The Adaptive Cruise Control for Curved Roads Using Archived Crow Search Algorithm

Based on Figure 1, the system is composed of two different control systems, adaptive cruise control (ACC) and steering control, which are continuously used to regulate the vehicle's behavior in order to provide comfort for the driver and enhance safety. The first subsystem, ACC, has the function of controlling the longitudinal speed of the vehicle on the road. Its role is to observe and respond to the movements of the vehicles in front of it to adjust the longitudinal speed to maintain a safe following distance [19]. This helps reduce the risk of potential accidents and relieves the driver's workload. On the other hand. The second subsystem, steering control, is responsible for controlling the lateral displacement of the vehicle when navigating curved roads. In this situation, steering control calculates the optimal steering angle based on the road curvature and the vehicle's speed [20]. This ensures that the vehicle can navigate curves smoothly and safely. The combination of these two subsystems allows the vehicle to operate more safely and efficiently [14].

As illustrated in Figure 1, the vehicle equipped with ACC and steering control is referred to as the ego car, while the vehicle in front is named the lead car. With steering control, the ego car can follow the centerline of the road, whereas the lead car, without steering control, still stays within the curved lane. CSA aims to optimize both ACC and steering control concurrently, seeking the global optimal solution, thereby enhancing the overall system performance. For the test, the curve radius of curvature configuration used is 760 meters with linewidth 4 meters.

To simulate vehicle movement on a curved road used vehicle dynamics on MATLAB to create a mathematical model of the vehicle that includes dynamic properties such as mass, mass distribution, inertia, suspension, and braking. The equations of motion for the vehicle, encompassing forces like gravity, friction, and aerodynamics, are numerically integrated within the MATLAB environment. Key parameters of the vehicle, such as mass and friction coefficient, play a crucial role in determining its dynamic characteristics. In this paper, the vehicle dynamics used come from the vehicle dynamics subsystem models the vehicle dynamics with the bicycle model - force input block from the automated driving toolbox.

The dynamics of the lead car and ego car equations transform the vehicle’s acceleration and steering into its actual position, yaw angle, lateral velocity, and longitudinal velocity. The dynamic equation for both cars can be expressed in Eq. (3). As written in Eq. (3), respectively, V_y is vehicle lateral velocity, ψ is vehicle yaw angle, ψ' is vehicle yaw angle rate, V_x is vehicle longitudinal velocity, V_x' is vehicle longitudinal acceleration, m is mass of the vehicle, l_f is longitudinal distance from center of gravity to front tires, I_z is yaw moment of inertia of vehicle, I_r is longitudinal distance from center of gravity to rear tires, τ is longitudinal time constant, C_f is cornering stiffness of front tires, and C_r is cornering stiffness of rear tires. The values for those vehicle parameters are written in Table 1.

Table 1. Vehicle parameter

The results obtained from the vehicle dynamics, including quantities such as longitudinal velocity V_x and lateral velocity V_y, are originally represented in a reference frame tied to the vehicle's body. To determine the path taken by the vehicle, these body-fixed coordinates are transformed into global coordinates using the following relationships, given by Eq. (1) and Eq. (2):

$X=V_x \cos (Y)-V_y \sin (Y)$            (1)

$Y=V_x \sin (Y)-V_y \cos (Y)$           (2)

$\frac{d}{d t}\left[\begin{array}{c}V_y \\ \Psi \\ \psi^{\prime} \\ V_x \\ V_x^{\prime}\end{array}\right]=\left[\begin{array}{ccccc}-\frac{2 C_f+2 C_r}{m V_x} & 0 & -V_x-\frac{2 C_f l_f-2 C_r l_r}{m V_x} & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ -\frac{2 C_f l_f-2 C_r l_r}{I_z V_x} & 0 & -\frac{2 C_f l_f^2+2 C_r l_r^2}{I_z V_x} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & -\frac{1}{\tau}\end{array}\right]\left[\begin{array}{c}V_y \\ \Psi \\ \psi^{\prime} \\ V_x \\ V_x^{\prime}\end{array}\right]+\left[\begin{array}{c}\frac{2 C_f}{m} \\ 0 \\ \frac{2 C_f l_f}{I_z} \\ 0 \\ 0\end{array}\right] \delta+\left[\begin{array}{c}0 \\ 0 \\ 0 \\ 0 \\ \frac{1}{\tau}\end{array}\right]+\left[\begin{array}{c}0 \\ 0 \\ 0 \\ V_y \psi^{\prime} \\ 0\end{array}\right]$       (3)

tu_pian_1.png

Figure 1. Schematic model of the system

2.1 System model and ACC

The ego vehicle is equipped with a sophisticated ACC system. This ego vehicle is also equipped with various sensors that can measure and monitor its distance from the vehicle in front of it, which is referred to as the main vehicle. The ACC system aims to ensure a safe distance from the vehicle in front by controlling the speed of the ego vehicle with a fast response time. It utilized sensor data to constantly gauge the gap between the lead vehicle and the ego vehicle, making necessary adjustments to ensure a safe distance. Figure 2 illustrates the case of ACC for a curved road discussed in this research.

As illustrated in Figure 3, ACC has two modes, namely speed control and distance control. These two modes change based on the relative distance (D_R) and safe distance (D_safe). Relative distance is the distance between the ACC vehicle and the car in front. Meanwhile, the safe distance is the safe distance between two vehicles. Speed control mode is active when the relative distance is greater than or equal to the safe distance because the goal of this mode is to match the maximum speed set by the user (V_set). Typically, the setpoint for ACC speed ranges from 30 km/h to 100 km/h. Conversely, distance control mode is active when the relative distance is below the safe distance with the aim of achieving a safe distance between the two vehicles. Generally, ACC will enter the distance control mode first until it approaches the safe distance, then enter the speed control mode to match the setpoint.

tu_pian_2.png

Figure 2. Illustration of ACC in curved road

tu_pian_3.png

Figure 3. Two control mode in ACC

tu_pian_4.png

Figure 4. Control system diagram of ACC

The controller design is carried out to integrate the CSA into ACC so that it can operate in a previously prepared simulation. This controller design stage involves several important steps such as determining the controller structure, determining the parameters to be optimized, establishing the objective function, and integrating CSA with the simulation platform. During this stage, the dynamics equations of the ACC system are implemented in the simulation platform. Figure 4 illustrates the elements related to ACC control, which consists of six main variables: (1) The set velocity (V_set), denoting the maximum speed determined by the driver, results in the ego car's speed being set at the specified velocity in m/s² when there is no other vehicle in front; (2) The time gap (T_gap) represents the preferred safe time interval between the ego car and the front vehicle, measured in seconds; (3) Longitudinal velocity (V_L) indicates the linear speed of the ego car in m/s; (4) Relative distance (D_R) signifies the gap between the ego car and the lead vehicle in meters; (5) Relative velocity (V_R) corresponds to the difference in speed between the ego car and the lead car. Finally, (6) Default spacing (D_d) refers to the minimum distance when the ego car is stationary, measured in meters. V_L, D_R, and V_R serve as independent variables in the calculation of the control signal, taking the form of acceleration from the ego car.

To adjust the program based on the distance between the lead car and the ego car, it can be demonstrated in Eq. (4).

$D_{s a f e}=D_d+T_{g a p} D_{e g o}$            (4)

In the classical ACC, there are three key parameters that have a significant impact on the system's outcomes. These three parameters are affecting two types of acceleration equation as follows:

1. The first parameter is known as v_err in Eq. (5). It is a variable multiplied by the gap between the desired velocity and the longitudinal velocity, producing the acceleration of the ego car, u₁ (first acceleration equation).

$u_1=\left(V_{s e t}-V_L\right) v_{e r r}$                 (5)

The second equation (u₂) comprises two components: x_err and V_x. Within this second type, the acceleration of the ego car is determined by multiplying V_x and the relative velocity, subtracted by x_err multiplied by the error distance, as formulated in Eqs. (6)-(7).

$u_2=V_R V_x-\left(D_{safe}-D_R\right) x_{err }$             (6)

$e_d=D_{safe}-D_R$           (7)

When the speed control mode is active (D_R) value greater than D_safe), the applied control signal will be selected from the smaller of the first and second parts, as indicated in Eq. (8). In contrast to speed control, the distance control mode only uses the second part, with the condition that the D_R value is less than D_safe, as shown in Eq. (9).

$a=\min \left(u_1, u_2\right)$           (8)

$a_E=u_2$           (9)

2.2 Steering control

A steering wheel is a central component in a motorized vehicle that gives the driver the ability to control the direction of movement of the vehicle. When a vehicle crosses a curved road, the role of the steering wheel becomes very significant to ensuring that the vehicle can navigate the curve safely and in accordance with applicable traffic rules. Meanwhile, ACC technology has successfully integrated automation functions that help vehicles maintain a safe distance from vehicles in front. However, ACC tends to focus more on aspects of controlling speed and distance to the vehicle in front, rather than on the ability to control steering functions, especially when negotiating corners.

The objective of the steering control is to ensure that the vehicle remains within its designated lane and effectively tracks the curved path of the road. This objective is accomplished by controlling the front steering angle. The lateral displacement error (e₁) towards zero shown in Eq. (10), illustrated in Figure 5, and the primary objective is to minimize the yaw angle error (e₂), as shown in Eq. (11). This involves adjusting the steering angle to maintain proper alignment with the desired path and lane position.

$e_1=V_x+e_2+V_y$          (10)

$e_2=\psi-\psi_{d e s}$                  (11)

Figure 6 illustrates the use of CSA to overcome problems that arise in efforts to integrate ACC with the steering wheel control function when a vehicle is crossing a curve. CSA, which takes inspiration from the social behavior of crows, is applied to optimize lateral displacement settings when the vehicle is moving through curves, with the aim of increasing efficiency and safety in dealing with such situations. The implementation of CSA in ACC has the potential to expand the role of this technology in controlling the steering wheel, which will ultimately have a positive impact on the level of safety and convenience when negotiating curves on the road.

In addition to CSA, a Proportional-Integral (PI) controller is employed to further enhance steering control. The PI controller has a general equation as written in Eq. (12).

$u(t)=k_p e(t)+k_i \int_0^t e(\tau) d \tau$               (12)

where, u(t) is the control input, e(t) is the error signal, k_p is the proportional gain, and k_i is the integral gain. These parameters play a crucial role in optimizing the steering angle. The CSA can be employed to fine-tune k_p and k_i for optimal performance, ensuring the minimization of yaw angle and lateral displacement errors.

tu_pian_5.png

Figure 5. Vehicle dynamic model for steering control

tu_pian_6.png

Figure 6. Steering control system diagram

2.3 Crow search algorithm tuning method

CSA describes a group of crows working together in search of food sources. Each bird sets a position that represents a potential solution in the search space [21]. The flight length process begins by calculating the fitness value for each solution based on the optimization objective. Through iteration, facilitated by the interchange of positions and the adoption of superior solutions, the algorithm encapsulates the essence of social dynamics within the crow population. The birds interact by switching between positions and updating a better solution. This concept reflects social behavior in the crow population.

CSA algorithm is a flock-based optimization algorithm inspired by the behavior of a flock of crows in searching for food [22]. In the context of controller parameter optimization, as explained in this paper, CSA used as a methodology for tuning crucial parameters such as v_err, x_err, and V_x parameters for straight roads as well as k_p and k_i for steering on ACC controller. The values for those CSA parameters are presented in Table 2.

After adjusting the parameters, determining the parameters in CSA, such as flock size, flight length or fl, iteration, and awareness probability or AP, becomes an important step. The value of fl affects the scope of the solution, where a larger value broadens the scope of the solution in general, while a smaller value narrows the scope of the solution to a local region. Determination of a smaller fl value is needed to obtain the optimal solution. A smaller AP determination drives more new positions on the crows, ultimately providing new solutions to increase the percentage of better solutions. The algorithm runs by making as many individuals as the population, then calculates using Integral Absolute Error (IAE). IAE measures the error between the setpoint (target) and the system output over the entire observation time period by calculating the total of the absolute value of the difference between the setpoint and the system output.

$\mathrm{IAE}=10^5 / \int_0^{\infty}|\mathrm{e}(\mathrm{t})| \mathrm{dt}$             (13)

Table 2. Parameter for CSA

2.4 Archived CSA algorithm

The pseudo-code of ACSA is presented to succinctly summarize the conceptual steps undertaken in the algorithm.

In stochastic traffic optimization, the computational time needed for CSA operators is typically insignificant compared to the time required for performance evaluation using simulation models. Thus, there's a critical need to improve the original CSA by reducing the overall computational time for assessing fitness functions. Recent advancements in computational intelligence have introduced an archive crow search algorithm, which employs a selection process utilizing a small population size alongside a large external archive. This external archive stores the best solutions found so far to effectively approximate the optimal solution in various applications. Leveraging the search history stored in the external archive, the selection process aims to minimize the number of function evaluations necessary for achieving convergence. ACSA has been implemented in our application and plays a pivotal role in finding optimal solutions.

In ACSA after updating the crow positions, the algorithm checks the feasibility of the new positions and determines the fitness values of these crows. Subsequently, the memory is updated for each crow, and information regarding fitness, memory, positions, and fitness values is sorted from best to worst. Only the first four rows of xn, x, ft, mem, and fit_mem are retained. In the optimization phase, the population size is reset to four in the first iteration, ensuring that only the top four individuals continue the search.

Overall, the computational efficiency of ACSA makes it a preferable algorithm for specific applications. The implementation of ACSA in this particular application context plays a central role in discovering optimal solutions within a complex search space.

This article describes quite promising results in the implementation of adaptive cruise control in curving road conditions using steering control, along with the optimization of its parameters through the use of the crow search algorithm as the optimization method. The results of this research indicate that the OCSA and ACSA have successfully improved the performance of ACC in accurately controlling distance and speed, thus providing a more comfortable and safe driving experience. Performance of both methods is similar. Moreover, ACC can now maintain a more stable distance in curved roads due to its integration with steering control, both of which have been optimized simultaneously using CSA.

However, significant challenges exist in translating simulation results into real-world applications. The implementation of ACC requires careful consideration of vehicle hardware requirements and calibration processes to ensure optimal performance and safety in various driving situations. Future research should focus more on overcoming these challenges to facilitate the deployment of optimized ACC in real-world driving scenarios.