Abstract
Considering the complexity of lane changing using automated vehicles and the frequency of turning lanes in city settings, this paper aims to generate an accelerated lane-changing trajectory using vehicle-to-vehicle collaboration (V2VC). Based on the characteristics of accelerated lane changing, we used a polynomial method and cooperative strategies for trajectory planning to establish a lane-changing model under different degrees of collaboration with the following vehicle in the target lane by considering vehicle kinematics and comfort requirements. Furthermore, considering the shortcomings of the traditional elliptical vehicle and round vehicle models, we established a rectangular vehicle model with collision boundary conditions by analysing the relationships between the possible collision points and the outline of the vehicle. Then, we established a simulation model for the accelerated lane-changing process in different environments under different degrees of collaboration. The results show that, by using V2VC, we can achieve safe accelerated lane-changing trajectories and simultaneously satisfy the requirements of vehicle kinematics and comfort control.
1. Introduction
In automated environments, the ability to sense surrounding traffic and the ability to control the trajectory of a vehicle in real time are significant factors for the safe and efficient operation of automated vehicles. The existing trajectory planning method is an extension of the path planning method in the field of robotics research. Trajectory planning can result in a smooth and continuous trajectory that satisfies the vehicle kinematics and comfort control requirements. The method proposes finding a global trajectory to connect the origin point to a target point. However, analysing changes in the driving environment requires significant time and is not suitable for handling emergency situations, making it difficult to achieve a global trajectory [1, 2]. The local trajectory is generated under the guidance of the global trajectory, which is a type of real-time trajectory that is created by sensing the environmental information and can thus be widely used in the trajectory planning of automated vehicles [3, 4].
The trajectory of an automated vehicle includes the vehicle following trajectory, the lane-changing trajectory, and the overtaking trajectory. Vehicle following research has rapidly developed, and relevant techniques have been applied in real vehicles. Since overtaking behaviour can be regarded as two occurrences of lane changing, lane-changing trajectory planning has become one of the key components of automated vehicles. Bézier [5, 6], spline [7, 8], and polynomial curves [9, 10] are currently the must utilized lane-changing trajectory planning methods for automated vehicles. The Bézier curve can continuously generate a lane-changing trajectory using the radius of curvature but is only applicable for static planning because it requires the selection of control points and cannot achieve real-time obstacle avoidance. The spline curve can be used to plan both the arc trajectory and sine trajectory while overcoming the problems associated with the original curve, such as abrupt changes in the curvature and discontinuities; however, spline curves cannot be used to achieve real-time control. The polynomial curve is rapidly calculated, has continuous curvature, and can achieve real-time control. In addition, polynomial curves can be used to obtain the comfort equation using the cubic derivatives.
In automated environments, it is difficult to change lanes because the majority of the vehicles are driving at a high speed, the velocities of the vehicles are stable, and the headway between the vehicles is small. Although overtaking and nonturning lane changes can be reduced or even avoided, mandatory lane changes at road entrances in cities and turning lane changes on urban roads are inevitable. However, the occurrence of vehicle-to-vehicle traffic can require different forms of lane changing because the vehicles can have the same velocity and maintain the same relative position to each other [11].
In this paper, based on automated environments, we generate a strategy using vehicle-to-vehicle collaboration (V2VC) in Section 2. Considering the real-time performance of the polynomial curve and the quantic polynomial obtained from the triple integral of the comfort function, we use the quantic polynomial for trajectory planning in Section 3. Then, using the safety requirements, kinematic requirements, comfort requirements, and cost function of the accelerated lane changing, we determine the constraint function of the optimal trajectory from Section 4 to Section 6. The simulation analysis of a scenario that utilizes the optimal lane-changing trajectory is shown in Section 7.
2. Strategy of V2VC
As shown in Figure 1, when the host vehicle () changes lanes from the present lane to the target lane, the other vehicles around it include the leading vehicle in the present lane (), the following vehicle in the present lane (), the leading vehicle in the target lane (), and the following vehicle in the target lane (). Among these vehicles, and have the greatest influence on when performs accelerated lane changes. The headway between and determines whether can accelerate, and the headway between and determines whether can move to the target lane.
When intends to change lanes, it will generate safe lane-changing trajectories according to the headway between and and the velocity difference between its current speed and the desired speed. At the same time, a lane-changing signal will be sent to and , which will both provide feedback according to their own driving conditions and the driving state of the nearby vehicles. This V2VC process is shown in Figure 2.
Before changing lanes, the headways between and and between and are and , respectively, and the vehicle speeds are , and . When and receive the lane-changing signal, they assess whether or not to accept the lane-changing request according to their current driving state based on the following criteria:
The leading vehicle in the present lane (): due to the limitations of the vehicles ahead of , it is difficult for to accelerate to provide lane-changing space for , but it can drive at a constant velocity because of the stable traffic flow in the automated environment. After receives the lane-changing signal from , it assesses whether it would collide with while maintaining a constant speed according to the surrounding environment. If it can guarantee that a collision will not occur while driving at a constant speed, it will accept the request; otherwise, it will refuse the request.
The following vehicle in the target lane (): when receives the lane-changing request from , the planned lane-changing trajectories of are also obtained by . determines whether a collision will occur in the lane-changing process of . If a collision will not occur, will accept the lane-changing request and drive at a constant speed; if a collision will occur, will consider the relative position of the following vehicle and the requirements of the collaboration degree, decelerate under an acceptable degree of collaboration, and assess whether will collide with collaborative deceleration. If collision will not occur, will accept the lane-changing request and drive at the predetermined deceleration degree of collaboration until completes the lane change; otherwise, will refuse the lane-changing request.
Under the effect of V2VC, when and receive and accept a lane-changing request, they will change their driving state. To simplify the calculation, we utilize the rectangular coordinate system and regarded the current driving direction of the vehicles as the -axis and the perpendicular direction of travel as the -axis. Then, we can acquire the driving trajectory of and under the lane-changing process of .
The driving trajectory of is defined as
and the driving trajectory of is defined aswhere and are the headways between and and between and , respectively; and are the initial speeds of and , respectively; is the horizontal distance from to , the value of which is equal to the value of the lane width; is the deceleration of ; and is the time it takes to decelerate from its current velocity to an acceptable speed . When maintains a constant velocity, and .
3. Lane-Changing Trajectory Planning of
The comfort degree is usually described using the jerk of the vehicle during research, and the comfort degree function is often formulated as a quadratic function of time [3, 5]. The vehicle trajectory is a 5-order polynomial function of time and can be derived from the triple integral of the comfort degree function.where and represent the longitudinal and lateral coordinates, respectively, which are quantic polynomials with independent variables of time; , and , are the parameters of the longitudinal and lateral coordinates of , respectively; and is the driving time.
Twelve unknown coefficients must be determined in the above functions. Considering the boundary conditions of the initial state and final state of the lane-changing process, we can solve the above equations as follows:where and are the current longitudinal and lateral coordinates of , respectively; and are the current longitudinal and lateral velocities of , respectively; and are the current longitudinal and lateral accelerations of , respectively; and are the longitudinal and lateral coordinates of , respectively, when it completes the lane-changing process; and are the longitudinal and lateral velocities of , respectively, when it completes the lane-changing process; and are the current longitudinal and lateral accelerations of , respectively, when it completes the lane-changing process; and is the lane-changing time. Among them, parameters and are the most difficult to determine, and they are required to solve the quantic polynomial lane-changing trajectory. The solution to this problem is determined in Section 7.
4. Safety Requirements for Lane Changing
Previous studies on lane changing have utilized a minimum safe space to ensure that can take emergency braking precautions to avoid a collision in the event of the sudden deceleration of the leading vehicle. A set of minimum safe distances is used which does not allow for lane changing when the headway between vehicles is too close; these distances are not appropriate in automated environments. The V2VC ensures that the vehicles around maintain a stable driving state and do not decelerate suddenly when they accept the request of . Therefore, the lane-changing gap can be reduced to the minimum gap, and the lane-changing probability in the automated environment can be increased considerably via V2VC.
4.1. Collision Type during Lane Changing
The safety requirements of the vehicles when changing lanes mainly refer to the collision avoidance requirements of the vehicles. For to successfully move to the target lane, a collision should be avoided during the lane-changing process if possible. In general, the methods presented in previous studies could not achieve real-time collision avoidance because they only considered the relative positions between the vehicles before lane changing and rarely considered whether collision would occur during the lane-changing process due to changes in the vehicle steering angle.
Vehicle collision accidents can be divided into frontal collisions, rear-end collisions, head-on side collisions, and oblique collisions. During the lane-changing process of , the most likely collisions are rear-end collisions with in the original line, oblique collisions with during the steering process, oblique collisions during the steering process, and rear-end collisions with in the target lane, as shown in Figure 3.
4.2. Selection of the Vehicle Model
During the lane-changing process, the geometric characteristics of the vehicles influence the vehicle collision angle and collision point; thus, an appropriate vehicle model is necessary. Previous studies have regarded the vehicle as a particle to simplify the modelling and calculation processes. Considering the geometric properties of a circle, a few scholars have used a circle around the vehicle to build a vehicle model, as shown in Figure 4(a). However, treating the length of the vehicle as a diameter results in a significant amount of space in the lateral direction which does not contain a vehicle, and the vehicle width can thus exceed that of a single lane; therefore, this approach is not appropriate for lane-changing maneuvering. A few scholars have used a dynamic circle to cover the entire vehicle, where the width of the vehicle is the diameter of the dynamic circle and the swept area of the dynamic circle is regarded as the vehicle area [9, 12], as shown in Figure 4(b). However, this method only considers the physical characteristics of the vehicle and neglects the different collision possibilities in terms of the different driving velocities and the different collision probabilities in the horizontal and vertical directions. Other scholars have used an ellipse to cover the vehicle, as shown in Figure 4(c), where the long half shaft of the ellipse changes with changes in the velocity. Furthermore, determining the appropriate length of the short half shaft is difficult, and steering during the lane-changing process is difficult to achieve [13, 14].
(a) Static circle model
(b) Dynamic circle model
(c) Dynamic ellipse model
(d) Rectangular vehicle model
Considering the advantages and disadvantages of various vehicle models and combining the requirements to achieve collision avoidance during the lane-changing process, we adopted the rectangular vehicle model, as shown in Figure 4(d). The rectangular vehicle model can accurately simulate the size and contour of the vehicle and is thus convenient for microscopic simulations. However, calculating all distances from the centre point of the vehicle model to the external contour is difficult. A previous analysis of the vehicle lane-changing process indicated that not all of the distances from the centre of the vehicle to the external contour of the vehicle must be calculated. We can assess whether lane changing is feasible as long as we ascertain the possible collision points of the external vehicle contour during the lane-changing process.
4.3. Safety Requirements for Lane Changing with the Rectangular Vehicle Model
During the accelerated lane-changing process, only is steering; the other vehicles are relatively stable. The potential collision points are shown in Figure 5: collision points 1 and 2 are possible collisions between and , and collision point 3 is a possible collision between and . To avoid collisions during the lane-changing process, we must assess the relationships between collision point 1 and line 1, collision point 2 and line 2, and collision point 3 and lines 3 and 4.
After analysing the above position relationships combined with the driving trajectories of , and , we can obtain the collision conditions between and :and the collision conditions between and arewhere and are the length and width of , respectively; and are the length and width of , respectively; is the radius of rotation along the centroid; is the steer angle of ; and is the angle of the vertex of the rectangle model with the horizontal direction.
5. Kinematic and Comfort Requirements of Accelerated Lane Changing
Considering the accelerated lane-changing characteristics and automated environment described in this paper, we combined previous studies and considered the following kinematic constraints.
(1) The Steering Angle. , where is the change in the vehicle steering angle with an increment of time (), is the change in the maximum vehicle steering angle with within the range of maintaining a safe driving performance; is the lateral velocity of at time ; and is the longitudinal velocity of at time .
(2) Curvature. , where is the curvature, is the turning radius, is the wheelbase of the vehicle, and is the maximum curvature of within the range of maintaining safe driving performance.
(3) Velocity. , where is the longitudinal velocity of at time and is the maximum longitudinal velocity of the road section.
(4) Position. , where is the lateral displacement of and is the lane width.
5.1. Comfort Requirements
Comfort is the subjective feeling of the passengers and is also the key factor for evaluating whether lane changing is reasonable in automated environments. In the model, comfort can be assessed based on the acceleration and jerk in the longitudinal and lateral directions.where and are the longitudinal acceleration and jerk, respectively; and are the lateral acceleration and jerk, respectively; and are the maximum longitudinal acceleration and jerk, respectively, for the comfort requirements; and and are the maximum lateral acceleration and jerk, respectively, for the comfort requirements.
6. Cost Function
When lane changing occurs, will have a certain impact on the other vehicles, pedestrians, and traffic flow. To minimize this impact, we established a cost function to evaluate the impact of the lane-changing trajectory. The cost function consists of three components: the comfort level, longitudinal displacement, and lane-changing time.
The comfort level of human beings is an important criterion for evaluating the optimal lane-changing trajectory. The comfort level of a human is defined as follows [15]:where and are the third-order derivatives of the lateral and longitudinal displacement functions of , respectively, indicating the relationship between the jerk and time in the lateral and longitudinal directions, and is a function of the evaluated comfort level, where a smaller value of indicates a higher degree of comfort.
The lane-changing longitudinal distance () is a traffic flow indicator that evaluates the degree of influence of the host vehicle on the surrounding traffic. A smaller value of indicates a lesser impact on the traffic flow. The lane-changing time () is related to the lane-changing efficiency. Considering the influential factors above, we establish the cost function as follows considering the uniform coordination of :where , , and are weighted coefficients with a sum of 1 and , , and are the comfort function, longitudinal lane-changing distance, and maximum lane-changing time in the feasible trajectory set, respectively.
When is used to select the deceleration coordination, its deceleration will have an effect on the surrounding traffic. Considering the negative effects, we establish a cost function as follows when is used to select the deceleration coordination:where , , , and are weighted coefficients with a sum of 1; is the deceleration collaborative distance of during the lane-changing time (); and is the theoretical uniform collaborative distance of during .
7. Scenario Simulation Analysis
Based on Matlab simulation software, the uniform coordination, and the deceleration coordination, we studied the differences between the two types of cooperative lane-changing trajectories using a scenario simulation. The main test parameters are as follows: the length and width of the vehicle are 4.5 m and 1.8 m, respectively; the initial velocity of is 30 km/h; the target velocity of is 40 km/h when completes the lane-changing process; and the current headway between and is 20 m. Considering that the difficulty of solving the quantic polynomial is determining the lane-changing time ( and the longitudinal displacement (, based on the real lane-changing situations, is varied from 0 s to 12 s. Previous studies [7, 16, 17] have found that when the influence of the comfort requirements is considered, the accelerated lane change will have an approximately constant acceleration process. Thus, the acceleration of can be obtained, and the longitudinal displacement of the lane-changing process can be determined by determining the lane-changing time () and speed difference between the initial and final states of .
7.1. Uniform Coordination of
When space in the target lane is sufficiently large, can result in uniform coordination. However, we must avoid a collision between and which could be caused by the sudden acceleration of . The test parameters are as follows: the velocity of is 40 km/h, the headway between and is 20 m, and the headway between and is 12 m. In this paper, we obtain the unconstrained lane-changing trajectory set based on the quantic polynomial, as shown in Figure 6(a). However, a few trajectories in the unconstrained lane-changing trajectory set do not meet the vehicle kinematic and comfort requirements; thus, we can obtain the constrained trajectory set by deleting these trajectories, as shown in Figure 6(b). At the same time, we acquire the relationship between the velocity and acceleration of the trajectory, as shown in Figure 7. Because of the influence of the comfort level, we can obtain the trajectory that satisfies the requirements when the lane-changing time exceeds 4.8 s in our simulation. On this basis, we can acquire a practical and feasible lane-changing trajectory set by finding the trajectory that avoids a collision, as shown in Figure 6(c). According to the cost function () obtained in this simulation, the optimal lane-changing time of is 7.0 s, and the optimal longitudinal displacement is 68.06 m, which is marked in black in the figure.
(a) Unconstrained lane-changing trajectory set
(b) Constrained lane-changing trajectory set
(c) The feasible lane-changing trajectory set
(a) Longitudinal velocity-time diagram (m/s)
(b) Longitudinal acceleration-time diagram (m2/s)
(c) Lateral velocity-time diagram (m/s)
(d) Lateral acceleration-time diagram (m2/s)
After obtaining the optimal trajectory, we conducted a dynamic simulation of the trajectory, as shown in Figure 8. Through the simulation, the host vehicle can achieve lane changing without collision with a relatively smooth lane-changing trajectory; this behaviour meets the comfort requirements of the passengers in the automated environment.
(a) Initial state of the lane-changing process
(b) Position relationship between and
(c) Position relationship between and
(d) Final state of the lane-changing process
7.2. Deceleration Coordination of
When there is insufficient space for the lane-changing process, must perform deceleration coordination. The test parameters are as follows: the velocity of is 48 km/h, the cooperative velocity of is 40 km/h, the headway between and is 20 m, and the headway between and is 12 m. The unconstrained trajectory set obtained from the quantic polynomial and the constrained trajectory set that satisfies the kinematic and comfort requirements are the same as those in Figure 6. Because the velocity of is too large, will collide with if maintains a uniform driving velocity. When detects this situation, it will decelerate according to the acceptable collaboration degree and follow the predetermined trajectory after accepting the lane-changing request. The feasible lane-changing trajectory set of is shown in Figure 9. According to the cost function (), we obtain an optimal lane-changing time for as 6.1 s in this simulation and an optimal longitudinal displacement of 59.31 m, which is marked in black in the figure. The optimal trajectory simulation of the deceleration coordination of is shown in Figure 10.
(a) Initial conditions of the lane-changing process
(b) Position relationship between and
(c) Position relationship between and
(d) Completion of the lane-changing process
7.3. Mixture of Uniform and Deceleration Coordination
When the headway between and exceeds a certain amount, the accelerated lane-changing trajectory set of is divided into two parts: one part is related to the uniform coordination of and the other part is related to the deceleration coordination of . In other words, a few trajectories may not be satisfied under the uniform coordination of but will be satisfied under the deceleration coordination of . Theoretically, the deceleration coordination will increase the lane-changing opportunity of the host vehicle (), but during the actual process of our simulation, the lane-changing time is generally increased due to the comfort requirements of automated environments; hence, the uniform coordination of can meet the short lane-changing time. However, when the lane-changing time is excessively long, because the speed of is higher than , the headway between and is not sufficiently large to satisfy the lane-changing process of with the uniform coordination of , as shown in Figure 11. Therefore, the optimal trajectory is located within the part of the lane-changing trajectory set of which involves the uniform collaboration of . The test parameters are as follows: the velocity of is 48 km/h, the collaboration degree is 40 km/h, and the headway between and is 30 m.
(a) Feasible lane-changing trajectory set of with the uniform collaboration of
(b) Feasible lane-changing trajectory set of with the deceleration collaboration of
8. Conclusion
In this paper, considering the difficulty of the lane-changing process and the accelerated lane-changing characteristics in an automated environment, we achieved lane changing using V2VC. We studied the accelerated lane-changing process of with the coordination of in an automated environment, and the process was described as a near-uniform acceleration process based on previous studies. Then, we generated an unconstrained lane-changing trajectory set using quantic polynomials, minimized the lane-changing set using actual vehicle kinematic constraints and comfort requirements, and obtained a range of actual trajectories. The results show that we can achieve trajectory planning with the uniform and deceleration collaboration of . We determined the optimal lane-changing trajectory by calculating the cost function, and the scenario simulations indicated that the generated optimal lane-changing trajectory could achieve safe and comfortable lane changing in an automated environment.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.