Time-dependent lane change trajectory optimisation considering comfort and efﬁciency for lateral collision avoidance

Lateral collision is one of the top two accidents in the world and often occurs in the lane change. Trajectory optimisation is an effective method to solve trafﬁc con-ﬂicts in the lane-changing process. However, current trajectory optimisation methods are not friendly to human-computer-based driving assistance. This study proposes a time-dependent lanechange trajectory optimisation considering comfort and efﬁciency. First, spacing constraints between the lane-changing vehicle and surrounding vehicles are determined and quantiﬁed. Second, lane-changing trajectory data are obtained by driving simulation experiments and extracted by data features. Third, a quintic multinomial model of lane-changing trajectory is proposed. The results reveal that the predicted trajectory is close to the observed value. Then, the obtained trajectory is optimised by the objective function considering lane-changing efﬁciency and comfort. Finally, a case study is used to demonstrate the application of the model. When a lane-changing vehicle changes a lane at an initial speed of 90 km/h to the faster lane at the terminal speed of 110 km/h, the optimal lane-changing time is 3.4 s, with the lateral acceleration 1.79 m/s 2 and the maximum yaw angle 0.081 rad.


FIGURE 1
Lane-changing process conditions [2,3] should be met to complete lane change and avoid lateral collisions.

Lane-changing trajectory model
The commonly used models of lane-changing trajectory include cosine curve [4], trapezoid lateral acceleration curve [5], hyperbolic tangent curve [6], B-spline curve [7], and so forth. The cosine curve has smooth curvature and the hyperbolic tangent curve has smooth lateral acceleration. Nevertheless, both of them cannot replan the trajectory according to a real-time traffic environment [8]. The trapezoid acceleration model worked for the lateral acceleration control but cannot adjust the longitudinal velocity of lane-changing vehicles. The lane-changing trajectory of the B-spline curve is smooth and continuous. However, due to the number of parameters, the solution may not be optimal. Piazzi and Bianco [9] proposed a quintic spline curve for trajectory planning [10]. They found that the flatness of the quintic polynomial curve could increase the robustness of the lane-changing trajectory. The multinomial curve [11,12] was adopted in the lane-changing trajectory because the multinomial function can adjust the order to achieve the desired performance [13]. For example, if the constant of the first derivative of a cubic polynomial trajectory was zero, the planned lateral velocity (in the geodetic coordinate system) was smooth [14]. If the constant of the first and second derivatives of the quintic polynomial was zero, the lateral curvature of the trajectory was smooth [15,16]. The higher-order multinomial function could smooth the variations of variables and, indeed, more input information was necessary. Additionally, the multinomial method could also realise the path re-planning of lane change [17]; therefore, it was more suitable for lane-changing trajectory function.
With the development of automatic driving, lane-changing trajectory optimisation has ushered in new development and application [18,19]. Displacement, lateral velocity, lateral acceleration and lane-changing time were key optimisation variables of lane change trajectory [20][21][22]. Most lane-changing trajectory planning methods based on numerical optimisation relied on other methods to find the optimal trajectory [23,24]. Zhang et al. [25] used the time-dependent cubic polynomial equation and presented a cost function considering the driving comfort and efficiency to optimise the lane-changing trajectory. Suh et al. [26] limited the maximum lateral acceleration of the vehicle in their model to ensure lateral comfort. Additionally, the maximum curvature was also a key variable to control the trajectory [27,28].
Although lane-changing trajectory was affected by many factors, the lateral velocity, lateral acceleration, displacement were not friendly for human-computer interaction [29]. The lateral velocity and acceleration of lane change were very small. Drivers were not sensitive to these variables and it was difficult to adjust operations accordingly. In contrast, time was a sensitive and easily acceptable indicator for people. As it happened, the lanechanging trajectory could be written as a time-dependent function. Time-based driving assistance and trajectory adjustment would be more practical. In the case of certain lateral displacement, drivers only need to be told when to start and end changing lanes to ensure the lane-changing track optimally.
The determination of the optimal lane-changing time is the key to determine the optimal lane-changing trajectory. In this study, a lane-changing trajectory optimisation method is proposed to avoid lateral collisions. First, the safety constraints between the lane-changing vehicle and surrounding vehicles before and after lane change need to be determined and quantified. Second, the lane-changing trajectory and related parameters will be obtained by driving simulation experiments. Third, a time-dependent lane-changing trajectory function is proposed and optimised by an objective optimisation function considering lane-changing comfort and efficiency. Finally, a case study is used to verify the application of the model. This study is organised as follows: Safety constraints in lane change are characterised in Section 2. Section 3 describes the data preparation of lane-changing data. The time-dependent lane-changing trajectory model and the objective optimisation function of trajectory are established in Sections 4 and 5, respectively. Section 6 demonstrates the results of a case study of the model and discusses the applications and limitations. Section 7 is the conclusion.

CHARACTERISATION OF SAFETY CONSTRAINTS IN LANE CHANGE
In the lane-changing analysis, the vehicle model can be simplified to an ellipse model [30,31] as shown in Figure 2 below. The major axis of the ellipse is L x , the minor axis is L w , the yaw angle is , the width of the vehicle is w and the length is L.
The three safety constraints of lane change are shown in Figures 3(a) to (c), respectively. In Figure 3 In Figure 3(b), the minimum safety distance between SV and LV is D MSS (SV, ALV), and the longitudinal displacement of SV is D(SV, ALV). At the collision point, the lateral displacement of the left front point of SV is H-w, where H is the lateral displacement of SV. The minimum safety distance shall meet the conditions as follows: In Figure 3(c), the minimum safety distance between SV and LV is D MSS (AFV, SV), and the longitudinal displacement of SV is D(AFV, SV). At the collision point, the lateral displacement of the left front point of SV is H-w+L sin . The minimum safety distance shall meet the conditions as follows: where t 0 , t c , t f denote lane-changing start time, critical collision time, lane-changing end time, respectively.X, v, a denote longitudinal displacement, velocity and acceleration, respectively. , denote integration variables.

Apparatus
The Tongji University driving simulator was applied to implement experiments and collect data. In the driving simulator, a fully instrumented Renault Megane III vehicle cab is mounted on an eight degree-of-freedom motion system with an X-Y range of 20 × 5 m ( Figure 4). An immersive five-projector system provides a front image view of 250 and 40 • at 1400 × 1050 resolution refreshed at 60 Hz. Liquid crystal display (LCD) monitors provide rear views at the central and side mirror positions. SCANeR™ studio software displays the simulated highway scenario and control a force feedback system that acquires data from the steering wheel, pedals and gear shift lever. Among these schemes, the free lane-changing task was directed by roadside sign, while the restricted lane-changing task was guided by conical buckets before and after the operation area. The experimental schemes were shown in the left column in Figure 5, and the scenarios are shown in the right column.

Participants
Thirty-four qualified participants (28 males, 6 females) were recruited from the university population, with 24 to 50 (mean = 30.2, S.D. = 4.5) years of age and six to 13 years of driving experience (mean = 7.2, S.D. = 3.6). The qualification included: (1) Driver license with at least 8000 km of driving experience; (2) average annual driving distance of at least 2000 km; (3) no criminal records, mental illness and drug use.

Driving tasks and pre-tests
Before the driving simulation experiment, the drivers were informed of the driving task: After identifying the sign or conical buckets, turn on the left or right turn signal and change lanes. Before the formal test, pilots had 10 min to conduct the pretest to adapt to the cockpit environment and control system. Drivers were observed whether having any discomfort reaction. If the driver was normal, the formal experiment would start after 5 min of rest. The formal experiment lasted for about 20 min. When drivers were experimenting, their operation data were collected simultaneously. The operation data included longitudinal and lateral velocity, acceleration, throttle and brake pedal percentage, steering angle, yaw angle and lane departure (Table 1).

Extraction of lane-changing data
The original data contained all driving states, including lanechanging and non-lane-changing states. To get the lanechanging law, the lane-changing data needs to be extracted from the original data first, but different indicators reflect different features of the lane change. Therefore, a feed-forward neural network is applied for pattern recognition to extract the lane-changing data. The operation parameters are illustrated in Figure 6. Among these parameters, the longitudinal speed variation is not affected by lane-changing mode, and there is no uniform law (acceleration, deceleration, first increase then decrease, first decrease then increase are all present). The lateral speed and steering wheel angle have apparent characteristics similar to sine or cosine function when changing lanes. Lane departure also shows distinct step-downs with a lane change. Therefore, the lateral operation parameters are used to identify the starting and ending points of lane change. The seven lateral parameters  Running the model in Figure 7 seven times, the recognition rates for the seven lateral parameters can be obtained in Table 2. The results indicate that the recognition rate of the indi-     The noise of lateral acceleration data is small compared to lateral velocity ( Figure 10). The data trends display an inverted pattern to the lateral velocity, in which the extreme values reach ±0.8 m/s 2 . The lateral acceleration suggests that the value of the lateral acceleration is 0 at the initial and terminal time.

The quintic multinomial lane-changing trajectory
The quintic multinomial lane change trajectory is defined as follows: { x(t ) = a 0 + a 1 t + a 2 t 2 + a 3 t 3 + a 4 t 4 + a 5 t 5 where x denotes longitudinal displacement, y denotes lateral displacement, a 0 , a 1 , … , a 5 and b 0 , b 1 , … , b 5 denote multinomial coefficient. The lateral velocity and acceleration of the lane change trajectory are given as follows: There are six constraints in y-direction and five constraints in x-direction. According to the 11 initial constraints, the constantsa i (i = 0 ∼ 5) and b i (i = 0 ∼ 5) are determined as follows: The constants are brought into Equation (4), and the lane change trajectory is rewritten as follows: The actual lane width w= 3.75 m and lane change time t f = 5 s are brought into Equation (7), lane change trajectory is plotted in Figure 11 below. It can be seen from the diagram that the model well describes the trajectory of the lane change.

Influence of lane-changing time on lane-changing parameters
The lane-changing trajectory of SV is constrained by LV, ALV and AFV. Therefore, the key to lateral collision avoidance is to seek an optimal lane change trajectory.
The key parameters are affected by the lane-changing trajectory including longitudinal and lateral displacement, lateral acceleration and the maximum yaw angle. According to Equation (

Influence on longitudinal displacement
The lane-changing trajectories under different t f are plotted in Figure 12, where longitudinal displacement and lateral displacement are set as x and y axes, respectively. Different from the previous three pictures, the line colours in the latter three pictures denote different lane-changing time. In Figure 12, with the increase of t f , the lateral displacement remains the value of lane width. When t f equals 2 s, the longitudinal displacement is only 70 m; when t f equals 8 s, the longitudinal displacement reaches 230 m.

Influence on lateral acceleration
According to Equation (8), the lateral acceleration a y of SV can be obtained and displayed in Figure 13. The image shows that  Table 3 and Figure 14. It reveals that the moment of max is slightly before the midpoint of lane-changing time. The maximum yaw max increases with the shortening of t f . When t f equals 2 s, max reaches 7.6 • , and when t f equals 8 s, max

Influence on D MSS
As the speed of ALV in the target lane is higher than the speed of SV in the lane-changing process, it is considered that ALV does not affect SV as long as there is a safe D(SV, ALV) at the beginning of lane change. However, LV and AFV still threaten the lane change of SV. The minimum safe lane-changing distance between SV and the two vehicles are obtained as follows (taking t f = 2s as an example):

5.2
The objective function of lane-changing time According   of t f is mostly given by experience, which is not conducive to the flexible application in the complex traffic environment. Lateral acceleration a y has an effect on lane-changing safety and comfort, and lane-changing time t f influences lane-changing efficiency. Therefore, the objective function J(t f ) is established to reduce the lateral acceleration and lane-changing time. Therefore, t f is optimised and the objective function is defined as follows: The maximum longitudinal acceleration a x , lateral acceleration a y and minimum safety distance of lane change D MSS should meet the constraint conditions reflecting comfort and safety, respectively. The constraint conditions are obtained as follows: where 1 , 2 are weight coefficient, and 1 + 2 =1.  In this case, the lane-changing trajectory function is as shown in Equation (8). Then, the value of 1 and 2 need to be calculated according to Equation (11). Take t f = 2 s as an example and the value of 2 is calculated in Equation (13) Thus, the value of 1 can also be obtained 1 =1 − 2 = 0.45. The optimisation problem is solved by iteration according to Equation (12). The results are illustrated in Figure 15 Table 5. It can be seen that the two safety distances at t f = 3.4 s are safer than those at t f = 2 s and more efficient than those at t f = 8 s. This trajectory optimisation method can automatically select the proportion of efficiency and comfort according to the spacing relationship with surrounding vehicles, which is not available in other methods. Therefore, the global optimal lane change time and trajectory parameters can be obtained. This method can be applied in the driving assistance system to assist the driver in decision-making and operation adjustment. In the future, if the driver's style can be added, this method will be more customised.

CONCLUSION
This study proposes a novel method of lane change trajectory optimisation, which aims to solve the traffic conflict caused by the contradiction between lane-changing safety and efficiency. The novel lane-changing trajectory optimisation model can be applied in driving assistance. It is the development trend that the contradiction between comfort and efficiency of lane-changing solved by machines and provide decision-making. In the future, the parameters in the model can be customised according to different driving styles (aggressive, ordinary and conservative), which can achieve better results.