# Boundary Scenario Generation for HAVs Based on Classification and Local Sampling

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## Abstract

**:**

## 1. Introduction

- (1)
- A novel approach is proposed to obtain a High-Performance Classifier (HPC) for efficiently classifying critical and non-critical scenarios without consuming significant resources. This approach combines the Gaussian Process Classification (GPC) and Support Vector Machine (SVM) algorithms, which mutually detect uncertain scenarios and improve each other’s performance iteratively.
- (2)
- A distance-based method is presented for identifying candidate scenarios using the HPC without executing scenarios. This approach enables the efficient identification of potential boundary scenarios.
- (3)
- This study also uses local sampling iteratively to generate diverse candidate scenarios based on a small set of them. This approach can be applied to generate high-dimensional candidate scenarios efficiently.

## 2. Related Works

## 3. Methodology

#### 3.1. A Distance-Based Approach

#### 3.2. Overall Framework

#### 3.3. Gaussian Processes

#### 3.4. Support Vector Machines

#### 3.5. Obtaining High-Performance Classifiers

#### 3.6. Searching for Candidate Scenarios Based on an HPC

#### 3.7. Generating Diverse Candidate Scenarios by Local Sampling

## 4. Simulation Experiments

#### 4.1. Testing Scenarios

#### 4.2. Intelligent Driver Model

#### 4.3. Stop Conditions

#### 4.4. Distance Thresholds for Determining Candidate or Boundary Scenarios

#### 4.5. Criticality Metric

## 5. Numerical Results and Discussions

#### 5.1. Results and Discussions of Car-Following Scenario Experiments

#### 5.2. Results and Discussions of Cut-In Scenario Experiments

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The given scenario and its adjacent, adverse scenarios and the nearest adverse scenario. Blue circles are adjacent scenarios; Green circles are adverse scenarios; The pink circle is the nearest adverse scenario to the given scenario. The red circle is the given scenario.

**Figure 2.**Sketch of soft classification and nonlinear classification based on SVM. (

**a**) Separation hyperplanes; (

**b**) Optimal hyperplanes. Blue and red dots are data belonging to different classes.

**Figure 6.**Flow chart of deriving more candidate scenarios based on a small set of candidate scenarios.

**Figure 7.**Sketches of the car-following scenario and the cut-in scenario. (

**a**) The car-following scenario; (

**b**) The cut-in scenario.

**Figure 8.**The performance of the four classifiers in classifying critical and non-critical car-following scenarios.

**Figure 10.**Size curves of training datasets for GSVM and GGPC in experiments involving car-following scenarios. The blue and red curves belong to GGPC and GSVM, respectively.

**Figure 11.**Candidate scenarios detected based on GGPC and their fitting surface. Red circles are critical scenarios, while blue circles are non-critical scenarios. In addition, the semi-transparent blue surface is the performance boundary.

**Figure 12.**The performance boundary and test samples. The blue surface is the performance boundary between critical and non-critical scenarios. Red circles are critical scenarios, while green circles are non-critical scenarios. To enhance the visualization, we only display 2% of non-critical and 20% of critical scenarios from the test dataset.

**Figure 13.**The performance boundary and the candidate scenarios on it. A total of 99.87% of them are found to be boundary scenarios. The mean distance between boundary scenarios and their nearest adverse scenarios is 0.0057, much smaller than the threshold of 0.02.

**Figure 14.**Size curves of the training datasets for GSVM and GGPC in experiments involving cut-in scenarios. The blue and red curves belong to GSVM and GGPC, respectively.

**Figure 16.**Derived, initial, and lonely candidate scenarios. Red circles are initial candidate scenarios. Green and red circles are derived from candidate and lonely scenarios, respectively. (

**a**) Iteration = 1; (

**b**) Iteration = 3; (

**c**) Iteration = 10; (

**d**) Iteration = 44.

**Figure 17.**The distribution of distances between candidate scenarios and their nearest adverse scenarios.

Name | Equation |
---|---|

Linear Kernel | $k\left(x,y\right)={x}^{\mathrm{T}}y+c$ |

Polynomial Kernel | $k\left(x,y\right)=a{x}^{\mathrm{T}}y+{c}^{d}$ |

Radial Basis Function | $k\left(x,y\right)=\mathrm{exp}(-\gamma {||x-y||}^{2})$ |

Gaussian Kernel | $k\left(x,y\right)=\mathrm{exp}(-\frac{{||x-y||}^{2}}{2{\sigma}^{2}})$ |

Exponential Kernel | $k\left(x,y\right)=\mathrm{exp}(-\frac{||x-y||}{2\sigma})$ |

Description | Key Parameter | Value Range |
---|---|---|

the initial distance between the two vehicles | ${S}_{x,0}$ | (15, 100) m |

the initial velocity of the ego vehicle | ${v}_{\mathrm{x},0}^{\mathrm{ego}}$ | (5, 40) m/s |

the initial velocity of the reference vehicle | ${v}_{\mathrm{x},0}^{\mathrm{ref}}$ | (5, 40) m/s |

Description | Key Parameter | Value Range |
---|---|---|

the initial longitudinal distance between two vehicles | ${S}_{x,0}$ | (15, 100 m |

initial lateral distance between two vehicles | ${S}_{y,0}$ | (1.9, 3.8) m |

initial longitudinal speed of the ego vehicle | ${v}_{x,0}^{\mathrm{ego}}$ | (10, 40) m/s |

initial lateral speed of the reference vehicle | ${v}_{y,0}^{\mathrm{ref}}$ | (0.5, 1.75) m/s |

initial longitudinal speed of the reference vehicle | ${v}_{x,0}^{\mathrm{ref}}$ | (10, 35) m/s |

Description | Key Parameter | Value |
---|---|---|

vehicle width | ${w}_{veh}$ | 1.8 m |

vehicle length | ${l}_{veh}$ | 5 m |

lane width | ${w}_{lane}$ | 3.8 m |

the maximum duration of the scenario | ${T}_{\mathrm{max}}$ | 10 m |

Description | Key Parameter | Typical Value |
---|---|---|

Desired velocity | ${v}_{d}$ | 29.8 m/s |

Safe time headway | $T$ | 1.6 s |

Maximum acceleration | $a$ | 2.62 m/s^{2} |

Desired deceleration | $b$ | 2.67 m/s^{2} |

Acceleration exponent | $\delta $ | 4 |

Jam distance 1 | ${s}_{0}$ | 1 m |

Jam distance 2 | ${s}_{1}$ | 2 m |

Vehicle length | $l$ | 5 m |

Metrics | GPC | SVM | GGPC | GSVM |
---|---|---|---|---|

True Positive Rate (TPR) | 66.02% | 82.90% | 99.66% | 98.75% |

True Negative Rate (TNR) | 93.86% | 100% | 99.87% | 98.77% |

False Positive Rate (FPR) | 6.14% | 0 | 0.13% | 1.23% |

False Negative Rate (FNR) | 33.98% | 17.10% | 0.34% | 1.25% |

Overall Accuracy | 91.42% | 98.50% | 99.85% | 98.77% |

Method | Number of Candidate Scenarios | Number of Boundary Scenarios | The Proportion of Boundary Scenarios | Mean Distance to the NAS |
---|---|---|---|---|

GPC | 7866 | 1346 | 17.11% | 0.017 |

SVM | 2120 | 1308 | 61.70% | 0.017 |

GGPC | 2329 | 2301 | 98.80% | 0.015 |

GSVM | 2348 | 1562 | 66.52% | 0.017 |

GPC | SVM | GGPC | GSVM | |
---|---|---|---|---|

True Positive Rate | 60.65% | 49.52% | 97.10% | 97.90% |

True Negative Rate | 88.11% | 100% | 99.47% | 99.45% |

False Positive Rate | 11.89% | 0 | 0.53% | 0.55% |

False Negative Rate | 39.35% | 50.48% | 2.90% | 2.10% |

Overall Right Rate | 86.43% | 96.91% | 99.33% | 99.36% |

Method | Candidate Scenarios | Boundary Scenarios | Proportion of Boundary Scenarios | Mean Distance to the NAS |
---|---|---|---|---|

GPC | 489 | 50 | 10.22% | 0.033 |

SVM | 52 | 17 | 32.69% | 0.037 |

GGPC | 111 | 88 | 79.28% | 0.031 |

GSVM | 106 | 97 | 91.51% | 0.031 |

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**MDPI and ACS Style**

Cai, J.; Deng, W.; Wang, Y.; Guang, H.; Li, J.; Ding, J.
Boundary Scenario Generation for HAVs Based on Classification and Local Sampling. *Machines* **2023**, *11*, 426.
https://doi.org/10.3390/machines11040426

**AMA Style**

Cai J, Deng W, Wang Y, Guang H, Li J, Ding J.
Boundary Scenario Generation for HAVs Based on Classification and Local Sampling. *Machines*. 2023; 11(4):426.
https://doi.org/10.3390/machines11040426

**Chicago/Turabian Style**

Cai, Jinkang, Weiwen Deng, Ying Wang, Haoran Guang, Jiangkun Li, and Juan Ding.
2023. "Boundary Scenario Generation for HAVs Based on Classification and Local Sampling" *Machines* 11, no. 4: 426.
https://doi.org/10.3390/machines11040426