Keywords

1 Introduction

Vehicle automation and driver assistance systems are pivotal advancements in modern vehicles, overseeing safety, performance, and driver comfort. The Lane Keeping System (LKS), a fundamental technology for path tracking and lateral control, necessitates a robust control strategy as well as accurate knowledge of the vehicle’s lateral dynamics characteristics.

Cornering stiffness, defining a tire’s resistance to lateral deformation under lateral forces, is a critical parameter in vehicle lateral dynamics. Due to its significance and challenges in estimation, cornering stiffness estimation has been extensively studied. Multiple estimation techniques have been tested for cornering stiffness estimation [1, 2]. Notably, an estimator utilizing a KF-based IMM shows rapid responsiveness and smoothness in output [3] but is limited by model selection and computational load. This study builds upon this technique for a more robust and cost-effective estimation.

Aside from estimator inaccuracy, measurement signal errors contribute to estimation errors. While some studies propose estimation methods without using lateral speed, these techniques often lead to under-determined systems or limited usage due to extensive assumptions [4, 5]. In this study, a cornering stiffness uncertainty estimator is developed to compensate for such inevitable estimation errors. The obtained estimation uncertainty boundaries are coupled with an RMPC, utilizing real-time cornering stiffness estimations and uncertainty boundaries. The effectiveness of the RMPC under parametric uncertainty is demonstrated well in [6]. In summary, this paper presents an adaptive control algorithm resilient to estimation errors, contributing:

  1. 1.

    A novel probability calculation method in the IMM for reduced estimation bias.

  2. 2.

    Development of a cornering stiffness uncertainty boundary estimator.

  3. 3.

    A robust lane-keeping RMPC controller utilizing real-time cornering stiffness estimations and uncertainty boundaries.

2 Cornering Stiffness Estimation

2.1 Estimator Vehicle Model

This section outlines the vehicle dynamics model used in the IMM’s Kalman Filters, based on a simplified 2-DOF bicycle model [7], assuming small slip angles and constant longitudinal speed.

$$\begin{aligned} \begin{array}{cl} m(\dot{v}_{y}+\gamma v_{x}) = C_f \alpha _f cos\delta +C_r \alpha _r, \quad I_z\dot{\gamma } = l_f C_f \alpha _f cos\delta -l_r C_r \alpha _r \end{array} \end{aligned}$$
(1)

\(m, I_z, l_f, l_r, C_f, C_r\) are vehicle parameters; \(\alpha _f\) and \(\alpha _r\) are front and rear tire slip angles; \(\gamma \) denotes the yaw rate, with \(\delta \) representing the front wheel steering angle. Front and rear tire slip angles are determined using vehicle geometry.

Cornering stiffness is represented by a modified linear tire model. The fluctuations and variations of the cornering stiffness \(C_f, C_r\) are denoted as a sum of its respective base values \(C_{f,base}, C_{r,base}\) and candidate variance values \(\varDelta C_f, \varDelta C_r\), allowing for estimations in the tire’s nonlinear regions.

IMM accuracy is often linked with model selection. To enhance filter robustness and reduce reliance on model selection, a novel Approximate IMM (AIMM) is introduced in this study. As long as the cornering stiffness variance values encompass the potential range of cornering stiffness, the base and variance cornering stiffness values can be flexibly chosen.

2.2 Approximate IMM-KF

The IMM filter assesses multiple model filter estimations to derive probabilities for each model and a combined output. The IMM is selected for its rapid responsiveness, smooth output, and capability to integrate multiple vehicle models. For this study, 5 candidate cornering stiffness variance values of each front and rear tire are combined into a total of 25 filter models employed in the IMM.

Despite its advantages, the traditional method of probability calculation in the IMM is found to introduce estimation bias under certain conditions. Below are the modifications implemented in the IMM to mitigate bias and enhance consistency.

Min-Max Scaled Probability Calculation. In the traditional IMM, the range of probability distribution differs upon the excitation of the system. High excitation leads to large estimation errors and extreme probability gaps, while low excitation yields indistinct probability distributions. Min-max scaling is implemented in probability calculations to ensure that probability distributions are independent of system excitation and accurately represent model feasibility.

Approximate Probability Calculation. In IMM model filters comprising two or more varying parameter values, erroneous combinations of parameter values can lead to low estimation errors, thereby yielding falsely high probabilities. The approximate method calculates each parameter’s probability individually, by fixing all other parameters as their approximate values. Subsequently, these calculated probabilities are combined to form the complete probability matrix for all model filters. In each subsequent cycle, previous approximations of each parameter are replaced with newly estimated parameter values, facilitating a gradual convergence toward the true value.

3 Cornering Stiffness Uncertainty Estimation

In this section, a novel strategy is introduced to isolate only the cornering stiffness uncertainty values from a basic disturbance observer. The proposed uncertainty estimator utilizes the probability distributions of each model from the AIMM to assess relative estimation certainty and determine upper and lower uncertainty proportions. A concentrated probability distribution indicates high estimation certainty, while a dispersed distribution suggests the opposite. The upper base uncertainty boundaries for each front and rear wheel are derived as the sum of products of positive cornering stiffness variance values and their corresponding probabilities, with similar construction for lower boundaries using negative variance values. Since the base uncertainty boundaries are proportional values, definitive uncertainty boundaries are expressed as the product of base uncertainty values \(\varDelta \widetilde{C}_{f, base}, \varDelta \widetilde{C}_{r, base}\) and their respective weight factors \(\rho _1, \rho _2\),

Weight factors are determined through the integration of front and rear dynamics equations. The disturbance observer with the denotation of uncertainties is described below. Cornering stiffness uncertainty is denoted using base uncertainty values and weight factors, while lateral speed and force disturbances are denoted by k.

$$\begin{aligned} &F_{yf}(1+k_f) = (\delta -\frac{V_y (1+k_v)}{V_x}-\frac{l_f\gamma }{V_x} )(\widehat{C}_f + \varDelta \widetilde{C}_{f, base}*\rho _{1}) \end{aligned}$$
(2)
$$\begin{aligned} &F_{yr}(1+k_r) = (-\frac{V_y (1+k_v)}{V_x}-\frac{l_r\gamma }{V_x} )(\widehat{C}_r + \varDelta \widetilde{C}_{r, base}*\rho _{2}) \end{aligned}$$
(3)

Because the errors in \(F_{yf}\) and \(F_{yr}\) are caused mostly due to changes of vehicle mass, coefficients \(k_f\) and \(k_r\) can be assumed to be of equal size. Dividing Eq. 2 by Eq. 3 gives us:

$$\begin{aligned} \frac{F_{yf}}{F_{yr}} = \frac{(\alpha _f-\frac{V_y}{V_x}k_v)(\widehat{C}_f + \varDelta \widetilde{C}_{f, base}*\rho _{1})}{(\alpha _r-\frac{V_y}{V_x}k_v)(\widehat{C}_r + \varDelta \widetilde{C}_{r, base}*\rho _{2})} \end{aligned}$$
(4)

because \(\left| \alpha _f\right| , \left| \alpha _r\right| >> \left| \frac{V_y}{V_x}k_v\right| \), the equation can be simplified to reveal the relationship between front and rear uncertainty boundaries. From where the weight factors and definitive uncertainty boundaries can be derived.

$$\begin{aligned} \frac{\widehat{C}_f + \varDelta \widetilde{C}_{f, base}*\rho _{1}}{\widehat{C}_r + \varDelta \widetilde{C}_{r, base}*\rho _{2}} = \frac{F_{yf}\alpha _r}{F_{yr}\alpha _f} \end{aligned}$$
(5)

4 Controller

The proposed lane-keeping system is controlled through a robust model predictive controller (RMPC). The vehicle model used for model prediction is identical to the bicycle model used in the estimation phase. Estimated cornering stiffness is fed to the controller to update the vehicle model. The cornering stiffness uncertainty boundaries are implemented into the RMPC via constraint tightening and smoothing techniques. The cost function comprises the steering input derivation and lateral error with its respective weight factors for optimal control input.

5 Simulation

The CARMAKER software is used to validate estimator and controller performance. The vehicle is driven on a clothoid-shaped path with the turning radius decreasing from 50 m to 20 m, designed to induce decreasing cornering stiffness and test path tracking abilities.

5.1 Estimator Performance

The cornering stiffness estimation accuracy is tested by comparing it with the true value, while the uncertainty boundary’s capability to encapsulate error is evaluated. As depicted in Fig. 1, the cornering stiffness is estimated with an average error of 9.5% in the front and 13% in the rear tires. While the result shows reliable accuracy and tracking capabilities, some errors are evident. However, the uncertainty boundaries effectively encapsulate the true value even under these errors, expanding during periods of low certainty and contracting during high certainty.

Fig. 1.
figure 1

Cornering stiffness estimation with uncertainty boundaries

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Fig. 2.
figure 2

LKS lateral deviation and steering angle

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5.2 Control Performance

The control performance of the proposed system’s RMPC is contrasted with that of a basic MPC lane-keeping system that shares the same vehicle model and cost function. However, the MPC model relies on a constant cornering stiffness value.

Figure 2 shows that the standard MPC deviates from the required curve at 12 seconds, while the RMPC maintains stability until 14 s due to accurate cornering stiffness updates. From 14 s onwards, the RMPC approaches the cornering stiffness uncertainty boundary constraints, prompting the generation of more aggressive control actions to sustain stability, as evidenced by the steering angle graph.

6 Conclusion

This paper presents the development of a robust Lane Keeping System (LKS) control system, integrating cornering stiffness estimation and uncertainty. The developed AIMM demonstrates reliable estimation capabilities, while the uncertainty estimator effectively compensates for potential estimation errors. Cornering stiffness values are utilized to update the vehicle model, while uncertainty boundaries serve as constraints in the RMPC, resulting in an adaptive and robust lane-keeping control system.

Simulation tests against a standard MPC underscore the proposed controller’s ability to maintain stability in tight corners. The results show that combining the uncertainty estimator with RMPC forms an effective control framework, adept at handling parameter uncertainty and estimation errors. This suggests potential for future research to explore applying this framework in various vehicle control systems, enhancing robustness against diverse parameter uncertainties.