant ways. First, it develops an extended second-order SMO (SOSMO) framework designed to enable the precise estimation of lateral tire forces in single-track road vehicles, ensuring a trade-off between estimation accuracy and computational efficiency. Second, the methodology is supplemented with two approaches for estimating the sideslip angle: one based on dynamic equations and the other employing an inverse model approach, incorporating a cutting-edge tire model that accurately captures nonlinear tire–road friction characteristics. It is worth noting that while the theoretical foundation builds upon the established SOSMO method, the supplementary techniques provide a robust alternative to conventional, time-consuming estimation methods, which often rely heavily on extensive knowledge of tire–road interactions. The proposed approach enhances reliability and cost-efficiency while achieving superior performance in estimating sideslip angles and lateral forces. The key contributions are outlined as follows:

Development of a class of SOSMO frameworks specifically designed for the precise estimation of lateral tire forces in single-track road vehicles.

Exploration of two distinct methodologies for sideslip angle estimation: one based on dynamic equations and the other innovatively employing an inverse model approach, incorporating an advanced tire model capable of capturing nonlinear tire–road friction characteristics.

Demonstration of the superior accuracy and efficiency of the proposed strategies compared to conventional observers, such as EKF and SDRE methods, particularly in estimating sideslip angles and lateral forces.

Presentation of significant improvements in estimation accuracy, validated through comprehensive analyses conducted across two practical scenarios.

Highlighting the potential of the proposed techniques as reliable and cost-effective alternatives to traditional estimation methods, which typically require extensive prior knowledge of tire–road interactions.

Extensive computer simulations were conducted using MATLAB software, where two practical scenarios were meticulously designed to assess the proposed methodologies. In the first scenario, the vehicle’s steady-state circular driving behavior was analyzed, maintaining a constant longitudinal speed while progressively increasing steering angles. This controlled environment provided a basis for evaluating the accuracy of yaw rate and lateral force estimations. In the second scenario, a transient lane change maneuver was performed under high-adhesion conditions, incorporating sudden steering inputs to assess the vehicle’s dynamic response. The precise tuning of the observer parameters was essential for the accuracy of these simulations, as it was calibrated to minimize the root mean square error (RMSE) in both scenarios. The results revealed that sideslip angle estimations exhibited exceptional precision, with the second estimation method achieving a significantly lower RMSE compared to traditional approaches.

The paper is structured systematically. The section “Modeling the single-track vehicle dynamics” provides an illustrative representation of a reference single-track vehicle model, emphasizing the application of the Pacejka Magic Formula, which employs four parameters to effectively model tire–road interactions. The section “Development of the second-order sliding mode observer” begins by elaborating on the model’s details and its significance in capturing the nonlinear characteristics of tire behavior under various conditions. This section then discusses the design aspects of the extended-state SOSMO and complementary observation algorithms. The section “Simulation results” demonstrates the efficacy of the observer through numerical validation, supported by simulation results that highlight the accuracy and reliability of the approaches. Finally, the section “Conclusion” concludes by summarizing the key findings of the study and outlining potential research directions, focusing on improvements to the observer framework and the exploration of new methods for estimating vehicle states in increasingly complex driving environments.

Modeling the single-track vehicle dynamics

This section focuses on the depiction of a conventional single-track vehicle, as shown in Figure 1, which includes a front steering axle positioned within the inertial reference frame denoted as Oxy. In addition, it introduces a body-fixed reference frame referred to as O′xy.

Figure 1.

Reference frames of a 2-degree-of-freedom vehicle model.

The single-track vehicle model described in equations (1a) and (1b) is used to represent the lateral dynamics of the vehicle, with particular emphasis on critical states, including the yaw rate and sideslip angle (Nguyen et al., 2024)

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 represents the total mass of the vehicle, r denotes the yaw rate, δ signifies the steering wheel angle of the front tires, $J_{z}$ stands for the vehicle’s moment of inertia in relation to the yaw motion, and the distances from the front and rear axles to the center of gravity are represented by $L_{1}$ and $L_{2}$ , respectively. As depicted, the vehicle’s sideslip angle β is the angle formed between the vehicle’s longitudinal axis and the direction of its velocity vector. This angle quantifies the vehicle’s lateral sliding, influenced by both the yaw rate and lateral acceleration. In addition, the lateral velocity and acceleration are represented 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 $V_{x}$ is the longitudinal velocity.

Remark 1: It is important to note that the yaw rate represents the vehicle’s rotational velocity around its vertical axis, which is influenced by the lateral forces acting on the tires. The relationship between yaw rate and lateral forces is detailed in equation (1b), derived from the vehicle’s rotational dynamics (Ben Moussa and Bakhti, 2024). In addition, lateral acceleration refers to the acceleration of the vehicle’s center of gravity in the lateral direction and depends on the yaw rate, vehicle velocity, and sideslip angle, as outlined in equation (2b).

As shown in equation (3), $F_{y 1}$ and $F_{y 2}$ represent the lateral interaction forces generated by tire slip angles and influenced by the steering angle, acting on the front and rear axles of the single-track vehicle, as described by the tire–road interaction model. This equation represents a simplified form of the Pacejka Magic Formula, as outlined in reference (Rabhi et al., 2024)

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

(3)

where B signifies the stiffness factor, C represents the form factor, and E is the curvature factor. Besides, 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 $F_{z 2} = \frac{mg L_{1}}{L_{1} + L_{2}}$ . Furthermore, the wheel slip angles $α_{i}$ associated with the front and rear axles, respectively, are defined as $α_{1} = atan (\frac{V_{y} + r L_{1}}{V_{x}}) - δ$ and $α_{2} = atan (\frac{V_{y} - r L_{2}}{V_{x}})$ .

Prior to addressing the estimation of lateral forces exerted by the tires and the vehicle’s sideslip angle, it is essential to outline widely accepted simplifying assumptions that serve as a practical foundation for observer design. The first assumption states that the longitudinal velocity remains constant ( ${\overset{\cdot}{V}}_{x} = 0$ ). The second assumption considers a small sideslip angle, where $1 + \tan^{2} β ≅ 1$ , ensuring it does not exceed $15^{°}$ , even in challenging driving conditions for typical drivers. This assumption is based on the fact that under normal driving conditions without traction loss, a vehicle’s sideslip angle usually remains small, typically under 15° (Puscul et al., 2024; Zhu et al., 2024). Larger sideslip angles often indicate an unstable vehicle state, which is beyond the study’s scope. To improve system performance under significant tire slip, more precise tire models and tire cornering stiffness estimation techniques for real-time adjustments in the nonlinear region can be employed (Chen et al., 2024). The third assumption addresses the availability of measurements for steering angle, longitudinal velocity, yaw rate, and lateral acceleration, which are generally accessible in automotive applications (Solmaz and Başlamışlı, 2012).

Remark 2: In steady-state conditions, such as highway driving, assuming constant velocity simplifies the analysis by reducing equation complexity. This assumption is valid when speeds are stable but may lead to inaccuracies during sudden acceleration or sudden braking.

Taking the first two assumptions into account, it can be deduced that $a_{y} = V_{x} (r + \overset{\cdot}{β})$ , leading to the following formulation of the vehicle’s state-space model

\overset{\cdot}{r} = - \frac{L_{2} m}{J_{z}} a_{y} + Δ_{1}

(4)

where $Δ_{1} = \cos δ \frac{L_{1} + L_{2}}{J_{z}} F_{y 1}$ . It can be demonstrated that $\overset{\cdot}{r}$ could also be redefined in relation to $F_{y 2}$ as follows

\overset{\cdot}{r} = \frac{L_{1} m}{J_{z}} a_{y} + Δ_{2}

(5)

in which $Δ_{2} = - \frac{L_{1} + L_{2}}{J_{z}} F_{y 2}$ . Recognizing the impact of these terms on estimation accuracy, an extended-state observer framework (Razmjooei et al., 2020a), in terms of ultimate boundedness, will be formulated based on the higher-order sliding mode observer approach, which has been widely utilized to enhance estimation precision.

Assumption 1: Acknowledging that these terms represent unknowns within the system, the lateral interaction forces $F_{y 1}$ and $F_{y 2}$ adhere to the Lipschitz condition (Razmjooei et al., 2022a). This condition, supported by the known Lipschitz constants $L_{Δ 1}$ and $L_{Δ 2}$ , guarantees that $| {\overset{\cdot}{Δ}}_{i} | \leq L_{Δ i}$ .

Development of the SOSMO

The observation algorithm implemented in this study leverages readily available data, including parameters such as the steering angle of the front wheels, vehicle speed relative to the ground, lateral acceleration, and yaw rate. To simultaneously estimate yaw rate and unknown lateral forces, extended-state SOSMOs given by equations (6) and (7) have been developed based on the standard SOSMO framework presented in literature (Davila et al., 2005; Razmjooei et al., 2022b)

\begin{array}{l} \overset{\cdot}{\hat{r}} = - \frac{L_{2} m}{J_{z}} a_{y} + {\hat{Δ}}_{1} - k_{01} L_{Δ 1}^{\frac{1}{2}} {| \hat{r} - r |}^{\frac{1}{2}} sign (\hat{r} - r) \\ {\hat{\overset{\cdot}{Δ}}}_{1} = - k_{11} L_{Δ 1} sign (k_{01} L_{Δ 1}^{\frac{1}{2}} {| \hat{r} - r |}^{\frac{1}{2}} sign (\hat{r} - r)) \end{array}

(6)

and

\begin{matrix} \overset{\cdot}{\hat{r}} = \frac{L_{1} m}{J_{z}} a_{y} + {\hat{Δ}}_{2} - k_{02} L_{Δ 2}^{\frac{1}{2}} {| \hat{r} - r |}^{\frac{1}{2}} sign (\hat{r} - r) \\ {\hat{\overset{\cdot}{Δ}}}_{2} = - k_{12} L_{Δ 2} sign (k_{02} L_{Δ 2}^{\frac{1}{2}} {| \hat{r} - r |}^{\frac{1}{2}} sign (\hat{r} - r)) \end{matrix}

(7)

where $\hat{r}$ , ${\hat{Δ}}_{1}$ and ${\hat{Δ}}_{2}$ are the estimations of r and the augmented lateral interaction forces $Δ_{1}$ and $Δ_{2}$ , respectively. Furthermore, $k_{0 i}$ and $k_{1 i}$ are both positive constants, and $sign (.)$ denoting the sign function. To evaluate the convergence of estimations and, consequently, the stability of estimation errors, the closed-loop systems of observation errors are first derived using equations (4) to (7), along with the definition of the observation errors as $e_{r} = \hat{r} - r$ , $e_{Δ_{1}} = {\hat{Δ}}_{1} - Δ_{1}$ , and $e_{Δ_{2}} = {\hat{Δ}}_{2} - Δ_{2}$

\begin{matrix} {\overset{\cdot}{e}}_{r} = e_{Δ_{1}} - k_{01} L_{Δ 1}^{\frac{1}{2}} {| e_{r} |}^{\frac{1}{2}} sign (e_{r}) \\ {\overset{\cdot}{e}}_{Δ_{1}} = - {\overset{\cdot}{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{matrix}

(8)

and

\begin{matrix} {\overset{\cdot}{e}}_{r} = e_{Δ_{2}} - k_{02} L_{Δ 2}^{\frac{1}{2}} {| e_{r} |}^{\frac{1}{2}} sign (e_{r}) \\ {\overset{\cdot}{e}}_{Δ_{2}} = - {\overset{\cdot}{Δ}}_{2} - k_{12} L_{Δ 2} sign (k_{02} L_{Δ 2}^{\frac{1}{2}} {| e_{r} |}^{\frac{1}{2}} sign (e_{r})) \end{matrix}

(9)

Accordingly, by taking into account the Lipschitz condition and ensuring that all differential inclusions comply with the upper semi-continuity property in the Filippov sense, the error dynamics are reformulated as follows

\begin{matrix} {\overset{\cdot}{e}}_{r} = e_{Δ_{1}} - k_{01} L_{Δ 1}^{\frac{1}{2}} {| e_{r} |}^{\frac{1}{2}} sign (e_{r}) \\ {\overset{\cdot}{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{matrix}

(10)

and

\begin{matrix} {\overset{\cdot}{e}}_{r} = e_{Δ_{2}} - k_{02} L_{Δ 2}^{\frac{1}{2}} {| e_{r} |}^{\frac{1}{2}} sign (e_{r}) \\ {\overset{\cdot}{e}}_{Δ_{2}} \in [- L_{Δ 2}, L_{Δ 2}] - k_{12} L_{Δ 2} sign (k_{02} L_{Δ 2}^{\frac{1}{2}} {| e_{r} |}^{\frac{1}{2}} sign (e_{r})) \end{matrix}

(11)

where the secondary formulations for ${\overset{\cdot}{e}}_{Δ_{1}}$ and ${\overset{\cdot}{e}}_{Δ_{2}}$ evolve into

\begin{matrix} {\overset{\cdot}{e}}_{Δ_{1}} \in [- (1 + k_{11}) L_{Δ 1}, (1 - k_{11}) L_{Δ 1}] \\ {\overset{\cdot}{e}}_{Δ_{2}} \in [- (1 + k_{12}) L_{Δ 2}, (1 - k_{12}) L_{Δ 2}] \end{matrix}

(12)

The stability analysis of the error dynamics is now concluded using the following lemma, which guarantees that the observation error converges to a small neighborhood around zero, even in the presence of uncertainties $F_{y 1}$ and $F_{y 2}$ .

Lemma 1 (Davila et al., 2005): The following dynamics, associated with a system possessing a relative degree of m, witness finite-time stability

\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} L sign (z_{m} - {\overset{\cdot}{z}}_{m - 1}) + [- L, L] \end{matrix} \end{matrix}

(13)

where, for $m = 1$ , the convergence of the error dynamics in equations (8) and (9) to small neighborhoods around zero within a finite time can be concluded, based on the stability condition presented in reference.

Remark 3: The estimations of the lateral tire forces can now be directly calculated using ${\hat{F}}_{y 1} = \frac{{\hat{Δ}}_{1} J_{z}}{(L_{1} + L_{2}) \cos δ}$ and ${\hat{F}}_{y 2} = \frac{- {\hat{Δ}}_{2} J_{z}}{L_{1} + L_{2}}$ , where ${\hat{Δ}}_{1}$ and ${\hat{Δ}}_{2}$ have already been estimated using the extended-state SOSMOs given by equations (6) and (7).

Two approaches are now considered to estimate the vehicle’s sideslip angle. The first approach uses dynamic formulations, where substituting equation (1a) into equation (2b) yields

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

(14)

where ${\hat{F}}_{y 1}$ and ${\hat{F}}_{y 2}$ have already been obtained. Then, an estimation of the sideslip angle is presented based on equation (2a) as follows

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

(15)

where the sideslip angle $\hat{β}$ is estimated without any actual knowledge of the tire model. An alternative approach is now proposed, incorporating an innovative, high-precision nonlinear tire model, as described in equation (16). This model represents a significant advancement in tire modeling techniques within automotive engineering, as it eliminates the reliance on linear tire models, avoids the simplified yet less accurate Dugoff tire model, and circumvents the use of complex nonlinear models, such as the Magic Formula

{\bar{F}}_{y 1} = a_{11} \tanh (a_{21} (- β + δ - \frac{L_{1}}{V_{x}} r))

(16a)

{\bar{F}}_{y 2} = a_{12} \tanh (a_{22} (- β + \frac{L_{2}}{V_{x}} r))

(16b)

where $a_{ij}$ , for $i, j = 1$ and 2, are model parameters. The parameter identification of the tire model is crucial for accurate sideslip estimation. A wide range of operating conditions was examined using 21,000 random values of experimental data for each tire. To ensure the safe conduct of the identification tests, experiments were performed on 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 combined with a differential global positioning system to record longitudinal velocity, yaw rate, sideslip angle, and lateral acceleration. It is important to note that not all this data serves as output variables; for example, the sideslip angle signal was used as reference data for comparative studies aimed at validation and identification.

Notably, the single-track vehicle model under investigation is equipped with the established Magic Formula tire model given by equation (3), while the proposed model given by equation (16) specifically targets accurate and effective tire behavior estimations. Considering the proposed model given by equation (16), it is evident that once the finite-time estimation of the uncertain terms $Δ_{i}$ and, subsequently, the lateral forces ${\hat{F}}_{yi}$ is established, the vehicle’s sideslip angle can be accurately estimated. This estimation utilizes the inverse model of equation (16), specifically equation (16b), while assuming ${\bar{F}}_{y 2} = {\hat{F}}_{y 2}$ , as demonstrated below

\hat{β} = \frac{L_{2}}{V_{x}} r - \frac{1}{a_{22}} \tanh^{- 1} (\frac{{\hat{F}}_{y 2}}{a_{12}})

(17)

The effectiveness of this approach, along with the innovative tire model that streamlines and ensures accurate estimations, is demonstrated in the subsequent evaluation section.

Simulation results

This section presents comprehensive computer simulations conducted using MATLAB software to compare the results obtained from the proposed frameworks with those from leading state-of-the-art benchmarks. To ensure a fair comparison, these methods are evaluated using the nominal parameters $L_{1} = L_{2} = 1.15 m$ , $m = 1600 kg$ , and $J_{z} = 1800 kg m^{2}$ . On one hand, there are no stringent constraints on determining the observer parameters, as a wide range of values meets the standard conditions. Nevertheless, to facilitate a fair comparison among the observers, their parameters were optimized using the particleswarm function in MATLAB R2018a. This optimization resulted in $k_{01} = k_{11} = 100$ and $k_{02} = k_{12} = 174$ , minimizing the RMSE and substantially improving the overall accuracy and reliability of the estimations. On the other hand, after collecting data with the instrumented vehicle and calculating the lateral interaction forces on the front and rear axles (designated as $F_{y 1}$ and $F_{y 2}$ ), the coefficients of the polynomials in equations (16) were determined using the least squares method. The polynomial orders were selected to be the smallest that produced an acceptable fitting error, ensuring model simplicity to avoid over-fitting. The chosen coefficients were $a_{11} = 1.4 \times 10^{5}$ , $a_{12} = 1.1 \times 10^{5}$ , $a_{21} = 55$ , and $a_{22} = 105$ , resulting in RMSE values of 1.193% for $F_{y 1}$ and 2.704% for $F_{y 2}$ . To validate the accuracy of the proposed model, a numerical comparative analysis was performed using the Arctangent-based Dugoff tire model (Gao et al., 2010) and the widely recognized Magic Formula tire model given by equation (3). This analysis focused on the front tire’s performance under a maximum lateral acceleration of $8 m / secon d^{2}$ . The results showed that the Arctangent model closely followed the Magic Formula curve when the front tire’s slip angle was below $3^{°}$ . However, its accuracy diminished as the slip angle exceeded $3^{°}$ . In contrast, the proposed H-T tire model demonstrated a broader accuracy range, aligning closely with the Magic Formula curve even at slip angles above $5^{°}$ , where the Magic Formula curve began to flatten. These analyses are based on an unshown figure, omitted due to illustration limits but available upon request. The improved accuracy of the proposed model is anticipated to facilitate more efficient and precise estimations, thereby enhancing overall automotive performance and safety through streamlined processes.

To maintain focus and prevent data overload, simulation results are presented for two specific scenarios (Ben Moussa and Bakhti, 2024; Nguyen et al., 2024; Strano and Terzo, 2017; Strano and Terzo, 2018). The first maneuver involves a gradual increase in the steering angle while maintaining a constant vehicle speed relative to the ground, which allows for the assessment of the vehicle’s behavior during circular driving. The second scenario features specific parameters for a lane-change maneuver, incorporating abrupt steering inputs for rapid lane changes, thereby providing insights into the vehicle’s lateral acceleration and yaw rate responses during dynamic maneuvers.

Scenario 1: Open loop steering pad maneuver

The first simulated maneuver is designed to assess the vehicle’s steady-state behavior during circular driving. This scenario involves operating at a constant longitudinal speed of $120 km / hour$ , with the steering angle gradually and linearly increasing over time, all conducted under high-adhesion conditions.

Figure 2 illustrates the effectiveness of the implemented extended-state SOSMO in estimating the yaw rate. Although this variable has been measured previously, the primary focus here is on assessing the proposed strategy’s ability to accurately estimate this measurable state variable. Notably, the estimation error for the yaw rate consistently remains within an exceptionally low range, highlighting the high accuracy achieved.

Figure 2.

Yaw rate estimation in the extended-state SOSMO.

Figure 3 demonstrates the excellent estimation performance of the lateral tire forces compared to the actual forces determined by the developed observer framework, revealing minimal estimation errors. This further affirms the extended-state SOSMO’s competence in estimating unknown variables. Regarding sideslip angle estimation, Figure 4 showcases the efficacy of the proposed estimation strategies. The dynamical model-based approach (depicted by the blue dashed curves), which employs a straightforward procedure, exhibits an acceptably small estimation error. In contrast, the second framework (represented by the red dashed-dotted curves), based on an innovative model, demonstrates remarkable performance in correlating the estimated sideslip angle with the actual state.

Figure 3.

Lateral forces estimation in the extended-state SOSMO.

Figure 4.

Sideslip estimation in the extended-state SOSMO.

Despite the indirect nature of the sideslip angle estimations, both methods successfully provide accurate estimates within a finite time frame. Building on the superior performance of the H-T model-based method, the estimation of lateral acceleration is expressed by the equation ${\hat{a}}_{y} = V_{x} (r + \overset{\cdot}{\hat{β}})$ , which depends on the previously determined sideslip angle (derived from the second approach) and the yaw rate (obtained using the extended-state SOSMO).

Figure 5 illustrates the estimation and associated error of the vehicle’s lateral acceleration, clearly indicating strong convergence of the estimated values toward the actual lateral acceleration. This underscores the exceptional accuracy achieved over time, reflecting a robust estimation process characterized by minimal deviations.

Figure 5.

Lateral acceleration estimation in the SOSMO.

To further assess the efficacy of the proposed approaches, two additional model-based estimation techniques, recognized for their high-performance capabilities—the SDRE filter discussed in the study by Strano and Terzo (2017) and the EKF outlined in other literatures (Li et al., 2014; May et al., 2023)—are employed for comparative analysis. Figure 6 illustrates the yaw rate estimation errors. The bar chart quantitatively depicts the RMSE performance index values, defined in terms of the infinity norm of the yaw rate observation error data. The developed SOSMO demonstrates the most favorable performance, achieving an RMSE of $1.33 \times 10^{- 30}$ , while the SDRE method yields an RMSE of $1.59 \times 10^{- 27}$ , and the EKF method shows an RMSE of $1.02 \times 10^{- 26}$ , indicating comparatively higher estimation errors.

Figure 6.

Comparative analysis of yaw rate estimation errors.

Furthermore, with regard to sideslip angle estimation, Figure 7 displays a bar chart illustrating the numerical values of the performance index, represented as the infinity norm of the sideslip angle observation error. Both proposed estimation frameworks exhibit high effectiveness; however, the H-T scheme outperforms the others, achieving an RMSE of $1.55 \times 10^{- 8}$ . Also, the first approach and the SDRE method show slightly lower performance, with RMSE values of $1.57 \times 10^{- 6}$ and $2.51 \times 10^{- 6}$ , respectively. The EKF method, on the contrary, records the least favorable results, with an RMSE of $0.273$ , indicating relatively higher estimation errors. Taken together, these results indicate that the proposed extended-state SOSMO, along with two other innovative approaches and the SDRE method, outperform the EKF method. A more comprehensive analysis will involve comparing the effectiveness of the proposed approaches with the SDRE method in a second, more challenging scenario.

Figure 7.

Comparative analysis of sideslip angle estimation errors.

Scenario 2: Lane change maneuver

This scenario, characterized by a transient maneuver, involves a lane change executed at a constant longitudinal speed of $120 km / hour$ , with a steering angle amplitude of 50° under high-adhesion conditions. The objective is to assess the vehicle’s dynamic response during lateral movements by utilizing specific parameters for the lane change maneuver and applying abrupt steering inputs to facilitate rapid lane changes.

Figure 8 presents the results of yaw rate estimation for both the extended-state SOSMO and SDRE methods. The data clearly demonstrate that the extended-state SOSMO method produces significantly better outcomes, characterized by notably smaller yaw rate estimation errors. In contrast, the SDRE method shows considerably larger estimation errors. To quantitatively evaluate overall performance, the norm of the yaw rate estimation errors was computed. The results demonstrate that the extended-state SOSMO method, with an RMSE of $6.49 \times 10^{- 30}$ , significantly surpasses the SDRE method, which has an RMSE of $1.35 \times 10^{- 28}$ .

Figure 8.

Comparative analysis of yaw rate estimation errors in the second scenario.

Furthermore, Figure 9 illustrates the performance of sideslip angle estimation along with the associated estimation errors for both the innovative methods and the SDRE framework. The findings reveal that the proposed techniques exhibit smaller error bounds, with an RMSE of $2.52 \times 10^{- 8}$ for the first method and $2.25 \times 10^{- 8}$ for the H-T based method. In contrast, the SDRE method demonstrates a significantly larger RMSE of $6.03 \times 10^{- 6}$ . This disparity emphasizes the considerable performance gap between the innovative approaches and the SDRE method, highlighting the substantial advantages of the proposed strategies in terms of precision and accuracy in estimating sideslip angles.

Figure 9.

Comparative analysis of sideslip angle estimation errors in the second scenario.

In alignment with another primary objective of this study—to provide an accurate estimation of lateral tire forces—Figure 10 illustrates the performance of the developed extended-state SOSMO method in estimating these forces. These comprehensive analyses underscore the significance of the proposed techniques, particularly the innovative second approach, in achieving precise estimation within the field of vehicle dynamics.

Figure 10.

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

Detailed comparative analysis of the proposed methodologies for vehicle dynamics estimation in challenging scenarios

The simulation results suggest that the proposed methods effectively manage abrupt changes in vehicle dynamics and are suitable for challenging scenarios, such as lane changes. However, it is important to note that EKF encounters difficulties due to its reliance on linear approximations. While the SDRE method demonstrates adaptive characteristics, it still produces larger estimation errors. EKF performs adequately in systems with small or gradual changes but faces significant challenges in scenarios involving rapid dynamics. This limitation is particularly evident during sharp steering maneuvers, such as lane changes, where the EKF fails to accurately capture the underlying dynamics in comparison to alternative methods. The SDRE method demonstrates improved performance over the EKF, with RMSE values of $1.59 \times 10^{- 27}$ versus $1.02 \times 10^{- 26}$ and $2.51 \times 10^{- 6}$ versus $0.273$ for yaw rate and sideslip angle estimation in the first scenario, respectively. However, it still falls short of the accuracy achieved by the developed SOSMO, particularly in highly dynamic conditions, with RMSE values of $6.03 \times 10^{- 6}$ compared to $2.52 \times 10^{- 8}$ for SOSMO and $2.25 \times 10^{- 8}$ for the H-T scheme in sideslip angle estimation. Moreover, SOSMO excels in estimating lateral tire forces, an area where both the EKF and SDRE have shown limitations.

Since the proposed methods demonstrated exceptional performance in handling abrupt vehicle dynamics, surpassing both EKF and SDRE in demanding scenarios such as rapid lane changes, their superior robustness to external disturbances, even though not fully illustrated here due to figure limitations, is detailed below.

Remark 4: The robustness of the proposed methods to external disturbances was also assessed, revealing a notable robustness in sideslip angle estimation accuracy. Specifically, the RMSE for the first method was $1.19 \times 10^{- 6}$ , while the H-T-based method achieved an RMSE of $1.035 \times 10^{- 6}$ , significantly outperforming the SDRE method, which exhibited an RMSE of $3.03 \times 10^{- 4}$ . These numerical results highlight the superior precision and robustness of the proposed methods in sideslip angle estimation, even under the influence of measurement noise.

Future research will prioritize real-world experimentation, expanding the study to account for factors such as time-varying speed, wet road surfaces, environmental fluctuations, real-world noise, and various uncertainties, including wind. Advanced and robust methodologies will be explored, including the second-order EKF (Wang et al., 2023), derived from the second-order Taylor expansion of a nonlinear system; hierarchical techniques for state and parameter estimation (Razmjooei and Safarinejadian, 2015), which effectively enhance modeling and estimation accuracy; robust data-driven learning algorithms (Zhao et al., 2024), which offer greater adaptability to diverse road and driving conditions by reducing reliance on idealized assumptions; time-varying estimation algorithms (Razmjooei et al., 2020b), which effectively push the state variables to converge to the actual state variables; distributed real-time architectures (Jleilaty et al., 2024), which enable real-time error correction and rapid responses to sudden driving events, such as sharp turns or abrupt braking; and adaptive estimation techniques (Razmjooei et al., 2023; Wang et al., 2024), which are capable of accommodating larger sideslip angles and addressing more critical practical challenges.

Conclusion

This study effectively addressed the challenge of accurately estimating the sideslip angle and lateral tire–road forces in single-track vehicles. The development of an extended-state SOSMO framework, supplemented by two additional estimation algorithms, facilitated the finite-time convergence of lateral tire force estimates. The proposed frameworks were rigorously evaluated under steady-state circular driving and lane change scenarios, significantly outperforming the SDRE and EKF methods with minimal errors. While the SOSMO approach demonstrated adaptability to varying conditions, unknown tire models, and external disturbances, its performance relied on the assumption of constant longitudinal speed. Even during demanding lane change maneuvers, accuracy remained high, confirming its effectiveness for practical applications. However, its dependence on calibrated parameters posed challenges for real-world applicability, emphasizing the need to improve the model’s accuracy and generalizability by incorporating adaptive techniques to enhance robustness in diverse operating environments. Motivated by these research gaps and inspired by findings that established the proposed methods as reliable and accurate solutions for vehicle state estimation, future research will emphasize real-world experimentation. This will involve expanding the study to include factors such as time-varying speeds, wet road surfaces, environmental variations, real-world noise, and various uncertainties, including wind. Advanced observers capable of accommodating larger sideslip angles and addressing these challenges will be prioritized. In addition, the model’s accuracy and generalizability must be enhanced through the integration of adaptive hierarchical techniques, robust data-driven learning algorithms, and distributed real-time architectures to improve robustness across diverse operating environments.

Footnotes

Acknowledgements

The authors acknowledge that a preprint version of this manuscript is available at .

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.

ORCID iD

Hamid Razmjooei

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

Alai

Zemouche

Rajamani

(2024) Vehicle trajectory estimation using a high-gain multi-output nonlinear observer. IEEE Transactions on Intelligent Transportation Systems 25: 5733–5742.

Antonov

Fehn

Kugi

(2011) Unscented Kalman filter for vehicle state estimation. Vehicle System Dynamics 49(9): 1497–1520.

Ben Moussa

Bakhti

(2024) Nonlinear tyre model-based sliding mode observer for vehicle state estimation. International Journal of Dynamics and Control, pp.1–14.

Bertipaglia

Alirezaei

Happee

, et al. (2024) An unscented Kalman filter-informed neural network for vehicle sideslip angle estimation. IEEE Transactions on Vehicular Technology 73: 12731–12746.

Chen

Yao

Gao

, et al. (2024) Estimation of lateral velocity and cornering stiffness in vehicle dynamics based on multi-source information fusion. SAE International Journal of Vehicle Dynamics, Stability, and NVH, Paper no 8(10–08–01–0003).

Cheng

Yan

, et al. (2019) Simultaneous estimation of tire side-slip angle and lateral tire force for vehicle lateral stability control. Mechanical Systems and Signal Processing 132: 168–182.

Davila

Fridman

Levant

(2005) Second-order sliding-mode observer for mechanical systems. IEEE Transactions on Automatic Control 50(11): 1785–1789.

Ding

Chen

Zhang

, et al. (2014) An extended Luenberger observer for estimation of vehicle sideslip angle and road friction. International Journal of Vehicle Design 66(4): 385–414.

Gadola

Chindamo

Romano

, et al. (2014) Development and validation of a Kalman filter-based model for vehicle slip angle estimation. Vehicle System Dynamics 52(1): 68–84.

10.

Gao

Neubeck

, et al. (2010) Sideslip angle estimation based on input–output linearisation with tire–road friction adaptation. Vehicle System Dynamics 48(2): 217–234.

11.

Han

Huh

(2011) Monitoring system design for lateral vehicle motion. IEEE Transactions on Vehicular Technology 60(4): 1394–1403.

12.

Jin

Yin

, et al. (2017) Improving vehicle handling stability based on combined AFS and DYC system via robust Takagi-Sugeno fuzzy control. IEEE Transactions on Intelligent Transportation Systems 19(8): 2696–2707.

13.

Jleilaty

Ammounah

Abdulmalek

, et al. (2024) Distributed real-time control architecture for electrohydraulic humanoid robots. Robotic Intelligence and Automation 44(4): 607–620.

14.

Kim

Min

Kim

, et al. (2020) Vehicle sideslip angle estimation using deep ensemble-based adaptive Kalman filter. Mechanical Systems and Signal Processing 144: 106862.

15.

Leanza

Mantriota

Reina

(2024) On the vehicle dynamics prediction via model-based observation. Vehicle System Dynamics 62(5): 1181–1202.

16.

Lenzo

Ottomano

Strano

, et al. (2020) September. A Physical-based observer for vehicle state estimation and road condition monitoring. IOP Conference Series: Materials Science and Engineering 922(1): 012005.

17.

Jia

Ran

, et al. (2014) A variable structure extended Kalman filter for vehicle sideslip angle estimation on a low friction road. Vehicle System Dynamics 52(2): 280–308.

18.

Lian

Zhao

, et al. (2015) Longitudinal collision avoidance control of electric vehicles based on a new safety distance model and constrained-regenerative-braking-strength-continuity braking force distribution strategy. IEEE Transactions on Vehicular Technology 65(6): 4079–4094.

19.

Liu

Xiong

Xia

, et al. (2018) Vehicle sideslip angle estimation: A review. SAE Technical Paper,2018-01-0569.

20.

May

Henning

Sawodny

(2023) Experimental validation of sensor fault estimation for vehicle dynamics with a nonlinear tire model. Control Engineering Practice 141: 105725.

21.

Nguyen

Frezzatto

Guerra

, et al. (2024) Cost-effective estimation of vehicle lateral tire-road forces and sideslip angle via nonlinear sampled-data observers: Theory and experiments. IEEE/ASME Transactions on Mechatronics 12: 31–42.

22.

Pacejka

(2005) Tire and vehicle dynamics. New York: Elsevier.

23.

Park

(2022) Vehicle sideslip angle estimation based on interacting multiple model Kalman filter using low-cost sensor fusion. IEEE Transactions on Vehicular Technology 71(6): 6088–6099.

24.

Puscul

Lex

Vignati

, et al. (2024) A literature survey on sideslip angle estimation using vehicle dynamics based methods. IEEE Access 12: 70263–70277.

25.

Rabhi

AED

Hartani

Guettaf

(2024) Estimation of vehicle tire effective rolling radius and vertical wheel/road contact force. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering. Epub ahead of print 11 March. DOI: 10.1177/09544070241232734

26.

Razmjooei

Safarinejadian

(2015) A novel algorithm for hierarchical state and parameter estimation in slowly time varying systems. Journal of Advanced and Applied Sciences (JAAS) 3(5): 189–200.

27.

Razmjooei

Palli

Abdi

, et al. (2022a) Finite-time continuous extended state observers: Design and experimental validation on electro-hydraulic systems ☆. Mechatronics 85: 102812.

28.

Razmjooei

Palli

Abdi

, et al. (2023) Design and experimental validation of an adaptive fast-finite-time observer on uncertain electro-hydraulic systems. Control Engineering Practice 131: 105391.

29.

Razmjooei

Palli

Abdi

(2022b) Continuous finite-time extended state observer design for electro-hydraulic systems. Journal of the Franklin Institute 359(10): 5036–5055.

30.

Razmjooei

Shafiei

Abdi

(2020a) A novel continuous finite-time extended state observer design for a class of uncertain nonlinear systems. IEEE Access 8: 228289–228302.

31.

Razmjooei

Shafiei

Palli

, et al. (2020b) Chattering-free robust finite-time output feedback control scheme for a class of uncertain non-linear systems. IET Control Theory & Applications 14(19): 3168–3178.

32.

Solmaz

Başlamışlı

SÇ

(2012) Simultaneous estimation of road friction and sideslip angle based on switched multiple non-linear observers. IET Control Theory & Applications 6(14): 2235–2247.

33.

Stéphant

Charara

Meizel

(2007) Evaluation of a sliding mode observer for vehicle sideslip angle. Control Engineering Practice 15(7): 803–812.

34.

Strano

Terzo

(2017) Vehicle sideslip angle estimation via a Riccati equation based nonlinear filter. Meccanica 52: 3513–3529.

35.

Strano

Terzo

(2018) Constrained nonlinear filter for vehicle sideslip angle estimation with no a priori knowledge of tyre characteristics. Control Engineering Practice 71: 10–17.

36.

Venhovens

PJT

Naab

(1999) Vehicle dynamics estimation using Kalman filters. Vehicle System Dynamics 32(2–3): 171–184.

37.

Wang

Zhao

Shen

, et al. (2024) A robust adaptive fault-tolerant estimator for sideslip angle and tire cornering stiffness with multiple missing data. IEEE/ASME Transactions on Mechatronics 29(6): 4800–4813.

38.

Wang

Wei

, et al. (2023) Robust estimation of vehicle dynamic state using a novel second-order fault-tolerant extended Kalman filter. SAE International Journal of Vehicle Dynamics, Stability, and NVH, Paper no 7(10–07–03–0019).

39.

Xia

Xiong

Lin

, et al. (2018) Vehicle sideslip angle estimation considering the tire pneumatic trail variation. SAE Technical Paper, 2018-01-0571.

40.

Zhao

Wang

, et al. (2024) Data-driven learning for h∞ control of adaptive cruise control systems. IEEE Transactions on Vehicular Technology 73(12): 18348–18362.

41.

Zhao

Yang

Cao

, et al. (2023) Observer-based discrete-time cascaded control for lateral stabilization of steer-by-wire vehicles with uncertainties and disturbances. IEEE Transactions on Circuits and Systems I: Regular Papers 70(8): 3347–3358.

42.

Zhu

Pang

Wang

, et al. (2024) Pose trajectory formation mechanism for corner module vehicles in highway lane-changing scenarios. IEEE Transactions on Intelligent Vehicles 1: 8952.

Development of sliding mode observers for estimating sideslip angle and lateral forces in road vehicles

Abstract

Keywords

Introduction

Modeling the single-track vehicle dynamics

Development of the SOSMO

Simulation results

Scenario 1: Open loop steering pad maneuver

Scenario 2: Lane change maneuver

Detailed comparative analysis of the proposed methodologies for vehicle dynamics estimation in challenging scenarios

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

Data availability

References