# Path Planning for Autonomous Platoon Formation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}, lateral jerk must be below 5 m/s

^{3}, all the while optimizing to minimize the time to join the platoon. The main focus of the work is the path planning required for such a process, and considerations for vehicle dynamics during the manoeuvres is discussed; this is achieved via the combination of a variant of RRT and MPC, which does away with the conventional problems of just using RRT [16].

## 2. Platoon Formation and Path Planning

- Raw path calculation: this first step generates a path that is valid within the configuration space: the path stays inside the road boundaries and prevents the hitting of any road obstacle while making its way to the target position as fast as possible.
- Path optimization: The path calculated in (1) does not guarantee that it is physically achievable for a vehicle. The average driver will not be fond of high acceleration or jerk peaks. This leads to the second operation: path optimization. From the path defined in (1), this step will output a dynamically realistic path for the vehicle to follow.

#### 2.1. Path Planning with Rapid-Exploring Random Trees (RRT)

- There is no need to explicitly characterize the configuration space, but instead probe the space and use collision detection on the go.
- They are incremental in nature and efficient which offers the potential for real-time implementation while retaining completeness guarantees.

#### 2.2. Biased RRT Star (RRT*)

#### 2.3. Informed RRT* (i-RRT*)

## 3. Path Optimization Using Model Predictive Control

- $y$ the lateral vehicle coordinate [m].

- ${y}_{r}$ the RRT* lateral coordinate [m] (Figure 8).

- ${x}^{*}$ the longitudinal distance between leader and slave [m].

- ${x}_{leader}$ the leader longitudinal coordinate [m].

- $\psi ,\dot{\psi},{\delta}_{f}$ the heading angle, the yaw rate and the steering angle, respectively [rad].

- ${v}_{x},{v}_{y}$ the slave lateral and longitudinal speed, respectively [m/s].

- ${q}_{i\in 1,2,3,4}$ the weights of the cost function [-].

- ${t}_{f}$ the prediction horizon (time over which the dynamic model is solved).

- ${\left(y-{y}_{r}\left({x}^{*}\right)\right)}^{2}$: deviation between slave lateral position and the RRT* lateral coordinate.

- ${x}^{{*}^{2}}$: distance between the slave and the goal. Minimizing this state variable is the main lever to reach the leader.

- ${j}^{2}$: slave jerk. Minimizing the jerk prevents being too demanding on it.

- ${\delta}_{f}^{2}:$ slave steering angle. Same justification as for the jerk.

## 4. Results

#### 4.1. Path Optimization

^{−2}, 5 m·s

^{−3}and 8°. The blue dashed line represents the jerk for the dynamic bicycle model. Beyond smoothing the jerk of raw RRT* path, the MPC method has provided us with the time profiles of each command variable to be adopted by the slave. The optimization solver finds a kinematically feasible solution while minimizing the deviation between the slave lateral position and RRT* lateral reference coordinate. The optimization has been run for a prediction horizon of 15 s but a 10 s horizon has turned out to be enough to reach the leader. In the early stages of the manoeuvre the acceleration and the jerk control variables hit their limits. Indeed, since the cost function includes the distance to the leader ${x}^{*}$, the solver minimizes this quantity over the complete prediction horizon. This results in reaching the leader position as quickly as possible and thus the control variables are asked to operate at their limits as soon as possible.

#### 4.2. Simulation Results

## 5. Discussion

^{2}, and lateral jerk below 5 m/s

^{3}) is feasible. This is made possible by the prediction horizon of the MPC of 10 units ahead in time. Furthermore, the trajectory is updated continuously within and thus can deal with varying environments (traffic vehicles changing lanes, for example). Therefore, these results should be taken into account when considering highly automated highway systems.

- Several scenarios are still to be studied (border cases), such as the “leader” being behind “slave” vehicles, or all three lanes being completely blocked by traffic.
- Although the RRT* algorithm provides us with an obstacle-free trajectory, the MPC controller yields a very close but still different trajectory. There is thus a risk of colliding with obstacles if the path planner frequency is too low. To that extent, the real-time performance in embedded systems is an important aspect to be tested, for example, with hardware in-the-loop modelling. Although the results in simulation show a high frequency path calculation, it must be real-time capable for a given hardware.
- The path planner implemented here considered no highway driving protocols, such as knowing the legal way to cross lanes, which gives scope for future study. Another important direction is the usage of informed-RRT* which should result in faster converging paths.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Weight | ${\mathit{q}}_{1}$ | ${\mathit{q}}_{2}$ | ${\mathit{q}}_{3}$ | ${\mathit{q}}_{4}$ |
---|---|---|---|---|

Value | 150 | 5 | 50 | 1 |

State Variable | v_{x} | a | $\dot{\mathit{\psi}}$ | j | ${\mathit{\delta}}_{\mathit{f}}$ |
---|---|---|---|---|---|

Minimum | 0 m/s | −5 m/s^{2} | −10°/s | −5 m/s^{3} | −10° |

Maximum | 30 m/s | 5 m/s^{2} | 10°/s | 5 m/s^{3} | 10° |

Bicycle Model | Computational Time |
---|---|

Kinematic | 1.3 s |

Dynamic | 1.6 s |

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**MDPI and ACS Style**

El Ganaoui-Mourlan, O.; Camp, S.; Hannagan, T.; Arora, V.; De Neuville, M.; Kousournas, V.A.
Path Planning for Autonomous Platoon Formation. *Sustainability* **2021**, *13*, 4668.
https://doi.org/10.3390/su13094668

**AMA Style**

El Ganaoui-Mourlan O, Camp S, Hannagan T, Arora V, De Neuville M, Kousournas VA.
Path Planning for Autonomous Platoon Formation. *Sustainability*. 2021; 13(9):4668.
https://doi.org/10.3390/su13094668

**Chicago/Turabian Style**

El Ganaoui-Mourlan, Ouafae, Stephane Camp, Thomas Hannagan, Vaibhav Arora, Martin De Neuville, and Vaios Andreas Kousournas.
2021. "Path Planning for Autonomous Platoon Formation" *Sustainability* 13, no. 9: 4668.
https://doi.org/10.3390/su13094668