﻿ 平台式惯导系统三点校的常值误差解析
 舰船科学技术  2017, Vol. 39 Issue (1): 118-121 PDF

Constant error analysis of gimbaled INS in three-point comprehensive calibration
YANG Xiao-dong, XIA Wei-xing
Department of Navigation and Communication, Submarine Academy, Qingdao 266199, China
Abstract: The influence caused by exoteric information error and platform horizontal error in two-point comprehensive calibration was studied in theory. The relation between maximum value of gyro drift error and calibration intervals was derived. It also represented the simulation of three-point calibration affected by exoteric information error. The study result shows that these two kinds of error are both linear with the estimation error of the constant gyro drift. In order to treat it as a high precision approach of calibration, it is important to increase the exoteric information precision and platform horizontal accuracy efficiently.
Key words: navigation     comprehensive calibration     three-point comprehensive calibration     constant error analysis     gimbaled INS
0 引言

1 基于误差的三点校原理

 $\Phi \left( t \right)=\Theta \left( t \right)+\psi \left( t \right),$ (1)

 $\Phi \left( t \right)={{\left[ \begin{matrix} \alpha \left( t \right) & \beta \left( t \right) & \gamma \left( t \right) \\ \end{matrix} \right]}^{\text{T}}}\text{,}$ (2)

Θt）在地理坐标系各轴向分量与经纬度误差的关系为：

 $\left\{ \begin{array}{*{35}{l}} {{\theta }_{X}}\left( t \right)=-\delta \varphi \left( t \right), \\ {{\theta }_{Y}}\left( t \right)=\delta \lambda \left( t \right)\cos \varphi , \\ {{\theta }_{Z}}\left( t \right)=\delta \lambda \left( t \right)\sin \varphi \\ \end{array} \right.$ (3)

 $P(t)=M\psi (t)-NW(t)\text{,}$ (4)

 $\psi ({{t}_{N+1}})=T({{t}_{N+1}},{{t}_{N}})\psi ({{t}_{N}})+U({{t}_{N+1}},{{t}_{N}})\frac{\varepsilon }{\Omega }\text{,}$ (5)

aN = ΩtN + 1-tN），则

 ${{T}}({t_{N + 1}}{\rm{,}}{t_N}) = \left[{\begin{array}{*{20}{c}} {\cos {a_N}} & 0 & {-\sin {a_N}}\\ 0 & 1 & 0\\ {\sin {a_N}} & 0 & {\cos {a_N}} \end{array}} \right]$，${{U}}({t_{N + 1}}{\rm{,}}{t_N}) = \left[{\begin{array}{*{20}{c}} {\sin {a_N}} & 0 & {-(1-\cos {a_N})}\\ 0 & {{a_N}} & 0\\ {1-\cos {a_N}} & 0 & {\sin {a_N}} \end{array}} \right]$

OEPQ 坐标系下陀螺漂移：

 ${{\varepsilon }} = {\left[{\begin{array}{*{20}{c}} {{\varepsilon _E}} & {{\varepsilon _P}} & {{\varepsilon _Q}} \end{array}} \right]^{\rm{T}}}$

 \begin{align} & P\left( {{t}_{1}} \right)=M\psi \left( {{t}_{1-}} \right)-NW\left( {{t}_{1}} \right)= \\ & {{S}_{1}}\left[ \begin{matrix} {{\psi }_{Q}}\left( {{t}_{1-}} \right) \\ \frac{\varepsilon }{\Omega } \\ \end{matrix} \right]-NW\left( {{t}_{1}} \right)\text{,} \\ \end{align} (6)

 \begin{align} & P\left( {{t}_{2}} \right)=M\psi \left( {{t}_{2-}} \right)-NW\left( {{t}_{2}} \right)= \\ & M\left[ T\left( {{t}_{2}},{{t}_{1}} \right)\psi \left( {{t}_{1+}} \right)+U\left( {{t}_{2}},{{t}_{1}} \right)\frac{\varepsilon }{\Omega } \right]-NW\left( {{t}_{2}} \right)= \\ & MT\left( {{t}_{2}},{{t}_{1}} \right)\psi \left( {{t}_{1+}} \right)+MU\left( {{t}_{2}},{{t}_{1}} \right)\frac{\varepsilon }{\Omega }-NW\left( {{t}_{2}} \right)\text{,} \\ \end{align} (7)
 $\left\{ \begin{array}{*{35}{l}} \Delta P(t)=M\psi ({{t}_{+}})-NW(t)\text{,} \\ P(t)=M\psi ({{t}_{-}})-NW(t) \\ \end{array} \right.$ (8)

 $\psi \left( {{t}_{+}} \right)=G\cdot \psi \left( {{t}_{-}} \right)+H\cdot \Delta P\left( t \right)+R\cdot W\left( t \right)\text{,}$ (9)

 \begin{align} & P\left( {{t}_{2}} \right)=M\psi \left( {{t}_{2-}} \right)-NW\left( {{t}_{2}} \right)= \\ & M\left[ T\left( {{t}_{2}},{{t}_{1}} \right) \right.\left. \left( G\cdot \psi \left( {{t}_{1-}} \right)+H\cdot \Delta P\left( {{t}_{1}} \right)+R\cdot W\left( {{t}_{1}} \right) \right) \right]+ \\ & \left. U\left( {{t}_{2}},{{t}_{1}} \right)\frac{\varepsilon }{\Omega } \right]-NW\left( {{t}_{2}} \right)=MT\left( {{t}_{2}},{{t}_{1}} \right)G\cdot \psi \left( {{t}_{1-}} \right)+ \\ & MU\left( {{t}_{2}},{{t}_{1}} \right)\frac{\varepsilon }{\Omega }-NW\left( {{t}_{2}} \right)+MT\left( {{t}_{2}},{{t}_{1}} \right)\left[ H\cdot \Delta P\left( {{t}_{1}} \right)+ \right. \\ & \left. R\cdot W\left( {{t}_{1}} \right) \right]={{S}_{2}}\left[ \begin{matrix} {{\psi }_{Q}}\left( {{t}_{1-}} \right) \\ [5pt]\frac{\varepsilon }{\Omega } \\ \end{matrix} \right]-NW\left( {{t}_{2}} \right)+MT\left( {{t}_{2}},{{t}_{1}} \right)\times \\ & \left. \left[ H\cdot \Delta P\left( {{t}_{1}} \right)+ \right.R\cdot W\left( {{t}_{1}} \right) \right] \\ \end{align} (10)

 $P\left( {{t}_{3}} \right)=MT\left( {{t}_{3}},{{t}_{2}} \right)\psi \left( {{t}_{2+}} \right)+MU\left( {{t}_{3}},{{t}_{2}} \right)\frac{\varepsilon }{\Omega }-NW\left( {{t}_{3}} \right)\text{,}$ (11)

 \begin{align} & P\left( {{t}_{3}} \right)={{S}_{3}}\left[ \begin{matrix} {{\psi }_{Q}}\left( {{t}_{1-}} \right) \\ \frac{\varepsilon }{\Omega } \\ \end{matrix} \right]-NW\left( {{t}_{3}} \right)+ \\ & MT\left( {{t}_{3}},{{t}_{2}} \right)\left[ G\cdot T\left( {{t}_{2}},{{t}_{1}} \right)\left[ H\cdot \Delta P\left( {{t}_{1}} \right)+R\cdot W\left( {{t}_{1}} \right) \right]+ \right. \\ & \left. \left( H\cdot \Delta P\left( {{t}_{2}} \right)+R\cdot W\left( {{t}_{2}} \right) \right) \right] \\ \end{align} (12)

${y_1} = \cos \left( {{a_1}} \right){\rho _1} + \omega \left[{\tan \left( \varphi \right)\sin \left( {{a_1}} \right){\rho _1} + {\chi _1}} \right]$，${y_2} = \cos \left( {{a_2}} \right){\rho _2}-\sin \left( {{a_1}} \right)\sin \left( {{a_2}} \right){\rho _1}$，${y_3} \!=\! \tan \left( \varphi \right)\sin \left( {{a_1}} \right)\left[{\cos \left( {{a_2}} \right) \!-\! 1} \right]{\rho _1}\! +\! \tan \left( \varphi \right)\sin \left( {{a_2}} \right){\rho _2} \!+\! {\chi _2}$,${\rho _i} = \Delta {\varphi _i} + {\alpha _i}$，${\chi _i} = \Delta {\lambda _i}-{\beta _i}\sec \varphi$。

2 三点校过程误差分析

E' 为综合校正中广义陀螺漂移的计算值，则

 ${{E}}' = \Omega {{C}}{{{F}}^{-1}}\left[{\begin{array}{*{20}{c}} {\delta \varphi '\left( {{t_{2-}}} \right) + \omega \delta \lambda '\left( {{t_{2-}}} \right)}\\ {\delta \varphi '\left( {{t_{3-}}} \right)}\\ {\delta \lambda '\left( {{t_{3-}}} \right)} \end{array}} \right]\text{,}$ (15)

 \begin{aligned} & \Delta { E} = {{ E}^\prime }-{ E} = \\ &-\Omega { C}{{ F}^{-1}}\left( {\left[{\begin{array}{*{20}{c}} {\Delta \varphi \left( {{t_2}} \right) + \omega \cdot \Delta \lambda \left( {{t_2}} \right)}\\ {\Delta \varphi \left( {{t_3}} \right)}\\ {\Delta \lambda \left( {{t_3}} \right)} \end{array}} \right] + x-y} \right) = \\ &-\Omega { C}{{ F}^{-1}}\left( {{ J}\left[{\begin{array}{*{20}{c}} {{\rho _1}}\\ {{\rho _2}}\\ {{\rho _3}} \end{array}} \right] + { K}\left[{\begin{array}{*{20}{c}} {{\chi _1}}\\ {{\chi _2}}\\ {{\chi _3}} \end{array}} \right]} \right)\text{。} \end{aligned} (16)

 ${{J}} = \left[{\begin{array}{*{20}{c}} {-\cos {a_1}-\omega \tan \varphi \sin {a_1}} & 1 & 0\\ {\sin {a_1}\sin {a_2}} & {-\cos {a_2}} & 1\\ {-\tan \varphi \sin {a_1}\left( {\cos {a_2}-1} \right)} & {-\tan \varphi \sin {a_2}} & 0 \end{array}} \right]$；${{K}} = \left[{\begin{array}{*{20}{c}} {-\omega } & \omega & 0\\ 0 & 0 & 0\\ 0 & {-1} & 1 \end{array}} \right]$

1） 重调时，外部测量误差和平台水平失调角均影响陀螺漂移的估计结果，产生误差，且二者与陀螺漂移估计误差呈线性关系；

2） 纬度误差 Δφt）和东向平台水平失调角 αt）对陀螺漂移估计误差的贡献相同，同理，经度误差 Δλt）和 βt）sec（φ）对陀螺漂移估计误差贡献也相同。

3） Δλt）或北向陀螺漂移有常值误差时，不会造成陀螺漂移的估计误差；

4） 引起陀螺漂移估计误差的因素不相同，EX 估计误差由 Δφt1），Δλt1），Δφt2），Δλt2），α1β1α2β2 造成；Δφt1），Δφt3），α1α3 产生 EY 的估计误差；EZ 的估计误差则由 Δφt1），Δλt3），α1β3 造成。

 $\begin{array}{l} \max \left( {\Delta E} \right) \!\!=\!\! {{abs}}\left( {\Omega C{F^{-1}}J} \right) \!\!\times \!\! \left[\!\!\!\!{\begin{array}{*{20}{c}} {\max \left( {\Delta \varphi \left( {{t_1}} \right) \!\!+\!\! {\alpha _1}} \right)}\\ {\max \left( {\Delta \varphi \left( {{t_2}} \right) \!\!+\!\! {\alpha _2}} \right)}\\ {\max \left( {\Delta \varphi \left( {{t_3}} \right) \!\!+\!\! {\alpha _3}} \right)} \end{array}} \!\!\!\!\right] + \\ {{abs}}\left( {\Omega C{F^{-1}}K} \right)\left[{\begin{array}{*{20}{c}} {\max \left( {\Delta \lambda \left( {{t_1}} \right)-\sec \left( \varphi \right){\beta _1}} \right)}\\ {\max \left( {\Delta \lambda \left( {{t_2}} \right)-\sec \left( \varphi \right){\beta _2}} \right)}\\ {\max \left( {\Delta \lambda \left( {{t_3}} \right)-\sec \left( \varphi \right){\beta _3}} \right)} \end{array}} \right] = \\ {{abs}}\left( {\Omega C{F^{-1}}J} \right) \cdot \left[{\begin{array}{*{20}{c}} {\max \left( {\Delta \varphi \left( {{t_1}} \right)} \right) + \max \left( {{\alpha _1}} \right)}\\ {\max \left( {\Delta \varphi \left( {{t_2}} \right)} \right) + \max \left( {{\alpha _2}} \right)}\\ {\max \left( {\Delta \varphi \left( {{t_3}} \right)} \right) + \max \left( {{\alpha _3}} \right)} \end{array}} \right] + \\ {{abs}}\left( {\Omega C{F^{-1}}K} \right)\left[\!\!\!\!{\begin{array}{*{20}{c}} {\max \left( {\Delta \lambda \left( {{t_1}} \right)} \right) \!\!+\!\! \sec \left( \varphi \right)\max \left( {{\beta _1}} \right)}\\ {\max \left( {\Delta \lambda \left( {{t_2}} \right)} \right) \!\!+\!\! \sec \left( \varphi \right)\max \left( {{\beta _2}} \right)}\\ {\max \left( {\Delta \lambda \left( {{t_3}} \right)} \right) \!\!+\!\! \sec \left( \varphi \right)\max \left( {{\beta _3}} \right)} \end{array}} \!\!\right] = \\ \left[{\begin{array}{*{20}{c}} {1.7}\\ {0.9}\\ {0.9} \end{array}} \right] \times {\rm{1}}{{\rm{0}}^{-3}}^{^ \circ } \cdot {{\rm{h}}^{-1}}\text{。} \end{array}$ (17)

3 仿真与分析

4 结语

 [1] 王光辉, 朱海, 莫军, 等. 一种基于天空光偏振特性的天文导航方式[J]. 中国航海, 2009, 32 (1):14–16. [2] 于堃, 李琳, 刘为任, 等. 舰船惯性导航系统海上无阻尼状态的校准[J]. 中国惯性技术学报, 2008, 16 (6):637–642. [3] SINOPOLI B, SCHENATO L, FRANCESCHEAI M, et al. Kalman filtering with intermittent observations[J]. IEEE Transactions on Automatic Control, 2004, 49 (9):1453–1464. DOI: 10.1109/TAC.2004.834121 [4] 程建华, 郝燕玲, 孙枫, 等. 自动补偿技术在平台式惯导系统综合校正中的应用研究[J]. 哈尔滨工程大学学报, 2008, 29 (1):40–44. [5] 关劲, 程建华, 吴磊, 等. 船用平台式惯导系统状态转换技术的应用[J]. 中国造船, 2008, 49 (2):75–80. [6] 钱山, 李鹏奎, 张士峰, 等. MIMU/GPS组合导航建模及GPS时间延迟补偿算法研究[J]. 系统工程与电子技术, 2009, 31 (6):1432–1435. [7] 张源, 许江宁, 卞鸿巍. GPS姿态测量系统对惯性导航系统误差修正能力分析[J]. 情报指挥控制系统与仿真技术, 2005, 27 (5):96–100. [8] 韩璐, 景占荣, 段哲民. SINS/GPS组合导航系统仿真研究[J]. 计算机仿真, 2009, 26 (9):32–36. [9] 杨艳娟, 卞鸿巍, 田蔚风, 等. 一种新的INS/GPS组合导航技术[J]. 中国惯性技术学报, 2004, 12 (2):23–26. [10] 单丹萍. 基于COM组件的平台式惯导系统模拟器设计与实现[D]. 哈尔滨:哈尔滨工程大学, 2007:23-26. http://cn.bing.com/academic/profile?id=2de6df28bf3575cc6b0857f7e8c87fd9&encoded=0&v=paper_preview&mkt=zh-cn [11] 牛其虎. 平台式惯导系统模拟器的设计及实现[D]. 哈尔滨:哈尔滨工程大学, 2006:25-28. http://cn.bing.com/academic/profile?id=2de6df28bf3575cc6b0857f7e8c87fd9&encoded=0&v=paper_preview&mkt=zh-cn [12] 冯跃强, 王彬, 蔡伯根. GPS/GLONASS组合定位导航技术的测试[J]. 北京交通大学学报, 2005, 29 (2):91–93. [13] 岳亚洲, 田宇, 张晓冬. 机载惯性/天文组合导航研究[J]. 光学与光电技术, 2008, 6 (2):1–5. [14] 何炬. 国外天文导航技术发展综述[J]. 舰船科学技术, 2005, 27 (5):91–96. [15] 方海斌. 平台式惯导系统仿真软件的设计[D]. 哈尔滨:哈尔滨工程大学, 2009:21-22.