# Optimal Guidance Laws for a Hypersonic Multiplayer Pursuit-Evasion Game Based on a Differential Game Strategy

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## Abstract

**:**

## 1. Introduction

## 2. Engagement Formulation

#### 2.1. Problem Statement

#### 2.2. Equations of Motion

#### 2.3. Linearized Equations of Motion

**Remark**

**1.**

#### 2.4. Timeline

## 3. Guidance Schemes

#### 3.1. Cost Function

#### 3.2. Cost Function

#### 3.3. Proof of Saddle-Point Condition

## 4. Simulation and Analysis

#### 4.1. Simulation Setup

#### 4.2. Numerical Examples

- The interceptor adopts PN guidance law, the defender adopts PN guidance law, and the target adopts LQOGL (PNvPNvLQOGL);
- The interceptor adopts PN guidance law, the defender adopts LQOGL, and the target adopts LQOGL (PNvLQOGLvLQOGL);
- The interceptor adopts LQOGL, the defender adopts LQOGL, and the target adopts LQOGL (LQOGLvLQOGLvLQOGL).

**Case**

**1.**

**Remark**

**2.**

**Case**

**2.**

**Case**

**3.**

## 5. Conclusions

- In this research, a set of guidance laws for a hypersonic multiplayer pursuit-evasion game is derived based on linear-quadratic differential strategy. The energy cost, control saturation, chattering phenomenon, and aerodynamics were considered simultaneously. The satisfaction of saddle-point condition in a differential game was also proven theoretically.
- Nonlinear numerical examples of the multiplayer game were presented to validate the analysis. The advantage and efficiency of the proposed guidance were verified by the results. The LQOGLs exactly reduce the maneuverability requirement of the target in the pursuit-evasion game. Compared with the norm differential strategy, the proposed guidance strategy reduces the energy cost, alleviates the saturation problem, and avoids the chattering phenomenon, which guarantees task accomplishment and increases guidance phase stability.
- The performance of the interceptor showed that the proposed optimal guidance approach is able to complete the intercept mission if the interceptor possesses superior maneuverability. It is important to note that the saturation problem cannot be avoided completely when all the adversaries employ the LQOGL, since maneuverability is the most important factor in determining whether they will win or lose in the game. The interceptor or the target pair should make their best effort to attack or defend by exhaustedly performing maneuvers.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Simulation results of Case 1: (

**a**) trajectory; (

**b**) ZEM; (

**c**) target’s AOA; (

**d**) defender’s AOA; (

**e**) interceptor’s AOA.

**Figure 4.**Simulation results of Case 2: (

**a**) trajectory; (

**b**) ZEM; (

**c**) target’s AOA; (

**d**) defender’s AOA.

Parameters | Adversary | ||
---|---|---|---|

Interceptor | Defender | Target | |

Latitude | 0 | 0 | 0 |

Longitude | 0 deg | 0.0031 deg | 0.023 deg |

Altitude | 40.5 km | 40.1 km | 40 km |

Horizonal velocity | −2000 m/s | 3000 m/s | 3000 m/s |

Vertical velocity | 0 | 0 | 0 |

Maximum AOA | 35 deg | 35 deg | 35 deg |

Rate of AOA change | 6 deg/s | 5 deg/s | 3 deg/s |

Time constant | 0.005 s | 0.01 s | 0.01 s |

Killing radius | 0.3 m | 0.5 m | 0.5 m |

Engagements | Case 1 | Case 2 | Case 3 |
---|---|---|---|

PNvPNvLQOGL | PNvLQOGLvLQOGL | LQOGLvLQOGLvLQOGL | |

$\mathbf{ZE}{\mathbf{M}}_{\mathbf{ID}}$ | 15.01 m | −15.03 m | 15.01 m |

$\mathbf{ZE}{\mathbf{M}}_{\mathbf{IM}}$ | −0.17 m | −28.07 m | −0.17 m |

Result | Target is intercepted by interceptor | Interceptor is expelled by the defender. | Target is intercepted by the interceptor. |

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**MDPI and ACS Style**

Liang, H.; Li, Z.; Wu, J.; Zheng, Y.; Chu, H.; Wang, J.
Optimal Guidance Laws for a Hypersonic Multiplayer Pursuit-Evasion Game Based on a Differential Game Strategy. *Aerospace* **2022**, *9*, 97.
https://doi.org/10.3390/aerospace9020097

**AMA Style**

Liang H, Li Z, Wu J, Zheng Y, Chu H, Wang J.
Optimal Guidance Laws for a Hypersonic Multiplayer Pursuit-Evasion Game Based on a Differential Game Strategy. *Aerospace*. 2022; 9(2):97.
https://doi.org/10.3390/aerospace9020097

**Chicago/Turabian Style**

Liang, Haizhao, Zhi Li, Jinze Wu, Yu Zheng, Hongyu Chu, and Jianying Wang.
2022. "Optimal Guidance Laws for a Hypersonic Multiplayer Pursuit-Evasion Game Based on a Differential Game Strategy" *Aerospace* 9, no. 2: 97.
https://doi.org/10.3390/aerospace9020097