Keywords

1 Introduction

Automated Driving Systems (ADS) are being developed for a variety of applications, aiming to enhance driving safety, efficiency, and convenience. However, it is crucial to prepare for scenarios where ADS may not function normally due to various reasons, such as sensor failures, actuator failures, software glitches, or unforeseen road conditions. To address such situations, fallback functions are being researched extensively. One notable fallback function is the Minimal Risk Maneuver (MRM) [3], which is designed to bring the vehicle to a safe, stable stop when the ADS encounters a failure.

Typically, the primary consideration during an MRM is a straight stop. This maneuver is performed when only longitudinal deceleration control is possible, and lateral control is compromised due to lane detection failures or steering actuator malfunctions. As a result, straight stops can lead to collisions with roadside structures or others in certain situations. Conversely, an in-lane stop allows for lateral control, enabling the vehicle to halt within its lane, even on curved roads.

Previous research has explored various methods to achieve safe stopping maneuvers under ADS failures. However, many of these methods focus on straight stops or rely on fully functional steering systems [1, 5]. There remains a significant gap in addressing the safe stopping of vehicles on curved roads when the steering system fails.

In this paper, a differential braking in-lane stop system is proposed to execute the in-lane stop maneuver on curved roads, specifically addressing scenarios involving steering system failures. The proposed method employs differential braking to generate the necessary lateral motion, thus enabling safer in-lane stops compared to straight stops. The proposed approach is distinctive in that it does not rely on the steering system, making it particularly useful in scenarios where steering control is lost. Simulation results demonstrate that the proposed approach effectively performs in-lane stops, enhancing safety during ADS failures.

2 Vehicle Model

The behavior of the vehicle and changes in the front wheel steering angle occur directly due to the differential longitudinal force [4].

The extended two-track model [2, 4] is used to consider the differential longitudinal force in the commonly used bicycle model. At the same time, the simplified steering model is utilized.

Fig. 1.
figure 1

Single-track model with vehicle geometric parameters, applied forces and motion characteristics.

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The geometric parameters of the vehicle and the forces generated at each wheel are illustrated in Fig. 1. \(F_{x,i}\) and \(F_{y,i}\) represent the longitudinal and lateral tire forces with \(i=\{f,r\}\) denoting the front and rear tires. u and v are the longitudinal and lateral velocity, \(\gamma \) is the yaw rate, \(\delta _f\) is the front wheel steering angle and \(\alpha _i\) are the tire slip angles. \(l_f\) and \(l_r\) are the distance of the front and rear axles from the vehicle’s center of gravity (CG).

The combined state space equation for the lateral dynamics motion model and the steering system model can be expressed as follows:

$$\begin{aligned} \dot{x} = A x + B u \end{aligned}$$
(1)
$$\begin{aligned} x = \left[ \begin{array}{cccc} v & \gamma & \delta _f & \dot{\delta } _f \end{array}\right] ^T, \quad u = \left[ \begin{array}{c} \varDelta F_{x,f} \\ \varDelta F_{x,r} \end{array}\right] \end{aligned}$$
(2)
$$\begin{aligned} A= \left[ \begin{array}{cccc} -\frac{C_f + C_r}{m u} & -\frac{l_f C_f - l_r C_r}{m u} - u & \frac{C_f}{m} & 0\\ -\frac{l_f C_f - l_r C_r}{J_z u} & -\frac{l_f^2 C_f + l_r^2 C_r}{J_z u} & \frac{l_f C_f}{J_z} & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & -\frac{k_{\delta }}{J_{\delta }} & -\frac{b_{\delta }}{J_{\delta }} \end{array}\right] , B = \left[ \begin{array}{cc} 0 & 0\\ \frac{w}{2 J_z} & \frac{w}{2 J_z}\\ 0 & 0\\ \frac{s_{eff}}{J_{\delta }} & 0 \end{array}\right] \end{aligned}$$
(3)

\(\varDelta F_{x,i}\) are the differential longitudinal forces, m is the vehicle mass, \(J_z\) is the moment of inertia about its z-axis and w is the track width of the vehicle. The coefficients of the tires slip, \(C_i\), are the cornering stiffness. In addition, \(J_\delta \), \(b_\delta \), and \(k_\delta \) are the effective moments of inertia, effective damping, and effective stiffness of the steering system, respectively. \(s_{eff}\) is the effective scrub radius, which is defined as the distance between the point of application of resultant longitudinal force and the steering axis at ground level.

3 Differential Braking In-Lane Stop System

The differential braking in-lane stop system consists of four modules: reference generator, state estimator, controller, and brake pressure distributor.

Fig. 2.
figure 2

Block diagram of the differential braking in-lane stop system.

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The system architecture is shown in Fig. 2. \(\gamma _{ref}\) is the reference yaw rate and \(\hat{x}\) is the estimated states. In addition, \(\varDelta F_{b,i}\) are the differential braking forces and \(P_{b,ij}\) are the braking pressures with \(j=\{L, R\}\) denoting the left and right tires.

The reference generator module determines a reference state based on lane detection. It calculates the reference yaw rate to reach the look-ahead point, which is determined based on the vehicle’s speed. Simultaneously, the state estimator module obtains the estimated states using a model-based state observer with a Kalman filter. Based on both the reference state and the estimated vehicle state, the controller then calculates the necessary differential braking amounts for the front and rear wheels. A Linear Quadratic Regulator (LQR) algorithm is utilized in the controller. The brake pressure distributor module calculates the appropriate signal for braking pressure.

4 Simulation Result

The proposed system is verified through a joint simulation using CarSim and MATLAB/Simulink. The simulation cycle is set to 1 ms, while the controller’s output cycle is set to 20 ms. The scenario assumes a steering system failure and includes an in-lane stop maneuver on curved roads. Despite the lack of torque input to the steering wheel due to the steering system failure, a front wheel steering angle can be generated by the differential braking.

Fig. 3.
figure 3

Vehicle Path for the in-lane stop maneuver on a curved road.

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Figure 3 shows the path of the vehicle as it comes to a stop on a curved road with the differential braking in-lane stop system active. The distance to the road center line after stopping is 0.047 m, successfully performing the in-lane stop maneuver without crossing the lane.

Fig. 4.
figure 4

Simulation results for the in-lane stop maneuver on a curved road.

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Figure 4 presents the speed, distance to the center line, yaw rate, front wheel steering angle and brake pressure inputs over time. After approximately 13 s, the vehicle speed is sufficiently reduced. At this point, differential braking is ceased, with brake pressure applied to all wheels to achieve a complete stop. The simulation results demonstrate that the in-lane stop maneuver is successfully performed without deviating from the road.

5 Conclusion

In this paper, we proposed a reliable system that utilizes differential braking to perform in-lane stop maneuvers in the event of a steering system failure. The reference state was calculated using road information to ensure the vehicle remains within the lane. To account for the steering angle generated by differential braking, the steering system model was integrated with the vehicle lateral motion model. An estimator and controller were designed using the integrated model.

Simulations confirmed that the proposed differential braking in-lane stop system can decelerate and stop the vehicle within the lane on curved roads without deviating. Our results demonstrate that differential braking is an effective method for maintaining vehicle control and safety when a steering system failure occurs. The integration of the steering system model with the vehicle lateral motion model allows for precise control of the vehicle’s trajectory, even on curved roads.

Future research is expected to extend this work to more complex road environments and different maneuvers, such as road shoulder stops. Additionally, experiments with actual vehicles will be conducted to validate the system’s performance in real-world conditions.