# Smooth Trajectory Planning at the Handling Limits for Oval Racing

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work

#### 1.2. Contribution and Outline of the Paper

- 1.
- The main idea is to treat the initial edges differently from the other edges in a spatio temporal graph. Our sampling-based approach for the generation of the initial edges uses jerk-optimal curves, whose smoothness reduces lateral acceleration deflections and thus increases vehicle stability, especially during braking maneuvers.
- 2.
- We propose a concept for the selection of the end conditions of the jerk-optimal curves in order to adapt the initial edges to the racing scenario and thus get closer to the handling limits of the vehicle. We introduce the concept using the already proposed graph structure described in [16].

## 2. Material and Methods

#### 2.1. Frenét Frame and Graph Structure

#### 2.2. Local Planning Framework

#### 2.3. Start State Identification

#### 2.4. Initial Edge Generation

#### 2.4.1. Coordinate Transformation

#### 2.4.2. Jerk-Optimal Movement

#### 2.4.3. Integration into Graph Structure

#### 2.4.4. Performance Improvement for Racing-Scenarios

^{2}for comfort reasons. In the context of a racing scenario, however, this is undesirable. For example, in an acceleration maneuver, the initial edges would have to adapt at an early stage to fulfill the end conditions, resulting in a loss of race performance. Simple equidistant sampling of the end acceleration and the end time is also not possible due to the curse of dimensionality and the accompanying computing time. Instead, reducing the number of samples without sacrificing race performance is desired.

#### 2.4.5. Stopping to Standstill

## 3. Results

#### 3.1. Autonomous Driving Software Architecture

#### 3.2. Simulation Results

#### 3.2.1. Single-Vehicle Driving

^{2}. The sampling-based approach, on the other hand, produces a transition from zero acceleration to higher accelerations. However, since only flying starts are decisive in the competition formats of IAC and AC@CES, this disadvantage at standing starts was not pursued further.

#### 3.2.2. Braking Maneuver

#### 3.2.3. Object Evasion

#### 3.3. Experimental Results

**Figure 12.**Predicted opponent velocity and planned ego motion for the time step marked in Figure 11. The end of the initial edge and the minimum planning horizon of 5 s are indicated by black vertical lines.

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AC@CES | Autonomous Challenge at CES |

IAC | Indy Autonomous Challenge |

IMS | Indianapolis Motor Speedway |

LVMS | Las Vegas Motor Speedway |

ODD | Operational Design Domain |

PVD | Path-Velocity Decomposition |

RRT | Rapidly Exploring Random Trees |

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**Figure 1.**The Dallara AV-21 race cars of the TUM Autonomous Motorsports and TII EuroRacing team during the AC@CES on the LVMS.

**Figure 2.**Used graph structure and initial edges. (

**a**) Spatial nodes (black dots) of the graph are placed on layers perpendicular to the reference line (dashed line). (

**b**) Initial edges ${E}_{0}$ connecting the start state to the initial layer ${L}_{\mathrm{init}}$ in both the spatial and temporal dimensions.

**Figure 4.**The start state of the trajectory generation is determined by projecting the estimated state onto the previously planned trajectory and keeping constant the part corresponding to the average planning calculation time.

**Figure 5.**Example initial edges (gray lines) connecting the start state (blue dot) with the initial layer and fulfilling the individual end pose of each spatial node (black dots). Coordinates are shown exemplarily for the second node.

**Figure 6.**Comparison of velocity profiles generated by our sampling-based approach and uniform acceleration. Exemplarily, several end velocities from 0 to 80 m/s are sampled with a start velocity of 50 m/s and a start acceleration of 10 m/s.

**Figure 7.**Comparison of our sampling-based approach and the uniform acceleration approach for a standing start and a flying lap on the LVMS.

**Figure 8.**Sampling-based approach and uniform acceleration approach comparison for a braking maneuver from 75 m/s to 30 m/s on the mainstretch of the LVMS.

**Figure 10.**Mean absolute lateral acceleration control error per run (dots) using the sampling-based approach and the uniform acceleration approach with different ideally assumed detection ranges for the evasion maneuver shown in Figure 9. The lines represent the average values of the means.

**Figure 11.**Last two overtaking maneuvers during the final of AC@CES on the LVMS. The permitted passing period is marked in red. The black vertical line indicates the time step shown in Figure 12.

Standing Start Lap | Flying Lap (Mean/Variance) | |
---|---|---|

Uniform acceleration | 47.814 s | 30.431 s/0.001 s^{2} |

Sampling-based approach | 51.854 s | 30.001 s/0.004 s^{2} |

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**MDPI and ACS Style**

Ögretmen, L.; Rowold, M.; Ochsenius, M.; Lohmann, B.
Smooth Trajectory Planning at the Handling Limits for Oval Racing. *Actuators* **2022**, *11*, 318.
https://doi.org/10.3390/act11110318

**AMA Style**

Ögretmen L, Rowold M, Ochsenius M, Lohmann B.
Smooth Trajectory Planning at the Handling Limits for Oval Racing. *Actuators*. 2022; 11(11):318.
https://doi.org/10.3390/act11110318

**Chicago/Turabian Style**

Ögretmen, Levent, Matthias Rowold, Marvin Ochsenius, and Boris Lohmann.
2022. "Smooth Trajectory Planning at the Handling Limits for Oval Racing" *Actuators* 11, no. 11: 318.
https://doi.org/10.3390/act11110318