# Real-Time Drift-Driving Control for an Autonomous Vehicle: Learning from Nonlinear Model Predictive Control via a Deep Neural Network

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Vehicle Dynamics Analysis

#### 2.1. Three-Degrees-of-Freedom Bicycle Model

#### 2.2. Brush Tire Model

#### 2.3. Drift Equilibrium State Analysis

## 3. Design of the Nonlinear Model Predictive Controller

#### 3.1. Vehicle State Prediction Model

#### 3.2. Nonlinear Model Predictive Controller Cost Function

#### 3.3. Nonlinear Model Predictive Controller System for Drift Driving

## 4. Drift-Driving Test of the Nonlinear Model Predictive Controller

#### 4.1. Test Scenario

#### 4.2. Drift Test Results

## 5. Design of the Neural Network Drift Controller

#### 5.1. Training Data Preprocess

#### 5.2. Neural-Network-Based Controller Architecture

#### 5.2.1. Deep Neural-Network-Based Controller for Steering Control

#### 5.2.2. Time Delay Neural-Network-Based Controller for Drift State Control

## 6. Simulation Results of the Neural Network Drift Controller

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Saturation conditions of the brush tire model: (

**a**) longitudinal and lateral forces and (

**b**) slip VS tire force.

**Figure 5.**Equilibrium states of (

**a**) sideslip angle (β), (

**b**) yaw rate (r), and (

**c**) rear wheel force (${F}_{xr}$) at a vehicle speed of 1.7 m/s.

**Figure 7.**3D maps of the equilibrium states of the (

**a**) sideslip angle ($\beta $), (

**b**) yaw rate ($r$), and (

**c**) rear-wheel force (${F}_{xr}$).

**Figure 10.**Sideslip angle ($\beta $) and yaw rate ($r$) of the vehicle during the drift maneuver. The dotted red lines and solid blue lines trace the control-target point of the drift control and the vehicle state, respectively.

**Figure 11.**Front and rear tire slip angles of the vehicle during the drift maneuver. The solid blue curves in the upper and lower panels represent the front and rear tire slip angles of the vehicle, respectively, and the dotted red lines show the upper and lower saturation limits of the tires.

**Figure 17.**Front and rear tire slip angles during the drift maneuver. The solid blue lines in the upper and lower panels present the slip angles of the front and rear wheels, respectively, and the dotted red line shows the tire saturation threshold.

**Figure 18.**Vehicle states during a drift maneuver (solid blue lines). The desired equilibrium points (dotted red lines) are plotted for reference.

Symbol | Meaning | Unit |
---|---|---|

${F}_{yf}$ | Front tire lateral force | N |

${F}_{yr}$ | Rear tire lateral force | N |

$v$ | Vehicle velocity | m/s |

${v}_{w}$ | Rear-wheel velocity | m/s |

${v}_{x}$ | Longitudinal velocity | m/s |

$\mathsf{\alpha}$ | Tire slip angle | rad |

${\alpha}_{f}$ | Front tire slip angle | rad |

${\alpha}_{r}$ | Rear tire slip angle | rad |

$\mu $ | Friction coefficient | - |

${\mu}_{s}$ | Friction coefficient of tire skids | - |

$r$ | Yaw rate | rad/s |

$\delta $ | Steering angle | rad |

$\beta $ | Sideslip angle | rad |

$m$ | Vehicle mass | kg |

${l}_{f}$ | Distance from the center of gravity (CG) to the front axle | M |

${l}_{r}$ | Distance from CG to the rear axle | M |

${I}_{zz}$ | Yaw moment of inertia | N·m/rad^{2} |

${C}_{xr}$ | Rear tire longitudinal slip angle | - |

$\kappa $ | Tire slip ratio | - |

Mean and Standard Deviation | Normalization Variables | ||||
---|---|---|---|---|---|

$\mathit{\mu}$ | $\mathit{\sigma}$ | Min | Max | ||

Input Data | ${x}_{e}$* | −0.0307 | 0.1819 | −0.2697 | 0.2674 |

$y$^{†} | 0.0080 | 0.2070 | −0.2688 | 0.2695 | |

$\beta $ | −0.4278 | 0.1314 | −0.5625 | −0.3534 | |

${\beta}_{eq}$^{o} | −0.4969 | 0.1213 | −0.6379 | −0.2630 | |

Output Data | $\delta $ | −0.1742 | 0.1216 | −0.3876 | 0.0651 |

^{†}Lateral position error with respect to the reference point;

^{o}Sideslip angle equilibrium point.

Mean and Standard Deviation | Normalization Variables | ||||
---|---|---|---|---|---|

$\mathit{\mu}$ | $\mathit{\sigma}$ | Min | Max | ||

Input Data | ${v}_{x}$ | −0.0307 | 0.1819 | −0.2697 | 0.2674 |

${v}_{y}$ | 0.0080 | 0.2070 | −0.2688 | 0.2695 | |

$\beta $ | −0.4278 | 0.1314 | −0.5625 | −0.3534 | |

$r$ | −0.4969 | 0.1213 | 0.6379 | 0.2630 | |

Output Data | ${v}_{w}$ * | −0.1742 | 0.1216 | −0.3876 | 0.0651 |

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**MDPI and ACS Style**

Lee, T.; Seo, D.; Lee, J.; Kang, Y.
Real-Time Drift-Driving Control for an Autonomous Vehicle: Learning from Nonlinear Model Predictive Control via a Deep Neural Network. *Electronics* **2022**, *11*, 2651.
https://doi.org/10.3390/electronics11172651

**AMA Style**

Lee T, Seo D, Lee J, Kang Y.
Real-Time Drift-Driving Control for an Autonomous Vehicle: Learning from Nonlinear Model Predictive Control via a Deep Neural Network. *Electronics*. 2022; 11(17):2651.
https://doi.org/10.3390/electronics11172651

**Chicago/Turabian Style**

Lee, Taekgyu, Dongyoon Seo, Jinyoung Lee, and Yeonsik Kang.
2022. "Real-Time Drift-Driving Control for an Autonomous Vehicle: Learning from Nonlinear Model Predictive Control via a Deep Neural Network" *Electronics* 11, no. 17: 2651.
https://doi.org/10.3390/electronics11172651