# Real-Time Modeling of Vehicle’s Longitudinal-Vertical Dynamics in ADAS Applications

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## Abstract

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## 1. Introduction

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- The longitudinal-vertical dynamics of an autonomous vehicle is modeled using an efficient semi-recursive multibody method. The dynamic properties of all components, e.g., the chassis, suspension, and tires, are considered and modeled.
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- The fork-arm removal technique is proposed using the rod-removal technique to further reduce the size of the equations of motion.
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- For ADAS applications, dynamic simulations based on a multibody model are executed in real-time on bumpy and sloping roads.
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- The accuracy and efficiency of the vehicle model are investigated in depth, and the model’s efficacy is confirmed.

## 2. Modeling of Vehicle Coupling Dynamics

## 3. Fork-Arm Removal Technique

## 4. Results in Accuracy and Efficiency

#### 4.1. Maneuver on a Bumpy Road

#### 4.2. Maneuver on a Sloping Road

#### 4.3. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 6.**The bumpy road environment for forward collision avoidance, adaptive cruise, or platooning.

**Figure 10.**The sloping road environment for forward collision avoidance, adaptive cruise, or platooning.

Loop-Closure Techniques | Decrease | Increase | ||||
---|---|---|---|---|---|---|

Constraints | Joints | Bodies | Relative Coordinates | Inertial Forces | External Forces | |

Cut a revolute joint | 5 | 1 | 0 | 1 | Null | Null |

Cut a spherical joint | 3 | 1 | 0 | 3 | Null | Null |

Remove a rigid rod | 1 | 2 | 1 | 6 | ✓ | ✓ |

Models | X-Axis (m and m/s) | Z-Axis (m and m/s) | Pitch (rad and rad/s) | |||
---|---|---|---|---|---|---|

Displacement | Velocity | Displacement | Velocity | Angle | Rate | |

14-DOF | 104.06 | 21.95 | 0.491 | −4.03 × 10${}^{-4}$ | −0.026 | −9.36 × 10${}^{-4}$ |

7-DOF | 104.06 | 21.95 | 0.491 | −4.04 × 10${}^{-4}$ | −0.026 | −9.37 × 10${}^{-4}$ |

7-DOF (fork-arm removal) | 103.79 | 21.88 | 0.491 | −3.91 × 10${}^{-4}$ | −0.026 | −1.50 × 10${}^{-3}$ |

Vehicle Models | |||
---|---|---|---|

14-DOF | 7-DOF | 7-DOF (Fork-Arm Removal) | |

CPU time (s) | 7.436 | 4.806 | 4.921 |

Time saving | 35.37% (↓) | 33.82% (↓) |

Models | X-Axis (m and m/s) | Z-Axis (m and m/s) | Pitch (rad and rad/s) | |||
---|---|---|---|---|---|---|

Displacement | Velocity | Displacement | Velocity | Angle | Rate | |

14-DOF | 74.89 | 14.95 | 0.4881 | −6.73 × 10${}^{-4}$ | −0.0367 | 7.08 × 10${}^{-4}$ |

7-DOF | 74.89 | 14.95 | 0.4881 | −6.7 4× 10${}^{-4}$ | −0.0367 | 7.08 × 10${}^{-4}$ |

7-DOF (fork-arm removal) | 74.89 | 14.94 | 0.4882 | −6.91 × 10${}^{-4}$ | −0.0367 | 6.86 × 10${}^{-4}$ |

Vehicle Models | |||
---|---|---|---|

14-DOF | 7-DOF | 7-DOF (Fork-Arm Removal) | |

CPU time (s) | 7.385 | 4.790 | 4.836 |

Time saving | 35.14% (↓) | 34.16% (↓) |

**Table 6.**Efficiency of a 14-DOF vehicle model with and without the fork-arm removal technique in 20 s simulation.

Techniques | Road Environment | |
---|---|---|

Bumpy Raod | Sloping Road | |

Without fork-arm removal | 7.436 s | 7.385 s |

With fork-arm removal | 7.268 s (2.26% ↓) | 7.237 s (2.00% ↓) |

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**MDPI and ACS Style**

Dai, W.; Pan, Y.; Min, C.; Zhang, S.-P.; Zhao, J.
Real-Time Modeling of Vehicle’s Longitudinal-Vertical Dynamics in ADAS Applications. *Actuators* **2022**, *11*, 378.
https://doi.org/10.3390/act11120378

**AMA Style**

Dai W, Pan Y, Min C, Zhang S-P, Zhao J.
Real-Time Modeling of Vehicle’s Longitudinal-Vertical Dynamics in ADAS Applications. *Actuators*. 2022; 11(12):378.
https://doi.org/10.3390/act11120378

**Chicago/Turabian Style**

Dai, Wei, Yongjun Pan, Chuan Min, Sheng-Peng Zhang, and Jian Zhao.
2022. "Real-Time Modeling of Vehicle’s Longitudinal-Vertical Dynamics in ADAS Applications" *Actuators* 11, no. 12: 378.
https://doi.org/10.3390/act11120378