# A Real-Time Trajectory Optimization Method for Hypersonic Vehicles Based on a Deep Neural Network

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Three-DOF Dynamic Model Development

#### 2.2. Problem Statement

#### 2.2.1. Dynamic Pressure Constraint

#### 2.2.2. Heat Flow Constraint

#### 2.2.3. Overload Constraint

#### 2.3. Research Ideas

## 3. Sample Data Generation Method Based on Chebyshev Pseudo-Spectral Method

#### 3.1. Chebyshev Pseudo-Spectral Method

#### 3.2. Training Data Generation

## 4. Neural Network Design and Training

Algorithm 1 Imitation learning |

1: Initialize network weighting values $\omega $ and $\alpha $ 2: Set $lr=0.0001,n\_epochs=30,batch\_size=256$ 3: for $epoch=1,n\_epochs$ do 4: for $batch\_index=1,n\_batches$ do 5: obtain the optimal sequence of pseudo-spectral method ballistic $\left[s,a\right]$ 6: $net\_in=\left[{s}_{0},{s}_{f},{s}_{current}\left],net\_out=\right[\alpha ,\sigma \right]$ data feature extraction and normalization 7: update network parameters using Adam algorithm: $loss=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}}{[f\left({x}_{i}\right)-{y}_{i}]}^{2}$ 8: end for 9: Randomly generate a ballistic path by pseudo-spectral method $\left[{s}_{1},{a}_{1}\right]$ set up data buffering ℜ 10: $if$$dx\_angle{0.1}^{\xb0}$ do 11: $\mathrm{use}\mathrm{neural}\mathrm{network},\mathrm{input}\left[{s}_{0},{s}_{f},{s}_{current}\right]$$,\mathrm{output}\left[\alpha ,\sigma \right]$ 12: $\mathrm{put}\left[\alpha ,\sigma \right]\mathrm{into}\mathrm{environment()},\mathrm{obtain}{s}_{current+1}$ 13: $\mathrm{store}\mathrm{samples}\left[{s}_{0},{s}_{f},{s}_{current}\right]$, $\left[\alpha ,\sigma \right]$ to $\Re $, update ${s}_{current}$ 14: end |

## 5. Simulations and Result Analysis

^{2}. The CAV-H had a high maximum lift-to-drag ratio of E* = 3.24, and the corresponding lift coefficient ${C}_{L}^{*}$ was 0.45. The pneumatic reference area was ${\mathrm{s}}_{\mathrm{ref}}$= 0.8. The gravitational acceleration was${g}_{0}$= 9.8 m/s

^{2}, and the Earth radius was considered to be ${R}_{0}$= 6378 km.

#### 5.1. Generation of the Training Data

#### 5.2. Training Process of the DNN

#### 5.3. Random Single Trajectory Error Analysis

#### 5.4. Validation with Vehicle Dynamics Model

#### 5.5. Monte Carlo Simulation Verification

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**State of training data. (

**a**) Height–time curve; (

**b**) Longitude–time curve; (

**c**) Latitude–time curve; (

**d**) velocity–time curve.

**Figure 4.**Control of training data. (

**a**) Generalized lift coefficient–time curve; (

**b**) Heeling Angle–time curve.

**Figure 5.**Training results of deep neural network. (

**a**) The training loss for sigmoid activation function epochs; (

**b**) the training loss for the ReLU activation function; (

**c**) the test loss for the sigmoid activation function; (

**d**) the test loss for the ReLU activation function.

**Figure 6.**Comparison of the predicted and expected values of the generalized coefficient of the lift.

**Figure 8.**Comparison of the predicted and expected values of the generalized coefficient of the lift.

Parameter | Value Range |
---|---|

Initial height h_{0} | 41 km~46 km |

Initial longitude θ_{0} | −2°~2° |

Initial latitude φ_{0} | −2°~2° |

Initial velocity V_{0} | 5300 m/s |

Initial track angle γ_{0} | 0° |

Initial course angle ψ_{0} | 90° |

Final longitude θ_{f} | 38°~42° |

Final latitude φ_{f} | 18°~22° |

Parameter | $\dot{\mathit{Q}}{\left(\frac{\mathbf{kW}}{{\mathit{m}}^{2}}\right)}_{\mathit{m}\mathit{a}\mathit{x}}$ | $\overline{\mathit{q}}{\left(\mathbf{kPa}\right)}_{\mathit{m}\mathit{a}\mathit{x}}$ | $\mathit{n}{\left({\mathit{g}}_{0}\right)}_{\mathit{m}\mathit{a}\mathit{x}}$ | Generalized Lift Coefficient | Heeling Angle (°) |
---|---|---|---|---|---|

Value | 2000 | 500 | 3 | $0\le \lambda \le 2$ | $-80\le \sigma \le 80$ |

Actual Vehicle Position | Predicted Vehicle Position | Position Error | |
---|---|---|---|

Altitude (m) | 30,151 | 30,940 | 789 |

Longitude (°) | 34.84 | 34.74 | 0.10 |

Latitude (°) | 18.16 | 18.19 | 0.03 |

Velocity (m/s) | 2267 | 2271 | 4 |

The Absolute Terminal Longitude | The Absolute Terminal Latitude | The Absolute Terminal Range Angle | |
---|---|---|---|

error (°) | 0.042 | 0.125 | 0.126 |

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**MDPI and ACS Style**

Wang, J.; Wu, Y.; Liu, M.; Yang, M.; Liang, H.
A Real-Time Trajectory Optimization Method for Hypersonic Vehicles Based on a Deep Neural Network. *Aerospace* **2022**, *9*, 188.
https://doi.org/10.3390/aerospace9040188

**AMA Style**

Wang J, Wu Y, Liu M, Yang M, Liang H.
A Real-Time Trajectory Optimization Method for Hypersonic Vehicles Based on a Deep Neural Network. *Aerospace*. 2022; 9(4):188.
https://doi.org/10.3390/aerospace9040188

**Chicago/Turabian Style**

Wang, Jianying, Yuanpei Wu, Ming Liu, Ming Yang, and Haizhao Liang.
2022. "A Real-Time Trajectory Optimization Method for Hypersonic Vehicles Based on a Deep Neural Network" *Aerospace* 9, no. 4: 188.
https://doi.org/10.3390/aerospace9040188