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Optimization of aerocapture orbit based on improved pigeon inspired optimization algorithms
WU Aiguo, GONG Zhihao
School of Mechanical Engineering and Automation, Harbin Institute of Technology, Shenzhen, Shenzhen 518055, China
Abstract: An improved pigeon-inspired algorithm is proposed to optimize the aerocapture orbits of Mars explorers. The terminal and process constraints imposed by successful aerocapture are first analyzed, followed by the introduction of an appropriate performance index for the optimization of the capture orbit according to the speed increment required by the orbit transfer from the capture orbit to the target orbit. Then, to overcome the shortcomings of the original pigeon-inspired algorithm, an improved version is proposed by introducing an exponential function. The functions of the parameters in the improved algorithm are analyzed. Finally, based on the dynamic equations of flight in the atmosphere, the aerocapture orbital optimization problem is transformed into a multi-parameter optimization problem which is solved by the proposed improved pigeon inspired algorithm. The effectiveness of this algorithm is verified by a simulation example.
Keywords: Mars explorers    aerocapture orbital    orbital optimization    improved pigeon inspired optimization algorithm    multi-parameter optimization

1 气动捕获轨道优化问题建模 1.1 大气内飞行动力学方程的建立

 $\left\{ {\begin{array}{*{20}{l}} {\dot r = v{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma }\\ {\dot v = - D - {g_{\rm{M}}}{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma }\\ {\dot \gamma = \frac{1}{v}L{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \sigma + \left( {\frac{v}{r} - \frac{{{g_{\rm{M}}}}}{v}} \right){\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma }\\ {\dot \lambda = \frac{{v{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi }}{{r{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta }}}\\ {\dot \theta = \frac{v}{r}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi }\\ {\dot \psi = \frac{{L{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \sigma }}{{v{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma }} + \frac{v}{r}{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma {\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta } \end{array}} \right.$ （1）

 $\left\{ {\begin{array}{*{20}{l}} {D = \rho {v^2}{S_{\rm{r}}}{C_D}/(2m)}\\ {L = \rho {v^2}{S_{\rm{r}}}{C_L}/(2m)}\\ {{g_{\rm{M}}} = \frac{\mu }{{{r^2}}}} \end{array}} \right.$

 $\rho = {\rho _0}{\rm{exp}}( - h/{h_{\rm{s}}})$ （2）

 $\left\{ {\begin{array}{*{20}{l}} {\dot r = v{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma }\\ {\dot v = - D - {g_{\rm{M}}}{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma }\\ {\dot \gamma = \frac{1}{v}L{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \sigma + \left( {\frac{v}{r} - \frac{{{g_{\rm{M}}}}}{v}} \right){\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma } \end{array}} \right.$ （3）
1.2 能量最优气动捕获轨道的确定

 $\Delta {V_1} = \sqrt {2\mu } \left( {\sqrt {\frac{1}{{{r_{{\rm{ca}}}}}} - \frac{1}{{{r_{{\rm{ca}}}} + {r_{{\rm{tp}}}}}}} - \sqrt {\frac{1}{{{r_{{\rm{ca}}}}}} - \frac{1}{{{r_{{\rm{ca}}}} + {r_{{\rm{cp}}}}}}} } \right)$ （4）
 $\Delta {V_2} = \sqrt {2\mu } \left( {\sqrt {\frac{1}{{{r_{{\rm{tp}}}}}} - \frac{1}{{{r_{{\rm{ta}}}} + {r_{{\rm{tp}}}}}}} - \sqrt {\frac{1}{{{r_{{\rm{tp}}}}}} - \frac{1}{{{r_{{\rm{ca}}}} + {r_{{\rm{tp}}}}}}} } \right)$ （5）

 $\Delta V = \Delta {V_1} + \Delta {V_2}$ （6）

 $\frac{{\partial (\Delta V)}}{{\partial ({r_{{\rm{ca}}}})}} = \frac{{\partial (\Delta {V_1})}}{{\partial ({r_{{\rm{ca}}}})}} + \frac{{\partial (\Delta {V_2})}}{{\partial ({r_{{\rm{ca}}}})}}$ （7）

 $\begin{array}{*{20}{c}} {\frac{{\partial (\Delta {V_1})}}{{\partial ({r_{{\rm{ca}}}})}} = \frac{{\sqrt {2\mu } }}{2}\sqrt {{r_{{\rm{cp}}}}} \frac{{2{r_{{\rm{ca}}}} + {r_{{\rm{cp}}}}}}{{{{(r_{{\rm{ca}}}^2 + {r_{{\rm{cp}}}}{r_{{\rm{ca}}}})}^{\frac{3}{2}}}}} - }\\ {\frac{{\sqrt 2 \mu }}{2}\sqrt {{r_{{\rm{tp}}}}} \frac{{2{r_{{\rm{ca}}}} + {r_{{\rm{tp}}}}}}{{{{(r_{{\rm{ca}}}^2 + {r_{{\rm{tp}}}}{r_{{\rm{ca}}}})}^{\frac{3}{2}}}}}} \end{array}$ （8）
 $\frac{{\partial (\Delta {V_2})}}{{\partial ({r_{{\rm{ca}}}})}} = \left\{ {\begin{array}{*{20}{l}} { - \frac{{\sqrt {2\mu } }}{2} \cdot \frac{{\sqrt {{r_{{\rm{tp}}}}} }}{{\sqrt {{r_{{\rm{ca}}}}} {{({r_{{\rm{ca}}}} + {r_{{\rm{tp}}}})}^{\frac{3}{2}}}}}}&{{r_{{\rm{ca}}}} < {r_{{\rm{ta}}}}}\\ {\frac{{\sqrt {2\mu } }}{2} \cdot \frac{{\sqrt {{r_{{\rm{tp}}}}} }}{{\sqrt {{r_{{\rm{ca}}}}} {{({r_{{\rm{ca}}}} + {r_{{\rm{tp}}}})}^{\frac{3}{2}}}}}}&{{r_{{\rm{ca}}}} > {r_{{\rm{ta}}}}} \end{array}} \right.$ （9）

 ${r_{{\rm{ta}}}}{v_{{\rm{ca}}}} = {r_{\rm{a}}}{v_{\rm{f}}}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\gamma _{\rm{f}}}$ （10）

 $\frac{{v_{{\rm{ca}}}^2}}{2} - \frac{\mu }{{{r_{{\rm{ta}}}}}} = \frac{{v_{\rm{f}}^2}}{2} - \frac{\mu }{{{r_{\rm{a}}}}}$ （11）

 $\frac{1}{2}{\left( {\frac{{{r_{\rm{a}}}}}{{{r_{{\rm{ta}}}}}}{v_{\rm{f}}}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\gamma _{\rm{f}}}} \right)^2} - \frac{\mu }{{{r_{{\rm{ta}}}}}} = \frac{{v_{\rm{f}}^2}}{2} - \frac{\mu }{{{r_{\rm{a}}}}}$ （12）

 ${v_{\rm{f}}} = \sqrt {\frac{{2\mu {r_{{\rm{ta}}}}({r_{{\rm{ta}}}} - {r_{\rm{a}}})}}{{{r_{\rm{a}}}[r_{{\rm{ta}}}^2 - {{({r_{\rm{a}}}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\gamma _{\rm{f}}})}^2}]}}}$ （13）

 $\Delta V = {v_{{\rm{ta}}}} - \sqrt {\frac{{2\mu {r_{{\rm{ta}}}}({r_{{\rm{ta}}}} - {r_{\rm{a}}})}}{{{r_{{\rm{ta}}}}[r_{{\rm{ta}}}^2 - {{({r_{\rm{a}}}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\gamma _{\rm{f}}})}^2}]}}} {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\gamma _{\rm{f}}}$ （14）

 $J = {\left\{ {{v_{{\rm{ta}}}} - \sqrt {\frac{{2\mu {r_{\rm{a}}}({r_{{\rm{ta}}}} - {r_{\rm{a}}})}}{{{r_{{\rm{ta}}}}(r_{{\rm{ta}}}^2 - {{({r_{\rm{a}}}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\gamma _{\rm{f}}})}^2})}}} {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\gamma _{\rm{f}}}} \right\}^2} + {({v_{\rm{f}}} - v_{\rm{f}}^*)^2}$ （15）

 $v_{\rm{f}}^* = \sqrt {\frac{{2\mu {r_{{\rm{ta}}}}({r_{{\rm{ta}}}} - {r_{\rm{a}}})}}{{{r_{\rm{a}}}(r_{{\rm{ta}}}^2 - {{({r_{\rm{a}}}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\gamma _{\rm{f}}})}^2})}}}$

 $\left\{ {\begin{array}{*{20}{l}} {{r_0} = {r_{\rm{a}}}}\\ {{v_0} = v_0^*}\\ {{\gamma _0} = \gamma _0^*} \end{array}} \right.$ （16）

 $\left\{ {\begin{array}{*{20}{l}} {{r_{\rm{f}}} = {r_{\rm{a}}}}\\ {0 < {\gamma _{\rm{f}}} < {\gamma _{{\rm{fmax}}}}} \end{array}} \right.$ （17）

 $\left\{ {\begin{array}{*{20}{l}} {n - {n_{{\rm{max}}}} \le 0}\\ {\dot Q - {{\dot Q}_{{\rm{max}}}} \le 0}\\ {{h_{{\rm{max}}}} - h \le 0}\\ {{\gamma _{\rm{f}}} - {\gamma _{{\rm{ fmax }}}} \le 0} \end{array}} \right.$ （18）

 ${\tau _1} = {\rm{cos}}(\pi l/n)\quad l = 0,1, \cdots ,n$ （19）

 $t = \left( {\frac{{{t_0} - {t_{\rm{f}}}}}{2}} \right)\tau + \left( {\frac{{{t_0} + {t_{\rm{f}}}}}{2}} \right)$ （20）

 ${\sigma _l} = \sigma (t({\tau _l}))\quad l = 0,1, \cdots ,n$

 $\mathit{\boldsymbol{\xi }} = {\left[ {\begin{array}{*{20}{l}} {{t_{\rm{f}}}}&{{\sigma _0}}&{{\sigma _1}}& \cdots &{{\sigma _n}} \end{array}} \right]^{\rm{T}}}$ （21）

2 鸽群算法的分析与改进 2.1 原始鸽群算法及其存在的问题

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{X}}_i} = \left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{x}}_{i1}}}&{{\mathit{\boldsymbol{x}}_{i2}}}& \cdots &{{\mathit{\boldsymbol{x}}_{im}}} \end{array}} \right]}\\ {{\mathit{\boldsymbol{V}}_i} = \left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{v}}_{i1}}}&{{\mathit{\boldsymbol{v}}_{i2}}}& \cdots &{{\mathit{\boldsymbol{v}}_{im}}} \end{array}} \right]} \end{array}} \right.$

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{V}}_i}(t) = {\mathit{\boldsymbol{V}}_i}(t - 1){{\rm{e}}^{ - Rt}} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{rand }} \cdot [\mathit{\boldsymbol{G}}(t - 1) - {\mathit{\boldsymbol{X}}_i}(t - 1)]}\\ {{\mathit{\boldsymbol{X}}_i}(t) = {\mathit{\boldsymbol{X}}_i}(t - 1) + {\mathit{\boldsymbol{V}}_i}(t)} \end{array}} \right.$ （22）

 $J(\mathit{\boldsymbol{G}}(t - 1)) = \mathop {{\rm{min}}}\limits_{i \in \{ 1,2, \ldots ,N\} } J({\mathit{\boldsymbol{X}}_i}(t - 1))$

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{X}}_{\rm{c}}}(t) = \frac{{\sum\nolimits_{i = 1}^{N(t)} {{\mathit{\boldsymbol{X}}_i}} (t - 1){\rm{fitness}}[{\mathit{\boldsymbol{X}}_i}(t - 1)]}}{{N(t)\sum\nolimits_{i = 1}^{N(t)} {{\rm{fitness}}} [{\mathit{\boldsymbol{X}}_i}(t - 1)]}}}\\ {{\mathit{\boldsymbol{X}}_i}(t) = {\mathit{\boldsymbol{X}}_i}(t - 1) + {\rm{rand}} [{\mathit{\boldsymbol{X}}_{\rm{c}}}(t) - {\mathit{\boldsymbol{X}}_i}(t - 1)]} \end{array}} \right.$ （23）

 $\left\{ {\begin{array}{*{20}{l}} {{\rm{ fitness }}[{\mathit{\boldsymbol{X}}_i}(t)] = \frac{1}{{J({\mathit{\boldsymbol{X}}_i}(t)) + \varepsilon }}}\\ {N(t) = \frac{{N(t - 1)}}{2}} \end{array}} \right.$

 图 1 R的取值对动态权值的影响 Fig. 1 Influence of R on dynamic weight

2.2 改进的鸽群算法

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{V}}_i}(t) = {\mathit{\boldsymbol{V}}_i}(t - 1)f(t) + {\rm{rand}} [G(t - 1) - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{X}}_i}(t - 1)]}\\ {{\mathit{\boldsymbol{X}}_i}(t) = {\mathit{\boldsymbol{X}}_i}(t - 1) + {\mathit{\boldsymbol{V}}_i}(t)} \end{array}} \right.$ （24）

 $f(t) = \frac{1}{{k + {{\rm{e}}^{\omega (t - b)}}}}$ （25）

 $0 \le f(t) \le 1$ （26）

 $\left\{ {\begin{array}{*{20}{l}} {f({t_1}) = {\alpha _1}}\\ {f({t_2}) = {\alpha _2}} \end{array}} \right.$ （27）

 $\left\{ {\begin{array}{*{20}{l}} {\frac{1}{{k + {{\rm{e}}^{\omega ({t_1} - b)}}}} = {\alpha _1}}\\ {\frac{1}{{k + {{\rm{e}}^{\omega ({t_2} - b)}}}} = {\alpha _2}} \end{array}} \right.$ （28）

 $\left\{ {\begin{array}{*{20}{l}} {\omega = \frac{1}{{{t_2} - {t_1}}}\left[ {{\rm{ln}}\left( {\frac{1}{{{\alpha _2}}} - k} \right) - {\rm{ln}}\left( {\frac{1}{{{\alpha _1}}} - k} \right)} \right]}\\ {b = \frac{1}{\omega } \cdot \frac{{{t_1}}}{{{t_2} - {t_1}}}\left[ {{\rm{ln}}\left( {\frac{1}{{{\alpha _2}}} - k} \right) - {\rm{ln}}\left( {\frac{1}{{{\alpha _1}}} - k} \right)} \right] - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{\omega }{\rm{ln}}\left( {\frac{1}{{{\alpha _1}}} - k} \right)} \end{array}} \right.$ （29）

 $\dot f = \frac{{ - {{\rm{e}}^{\omega (t - b)}}\omega }}{{{{[k + {{\rm{e}}^{\omega (t - b)}}]}^2}}}$ （30）

k≥1时，对式(25)取极限可得

 $\left\{ {\begin{array}{*{20}{l}} {{\rm{li}}{{\rm{m}}_{t \to \infty }}\frac{1}{{k + {{\rm{e}}^{\omega (t - b)}}}} = 0}\\ {{\rm{li}}{{\rm{m}}_{t \to - \infty }}\frac{1}{{k + {{\rm{e}}^{\omega (t - b)}}}} = \frac{1}{k} \le 1} \end{array}} \right.$ （31）

k < 1时，此时k的取值与(t1, α1)、(t2, α2)相互制约，由式(26)可得

 $0 < \frac{1}{{k + {{\rm{e}}^{ - \omega b}}}} \le 1$ （32）

 $1 - {{\rm{e}}^{ - \omega b}} \le k$ （33）

 ${{\rm{e}}^{ - \omega b}} = \frac{{{{\left( {\frac{1}{{{\alpha _1}}} - k} \right)}^{1 + c}}}}{{{{\left( {\frac{1}{{{\alpha _2}}} - k} \right)}^c}}}$ （34）

 $k \ge 1 - \frac{{{{\left( {\frac{1}{{{\alpha _1}}} - k} \right)}^{1 + c}}}}{{{{\left( {\frac{1}{{{\alpha _2}}} - k} \right)}^c}}}$ （35）

 $k \ge 1 - \frac{{{{\left( {\frac{1}{{{\alpha _1}}} - k} \right)}^2}}}{{\frac{1}{{{\alpha _2}}} - k}}$ （36）

 $\frac{1}{{{\alpha _2}}}k - {k^2} \ge \frac{1}{{{\alpha _2}}} - k - \left[ {{{\left( {\frac{1}{{{\alpha _1}}}} \right)}^2} - \frac{{2k}}{{{\alpha _1}}} + {k^2}} \right]$ （37）

 $\left( {1 - \frac{2}{{{\alpha _1}}} + \frac{1}{{{\alpha _2}}}} \right)k \ge \frac{1}{{{\alpha _2}}} - {\left( {\frac{1}{{{\alpha _1}}}} \right)^2}$ （38）

 $\left\{ {\begin{array}{*{20}{l}} {k < 1}\\ {1 - \frac{2}{{{\alpha _1}}} + \frac{1}{{{\alpha _2}}} > \frac{1}{{{\alpha _2}}} - {{\left( {\frac{1}{{{\alpha _1}}}} \right)}^2}} \end{array}} \right.$ （39）

$1 - \frac{2}{{{a_1}}} + \frac{1}{{{a_2}}} > 0$时，有

 $\left\{ {\begin{array}{*{20}{l}} {{\alpha _1} > \frac{{2{\alpha _2}}}{{1 + {\alpha _2}}}}\\ {k \ge \frac{{\frac{1}{{{\alpha _2}}} - {{\left( {\frac{1}{{{\alpha _1}}}} \right)}^2}}}{{1 - \frac{2}{{{\alpha _1}}} + \frac{1}{{{\alpha _2}}}}}} \end{array}} \right.$ （40）

$1 - \frac{2}{{{a_1}}} + \frac{1}{{{a_2}}} < 0$时，有

 $\left\{ {\begin{array}{*{20}{l}} {{\alpha _1} < \frac{{2{\alpha _2}}}{{1 + {\alpha _2}}}}\\ {k < 1} \end{array}} \right.$ （41）

α2=α12时，有

 ${\alpha _1} - \frac{{2{\alpha _2}}}{{1 + {\alpha _2}}} = \frac{{{\alpha _1}}}{{1 + \alpha _1^2}}{(1 - {\alpha _1})^2} > 0$ （42）

 $f(t) = \frac{1}{{{{\rm{e}}^{\omega t}}}}$ （43）

 图 2 动态权值随迭代次数的变换 Fig. 2 Evolution of dynamic weights

3 气动捕获轨道优化

 $\mathit{\boldsymbol{X}} = \left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{X}}_1}}&{{\mathit{\boldsymbol{X}}_2}}& \cdots &{{\mathit{\boldsymbol{X}}_m}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{t_{f1}}}&{{t_{f2}}}& \cdots &{{t_{fm}}}\\ {{\sigma _{01}}}&{{\sigma _{02}}}& \cdots &{{\sigma _{0m}}}\\ {{\sigma _{11}}}&{{\sigma _{12}}}& \cdots &{{\sigma _{1m}}}\\ \vdots & \vdots & \ddots & \vdots \\ {{\sigma _{n1}}}&{{\sigma _{n2}}}& \cdots &{{\sigma _{nm}}} \end{array}} \right]$ （44）

 $\begin{array}{*{20}{l}} {J = {p_1}{{\left\{ {{v_{{\rm{ta}}}} - \sqrt {\frac{{2\mu ({r_{{\rm{ta}}}} - {r_{\rm{a}}})}}{{{r_{{\rm{ta}}}}[r_{{\rm{ta}}}^2 - {{({r_{\rm{a}}}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\gamma _{\rm{f}}})}^2}]}}} {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\gamma _{\rm{f}}}} \right\}}^2} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {p_2}{{({v_{\rm{f}}} - v_{\rm{f}}^*)}^2} + {p_3}\sum\limits_{i = 1}^4 {{\rm{max}}} (0,{g_i}(x))} \end{array}$

 $\begin{array}{*{20}{l}} {J = {p_1}{{\left[ {1 - \sqrt {\frac{{2{r_{\rm{a}}}({r_{{\rm{ta}}}} - {r_{\rm{a}}})}}{{r_{{\rm{ta}}}^2 - {{({r_{\rm{a}}}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\gamma _{\rm{f}}})}^2}}}} {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\gamma _{\rm{f}}}} \right]}^2} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {p_2}{{({v_{\rm{f}}} - v_{\rm{f}}^*)}^2} + {p_3}\sum\limits_{i = 1}^4 {{\rm{max}}} (0,{g_i}(x))} \end{array}$ （45）
4 仿真分析

 m/kg rn S/m2 CL CD 2 804 0.66 15.9 0.36 1.45

 半径/km 大气高度/km 引力常数/(m3·s-2) 3 395 125 15.9

 图 3 气动捕获轨道优化结果 Fig. 3 Performance of optimized aerocapture orbit
5 结论

1) 在对火星探测器的气动捕获过程分析的基础上，从脉冲变轨的角度给出了气动捕获轨道的优化指标。

2) 对原始鸽群算法的局限性进行分析，提出一种能够平衡收敛速度与全局搜索关系的改进算法。

3) 将气动捕获轨道优化问题转化为多参数优化问题，并利用提出的改进个鸽群算法对其进行优化。

 [1] HOWARD S. Change of satellite orbit plane by aerodynamic maneuvering[J]. Journal of the Aerospace Sciences, 1962, 29(3): 323-332. Click to display the text [2] AI Y H, CUI H T, ZHENG Y Y. Identifying method of entry and exit conditions for aerocapture with near Minimum fuel consumption[J]. Aerospace Science and Technology, 2016, 58: 582-593. Click to display the text [3] BETTS J T. Survey of numerical methods for trajectory optimization[J]. Journal of Guidance, Control, and Dynamics, 1998, 21(2): 193-207. Click to display the text [4] JORRIS T R, COBB R G. Multiple method 2-D trajectory optimization satisfying waypoints and nofly zone constraints[J]. Journal of Guidance, Control, and Dynamics, 2008, 31(3): 543-553. Click to display the text [5] 周浩, 周韬, 陈万春, 等. 高超声速滑翔飞行器引入段弹道优化[J]. 宇航学报, 2006, 27(5): 970-973. ZHOU H, ZHOU T, CHEN W C, et al. Trajectory optimization in injection phase for hypersonic gliding vehicle[J]. Journal of Astronautics, 2006, 27(5): 970-973. (in Chinese) Cited By in Cnki (63) | Click to display the text [6] 陈功, 傅瑜, 郭继峰. 飞行器轨迹优化方法综述[J]. 飞行力学, 2011, 29(4): 1-5. CHEN G, FU Y, GUO J F. Survey of aircraft trajectory optimization methods[J]. Flight Dynamics, 2011, 29(4): 1-5. (in Chinese) Cited By in Cnki (78) | Click to display the text [7] HARGRAVES C, JOHNSON F, PARIS S, et al. Numerical computation of optimal atmospheric trajectories[J]. Journal of Guidance, Control and Dynamics, 1981, 4(4): 406-414. Click to display the text [8] HARGRAVES C, JOHNSON F, PARIS S, et al. Direct trajectory optimization using nonlinear programming and collocation[J]. Journal of Guidance, Control, and Dynamics, 1987, 10(4): 338-342. Click to display the text [9] TIAN B L, ZONG Q. Optimal guidance for reentry vehicles based on indirect legendre pseudospectral method[J]. Acta Astronautica, 2011, 68(7-8): 1176-1184. Click to display the text [10] HARGRAVES C, JOHNSON F, PARIS S, et al. Numerical computation of optimal atmospheric trajectories[J]. Journal of Guidance, Control and Dynamics, 1981, 4(4): 406-414. Click to display the text [11] 王铀, 赵辉, 惠百斌, 等. 利用Radau伪谱法求解UCAV对地攻击轨迹研究[J]. 电光与控制, 2012, 19(10): 50-53. WANG Y, ZHAO H, HUI B B, et al. Air-to-ground trajectory planning for UCAVs using a radau pseudo-spectral method[J]. Electronics Optics & Control, 2012, 19(10): 50-53. (in Chinese) Cited By in Cnki (8) | Click to display the text [12] RAHIMI A, DEV K K, ALIGHANBARI, et al. Swarm optimization applied to spacecraft reentry trajectory[J]. Journal of Guidance, Control, and Dynamics, 2012, 36(1): 307-310. Click to display the text [13] PONTANI M, CONWAY B A. Particle swarm optimization applied to space trajectories trajectory[J]. Journal of Guidance, Control, and Dynamics, 2010, 33(5): 1429-1441. Click to display the text [14] YOKOYAMA N, SUZUKI S. Modifled genetic algorithm for constrained trajectory optimization[J]. Journal of Guidance, Control, and Dynamics, 2005, 28(1): 139-144. Click to display the text [15] LI Y T, WU Y, QU X J. Chicken swarm-based method for ascent trajectory optimization of hypersonic vehicles[J]. Journal of Aerospace Engineering, 2017, 30(5): 04017043. Click to display the text [16] DUAN H B, QIAO P X. Pigeon-inspired optimization:A new swarm intelligence optimizer for air robot path planning[J]. International Journal of Intelligence Computing and Cybernetics, 2014, 7: 24-37. Click to display the text [17] 段海滨, 邱华鑫, 范彦铭. 基于捕食逃逸鸽群优化的无人机紧密编队协调控制[J]. 中国科学:技术科学, 2015, 45(6): 559-572. DUAN H B, QIU H X, FAN Y M. Unmanned aerial vehicle close formation cooperative control based on predatory escaping pigeon-inspired optimization[J]. Scientia Sinica Technologica, 2015, 45(6): 559-572. (in Chinese) Cited By in Cnki | Click to display the text [18] 华冰, 刘睿鹏, 孙胜刚, 等. 一种基于自适应种群变异鸽群优化的航天器集群轨道规划方法[J]. 中国科学:技术科学, 2020, 50(4): 453-460. HUA B, LIU R P, SUN S G, et al. Spacecraft cluster orbit planning method based on adaptive population mutated pigeon group optimization[J]. Scientia Sinica Technologica, 2020, 50(4): 453-460. (in Chinese) Cited By in Cnki | Click to display the text [19] SUSHNIGDHA G, JOSHI A. Trajectory design of re-entry vehicles using combined pigeon inspired optimization and orthogonal collocation method[J]. IFAC Papers OnLine, 2018, 51(1): 656-662. Click to display the text [20] 徐博, 张大龙. 基于量子行为鸽群优化的无人机紧密编队控制[J]. 航空学报, 2020, 41(8): 323722. XU B, ZHANG D L. Close formation control of UAV Based on pigeon group optimization of quantum behavior[J]. Acta Aeronautica et Astronautica Sinica, 2020, 41(8): 323722. (in Chinese) Cited By in Cnki | Click to display the text [21] VINH N X, BUSEMANN A, CULP R D. Hypersonic and planetary entry flight mechanics: NASA STI/RECON Technical Report A[R].Wshington, D.C.: NASA, 1980.
http://dx.doi.org/10.7527/S1000-6893.2020.24292

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#### 文章信息

WU Aiguo, GONG Zhihao

Optimization of aerocapture orbit based on improved pigeon inspired optimization algorithms

Acta Aeronautica et Astronautica Sinica, 2020, 41(9): 324292.
http://dx.doi.org/10.7527/S1000-6893.2020.24292