# Research on Lane-Change Decision and Planning in Multilane Expressway Scenarios for Autonomous Vehicles

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## Abstract

**:**

## 1. Introduction

## 2. Considering the Driving Decisions for Multiple Lanes

#### 2.1. Willingness to Change Lanes Based on Fuzzy Theory

_{v}is defined as shown in Equation (1):

_{t}is the speed of the self-driving car, v

_{c}is the desired speed, and v

_{z}is the vehicle in front’s speed.

_{safe}is the desired safe distance, it is the maximum acceleration of the self-driving vehicle, take a

_{t}as 4 m/s, a

_{z}is the maximum deceleration of the front vehicle, take a

_{z}as −5 m/s, v

_{rel}is the relative speed of the self-driving vehicle and the front vehicle, t

_{1}and t

_{2}are the system reaction and braking delay time, respectively, take t

_{1}as 0.1 s and t

_{2}as 0.5 s, d is the minimum stopping distance, take d as 5 m.

_{D}as shown in Equation (3):

_{h}was positively correlated with ψ

_{v}and negatively correlated with ψ

_{D}. The fuzzy rules are shown in Table 1.

#### 2.2. Adjacent Lane Safety Posture Determination

#### 2.2.1. Classification of Surrounding Vehicles

_{m}) and (i, r

_{m}) denote the mth vehicle behind the vehicle and the mth vehicle in front of the vehicle in lane i, respectively. The range of lateral distances Δe between the self-driving vehicle and the surrounding vehicles can be calculated by Equation (4):

_{ov}, d

_{v}, and d

_{1}denote the width of the other vehicle, the width of the self-driving vehicle, and the width of the lane, respectively.

#### 2.2.2. Division of Surrounding Vehicle Behavior

_{0}is the vehicle acceleration at the initial moment and r

_{0}is the rate of change of the vehicle heading angle at the initial moment. v

_{0}and ψ

_{0}denote the predicted vehicle velocity and the heading angle at the initial moment, respectively.

_{xt}and v

_{yt}are the vehicle’s velocity in longitudinal and transverse directions, respectively, at the prediction moment t.

_{0}and y

_{0}, respectively, and the integration of the velocity v gives the vehicle’s trajectory at the future moment t, as shown in Equation (7):

_{n}(i,j

_{n}), as shown in Equation (9):

#### 2.2.3. Division of Surrounding Vehicle Behavior and External Factors

- There needs to be more space in the adjacent lane for the self-driving car to change lanes;
- There is sufficient space in the adjacent lane to make a lane change and no vehicles in the second to adjacent lane need to be considered for lateral movement;
- There is sufficient space in the adjacent lane for a lane change and the vehicle in front of the vehicle in the second to adjacent lane needs to consider lateral movements;
- There is sufficient space in the adjacent lane for a lane change and the vehicle behind the vehicle in the second to adjacent lane needs to be considered for lateral movement;
- There is sufficient space in the adjacent lane for a lane change and the vehicles in front of and behind the vehicle in the second to adjacent lane need to be considered for lateral movement.

- If there is no space for a lane change in the adjacent lane on the left, the safety level of the target lane is recorded as 1;
- If there is space to change lanes in the adjacent lane on the left and an associated vehicle is changing lanes into the target lane, the safety level of the target lane is recorded as 2;
- If there is space for a lane change in the adjacent lane on the left and the associated vehicle is in a lane departure, the safety level of the target lane is recorded as 3;
- If there is space to change lanes in the adjacent lane on the left and there is no associated vehicle in a lane departure, the safety level of the target lane is recorded as 4;

_{L}represents the safety level rating of the adjacent lane on the left side of the ego vehicle. Similarly, D

_{R}denotes the rating of the adjacent lane on the right side. A higher value of the safety level parameter D

_{L}or D

_{R}indicates a safer surrounding environment for the ego vehicle.

_{h}is derived, a judgement in made to determine whether the vehicle needs to change lanes. A D

_{L}/D

_{R}value of 4 means that the left and right adjacent lanes are safe for lane changing, whereas a D

_{L}/D

_{R}value of 3 means that a lane change is required based on the driver’s style. No lane change is required if the driver style parameter D

_{dr}value is 0. If the value of D

_{dr}is 1, the next step is to determine if the associated vehicle has not changed lanes, then there is room for the self-driving vehicle to change lanes. Otherwise, no lane change is allowed. The fuzzy control table can set the driver style in the fuzzy lane-change intention controller. To ensure the safety of the self-driving vehicle throughout the lane change process, the vehicle’s safety posture monitoring of the target lane is permanently active.

## 3. Intelligent Vehicle Trajectory Planning and Control

#### 3.1. Intelligent Lane-Change Trajectory Planning

_{f}is vehicle longitudinal position, V

_{x}

_{,f}is velocity, and a

_{x}is acceleration; the result can be obtained by solving the 12 linearly independent equations.

_{C}is the moment when the vehicle leaves the current lane; x

_{AV}, L

_{AV}, and W

_{AV}denote the longitudinal position, length, and width of the self-driving vehicle, respectively; X

_{DA}and L

_{DA}denote the longitudinal position and length of the vehicle in front, respectively; and ψ is the heading angle of the self-driving vehicle, take L

_{DA}= L

_{AV}.

_{x}

_{,AV}and V

_{x}

_{,DA}denote the longitudinal velocity of the self-driving vehicle AV and the preceding vehicle DA, respectively; b

_{AV}

_{,max}and b

_{DA}

_{,max}are the maximum acceleration values for AV and DA, respectively; and Ts is the reaction time.

_{f}, the velocity V

_{x}

_{,f}, and the acceleration a

_{x}.

#### 3.1.1. Equally Spaced Sampling

_{min}, and the longest lane change time, T

_{max}, and then optimizes the trajectory according to the different lane change times. The sampling time during the lane change is shown in Equation (20):

#### 3.1.2. Boundary Conditions

#### 3.1.3. Cost Function and Its Optimal Solution

_{x}and j

_{y}denote the longitudinal and lateral accelerations of the self-driving car, respectively, and w

_{x}and w

_{y}are the weights.

_{k}to determine the trajectory of the lane change.

#### 3.2. Stability-Based Trajectory Tracking Control

#### 3.2.1. Vehicle Dynamics Model

_{x}and v

_{y}are denoted as the longitudinal and lateral velocities of the vehicle body’s center, m is the vehicle’s center of gravity mass, m

_{s}is the mass on the springs, γ is the vehicle transverse angular velocity, φ is the vehicle heading angle, β is the vehicle center of mass lateral deflection angle, □

_{f}is the front wheel turning angle, I

_{x}and I

_{z}denote the body rotational inertia around the x and z axes, respectively, L

_{f}and L

_{r}are the distances from the vehicle’s center of gravity to the front and rear axles, respectively, Tr and h are the wheelbase and center-of-mass height, F

_{yf}and F

_{yr}are the lateral forces of the front and rear wheels, and ϕ is the lateral camber angle of the vehicle. It is the curvature of the road. K

_{ϕ}and D

_{ϕ}are the vehicle lateral stiffness coefficient and damping coefficient, respectively, and e

_{y}and e

_{φ}are the lateral deviations and heading angle deviation of the vehicle from the reference path, respectively.

_{y}is the shape factor, B

_{y}is the stiffness factor, D

_{y}is the peak value, and E

_{y}is the curvature value; the above parameters are related to the magnitude of the tire’s droop force. α is the tire slip angle. μ is the road adhesion force. To obtain the real-time nature of the calculations, the magic tire model was linearized; the results are shown in Equation (26):

#### 3.2.2. Vehicle Stability Constraints

- Vehicle Lateral Stability Constraint

_{x}is 98 km/h and μ is 0.85, the phase-plane diagrams for front wheel steering angles □

_{f}of 0 rad, −0.12 rad, −0.20 rad, and −0.31 rad are shown in Figure 7. In the diagrams, the balance points are indicated by red dots and the saddle points by red triangles.

_{s}

_{,r}is the maximum lateral deflection angle of the front wheels.

_{x}of the vehicle remains constant, a stability boundary for the lateral stability of the vehicle can be derived by considering the maximum/minimum transverse sway speed near the saddle point, the maximum/minimum range of lateral deflection angles determined for the front and rear wheels, and the maximum/minimum stable steering angle of the front wheels at the current vehicle speed. This stability domain can be expressed by the inequality shown in Equation (30):

- 2.
- Vehicle Lateral Stability Constraint

_{ZMP}of the zero-moment point concerning the wheelbase is shown in Equation (32):

_{ZMP}can be expressed in the form shown in Equation (33):

#### 3.3. Model Predictive Controller Design

_{k}, is taken as the control input to the vehicle to complete the control of the vehicle.

## 4. Simulation Experiments

#### 4.1. Follow the Driving Conditions

_{safe}and is therefore a dangerous traveling distance. At the same time, the longitudinal distance between the vehicle and vehicle 2 and vehicle 3 is so close that it is impossible to change lanes. In this case, the autocar can only brake and slow down, using the speed and relative longitudinal distance of vehicle 1 as a reference for following the vehicle. At 7 s, the speed of the self-driving car is equal to the speed of vehicle 1, at which point the distance between the two vehicles remains at around 71 m, as shown in Figure 11b. Until the end of 10 s, there is no rear-end collision between the self-driving vehicle and vehicle 1 and the minimum safe distance model L

_{f}≥ D

_{safe}is satisfied, as shown in Figure 11c.

^{2}, which does not affect the smoothness of the vehicle too much and also satisfies the minimum safe distance model. In addition, the speed tracking controller can track the reference speed in time. Figure 12c reflects the safety level of the adjacent lane of the self-propelled vehicle, which is always 1, as there is no space for lane changing on either side.

#### 4.2. Constant Speed Lane-Change Conditions

_{f}between the car and the car in front of it is equal to the minimum safe distance D

_{safe}. As there is no space to change lanes on the right side of the car, the car can only change lanes to the left. After 7.5 s, the car has completed the lane change and enters lane 2. According to Figure 13c, the relative longitudinal distance between the car and the car before and after the lane change is greater than the minimum safe distance D

_{safe}, indicating that the planned lane-change trajectory meets the obstacle avoidance requirements. According to Figure 13d, the main car completed the entire lane change process.

#### 4.3. Simultaneous Lane-Change Conditions

_{f}of car 1 meet the minimum safety distance requirement of Lr and L

_{f}≤ D

_{safe}, the safety level of the surrounding adjacent lane is 2; therefore, the safety conditions for lane change are not met. Therefore, the self-driving car decides to slow down and maintain the current lane until it is equal to the speed of the one car ahead. When the third car fully enters the second lane, the safety level of the adjacent lane of the self-driving car will all change to 1. By the 7th second, the longitudinal relative distance L

_{f}between the self-driving car and the third car reaches a safe distance and the safety level on the left side of the self-driving car is raised to 4. At this point, the self-driving car changes lanes to overtake the first car and enters the second lane; the safety level of the adjacent lane of the self-driving car is shown in Figure 16d. The intermediate process of lane change is shown in Figure 15c. According to Figure 15d, the main car completed the entire lane change process.

#### 4.4. Slow Lane Change

_{safe}, whereas the safety level of lane 1 on the left of the car is raised to 3 due to car 2 deviating from the lane and the safety level of the lane on the right is 3. Therefore, the decision level determines that it cannot overtake the car and the car brakes to slow down. Car 2 completes the lane change into lane 2 at the 4th second, at which point the safety level on the left side of the car reverts to 1. The car cruises until the 10th second with the front car’s speed as the reference speed, at which point the right lane meets the lane-change condition and the safety level is raised to 4, whereas the safety level on the left side continues to remain at 1 due to car 2.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Vehicle Relations with Surrounding Vehicles. (purple represents the current vehicle, blue represents the same lane vehicle, red represents adjacent lane vehicles, and yellow represents adjacent second lane vehicles. The vehicle to the left of the red dotted line is after the longitudinal position of the main car. For the convenience of illustration, the vehicle is marked and its designation is shown in the figure.)

**Figure 7.**Phase plan under different front wheel angles, which are (

**a**) 0 rad, (

**b**) 0.12 rad, (

**c**) 0.20 rad and (

**d**) 0.31 rad.

**Figure 8.**Stability boundary under different front wheel angles, which are (

**a**) 0 rad, (

**b**) 0.12 rad, (

**c**) 0.20 rad and (

**d**) 0.31 rad, at 98 km/h.

**Figure 11.**Follow the driving conditions. The subfigure (

**a**) donates the initial status of the simulation results. The subfigure (

**b**) donates the intermediate process of the simulation results. The subfigure (

**c**) donates the end status of the simulation results.

**Figure 12.**Simulation results for following the driving conditions. The subfigures (

**a**–

**e**) donate the change of intention, safety rating, relative longitudinal distance, velocity and acceleration, respectively.

**Figure 13.**Constant speed lane change condition. The subfigure (

**a**) donates the initial status of the simulation results. The subfigures (

**b**,

**c**) donate the intermediate process of the simulation results. The subfigure (

**d**) donates the end status of the simulation results.

**Figure 14.**Simulation results for the constant speed lane change conditions. Simulation results for constant speed lane change conditions have been given as Figure 14. The subfigures (

**a**–

**e**) donate the change of intention, safety rating, relative longitudinal distance, lateral position and yaw angel and yaw rate, respectively.

**Figure 15.**Simultaneous lane change condition. The subfigure (

**a**) donates the initial status of the simulation results. The subfigures (

**b**,

**c**) donate the intermediate process of the simulation results. The subfigure (

**d**) donates the end status of the simulation results.

**Figure 16.**Simulation results for simultaneous lane change conditions. Simulation results for simultaneous lane change conditions have been given as Figure 16. The subfigures (

**a**–

**e**) donate the change of intention, safety rating, relative longitudinal distance, velocity and lateral position, respectively.

**Figure 17.**Slow lane change conditions. The subfigure (

**a**) donates the initial status of the simulation results. The subfigures (

**b**–

**d**) donate the intermediate process of the simulation results. The subfigure (

**e**) donates the end status of the simulation results.

**Figure 18.**Simulation results of slow lane change conditions. Simulation results for slow lane change conditions have been given as Figure 18. The subfigures (

**a**–

**e**) donate the change of intention, safety rating, relative longitudinal distance, velocity and lateral position, respectively.

ψ_{D} | NB | NM | NS | ZO | PS | PM | PB | |
---|---|---|---|---|---|---|---|---|

ψ_{v} | ||||||||

NB | NS | NS | NM | NM | NB | NB | NB | |

NM | NS | NS | NM | NM | NM | NB | NB | |

NS | ZO | ZO | NS | NS | NS | NM | NM | |

ZO | PM | PM | PS | PS | ZO | NS | NM | |

PS | PM | PM | PS | PS | ZO | ZO | NS | |

PM | PB | PB | PM | PM | PS | ZO | NS | |

PB | PB | PB | PB | PM | PM | PS | PS |

Lane-Changing Willingness Value (ψ_{h}) | Vehicle Lane-Changing Decision |
---|---|

0.71 < ψ_{h} ≤ 0.51 | no lane change |

0.51 < ψ_{h} ≤ 0.71 | waiting for lane change |

0.71 < ψ_{h} ≤ 1.00 | executing lane change |

Factor | Correction Factor | The Range of Real Factor Value |
---|---|---|

Rain | q_{rain} ∈ [0, 1, 2, 3] | 0~50 (mm/h) |

Wind | q_{wind} ∈ [0, 1, 2, 3] | 0~12 (Beaufort scale) |

Fog | q_{fog} ∈ [0, 1, 2, 3] | 0~6.2 (mile) |

Road | q_{road} ∈ [0, 1, 2, 3] | Dry/Damp/Stagnant water/Snow and ice cover/Muddy |

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## Share and Cite

**MDPI and ACS Style**

Tang, C.; Pan, L.; Xia, J.; Fan, S.
Research on Lane-Change Decision and Planning in Multilane Expressway Scenarios for Autonomous Vehicles. *Machines* **2023**, *11*, 820.
https://doi.org/10.3390/machines11080820

**AMA Style**

Tang C, Pan L, Xia J, Fan S.
Research on Lane-Change Decision and Planning in Multilane Expressway Scenarios for Autonomous Vehicles. *Machines*. 2023; 11(8):820.
https://doi.org/10.3390/machines11080820

**Chicago/Turabian Style**

Tang, Chuanyin, Lv Pan, Jifeng Xia, and Shi Fan.
2023. "Research on Lane-Change Decision and Planning in Multilane Expressway Scenarios for Autonomous Vehicles" *Machines* 11, no. 8: 820.
https://doi.org/10.3390/machines11080820