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Analytical solutions of steady glide trajectory for hypersonic vehicle and planning application
HU Jinchuan , ZHANG Jing , CHEN Wanchun
School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2015-05-22; Accepted: 2015-09-11; Published online: 2016-05-25 12: 00
Corresponding author. Tel.:010-82339769 E-mail::wanchun_chen@buaa.edu.cn
Abstract: A kind of analytical solving method for the three-dimensional steady glide trajectory of the hypersonic vehicle is presented in this paper for the online planning problem. Firstly, the lift coefficient is separated into three components of lateral component, steady glide normal component and equilibrium glide normal component. Then, the longitudinal trajectory, lateral trajectory and velocity are decoupled in the dynamics. After that, the solutions of height, downrange and velocity azimuth angle are obtained by the analytical integration and the regular perturbation method, while the solutions of longitude, latitude and velocity are obtained by the Gaussian quadrature and single-step Runge-Kutta integration separately, and the accuracy of those solutions are improved by increasing the computational steps. Finally, a rapid steady glide trajectory programming algorithm is proposed based on the above analytical solutions, and the planning parameters are the lateral lift coefficient and steady glide normal lift coefficient. The simulation results show that those analytical solutions are more accurate than the Bell's solutions, and the planning algorithm only needs a small amount of calculation and obtains the result quickly, which can be used for online trajectory planning.

1 数学模型 1.1 坐标系定义

 图 1 广义赤道及坐标系 Fig. 1 Generalized equator and coordinate
 (1)

l11=cosωcosicosΩ-sinωsinΩ

l12=cosωcosisinΩ+sinωcosΩ

l13=-cosωsini

l21=-sinωcosicosΩ-cosωsinΩ

l22=-sinωcosisinΩ+cosωcosΩ

l23=sinωsini

l31=sinicosΩ

l32=sinisinΩ

l33=cosi

1.2 运动方程

 (2)
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1.3 运动方程解耦和分段求解策略

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 图 2 平稳滑翔弹道的分段 Fig. 2 Segment of steady glide trajectory
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 (13)
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2 滑翔弹道解析解

2.1 滑翔高度解析解

CN1为常数,对式(13)进行积分可得

 (19)

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2.2 射程解析解

 (22)

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2.3 基于正则摄动的弹道偏角解析解

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2.4 基于高斯积分公式的经度和纬度解析解

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2.5 基于单步龙格-库塔积分的滑翔速度解

 (44)

 图 3 归一化速度fv与弹道倾角γ的关系 Fig. 3 Relationship between normalized velocity fv and flight path angle γ
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 (46)

 (47)

wi0=ln Vi0

wif=ln Vif

Δγ=γif-γi0

k1=fK(wi0,γi0)

k2=fK(wi0+k1Δγ/2,γi0/2+γif/2)

k3=fK(wi0+k2Δγ/2,γi0/2+γif/2)

k4=fK(wi0+k3Δγ,γif)

 (48)
2.6 最优初始下滑角度解析解

 (49)

3 解析解精度校验 3.1 解析解与数值积分的精度对比

 参数 数值 h0/km 60 V0/(m·s-1) 7 000 θ0/(°) 0 ø0/(°) 0 ψ0/(°) π/2 s0/km 0 hf/km 30 γ0/(°) -0.068 4 CN1 -0.001 1 CY 0.4

 图 4 速度解析解与数值解对比 Fig. 4 Comparison between velocity analytical solution and numerical solution
 图 5 纵向弹道解析解与数值解对比 Fig. 5 Comparison between longitudinal trajectory analytical solution and numerical solution
 图 6 横向弹道解析解与数值解对比 Fig. 6 Comparison between lateral trajectory analytical solution and numerical solution
3.2 本文解析解与Bell解析解的精度对比

 图 7 弹道分为1段时的解析解精度对比 Fig. 7 Precision comparison of analytical solutions for 1 trajectory phase
 图 8 弹道分为2段时的解析解精度对比 Fig. 8 Precision comparison of analytical solutions for 2 trajectory phases
 图 9 弹道分为5段时的解析解精度对比 Fig. 9 Precision comparison of analytical solutions for 5 trajectory phases

 段数 CY 本文终端位置误差/km Bell终端位置误差/km 本文终端速度误差/(m·s-1) Bell终端速度误差/(m·s-1) 1 0.1 171.11 811.56 -1 940.12 -631.06 2 0.1 47.01 410.28 -642.10 -631.06 5 0.1 9.70 237.18 -44.41 -631.06 1 0.6 85.83 656.80 -90.72 -700.74 2 0.6 29.34 519.40 -41.89 -700.74 5 0.6 1.79 432.99 -3.89 -700.74

4 基于解析解的快速弹道规划

 参数 数值 hf/km 30 Vf/(m·s-1) 2 000 目标1经纬度/(°) (100,0) 目标2经纬度/(°) (115,0) 目标3经纬度/(°) (130,0) 500 qmax/Pa 60 000 nmax 2.5

 禁飞区 中心坐标/(°) 半径/km 禁飞区1 (40,-6) 891 禁飞区2 (90,6) 1 002

 图 10 滑翔段机动弹道规划流程 Fig. 10 Flowchart of gliding maneuvering trajectory planning

 图 11 不同任务下的攻角曲线 Fig. 11 Angle of attack curves for different missions
 图 12 不同任务下的倾侧角曲线 Fig. 12 Bank angle curves of different missions
 图 13 不同任务下的高度曲线 Fig. 13 Altitude curves of different missions
 图 14 不同任务下的速度曲线 Fig. 14 Velocity curves of different missions
 图 15 不同任务下的横向弹道 Fig. 15 Lateral trajectories of different missions
 图 16 不同任务下的再入走廊 Fig. 16 Reentry corridors of different missions

5 结 论

1) 提出了一种升力系数的分解方法,将其分解为横向分量、平衡滑翔纵向分量和平稳滑翔纵向分量3个部分,从而实现了对动力学方程中纵向运动方程、横向运动方程和速度方程的解耦。

2) 利用解耦的运动方程,首先通过解析积分获得了高度和射程的精确解析解,然后采用正则摄动法获得了较为精确的弹道偏角解析解,并在此基础上利用高斯积分公式获得了经度和纬度的解析解,最后采用单步龙格-库塔积分获得了速度的解。

3) 提出了分段逐步求解的方法来提高解析解的精度。仿真校验表明,在分段逐步求解的情况下,本文解析解的精度要大于Bell解析解的精度。

4) 采用本文解析解进行滑翔段弹道规划,不需要进行弹道积分,同时采用解析解还可先规划出符合突防要求的弹道,之后再进行速度校正,大大提高了弹道规划的效率。仿真校验表明,在普通台式机上规划1条平稳滑翔机动弹道的时间小于0.3 s。

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#### 文章信息

HU Jinchuan, ZHANG Jing, CHEN Wanchun

Analytical solutions of steady glide trajectory for hypersonic vehicle and planning application

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(5): 961-968
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0330