﻿ 主动防御飞行器的范数型微分对策制导律<sup>*</sup>
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1. 海军航空大学 岸防兵学院, 烟台 264001;
2. 北京航天长征飞行器研究所, 北京 100076

Norm differential game guidance law for active defense aircraft
GUO Zhiqiang1, SUN Qilong2, ZHOU Shaolei1, YAN Shi1
1. Costal Defense Academy, Naval Aviation University, Yantai 264001, China;
2. Beijing Institute of Space Long March Vehicle, Beijing 100076, China
Received: 2018-12-18; Accepted: 2019-01-18; Published online: 2019-01-23 09:53
Foundation item: National Natural Science Foundation of China (61273058)
Corresponding author. ZHOU Shaolei, E-mail:zhouslsd@sina.com
Abstract: For the pursuit-evasion problems between an active defense aircraft which can launch a defending missile from itself and an attacking missile, a guidance law for aircraft and defending missile is derived and analyzed based on the differential game theory. First, Optimal guidance strategies with bounded lateral controls of aircraft, defending missile and attacking missile are derived for the three players by using a norm performance index. Second, the conditions of a successful evasion for the aircraft and a successful interception for the defender are deduced, and the minimal evasion maneuver of the aircraft and the minimal interception maneuver of the defender are obtained. Finally, Nonlinear simulations are carried out to validate the guidance law proposed. It is verified that the aircraft can evade the attacking missile if its maneuver is equal or greater than the minimal evasion maneuver, and the defender can intercept the attacking missile if its maneuver is equal or greater than the minimal interception maneuver.
Keywords: aircraft     active defense     norm     differential game     guidance law

1 问题描述与运动学建模 1.1 协同追逃对策问题

1) 对策三方在运动中均可视为质点，且速度均为常值。

2) 对策三方之间的相对运动可在初始视线方向附近线性化。

3) 重力因素可忽略，且制导问题可以在俯仰平面和偏航平面内解耦。

 图 1 对策三方的平面相对运动关系 Fig. 1 Relative in-plane movement relationship of three players

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1.2 运动学建模与系统降阶

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uT=uTNuD=uDNuM=uMN，则系统在垂直于初始视线方向上的运动方程可写为

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2 范数型追逃对策 2.1 范数型的最优对策

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2.2 对策空间分析

2.2.1 飞行器-攻击弹的最优对策空间

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 图 2 飞行器-攻击弹的对策空间 Fig. 2 Aircraft-attacking missile game space

2.2.2 防御弹-攻击弹的最优对策空间

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 图 3 防御弹-攻击弹的对策空间(uDmax＜uMmax) Fig. 3 Defender-attacting missile game space (uDmax < uMmax)
3 飞行器和防御弹的对策条件

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1) 情况1

sign(ZMD(t))=－sign(ZMT(t))。此时飞行器、防御弹与攻击弹的视线角速度的符号相反，攻击弹的最优追踪策略(相对于飞行器)和最优逃逸策略(相对于防御弹)的符号也相反，攻击弹可在对飞行器进行追踪的同时对防御弹进行逃逸，即

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 图 4 2个零控脱靶量异号时的变化 Fig. 4 Evolution of two ZEM in case of opposite signs

2) 情况2

sign(ZMT(t))=sign(ZMD(t))。此时2个视线角速度的符号相同，根据式(24)第3式可得ttfMD时攻击弹的最优对策为uM*(t)=0，但这不意味着攻击弹不会采取机动，其意义是：①攻击弹可选择最优逃逸策略uMe*(t)=uMmax·sign(ZMD(t))躲避防御弹，攻击弹与防御弹间的脱靶量将增大，但其与飞行器之间的脱靶量也将同时增大；②攻击弹也可直接采取最优追踪策略uMp*(t)=uMmaxsign(ZMT(t))来攻击飞行器，此时攻击弹与飞行器间的脱靶量将减小，但其与防御弹之间的脱靶量也将同时减小。下面根据攻击弹的对策情况分别进行分析。

① 攻击弹首先采用最优逃逸策略而忽略对飞行器的拦截。此时攻击弹的制导策略为

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 图 5 攻击弹首先采用最优逃逸策略时的零控脱靶量 Fig. 5 Evolution of ZEM while attacking missile performs optimal evasion first

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② 攻击弹直接采用最优追踪策略而忽略对防御弹的躲避。此时攻击弹的制导策略为

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 图 6 攻击弹直接采用最优追踪策略时的零控脱靶量 Fig. 6 Evolution of ZEM while attacking missile performs optimal pursuit directly

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4 仿真验证

4.1 攻击弹首先采用最优逃逸策略

 仿真参数 飞行器 防御弹 攻击弹 初始位置/m (0, 0) (0, 0) (9 000, -200) 初始航向/(°) 60 20 0 速度/(m·s-1) 300 800 700

1) 假设飞行器、防御弹和攻击弹的最大加速度指令分别为uTmax=50 m/s2uDmax=80 m/s2uMmax=180 m/s2。基于上述参数，在仿真中可以发现飞行器的控制指令满足式(36)的约束，此时三方的运动轨迹如图 7所示。从图 7可以明显看出，当飞行器的最大加速度指令满足最小机动条件时可以成功逃脱攻击弹，飞行器并不严格依赖于防御弹提前将攻击弹拦截。这一点与文献[8]不同，该文中当飞行器速度小于攻击弹速度时，飞行器始终会被攻击弹命中，飞行器严格依赖防御弹的存在。图 8为2个剩余对策时间随时间变化的示意图。可以看出，由于攻击弹先采用逃逸策略，剩余对策时间tgoMT与时间t已完全不是线性的关系，即使攻击弹切换成最优追踪策略，由于飞行器满足实现逃逸的最小机动条件，新的碰撞三角形无法建立，tgoMT与时间t仍然不具有线性关系。

 图 7 攻击弹首先逃逸时的三方运动轨迹(uTmax大于其最小逃逸机动) Fig. 7 Trajectories of three players while attacking missile evades first (uTmaxis larger than minimal evasion maneuver)
 图 8 攻击弹首先逃逸时2个剩余对策时间的变化(uTmax大于其最小逃逸机动) Fig. 8 Evolution of two time-to-go while attacking missile evades first (uTmax is larger than minimal evasion maneuver)

2) 假设飞行器、防御弹和攻击弹的最大加速度指令分别为uTmax=50 m/s2uDmax=80 m/s2uMmax=200 m/s2。基于上述参数，在仿真过程中可以发现飞行器的控制指令不满足式(36)的约束，此时三方的运动轨迹如图 9所示。从图 9可以看出，由于攻击弹的最大加速度指令增大，导致飞行器的最大加速度不满足式(36)所示的最小机动条件，使得飞行器不能逃脱攻击弹的拦截(攻击弹对飞行器最终的脱靶量为yMT=4.481 m＜RM)。

 图 9 攻击弹首先逃逸时的三方运动轨迹(uTmax小于其最小逃逸机动) Fig. 9 Trajectories of three players while attacking missile evades first (uTmax is smaller than minimal evasion maneuver)

 图 10 攻击弹首先逃逸时2个剩余对策时间的变化(uTmax小于其最小逃逸机动) Fig. 10 Evolution of two time-to-go while attacking missile evades first (uTmax is smaller than minimal evasion maneuver)
4.2 攻击弹直接采用最优追踪策略

 仿真参数 飞行器 防御弹 攻击弹 初始位置/m (0, 0) (0, 0) (3 000, -200) 初始航向/(°) 12 7 0 速度/(m·s-1) 300 800 700

1) 假设飞行器、防御弹和攻击弹的最大加速度指令分别为uTmax=50 m/s2uDmax=80 m/s2uMmax=160 m/s2。基于上述参数，在仿真过程中可以发现防御弹的加速度指令满足式(42)的约束，此时三方的运动轨迹如图 11所示。从图 11可以看出，防御弹在攻击弹命中飞行器之前成功将攻击弹拦截(脱靶量为yMD=5.902 m＜RD)。图 12为2个剩余对策时间随时间变化的示意图。可以看出，2个剩余对策时间与时间几乎呈线性关系，而剩余对策时间tgoMD小于tgoMT，防御弹在攻击弹命中飞行器之前对其实施了拦截。

 图 11 攻击弹直接追踪时的三方运动轨迹(uDmax大于其最小拦截机动) Fig. 11 Trajectories of three players while attacking missile pursues directly (uDmax is larger than minimal interception maneuver)
 图 12 攻击弹直接追踪时2个剩余对策时间的变化(uDmax大于其最小拦截机动) Fig. 12 Evolution of two time-to-go while attacking missile pursues directly (uDmax is larger than minimal interception maneuver)

2) 假设飞行器、防御弹和攻击弹的最大加速度指令分别为uTmax=40 m/s2uDmax=40 m/s2uMmax=120 m/s2。基于上述参数，在仿真过程中可以发现防御弹的加速度指令不满足式(42)的约束，此时三方的运动轨迹如图 13所示。可以看出，防御弹未能在攻击弹命中飞行器之前将其拦截(防御弹对攻击弹脱靶量为yMD=15.199 m>RD)。图 14为2个剩余对策时间与时间的变化关系。

 图 13 攻击弹直接追踪时的三方运动轨迹(uDmax小于其最小拦截机动) Fig. 13 Trajectories of three players while attacking missile pursues directly (uDmax is smaller than minimal interception maneuver)
 图 14 攻击弹直接追踪时2个剩余对策时间的变化(uDmax小于其最小拦截机动) Fig. 14 Evolution of two time-to-go while attacking missile pursues directly (uDmax is smaller than minimal interception maneuver)
5 结论

1) 针对具有主动防御能力的飞行器面对来袭导弹发射一枚防御弹进行防御的制导问题，给出了一种基于范数型性能指标的最优制导律。

2) 对攻击弹采用不同制导策略的情况，分别推导了飞行器能够逃离攻击弹和防御弹能够拦截攻击弹的条件。

3) 攻击弹采取先逃逸后追踪的策略时，飞行器若满足其最小逃逸机动条件则可逃脱攻击弹的拦截，否则飞行器将被攻击弹拦截。

4) 攻击弹直接采取追踪策略时，防御弹若满足其最小拦截机动条件则可拦截攻击弹，否则防御弹将无法在飞行器被命中前将攻击弹拦截。

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文章信息

GUO Zhiqiang, SUN Qilong, ZHOU Shaolei, YAN Shi

Norm differential game guidance law for active defense aircraft

Journal of Beijing University of Aeronautics and Astronsutics, 2019, 45(9): 1787-1796
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0738