﻿ 一种改进的轨道动力学模型
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 力学学报  2015, Vol. 47 Issue (1): 154-162  DOI: 10.6052/0459-1879-14-298 0

### 引用本文 [复制中英文]

[复制中文]
Yang Mengjie, Yuan Jianping. AN IMPROVED MODEL OF ORBITAL DYNAMICS[J]. Chinese Journal of Ship Research, 2015, 47(1): 154-162. DOI: 10.6052/0459-1879-14-298.
[复制英文]

### 文章历史

2014-09-28收稿
2014-10-27录用
2014-11-18网络版发表

1 新轨道动力学方程组 1.1 矢径分解

 $\frac{{{{\text{d}}^2}r}}{{{\text{d}}{t^2}}} + \frac{u}{{{r^3}}}r = P$ (1)

 $dt = \frac{{{r^2}}}{h}{\text{ }}d\theta$ (2)

 $d t = \dfrac{r^2}{\tilde {h}}d s$ (3)

 ${\left( {\;\;} \right)^ \bullet } = \frac{{\tilde h}}{{{r^2}}}{\left( {\;\;} \right)^\prime }$ (4)
 ${\left( {\;\;} \right)^{ \bullet \bullet }} = \frac{{{{\tilde h}^2}}}{{{r^4}}}{\text{(}}\;){\text{}} + \left( {\frac{{\tilde h'}}{{\tilde h}} - 2\frac{{r'}}{r}} \right){\left( {\;\;} \right)^\prime }{\text{]}}$ (5)

 ${x}" + \dfrac{\tilde {h}'}{h}{x}' + \left[{\dfrac{r"}{r} + \left( {\dfrac{\tilde {h}'}{h} - \dfrac{2r'}{r}} \right)\dfrac{r'}{r} + \dfrac{ur}{\tilde {h}^2}} \right] {x} = \dfrac{r^3}{\tilde {h}^2}{P}$ (6)

 $\ddot {r} + \dot {r}^2 = \left( {\ddot {r},{r}} \right)+ \left( {\dot {r},\dot {r}}\right)$ (7)

 $\dfrac{r"}{r} + \left( {\dfrac{\tilde {h}'}{h} - \dfrac{2r'}{r}}\right)\dfrac{r'}{r} + \dfrac{ur}{\tilde {h}^2} = 1 + \dfrac{r^2}{\tilde{h}^2}\left( {{P},{r}} \right)$ (8)

 $\rho" + \rho = \dfrac{u}{\tilde {h}^2} - \dfrac{1}{\tilde {h}^2\rho ^2}\left( {{P},{x}} \right) - \dfrac{\tilde {h}'}{\tilde {h}}\rho'$ (9)
 $x{\text{}} + x = \frac{1}{{{{\tilde h}^2}{\rho ^3}}}\left( {P - \left( {{\text{p}},x} \right)x} \right) - \frac{{\tilde h'}}{{\tilde h}}x'$ (10)

 ${h}' = \dfrac{1}{\tilde {h}\rho ^3}\left( {{P},{x}'} \right)$ (11)

1.2 第1组动力学方程

 $x = \frac{r}{r},\;\;y = \frac{{{\text{d}}x}}{{ds}} = x',\;\;z = x \times y$

 $P = \left( {P,x} \right)x + \left( {P,y} \right)y + \left( {P,z} \right)z$ (12)

 $rho = uc_0^2 + c_1 \cos s + c_2 \sin s$ (13)

 $\dfrac{d\rho }{d s} = 2uc_0 c'_0 - c_1 \sin s + c'_1 \cos s + c_2 \cos s +c'_2 \sin s$

 $\frac{{{\text{d}}\rho }}{{{\text{d}}s}} = - {c_1}\sin s + {c_2}\cos s$ (14)

 $\frac{{{{\text{d}}^2}\rho }}{{{\text{d}}{s^2}}} = - {c_1}\cos s - {c'_1}\sin s - {c_2}\sin s + {c'_2}{\text{ }}\cos s$ (15)

 $\frac{{{\text{d}}{c_1}}}{{{\text{d}}s}} = \frac{{c_0^2}}{{{\rho ^2}}}{P_x}\sin s - \frac{{{{c'}_0}}}{{{c_0}}}\left[{\left( {\rho + uc_0^2} \right)\cos s - {c_1}} \right]$ (16)
 $\frac{{d{c_2}}}{{{\text{d}}s}} = - \frac{{c_0^2}}{{{\rho ^2}}}{P_x}\cos s - \frac{{{{c'}_0}}}{{{c_0}}}\left[{\left( {\rho + uc_0^2} \right)\sin s - {c_2}} \right]$ (17)

 $\frac{{{\text{d}}{c_0}}}{{{\text{d}}s}} = - \frac{{\tilde h}}{{{{\tilde h}^2}}} = - \frac{{c_0^3}}{{{\rho ^3}}}{P_y}$ (18)
$\left( {c_0 ,c_1 ,c_2 } \right)$为描述航天器矢径模的新轨道变量. 式(16),(17)和式(18)为第1组轨道变量的变化规律.

 $\varepsilon = \frac{1}{2}\left( {{{\dot r}^2} + \frac{{{{\tilde h}^2}}}{{{r^2}}}} \right) - \frac{\mu }{r} = \frac{1}{2}{\tilde h^2}({\rho '^2} + {\rho ^2}) - \mu \rho$

 $\frac{{{\text{d}}\varepsilon }}{{{\text{d}}s}} = \frac{1}{{{\rho ^2}}}(\rho {P_y} - \rho '{P_x})$ (19)

 $\varepsilon = \frac{{c_1^2 + c_2^2 - {u^2}c_0^4}}{{2c_0^2}}$

1.3 第2组动力学方程

 ${x}" + {x} = \dfrac{1}{\tilde {h}^2\rho ^3}\left( {{P},{z}} \right) {z}$ (20)

 $\left. \begin{gathered} x = \left[{\begin{array}{*{20}{c}} {q_0^2 + q_1^2 - q_2^2 - q_3^22{q_1}{q_2} + 2{q_0}{q_3} - 2{q_0}{q_2} + 2{q_1}{q_3}} \end{array}} \right] \hfill \\ y = \left[{\begin{array}{*{20}{c}} {2{q_1}{q_2} - 2{q_0}{q_3}q_0^2 - q_1^2 + q_2^2 - q_3^22{q_0}{q_1} + 2{q_2}{q_3}} \end{array}} \right] \hfill \\ z = \left[{\begin{array}{*{20}{c}} {2{q_0}{q_2} + 2{q_1}{q_3} - 2{q_0}{q_1} + 2{q_2}{q_3}q_0^2 - q_1^2 - q_2^2 + q_3^2} \end{array}} \right] \hfill \\ \end{gathered} \right\}$ (21)

 $A\left( q \right) = \left( \begin{gathered} {q_1}{q_2} - {q_3}{q_0} \hfill \\ {q_2}{q_1}{q_0}{q_3} \hfill \\ {q_3} - {q_0}{q_1} - {q_2} \hfill \\ \end{gathered} \right)$
 $q = \left( \begin{gathered} {q_1} \hfill \\ {q_2} \hfill \\ {q_3} \hfill \\ {q_0} \hfill \\ \end{gathered} \right),\zeta = \left( \begin{gathered} {q_2} \hfill \\ - {q_1} \hfill \\ {q_0} \hfill \\ - {q_3} \hfill \\ \end{gathered} \right)$
 $v = \left( \begin{gathered} {q_3} \hfill \\ - {q_0} \hfill \\ - {q_1} \hfill \\ {q_2} \hfill \\ \end{gathered} \right),\xi = \left( \begin{gathered} - {q_{_0}} \hfill \\ - {q_3} \hfill \\ {q_2} \hfill \\ {q_1} \hfill \\ \end{gathered} \right)$

 $x = A\left( q \right)q,\;\;y = A\left( q \right)\varsigma ,\;\;z = A\left( q \right)v$ (22)

 $\varsigma ' = \frac{1}{2}\left( {\frac{1}{{{{\tilde h}^2}{\rho ^3}}}{P_z}v - q} \right)$ (23)

 $\left( {\begin{array}{*{20}{c}} \begin{gathered} {{q'}_1} \hfill \\ {{q'}_2} \hfill \\ {{q'}_3} \hfill \\ {{q'}_0} \hfill \\ \end{gathered} \end{array}} \right) = \frac{1}{2}\left( \begin{gathered} 010{w_x} \hfill \\ - 10{w_x}0 \hfill \\ 0 - {w_x}01 \hfill \\ - {w_x}0 - 10 \hfill \\ \end{gathered} \right)\left( {\begin{array}{*{20}{c}} \begin{gathered} {q_1} \hfill \\ {q_2} \hfill \\ {q_3} \hfill \\ {q_0} \hfill \\ \end{gathered} \end{array}} \right)$

 $w_x = \dfrac{1}{\tilde {h}^2\rho ^3}\left( { {P},{z}} \right)$ (24)

 $w_z = 1 ,\ \ \ w_y = 0$

 $\begin{gathered} {q_1}(s) = - {q_1}(0)\cos \frac{{ws}}{2} - \frac{{{q_2}(0) + {w_x}{q_0}(0)}}{w}\sin \frac{{ws}}{2} \hfill \\ {q_2}(s) = - {q_2}(0)\cos \frac{{ws}}{2} - \frac{{ - {q_1}(0) + {w_x}{q_3}(0)}}{w}\sin \frac{{ws}}{2} \hfill \\ {q_3}(s) = - {q_3}(0)\cos \frac{{ws}}{2} - \frac{{{q_0}(0) - {w_x}{q_2}(0)}}{w}\sin \frac{{ws}}{2} \hfill \\ {q_0}(s) = - {q_0}(0)\cos \frac{{ws}}{2} + \frac{{{q_3}(0) + {w_x}{q_1}(0)}}{w}\sin \frac{{ws}}{2} \hfill \\ \end{gathered}$ (25)
$\left( {q_0 ,q_1 ,q_2 ,q_3 } \right)$为描述矢径单位方向的一组新变量，进而也就确定了轨道平面在空间的位置取向.

1.4 新动力学方程组

 $\begin{gathered} {\text{d}}t = {c_0}{r^2}{\text{d}}s \hfill \\ \frac{{{\text{d}}{c_0}}}{{ds}} = - \frac{{c_0^3}}{{{\rho ^3}}}{P_y} \hfill \\ \frac{{{\text{d}}{c_1}}}{{{\text{d}}s}} = \frac{{c_0^2}}{{{\rho ^2}}}{P_x}\sin s - \frac{{{{c'}_0}}}{{{c_0}}}\left[{\left( {\rho + uc_0^2} \right)\cos s - {c_1}} \right] \hfill \\ \frac{{d{c_2}}}{{ds}} = - \frac{{c_0^2}}{{{\rho ^2}}}{P_x}\cos s - \frac{{{{c'}_0}}}{{{c_0}}}\left[{\left( {\rho + uc_0^2} \right)\sin s - {c_2}} \right] \hfill \\ {q_1}(s) = - {q_1}(0)\cos \frac{{ws}}{2} - \frac{{{q_2}(0) + {w_x}{q_0}(0)}}{w}\sin \frac{{ws}}{2} \hfill \\ {q_2}(s) = - {q_2}(0)\cos \frac{{ws}}{2} - \frac{{ - {q_1}(0) + {w_x}{q_3}(0)}}{w}\sin \frac{{ws}}{2} \hfill \\ {q_3}(s) = - {q_3}(0)\cos \frac{{ws}}{2} - \frac{{{q_0}(0) - {w_x}{q_2}(0)}}{w}\sin \frac{{ws}}{2} \hfill \\ {q_0}(s) = - {q_0}(0)\cos \frac{{ws}}{2} + \frac{{{q_3}(0) + {w_x}{q_1}(0)}}{w}\sin \frac{{ws}}{2} \hfill \\ \end{gathered}$ (26)

2 新轨道参数与惯性系下位置速度的相互 2.1 新参数向位置速度的转换

 ${r} = \left[{r_X } \ \ {r_Y } \ \ {r_Z } \right]^{\rm T} ,\ \ \ v = \left[{v_X } \ \ {v_Y } \ \ {v_Z } \right]^{\rm T}$

 $Q = {\left[\begin{gathered} q_0^2 + q_1^2 - q_2^2 - q_3^22{q_1}{q_2} + 2{q_0}{q_3} - 2{q_0}{q_2} + 2{q_1}{q_3} \hfill \\ 2{q_1}{q_2} + 2{q_0}{q_3}q_0^2 - q_1^2 + q_2^2 - q_3^22{q_0}{q_1} + 2{q_2}{q_3} \hfill \\ 2{q_0}{q_2} + 2{q_1}{q_3} - 2{q_0}{q_1} + 2{q_2}{q_3}q_0^2 - q_1^2 - q_2^2 + q_3^2 \hfill \\ \end{gathered} \right]^T}$

 $\begin{gathered} {\left[{{r_X}\;\;{r_Y}\;\;{r_Z}} \right]^{\text{T}}} = Q{\left[{r\;\;0\;\;0} \right]^{\text{T}}} \hfill \\ {\left[{{v_X}\;\;{v_Y}\;\;{v_Z}} \right]^{\text{T}}} = Q{\left[{{v_x}\;\;{v_y}\;\;0} \right]^{\text{T}}} \hfill \\ \end{gathered}$

 $v$ (27)
 ${v_x} = \frac{{dr}}{{dt}} = - \frac{{\rho '}}{{{c_0}}} = \frac{1}{{{c_0}}}\left( {{c_1}{\text{ sins}} - {c_2}\cos s} \right)$ (28)
 ${v_y} = \frac{{\tilde h}}{r} = \frac{1}{{{c_0}}}\rho = u{c_0} + \frac{1}{{{c_0}}}({c_1}\cos s + {c_2}\sin s)$ (29)

 $\begin{gathered} {r_X} = \frac{{r{r_{{X_0}}}}}{{{r_0}{w^2}}}\cos (ws) + \frac{{2r}}{w}\left( {{q_1}(0){q_2}(0) - {q_0}(0){q_3}(0)} \right)\sin (ws) + \hfill \\ \frac{{w_x^2}}{{{w^2}}}\frac{{r{r_{{X_0}}}}}{{{r_0}}} + \frac{{2{w_x}r}}{{{w^2}}}\left( {{q_1}(0){q_3}(0) + {q_2}(0){q_0}(0)} \right)(1 - \cos (ws)) \hfill \\ \end{gathered}$
 $\begin{gathered} {r_Y} = \frac{{r{r_{{Y_0}}}}}{{{r_0}{w^2}}}\cos (ws) + \frac{r}{w}\left( {q_0^2(0) + q_2^2(0) - q_1^2(0) - q_3^2(0)} \right) \hfill \\ \sin (ws) + \frac{{w_x^2}}{{{w^2}}}\frac{{r{r_{{Y_0}}}}}{{{r_0}}} - \frac{{2{w_x}r}}{{{w^2}}}\left( {{q_0}(0){q_1}(0) - {q_2}(0){q_3}(0)} \right)(1 - \cos (ws)) \hfill \\ \end{gathered}$
 $\begin{gathered} {r_Z} = \frac{{r{r_{{Z_0}}}}}{{{r_0}{w^2}}}\cos (ws) + \frac{{2r}}{w}\left( {{q_2}(0){q_3}(0) + {q_0}(0){q_1}(0)} \right)\sin (ws) + \hfill \\ \frac{{w_x^2}}{{{w^2}}}\frac{{r{r_{{Z_0}}}}}{{{r_0}}} - \frac{{{w_x}r}}{{{w^2}}}\left( {q_1^2(0) + q_2^2(0) - q_3^2(0) - q_0^2(0)} \right)(1 - \cos (ws)) \hfill \\ \end{gathered}$
 $v_X = \left( {2q_1 (0)q_2 (0) - 2q_0 (0)q_3 (0)} \right)\left( {\dfrac{\sin(ws)}{w}v_x + v_y \cos (ws)} \right)$
 $\frac{1}{w}\left( { - {w_x}(2{q_1}(0){q_3}(0) + 2{q_2}(0){q_0}(0)) + \frac{{{r_{{X_0}}}}}{{{r_0}}}} \right)\left( {\frac{{\cos (ws)}}{w}{v_x} - {v_y}\sin (ws)} \right) +$
 $\qquad \left( {2q_1 (0)q_3 (0) + 2q_2 (0)q_0 (0) + w_x \dfrac{r_{X_0 } }{r_0 }}\right)\dfrac{w_x }{w^2}v_x$
 $\begin{gathered} {v_Y} = \left( {q_0^2(0) + q_2^2(0) - q_1^2(0) - q_3^2(0)} \right)\left( {\frac{{\sin (ws)}}{w}{v_x} + {v_y}\cos (ws)} \right) + \hfill \\ \frac{1}{w}\left( {\left( {2{q_0}(0){q_1}(0) - 2{q_2}(0){q_3}(0)} \right){w_x} + \frac{{{r_{{Y_0}}}}}{{{r_0}}}} \right)\left( {\frac{{\cos (ws)}}{w}{v_x} - {v_y}\sin (ws)} \right) \hfill \\ + ( - 2{q_0}(0){q_1}(0) + 2{q_2}(0){q_3}(0) + {w_x}\frac{{{r_{{Y_0}}}}}{{{r_0}}})\frac{{{w_x}}}{{{w^2}}}{v_x} \hfill \\ \end{gathered}$
 $\begin{gathered} {v_Z} = \left( {2{q_2}(0){q_3}(0) + 2{q_0}(0){q_1}(0)} \right)\left( {\frac{{\sin (ws)}}{w}{v_x} + {v_y}\cos (ws)} \right) + \hfill \\ \frac{1}{w}\left( { - {w_x}(q_0^2(0) - q_1^2(0) - q_2^2(0) + q_3^2(0)) + \frac{{{r_{{Z_0}}}}}{{{r_0}}}} \right)\left( {\frac{{\cos (ws)}}{w}{v_x} - {v_y}\sin (ws)} \right) + \hfill \\ \left( {q_0^2(0) - q_1^2(0) - q_2^2(0) + q_3^2(0) + {w_x}\frac{{{r_{{Z_0}}}}}{{{r_0}}}} \right)\frac{{{w_x}}}{{{w^2}}}{v_x} \hfill \\ \end{gathered}$

2.2 位置速度向新参数的转换

 $\begin{gathered} {c_0} = \frac{1}{{\tilde h}} \hfill \\ {c_1} = \frac{1}{{\tilde h}}\left( {{v_y} - \frac{u}{{\tilde h}}} \right)\cos s + \frac{{{v_x}}}{{\tilde h}}\sin s \hfill \\ {c_2} = \frac{1}{{\tilde h}}\left( {{v_y} - \frac{u}{{\tilde h}}} \right)\sin s - \frac{{{v_x}}}{{\tilde h}}\cos s \hfill \\ \end{gathered}$

 $\begin{gathered} \tilde h = \sqrt {h_X^2 + h_Y^2 + h_Z^2} \hfill \\ {h_X} = {r_Y}{v_Z} - {r_Z}{v_Y},{h_Y} = {r_Z}{v_X} - {r_X}{v_Z},{h_Z} = {r_X}{v_Y} - {r_Y}{v_X} \hfill \\ \end{gathered}$

 $Q = \left[\begin{gathered} \frac{{{r_X}}}{r}\frac{{{r^2}{v_X} - {r_X}({r_X}{v_X} + {r_Y}{v_Y} + {r_Z}{v_Z})}}{{\tilde hr}}\frac{{{h_X}}}{{\tilde h}} \hfill \\ \frac{{{r_Y}}}{r}\frac{{{r^2}{v_Y} - {r_Y}({r_X}{v_X} + {r_Y}{v_Y} + {r_Z}{v_Z})}}{{\tilde hr}}\frac{{{h_Y}}}{{\tilde h}} \hfill \\ \frac{{{r_Z}}}{r}\frac{{{r^2}{v_Z} - {r_Z}({r_X}{v_X} + {r_Y}{v_Y} + {r_Z}{v_Z})}}{{\tilde hr}}\frac{{{h_Z}}}{{\tilde h}} \hfill \\ \end{gathered} \right]$

 $\begin{gathered} {q_0} = \pm \frac{{\sqrt {1{\text{ + }}{Q_{11}} + {Q_{22}} + {Q_{33}}} }}{2} \hfill \\ {q_1} = \frac{{{Q_{32}} - {Q_{23}}}}{{4{q_0}}} \hfill \\ {q_2} = \frac{{{Q_{13}} - {Q_{31}}}}{{4{q_0}}} \hfill \\ {q_3} = \frac{{{Q_{21}} - {Q_{12}}}}{{4{q_0}}} \hfill \\ \end{gathered}$

3 新轨道参数与传统轨道六要素的相互转化 3.1 新参数向轨道六要素的转换

 $v_x = \dfrac{d r}{d t} = \dfrac{u}{\tilde {h}}e\sin \theta$

 $\tilde h = \frac{1}{{{c_0}}},e = \frac{1}{{uc_0^2}}\sqrt {c_1^2 + c_2^2}$

 $\begin{array}{*{20}{l}} {L = \left[\begin{gathered} \cos u\cos {\text{ }}\Omega - \sin u\cos i\sin {\text{ }}\Omega \cos u\sin {\text{ }}\Omega + \sin u\cos i\cos {\text{ }}\Omega \sin u\sin i \hfill \\ - \sin u\cos {\text{ }}\Omega - \cos u\cos i\sin {\text{ }}\Omega - \sin u\sin {\text{ }}\Omega + \cos u\cos i\cos {\text{ }}\Omega \cos u\sin i \hfill \\ \sin i\sin {\text{ }}\Omega - \sin i\cos {\text{ }}\Omega \cos i \hfill \\ \end{gathered} \right]} \end{array}$

 $\begin{array}{*{20}{l}} \begin{gathered} i = \arccos \left( {q_0^2 - q_1^2 - q_2^2 + q_3^2} \right) \hfill \\ \omega = \arctan (\frac{{ - {q_0}{q_2} + {q_1}{q_3}}}{{{q_0}{q_1} + {q_2}{q_3}}}) - \theta \hfill \\ \Omega = \arctan (\frac{{{q_0}{q_2} + {q_1}{q_3}}}{{{q_0}{q_1} - {q_2}{q_3}}}) \hfill \\ \end{gathered} \end{array}$

 $\theta = \arctan \left( {\dfrac{v_x }{v_y - uc_0 }} \right)$

3.2 轨道六要素向新参数的转换

 $\begin{gathered} {c_0} = \frac{1}{{\tilde h}} \hfill \\ {c_1} = \frac{1}{{\tilde h}}\left( {\frac{u}{{\tilde h}}e\cos \theta } \right)\cos s + \frac{{ue\sin \theta }}{{{{\tilde h}^2}}}\sin s \hfill \\ {c_2} = \frac{1}{{\tilde h}}\left( {\frac{u}{{\tilde h}}e\cos \theta } \right)\sin s - \frac{{ue\sin \theta }}{{{{\tilde h}^2}}}\cos s \hfill \\ \end{gathered}$

 $\begin{array}{*{20}{l}} \begin{gathered} {q_0} = \pm \frac{{\sqrt {1 + {L_{11}} + {L_{22}} + {L_{33}}} }}{2} \hfill \\ {q_1} = \frac{{{L_{23}} - {L_{32}}}}{{4{q_0}}} \hfill \\ {q_2} = \frac{{{L_{31}} - {L_{13}}}}{{4{q_0}}} \hfill \\ {q_3} = \frac{{{L_{12}} - {L_{21}}}}{{4{q_0}}} \hfill \\ \end{gathered} \end{array}$

4 仿真校验

 $\begin{gathered} a = 7178.145{\text{km}},\;\;e = 0,\;\;i = 0^\circ \hfill \\ \Omega = 10^\circ ,\;\;\omega = 20^\circ ,\theta = 60^\circ \hfill \\ \end{gathered}$

 $\begin{gathered} a = 7178.145{\text{km}},\;\;e = 0,\;\;i = 60^\circ \hfill \\ \Omega = 45^\circ ,\;\;\omega = 15^\circ ,\theta = 30^\circ \hfill \\ \end{gathered}$

 图 1 $P_{x} =0.5$ m/s$^{2}$时，轨道高度的变化 Fig.1 $P_{x} =0.5$ m/s$^{2}$，changes in radius vector
 图 2 $P_{x}=0.5$ m/s$^{2}$时，航天器运动轨迹 Fig.2 $P_{x}=0.5$ m/s$^{2}$，spacecraft trajectories
 图 3 $P_{y} =0.1$ m/s$^{2}$时，航天器的运动轨迹 Fig.3 $P_{y} =0.1$ m/s$^{2}$，spacecraft trajectories

 图 4 $P_{z}=-1$ m/s$^{2}$时，航天器的运动轨迹 Fig.4 $P_{z}=-1$ m/s$^{2}$，spacecraft trajectories

 图 5 非赤道平面J2束缚轨道 Fig.5 Anatomy of J2 bounded orbits in non-equatorial plane

5 结论

(1) 改进后的模型体现了轨道动力学与刚体动力 学的联系.在求解单位向量的动力学方程时，将轨道坐标系视为虚拟刚体，轨道坐标系的变化，可视为虚拟刚体相对于惯性空间的姿态变化，可用四元数描述.新模型适用于任意形式的推力或摄动，物理意义更加明显，方便进行轨道设计.

(2) 分别在常值径向力、法向力、切向力以及变摄动力的情况下，对新的动力学方程组进行了验证，与以往的结论相符.同时，新模型的数值稳定性、积分精度都好于惯性系下的轨道动力学模型.

(3) KS变换下，一组物理坐标对应于多组参数坐标，会导致参数混乱. 该模型不具有此弊端；相对于传统的轨道动力学模型，新模型避免了奇异. 当无摄运动时，方程为谐振动方程，可以求得解析解，与KS变换达到的效果一致.

(4) 将传统轨道六要素$\left( {h,e,i,\varOmega ,\omega ,\theta }\right)$转化为新参数$\left( {c_0 ,c_1 ,c_2 ,q_0 ,q_1 ,q_2 ,q_3 }\right)$，其中$\left( {c_0 ,c_1 ,c_2 }\right)$描述航天器在轨道上的位置信息，$\left( {q_0 ,q_1 ,q_2 ,q_3 }\right)$描述轨道在空间的位置信息.

(5) 使用四元数描述轨道的空间方位，体现了轨道动力学和姿态动力学的联系，今后可利用姿态控制的设计方法来设计轨道机动的需用控制力.

 [1] 袁建平, 朱战霞. 空间操作与非开普勒运动. 宇航学报, 2009, 30(1): 42-46 (Yuan Jianping, Zhu Zhanxia. Space operations and non-Keplerian orbit motion. Journal of Astronautics, 2009, 30(1): 42-46 (in Chinese)) [2] 王萍, 袁建平, 范剑峰. 关于非开普勒轨道的讨论. 宇航学报, 2009, 30(1): 37-41 (Wang Ping, Yuan Jiangping, Fan Jianfeng. A discussion on non-Keplerian orbit. Journal of Astronautics, 2009, 30(1): 37-41 (in Chinese)) [3] Stiefel EL, Scheifele G. Linear and Regular Celestial Mechanics. Berlin, New York, Springer-Verlag, 1971: 18-33 [4] Vitins M. Kepler motion and gyration. Celestial Mechanics, 1978, 17(2): 173-192 [5] 曹静, 袁建平, 罗建军. 陀螺进动与强迫进动轨道. 力学学报, 2013, 45(3): 406-411(Cao Jing, Yuan Jianping, Luo Jianjun. Gyroscopic precession and forced precession orbit. Chinese Journal of Theoretical and Applied Mechanics, 2013, 45(3): 406-411 (in Chinese)) [6] Peláez J, Hedo JM, de Andrés PR. A special perturbation method in orbital dynamics. Celest Mech Dyn Astr, 2007, 97(2): 131-150 [7] Baú G, Bombardelli C, Peláez J. A new set of integrals of motion to propagate the perturbed two-body problem. Celest Mech Dyn Astr, 2013, 116(3): 53-78 [8] Junkins JL, Turner JD. On the analogy between orbital dynamics and rigid body dynamics. The Journal of the Astronautical Science, 1979, 17(4): 345-358. [9] Battin RH. An introduction to the mathematics and methods of astrodynamics. AIAA, 2001: 408-415 [10] 章仁为. 卫星轨道姿态动力学与控制. 北京: 北京航空航天大学出版社, 1998: 157-176 (Zhang Renwei. Satellite Attitude Dynamics and Control. Beijing: Beijing University of Aeronautics and Astronautics Press, 1998: 157-176 (in Chinese)) [11] 赵育善, 师鹏. 航天器飞行动力学建模理论与方法. 北京: 北京航空航天大学出版社, 2012: 5-27 (Zhao Yushan, Shi Peng. Spacecraft Flight Dyna- mics Modeling Theory and Methods. Beijing: Beijing University of Aeronautics and Astronautics Press, 2012: 5-27 (in Chinese)) [12] 袁建平, 李俊峰, 和兴锁等. 航天器相对运动轨道动力学. 北京: 中国宇航出版社, 2013: 101-139 (Yuan Jianping, Li Junfeng, He Xingsuo, et al. Relative Orbit Dynamics of Spacecraft. Beijing: China Aerospace Press. 2013: 101-139(in Chinese)) [13] 刘暾, 赵钧. 空间飞行器动力学. 哈尔滨: 哈尔滨工业大学出版社, 2007: 12-14, 102-114 (Liu Dun, Zhao Jun. Spacecraft Dynamics. Harbin: Harbin Institute of Technology Press. 2007: 12-14, 102-114(in Chinese)) [14] 肖业伦. 航天器飞行动力学原理. 北京: 宇航出版社, 1995: 26-45 (Xiao Yelun. Spacecraft Dynamics Theory. Beijing: Aerospace Press. 1995: 26-45 (in Chinese)) [15] 王伟, 袁建平, 罗建军等. 赤道平面J2束缚轨道研究. 中国科学: 物理学, 力学, 天文学, 2013, 43(3): 309-317 (Wang Wei, Yuan Jianping, Luo Jianjun, et al. Anatomy of J2 bounded orbits in equatorial plane. Scientia Sinica Physica, Mechanica & Astronomica, 2013, 43(3): 309-317 (in Chinese)) [16] Robert JM, Malcolm M, James B, et al. Survey of highly-non-keplerian orbits with low-thrust propulsion. Journal of Guidance, Control, and Dynamics, 2011, 34(3): 645-666 [17] 程国采. 四元数法及其应用. 长沙: 国防科技大学出版社, 1991 (Cheng Guocai. Quaternion Theory and Application. Changsha: National University of Defense Technology Press, 1991 (in Chinese)) [18] 曹静, 袁建平. 空间飞行器轨道相对运动动力学及应用研究.[博士论文]. 西安: 西北工业大学, 2013 (Cao Jing, Yuan Jianping. Kinetics and applied research of spacecraft orbit relative motion.[PhD Thesis]. Xi'an: Northwestern Polytechnical University. 2013 (in Chinese))
AN IMPROVED MODEL OF ORBITAL DYNAMICS
Yang Mengjie, Yuan Jianping
National Key Laboratory of Aerospace Flight Dynamics, Northwestern Polytechnical University, Xi'an 710072, China
Abstract: The radius vector of spacecraft can be decomposed into the product of the mold and the unit vector. Using this property, the traditional orbital dynamics equation can be transformed into two equations which describe the mold's and direction's motions separately. The mold's equation can be converted to a linear equation without singularity by introducing the inverse of the mold; and using the variation of constants method, the linear equation can be reduced to one-order. As for the direction's equation, the quaternion description is suitable. This equation can be completely solved. Through the above handling methods, we obtain a new orbital dynamic model which contains seven equations. In the sense of the virtual time, the angular velocity of the spacecraft depends only on the normal force. This new orbital model is applicable to any form of thrust or perturbation. At the same time, we get seven new stable variables which completely equivalent to the kepler elements. And the transforming relationship has been established. In the end of this article, we verify the accuracy and applicability of the new model in the cases of constant and variable thrusts.
Key words: orbital model    continuous thrust    orbital maneuvering    changing constant    quaternion