# Research on Safety Lane Change Warning Method Based on Potential Angle Collision Point

Authors: Tongqiang Ding; Xiaorong Li; Lili Zheng; Zhiguo Hao
Journal: Journal of Advanced Transportation
 (2019)

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## Abstract

In order to ensure safe lane change and avoid traffic accidents, an effective lane change assist system is required. In a lane change assist system, it is very important to obtain the following elements in time, such as actual distance between vehicles, minimum safety distance, and warning signal. To this end, this paper analyzed four kinds of lane change angle collision scenes. Initial position, initial velocity, acceleration, heading angle, and kinematics of vehicles were used to calculate the position of potential angle collision points between lane change vehicle and obstacle vehicles. Then, actual distance model was constructed based on potential angular collision points. The minimum safety distance model was also established under the two most unfavorable conditions. In order to achieve the lane change warning, three early warning rules were formulated. We verified the validity of models and early warning rules using vehicle driving video data of Interstate 80 in California. Models and early warning rules constructed in our research can be applied to the advanced active safety systems of vehicle, such as vehicle lane change assist system and active collision early warning system, which can improve the active safety and reduce traffic accidents.


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## Body

## 1. Introduction

Lane change is a relatively complicated process for driving vehicles. Drivers’ improper operation during lane change may cause traffic delays and accidents. According to incomplete statistics, lane change accidents account for 4-10% of all traffic accidents [1]. Safety is affected by the type, position, speed, acceleration, and deceleration of obstacle vehicles in the present lane and the target lane. In order to ensure safe lane change and avoid angle accidents, the vehicle lane change assist system needs to calculate actual distance between lane change vehicle and obstacle vehicles and make decisions and early warning by comparing the minimum safety distance [2–7].In terms of actual distance estimation between lane change vehicle and obstacle vehicles, C. L. Wang [8] processed data using statistical mean model and then applied RSSI (Received Signal Strength Indication) to estimate the distance between the two vehicles. C. Wang [9] determined actual distance between these two vehicles by calculating the distance between the microwave radar in the rear bumper of preceding vehicle and that in the preceding bumper of rear vehicle, as well as the included angle of axle wire. J. Feng et al. [10] assumed that the lateral acceleration of lane change vehicle satisfied the positive and negative trapezoidal constraints and determined the position of lane change vehicle by predicting its lane change path. Meanwhile, the distance between those vehicles is estimated by considering their length. G. Xu et al. [11] predicted the lane change trajectory to determine positions of lane change vehicles by trapezoidal lateral acceleration, spline curve, commutation trajectory of convoluted curve, polynomial function, and artificial potential field. Parameters such as the length of vehicles were considered in estimating the distance between vehicles. Y. G. Luo et al. [12] achieved actual distance measurement with other vehicles by establishing a five-time polynomial lane change trajectory tracking model for time. H. Z. Zhu et al. [13] proposed that vehicles equipped with RUPS (relative urban positioning system) can perceive their trajectory information and then perceive the relative distance between vehicles by exchanging, comparing, and identifying overlapping positions between their own and adjacent vehicle trajectories. Y. Zhu et al. [14] detected the position of the corresponding vehicle using fish swarm algorithm. After simplifying vehicle coordinates, the corresponding actual distance was obtained by calculating their Euclidean distance. C. Olaverri-Monreal et al. [15] stereoscopically captured and processed images through the rear camera to obtain positional relationships between vehicles, with the distance between the preceding and rear vehicles calculated.In the past studies, actual distance between lane change vehicle and obstacle vehicles was calculated by considering the coordinates of the centre points and the length of the vehicle bodies as well as the angle coordinate of lane change vehicle. In actual lane change, the collision of two vehicles depends on the space between potential angle collision points. Based on this, our research analyzed coordinates of potential angle collision points in lane change and established actual distance model for lane change vehicle and the adjacent obstacle vehicle by considering initial position, initial speed, acceleration, and heading angle of vehicle.In terms of minimum safety distance, Q. Luo et al. [16] considered the gradual process of state and speed of preceding vehicle and established the minimum safety distance model in the motionlessness, uniform speed, and uniform deceleration of the preceding vehicle. S. M. Wang et al. [17] regarded the lateral acceleration as a sinusoidal characteristic function of time and derived the critical position of vehicle collision by derivation. Then the minimum safe distance condition was obtained to avoid vehicle collision. R. S. Tomar et al. [18] extracted the relative velocity, acceleration, and other parameters between lane change vehicle and the surrounding vehicle through support vector machine, which was used to calculate the minimum safety distance. In actual lane change, the minimum safety distance changes with the running state of vehicle, and most of the above studies cannot reflect real-time running state of vehicle [19]. A. Kanaris et al. [20] proposed that the preceding vehicle braking should be considered to calculated the minimum safety distance, so the remaining vehicles have enough time and space to adjust and avoid collision. Based on it, this paper analyzed the most unfavorable situations in lane change and combined the real-time running state of vehicle to establish the minimum safety distance model between the preceding and rear vehicles in the emergency braking of the preceding vehicle as well as the minimum safety distance model when the speed of rear vehicle is higher than that of preceding vehicle.Different from the existing research, this paper studied functions of potential angle collision point positions under four lane change scenes to determine the actual distance model of lane change vehicles and surrounding obstacle vehicles. And then, according to the minimum safety distance, lane change risks were classified and corresponding warning rules were formulated, which assisted driver to adjust vehicle’s state in time and improve vehicle’s active safety.


## 2. Methods

### 2.1. Relevant Vehicle Definitions and Scenes of Angle Collision during Lane Change

#### 2.1.1. Relevant Vehicle Definitions Involved in the Lane Change

In Figure1, it is defined as follows:CL-car is the lane change vehicle;P-front the preceding obstacle vehicle before lane change;P-back the rear obstacle vehicle before lane change;T-front the preceding obstacle vehicle after lane change;T-back the rear obstacle vehicle after lane change.Figure 1
Vehicle definition.


#### 2.1.2. Scenes of Angle Collision during Lane Change

Figure2 shows the scenes of angle collision during lane change, that is, the angle collision ofCL-carandP-front, that ofCL-car andP-back, that ofCL-car andT-front, and that ofCL-car andT-back. Our research analyzed positions of potential angle collisions of lane change vehicles in the four scenes and established relevant models.Figure 2
Scenes of angle collisions during lane change.
(a)
The angle collision betweenT-back andCL-car (b)
The angle collision betweenT-front andCL-car (c)
The angle collision betweenP-backandCL-car (d)
The angle collision betweenP-front andCL-car


### 2.2. Actual Distance Model between Two Vehicles

#### 2.2.1. Assumptions

This paper has made the following assumptions:( 1 ) T-back,P-back,T-front, andP-front drive straight on the respective roads without lane change, and the speed and acceleration are in a horizontal state.( 2 ) When the position of angle collision point is determined, lane change vehicles are regarded as a rectangle, and its coordinates are simplified to the geometric centre of the rectangle.


#### 2.2.2. Lane Change Parameters and Coordinate Definition

Without loss of generality, modeling and analysis were carried out by taking the left lane change as an example. The lane change process was positioned by the absolute Cartesian coordinate system. The direction of the vehicle along the lane was defined as the positive direction of X-axis; vertical to the lane and upward were defined as the positive direction of Y-axis (see Figure3). The right edge line of the current lane is defined as X-axis, and Y-axis is located at the geometric centre of the end obstacle vehicle at the beginning of lane change. Then the origin of coordinates can be obtained (see Figure 3). After the definition is completed, the coordinate system does not change with the change of vehicle position during lane change.Figure 3
Definition of coordinate axes and coordinate origins.As shown in Figure4, the position of each vehicle is (xj,yj), with the unit ofm; vj is the speed of each vehicle, with the unit ofm/s; aj is the acceleration of each vehicle, with the unit of m/s2; Lj is the length of the vehicle, with the unit ofm; Wj is the vehicle width, where j∈[T-back, P-back, T-front, P-front, CL-car].Figure 4
Lane change parameters.Figure4 shows that the speed and acceleration of each vehicle are constantly changing during driving, so the start time of the lane change is defined to be zero, and the position of each vehicle can be expressed by(1)xj=x0j+∫v0j,x+∫aj,xdtdt,yj=y0j+∫v0j,y+∫aj,ydtdtwhere x0j represents the initial position of the vehicle j in the X direction; v0j,x is the initial speed of the vehicle j in the X direction; aj,x is the acceleration of the vehicle j in the X direction. Similarly, the meanings of y0j, v0j,y and aj,y are similar.When the lane change vehicleCL-car is driving, the heading angle is constantly changing. In Figure 5, α is the heading angle of vehicle; β is the shape parameter angle of vehicle.Figure 5
Angles of the lane change vehicle.The anglesα and β can be calculated using(2)α=arctan⁡vCL-car,yvCL-car,x=arctan⁡v0CL-car,y+∫aCL-car,ydtv0CL-car,x+∫aCL-car,xdt,β=arctan⁡WCL-carLCL-car


#### 2.2.3. Calculation of the Preceding and Rear Angular Coordinates of Obstacle Vehicles

Figure6 shows the preceding and rear angles of obstacle vehicles that are B1,i,B2,i,B3,i,B4,i. And coordinates can be calculated from the position, length, and width of obstacle vehicles (see Table 1), wherein i∈[T-back,P-back,T-front,P-front].Table 1
Preceding and rear point coordinates of obstacle vehicles.
Preceding and rear angles of obstacle vehicles Abscissa Ordinate B 1 , i xi-Li2 y i - W i 2 B 2 , i xi-Li2 y i + W i 2  B 3 , i xi+Li2 y i - W i 2  B 4 , i xi+Li2 y i + W i 2Figure 6
Preceding and rear angles of obstacle vehicles.


#### 2.2.4. Coordinate Calculation of Potential Angle Collision Points of Lane Change Vehicles

The preceding and rear angles of lane change vehicles are defined asA1,A2,A3,A4, and the potential angle collision points of lane change vehicles are calculated by combining B1,i,B2,i,B3,i,B4,i.( 1 ) In the lane change scene ofCL-car andP-front, the potential collision coordinates ofCL-car are calculated as follows.Figure7 shows that, in the lane change process 1,CL-car andP-front generate a potential angle collision point ①, and the potential collision position ofCL-car is at point A1. The coordinates are as follows:(3)xA1=xCL-car+LCL-car2+WCL-car22cos⁡α-β(4)yA1=yCL-car+LCL-car2+WCL-car22sin⁡α-βIn the lane change process 2,CL-car andP-front generate a potential angle collision point ②, and the potential collision position ofCL-car is located within A1→A2. The coordinates are(5)xA1→A2=xA2+ΔxCL-car,P-front(6)yA1→A2=yB2,P-frontwhere(7)xA2=xCL-car-LCL-car2+WCL-car22cos⁡α+β(8)ΔxCL-car,P-front=ΔyCL-car,P-frontcot⁡α(9)ΔyCL-car,P-front=yB2,P-front-yA2(10)yA2=yCL-car-LCL-car2+WCL-car22sin⁡α+β(2) In the lane change scene ofCL-car andP-back, the potential collision position coordinates ofCL-car are calculated as  follows.Figure 7
Potential angle collision point positions in different lane change processes betweenCL-carandP-front.Figure8 shows that in the lane change process 1,CL-car andP-back generate a potential angle collision point ①, and the potential collision position ofCL-car is at A3. The coordinates are(11)xA3=xCL-car-LCL-car2+WCL-car22cos⁡α-β(12)yA3=yCL-car-LCL-car2+WCL-car22sin⁡α-βAs shown in Figure 8, in the lane change process 2,CL-carandP-back generate a potential angle collision point ②, and the potential collision position ofCL-car is located within A2→A3. The coordinates are(13)xA2→A3=xA2-ΔxP-back,CL-car(14)yA2→A3=yB4,P-backwhere(15)ΔxP-back,CL-car=ΔyP-back,CL-cartan⁡α(16)ΔyP-back,CL-car=yB4,P-back-yA2(3) In the lane change scene ofCL-car andT-front, the potential collision position coordinates ofCL-car are calculated as  follows.Figure 8
Potential angle collision point positions in different lane change processes betweenCL-car andP-back.As shown in Figure9, in the lane change process 1,CL-car andT-frontgenerate a potential angle collision point ①, and the potential collision position ofCL-car is located within A4→A1. The coordinates are(17)xA4→A1=xA1-ΔxCL-car,T-front(18)yA4→A1=yB1,T-frontwhere(19)ΔxCL-car,T-front=ΔyCL-car,T-fronttan⁡α(20)ΔyCL-car,T-front=yB1,T-front-yA1In the lane change process 2,CL-car andT-front generate a potential angle collision point ②; the potential collision position ofCL-car is at A1; its coordinate is (xA1,yA1).Figure 9
Potential angle collision point positions in different lane change processes betweenCL-car andT-front.( 4 ) In the lane change scene ofCL-car andT-back, the potential collision position coordinates ofCL-car are calculated as  follows.Figure10 shows that, in the lane change process 1,CL-car andT-back generate a potential angle collision point ①, and the potential collision position ofCL-car is located within A3→A4. The coordinates are(21)xA3→A4=xA4-ΔxT-back,CL-car(22)yA3→A4=yB3,T-backwhere(23)xA4=xCL-car+LCL-car2+WCL-car22cos⁡α+β(24)ΔxT-back,CL-car=ΔyT-back,CL-carcot⁡α(25)ΔyT-back,CL-car=yA4-yB3,T-back(26)yA4=yT-back+LCL-car2+WCL-car22sin⁡α+βFigure 10 shows, in the lane change process 2,CL-car andT-back generate a potential angle collision point 2; the potential collision position ofCL-car is at A3; its coordinate is (xA3,yA3).Figure 10
Potential angle collision point positions in different lane change processes betweenCL-car andT-back.


#### 2.2.5. Determination of Actual Distance

According to the preceding and rear angular coordinates of obstacle vehicles and the coordinates of potential angle collision points of lane change vehicle, we calculate actual distance between lane change vehicle and obstacle vehicles during the two-stage lane change in four scenes. Figure11 shows that when yB1,P-front<yA1<yB2,P-front, the potential angle collision position ofCL-car is at A1, and the actual distance SP-front,CL-car between the two vehicles can be calculated by (27)SP-front,CL-car=xB2,P-front-xA1Similarly, actual distance between lane change vehicle and obstacle vehicles in other scenes can be obtained (see Table 2).Table 2
Actual distance during the two-stage lane change in four scenes.
Angle collision scene Angle collision point Conditions for angle collision point Actual distance Remarks CL-car and P-front ① yB1,P-front<yA1<yB2,P-front xB2,P-front-xA1 SP-front,CL-car ② y A 1 > y B 2 , P - f r o n t,yA2<yB2,P-front xB2,P-front-xA1→A2  CL-car and P-back ① yB3,P-back<yA3<yB4,P-back xA3-xB4,P-back SCL-car,P-back ② y A 3 > y B 4 , P - b a c k,yA2<yB4,P-back xA2→A3-xB4,P-back  CL-car and T-front ① y A 1 < y B 1 , T - f r o n t,yA4>yB1,T-front xB1,T-front-xA4→A1 ST-front,CL-car, ② yB2,T-front>yA1>yB1,T-front xB1,T-front-xA1  CL-car and T-back ① y A 3 < y B 3 , T - b a c k,yA4>yB3,T-back xA3→A4-xB3,T-back SCL-car,T-back ② yB4,T-back>yA3>yB3,T-back xA3-xB3,T-backFigure 11
Actual distance diagram.


### 2.3. Minimum Safety Distance Model

For the assisted driving system, after measuring actual distance between lane change vehicle and obstacle vehicles, it is necessary to determine the corresponding minimum safety distance for safe lane change. Decision and early warning are made by comparison [21]. The minimum safe distance refers to the necessary separation distance between the rear and preceding vehicles to avoid collision.In lane change, there are two most unfavorable scenes: one is emergency braking of the preceding vehicle and the other is the speed of rear vehicle which is greater than that of preceding vehicle. For the first type, when preceding vehicle is parked under emergency braking, the rear vehicle should adjust its deceleration and speed to make the vehicle stop within the shortest distance for safety [22]. The second type is that, during lane change of continuous driving, the rear vehicle needs to decelerate to the same speed as preceding vehicle for safety.In view of the above-mentioned most unfavorable scenes, based on kinematics principle of vehicle, this paper established the minimum safety distance model between preceding and rear vehicles when the preceding vehicle is in emergency braking and the minimum safety distance model between preceding and rear vehicles when the rear vehicle is adjusted to the same speed as the preceding vehicle.


#### 2.3.1. Minimum Safety Distance Model between the Preceding and Rear Vehicles during Emergency Braking of the Preceding Vehicle

The vehicle braking time consists of driver response time, brake coordination time, brake acceleration growth time, and continuous braking time [23], wherein vehicle tires, suspension, and ground have the impact on brake coordination time [24, 25]. The sum of driver response time and brake coordination time is generally 0.8-1.0 s [26, 27]. The deceleration build-up time of the preceding vehicle is generally 0.1 to 0.2 s, and tfront,c is the braking time of preceding vehicle, with the unit ofs. tback is the deceleration build-up time of the rear car, generally 0.1-0.2 s, and tback,c is the rear brake time, with the unit of s [28]. The braking principles of different vehicles are different [29, 30], and the maximum braking force is not exactly the same. The maximum average braking acceleration in the horizontal direction of the vehicle during emergency braking is determined as 7m/s2 [27, 31]. Since the driver of the rear vehicle takes action after the preceding vehicle brakes, the change of the braking acceleration of rear vehicle will be delayed relative to the preceding vehicle. When the minimum safety distance LBback,front is calculated, the acceleration changes of preceding and rear vehicles are shown in Figures 12 and 13.Figure 12
Braking deceleration changes of preceding vehicle.Figure 13
Braking deceleration changes of rear vehicle.Then the braking distanceLback of the rear vehicle is(28)Lback=vback,xtr1+tr2+tback2-124amaxback,xtback2+vback,x22amaxback,xwhere vback,x represents the speed of the rear vehicle in X direction; amaxback,x is the maximum braking acceleration of the rear vehicle in X direction.Similarly,(29)LCL-car,back=vCL-car,xtr1+tr2+tback2-124amaxCL-car,xtback2+vCL-car,x22amaxCL-car,x(30)LP-back=vP-back,xtr1+tr2+tback2-124amaxP-back,xtback2+vP-back,x22amaxP-back,x(31)LT-back=vT-back,xtr1+tr2+tback2-124amaxT-back,xtback2+vT-back,x22amaxT-back,xThe braking distance of the preceding vehicle is(32)Lfront=12vfront,xtfront-124amaxfront,xtfront2+vfront,x22amaxfront,xwhere vfront,x represents the speed of the preceding vehicle in X direction; amaxfront,x is the maximum braking acceleration of the preceding vehicle in X direction.Similarly,(33)LP-front=12vP-front,xtfront-124amaxP-front,xtfront2+vP-front,x22amaxP-front,x(34)LT-front=12vT-front,xtfront-124amaxT-front,xtfront2+vT-front,x22amaxT-front,x(35)LCL-car,front=12vCL-car,xtfront-124amaxCL-car,xtfront2+vCL-car,x22amaxCL-car,xThe minimum safety distance LBback,front under emergency braking of the preceding vehicle is(36)LBback,front=Lback-LfrontSimilarly,(37)LBCL-car,P-front=LCL-car,back-LP-front(38)LBCL-car,T-front=LCL-car,back-LT-front(39)LBP-back,CL-car=LP-back-LCL-car,front(40)LBT-back,CL-car=LT-back-LCL-car,front


#### 2.3.2. Minimum Safety Distance Model between Preceding and Rear Vehicles When the Rear Vehicle Is Adjusted to the Same Speed as the Preceding Vehicle

If the speedvfront,x of the preceding vehicle is greater than the speed vback,x of the rear vehicle, the actual distance between the two vehicles increases with time, and no angle collision and rear-end collision occur. Therefore, the minimum safety distance LSback,front is zero [32]. If the speed vfront,x of the preceding vehicle is less than the speed vback,x of the rear vehicle, the minimum safe distance is(41)LSback,front=vback,x2-vfront,x22amaxback,xSimilarly,(42)LSCL-car,P-front=0vP-front,x≥vCL-car,xvCL-car,x2-vP-front,x22amaxCL-car,xvP-front,x<vCL-car,x(43)LSCL-car,T-front=0vT-front,x≥vCL-car,xvCL-car,x2-vT-front,x22amaxCL-car,xvT-front,x<vCL-car,x(44)LSP-back,CL-car=0vCL-car,x≥vP-back,xvP-back,x2-vCL-car,x22amaxP-back,xvCL-car,x<vP-back,x(45)LST-back,CL-car=0vCL-car,x≥vT-back,xvT-back,x2-vCL-car,x22amaxT-back,xvCL-car,x<vT-back,x


### 2.4. Definition of Early Warning Rules

According to the position of the potential angle collision point, we can obtain the actual distanceSback,front between the rear vehicle and preceding vehicle, the minimum safety distance LBback,front under the emergency braking of the preceding vehicle, and the minimum safety distance LSback,front between preceding and rear vehicles when the rear vehicle is adjusted to the same speed as the preceding vehicle. The comparison of Sback,front, LBback,front, and LSback,front divides the early warning into no early warning, mild early warning, and high early warning, and the colors of the early warning lights are green, yellow, and red, respectively. Specific definitions are shown in Table 3.Table 3
Introduction to early warning rules.
Early warning rule Condition Status description Display color No early warning Sback,front>LBback,front No need to adjust the speed  Mild early warning LSback,front<Sback,front≤LBback,front If the preceding vehicle does not take emergency braking, the distance between the vehicles meets the safety needs.  Severe early warning Sback,front≤LSback,front When the speed of the vehicle is not adjusted in time, a angle collision accident will occur.


## 2.1. Relevant Vehicle Definitions and Scenes of Angle Collision during Lane Change

### 2.1.1. Relevant Vehicle Definitions Involved in the Lane Change

In Figure1, it is defined as follows:CL-car is the lane change vehicle;P-front the preceding obstacle vehicle before lane change;P-back the rear obstacle vehicle before lane change;T-front the preceding obstacle vehicle after lane change;T-back the rear obstacle vehicle after lane change.Figure 1
Vehicle definition.


### 2.1.2. Scenes of Angle Collision during Lane Change

Figure2 shows the scenes of angle collision during lane change, that is, the angle collision ofCL-carandP-front, that ofCL-car andP-back, that ofCL-car andT-front, and that ofCL-car andT-back. Our research analyzed positions of potential angle collisions of lane change vehicles in the four scenes and established relevant models.Figure 2
Scenes of angle collisions during lane change.
(a)
The angle collision betweenT-back andCL-car (b)
The angle collision betweenT-front andCL-car (c)
The angle collision betweenP-backandCL-car (d)
The angle collision betweenP-front andCL-car


## 2.1.1. Relevant Vehicle Definitions Involved in the Lane Change

In Figure1, it is defined as follows:CL-car is the lane change vehicle;P-front the preceding obstacle vehicle before lane change;P-back the rear obstacle vehicle before lane change;T-front the preceding obstacle vehicle after lane change;T-back the rear obstacle vehicle after lane change.Figure 1
Vehicle definition.


## 2.1.2. Scenes of Angle Collision during Lane Change

Figure2 shows the scenes of angle collision during lane change, that is, the angle collision ofCL-carandP-front, that ofCL-car andP-back, that ofCL-car andT-front, and that ofCL-car andT-back. Our research analyzed positions of potential angle collisions of lane change vehicles in the four scenes and established relevant models.Figure 2
Scenes of angle collisions during lane change.
(a)
The angle collision betweenT-back andCL-car (b)
The angle collision betweenT-front andCL-car (c)
The angle collision betweenP-backandCL-car (d)
The angle collision betweenP-front andCL-car


## 2.2. Actual Distance Model between Two Vehicles

### 2.2.1. Assumptions

This paper has made the following assumptions:( 1 ) T-back,P-back,T-front, andP-front drive straight on the respective roads without lane change, and the speed and acceleration are in a horizontal state.( 2 ) When the position of angle collision point is determined, lane change vehicles are regarded as a rectangle, and its coordinates are simplified to the geometric centre of the rectangle.


### 2.2.2. Lane Change Parameters and Coordinate Definition

Without loss of generality, modeling and analysis were carried out by taking the left lane change as an example. The lane change process was positioned by the absolute Cartesian coordinate system. The direction of the vehicle along the lane was defined as the positive direction of X-axis; vertical to the lane and upward were defined as the positive direction of Y-axis (see Figure3). The right edge line of the current lane is defined as X-axis, and Y-axis is located at the geometric centre of the end obstacle vehicle at the beginning of lane change. Then the origin of coordinates can be obtained (see Figure 3). After the definition is completed, the coordinate system does not change with the change of vehicle position during lane change.Figure 3
Definition of coordinate axes and coordinate origins.As shown in Figure4, the position of each vehicle is (xj,yj), with the unit ofm; vj is the speed of each vehicle, with the unit ofm/s; aj is the acceleration of each vehicle, with the unit of m/s2; Lj is the length of the vehicle, with the unit ofm; Wj is the vehicle width, where j∈[T-back, P-back, T-front, P-front, CL-car].Figure 4
Lane change parameters.Figure4 shows that the speed and acceleration of each vehicle are constantly changing during driving, so the start time of the lane change is defined to be zero, and the position of each vehicle can be expressed by(1)xj=x0j+∫v0j,x+∫aj,xdtdt,yj=y0j+∫v0j,y+∫aj,ydtdtwhere x0j represents the initial position of the vehicle j in the X direction; v0j,x is the initial speed of the vehicle j in the X direction; aj,x is the acceleration of the vehicle j in the X direction. Similarly, the meanings of y0j, v0j,y and aj,y are similar.When the lane change vehicleCL-car is driving, the heading angle is constantly changing. In Figure 5, α is the heading angle of vehicle; β is the shape parameter angle of vehicle.Figure 5
Angles of the lane change vehicle.The anglesα and β can be calculated using(2)α=arctan⁡vCL-car,yvCL-car,x=arctan⁡v0CL-car,y+∫aCL-car,ydtv0CL-car,x+∫aCL-car,xdt,β=arctan⁡WCL-carLCL-car


### 2.2.3. Calculation of the Preceding and Rear Angular Coordinates of Obstacle Vehicles

Figure6 shows the preceding and rear angles of obstacle vehicles that are B1,i,B2,i,B3,i,B4,i. And coordinates can be calculated from the position, length, and width of obstacle vehicles (see Table 1), wherein i∈[T-back,P-back,T-front,P-front].Table 1
Preceding and rear point coordinates of obstacle vehicles.
Preceding and rear angles of obstacle vehicles Abscissa Ordinate B 1 , i xi-Li2 y i - W i 2 B 2 , i xi-Li2 y i + W i 2  B 3 , i xi+Li2 y i - W i 2  B 4 , i xi+Li2 y i + W i 2Figure 6
Preceding and rear angles of obstacle vehicles.


### 2.2.4. Coordinate Calculation of Potential Angle Collision Points of Lane Change Vehicles

The preceding and rear angles of lane change vehicles are defined asA1,A2,A3,A4, and the potential angle collision points of lane change vehicles are calculated by combining B1,i,B2,i,B3,i,B4,i.( 1 ) In the lane change scene ofCL-car andP-front, the potential collision coordinates ofCL-car are calculated as follows.Figure7 shows that, in the lane change process 1,CL-car andP-front generate a potential angle collision point ①, and the potential collision position ofCL-car is at point A1. The coordinates are as follows:(3)xA1=xCL-car+LCL-car2+WCL-car22cos⁡α-β(4)yA1=yCL-car+LCL-car2+WCL-car22sin⁡α-βIn the lane change process 2,CL-car andP-front generate a potential angle collision point ②, and the potential collision position ofCL-car is located within A1→A2. The coordinates are(5)xA1→A2=xA2+ΔxCL-car,P-front(6)yA1→A2=yB2,P-frontwhere(7)xA2=xCL-car-LCL-car2+WCL-car22cos⁡α+β(8)ΔxCL-car,P-front=ΔyCL-car,P-frontcot⁡α(9)ΔyCL-car,P-front=yB2,P-front-yA2(10)yA2=yCL-car-LCL-car2+WCL-car22sin⁡α+β(2) In the lane change scene ofCL-car andP-back, the potential collision position coordinates ofCL-car are calculated as  follows.Figure 7
Potential angle collision point positions in different lane change processes betweenCL-carandP-front.Figure8 shows that in the lane change process 1,CL-car andP-back generate a potential angle collision point ①, and the potential collision position ofCL-car is at A3. The coordinates are(11)xA3=xCL-car-LCL-car2+WCL-car22cos⁡α-β(12)yA3=yCL-car-LCL-car2+WCL-car22sin⁡α-βAs shown in Figure 8, in the lane change process 2,CL-carandP-back generate a potential angle collision point ②, and the potential collision position ofCL-car is located within A2→A3. The coordinates are(13)xA2→A3=xA2-ΔxP-back,CL-car(14)yA2→A3=yB4,P-backwhere(15)ΔxP-back,CL-car=ΔyP-back,CL-cartan⁡α(16)ΔyP-back,CL-car=yB4,P-back-yA2(3) In the lane change scene ofCL-car andT-front, the potential collision position coordinates ofCL-car are calculated as  follows.Figure 8
Potential angle collision point positions in different lane change processes betweenCL-car andP-back.As shown in Figure9, in the lane change process 1,CL-car andT-frontgenerate a potential angle collision point ①, and the potential collision position ofCL-car is located within A4→A1. The coordinates are(17)xA4→A1=xA1-ΔxCL-car,T-front(18)yA4→A1=yB1,T-frontwhere(19)ΔxCL-car,T-front=ΔyCL-car,T-fronttan⁡α(20)ΔyCL-car,T-front=yB1,T-front-yA1In the lane change process 2,CL-car andT-front generate a potential angle collision point ②; the potential collision position ofCL-car is at A1; its coordinate is (xA1,yA1).Figure 9
Potential angle collision point positions in different lane change processes betweenCL-car andT-front.( 4 ) In the lane change scene ofCL-car andT-back, the potential collision position coordinates ofCL-car are calculated as  follows.Figure10 shows that, in the lane change process 1,CL-car andT-back generate a potential angle collision point ①, and the potential collision position ofCL-car is located within A3→A4. The coordinates are(21)xA3→A4=xA4-ΔxT-back,CL-car(22)yA3→A4=yB3,T-backwhere(23)xA4=xCL-car+LCL-car2+WCL-car22cos⁡α+β(24)ΔxT-back,CL-car=ΔyT-back,CL-carcot⁡α(25)ΔyT-back,CL-car=yA4-yB3,T-back(26)yA4=yT-back+LCL-car2+WCL-car22sin⁡α+βFigure 10 shows, in the lane change process 2,CL-car andT-back generate a potential angle collision point 2; the potential collision position ofCL-car is at A3; its coordinate is (xA3,yA3).Figure 10
Potential angle collision point positions in different lane change processes betweenCL-car andT-back.


### 2.2.5. Determination of Actual Distance

According to the preceding and rear angular coordinates of obstacle vehicles and the coordinates of potential angle collision points of lane change vehicle, we calculate actual distance between lane change vehicle and obstacle vehicles during the two-stage lane change in four scenes. Figure11 shows that when yB1,P-front<yA1<yB2,P-front, the potential angle collision position ofCL-car is at A1, and the actual distance SP-front,CL-car between the two vehicles can be calculated by (27)SP-front,CL-car=xB2,P-front-xA1Similarly, actual distance between lane change vehicle and obstacle vehicles in other scenes can be obtained (see Table 2).Table 2
Actual distance during the two-stage lane change in four scenes.
Angle collision scene Angle collision point Conditions for angle collision point Actual distance Remarks CL-car and P-front ① yB1,P-front<yA1<yB2,P-front xB2,P-front-xA1 SP-front,CL-car ② y A 1 > y B 2 , P - f r o n t,yA2<yB2,P-front xB2,P-front-xA1→A2  CL-car and P-back ① yB3,P-back<yA3<yB4,P-back xA3-xB4,P-back SCL-car,P-back ② y A 3 > y B 4 , P - b a c k,yA2<yB4,P-back xA2→A3-xB4,P-back  CL-car and T-front ① y A 1 < y B 1 , T - f r o n t,yA4>yB1,T-front xB1,T-front-xA4→A1 ST-front,CL-car, ② yB2,T-front>yA1>yB1,T-front xB1,T-front-xA1  CL-car and T-back ① y A 3 < y B 3 , T - b a c k,yA4>yB3,T-back xA3→A4-xB3,T-back SCL-car,T-back ② yB4,T-back>yA3>yB3,T-back xA3-xB3,T-backFigure 11
Actual distance diagram.


## 2.2.1. Assumptions

This paper has made the following assumptions:( 1 ) T-back,P-back,T-front, andP-front drive straight on the respective roads without lane change, and the speed and acceleration are in a horizontal state.( 2 ) When the position of angle collision point is determined, lane change vehicles are regarded as a rectangle, and its coordinates are simplified to the geometric centre of the rectangle.


## 2.2.2. Lane Change Parameters and Coordinate Definition

Without loss of generality, modeling and analysis were carried out by taking the left lane change as an example. The lane change process was positioned by the absolute Cartesian coordinate system. The direction of the vehicle along the lane was defined as the positive direction of X-axis; vertical to the lane and upward were defined as the positive direction of Y-axis (see Figure3). The right edge line of the current lane is defined as X-axis, and Y-axis is located at the geometric centre of the end obstacle vehicle at the beginning of lane change. Then the origin of coordinates can be obtained (see Figure 3). After the definition is completed, the coordinate system does not change with the change of vehicle position during lane change.Figure 3
Definition of coordinate axes and coordinate origins.As shown in Figure4, the position of each vehicle is (xj,yj), with the unit ofm; vj is the speed of each vehicle, with the unit ofm/s; aj is the acceleration of each vehicle, with the unit of m/s2; Lj is the length of the vehicle, with the unit ofm; Wj is the vehicle width, where j∈[T-back, P-back, T-front, P-front, CL-car].Figure 4
Lane change parameters.Figure4 shows that the speed and acceleration of each vehicle are constantly changing during driving, so the start time of the lane change is defined to be zero, and the position of each vehicle can be expressed by(1)xj=x0j+∫v0j,x+∫aj,xdtdt,yj=y0j+∫v0j,y+∫aj,ydtdtwhere x0j represents the initial position of the vehicle j in the X direction; v0j,x is the initial speed of the vehicle j in the X direction; aj,x is the acceleration of the vehicle j in the X direction. Similarly, the meanings of y0j, v0j,y and aj,y are similar.When the lane change vehicleCL-car is driving, the heading angle is constantly changing. In Figure 5, α is the heading angle of vehicle; β is the shape parameter angle of vehicle.Figure 5
Angles of the lane change vehicle.The anglesα and β can be calculated using(2)α=arctan⁡vCL-car,yvCL-car,x=arctan⁡v0CL-car,y+∫aCL-car,ydtv0CL-car,x+∫aCL-car,xdt,β=arctan⁡WCL-carLCL-car


## 2.2.3. Calculation of the Preceding and Rear Angular Coordinates of Obstacle Vehicles

Figure6 shows the preceding and rear angles of obstacle vehicles that are B1,i,B2,i,B3,i,B4,i. And coordinates can be calculated from the position, length, and width of obstacle vehicles (see Table 1), wherein i∈[T-back,P-back,T-front,P-front].Table 1
Preceding and rear point coordinates of obstacle vehicles.
Preceding and rear angles of obstacle vehicles Abscissa Ordinate B 1 , i xi-Li2 y i - W i 2 B 2 , i xi-Li2 y i + W i 2  B 3 , i xi+Li2 y i - W i 2  B 4 , i xi+Li2 y i + W i 2Figure 6
Preceding and rear angles of obstacle vehicles.


## 2.2.4. Coordinate Calculation of Potential Angle Collision Points of Lane Change Vehicles

The preceding and rear angles of lane change vehicles are defined asA1,A2,A3,A4, and the potential angle collision points of lane change vehicles are calculated by combining B1,i,B2,i,B3,i,B4,i.( 1 ) In the lane change scene ofCL-car andP-front, the potential collision coordinates ofCL-car are calculated as follows.Figure7 shows that, in the lane change process 1,CL-car andP-front generate a potential angle collision point ①, and the potential collision position ofCL-car is at point A1. The coordinates are as follows:(3)xA1=xCL-car+LCL-car2+WCL-car22cos⁡α-β(4)yA1=yCL-car+LCL-car2+WCL-car22sin⁡α-βIn the lane change process 2,CL-car andP-front generate a potential angle collision point ②, and the potential collision position ofCL-car is located within A1→A2. The coordinates are(5)xA1→A2=xA2+ΔxCL-car,P-front(6)yA1→A2=yB2,P-frontwhere(7)xA2=xCL-car-LCL-car2+WCL-car22cos⁡α+β(8)ΔxCL-car,P-front=ΔyCL-car,P-frontcot⁡α(9)ΔyCL-car,P-front=yB2,P-front-yA2(10)yA2=yCL-car-LCL-car2+WCL-car22sin⁡α+β(2) In the lane change scene ofCL-car andP-back, the potential collision position coordinates ofCL-car are calculated as  follows.Figure 7
Potential angle collision point positions in different lane change processes betweenCL-carandP-front.Figure8 shows that in the lane change process 1,CL-car andP-back generate a potential angle collision point ①, and the potential collision position ofCL-car is at A3. The coordinates are(11)xA3=xCL-car-LCL-car2+WCL-car22cos⁡α-β(12)yA3=yCL-car-LCL-car2+WCL-car22sin⁡α-βAs shown in Figure 8, in the lane change process 2,CL-carandP-back generate a potential angle collision point ②, and the potential collision position ofCL-car is located within A2→A3. The coordinates are(13)xA2→A3=xA2-ΔxP-back,CL-car(14)yA2→A3=yB4,P-backwhere(15)ΔxP-back,CL-car=ΔyP-back,CL-cartan⁡α(16)ΔyP-back,CL-car=yB4,P-back-yA2(3) In the lane change scene ofCL-car andT-front, the potential collision position coordinates ofCL-car are calculated as  follows.Figure 8
Potential angle collision point positions in different lane change processes betweenCL-car andP-back.As shown in Figure9, in the lane change process 1,CL-car andT-frontgenerate a potential angle collision point ①, and the potential collision position ofCL-car is located within A4→A1. The coordinates are(17)xA4→A1=xA1-ΔxCL-car,T-front(18)yA4→A1=yB1,T-frontwhere(19)ΔxCL-car,T-front=ΔyCL-car,T-fronttan⁡α(20)ΔyCL-car,T-front=yB1,T-front-yA1In the lane change process 2,CL-car andT-front generate a potential angle collision point ②; the potential collision position ofCL-car is at A1; its coordinate is (xA1,yA1).Figure 9
Potential angle collision point positions in different lane change processes betweenCL-car andT-front.( 4 ) In the lane change scene ofCL-car andT-back, the potential collision position coordinates ofCL-car are calculated as  follows.Figure10 shows that, in the lane change process 1,CL-car andT-back generate a potential angle collision point ①, and the potential collision position ofCL-car is located within A3→A4. The coordinates are(21)xA3→A4=xA4-ΔxT-back,CL-car(22)yA3→A4=yB3,T-backwhere(23)xA4=xCL-car+LCL-car2+WCL-car22cos⁡α+β(24)ΔxT-back,CL-car=ΔyT-back,CL-carcot⁡α(25)ΔyT-back,CL-car=yA4-yB3,T-back(26)yA4=yT-back+LCL-car2+WCL-car22sin⁡α+βFigure 10 shows, in the lane change process 2,CL-car andT-back generate a potential angle collision point 2; the potential collision position ofCL-car is at A3; its coordinate is (xA3,yA3).Figure 10
Potential angle collision point positions in different lane change processes betweenCL-car andT-back.


## 2.2.5. Determination of Actual Distance

According to the preceding and rear angular coordinates of obstacle vehicles and the coordinates of potential angle collision points of lane change vehicle, we calculate actual distance between lane change vehicle and obstacle vehicles during the two-stage lane change in four scenes. Figure11 shows that when yB1,P-front<yA1<yB2,P-front, the potential angle collision position ofCL-car is at A1, and the actual distance SP-front,CL-car between the two vehicles can be calculated by (27)SP-front,CL-car=xB2,P-front-xA1Similarly, actual distance between lane change vehicle and obstacle vehicles in other scenes can be obtained (see Table 2).Table 2
Actual distance during the two-stage lane change in four scenes.
Angle collision scene Angle collision point Conditions for angle collision point Actual distance Remarks CL-car and P-front ① yB1,P-front<yA1<yB2,P-front xB2,P-front-xA1 SP-front,CL-car ② y A 1 > y B 2 , P - f r o n t,yA2<yB2,P-front xB2,P-front-xA1→A2  CL-car and P-back ① yB3,P-back<yA3<yB4,P-back xA3-xB4,P-back SCL-car,P-back ② y A 3 > y B 4 , P - b a c k,yA2<yB4,P-back xA2→A3-xB4,P-back  CL-car and T-front ① y A 1 < y B 1 , T - f r o n t,yA4>yB1,T-front xB1,T-front-xA4→A1 ST-front,CL-car, ② yB2,T-front>yA1>yB1,T-front xB1,T-front-xA1  CL-car and T-back ① y A 3 < y B 3 , T - b a c k,yA4>yB3,T-back xA3→A4-xB3,T-back SCL-car,T-back ② yB4,T-back>yA3>yB3,T-back xA3-xB3,T-backFigure 11
Actual distance diagram.


## 2.3. Minimum Safety Distance Model

For the assisted driving system, after measuring actual distance between lane change vehicle and obstacle vehicles, it is necessary to determine the corresponding minimum safety distance for safe lane change. Decision and early warning are made by comparison [21]. The minimum safe distance refers to the necessary separation distance between the rear and preceding vehicles to avoid collision.In lane change, there are two most unfavorable scenes: one is emergency braking of the preceding vehicle and the other is the speed of rear vehicle which is greater than that of preceding vehicle. For the first type, when preceding vehicle is parked under emergency braking, the rear vehicle should adjust its deceleration and speed to make the vehicle stop within the shortest distance for safety [22]. The second type is that, during lane change of continuous driving, the rear vehicle needs to decelerate to the same speed as preceding vehicle for safety.In view of the above-mentioned most unfavorable scenes, based on kinematics principle of vehicle, this paper established the minimum safety distance model between preceding and rear vehicles when the preceding vehicle is in emergency braking and the minimum safety distance model between preceding and rear vehicles when the rear vehicle is adjusted to the same speed as the preceding vehicle.


### 2.3.1. Minimum Safety Distance Model between the Preceding and Rear Vehicles during Emergency Braking of the Preceding Vehicle

The vehicle braking time consists of driver response time, brake coordination time, brake acceleration growth time, and continuous braking time [23], wherein vehicle tires, suspension, and ground have the impact on brake coordination time [24, 25]. The sum of driver response time and brake coordination time is generally 0.8-1.0 s [26, 27]. The deceleration build-up time of the preceding vehicle is generally 0.1 to 0.2 s, and tfront,c is the braking time of preceding vehicle, with the unit ofs. tback is the deceleration build-up time of the rear car, generally 0.1-0.2 s, and tback,c is the rear brake time, with the unit of s [28]. The braking principles of different vehicles are different [29, 30], and the maximum braking force is not exactly the same. The maximum average braking acceleration in the horizontal direction of the vehicle during emergency braking is determined as 7m/s2 [27, 31]. Since the driver of the rear vehicle takes action after the preceding vehicle brakes, the change of the braking acceleration of rear vehicle will be delayed relative to the preceding vehicle. When the minimum safety distance LBback,front is calculated, the acceleration changes of preceding and rear vehicles are shown in Figures 12 and 13.Figure 12
Braking deceleration changes of preceding vehicle.Figure 13
Braking deceleration changes of rear vehicle.Then the braking distanceLback of the rear vehicle is(28)Lback=vback,xtr1+tr2+tback2-124amaxback,xtback2+vback,x22amaxback,xwhere vback,x represents the speed of the rear vehicle in X direction; amaxback,x is the maximum braking acceleration of the rear vehicle in X direction.Similarly,(29)LCL-car,back=vCL-car,xtr1+tr2+tback2-124amaxCL-car,xtback2+vCL-car,x22amaxCL-car,x(30)LP-back=vP-back,xtr1+tr2+tback2-124amaxP-back,xtback2+vP-back,x22amaxP-back,x(31)LT-back=vT-back,xtr1+tr2+tback2-124amaxT-back,xtback2+vT-back,x22amaxT-back,xThe braking distance of the preceding vehicle is(32)Lfront=12vfront,xtfront-124amaxfront,xtfront2+vfront,x22amaxfront,xwhere vfront,x represents the speed of the preceding vehicle in X direction; amaxfront,x is the maximum braking acceleration of the preceding vehicle in X direction.Similarly,(33)LP-front=12vP-front,xtfront-124amaxP-front,xtfront2+vP-front,x22amaxP-front,x(34)LT-front=12vT-front,xtfront-124amaxT-front,xtfront2+vT-front,x22amaxT-front,x(35)LCL-car,front=12vCL-car,xtfront-124amaxCL-car,xtfront2+vCL-car,x22amaxCL-car,xThe minimum safety distance LBback,front under emergency braking of the preceding vehicle is(36)LBback,front=Lback-LfrontSimilarly,(37)LBCL-car,P-front=LCL-car,back-LP-front(38)LBCL-car,T-front=LCL-car,back-LT-front(39)LBP-back,CL-car=LP-back-LCL-car,front(40)LBT-back,CL-car=LT-back-LCL-car,front


### 2.3.2. Minimum Safety Distance Model between Preceding and Rear Vehicles When the Rear Vehicle Is Adjusted to the Same Speed as the Preceding Vehicle

If the speedvfront,x of the preceding vehicle is greater than the speed vback,x of the rear vehicle, the actual distance between the two vehicles increases with time, and no angle collision and rear-end collision occur. Therefore, the minimum safety distance LSback,front is zero [32]. If the speed vfront,x of the preceding vehicle is less than the speed vback,x of the rear vehicle, the minimum safe distance is(41)LSback,front=vback,x2-vfront,x22amaxback,xSimilarly,(42)LSCL-car,P-front=0vP-front,x≥vCL-car,xvCL-car,x2-vP-front,x22amaxCL-car,xvP-front,x<vCL-car,x(43)LSCL-car,T-front=0vT-front,x≥vCL-car,xvCL-car,x2-vT-front,x22amaxCL-car,xvT-front,x<vCL-car,x(44)LSP-back,CL-car=0vCL-car,x≥vP-back,xvP-back,x2-vCL-car,x22amaxP-back,xvCL-car,x<vP-back,x(45)LST-back,CL-car=0vCL-car,x≥vT-back,xvT-back,x2-vCL-car,x22amaxT-back,xvCL-car,x<vT-back,x


## 2.3.1. Minimum Safety Distance Model between the Preceding and Rear Vehicles during Emergency Braking of the Preceding Vehicle

The vehicle braking time consists of driver response time, brake coordination time, brake acceleration growth time, and continuous braking time [23], wherein vehicle tires, suspension, and ground have the impact on brake coordination time [24, 25]. The sum of driver response time and brake coordination time is generally 0.8-1.0 s [26, 27]. The deceleration build-up time of the preceding vehicle is generally 0.1 to 0.2 s, and tfront,c is the braking time of preceding vehicle, with the unit ofs. tback is the deceleration build-up time of the rear car, generally 0.1-0.2 s, and tback,c is the rear brake time, with the unit of s [28]. The braking principles of different vehicles are different [29, 30], and the maximum braking force is not exactly the same. The maximum average braking acceleration in the horizontal direction of the vehicle during emergency braking is determined as 7m/s2 [27, 31]. Since the driver of the rear vehicle takes action after the preceding vehicle brakes, the change of the braking acceleration of rear vehicle will be delayed relative to the preceding vehicle. When the minimum safety distance LBback,front is calculated, the acceleration changes of preceding and rear vehicles are shown in Figures 12 and 13.Figure 12
Braking deceleration changes of preceding vehicle.Figure 13
Braking deceleration changes of rear vehicle.Then the braking distanceLback of the rear vehicle is(28)Lback=vback,xtr1+tr2+tback2-124amaxback,xtback2+vback,x22amaxback,xwhere vback,x represents the speed of the rear vehicle in X direction; amaxback,x is the maximum braking acceleration of the rear vehicle in X direction.Similarly,(29)LCL-car,back=vCL-car,xtr1+tr2+tback2-124amaxCL-car,xtback2+vCL-car,x22amaxCL-car,x(30)LP-back=vP-back,xtr1+tr2+tback2-124amaxP-back,xtback2+vP-back,x22amaxP-back,x(31)LT-back=vT-back,xtr1+tr2+tback2-124amaxT-back,xtback2+vT-back,x22amaxT-back,xThe braking distance of the preceding vehicle is(32)Lfront=12vfront,xtfront-124amaxfront,xtfront2+vfront,x22amaxfront,xwhere vfront,x represents the speed of the preceding vehicle in X direction; amaxfront,x is the maximum braking acceleration of the preceding vehicle in X direction.Similarly,(33)LP-front=12vP-front,xtfront-124amaxP-front,xtfront2+vP-front,x22amaxP-front,x(34)LT-front=12vT-front,xtfront-124amaxT-front,xtfront2+vT-front,x22amaxT-front,x(35)LCL-car,front=12vCL-car,xtfront-124amaxCL-car,xtfront2+vCL-car,x22amaxCL-car,xThe minimum safety distance LBback,front under emergency braking of the preceding vehicle is(36)LBback,front=Lback-LfrontSimilarly,(37)LBCL-car,P-front=LCL-car,back-LP-front(38)LBCL-car,T-front=LCL-car,back-LT-front(39)LBP-back,CL-car=LP-back-LCL-car,front(40)LBT-back,CL-car=LT-back-LCL-car,front


## 2.3.2. Minimum Safety Distance Model between Preceding and Rear Vehicles When the Rear Vehicle Is Adjusted to the Same Speed as the Preceding Vehicle

If the speedvfront,x of the preceding vehicle is greater than the speed vback,x of the rear vehicle, the actual distance between the two vehicles increases with time, and no angle collision and rear-end collision occur. Therefore, the minimum safety distance LSback,front is zero [32]. If the speed vfront,x of the preceding vehicle is less than the speed vback,x of the rear vehicle, the minimum safe distance is(41)LSback,front=vback,x2-vfront,x22amaxback,xSimilarly,(42)LSCL-car,P-front=0vP-front,x≥vCL-car,xvCL-car,x2-vP-front,x22amaxCL-car,xvP-front,x<vCL-car,x(43)LSCL-car,T-front=0vT-front,x≥vCL-car,xvCL-car,x2-vT-front,x22amaxCL-car,xvT-front,x<vCL-car,x(44)LSP-back,CL-car=0vCL-car,x≥vP-back,xvP-back,x2-vCL-car,x22amaxP-back,xvCL-car,x<vP-back,x(45)LST-back,CL-car=0vCL-car,x≥vT-back,xvT-back,x2-vCL-car,x22amaxT-back,xvCL-car,x<vT-back,x


## 2.4. Definition of Early Warning Rules

According to the position of the potential angle collision point, we can obtain the actual distanceSback,front between the rear vehicle and preceding vehicle, the minimum safety distance LBback,front under the emergency braking of the preceding vehicle, and the minimum safety distance LSback,front between preceding and rear vehicles when the rear vehicle is adjusted to the same speed as the preceding vehicle. The comparison of Sback,front, LBback,front, and LSback,front divides the early warning into no early warning, mild early warning, and high early warning, and the colors of the early warning lights are green, yellow, and red, respectively. Specific definitions are shown in Table 3.Table 3
Introduction to early warning rules.
Early warning rule Condition Status description Display color No early warning Sback,front>LBback,front No need to adjust the speed  Mild early warning LSback,front<Sback,front≤LBback,front If the preceding vehicle does not take emergency braking, the distance between the vehicles meets the safety needs.  Severe early warning Sback,front≤LSback,front When the speed of the vehicle is not adjusted in time, a angle collision accident will occur.


## 3. Result and Discussion

### 3.1. Introduction to Data Sources

The model validation data in our research was obtained from vehicle travel video data of Interstate 80 in Emeryville, California, collected by the Berkeley Highway Laboratory, from 4:00 pm to 4:15 pm on April 13, 2005. The US federal highway management software NGSIM was used to transcribe video data at 10 frames per second and then obtain data such as the length, width, coordinates, acceleration, and other data of vehicle in each frame.


### 3.2. Model Verification Parameters and Preprocessing

This paper analyzed the lane change video of the vehicle 1078, which considers preceding and rear obstacle vehicles before and after lane change. Thus, vehicle 1078 is defined asCL-car, vehicle 1062 asP-front, vehicle 1077 asT-front, vehicle 1083 asT-back, and vehicle 1084 asP-back (see Figure 14). According to the definition of coordinates origin and coordinate axis in Section 2.2.2, at the beginning of lane change, the end obstacle vehicle is vehicle 1083, so Y-axis in coordinate system is located at the geometric centre of vehicle 1083, and X-axis is on the right edge of the lane where vehicle 1084 and vehicle 1062 are located. Table 4 shows the data involved in model validation.Table 4
Vehicle basic data.
Vehicle L j(m) W j(m) x 0 j(m) y 0 j(m) v 0 j , x(m/s) v 0 j , y(m/) CL-car 4.20624 2.22504 12.8784096 2.2856216 11.3011712 0 P-front 18.19656 2.59080 41.1053280 1.3757936 8.9631520 0 T-front 4.05384 2.07264 17.5287432 5.8575728 16.6254176 0 P-back 5.15112 1.79832 1.67365680 1.8046472 11.0150656 0 T-back 4.81584 2.10312 0 6.0776384 15.6151072 0Figure 14
Data extraction scene.Since the data is extracted by frame, it is necessary to fit the acceleration of vehicle. MATLAB is used to determine the acceleration of lane change vehicle and obstacle vehicles. Figures15–20 show the vehicle acceleration versus time curve, and Table 5 shows the corresponding fitting effect indicators.Table 5
Fitting effect indicators.
R-square Adjusted R-square RMSE a C L - c a r , x 0.993 0.982 0.221 a C L - c a r , y 0.991 0.978 0.014 a P - f r o n t , x 0.993 0.981 0.184 a T - f r o n t , x 0.990 0.974 0.245 a P - b a c k , x 0.994 0.985 0.156 a T - b a c k , x 0.994 0.983 0.162Figure 15
Acceleration curve ofCL-carin the positive direction of X-axis with time.Figure 16
Acceleration curve ofCL-carin the positive direction of Y-axis with time.Figure 17
Acceleration curve ofT-front in the positive direction of X-axis with time.Figure 18
Acceleration curve ofP-front in the positive direction of X-axis with time.Figure 19
Acceleration curve ofP-backin the positive direction of X-axis with time.Figure 20
Acceleration curve ofT-backin the positive direction of X-axis with time.


### 3.3. Early Warning Analysis

According to the established actual distance model, minimum safety distance model, and early warning rule definition, this paper drew the time variation curves of actual distance and the minimum safety distance between the lane change vehicle 1078 and its adjacent obstacle vehicles, and the curve of vehicles speed changes with time (see Figures21–24). Through comparing with video recording, the actual distance variation curve between vehicles is consistent with the change of the distance between vehicles in the video, and the validity of the model constructed in the work is verified. At the same time, according to early warning rules, warnings and prompts can be given to potential conflict vehicles. The early warning analysis is as follows.Figure 21
Distance and speed change betweenCL-car andP-front.Figure 22
Distance and speed change betweenP-back andCL-car.Figure 23
Distance and speed change betweenCL-car andT-front.Figure 24
Distance and speed change betweenT-back andCL-car.


#### 3.3.1. Early Warning Analysis ofCL-car andP-front

Figure21 shows in the early stage of the potential angle collision point ① that vCL-car,x(t) is always greater than vP-front,x(t), resulting in the decrease of SCL-car,P-front(t) and the increase of LSCL-car,P-front(t) and LBCL-car,P-front(t). Since SCL-car,P-front(t) is always greater than LBCL-car,P-front(t) and LSCL-car,P-front(t) at this stage, which satisfies the safety conditions,CL-carneeds no early warning.In the middle and late stages of the potential angle collision point ①,vCL-car,x(t) is always greater than vP-front,x(t), resulting in the decrease of SCL-car,P-front(t) and the increase of LSCL-car,P-front(t) and LBCL-car,P-front(t). SCL-car,P-front(t) is always greater than LSCL-car,P-front(t) and less than LBCL-car,P-front(t). Then,CL-car should be given the mild early warning and pay attention to the emergency brake of preceding vehicle with appropriate deceleration.During the potential angle collision point ②,vCL-car,x(t) is always greater than vP-front,x(t), but the change of potential angle collision point increases SCL-car,P-front(t) and decreases LSCL-car,P-front(t) and LBCL-car,P-front(t). Since SCL-car,P-front(t) is always greater than LSCL-car,P-front(t) and less than LBCL-car,P-front(t),CL-car should be given mild early warning and pay attention to the emergency braking of preceding vehicle with appropriate deceleration. When there is no potential angle collision point between two vehicles, the early warning is terminated.


#### 3.3.2. Early Warning Analysis ofCL-car andP-back

Figure22 shows that sinceCL-car travels to the left road edge line in the lane, there is potential angle collision point ② between vehicles instead of potential angle collision point ①. In the early stage of potential angle collision point ②, SP-back,CL-car(t) is less than LBP-back,CL-car(t) and greater than LSP-back,CL-car(t).P-back can be given mild early warning and pay attention to the emergency braking of preceding vehicle with appropriate deceleration. The speed vCL-car,x(t) of the lane change vehicleCL-car is basically in increase trend; there is a slight increase for vP-back,x(t) in the early period, but the sharp decline in the later period. The speed difference between the two vehicles gradually increases; the actual distance SP-back,CL-car(t) between the two vehicles increases; LBP-back,CL-car(t) is in the decrease trend; LSP-back,CL-car(t) is zero and remains unchanged. When SP-back,CL-car(t) is greater than LBP-back,CL-car(t), which satisfies the safety conditions, the early warning forP-back ends.


#### 3.3.3. Early Warning Analysis ofCL-car andT-front

Figure23 shows in the early and middle stage of potential angle collision point ① that since vT-front,x(t) is always larger than vCL-car,x(t), SCL-car,T-front(t) increases, and LSCL-car,T-front(t) is zero and remains unchanged. Because SCL-car,T-front(t) is less than the minimum safety distance LBCL-car,T-front(t) and greater than LSCL-car,T-front(t),CL-carshould be given the mild early warning and pay attention to the emergency brake of preceding vehicle with appropriate deceleration.In the late stage of the potential angle collision point ①,vT-front,x(t) decreases sharply with a large deceleration. Then SCL-car,T-front(t) becomes small, and LBCL-car,T-front(t) and LSCL-car,T-front(t) become large, reflecting the sensitivity and effectiveness of the minimum safety distance model in the work. Since SCL-car,T-front(t) is always greater than LSCL-car,T-front(t) and less than LBCL-car,T-front(t),CL-carshould be given the mild early warning at this stage and pay attention to the emergency braking of preceding with appropriate deceleration.In the early stage of the potential angle collision ②, for the sudden decrease ofvT-front,x(t) in the later period of potential angle collision point ①,CL-car reacts rapidly and then decelerates, and vT-front,x(t) increases with a larger acceleration and exceeds vCL-car,x(t). It makes SCL-car,T-front(t) increase, LBCL-car,T-front(t) decrease, and LSCL-car,T-front(t) become zero. Since SCL-car,T-front(t) is greater than LBCL-car,T-front(t), which meets safety conditions,CL-car needs no early warning. In the middle and late stages of the potential angle collision point②, since vT-front,x(t) is always larger than vCL-car,x(t), the early warning for theCL-car is changed from the mild early warning to no early warning.


#### 3.3.4. Early Warning Analysis ofCL-car andT-back

Figure24 shows thatCL-carappears as the preceding vehicle and the vehicle T-back as the rear vehicle. At the process of potential angle collision point ①, vT-back,x(t) is always greater than vCL-car,x(t), which decreases ST-back,CL-car(t) and increases LST-back,CL-car(t) and LBT-back,CL-car(t). Since ST-back,CL-car(t) is always greater than LST-back,CL-car(t) and less than LBT-back,CL-car(t).T-back should be given the mild early warning and pay attention to the emergency brake of the rear vehicle with appropriate deceleration.In the early stage of the potential angle collision ②,vT-back,x(t) is still higher than vCL-car,x(t). It makes ST-back,CL-car(t) even less than LST-back,CL-car(t).T-back should be given the severe early warning and decelerate immediately for safety.In the middle and late stage of the potential angle collision point ②, for the sudden decrease ofvCL-car,x(t) in the early stage,T-back perceives and then decelerates, which makes vT-back,x(t) less than vCL-car,x(t) and LST-back,CL-car(t) become zero. Then the two cars drive at a relatively stable speed difference, with the little change of ST-back,CL-car(t) and LBT-back,CL-car(t). Since ST-back,CL-car(t) is greater than LST-back,CL-car(t) and less than LBT-back,CL-car(t), T-back should be given mild early warning and pay attention to the front emergency brake with appropriate deceleration.


## 3.1. Introduction to Data Sources

The model validation data in our research was obtained from vehicle travel video data of Interstate 80 in Emeryville, California, collected by the Berkeley Highway Laboratory, from 4:00 pm to 4:15 pm on April 13, 2005. The US federal highway management software NGSIM was used to transcribe video data at 10 frames per second and then obtain data such as the length, width, coordinates, acceleration, and other data of vehicle in each frame.


## 3.2. Model Verification Parameters and Preprocessing

This paper analyzed the lane change video of the vehicle 1078, which considers preceding and rear obstacle vehicles before and after lane change. Thus, vehicle 1078 is defined asCL-car, vehicle 1062 asP-front, vehicle 1077 asT-front, vehicle 1083 asT-back, and vehicle 1084 asP-back (see Figure 14). According to the definition of coordinates origin and coordinate axis in Section 2.2.2, at the beginning of lane change, the end obstacle vehicle is vehicle 1083, so Y-axis in coordinate system is located at the geometric centre of vehicle 1083, and X-axis is on the right edge of the lane where vehicle 1084 and vehicle 1062 are located. Table 4 shows the data involved in model validation.Table 4
Vehicle basic data.
Vehicle L j(m) W j(m) x 0 j(m) y 0 j(m) v 0 j , x(m/s) v 0 j , y(m/) CL-car 4.20624 2.22504 12.8784096 2.2856216 11.3011712 0 P-front 18.19656 2.59080 41.1053280 1.3757936 8.9631520 0 T-front 4.05384 2.07264 17.5287432 5.8575728 16.6254176 0 P-back 5.15112 1.79832 1.67365680 1.8046472 11.0150656 0 T-back 4.81584 2.10312 0 6.0776384 15.6151072 0Figure 14
Data extraction scene.Since the data is extracted by frame, it is necessary to fit the acceleration of vehicle. MATLAB is used to determine the acceleration of lane change vehicle and obstacle vehicles. Figures15–20 show the vehicle acceleration versus time curve, and Table 5 shows the corresponding fitting effect indicators.Table 5
Fitting effect indicators.
R-square Adjusted R-square RMSE a C L - c a r , x 0.993 0.982 0.221 a C L - c a r , y 0.991 0.978 0.014 a P - f r o n t , x 0.993 0.981 0.184 a T - f r o n t , x 0.990 0.974 0.245 a P - b a c k , x 0.994 0.985 0.156 a T - b a c k , x 0.994 0.983 0.162Figure 15
Acceleration curve ofCL-carin the positive direction of X-axis with time.Figure 16
Acceleration curve ofCL-carin the positive direction of Y-axis with time.Figure 17
Acceleration curve ofT-front in the positive direction of X-axis with time.Figure 18
Acceleration curve ofP-front in the positive direction of X-axis with time.Figure 19
Acceleration curve ofP-backin the positive direction of X-axis with time.Figure 20
Acceleration curve ofT-backin the positive direction of X-axis with time.


## 3.3. Early Warning Analysis

According to the established actual distance model, minimum safety distance model, and early warning rule definition, this paper drew the time variation curves of actual distance and the minimum safety distance between the lane change vehicle 1078 and its adjacent obstacle vehicles, and the curve of vehicles speed changes with time (see Figures21–24). Through comparing with video recording, the actual distance variation curve between vehicles is consistent with the change of the distance between vehicles in the video, and the validity of the model constructed in the work is verified. At the same time, according to early warning rules, warnings and prompts can be given to potential conflict vehicles. The early warning analysis is as follows.Figure 21
Distance and speed change betweenCL-car andP-front.Figure 22
Distance and speed change betweenP-back andCL-car.Figure 23
Distance and speed change betweenCL-car andT-front.Figure 24
Distance and speed change betweenT-back andCL-car.


### 3.3.1. Early Warning Analysis ofCL-car andP-front

Figure21 shows in the early stage of the potential angle collision point ① that vCL-car,x(t) is always greater than vP-front,x(t), resulting in the decrease of SCL-car,P-front(t) and the increase of LSCL-car,P-front(t) and LBCL-car,P-front(t). Since SCL-car,P-front(t) is always greater than LBCL-car,P-front(t) and LSCL-car,P-front(t) at this stage, which satisfies the safety conditions,CL-carneeds no early warning.In the middle and late stages of the potential angle collision point ①,vCL-car,x(t) is always greater than vP-front,x(t), resulting in the decrease of SCL-car,P-front(t) and the increase of LSCL-car,P-front(t) and LBCL-car,P-front(t). SCL-car,P-front(t) is always greater than LSCL-car,P-front(t) and less than LBCL-car,P-front(t). Then,CL-car should be given the mild early warning and pay attention to the emergency brake of preceding vehicle with appropriate deceleration.During the potential angle collision point ②,vCL-car,x(t) is always greater than vP-front,x(t), but the change of potential angle collision point increases SCL-car,P-front(t) and decreases LSCL-car,P-front(t) and LBCL-car,P-front(t). Since SCL-car,P-front(t) is always greater than LSCL-car,P-front(t) and less than LBCL-car,P-front(t),CL-car should be given mild early warning and pay attention to the emergency braking of preceding vehicle with appropriate deceleration. When there is no potential angle collision point between two vehicles, the early warning is terminated.


### 3.3.2. Early Warning Analysis ofCL-car andP-back

Figure22 shows that sinceCL-car travels to the left road edge line in the lane, there is potential angle collision point ② between vehicles instead of potential angle collision point ①. In the early stage of potential angle collision point ②, SP-back,CL-car(t) is less than LBP-back,CL-car(t) and greater than LSP-back,CL-car(t).P-back can be given mild early warning and pay attention to the emergency braking of preceding vehicle with appropriate deceleration. The speed vCL-car,x(t) of the lane change vehicleCL-car is basically in increase trend; there is a slight increase for vP-back,x(t) in the early period, but the sharp decline in the later period. The speed difference between the two vehicles gradually increases; the actual distance SP-back,CL-car(t) between the two vehicles increases; LBP-back,CL-car(t) is in the decrease trend; LSP-back,CL-car(t) is zero and remains unchanged. When SP-back,CL-car(t) is greater than LBP-back,CL-car(t), which satisfies the safety conditions, the early warning forP-back ends.


### 3.3.3. Early Warning Analysis ofCL-car andT-front

Figure23 shows in the early and middle stage of potential angle collision point ① that since vT-front,x(t) is always larger than vCL-car,x(t), SCL-car,T-front(t) increases, and LSCL-car,T-front(t) is zero and remains unchanged. Because SCL-car,T-front(t) is less than the minimum safety distance LBCL-car,T-front(t) and greater than LSCL-car,T-front(t),CL-carshould be given the mild early warning and pay attention to the emergency brake of preceding vehicle with appropriate deceleration.In the late stage of the potential angle collision point ①,vT-front,x(t) decreases sharply with a large deceleration. Then SCL-car,T-front(t) becomes small, and LBCL-car,T-front(t) and LSCL-car,T-front(t) become large, reflecting the sensitivity and effectiveness of the minimum safety distance model in the work. Since SCL-car,T-front(t) is always greater than LSCL-car,T-front(t) and less than LBCL-car,T-front(t),CL-carshould be given the mild early warning at this stage and pay attention to the emergency braking of preceding with appropriate deceleration.In the early stage of the potential angle collision ②, for the sudden decrease ofvT-front,x(t) in the later period of potential angle collision point ①,CL-car reacts rapidly and then decelerates, and vT-front,x(t) increases with a larger acceleration and exceeds vCL-car,x(t). It makes SCL-car,T-front(t) increase, LBCL-car,T-front(t) decrease, and LSCL-car,T-front(t) become zero. Since SCL-car,T-front(t) is greater than LBCL-car,T-front(t), which meets safety conditions,CL-car needs no early warning. In the middle and late stages of the potential angle collision point②, since vT-front,x(t) is always larger than vCL-car,x(t), the early warning for theCL-car is changed from the mild early warning to no early warning.


### 3.3.4. Early Warning Analysis ofCL-car andT-back

Figure24 shows thatCL-carappears as the preceding vehicle and the vehicle T-back as the rear vehicle. At the process of potential angle collision point ①, vT-back,x(t) is always greater than vCL-car,x(t), which decreases ST-back,CL-car(t) and increases LST-back,CL-car(t) and LBT-back,CL-car(t). Since ST-back,CL-car(t) is always greater than LST-back,CL-car(t) and less than LBT-back,CL-car(t).T-back should be given the mild early warning and pay attention to the emergency brake of the rear vehicle with appropriate deceleration.In the early stage of the potential angle collision ②,vT-back,x(t) is still higher than vCL-car,x(t). It makes ST-back,CL-car(t) even less than LST-back,CL-car(t).T-back should be given the severe early warning and decelerate immediately for safety.In the middle and late stage of the potential angle collision point ②, for the sudden decrease ofvCL-car,x(t) in the early stage,T-back perceives and then decelerates, which makes vT-back,x(t) less than vCL-car,x(t) and LST-back,CL-car(t) become zero. Then the two cars drive at a relatively stable speed difference, with the little change of ST-back,CL-car(t) and LBT-back,CL-car(t). Since ST-back,CL-car(t) is greater than LST-back,CL-car(t) and less than LBT-back,CL-car(t), T-back should be given mild early warning and pay attention to the front emergency brake with appropriate deceleration.


## 3.3.1. Early Warning Analysis ofCL-car andP-front

Figure21 shows in the early stage of the potential angle collision point ① that vCL-car,x(t) is always greater than vP-front,x(t), resulting in the decrease of SCL-car,P-front(t) and the increase of LSCL-car,P-front(t) and LBCL-car,P-front(t). Since SCL-car,P-front(t) is always greater than LBCL-car,P-front(t) and LSCL-car,P-front(t) at this stage, which satisfies the safety conditions,CL-carneeds no early warning.In the middle and late stages of the potential angle collision point ①,vCL-car,x(t) is always greater than vP-front,x(t), resulting in the decrease of SCL-car,P-front(t) and the increase of LSCL-car,P-front(t) and LBCL-car,P-front(t). SCL-car,P-front(t) is always greater than LSCL-car,P-front(t) and less than LBCL-car,P-front(t). Then,CL-car should be given the mild early warning and pay attention to the emergency brake of preceding vehicle with appropriate deceleration.During the potential angle collision point ②,vCL-car,x(t) is always greater than vP-front,x(t), but the change of potential angle collision point increases SCL-car,P-front(t) and decreases LSCL-car,P-front(t) and LBCL-car,P-front(t). Since SCL-car,P-front(t) is always greater than LSCL-car,P-front(t) and less than LBCL-car,P-front(t),CL-car should be given mild early warning and pay attention to the emergency braking of preceding vehicle with appropriate deceleration. When there is no potential angle collision point between two vehicles, the early warning is terminated.


## 3.3.2. Early Warning Analysis ofCL-car andP-back

Figure22 shows that sinceCL-car travels to the left road edge line in the lane, there is potential angle collision point ② between vehicles instead of potential angle collision point ①. In the early stage of potential angle collision point ②, SP-back,CL-car(t) is less than LBP-back,CL-car(t) and greater than LSP-back,CL-car(t).P-back can be given mild early warning and pay attention to the emergency braking of preceding vehicle with appropriate deceleration. The speed vCL-car,x(t) of the lane change vehicleCL-car is basically in increase trend; there is a slight increase for vP-back,x(t) in the early period, but the sharp decline in the later period. The speed difference between the two vehicles gradually increases; the actual distance SP-back,CL-car(t) between the two vehicles increases; LBP-back,CL-car(t) is in the decrease trend; LSP-back,CL-car(t) is zero and remains unchanged. When SP-back,CL-car(t) is greater than LBP-back,CL-car(t), which satisfies the safety conditions, the early warning forP-back ends.


## 3.3.3. Early Warning Analysis ofCL-car andT-front

Figure23 shows in the early and middle stage of potential angle collision point ① that since vT-front,x(t) is always larger than vCL-car,x(t), SCL-car,T-front(t) increases, and LSCL-car,T-front(t) is zero and remains unchanged. Because SCL-car,T-front(t) is less than the minimum safety distance LBCL-car,T-front(t) and greater than LSCL-car,T-front(t),CL-carshould be given the mild early warning and pay attention to the emergency brake of preceding vehicle with appropriate deceleration.In the late stage of the potential angle collision point ①,vT-front,x(t) decreases sharply with a large deceleration. Then SCL-car,T-front(t) becomes small, and LBCL-car,T-front(t) and LSCL-car,T-front(t) become large, reflecting the sensitivity and effectiveness of the minimum safety distance model in the work. Since SCL-car,T-front(t) is always greater than LSCL-car,T-front(t) and less than LBCL-car,T-front(t),CL-carshould be given the mild early warning at this stage and pay attention to the emergency braking of preceding with appropriate deceleration.In the early stage of the potential angle collision ②, for the sudden decrease ofvT-front,x(t) in the later period of potential angle collision point ①,CL-car reacts rapidly and then decelerates, and vT-front,x(t) increases with a larger acceleration and exceeds vCL-car,x(t). It makes SCL-car,T-front(t) increase, LBCL-car,T-front(t) decrease, and LSCL-car,T-front(t) become zero. Since SCL-car,T-front(t) is greater than LBCL-car,T-front(t), which meets safety conditions,CL-car needs no early warning. In the middle and late stages of the potential angle collision point②, since vT-front,x(t) is always larger than vCL-car,x(t), the early warning for theCL-car is changed from the mild early warning to no early warning.


## 3.3.4. Early Warning Analysis ofCL-car andT-back

Figure24 shows thatCL-carappears as the preceding vehicle and the vehicle T-back as the rear vehicle. At the process of potential angle collision point ①, vT-back,x(t) is always greater than vCL-car,x(t), which decreases ST-back,CL-car(t) and increases LST-back,CL-car(t) and LBT-back,CL-car(t). Since ST-back,CL-car(t) is always greater than LST-back,CL-car(t) and less than LBT-back,CL-car(t).T-back should be given the mild early warning and pay attention to the emergency brake of the rear vehicle with appropriate deceleration.In the early stage of the potential angle collision ②,vT-back,x(t) is still higher than vCL-car,x(t). It makes ST-back,CL-car(t) even less than LST-back,CL-car(t).T-back should be given the severe early warning and decelerate immediately for safety.In the middle and late stage of the potential angle collision point ②, for the sudden decrease ofvCL-car,x(t) in the early stage,T-back perceives and then decelerates, which makes vT-back,x(t) less than vCL-car,x(t) and LST-back,CL-car(t) become zero. Then the two cars drive at a relatively stable speed difference, with the little change of ST-back,CL-car(t) and LBT-back,CL-car(t). Since ST-back,CL-car(t) is greater than LST-back,CL-car(t) and less than LBT-back,CL-car(t), T-back should be given mild early warning and pay attention to the front emergency brake with appropriate deceleration.


## 4. Conclusions and Outlook

The innovation of this paper is that we analyzed more comprehensive lane change scenes, and, based on this, position functions of potential angle collision points on the sides, front, and rear of lane change vehicles and obstacle vehicles were studied in this paper. At the same time, different early warning rules were formulated according to the relationship between the actual distance and the two kinds of minimum safety distances.Under assumptions, this paper can use the constructed models to judge the potential collision risk of different degrees in the process of changing lanes in a timely and effective manner. The driver’s behavior can be guided in real time according to early warning rules. Models and early warning rules established in this research could be applied to smart cars equipped with IOT (Internet of Things). The parameters such as speed and acceleration required by models were obtained through data acquisition and data interaction. Then, through the calculation and analysis of vehicle intelligent system, drivers were assisted to complete the safe lane change. The research content of this paper had good application value and practical significance for the development of advanced vehicle active safety system and the reduction of traffic accidents caused by lane change.Models in this paper are mainly applicable to the analysis and early warning of lane change in long straight sections and longitudinal slope sections. Therefore, in the follow-up research, it is necessary to analyze the flat and curved sections. At the same time, the follow-up research will consider the influence of driver’s psychological role and the external environment on the minimum safe distance model and then modify the model.



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*Source: 1281425-2019-06-27.xml*