基于Hamilton-Jacobi方程的飞行器机动动作可达集分析

刘瑛; 杜光勋; 全权; 田云川

doi:10.16383/j.aas.2016.c140888

基于Hamilton-Jacobi方程的飞行器机动动作可达集分析

doi: 10.16383/j.aas.2016.c140888

刘瑛^1,2, ,,
杜光勋^1,,
全权^1,,
田云川^2,

1.
北京航空航天大学自动化科学与电气工程学院北京 100191
2.
中国人民解放军95949部队沧州 061000

基金项目:

国家自然科学基金 61473012

详细信息

作者简介:
杜光勋北京航空航天大学自动化学院博士后.2009年获得北京航空航天大学学士学位.2015年获得北京航空航天大学博士学位.主要研究方向为可靠飞行控制, 飞行安全, 可达集分析.E-mail:dgx@buaa.edu.cn

全权北京航空航天大学自动化学院副教授.2004年获得北京航空航天大学学士学位.2010年获得北京航空航天大学博士学位.主要研究方向为视觉导航, 可靠飞行控制.E-mail:qq_buaa@buaa.edu.cn

田云川 95949部队飞行员.2002年获得空军航空大学学士学位.主要研究方向为飞行安全.E-mail:tianyunchuan@sohu.com

通讯作者:
刘瑛北京航空航天大学自动化学院博士后.2004年获得空军工程大学学士学位.2007年获得空军工程大学硕士学位.2014年获得天津大学博士学位.主要研究方向为飞行安全, 可达集分析, 航迹控制.本文通信作者E-mail:liuying204@buaa.edu.cn

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出版历程
- 收稿日期: 2014-12-29
- 录用日期: 2016-01-04
- 刊出日期: 2016-03-01

Reachability Calculation for Aircraft Maneuver Using Hamilton-Jacobi Function

LIU Ying^{1,2
, ,},
DU Guang-Xun^1
,,
QUAN Quan^1
,,
TIAN Yun-Chuan^2
,

1.
School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191
2.
95949 Army of People's Liberation Army Air Force, Cangzhou 061000

Funds:

National Natural Science Foundation of China 61473012

More Information

Author Bio:
Postdoctor at Beihang University. He received his bachelor and Ph.D. degrees from Beihang University in 2009 and 2015, respectively. His research interest covers reliable flight control, flight safety, and reachability analysis. E-mail:

Associate professor at Beihang University. He received his bachelor and Ph.D. degrees from Beihang University in 2004 and 2010, respectively. His research interest covers vision-based navigation and reliable flight control.E-mail:

Pilot at the 95949 Army of People's Liberation Army Air Force. He received his bachelor degree from Aviation University of Air Force in 2002. His main research interest is flight safety..E-mail:

Corresponding author: LIU Ying Postdoctor at Beihang University. She received her bachelor and master degrees from Air Force Engineering University in 2004 and 2007, respectively. She received her Ph.D. degree in Tianjin University in 2014. Her research interest covers flight safety, reachability analysis, and trajectory control. Corresponding author of this paper.E-mail:liuying204@buaa.edu.cn

摘要

摘要: 为了给驾驶员完成标准机动动作提供决策支持, 提出一种使用哈密尔顿-雅克比(Hamilton-Jacobi)方程求解机动动作可行状态空间的研究方法.使用关键点将机动动作划分为不同阶段, 将各关键点的标准状态约束作为目标集, 逆时间求解目标集对应的可达集得到各阶段的边界状态范围, 目标集和可达集均由零水平集表示.使用该方法得到斤斗动作三维度运动模型下各阶段的可达集及斤斗动作的可行状态空间, 为了使运动模型的控制量与驾驶员实际操纵更为接近, 构建了以迎角变化率为控制量的四维度运动模型, 在此基础上对斤斗动作各阶段的可达集进行了分析.
- 机动动作 /
- 可达集 /
- 哈密尔顿-雅克比方程 /
- 水平集方法 /
- 斤斗动作
Abstract: To help the pilots make decisions for aircraft maneuvers, a reachability analysis method using Hamilton-Jacobi partial differential equation is proposed in this paper. The aircraft maneuver is divided into several phases based on the key points. The restricted set of aircraft states at each key point is seen as the target set, and then a reachable set can be obtained by solving the Hamilton-Jacobi partial differential equation. The target set and the reachable set are both described by zero level set. For the three dimensional dynamical model of the aircraft, the reachable set of each key point and the feasible state space of the whole loop maneuver have been achieved. In order to make the analysis results more easy to use in practice, a four dimensional dynamical model of the aircraft which takes the rate of the angle of attack as the control parameter is formulated. Based on the four dimensional dynamical model, the reachable set is analyzed by means of the proposed method.
- Maneuver /
- reachable set /
- Hamilton-Jacobi equation /
- level set method /
- loop maneuver

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图 1 目标集与可达集的关系

Fig. 1 The relationship between target set and reachable set of the stages

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图 2 斤斗动作阶段划分

Fig. 2 The stages division of the loop maneuver

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图 3 三维度可达集顶视图及前视图

Fig. 3 The three dimensions reachable set of each stage and the corresponded top view and front view

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图 4 斤斗动作各阶段的安全状态空间

Fig. 4 The safe state space of the loop maneuver

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图 5 第一阶段四维度可达集

Fig. 5 The reachable set of the first stage under the condition of four dimension motion equation

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图 6 第二阶段四维度可达集

Fig. 6 The reachable set of the second stage under the condition of four dimension motion equation

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图 7 第三阶段四维度可达集

Fig. 7 The reachable set of the third stage under the condition of four dimension motion equation

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表 1 关键点状态约束条件

Table 1 quad The range of state variables at the key points

Key points	Three dimensional dynamical model	Four dimensional dynamical model
1	$ 213 {\rm {m/s}}\leq V\leq 231 m/s$ $ 30^{\circ}\leq\gamma\leq40^{\circ}$ $ 2 450 {\rm m}\leq h\leq2 550 {\rm m}$	$ 213 {\rm {m/s}}\leq V\leq 231 $ m/s $ 30^{\circ}\leq\gamma\leq 40^{\circ}$ $ 2 450 {\rm m}\leq h\leq 2 550 $ m $ 6^{\circ}\leq\alpha\leq 10^{\circ}$
2	$ 213 {\rm {m/s}}\leq V\leq231$ m/s $ 110^{\circ}\leq\gamma\leq120^{\circ}$ $ 3 500 {\rm m}\leq h\leq3 700 $ m	$ 213 {\rm {m/s}}\leq V\leq231 $ m/s $ 110^{\circ}\leq\gamma\leq120^{\circ}$ $ 3 500 {\rm m}\leq h\leq3 700 $ m $ 10^{\circ}\leq\alpha\leq13^{\circ}$
3	$ 110 {\rm {m/s}}\leq V\leq130$ m/s $ 170^{\circ}\leq\gamma\leq180^{\circ}$ $ 3 900 {\rm m}\leq h\leq4 000 {\rm m}$	$ 110 {\rm {m/s}}\leq V\leq130$ m/s $ 170^{\circ}\leq\gamma\leq180^{\circ}$ $ 3 900 {\rm m}\leq h\leq4 000$ m $ 8^{\circ}\leq\alpha\leq11^{\circ}$

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表 2 斤斗动作飞行包线及控制量的取值范围

Table 2 Aerodynamic envelope and the range of control variables

	Aerodynamic envelope	Control
Three dimensional dynamical model	$ 90 {\rm {m/s}}\leq V\leq240$ m/s $ 0^{\circ}\leq\gamma\leq180^{\circ}$ $ 1 800 {\rm m}\leq h\leq4 200$ m	$-2^{\circ}\leq\alpha\leq21^{\circ}$
Four dimensional dynamical model	$ 90 {\rm {m/s}}\leq V\leq240$ m/s $ 0^{\circ}\leq\gamma\leq180^{\circ}$ $ 1 800 {\rm m}\leq h\leq4 200$ m $ -2^{\circ}\leq\alpha\leq21^{\circ}$	$ -0. 35\leq c\leq0. 35$

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表 3 状态空间网格划分

Table 3 Grid division of the state space

Parameters	Speed	Flight path angle	Height	AOA
Range	$ 90 {\rm {m/s}}\leq V\leq240$ m/s	$ 0^{\circ}\leq\gamma\leq180^{\circ}$	$ 1 800 {\rm m}\leq h\leq4 200$ m	$ -2^{\circ}\leq\alpha\leq21^{\circ}$
Grid number	250	180	500	24
Step size	1	1	4.8	1

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表 4 可达集对应的状态参数范围

Table 4 The range of state variables corresponding with the reachable set

Stage	Range of the state parameters
1	$ 213 {\rm {m/s}}\leq V\leq236$ m/s $ 25^{\circ}\leq\gamma\leq40^{\circ}$ $ 2 440 {\rm m}\leq h\leq2 550 $ m
2	$ 150 {\rm {m/s}}\leq V\leq235$ m/s $ 90^{\circ}\leq\gamma\leq120^{\circ}$ $ 3 410 {\rm m}\leq h\leq3 700$ m
3	$ 110 {\rm {m/s}}\leq V\leq130$ m/s $ 170^{\circ}\leq\gamma\leq180^{\circ}$ $ 3 900 {\rm m}\leq h\leq4 000$ m

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