﻿ 惯性系统仿真试验平台中的综合校准技术
 舰船科学技术  2018, Vol. 40 Issue (6): 117-119 PDF

Comprehensive calibration technology of the inertial system simulation test platform
MA Chi-xin, LIU Su-zhen
No. 92941 Unit of PLA, Huludao 125001, China
Abstract: Comprehensive calibration error model of inertial navigation system was analyzed. The project of comprehensive position and heading accuracy calibration was designed. Navigation benchmark information was simulated by parameter settings and system resetting, which provide the initial conditions for INS simulation calculating, INS information generating and comparing.
Key words: comprehensive calibration     error model     navigation benchmark
0 引　言

1 综合校准技术

1.1 综合校准误差模型

 图 1 qep坐标系与NED坐标系的关系 Fig. 1 The relation between qep axis and NED axis

q轴为赤道轴，平行于赤道平面与本地子午面的交线，由地轴指向外；

e轴为纬度轴，与等纬度圈相切，指向东；

p为极轴，平行于地球自转轴。

 ${\bf C}_{NED}^{qep} = \left[ {\begin{array}{*{20}{c}}{ - \sin L} & 0 & { - \cos L}\\0 & 1 & 0\\{\cos L} & 0 & { - \sin L}\end{array}} \right]\text{。}$ (1)

 ${\dot \psi ^p} = - [\omega _{ip}^p]{\psi ^p} + \delta \omega _d^p\text{，}$ (2)

 \begin{align}&{\psi _q}\left( t \right) = {\psi _q}\left( {{t_0}} \right)\cos {\omega _{ie}}\left( {t - {t_0}} \right) + {\psi _e}\left( {{t_0}} \right)\sin {\omega _{ie}}\left( {t - {t_0}} \right)+\\ & \frac{{{\varepsilon _q}}}{{{\omega _{ie}}}}\sin {\omega _{ie}}\left( {t - {t_0}} \right) + \frac{{{\varepsilon _e}}}{{{\omega _{ie}}}}\left( {1 - \cos {\omega _{ie}}\left( {t - {t_0}} \right)} \right)\text{，}\\[8pt] &{\psi _e}\left( t \right) = - {\psi _q}\left( {{t_0}} \right)\sin {\omega _{ie}}\left( {t - {t_0}} \right) + {\psi _e}\left( {{t_0}} \right)\cos {\omega _{ie}}\left( {t - {t_0}} \right)-\\ & \frac{{{\varepsilon _q}}}{{{\omega _{ie}}}}\left( {1 - \cos {\omega _{ie}}\left( {t - {t_0}} \right)} \right) + \frac{{{\varepsilon _e}}}{{{\omega _{ie}}}}\sin {\omega _{ie}}\left( {t - {t_0}} \right)\text{，}\\[8pt] &{\psi _p}\left( t \right) = {\psi _p}\left( {{t_0}} \right) + {\varepsilon _p}\left( {t - {t_0}} \right)\text{。}\end{align} (3)

 $\left[ {\begin{array}{*{20}{c}} {\delta l} \\ {\delta L} \\ {{\varphi _d}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\tan L}&0&{ - 1} \\ 0&1&0 \\ { - 1/{\rm cos}L}&0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\psi _q}} \\ {{\psi _e}} \\ {{\psi _p}} \end{array}} \right]\text{。}$ (4)

1.2 GPS三点校方案

 ${z_i} = {H_1}\Psi \left( {{t_i}} \right) + {v_i}\text{。}$ (5)

t1t2t3表示可以由GPS获得外部参考位置的时刻，假设在t1时刻舰艇获得第1点参考位置， ${z_1} = {H_1}\varPsi \left( {{t_1}} \right)$

 ${z_2} = {H_2}\varPsi \left( {{t_2}} \right)\text{。}$ (6)

 $\varPsi \left( {{t_2}} \right) = {\varPhi} \left( {{t_2},{t_1}} \right)\varPsi \left( {{t_1}} \right) + { B}\left( {{t_2},{t_1}} \right)\varepsilon \left( {{t_1}} \right)\text{。}$ (7)

 ${z_3} = {H_3}\varPsi \left( {{t_3}} \right)\text{。}$ (8)
 \begin{align}& \varPsi \left( {{t_3}} \right) = {\varPhi} \left( {{t_3},{t_1}} \right)\varPsi \left( {{t_1}} \right) + {\varPhi} \left( {{t_3},{t_2}} \right){ B}\left( {{t_2},{t_1}} \right)\varepsilon \left( {{t_1}} \right) +\\ & { B}\left( {{t_3},{t_2}} \right)\varepsilon \left( {{t_2}} \right)\text{。}\end{align} (9)

 $\begin{split}&\left[ {\begin{array}{*{10}{c}}{\tan L} \!\!\!&\!\!\! 0 \!\!\!& \!\!\!{ - 1} \!\!\!& \!\!\!0 \!\!\!&\!\!\! 0 \!\!\!& \!\!\!0\\0 \!\!\!& \!\!\!1 & \!\!\!0 &\!\!\! 0 \!\!\!& \!\!\!0 \!\!\!& \!\!\!0\\{\tan L\cos {\omega _{ie}}\left( {{t_2} - {t_1}} \right)}\!\!\! &\!\!\! {\tan L\sin {\omega _{ie}}\left( {{t_2} - {t_1}} \right)} \!\!\!&\!\!\! { - 1}\!\!\! &\!\!\! {\tan L\sin {\omega _{ie}}\left( {{t_2} - {t_1}} \right)}\!\!\! &\!\!\!{\tan L\left[ {1 - \cos {\omega _{ie}}\left( {{t_2} - {t_1}} \right)} \right]}\!\!\! & \!\!\!{ - {\omega _{ie}}\left( {{t_2} - {t_1}} \right)}\\{ - \sin {\omega _{ie}}\left( {{t_2} - {t_1}} \right)} \!\!\!& \!\!\!{\cos {\omega _{ie}}\left( {{t_2} - {t_1}} \right)} \!\!\!& \!\!\!0\!\!\! &\!\!\! { - 1 + \cos {\omega _{ie}}\left( {{t_2} - {t_1}} \right)} \!\!\!& \!\!\!{\sin {\omega _{ie}}\left( {{t_2} - {t_1}} \right)} \!\!\!&\!\!\! 0\\{\tan L\cos {\omega _{ie}}\left( {{t_3} - {t_1}} \right)} \!\!\!& \!\!\!{\tan L\sin {\omega _{ie}}\left( {{t_3} - {t_1}} \right)} \!\!\!&\!\!\! { - 1} \!\!\!& \!\!\!{\tan L\sin {\omega _{ie}}\left( {{t_3} - {t_1}} \right)} \!\!\!&\!\!\! {\tan L\left[ {1 - \cos {\omega _{ie}}\left( {{t_3} - {t_1}} \right)} \right]}\!\!\! &\!\!\! { - {\omega _{ie}}\left( {{t_3} - {t_1}} \right)}\\{ - \sin {\omega _{ie}}\left( {{t_3} - {t_1}} \right)} \!\!\!&\!\!\! {\cos {\omega _{ie}}\left( {{t_3} - {t_1}} \right)} \!\!\!& \!\!\!0 \!\!\!& \!\!\!{ - 1 + \cos {\omega _{ie}}\left( {{t_3} - t} \right)}\!\!\! &\!\!\! {\sin {\omega _{ie}}\left( {{t_3} - {t_1}} \right)}\!\!\! &\!\!\! 0\end{array}} \right] \times \\&\left[ {\begin{array}{*{20}{c}}{{\varPsi _q}\left( {{t_1}} \right)}\\{{\varPsi _e}\left( {{t_1}} \right)}\\{{\varPsi _p}\left( {{t_1}} \right)}\\{{\varepsilon _q}\left( {{t_1}} \right)/{\omega _{ie}}}\\{{\varepsilon _e}\left( {{t_1}} \right)/{\xi _{ie}}}\\{{\varepsilon _p}\left( {{t_1}} \right)/{\omega _{ie}}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\delta {l_1}}\\{\delta {l_1}}\\{\delta {l_2}}\\{\delta {l_2}}\\{\delta {l_3}}\\{\delta {l_3}}\end{array}} \right]\text{。}\end{split}$ (10)

1.3 外参考速度校准方案

 \begin{align}& \delta {v_N} = - R{\omega _{ie}}\left[ {\left( {\varPsi _x^e\left( 0 \right) + \Delta {\varepsilon _{1x}}t} \right)\cos {\gamma _2}\left( t \right) + } \right.\\ &\left. {\left( {\varPsi _y^e\left( 0 \right) + {\Delta _{1y}}t} \right)\sin {\gamma _2}\left( t \right)} \right]\text{，}\\ &\delta {v_E} = R{\omega _{ie}}\sin L\left[ { - \left( {\varPsi _x^e\left( 0 \right) + \Delta {\varepsilon _{1x}}t} \right)\sin {\gamma _2}\left( t \right) + } \right.\\&\left. {\left( {\varPsi _y^e\left( 0 \right) + \Delta {\varepsilon _{1y}}t} \right)\cos {\gamma _2}(t)} \right] - R\Delta {\varepsilon _{2z}}\cos L\text{。}\end{align} (11)

 $\begin{split}& \delta \left[ {\begin{array}{*{20}{c}}{\cos {\gamma _2}\left( {{t_1}} \right)} & {\sin {\gamma _2}\left( {{t_1}} \right)}\\ \vdots & \vdots \\{\cos {\gamma _2}\left( {{t_1} + n\Delta t} \right)} & {\sin {\gamma _2}\left( {{t_1} + n\Delta t} \right)}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{\varPsi _{x0}^e}\\{\varPsi _{y0}^e}\end{array}} \right] = \\ & \left[ {\begin{array}{*{20}{c}}{ - \frac{{\delta {v_N}\left( {{t_1}} \right)}}{{{R_{{w_{ie}}}}}}}\\ \vdots \\{ - \frac{{\delta {v_N}\left( {{t_1} + n\Delta t} \right)}}{{{R_{{\omega _{ie}}}}}}}\end{array}} \right]\text{。}\end{split}$ (12)

 $\begin{array}{l}\left[ {\begin{array}{*{20}{c}}{\overline {\varPsi _{x0}^e} }\\[8pt]{\overline {\varPsi _{y0}^e} }\end{array}} \right] = \left[\!\!\! {\begin{array}{*{20}{c}}{\displaystyle\sum\limits_{i = 0}^n {{{\cos }^2}{\gamma _2}\left( {{t_1} + i\Delta t} \right)} } & {\displaystyle\frac{1}{2}\sum\limits_{i = 0}^n {2{\gamma _2}\left( {{t_1} + i\Delta t} \right)} }\\ \vdots & \vdots \\{\displaystyle\frac{1}{2}\sum\limits_{i = 0}^n {\sin 2{\gamma _2}\left( {{t_1} + i\Delta t} \right)} } & {\displaystyle\sum\limits_{i = 0}^n {{{\sin }^2}{\gamma _2}\left( {{t_1} + i\Delta t} \right)} }\end{array}} \!\!\!\right]\times \\[10pt] \left[ {\begin{array}{*{20}{c}}{ - \displaystyle\sum\limits_{i = 0}^n {\displaystyle\frac{{\delta {v_N}\left( {{t_1} + i\Delta t} \right)\cos {\gamma _2}\left( {{t_1} + i\Delta t} \right)}}{{{R_{{w_{ie}}}}}}} }\\ \vdots \\{ - \displaystyle\sum\limits_{i = 0}^n {\displaystyle\frac{{\delta {v_N}\left( {{t_1} + i\Delta t} \right)\sin {\gamma _2}\left( {{t_1} + i\Delta t} \right)}}{{{R_{{\omega _{ie}}}}}}} }\end{array}} \right]\text{。}\quad\quad\quad\quad\ \ (13)\end{array}$

 $\begin{array}{l}\Delta {{\hat \varepsilon }_{1x}} = 2\overline {\varPsi _{x0}^e} /\left( {{t_1} + {t_2}} \right)\text{，}\\[7pt]\Delta {{\hat \varepsilon }_{1y}} = 2\overline {\varPsi _{y0}^e} /\left( {{t_1} + {t_2}} \right)\text{。}\end{array}$ (14)

 $\begin{split}& \overline {\Delta {{\hat \varepsilon }_{2x}}} = - \displaystyle\frac{1}{{\left( {n + 1} \right)R\cos L}}\sum\limits_{i = 0}^n {\left[ {\delta {v_E}\left( {{t_1} + i\Delta t} \right) + {R{\omega _{ie}}\sin L}} \right.} \\[3pt]& \left. {\left( {\overline {\varPsi _{x0}^e} \sin {\gamma _2}\left( {{t_1} + i\Delta t} \right) - \overline {\varPsi _{y0}^e} \cos {\gamma _2}\left( {{t_1} + i\Delta t} \right)} \right)} \right]\text{。}\end{split}$ (15)

 $\!\!\!\begin{array}{l}\left\{\begin{array}{l}\delta l\tilde L\left( {{t_c}} \right) = - \delta \tilde \varPsi _x^e\sin {\gamma _2}\left( {{t_c}} \right) + \delta \tilde \varPsi _y^e\cos {\gamma _2}\left( {{t_c}} \right)\text{，}\\[5pt]\delta \tilde l\left( {{t_c}} \right) = \tan L\left[ {\delta \tilde \varPsi _x^e\cos {\gamma _2}\left( {{t_c}} \right) + \delta \tilde \varPsi _y^e\sin {\gamma _2}\left( {{t_c}} \right)} \right] - \\ \Delta {{\hat \varepsilon }_{2z}}{t_c}\text{。}\end{array}\right.\end{array}$ (16)

 $\begin{array}{l}\left\{\begin{array}{l}\Delta {{\tilde \varepsilon }_{1x}} = \Delta {{\tilde \varepsilon }_{1Y}} = \displaystyle\frac{{\delta \tilde L\left( {{t_c}} \right)}}{{\left( {\cos {\gamma _2}\left( {{t_c}} \right) - \sin {\gamma _2}\left( {{t_c}} \right)} \right){t_c}}}\text{，}\\[8pt]\Delta {{\tilde \varepsilon }_{2x}} = \left\{ {\delta \tilde L\left( {{t_c}} \right)\tan L\left[ {\displaystyle\frac{{1 + \sin 2{\gamma _2}\left( {{t_c}} \right)}}{{\cos 2{\gamma _2}\left( {{t_c}} \right)}}} \right] - \delta \tilde l\left( {{t_c}} \right)/{t_c}} \right\}\text{。} (17)\end{array}\right.\end{array}$
2 结　语

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