# Combining Event-Based Maneuver Selection and MPC Based Trajectory Generation in Autonomous Driving

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Maneuver Planning

#### 2.1. Maneuver Generation

#### 2.2. Safety Criteria

#### 2.2.1. TTC

#### 2.2.2. TIV

#### 2.3. Maneuver Selection

#### 2.3.1. Lane-Choosing Maneuver in the Lateral Direction

- Remove the lateral maneuvers that will cause the road edge constraints to be violated. For instance, if the vehicle is in the rightmost lane of the road, the maneuver of changing to the right lane (LCR) is inadmissible.
- Exclude the lateral maneuvers with which the vehicle is not heading towards the goal lane. Here, we consider two cases: (a) the goal lane is the current lane; (b) the goal lane is a different lane. In case (a), we simply remove the lane-changing maneuvers, LCL and LCR. Case (b) is more complicated. If the goal lane is the lane at the left (right) side of the current lane, we first exclude the lateral maneuver with which the vehicle turns to the opposite/wrong direction, LCR (LCL) is therefore removed. Then, we consider whether the lane-changing maneuver is satisfied or not. If the conditions for changing lanes are satisfied, we exclude the LK maneuver; otherwise the lane-changing maneuver LCL (LCR) is removed.

#### 2.3.2. Speed Generating Maneuver in the Longitudinal Direction

- The EV is behind the OV; $\Delta {x}_{\mathrm{ev},\mathrm{ov}}<0$.
- $\Delta {v}_{\mathrm{ev},\mathrm{ov}}<0$. The velocity of the EV is smaller than that of the OV:
- -
- When the EV maintains current longitudinal speed (CS), the distance between the two vehicles increases. Thus, both TTC and TIV increase as time goes on;
- -
- The EV’s choice of deceleration (DE) will cause the distance between the two vehicles ($\left|\Delta {x}_{\mathrm{ev},\mathrm{ov}}\right|$) to increase, but also cause the relative velocity ($\left|\Delta {v}_{\mathrm{ev},\mathrm{ov}}\right|$) to increase. Thus, the effect of deceleration (DE) on TTC is unknown;
- -
- Choosing acceleration (AC) will cause the TIV to decrease. Thus, we will not keep the AC.

Therefore, maintaining current longitudinal speed (CS) is the best choice. - $\Delta {v}_{\mathrm{ev},\mathrm{ov}}>0$. The velocity of the EV is greater than that of the OV, which is dangerous:
- -
- Both maintaining current speed (CS) and acceleration (AC) will definitely cause a decrease of the TTC and TIV;
- -
- However, by executing deceleration (DE), TTC and TIV will probably trend upward.

Consequently, deceleration (DE) is selected as the longitudinal maneuver performed in this situation. - $\Delta {v}_{\mathrm{ev},\mathrm{ov}}=0$. The two vehicles have the same longitudinal speed:
- -
- Under this circumstance, only TIV is used to estimate the risk;
- -
- Whether the distance between the two vehicles is small or not, decreasing the speed of the EV is a safe maneuver, which contributes to obtaining a greater TIV.

Thus, deceleration (DE) is selected for the EV in this situation.

- The EV is in front of the OV; $\Delta {x}_{\mathrm{ev},\mathrm{ov}}>0$.
- $\Delta {v}_{\mathrm{ev},\mathrm{ov}}<0$. The velocity of the EV is less than that of the OV:
- -
- Not only deceleration (DE), but also maintaining current speed (CS) will cause a decreasing trend of the TTC and TIV;
- -
- Additionally, when choosing acceleration (AC), TIV will experience a rapid drop, while TTC might show an upward trend.

This is not a safe situation, but increasing the speed of the EV will probably contribute to avoiding possible collisions, so AC is taken as the longitudinal maneuver. - $\Delta {v}_{\mathrm{ev},\mathrm{ov}}>0$. The velocity of the EV is greater than that of the OV:
- -
- Both decelerating (DE) and maintaining current speed (CS) are beneficial to increasing TTC and TIV, so they can be taken as candidate maneuvers;
- -
- Furthermore, DE has a negative effect on the efficiency and smoothness.

Therefore, we finally keep CS as the longitudinal maneuver in this situation. - $\Delta {v}_{\mathrm{ev},\mathrm{ov}}=0$. The EV and OV drive at same speed:
- -
- In this situation, TTC is not calculated and only TIV is considered as a safety criterion;
- -
- Increasing the speed (AC) will help obtain greater TIV;
- -
- Neither decreasing (DE), nor maintaining current speed (CS) will contribute to increasing TIV.

Therefore, acceleration (AC) is selected.

## 3. Model Predictive Control

#### 3.1. Optimization Problem of the MPC Controller

#### 3.2. Vehicle Model

#### 3.3. Constraints

#### 3.3.1. State Constraints

- Constraints for Traffic Rules:$$x\left(k\right)\in [0,+\infty ]$$$$y\left(k\right)\in [{w}_{\mathrm{veh}}/2,m{w}_{\mathrm{lane}}-{w}_{\mathrm{veh}}/2]$$$${v}_{x}\left(k\right)\in [{v}_{{x}_{\mathrm{min}}},{v}_{{x}_{\mathrm{max}}}]$$$${v}_{y}\left(k\right)\in [{v}_{{y}_{\mathrm{min}}},{v}_{{y}_{\mathrm{max}}}]$$
- Safety constraint:In order to avoid collisions, a region around each vehicle is defined that other vehicles are not allowed to enter [34,38]. The region can be any convex shape larger than the shape of the vehicle [38]. As in [34,39,40,41], an elliptic region is chosen for our implementation, as shown in Figure 4.This safety constraint is then realized by the following inequality:$$\frac{\Delta {x}_{\mathrm{ev},\mathrm{ov}}^{2}}{{a}^{2}}+\frac{\Delta {y}_{\mathrm{ev},\mathrm{ov}}^{2}}{{b}^{2}}>1,$$For safety, the ellipse-shaped region has to be large enough that the vehicle shape is covered by the ellipse for all possible vehicle orientations. In order to find appropriate a and b, we consider the vehicle turning left or right, see Figure 4. If the vehicle is covered by the ellipse here, it will also be covered for all other orientations. Let ${l}_{\mathrm{veh}}$ and ${w}_{\mathrm{veh}}$ denote the length and the width of a vehicle, respectively. Then it holds for a and b:$$0<\sqrt{{\left(\frac{{l}_{\mathrm{veh}}}{2}\right)}^{2}+{\left(\frac{{w}_{\mathrm{veh}}}{2}\right)}^{2}}+\delta <b<a$$

#### 3.3.2. Input Constraints

#### 3.4. Cost Function

#### 3.5. MPC-Based Reference Trajectory Generation

## 4. Simulation Results

#### 4.1. Scenario Description

- ${S}_{0}$: The EV is driving behind the OV in ${L}_{0}$ with a greater longitudinal velocity.
- ${S}_{1}$: The EV is following the OV in ${L}_{0}$ with a smaller longitudinal velocity.
- ${S}_{2}$: The EV is driving in lane ${L}_{1}$, waiting for a chance to change to ${L}_{2}$. This is a transition stage.
- ${S}_{3}$: The EV reaches lane ${L}_{2}$ and drives in ${L}_{2}$ before overtaking the OV from the left, looking for an opportunity to go back to lane ${L}_{1}$. In this stage, lane keeping also occurs.
- ${S}_{4}$: The EV reaches lane ${L}_{1}$, and is preparing for moving to lane ${L}_{0}$.
- ${S}_{5}$: The EV drives in lane ${L}_{0}$.

#### 4.2. Maneuver Planning

- 1—The vehicle will reach the edge of the road if it turns left (or right).
- 2—The goal is to change lane but the the safety constraints for changing lane is not satisfied.
- 3—By conducting these maneuvers, the vehicle cannot move to the target lane even though the conditions for changing lane are fulfilled.
- 4—These maneuvers contribute less/nothing to increasing TTC and TIV.
- 5—These maneuvers contribute less to improving efficiency.
- 6—Selecting these maneuvers has a negative or no positive impact on smoothness of driving behaviors.

#### 4.2.1. Maneuver Selection in Car-Following Scenario: ${S}_{0}\to {S}_{1}$

#### 4.2.2. Maneuver Selection in Overtaking Scenario: ${S}_{0}\to {S}_{2}\to {S}_{3}\to {S}_{4}\to {S}_{5}$

#### 4.3. MPC-Based Trajectory Control

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MPC | model predictive control |

TTC | time to collision |

TIV | intervehicular time |

DE | decelerating |

CS | maintaining current speed |

AC | accelerating |

LCL | changing to the left lane |

LCR | changing to the right lane |

LK | keeping moving in the current lane |

EV | ego vehicle |

OV | object vehicle |

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**Figure 2.**Nine feasible combined maneuvers. Boxes with the same color contain three different maneuvers in the longitudinal direction: decelerating (DE), staying at/maintaining current speed (CS), and accelerating (AC). Boxes with different colors represent distinct maneuvers in the lateral direction. The red, yellow, and green boxes illustrate changing to left lane (LCL), keeping/continuing moving in current lane (LK), and changing to right lane (LCR), respectively.

**Figure 6.**Maneuver Selection: The box displays the 9 possible maneuvers: LCL+DE, LCL+CS, LCL+AC, LK+DE, LK+CS, LK+AC, LCR+DE, LCR+CS, and LCR+AC.

**Figure 8.**Five stages for the overtaking scenario: ${S}_{0}$, ${S}_{2}$, ${S}_{3}$, ${S}_{4}$, ${S}_{5}$.

$\mathbf{\Delta}{\mathit{x}}_{\mathbf{ev},\mathbf{ov}}$ | $\mathbf{\Delta}{\mathit{v}}_{\mathbf{ev},\mathbf{ov}}$ | Possible Maneuver | $\left|\mathbf{\Delta}{\mathit{x}}_{\mathbf{ev},\mathbf{ov}}\right|$ | $\left|\mathbf{\Delta}{\mathit{v}}_{\mathbf{ev},\mathbf{ov}}\right|$ | ${\mathit{v}}^{*}$ | TTC | TIV | Result |
---|---|---|---|---|---|---|---|---|

DE | ↑ | ↑ | ↓ | ? | ↑ | |||

$\Delta {v}_{\mathrm{ev},\mathrm{ov}}<0$ | CS | ↑ | − | − | ↑ | ↑ | CS | |

AC | ↑ | ↓ | ↑ | ↑ | ↓ | |||

DE | ↓ | ↓ | ↓ | ? | ? | |||

$\Delta {x}_{\mathrm{ev},\mathrm{ov}}<0$ | $\Delta {v}_{\mathrm{ev},\mathrm{ov}}>0$ | CS | ↓ | − | − | ↓ | ↓ | DE |

AC | ↓ | ↑ | ↑ | ↓ | ↓ | |||

DE | ↑ | ↑ | ↓ | / | ↑ | |||

$\Delta {v}_{\mathrm{ev},\mathrm{ov}}=0$ | CS | − | − | − | / | − | DE | |

AC | ↓ | ↑ | ↑ | / | ↓ | |||

DE | ↓ | ↑ | − | ↓ | ↓ | |||

$\Delta {v}_{\mathrm{ev},\mathrm{ov}}<0$ | CS | ↓ | − | − | ↓ | ↓ | AC | |

AC | ↓ | ↓ | − | ? | ↓ | |||

DE | ↑ | ↓ | − | ↑ | ↑ | |||

$\Delta {x}_{\mathrm{ev},\mathrm{ov}}>0$ | $\Delta {v}_{\mathrm{ev},\mathrm{ov}}>0$ | CS | ↑ | − | − | ↑ | ↑ | DE/CS |

AC | ↑ | ↑ | − | ? | ↑ | |||

DE | ↓ | ↑ | − | / | ↓ | |||

$\Delta {v}_{\mathrm{ev},\mathrm{ov}}=0$ | CS | − | − | − | / | − | AC | |

AC | ↑ | ↑ | − | / | ↑ |

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**MDPI and ACS Style**

Dang, N.; Brüdigam, T.; Leibold, M.; Buss, M. Combining Event-Based Maneuver Selection and MPC Based Trajectory Generation in Autonomous Driving. *Electronics* **2022**, *11*, 1518.
https://doi.org/10.3390/electronics11101518

**AMA Style**

Dang N, Brüdigam T, Leibold M, Buss M. Combining Event-Based Maneuver Selection and MPC Based Trajectory Generation in Autonomous Driving. *Electronics*. 2022; 11(10):1518.
https://doi.org/10.3390/electronics11101518

**Chicago/Turabian Style**

Dang, Ni, Tim Brüdigam, Marion Leibold, and Martin Buss. 2022. "Combining Event-Based Maneuver Selection and MPC Based Trajectory Generation in Autonomous Driving" *Electronics* 11, no. 10: 1518.
https://doi.org/10.3390/electronics11101518