﻿ 基于自适应伪谱法的高超声速飞行器再入轨迹优化<sup>*</sup>
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Reentry trajectory optimization for hypersonic vehicle based on adaptive pseudospectral method
REN Pengfei, WANG Hongbo, ZHOU Guofeng
China Academy of Launch Vehicle Technology, Beijing 100076, China
Received: 2019-04-15; Accepted: 2019-06-29; Published online: 2019-07-09 16:02
Corresponding author. WANG Hongbo, E-mail: caltwang2017@126.com
Abstract: In order to solve the reentry trajectory optimization for hypersonic vehicle, a three-degree-of-freedom reentry kinematic equation considering the earth rotation was established, and a reentry constraint model was built, which took America's universal space vehicle as the object. Firstly, Legendre-Gauss-Radau points were employed to transform the continuous-time optimal control problem into a nonlinear programming problem, and three typical optimization problems including maximum downrange, maximum crossrange and minimum change rate of path angle were discretized. Secondly, an estimation relational expression of relative error relying on decay rate of Legendre polynomial approximation was established, and an effective adaptive mesh refinement strategy was proposed. Finally, three typical reentry trajectory optimization problems were well solved. The simulation results show that the result solved by the proposed method is consistent with the integral of variable-step-size Runge-Kutta-Fehlberg method. Compared to the traditional adaptive pseudospectral method, the proposed method achieves more reasonable mesh refinement, less mesh iteration numbers, faster computation speed and less sensitivity to the user-specified parameters.
Keywords: hypersonic vehicle     complex constraints     trajectory optimization     adaptive pseudospectral method     Legendre coefficient

1 再入轨迹优化问题 1.1 三自由度再入运动方程

 (1)

 (2)

 (3)
1.2 约束条件

1) 端点约束

 (4)

2) 过程约束

 (5)

3) 控制约束

 (6)

1.3 目标函数

 (7)

2 Bolza型连续时间最优控制问题

 (8)

 (9)

3 多区间LGR伪谱法

 (10)

LGR伪谱法的配点采用τ∈[－1, 1]上的N个LGR点或反转的LGR点，本文选用前者。LGR点为(τ+1)(PN(τ)+PN－1(τ))的根，其中:PN(τ)为N阶Legendre多项式。因此对t进行区间转化，可使每个子区间满足Legendre正交多项式的定义区间。

 (11)

 (12)

 (13)

 (14)

 (15)

 (16)

 (17)
 (18)

 (19)
4 自适应网格重构技术

4.1 误差评估

 (20)

 (21)

 (22)

Sk内最大相对误差为

 (23)
4.2 Legendre多项式近似

 (24)

 (25)

 (26)

 (27)

 (28)

 (29)

 (30)

 (31)

 (32)

 (33)
4.3 光滑性判定

4.4 多项式阶数更新

υ>υε时，根据式(34)可计算当前网格的求解误差：

 (34)

 (35)

 (36)

 (37)
4.5 区间更新

υυε时，考虑到υ接近于0时，由式(33)确定区间分段数会产生很大的区间数量，从而导致NLP问题的规模过大，使得求解效率降低。因此采用υε代替υ，根据式(36)估计所需的多项式阶数：

 (38)

 (39)
4.6 网格缩减

4.7 算法流程

 图 1 自适应伪谱法流程图 Fig. 1 Flowchart of adaptive pseudospectral method
5 算例分析

 算例 状态 权重系数 1 最大纵程 w1=1，w2=0，w3=0 2 最大横程 w1=0，w2=1，w3=0 3 最小航迹角变化 w1=0，w2=0，w3=1

 参数 初始 末端 高度h/km 60 20 经度θ/(°) 0 自由 纬度ϕ/(°) 0 自由 速度V/(km·s-1) 5 1 航迹角γ/(°) 0 5 航向角ψ/(°) 90 自由

 图 2 3D轨迹 Fig. 2 Three-dimensional trajectory
 图 3 航迹角变化 Fig. 3 Path angle versus time
 图 4 再入走廊边界 Fig. 4 Boundary of reentry corridor
 图 5 迎角与倾侧角变化 Fig. 5 Angle of attack and heeling angle versus time
 图 6 迎角与倾侧角变化率变化 Fig. 6 Change rate of angle of attack and heeling angle versus time

 算法 算例 误差容限ε 衰减容限υε 区间配点限制 配点总数 区间总数 网格迭代次数 平均求解时间(5次)/s 性能指标 hpL 1 10-4 0.25 [3, 10] 130 33 3 3.520 1 -0.112 9 hpL 1 10-4 0.5 [3, 10] 122 37 4 3.878 5 -0.112 8 hpL 1 10-5 0.25 [3, 10] 209 30 5 7.467 0 -0.112 8 hp 1 10-4 [3, 10] 132 39 5 4.462 8 -0.112 8 hp 1 10-5 [3, 10] 250 58 13 56.272 2 -0.112 8 ph 1 10-4 [3, 10] 105 20 4 42.716 9 -0.112 6 ph 1 10-4 [3, 6] 114 28 4 4.461 6 -0.112 6 ph 1 10-5 [3, 6] 273 67 7 10.378 1 -0.112 9 hpL 2 10-4 0.25 [3, 10] 79 20 4 3.270 8 -0.014 8 hpL 2 10-4 0.5 [3, 10] 96 26 4 3.784 4 -0.014 8 hpL 2 10-5 0.25 [3, 10] 126 23 3 2.997 0 -0.014 8 hp 2 10-4 [3, 10] 82 23 4 3.059 3 -0.014 8 hp 2 10-5 [3, 10] 155 40 7 10.297 3 -0.014 8 ph 2 10-4 [3, 10] 79 20 4 6.972 8 -0.014 8 ph 2 10-4 [3, 6] 81 22 4 5.901 6 -0.014 8 ph 2 10-5 [3, 6] 152 34 5 5.333 2 -0.014 8 hpL 3 10-4 0.25 [3, 10] 65 20 4 11.745 5 0.467 8×10-4 hpL 3 10-4 0.5 [3, 10] 65 20 4 12.306 4 0.4678×10-4 hpL 3 10-5 0.25 [3, 10] 91 21 6 15.380 1 0.467 2×10-4 hp 3 10-4 [3, 10] 70 22 4 34.250 6 0.467 2×10-4 hp 3 10-5 [3, 10] 86 26 8 19.334 9 0.467 2×10-4 ph 3 10-4 [3, 10] 66 20 3 83.696 5 0.467 8×10-4 ph 3 10-4 [3, 6] 69 22 4 21.790 3 0.467 8×10-4 ph 3 10-5 [3, 6] 89 26 5 17.796 7 0.467 2×10-4

6 结论

1) 本文算法结果与变步长Runge-Kutta-Fehlberg法积分结果一致。

2) 本文算法较2种传统自适应伪谱法，配点与区间分配更合理，计算效率更高，且对于人工参数不敏感，具有较好的工程应用价值。

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#### 文章信息

REN Pengfei, WANG Hongbo, ZHOU Guofeng

Reentry trajectory optimization for hypersonic vehicle based on adaptive pseudospectral method

Journal of Beijing University of Aeronautics and Astronsutics, 2019, 45(11): 2257-2265
http://dx.doi.org/10.13700/j.bh.1001-5965.2019.0165