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 力学学报  2015, Vol. 47 Issue (5): 799-806  DOI: 10.6052/0459-1879-14-386 0

### 引用本文 [复制中英文]

[复制中文]
Wang Wei, Yuan Jianping, Luo Jianjun. OPTIMAL SINGLE IMPULSE MANEUVER FOR SPACECRAFT CLUSTER FLIGHT[J]. Chinese Journal of Ship Research, 2015, 47(5): 799-806. DOI: 10.6052/0459-1879-14-386.
[复制英文]

### 文章历史

2014-12-04 收稿
2015-07-14 录用
2015–07–15 网络版发表

1. 西北工业大学航天学院, 西安710072;
2. 航天飞行动力学技术重点实验室, 西安710072

1 动力学模型

 $\left. {\begin{array}{*{20}{l}} \begin{array}{l} {{\ddot x}_i} - 2{{\dot f}_{\rm{C}}}{{\dot y}_i} - \dot f_{\rm{C}}^2{x_i} - {{\ddot f}_{\rm{C}}}{y_i} = \frac{\mu }{{r_{\rm{C}}^2}} - \frac{{\mu \left( {{r_{\rm{C}}} + {x_i}} \right)}}{{{{\left[ {{{\left( {{r_{\rm{C}}} + {x_i}} \right)}^2} + y_i^2 + z_i^2} \right]}^{{\textstyle{3 \over 2}}}}}}\\ {{\ddot y}_i} + 2{{\dot f}_{\rm{C}}}{{\dot x}_i} + {{\ddot f}_{\rm{C}}}{x_i} - \dot f_{\rm{C}}^2{y_i} = - \frac{{\mu {y_i}}}{{{{\left[ {{{\left( {{r_{\rm{C}}} + {x_i}} \right)}^2} + y_i^2 + z_i^2} \right]}^{{\textstyle{3 \over 2}}}}}}\\ {{\ddot z}_i} = - \frac{{\mu {z_i}}}{{{{\left[ {{{\left( {{r_{\rm{C}}} + {x_i}} \right)}^2} + y_i^2 + z_i^2} \right]}^{{\textstyle{3 \over 2}}}}}} \end{array} \end{array}} \right\}$ (1)

 $r_{\rm C} = \left\| { {\pmb r}_{\rm C} } \right\|_2 = \dfrac{a_{\rm C} \left( {1 - e_{\rm C}^2 } \right)}{1 + e_{\rm C} \cos f_{\rm C} }$ (2)

 $\dot {f}_{\rm C} = \dfrac{n_{\rm C} \left( {1 + e_{\rm C} \cos f_{\rm C} } \right)^2}{\left( {1 - e_{\rm C}^2 } \right)^{\tfrac{3}{2}}}$ (3)

 $\ddot {f}_{\rm C} = \dfrac{ - 2e_{\rm C} n_{\rm C}^2 \left( {1 + e_{\rm C} \cos f_{\rm C} } \right)^2\sin f_{\rm C} }{\left( {1 - e_{\rm C}^2 } \right)^3}$ (4)

2 最优单脉冲机动

 ${\pmb d} = d_n \hat{\pmb n } + d_t \hat{\pmb t } + d_h \hat{\pmb h } = \left[{d_n },\ \ {d_t },\ \ {d_h } \right]^{\rm T}$ (5)

 $\dfrac{{\rm{d}} a}{{\rm{d}} t} = \dfrac{2a^2v}{\mu }d_t$ (6)

 $\mathop {\lim }\limits_{ \Delta t \to 0} \dfrac{{\rm{d}} a}{{\rm{d}} t} \cdot {\Delta }t = {\Delta }a\,,\ \ \mathop {\lim }\limits_{ \Delta t \to 0} d_t \cdot \Delta t = \Delta v_t$ (7)

 $\Delta a = \dfrac{2a^2v}{\mu } \Delta v_t$ (8)

2.1 机动时刻给定

 $J^2\left( {\tilde {a}^ * } \right) = \min \sum {} _{i = 1}^n {\tilde {k}_i \left\| {\left[{ \Delta {\pmb v}_i } \right]_{{\cal T}_i } } \right\|_2^2 }$ (9)

 $J^2\left( \tilde {a} \right) = \sum {} _{i = 1}^n {\tilde {k}_i \dfrac{\mu ^2}{4a_i^4 v_i^2 }\left( {\tilde {a} - a_i } \right)^2}$ (10)

 $\dfrac{\partial J^2\left( \tilde {a} \right)}{\partial \tilde {a}} = 0$ (11)

 $\tilde {a}^ * = \dfrac{\sum {} _{i = 1}^n {\left( {\tilde {k}_i a_i \cdot \prod\limits_{j = 1,j \ne i}^n {a_j^4 v_j^2 } } \right)} }{\sum {} _{i = 1}^n {\left( {\tilde {k}_i \cdot \prod\limits_{j = 1,j \ne i}^n {a_j^4 v_j^2 } } \right)} }$ (12)

 $\varepsilon _i = \dfrac{1}{2}\left\| { {\pmb v }_i } \right\|_2^2 - \dfrac{\mu }{r_i } = - \dfrac{\mu }{2a_{\rm C} } = \varepsilon _{\rm C}$ (13)

 $\left[{{\pmb v}_i } \right]_{\mathcal{L}_C } = \left[\dot {x}_i + \dot {r}_{\rm C} - \dot {f}_{\rm C} y_i ,\ \ \dot {y}_i + \dot {f}_{\rm C} x_i + \dot {f}_{\rm C} r_{\rm C} ,\ \ \dot {z}_i \right]^{\rm T}$ (14)

 $\left[{{\pmb r}_i } \right]_{\mathcal{L}_{\rm C} } = \left[{ r_{\rm C} + x_i ,\ \ y_i ,\ \ z_i } \right]^{\rm T}$ (15)

 ${\varPhi } = \sum {} _{i = 1}^n {\tilde {k}_i \left\| {\left[ { \Delta {\pmb v}_i } \right]_{\mathcal{L}_C } } \right\|_2^2 } + \sum {} _{i = 1}^n {\lambda _i \left( {\varepsilon _i^ + + \dfrac{\mu }{2\tilde {a}}} \right)}$ (16)

 $\dfrac{\partial {\varPhi }}{\partial \tilde {a}} = 0$ (17)

 $\sum {} _{i = 1}^n {\lambda _i } = 0$ (18)

 ${\dfrac{\partial {\varPhi }}{\partial \lambda _i } = 0,} \ \ {\dfrac{\partial {\varPhi }}{\partial \left[{ {\Delta }{\pmb v}_i } \right]_{\mathcal{L}_C } } = {\bf 0 }_{3\times 1} ,} \ \ {\dfrac{\partial ^2 {\varPhi }}{\partial \left[{ {\Delta }{\pmb v }_i } \right]_{\mathcal{L}_C }^2 } > {\bf 0 }_{3\times 3} }$ (19)

 $\sum {} _{i = 1}^n {\sqrt {\dfrac{\left( {2a_i - r_i } \right)\tilde {a}^ * }{\left( {2\tilde {a}^ * - r_i } \right)a_i }} = } \sum {} _{i = 1}^n {\tilde {k}_i }$ (20)

 ${\left[ {\Delta {v_i}} \right]_{{\Gamma _i}}} = \left[ {\begin{array}{*{20}{c}} {\cos {\gamma _i}}&{\sin {\gamma _i}}&0\\ { - \sin {\gamma _i}}&{\cos {\gamma _i}}&0\\ 0&0&1 \end{array}} \right]{\left[ {\Delta {v_i}} \right]_{{\Gamma _i}}}$ (21)

 $\cos \gamma _i = \dfrac{1 + e_i \cos f_i }{\sqrt {1 + 2e_i \cos f_i + e_i^2 } }$ (22)

2.2 机动时刻未定

 $J^2\left( {\bar {a}^ * ,\bar {t}^ * } \right) = \min \sum {} _{i = 1}^n {\bar {k}_i \left\| {\left[{ {\Delta }{\pmb v}_i } \right]_{\mathcal{T}_i } } \right\|_2^2 }$ (23)

 $\sin E_i = \dfrac{\sqrt {1 - e_i^2 } \sin f_i }{1 + e_i \cos f_i }$ (24)

 $M_i = E_i - e_i \sin E_i = n_i \left( {\bar {t} - \bar {t}_{0_i } } \right)$ (25)

 $f_i = M_i + 2\sum {} _{\kappa = 1}^\infty {\dfrac{1}{\kappa }} \left[ {\sum {} _{\eta = - \infty }^{ + \infty } { {\rm J}_\nu \left( { - \kappa e_i } \right)\beta _i^{\left| {\kappa + \eta } \right|} } } \right]\sin \left( {\kappa M_i } \right)$ (26)

 ${\rm J}_\eta \left( \nu \right) = \sum {} _{j = 0}^\infty {\dfrac{\left( { - 1} \right)^j\left( \nu \right)^{\eta + 2j}}{2^{\eta + 2j}{ j}!\left( {\eta + j} \right)!}}$ (27)

 $\beta _i = \dfrac{1 - \sqrt {1 - e_i^2 } }{e_i }$ (28)

 $\cos f_i = - e_i + \dfrac{2\left( {1 - e_i^2 } \right)}{e_i }\sum {} _{\kappa = 1}^\infty {{\rm J}_\kappa \left( {\kappa e_i } \right)\cos \left( {\kappa M_i } \right)}$ (29)

 $f_i \approx M_i + 2e_i \sin M_i + \dfrac{5}{4}e_i^2 \sin 2M_i$ (30)

 $\cos f_i \approx - e_i + \left( {1 - \dfrac{9}{8}e_i^2 } \right)\cos M_i + e_i \cos 2M_i + \dfrac{9}{8}e_i^2 \cos 3M_i$ (31)

 $\dfrac{\partial J^2\left( {\bar {a},\bar {t}} \right)}{\partial \bar {a}} = 0$ (32)

 $\begin{array}{l} {{\bar a}^*} = \sum {_{i = 1}^n} {{\bar k}_i}n_i^2\left( {1 - e_i^2} \right){a_i}\prod\limits_{j = 1,j \ne i}^n {[1 - e_j^2 + } \\ \;\;\;\;\;\;\;2{e_j}\cos \left( {{n_j}{{\bar t}^*} - {{\tilde M}_{j0}}} \right) + \\ \;\;\;\;\;\;\;2e_j^2\cos (2{n_j}{{\bar t}^*} - 2{{\tilde M}_{j0}})]/\\ \;\;\;\;\;\;\;\{ \sum {_{i = 1}^n} {{\bar k}_i}n_i^2(1 - e_i^2)\prod\limits_{j = 1,j \ne i}^n {[1 - e_j^2 + } \\ \;\;\;\;\;\;\;2{e_j}\cos \left( {{n_j}{{\bar t}^*} - {{\tilde M}_{j0}}} \right) + \\ \;\;\;\;\;\;\;2e_j^2\cos \left( {2{n_j}{{\bar t}^*} - 2{{\tilde M}_{j0}}} \right)]\} \end{array}$ (33)

 $\dfrac{\partial J^2\left( {\bar {a},\bar {t}} \right)}{\partial \bar {t}} = 0$ (34)

 $\sum {} _{i = 1}^n \bar {k}_i n_i^3 e_i \left( {1 - e_i^2 } \right)\left( {\bar {a}^ * - a_i } \right)^2\Big [\sin \left( {n_i \bar {t}^ * - \tilde {M}_{i0} } \right) +\\ \qquad 2e_i \sin \left( {2n_i \bar {t}^ * - 2\tilde {M}_{i0} } \right) \Big] \Bigg / \\ \qquad \Big [ \left( {1 - e_i^2 } \right) + 2e_i \cos \left( {n_i \bar {t}^ * - \tilde {M}_{i0} } \right) + \\ \qquad 2e_i^2 \cos \left( {2n_i \bar {t}^ * - 2\tilde {M}_{i0} } \right) \Big]^2 = 0$ (35)

3 数值仿真

3.1 机动时刻给定

 $\begin{array}{l} {\left[ {\Delta v_1^*} \right]_{{\mathcal{L}_{\rm{C}}}}} = {\left[ { - 0.000{\mkern 1mu} 21,\;\;0.892{\mkern 1mu} 28,\;\;0.001{\mkern 1mu} 01} \right]^{\rm{T}}}{\mkern 1mu} {\rm{mm/s}}\\ {\left[ {\Delta v_2^*} \right]_{{\mathcal{L}_{\rm{C}}}}} = {\left[ { - 0.002{\mkern 1mu} 82,\;\;5.002{\mkern 1mu} 22,\;\;0.005{\mkern 1mu} 66} \right]^{\rm{T}}}{\mkern 1mu} {\rm{m/s}}\\ {\left[ {\Delta v_3^*} \right]_{{\mathcal{L}_{\rm{C}}}}} = {\left[ { - 0.000{\mkern 1mu} 51,\;\; - 5.001{\mkern 1mu} 79,\;\; - 0.005{\mkern 1mu} 65} \right]^{\rm{T}}}{\mkern 1mu} {\rm{m/s}} \end{array}$

 $J\left( {\tilde {a}^ * } \right) = \sqrt {\left\| {\left[{ {\Delta }{\pmb v}_1^ * } \right]_{\mathcal{L}_{\rm C} } } \right\|_2^2 + \left\| {\left[{ {\Delta }{\pmb v}_2^ * } \right]_{\mathcal{L}_{\rm C} } } \right\|_2^2 + \left\| {\left[{ {\Delta }{\pmb v}_3^ * } \right]_{\mathcal{L}_{\rm C} } } \right\|_2^2 } =\\ \qquad 7.073\,91\,{\rm m/s}$

 图 1 不同参考长半轴取对应的总的速度脉冲增量 Fig. 1 The overall velocity impulse varying with different semimajor axes
 图 2 拉格朗日乘子之和随参考长半轴的变化 Fig. 2 The sum of Lagrange multipliers varying with different semimajor axes

 $L \triangleq \sum {} _{i,j = 1,i \ne j}^3 {\left\| { {\pmb r}_i-{\pmb r}_j } \right\|_2 }$ (36)

 图 3 $L^ +$与$L^ -$的比较 Fig. 3 The comparison between $L^ +$ and $L^ -$
 图 4 $L^ +$ 和$L^ -$的放大图 Fig. 4 The magnified view of $L^ +$ and $L^ -$

3.2 机动时刻未定

 图 5 $J\left( {\bar {a},\bar {t}} \right)$随参考长半轴以及机动时刻的变化曲线 Fig. 5 The contour lines of $J\left( {\bar {a},\bar {t}} \right)$
 图 6 $J\left( {\bar {a},\bar {t}} \right)$在$\left( {\bar {t}\sim \bar {a}} \right)$平面的投影 Fig. 6 The projected contour lines of total velocity impulse consumption in $\left( {\bar {t}\sim \bar {a}} \right)$ plane

 图 7 在7.78小时内$J\left( {\bar {a},\bar {t}} \right)$随参考长半轴和机动时刻的变化曲线 Fig. 7 Contour lines of $J\left( {\bar {a},\bar {t}} \right)$ within 7.78 hours

4 结 论

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OPTIMAL SINGLE IMPULSE MANEUVER FOR SPACECRAFT CLUSTER FLIGHT
Wang Wei, Yuan Jianping, Luo Jianjun
1. School of Astronautics, Northwestern Polytechnical University, Xi'an 710072, China;
2. National Key Laboratory of Aerospace Flight Dynamics, Xi'an 710072, China
Fund: The project was supported by the National Natural Science Foundation of China (11072194,11472213).
Abstract: The optimal single-impulse maneuver for spacecraft cluster flight is studied in this paper. Based on Gauss' variational equations, the optimal conditions and solutions for time-fixed and time-unfixed case are provided via the periodic condition of nonlinear relative motion. For the time-fixed case, the problem is also treated from the perspective of energy matching condition. In this case, the problem is transformed into seeking for the extremum of a quadratic equation or the solution of a single-root algebraic equation, and the optimal semimajor axis is derived, as well as the optimal velocity impulse. For the time-unfixed case, the problem can be solved by numerical algorithms or transformed into pursuing for the solution of a multi-root algebraic equation with the aid of Fourier-Bessel functions. In the end, the proposed method is validated by several carried out illustrating examples.
Key words: cluster flight    single impulse maneuver    periodic condition    spacecraft relative motion