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初值约束与两点边值约束轨道动力学方程的快速数值计算方法

张哲 代洪华 冯浩阳 汪雪川 岳晓奎

张哲, 代洪华, 冯浩阳, 汪雪川, 岳晓奎. 初值约束与两点边值约束轨道动力学方程的快速数值计算方法. 力学学报, 2022, 54(2): 1-14 doi: 10.6052/0459-1879-21-336
引用本文: 张哲, 代洪华, 冯浩阳, 汪雪川, 岳晓奎. 初值约束与两点边值约束轨道动力学方程的快速数值计算方法. 力学学报, 2022, 54(2): 1-14 doi: 10.6052/0459-1879-21-336
Zhang Zhe, Dai Honghua, Feng Haoyang, Wang Xuechuan, Yue Xiaokui. Efficient numerical method for orbit dynamic functions with initial value and two-point boundary-value constraints. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(2): 1-14 doi: 10.6052/0459-1879-21-336
Citation: Zhang Zhe, Dai Honghua, Feng Haoyang, Wang Xuechuan, Yue Xiaokui. Efficient numerical method for orbit dynamic functions with initial value and two-point boundary-value constraints. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(2): 1-14 doi: 10.6052/0459-1879-21-336

初值约束与两点边值约束轨道动力学方程的快速数值计算方法

doi: 10.6052/0459-1879-21-336
基金项目: 国家自然科学基金(12072270,U2013206)和科技部重点研发计划(2021 YFA0717100)资助项目
详细信息
    作者简介:

    代洪华, 教授, 主要研究方向: 在轨服务动力学仿真与高性能计算, 航天器动力学与控制. E-mail:hhdai@nwpu.edu.cn

  • 中图分类号: V412.4 + 1

EFFICIENT NUMERICAL METHOD FOR ORBIT DYNAMIC FUNCTIONS WITH INITIAL VALUE AND TWO-POINT BOUNDARY-VALUE CONSTRAINTS

  • 摘要: 轨道动力学快速计算是航天工程中的基础问题, 广泛存在于轨道设计、空间抓捕以及深空探测等任务中. 基于有限差分原理的经典数值积分算法, 由于精度严重依赖小积分步长, 难以满足航天器在轨快速计算需求. 针对该问题, 提出一种局部配点反馈迭代算法, 该算法能高效解算受到初值约束和两点边值约束的轨道动力学方程. 基于Picard迭代公式建立数值算法以避免计算雅可比矩阵, 引入误差反馈以加快迭代收敛速度. 通过时域配点法消除迭代公式推导中的复杂符号运算, 使迭代公式更为简洁. 结合拟线性化法及叠加法, 算法能对两点边值约束下的Lambert问题实现高效解算. 基于ph网格细化法建立计算参数自适应调节算法, 能进一步增强局部配点反馈迭代法的大步长计算优势. 通过求解二体动力学模型下的递推轨道, 摄动Lambert问题以及限制性三体动力学模型下的转移轨道验证了算法有效性. 仿真结果表明, 在相同计算精度下, 局部配点反馈迭代算法计算速度比拟线性化-局部变分迭代法提高1.5倍以上, 引入变参数方案能够进一步将算法计算速度提高6倍以上.

     

  • 图  1  局部配点反馈迭代法示意图

    Figure  1.  Illustration of LCFI

    图  2  LCFI与LVIM所得递推轨道对比

    Figure  2.  Comparison between orbits propagated via LCFI and LVIM

    图  3  LCFI与LVIM所得递推轨道绝对误差

    Figure  3.  Absolute error of the orbits propagated via LCFI and LVIM

    图  4  LCFI与LVIM所得递推轨道相对误差

    Figure  4.  Relative error of the orbits propagated via LCFI and LVIM

    图  5  LCFI与LVIM所得Lambert转移轨道结果对比

    Figure  5.  Comparison between Lambert orbits obtained via LCFI and LVIM

    图  6  LCFI与LVIM所得Lambert转移轨道绝对误差

    Figure  6.  Absolute error of Lambert orbits obtained via LCFI and LVIM

    图  7  LCFI与LVIM所得Lambert转移轨道相对误差

    Figure  7.  Relative error of Lambert orbits obtained via LCFI and LVIM

    图  8  LCFI与LVIM轨道计算结果对比

    Figure  8.  Comparison between orbits obtained via LCFI and LVIM

    图  9  LCFI与LVIM对平面圆形限制性三体模型约束下转移轨道计算绝对误差

    Figure  9.  Absolute error of transfer orbits under the constraint of circular restricted three body model obtained via LCFI and LVIM

    图  10  LCFI与LVIM对平面圆形限制性三体模型约束下转移轨道计算相对误差

    Figure  10.  Relative error of transfer orbits under the constraint of circular restricted three body model obtained via LCFI and LVIM

    图  11  变参数局部变分迭代算法流程图

    Figure  11.  Flow chart of adaptive local collocation feedback iteration method

    图  12  变参数LCFI及定参数LCFI圆轨道递推计算结果

    Figure  12.  Circular orbit obtained via Adaptive LCFI and LCFI

    图  13  变参数LCFI及定参数LCFI椭圆轨道递推计算结果

    Figure  13.  Elliptical orbit obtained via adaptive LCFI and LCFI

    图  14  变参数LCFI圆轨道递推计算步长变化

    Figure  14.  Step size of ALCFI in the propagation of a circular orbit

    图  15  变参数LCFI圆轨道递推计算配点数变化

    Figure  15.  Variation of the collocation point number in the propagation of a circular orbit

    图  16  椭圆轨道递推计算步长变化

    Figure  16.  Step size of ALCFI in the propagation of an elliptical orbit

    图  17  椭圆轨道递推计算配点数变化

    Figure  17.  Variation of the collocation point number in the propagation of an elliptical orbit

    表  1  递推轨道计算参数

    Table  1.   Calculation parameters of LCFI in orbit propagation

    Initial valueParameter
    r0/mv0/(m·s−1)tf /dMΔt/stol/m
    $\left(\begin{array}{c}-0.3889 \\7.7388 \\0.6736\end{array}\right) \times 10^{6} $$\left(\begin{array}{c}-4.2953 \\0 \\7.4396\end{array}\right) \times 10^{3}$756010-5
    注: 表中r0为起始位置, v0为起始速度, tf递推时长, M为配点个数, Δt为计算区间步长, tol为迭代误差限.
    Note: r0 is the initial position, v0 is the initial velocity, tf is the duration of global time domain, M is the number of collocation point, Δt is the duration of a single subinterval, tol is the error tolerance of iteration.
    下载: 导出CSV

    表  2  轨道递推计算效率对比

    Table  2.   Comparation of efficiency on orbit propagation

    MethodLCFILVIM
    computing time/s1.3152442.347178
    下载: 导出CSV

    表  3  摄动Lambert问题计算效率对比

    Table  3.   Comparation of efficiency on Lambert problem

    MethodLCFILVIM
    computing time/s0.0204640.054941
    下载: 导出CSV

    表  4  摄动Lambert转移轨道计算参数

    Table  4.   Calculation parameters of LCFI in perturbed Lambert problem

    Boundary valueParameter
    r0/mrf/mtf /sMΔt/s$ to{l_{\tilde x}}/m $$ to{l_s}/m $
    $\left(\begin{array}{c}-1.4000 \\2.1000 \\2.4249\end{array}\right) \times 10^{7} $$\left(\begin{array}{c}-3.1497 \\-0.0462 \\5.4554\end{array}\right) \times 10^{7} $2500010250010-710-5
    注: 表中rf为终点位置, tf为转移时长, $ to{l_{\tilde x}} $为迭代求解微分方程(37)与(38)迭代误差限, tols为叠加法求解式(35)的迭代误差限
    Note: rf is the final position, tf is the time length of orbit transfer, $ to{l_{\tilde x}} $ is the error tolerance of equation (37) and (38), tols is the error tolerance of equation (35)
    下载: 导出CSV

    表  5  圆型限制性三体模型约束下转移轨道计算参数

    Table  5.   Calculation parameters of LCFI in transfer orbit calculation problem with the constraint of circular restricted three-body model

    Boundary valueParameter
    r0rfMΔt$t o l_{\tilde{x}} $tols
    $\left(\begin{array}{c}4.9494300 \\0 \\0\end{array}\right) \times 10^{-3} $$\left(\begin{array}{c}0.2955171 \\3.7059066 \\0\end{array}\right) \times 10^{-1}$50$ \dfrac{\pi }{{40}} $10−710−5
    下载: 导出CSV

    表  6  圆形限制性三体模型约束下转移轨道计算效率对比

    Table  6.   Comparation of efficiency on transfer orbit calculation problem with the constraint of three-body model

    MethodLCFILVIM
    computing time/s0.1880861.301782
    下载: 导出CSV

    表  7  圆轨道递推初始计算参数

    Table  7.   Calculation parameters of orbit propagation

    Initial valueParameter
    r0/mv0/(m·s−1)tf /dMΔt/stol/m
    $ \left( {\begin{array}{*{20}{c}} { - 0.3889} \\ {7.7388} \\ {0.6736} \end{array}} \right) \times {10^6} $$ \left( {\begin{array}{*{20}{c}} { - 3.5794} \\ 0 \\ {6.1997} \end{array}} \right) \times {10^3} $15340010−6
    下载: 导出CSV

    表  8  椭圆轨道递推初始计算参数

    Table  8.   Calculation parameters of orbit propagation

    Initial valueParameter
    r0/mv0/(m·s−1)tf /dMΔt/stol/m
    $ \left( {\begin{array}{*{20}{c}} {4.05} \\ 0 \\ { - 7.0148} \end{array}} \right) \times {10^6} $$ \left( {\begin{array}{*{20}{c}} 0 \\ {9.1464} \\ 0 \end{array}} \right) \times {10^3} $15340010−6
    下载: 导出CSV

    表  9  变参数LCFI递推轨道计算参数

    Table  9.   Calculation parameters of Adaptive LCFI in orbit propagation

    Nmaxe1e2MminMmaxabcd
    301.0 × 10−49.0 × 10−7380.750.53.53.5
    注: 表9 中各参数含义见图2
    Note: please refer to figure 2 for the meaning of parameters in table 9
    下载: 导出CSV

    表  10  不同算法计算时间

    Table  10.   Time cost of different numerical method

    Circular orbitElliptical orbit
    LCFI ALCFI LCFI ALCFI
    740.02 s 112.69 s 740.02 s 112.69 s
    下载: 导出CSV
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  • 收稿日期:  2021-07-13
  • 录用日期:  2021-12-17
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