# A Double-Layer Model Predictive Control Approach for Collision-Free Lane Tracking of On-Road Autonomous Vehicles

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## Abstract

**:**

## 1. Introduction

- (1)
- This paper proposes a novel double-layer structure MPC algorithm to achieve lane tracking for on-road autonomous vehicles. The double-layer structure separates the trajectory planning and tracking tasks. The upper layer module uses a simplified vehicle model to calculate collision-free trajectories in real-time, while the lower layer module uses a higher fidelity vehicle model to track the trajectory.
- (2)
- This paper proposes a simplified vehicle model based on the second-order norm friction cone in the upper layer. Introducing the second-order norm friction cone simplifies the trajectory planning difficulty while capturing the characteristics of tire friction. Furthermore, this paper proposes a polygonal distance-based dynamic obstacle avoidance method. The polygonal calculation formula calculates the distance between the ego vehicle and obstacles precisely, and neighboring vehicles’ states are augmented into the ego vehicle’s state space to plan the dynamic collision-free trajectory.
- (3)
- This paper introduces the calculation of the vertical load of tires to design the anti-rollover constraint in the lower layer, which represents the contact situation between tires and the ground. This constraint ensures that the vertical load of each tire is higher than the preset value to avoid rollover.

## 2. Preliminaries

#### 2.1. Model Predictive Control

#### 2.2. Lane Waypoint Selection

**Remark**

**1.**

## 3. Double-Layer MPC-Based Lane Tracking Method

#### 3.1. Vehicle Model Design

#### 3.2. Dynamic Obstacles Avoidance Method Design

_{k}and b

_{k}requires information about neighboring vehicles. However, without vehicle-to-vehicle communication architecture, it is not possible to acquire the trajectory information of surrounding vehicles. However, the real-time speed and position of adjacent vehicles can be obtained through the environment perception module of the autonomous vehicle. Within the prediction horizon of the MPC, this paper assumes that the velocity ${v}_{k}$ and yaw angle ${\psi}_{k}$ of neighboring vehicles remain constant. The dynamic equation for the kth vehicle is presented in Equation (12).

**Remark**

**2.**

#### 3.3. Rollover Avoidance Constraint Design

#### 3.4. Detailed Formulation of the Double-Layer MPC-Based Lane Tracking Method

**Remark**

**3.**

_{t}, ${b}_{term}={3F}_{z\_offset}$

_{t}. The hyperbolic tangent function ($tanh$) is introduced to ensure smoothness.

## 4. Numerical Simulations

## 5. Conclusions

- (1)
- The double-layer structure, which separates the trajectory planning and trajectory tracking tasks, making the motion control task more straightforward and easily solvable by MPC. In contrast to the reports in the literature [13,14], the double-layer structure proposed in this paper avoids the long trajectory planning time using a high-fidelity model in the single-layer MPC proposed in the literature [13].
- (2)
- The dynamic collision avoidance constraint design uses the smooth and non-approximate calculation of obstacle distances and the augmentation of the states of neighboring vehicles. Due to this calculation method’s smoothness and strong duality, introducing this collision avoidance constraint in the upper layer module does not significantly increase the computational complexity. In contrast to the tecniques in the literature [13,14], these methods can effectively ensure dynamic collision avoidance in narrow environments.
- (3)

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Symbol | Value | Unit |
---|---|---|

M | 2600 | kg |

I | 3989 | kg·m^{2} |

L_{f}, L_{r} | 1.5, 1.7 | m |

W | 1.5 | m |

ΔT | [0.05, 0.1] | s |

N | 30 | - |

μ_{zx} | 800 | N/(m/s^{2}) |

μ_{zyf} | 679 | N/(m/s^{2}) |

μ_{zyr} | 1079 | N/(m/s^{2}) |

[Jerk_min, Jerk_max] | [−5, 5] | m/s^{3} |

[vx_min, vx_max] | [0, 25] | m/s |

[Str_min, Str_max] | [−5, 5] | deg/s |

[δ_min, δ_max] | [−30, 30] | deg |

F_threshold | 1000 | N |

a_term | 1270 | N |

b_term | 90 | N |

W_{lift_load} | 0.05 | - |

${Q}_{z}^{HL},{Q}_{t}^{HL}$ | Diag([0.01, 0.01]) | - |

${Q}_{u}^{HL}$ | Diag([0.05, 0.05, 0.05, 0.05, 0.005]) | - |

${Q}_{z}^{LL},{Q}_{t}^{LL}$ | Diag([0.025, 0.025]) | - |

${Q}_{u}^{HL}$ | Diag([0.015, 0.015] | - |

${W}_{pos}^{HL},{W}_{t}^{HL}$ | 0.02 | - |

${W}_{pos}^{LL},{W}_{t}^{LL}$ | 0.03 | - |

${W}_{u}^{HL},{W}_{u}^{LL}$ | 0.01 | - |

Pred_dist | 50 | m |

Safety_dist | 0.3 | m |

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**MDPI and ACS Style**

Yang, W.; Chen, Y.; Su, Y.
A Double-Layer Model Predictive Control Approach for Collision-Free Lane Tracking of On-Road Autonomous Vehicles. *Actuators* **2023**, *12*, 169.
https://doi.org/10.3390/act12040169

**AMA Style**

Yang W, Chen Y, Su Y.
A Double-Layer Model Predictive Control Approach for Collision-Free Lane Tracking of On-Road Autonomous Vehicles. *Actuators*. 2023; 12(4):169.
https://doi.org/10.3390/act12040169

**Chicago/Turabian Style**

Yang, Weishan, Yuepeng Chen, and Yixin Su.
2023. "A Double-Layer Model Predictive Control Approach for Collision-Free Lane Tracking of On-Road Autonomous Vehicles" *Actuators* 12, no. 4: 169.
https://doi.org/10.3390/act12040169