Research on Trajectory Planning Method Based on Bézier Curves for Dynamic Scenarios

Abstract

With the increase in car ownership, traffic congestion, and frequent accidents, autonomous driving technology, especially for dynamic driving scenarios in the whole domain, has become a technological challenge for today’s researchers. Trajectory planning, as a crucial component of the autonomous driving technology framework, is gradually becoming a hot topic in intelligent research. In response to the challenges of planning lane-changing trajectories in complex dynamic driving scenarios under emergency evasive maneuvers, where it is difficult to consider surrounding vehicles and achieve dynamic adaptability, this paper proposes a dynamic adaptive trajectory planning method based on Bézier curves. Firstly, a mathematical model of Bézier curves is established and its curve characteristics are analyzed, which facilitates the correlation between the trajectory control points and the vehicle and the surrounding obstacles. Secondly, a mathematical function representing the Bézier curve is formulated, where the control points serve as the input and the lane-changing control curve as the output. Finally, the proposed method is validated through simulations on a jointly established simulation platform. The results indicate that the proposed method can plan lane-changing trajectories that are both safe and efficient under emergency evasive maneuvers, considering both static and complex dynamic conditions. This provides a novel solution for lane-changing trajectory planning in emergency evasive maneuvers for autonomous vehicles and holds significant theoretical research value.

1. Introduction

As the global new energy vehicle industry and internet technology continue to advance, autonomous driving technology has experienced rapid development. Trajectory planning, as an essential and challenging aspect of achieving autonomous driving, is both a research focus and a difficulty. The key issue lies in effectively planning a smooth trajectory that ensures the vehicle can safely and smoothly avoid obstacles in dynamically complex road conditions. This has become a hot topic in the field of autonomous driving research. The ability to accurately predict and plan vehicle paths is crucial for the success of autonomous systems, as it directly impacts safety, efficiency, and the overall user experience. Therefore, ongoing research is focused on developing advanced trajectory planning algorithms that can adapt to various scenarios and optimize vehicle performance [1].

A variety of trajectory planning methods exist for local paths, each characterized by its distinct features. Among the more prevalent trajectory planning techniques currently employed, those based on lateral acceleration stand out [2]: a trajectory planning methodology grounded in network search algorithms [3], a sampling-based planning methodology [4,5], and an artificial potential field method [6,7]. In this context, S. Deolasee et al. [8] proposed using trapezoidal-prism-shaped corridors for optimization, which significantly enlarges the solution space compared to the existing cuboidal-corridor-based method. However, this method does not take into account the bend section condition or the situation where there are surrounding obstacles. Wulfmeier M et al. [9] presented an approach for creating spatial traversability maps for driving in complex urban settings. This approach leverages a comprehensive dataset showcasing expert humans’ driving habits. By establishing a direct mapping from raw input data to cost, this method circumvents the need for manually designing certain components of the process. It effectively utilizes a vast amount of data samples. Additionally, this method can be used to enhance manually crafted cost maps, which are typically based on manually engineered features. However, this method has the problem of excessive computational complexity when planning trajectories for large areas. R. Cuesta and J. Alvarez. [10] proposed a method for designing closed curves using second-order Bézier curves for trajectory planning of mobile robots. However, second-order Bézier curves are not sufficiently applicable in complex scenarios. X Li et al. [11] presented a framework for local trajectory generation for autonomous vehicles navigating along a given reference path. This framework utilizes a two-phase motion planning approach. Initially, a Support Vector Machine technique is employed to enhance the reference path by maximizing the lateral clearance from the corridor boundaries while maintaining curvature continuity. Subsequently, a series of terminal states are sampled and correlated with the refined reference path. To adhere to system constraints, a model predictive path generation technique is then applied to create multiple viable paths, each linking the current vehicle state to the sampled terminal states. Nonetheless, the model predictive path generation method employed in the second step, while capable of considering system constraints and generating multiple path candidates, may be impacted in terms of performance and accuracy by factors such as model precision, computational resources, and time constraints. Guo Minghao et al. [12] proposed an enhanced artificial potential field method for autonomous driving trajectory planning. This improvement involves incorporating factors such as distance adjustment, dynamic road repulsion, velocity repulsion, and acceleration repulsion. To address the shortcomings of the traditional artificial potential field method, they introduce the invasive weed algorithm. Additionally, they develop a prediction model by linearizing and discretizing a vehicle dynamics model. This model includes predefined constraint variables and an optimized objective function, forming the foundation of an MPC model controller designed to achieve precise trajectory tracking. However, the method may easily encounter local optimal solutions during the solution process, leading to deadlocks.

Bézier curves are generated based on predefined control points, yielding continuous and smooth curves. They have been widely applied in various fields. Among them, H Li et al. [13] presented a path planning method based on a fifth-order Bézier curve for lane changing with a vehicle in front. However, the proposed method does not take into account the surrounding obstacle vehicles, Y Chen et al. [14] presented an accurate and efficient clothoid approximation approach using Bézier curves based on the minimization of curvature profile difference. Compared with existing methods, the proposed approach can guarantee higher-order geometric continuity with smaller approximation errors in terms of position, orientation, and curvature. Nevertheless, the proposed method lacks simulation validation to demonstrate practicality in reality. J Chen et al. [15] generated a path based on a piecewise quadratic Bezier curve, and the maximum curvature of the Bezier curve was calculated to verify whether an autonomous vehicle could follow the path. However, there is a shortage of consideration of speed variation scenarios for surrounding vehicles. J. Moreau et al. [16] proposed a Bézier curve optimization method to cope with these constraints, and the autonomous vehicles under study were considered equipped with all the necessary sensors for obstacle detection. In this way, the obstacle avoidance problem is transformed into an optimization problem under equality constraints. However, the computational efficiency of the method is inefficient, and its real-time applicability is limited in practice.

The studies described above indicate that by optimizing the positioning and quantity of control points, Bézier curves can flexibly adapt to various complex road conditions and driving requirements, thereby achieving the goal of ensuring smooth and safe vehicle trajectories during operation. Furthermore, Bézier curves possess superior mathematical properties, which enable highly precise calculations and optimizations in trajectory planning. Through the adjustment and optimization of curve parameters, the smoothness and precision of the trajectory can be further enhanced, thereby bolstering the performance and safety of autonomous vehicles.

This paper proposes a dynamic and adaptive trajectory planning method based on Bézier curves. Firstly, a mathematical model of Bézier curves is established, and the correlation characteristics between the curves and their control points are analyzed. Secondly, the trajectory control points are associated with the vehicle and surrounding obstacle vehicles, and a mathematical expression of the Bézier curve is established to output lane-changing trajectory curves as inputs from the control points. Finally, the proposed method is simulated and validated on a joint simulation platform. The results indicate that the proposed method can plan safe and comfortable lane-changing trajectories under both static and dynamic complex lane-changing conditions. This provides a new solution for autonomous driving trajectory planning.

2. Bézier Curve Model and Its Characteristic Analysis

2.1. Bézier Curve Model

The Bézier curve was introduced in 1962, and its characteristic of defining its shape through control points endows it with a high degree of flexibility and controllability in describing complex smooth curves. Due to their smoothness and controllability, Bézier curves have been widely applied in trajectory curve planning within autonomous driving systems [17,18]. The n-degree Bézier curve defined by the given points $p_0$, $p_1$, …, $p_n$ can be expressed in the general form as shown in Equation (1) [19].

$$\begin{array}{c}B\left(t\right)={\displaystyle \sum _{i=0}^n}\left(\begin{array}{c}n\\ i\end{array}\right)p_i{\left(1-t\right)}^{n-i}{t}^i=\left(\begin{array}{c}n\\ 0\end{array}\right)p_0{\left(1-t\right)}^n{t}^0+\left(\begin{array}{c}n\\ 1\end{array}\right)p_1{\left(1-t\right)}^{n-1}{t}^1+\dots \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +\left(\begin{array}{c}n\\ n-1\end{array}\right)p_{n-1}{\left(1-t\right)}^1{t}^{n-1}+\left(\begin{array}{c}n\\ n\end{array}\right)p_n{\left(1-t\right)}^0{t}^n\hfill \end{array}$$

(1)

The lateral displacement $y$ and longitudinal displacement $x$ are expressed as functions of the parameter t; hence, the cubic Bézier curve can be represented by Equation (2).

$$\left\{\begin{array}{c}x=x_1{\left(1-t\right)}^3+3x_{2}t{\left(1-t\right)}^2+3x_3{t}^2\left(1-t\right)+x_4{t}^3\\ y=y_1{\left(1-t\right)}^3+3y_{2}t{\left(1-t\right)}^2+3y_3{t}^2\left(1-t\right)+y_4{t}^3\end{array}\right.$$

(2)

In the equation, ($x_i$,$y_i$), $i=1,2,3,4$, represents the coordinate information of the selected points, $y_i$ denotes the lateral displacement during the lane-changing process, and $x_i$ corresponds to the longitudinal displacement, and $t$ is a curve parameter; $t\in \left[0,1\right]$.

2.2. Bézier Curve Characteristic Analysis

From the illustrative diagram of the Bézier curve (Figure 1), it can be observed that, based on the function expression of the Bézier curve, a smooth and feasible curve can be generated using the four control points $P0$, $P1$, $P2$, and $P3$. The planned curve precisely passes through the initial and final points $P0$ and $P3$, while also avoiding the intermediate points $P1$ and $P2$.

To further analyze the impact of control points on the Bézier curve, the control points are denoted as $P0\left(x_0,y_0\right)$, $P1\left(x_1,y_1\right)$, $P2\left(x_2,y_2\right)$, and $P3\left(x_3,y_3\right)$. Firstly, an analysis is conducted to investigate the influence of different horizontal coordinates $x_1$ of point $P1$ on the Bézier curve. Values for $x_1$ are selected ranging from 2.8 to 2, with an interval of 0.2, while maintaining the original coordinates of $P0\left(0,0\right)$, $P2\left(2,3.5\right)$, and $P3\left(4,3.5\right)$ unchanged. Consequently, a cluster of curves is generated as depicted in Figure 2a. Similarly, an analysis is conducted to examine the influence of different $x_2$ values, which are the horizontal coordinates of point $P2$, on the curve. Values for $x_2$ are taken from 2 to 2.8, with an interval of 0.2, while keeping the original coordinates of $P0\left(0,0\right)$, $P2\left(2,3.5\right)$, and $P3\left(4,3.5\right)$ unchanged. This results in the generation of a cluster of curves, as illustrated in Figure 2b. Through analysis, it can be observed that as the value of $x_1$ gradually decreases, the generated Bézier curve shifts towards the negative half of the x-axis, meaning that the curve shifts in the same direction as the point $P1$ shifts. Similarly, as the value of $x_2$ gradually increases, the generated Bézier curve shifts towards the positive half of the x-axis, indicating that the curve shifts in the same direction as the point $P2$ shifts. Therefore, it can be concluded that the Bézier curve has the characteristic of changing with the position of its control points, providing theoretical support for the feasibility of trajectory planning in subsequent dynamic scenarios. Through analysis, it can be seen that as the value of $x_1$ gradually decreases, the generated Bézier curve gradually shifts towards the negative half of the x-axis, meaning that the curve shifts in the same direction as the point $P1$ shifts. Similarly, as the value of $x_2$ gradually increases, the generated Bézier curve gradually shifts towards the positive half of the x-axis, indicating that the curve shifts in the same direction as the point $P2$ shifts. Therefore, it can be concluded that the Bézier curve has the characteristic of changing with the position of its control points, providing theoretical support for the feasibility of trajectory planning in subsequent dynamic scenarios.

3. Bézier Curve Trajectory Planning

3.1. Establishment of Control Points for Bézier Curve

In dynamic and complex lane-changing scenarios, the lane-changing trajectory often needs to consider not only the obstacle vehicle ahead but also the approaching vehicle from the side and rear, as shown in Figure 3. An ideal collision avoidance lane-changing trajectory should not only safely bypass the vehicle ahead that is about to collide but also ensure a safe distance from the vehicle behind during the lane-changing process.

Based on the analysis of the characteristics of Bézier curves, it is known that a cubic Bézier curve can not only avoid the control points other than the start and end points but also shift in the same direction as the offset of that control point. This characteristic precisely meets the theoretical requirements of lane-changing trajectory planning to avoid the vehicle ahead and the vehicle from the side and rear while maintaining a safe distance from them. To apply Bézier curves in lane-changing trajectory planning, the initial and final points $P0$ and $P3$ are set as the initial position of the ego vehicle and the target position after lane changing, respectively. The intermediate points $P1$ and $P2$ are set as the coordinates of the obstacle vehicle or coordinates related to the obstacle vehicle. In this study, the position coordinates of the ego vehicle $P0\left(x_0,y_0\right)$ are taken as the starting point of the lane change, with $x_0=0$, $y_0=0$; $P1\left(x_1,y_1\right)$ is a control point related to the obstacle vehicle ahead. To increase the safety distance and reduce the risk of collision, the position of the obstacle vehicle ahead is not directly used as the coordinates of the control point $P1$. Instead, this study uses the midpoint coordinates between the ego vehicle and the obstacle vehicle ahead as the coordinates of point $P1$. This allows the lane-changing curve planned by the Bézier curve to have a larger distance redundancy from the obstacle vehicle while still avoiding the control point.

Given the coordinates of the obstacle vehicle ahead as $\left(d_1,0\right)$, have $x_1={\displaystyle \frac{d_1}{2}}$, and thus the coordinates of $P1$ are $\left({\displaystyle \frac{d_1}{2}},0\right)$. Similarly, using the midpoint coordinates between the obstacle vehicle ahead and the obstacle vehicle behind as $P2\left(x_2,y_2\right)$, with the lane width being $y_d$ and the coordinates of the obstacle vehicle behind being $\left(d_2,y_d\right)$, have $x_2={\displaystyle \frac{d_1+d_2}{2}}$. Therefore, the coordinates of $P2$ are $\left(x_2={\displaystyle \frac{d_1+d_2}{2}},y_d\right)$. Taking the intersection of the horizontal coordinate of the obstacle vehicle ahead and the vertical coordinate of the obstacle vehicle behind as the final point $P3\left(x_3,y_3\right)$ of the lane change, have $x_3=d_1$, $y_3=y_d$. Thus, the coordinates of the final point $P3$ of the lane change are $\left(d_1,y_d\right)$. The corresponding table of control points and their coordinates is shown in

Table 1. Control points and coordinate correspondence.

$\mathit{P}0$	$\mathit{P}1$	$\mathit{P}2$	$\mathit{P}3$
$\left(0,0\right)$	$\left({\displaystyle \frac{d_1}{2}},0\right)$	$\left(x_2={\displaystyle \frac{d_1+d_2}{2}},y_d\right)$	$\left(d_1,y_d\right)$

3.2. Bézier Curve Trajectory Planning Method in Static Scenarios

In the case where the road width $y_d$ is a constant, the variables that affect the coordinates of the control points of the Bézier curve are only $d_1$ and $d_2$, meaning that the generated Bézier curve will change with the horizontal coordinates of the preceding and following vehicles. Firstly, analyze the impact of different $d_1$ and $d_2$ values on the resulting Bézier curve trajectories. Select a cluster of curves generated by different $d_1$ values (taking a lane-changing lateral displacement of 3.5 m, with $d_2$ fixed at −10 m, and $d_1$ ranging from 60 m to 45 m, with an interval of 5 m) and another cluster generated by different $d_2$ values (taking a lane-changing lateral displacement of 3.5 m, with $d_1$ fixed at 60 m, and $d_2$ ranging from −20 m to −2 m, with an interval of 6 m). The plots corresponding to these clusters are shown in Figure 4a and Figure 4b, respectively.

From Figure 4a, it can be observed that as $d_1$ gradually decreases, the planned Bézier trajectory curve exhibits a trend of shifting towards the negative half-axis of the x-axis. This means that when the distance between the ego vehicle and the obstacle vehicle ahead decreases, the planned Bézier lane-changing trajectory curve shifts away from the direction of the obstacle vehicle ahead. Consequently, the risk of collision with the preceding vehicle during the lane-changing process decreases.

From Figure 4b, it can be seen that as $d_2$ gradually decreases, the planned Bézier trajectory curve shows a trend of shifting towards the positive half-axis of the x-axis. This indicates that when the distance between the ego vehicle and the obstacle vehicle behind decreases, the planned Bézier lane-changing trajectory curve shifts away from the direction of the obstacle vehicle behind. Therefore, the risk of collision with the following vehicle during the lane-changing process decreases.

In addition to ensuring the safety of the lane-changing process, a high-quality lane-changing trajectory planning method should also strive to ensure the smoothness and comfort of the lane-changing trajectory in non-emergency collision avoidance scenarios. By comparing the Bézier lane-changing trajectory curves dynamically generated under the condition of gradually increasing $d_1$ (with a lateral displacement for lane changing set at 3.5 m, $d_2$ fixed at −10 m, and $d_1$ ranging from 60 m to 120 m with an interval of 20 m), which corresponds to a situation where the distance between the ego vehicle and the obstacle vehicle ahead is relatively large and lane changing is not for emergency collision avoidance, as shown in Figure 5, it can be observed that as $d_1$ increases, the lane-changing trajectory generated by the Bézier curve becomes smoother, and the longitudinal displacement required to complete the lane changing also increases. Therefore, under the same vehicle speed and lateral displacement, the lateral acceleration generated during the lane-changing maneuver will gradually decrease, making the lane-changing process more comfortable.

3.3. Trajectory Planning Using Bézier Curves in Dynamic Scenarios

In the process of lane changing in dynamic scenarios, in addition to considering the positions of surrounding obstacle vehicles at the initial moment of lane changing to plan a lane-changing curve, it is also necessary to take into account the changes in the positions of surrounding vehicles during the ongoing lane-changing process, such as sudden acceleration or deceleration of the preceding or following vehicles. Therefore, during the lane-changing process, it is required to dynamically plan new lane-changing trajectories in real time based on the real-time positions of surrounding obstacle vehicles to ensure the safety of the lane-changing process.

With the initial position of the ego vehicle set as $P0(0,0)$, the initial longitudinal distance between the ego vehicle and the obstacle vehicle ahead is 20 m, and the initial longitudinal distance between the ego vehicle and the obstacle vehicle behind in the adjacent lane is 10 m. The lane width is 3.5 m. According to the coordinate correspondence table in Table 1, the control points can be obtained as $P1(10,0)$, $P2(5,3.5)$, and $P3(30,3.5)$, as shown in Figure 6.

Let the longitudinal relative distance between the ego vehicle and the preceding vehicle decrease at a speed of 0.5 m per unit of time and the longitudinal relative distance between the ego vehicle and the vehicle behind on the adjacent lane decrease at a speed of 0.6 m per unit of time (this simulates the scenario where the relative positions of surrounding vehicles change dynamically in real-time during the lane-changing process).

Firstly, a larger curve update cycle is selected to more clearly demonstrate the situation where the Bézier lane-changing trajectory curve is updated in real-time according to the dynamic changes of the control points during the lane-changing process, as shown in Figure 7a.

From Figure 7a, it can be seen that during the lane-changing process, a new trajectory curve is generated every cycle until the lane-changing action is completed. However, due to the large update cycle selected, the lane-changing trajectory is not smooth, which is not suitable for real vehicle operating conditions. In practical applications, the update cycle should be shortened to make the final trajectory curve smooth and ensure the comfort of lane changing, as shown in Figure 7b.

In summary, for static scenarios, the Bézier curve trajectory planning method can adjust the lane-changing trajectory curve according to the different coordinates of the vehicles ahead and behind at the beginning of lane changing. When the obstacle vehicles ahead or behind are close, the planned trajectory curve can deviate in the direction away from the obstacle vehicles, better controlling the longitudinal distance between the ego vehicle and the obstacle vehicles, significantly reducing the risk of collision with obstacle vehicles during the lane-changing process and ensuring the safety of lane changing. In non-emergency lane-changing and collision avoidance situations, the planned trajectory curve is smoother, ensuring the comfort of lane changing in non-emergency traffic environments.

For dynamic driving scenarios, the Bézier curve trajectory planning method can adaptively adjust the trajectory curve according to the real-time position changes of surrounding vehicles under dynamic and complex lane-changing conditions, improving the safety of lane changing and providing guidance and a basis for dynamic collision avoidance trajectory planning in complex traffic situations.

4. Simulation Analysis

4.1. Simulation Analysis of Bézier Curve Trajectory Planning Method in Static Scenarios

To validate the proposed method, this study utilized a co-simulation platform built based on MATLAB (R2021B) and Carsim (2019.0) using Logitech G29. As shown in Figure 8, Carsim outputs the position coordinates of the ego vehicle and obstacle vehicles to the MATLAB algorithm. The algorithm then calculates the Bézier curve lane-changing trajectory and inputs it back into Carsim to control the ego vehicle to perform the lane-changing maneuver.

For the static lane-changing scenarios where obstacles are present both ahead and to the rear side, the following two conditions are simulated and analyzed separately.

Condition 1 is the ego vehicle is initially travelling straight at a constant speed of 60 km/h. A stationary vehicle appears 40 m ahead, necessitating a lane change to avoid it. Additionally, there is a stationary vehicle 15 m behind on the left lane. This simulation condition focuses on the Bézier curve lane-changing trajectory planning simulation analysis when there is a relatively distant stationary obstacle ahead and another obstacle behind on the side-rear, resulting in a relatively smooth lane-changing maneuver. Figure 9 depicts the initial state of Condition 1. The red and yellow line in Figure 9 indicates that the vehicle’s sensors have sensed the vehicle ahead and behind it, respectively.

From the simulation results in Figure 10a,b, it can be observed that the planned lane-changing trajectory is relatively smooth when the distance to the preceding vehicle is far. Additionally, the maximum lateral acceleration during the entire lane-changing process is less than 1.5 m/s² [20], ensuring good lane-changing comfort. Figure 10c shows that the minimum distance between the ego vehicle and the preceding vehicle is 15 m, eliminating the risk of collision and thus ensuring good safety [21]. In summary, under the condition of a relatively far distance to the preceding vehicle and a relatively smooth lane-changing scenario, the Bézier curve trajectory planning method can plan a comfortable lane-changing trajectory while ensuring the safety of the lane-changing process.

Condition 2 describes a scenario where the ego vehicle is initially travelling straight at a constant speed of 60 km/h. A stationary vehicle suddenly appears 20 m ahead, necessitating an emergency lane change to avoid it. Additionally, there is another stationary vehicle 15 m behind in the left lane. This simulation focuses on the Bézier curve lane-changing trajectory planning analysis when the ego vehicle encounters a stationary obstacle close ahead under urgent lane-changing conditions and another obstacle is present behind in the adjacent lane. Figure 11a depicts a situation where the ego vehicle is travelling normally and has not yet detected the stationary vehicle ahead; Figure 11b shows the ego vehicle in the process of lane changing along the planned Bézier curve trajectory. The red and yellow line in Figure 11a indicates that the vehicle’s sensors have sensed the vehicle ahead and behind it, respectively. The red dotted line in Figure 11b shows the lane change trajectory.

Based on the simulation results, the vehicle successfully avoids the slower vehicle ahead and does not collide with the vehicle behind. From Figure 12a, it can be seen that under the more urgent lane-changing condition with a closer distance to the preceding vehicle, the planned lane-changing trajectory curve is more aggressive compared to Condition 1. Figure 12b, which compares the lateral acceleration generated during lane changing between Condition 1 and Condition 2, shows that under the more urgent lane-changing scenario of Condition 2, the maximum lateral acceleration along the generated trajectory is greater than that in the more relaxed Condition 1. However, the time required to complete the lane change in Condition 2 is only 3.2 s, significantly less than the 4.3 s required in Condition 1. Additionally, Figure 12c indicates that the minimum distance between the ego vehicle and the preceding vehicle is approximately 16 m, posing no risk of collision. Therefore, under urgent lane-changing conditions, the Bézier curve trajectory planning method can plan a safe and efficient lane-changing trajectory, ensuring the safety of the lane change.

In summary, the Bézier curve lane-changing trajectory planning method can adaptively plan appropriate lane-changing trajectories under both relaxed and urgent lane-changing conditions, effectively completing collision avoidance operations. While ensuring safety, it also maximizes the comfort or efficiency of the lane change as much as possible.

4.2. Simulation Analysis of Bézier Curve Trajectory Planning in Dynamic Scenarios

Based on Carsim/Simulink, a comprehensive simulation is conducted for dynamic lane-changing scenarios where there are moving and variable-speed obstacle vehicles both ahead and to the rear side. The simulation is divided into three conditions: Condition 1 is when the surrounding vehicles maintain a constant speed; Condition 2 is when a vehicle from the rear side accelerates and approaches as the ego vehicle begins to change lanes; and Condition 3 is when the preceding vehicle decelerates and approaches when the ego vehicle begins to change lanes.

This simulation compares scenarios where the vehicle behind on the side begins to accelerate and approach as the ego vehicle initiates a lane change. Figure 13a depicts the situation during the entire lane-changing operation when the vehicle ahead and the vehicle behind on the side are both travelling at a constant speed, with the ego vehicle travelling at 110 km/h and both the preceding vehicle and the vehicle behind on the side travelling at 100 km/h. Figure 13b shows a scenario where the vehicle behind accelerates and approaches midway through the lane-changing operation, with the ego vehicle maintaining a constant speed of 110 km/h, the preceding vehicle maintaining a constant speed of 100 km/h, and the vehicle behind on the side accelerating from 110 km/h to 120 km/h when the ego vehicle starts the lane change. Figure 13c illustrates a situation where the preceding vehicle decelerates and approaches during the lane-changing operation, with the ego vehicle maintaining a constant speed of 110 km/h, the vehicle behind maintaining a constant speed of 100 km/h, and the preceding vehicle decelerating from 110 km/h to 90 km/h when the ego vehicle starts the lane change. The red dotted line in Figure 13 indicates the schematic of the lane change trajectory in the simulation scenario of the corresponding subfigure.

From the comparison of lane-changing trajectories planned under the three different conditions shown in Figure 14, it can be observed that each condition results in a lane-changing trajectory curve with a different degree of inclination. In the simulation of Condition 2, where the vehicle behind accelerates, due to the approaching and accelerating vehicle from the rear side, if the ego vehicle were to change lanes with a large lateral speed at this time, it would be prone to a collision with the approaching vehicle. Therefore, the dynamic Bézier curve plans a trajectory that is more inclined to the right compared to other conditions, based on changes in the control points (with control point P2 gradually moving to the right). This, combined with the appropriate acceleration of the ego vehicle, ensures a safe distance from the approaching vehicle from behind. In the simulation of Condition 3, where the preceding vehicle decelerates, if the ego vehicle were to change lanes without a significant lateral speed at this time, it would be prone to a collision with the preceding vehicle. Hence, the dynamic Bézier curve plans a trajectory that is more inclined to the left compared to other conditions, based on changes in the control points (with control point P1 gradually moving to the left), allowing for a quicker lane change to the side and avoiding a collision with the preceding vehicle.

The conventional Bézier curve trajectory planning method merely generates a single path at the initiation of a lane change, without incorporating adaptive adjustments to the evolving relative positions between the ego vehicle and obstacle vehicles throughout the lane-changing process. To evaluate the proposed dynamic adaptive Bézier curve trajectory planning method, simulation experiments are conducted under Condition 2 and Condition 3, respectively, in comparison with the conventional Bézier curve trajectory planning method. The comparison focuses on the trend of longitudinal distance changes to the front and rear vehicle from the start of the lane change until the midpoint, i.e., when the lateral displacement of the ego vehicle reaches half of the lane width, in both conditions.

As shown in Figure 15, in the simulation of Condition 2, when the rear vehicle accelerates and approaches, the reduction in the distance between the ego vehicle and the rear vehicle using the adaptive control method proposed in this paper is obviously smaller than that using the traditional non-adaptive control method. Similarly, as shown in Figure 16, in the simulation of Condition 3, when the front vehicle is approaching with deceleration, the reduction in the distance between the ego vehicle and the front vehicle with the adaptive control method proposed in this paper is also notably smaller than that with the traditional non-adaptive control method. This indicates that the lane change trajectory planned by the method proposed in this paper has a better ability to maintain a safe distance compared with the traditional method, so it can increase the redundancy of the distance between the front and rear vehicles and significantly reduce the risk of collision with them.

5. Conclusions

This study proposes an adaptive trajectory planning method based on Bézier curves for lane-changing scenarios with obstacle vehicles in the vicinity. Firstly, the characteristics of Bézier curves and the impact of changes in control points on the curves are analyzed. Then, using a cubic Bézier curve as the basis for trajectory planning, control points related to the ego vehicle and obstacle vehicles are established for lane-changing scenarios. Subsequently, functions are developed to output static single and dynamic real-time updated and iterated Bézier curves, with static and dynamic control points as inputs, respectively. Finally, the proposed method is verified through simulation, and the results indicate that the static Bézier curve trajectory planning method can adaptively generate a smooth and safe lane-changing trajectory based on the position of obstacle vehicles, ensuring a safe distance from static obstacle vehicles.

Through the research in this paper, it is found that the dynamic Bézier curve trajectory planning method can dynamically adjust and generate lane-changing trajectory curves based on the coordinates of vehicles in front and behind. When obstacle vehicles in front or behind gradually approach, the dynamically planned trajectory curves can make real-time adjustments compared to static trajectory planning, significantly reducing the risk of collisions with obstacle vehicles during the lane-changing process and ensuring safety in complex traffic environments. The proposed method can use fewer control points to achieve adaptive control of lane-changing trajectories under both static and complex dynamic lane-changing conditions, which effectively improves the computational efficiency and provides a new solution for dynamic lane-changing trajectory planning of driverless cars under emergency collision avoidance conditions. The method will be optimized for more complex and changing driving scenarios in further research.