# A Hierarchical Lane-Changing Trajectory Planning Method Based on the Least Action Principle

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Trajectory Planning Framework

## 3. Hierarchical Lane-Changing Trajectory Planning Method

#### 3.1. Generation of Candidate Lane-Changing Trajectories

#### 3.1.1. Coordinate Transformation

#### 3.1.2. Lane-Changing Path Planning Based on B-Spline Curves

- (1)
- Selection of a feasible area for the lane-changing endpoint

- (2)
- Generation of candidate lane-changing paths

- (3)
- Collision checking for static obstacle

#### 3.1.3. Velocity Planning Using DP and QP

- (1)
- Establishment of temporal and spatial graphs

- 1.
- Trajectory prediction of surrounding vehicles

- 2.
- Potential conflict detection for a lane-changing path

- 3.
- Discretization of the temporal and spatial graph

- (2)
- Velocity generation and smoothing

- 1.
- Velocity generation based on DP

- 2.
- Velocity smoothing based on QP

- (3)
- Curvature checking of a lane-changing trajectory

#### 3.2. Decision for Lane-Changing Target Trajectory

#### 3.2.1. Definition of Trajectory Performance Evaluation Function

- (1)
- Dynamic driving risk modeling based on the DSF

- (2)
- Definition of the evaluation function based on the LAP

#### 3.2.2. Selection of Lane-Changing Trajectory

#### 3.2.3. Collision Checking for Moving Obstacles

## 4. Simulation Analysis and Discussion

#### 4.1. Simulation Scenario 1: Lane-Changing and Following

^{2}, moving with constant acceleration. The requirements for the lane-changing process are to avoid collision with the vehicle SF and maintain an appropriate following distance with the leading vehicle, LF, on the target lane. The vehicle’s heading angle should align with the centerline of the lane at the beginning and end of the lane-changing.

#### 4.2. Simulation Scenario 2: Lane-Changing and Overtaking

^{2}, moving with constant acceleration. The requirements for the ego vehicle during the lane-changing process are to avoid a collision with the front vehicle SF and maintain an appropriate safe distance with the rear vehicle LR in the target lane. At the start and end times of the lane-changing, the ego vehicle’s heading angle is required to align with the centerline of the lane.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Illustration of a coordinate transformation [15].

**Figure 8.**Principles of establishing temporal and spatial graphs: (

**a**) analysis of potential conflict and (

**b**) description of what is contained in the ST graph.

**Figure 11.**Dynamic driving risk modeling of lane-changing: (

**a**) before lane-changing and (

**b**) after lane-changing.

**Figure 12.**Lane-changing scenario in different coordinate systems: (

**a**) Cartesian coordinate system and (

**b**) Frenet coordinate system.

**Figure 13.**Illustration of the simulation scenario: (

**a**) lane-changing and following and (

**b**) lane-changing and overtaking.

**Figure 17.**Velocity planning for scenario 1: (

**a**) velocity generation based on DP and (

**b**) velocity smoothing based on QP.

**Figure 20.**The trajectories tracking error based on MPC: (

**a**) lane-changing and following and (

**b**) lane-changing and overtaking.

**Figure 22.**Velocity planning for scenario 2. (

**a**) Lane-changing and following; (

**b**) Velocity smoothing based on QP.

Type | Parameter | Value |
---|---|---|

Path optimization | ω_{1} | 0.5 |

ω_{2} | 0.5 | |

Dynamic programming | ω_{11} | 0.2 |

ω_{12} | 0.2 | |

ω_{D}_{2} | 0.5 | |

ω_{D}_{3} | 5 | |

Quadratic programming | ω_{S}_{1} | 20 |

ω_{S}_{2} | 20 | |

ω_{S}_{3} | 300 |

Parameter | Value |
---|---|

Ego vehicle velocity | 10 m/s |

Ego vehicle acceleration | 0 |

Coordinate of ego vehicle position | (30, 0) |

Vehicle SF velocity | 0 |

Vehicle SF acceleration | 0 |

Coordinate of vehicle SF position | (70, 0) |

Vehicle LF velocity | 10 m/s |

Vehicle LF acceleration | 0.5 m/s^{2} |

Coordinate of vehicle LF position | (40, D) |

Number of Trajectories | Lane-Changing Duration (s) |
---|---|

1 | 3.00 |

2 | 3.50 |

3 | 4.00 |

4 | 4.40 |

5 | 4.80 |

6 | 5.20 |

7 | 5.65 |

Parameter | Value |
---|---|

Ego vehicle velocity | 10 m/s |

Ego vehicle acceleration | 0 |

Coordinate of ego vehicle position | (30, 0) |

Vehicle SF velocity | 0 |

Vehicle SF acceleration | 0 |

Coordinate of vehicle SF position | (70, 0) |

Vehicle LR velocity | 10 m/s |

Vehicle LR acceleration | 0.5 m/s^{2} |

Coordinate of vehicle LR position | (20, D) |

Number of Trajectories | Lane-Changing Duration (s) |
---|---|

1 | 2.50 |

2 | 3.00 |

3 | 3.40 |

4 | 3.85 |

5 | 4.30 |

6 | 4.75 |

7 | 5.20 |

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**MDPI and ACS Style**

Liu, K.; Wen, G.; Fu, Y.; Wang, H.
A Hierarchical Lane-Changing Trajectory Planning Method Based on the Least Action Principle. *Actuators* **2024**, *13*, 10.
https://doi.org/10.3390/act13010010

**AMA Style**

Liu K, Wen G, Fu Y, Wang H.
A Hierarchical Lane-Changing Trajectory Planning Method Based on the Least Action Principle. *Actuators*. 2024; 13(1):10.
https://doi.org/10.3390/act13010010

**Chicago/Turabian Style**

Liu, Ke, Guanzheng Wen, Yao Fu, and Honglin Wang.
2024. "A Hierarchical Lane-Changing Trajectory Planning Method Based on the Least Action Principle" *Actuators* 13, no. 1: 10.
https://doi.org/10.3390/act13010010