# 基于平台调制的半解析式惯导系统高纬度工作研究Research on High-Latitude Work of Semi-Analytical Inertial Navigation System Based on Platform Modulation

Abstract: Aiming at the problem that meridian convergence in high latitude area leads to the failure of navigation algorithm with true north direction as heading reference, combined with the performance analysis of the traditional mechanical arrangement method in the polar region of the semi-analytical inertial navigation system, taking the problem that azimuth axis gyrotorquing under the inertial navigation system with local horizontal north pointing mechanical arrangement method is difficult to work in the polar region as an example, the special mechanical arrangement method of the inertial navigation system with azimuth axis gyrotorquing, i.e. wandering azimuth inertial navigation system, is introduced in detail. This paper proposes a semi-analytical inertial navigation system with vertical axis modulation of the inertial platform as the research object. Compared with the traditional semi-analytical inertial navigation system at high latitudes, the semi-analytical inertial navigation system with platform modulation function can modulate the zero bias error of the horizontal gyroscope and the horizontal accelerometer to avoid the divergence of the inertial navigation error. This paper deduces the high-latitude mechanical arrangement method of the semi-analytical inertial navigation system based on the vertical axis modulation of the inertial platform and proposes a polar region navigation algorithm based on the grid navigation mechanics arrangement. Through the analysis and simulation verification of the working principles of the two kinds of navigation mechanical arrangements, a semi-analytical inertial navigation scheme with a platform modulation function is established.

1. 引言

2. 半解析式惯导工作原理

2.1. 半解析式惯导系统平台结构及功能

Figure 1. Schematic diagram of semi-analytical inertial navigation system

2.2. 平台系统伺服回路的耦合

Figure 2. Schematic diagram of platform frame in neutral position

$\begin{array}{l}{\omega }_{ix}={\omega }_{px}\mathrm{cos}k-{\omega }_{py}\mathrm{sin}k\\ {\omega }_{0y}={\omega }_{px}\mathrm{sin}k+{\omega }_{py}\mathrm{cos}k\end{array}\right\}$ (1)

$\begin{array}{l}{\omega }_{px}={\omega }_{ix}\mathrm{cos}k+{\omega }_{0y}\mathrm{sin}k\\ {\omega }_{py}=-{\omega }_{ix}\mathrm{sin}k+{\omega }_{0y}\mathrm{cos}k\\ {\omega }_{pz}={\omega }_{pz}\end{array}\right\}$ (2)

${C}_{P}^{I}=\left[\begin{array}{ccc}\mathrm{cos}k& \mathrm{sin}k& 0\\ -\mathrm{sin}k& \mathrm{cos}k& 0\\ 0& 0& 1\end{array}\right]$ (3)

3. 游移方位编排

Figure 3. Walking azimuth inertial navigation system coordinate system

${C}_{t}^{n}=\left[\begin{array}{ccc}\mathrm{cos}\alpha & \mathrm{sin}\alpha & 0\\ -\mathrm{sin}\alpha & \mathrm{cos}\alpha & 0\\ 0& 0& 1\end{array}\right]$ (4)

$\left[\begin{array}{c}{x}_{n}\\ {y}_{n}\\ {z}_{n}\end{array}\right]=\left[\begin{array}{c}{x}_{e}\\ {y}_{e}\\ {z}_{e}\end{array}\right]{C}_{e}^{n}$ (5)

$\begin{array}{c}{C}_{e}^{n}={C}_{t}^{n}{C}_{e}^{t}=\left[\begin{array}{ccc}\mathrm{cos}\alpha & \mathrm{sin}\alpha & 0\\ -\mathrm{sin}\alpha & \mathrm{cos}\alpha & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}-\mathrm{sin}\lambda & \mathrm{cos}\lambda & 0\\ -\mathrm{sin}L\mathrm{cos}\lambda & -\mathrm{sin}L\mathrm{sin}\lambda & \mathrm{cos}L\\ \mathrm{cos}L\mathrm{cos}\lambda & \mathrm{sin}\lambda & \mathrm{sin}L\end{array}\right]\\ =\left[\begin{array}{ccc}-\mathrm{cos}\alpha \mathrm{sin}\lambda -\mathrm{sin}\alpha \mathrm{sin}L\mathrm{cos}\lambda & \mathrm{cos}\alpha \mathrm{cos}\lambda -\mathrm{sin}\alpha \mathrm{sin}L\mathrm{sin}\lambda & \mathrm{sin}\alpha \mathrm{cos}L\\ \mathrm{sin}\alpha \mathrm{sin}\lambda -\mathrm{cos}\alpha \mathrm{sin}L\mathrm{cos}\lambda & -\mathrm{sin}\alpha \mathrm{cos}\lambda -\mathrm{cos}\alpha \mathrm{sin}L\mathrm{sin}\lambda & \mathrm{cos}\alpha \mathrm{cos}L\\ \mathrm{cos}L\mathrm{cos}\lambda & \mathrm{cos}L\mathrm{sin}\lambda & \mathrm{sin}L\end{array}\right]\\ \text{ }=\left[\begin{array}{ccc}{C}_{11}& {C}_{12}& {C}_{13}\\ {C}_{21}& {C}_{22}& {C}_{23}\\ {C}_{31}& {C}_{32}& {C}_{33}\end{array}\right]\end{array}$ (6)

$\left\{\begin{array}{l}{\lambda }_{主}=\mathrm{arctan}\frac{{C}_{32}}{{C}_{31}}\\ L=\mathrm{arcsin}{C}_{33}\\ {\alpha }_{主}=\mathrm{arctan}\frac{{C}_{13}}{{C}_{23}}\end{array}$ (7)

${\omega }_{ie}^{n}={C}_{e}^{n}{\omega }_{ie}^{e}=\left[\begin{array}{ccc}{C}_{11}& {C}_{12}& {C}_{13}\\ {C}_{21}& {C}_{22}& {C}_{23}\\ {C}_{31}& {C}_{32}& {C}_{33}\end{array}\right]\left[\begin{array}{c}0\\ 0\\ \Omega \end{array}\right]=\left[\begin{array}{c}{C}_{13}\Omega \\ {C}_{23}\Omega \\ {C}_{33}\Omega \end{array}\right]=\left[\begin{array}{c}\Omega \mathrm{sin}\alpha \mathrm{cos}L\\ \Omega \mathrm{cos}\alpha \mathrm{cos}L\\ \Omega \mathrm{sin}L\end{array}\right]$ (8)

$\left[\begin{array}{c}{V}_{E}\\ {V}_{N}\end{array}\right]=\left[\begin{array}{cc}\mathrm{cos}\alpha & -\mathrm{sin}\alpha \\ \mathrm{sin}\alpha & \mathrm{cos}\alpha \end{array}\right]\left[\begin{array}{c}{V}_{x}\\ {V}_{y}\end{array}\right]=\left[\begin{array}{c}{V}_{x}\mathrm{cos}\alpha -{V}_{y}\mathrm{sin}\alpha \\ {V}_{x}\mathrm{sin}\alpha +{V}_{y}\mathrm{cos}\alpha \end{array}\right]$ (9)

$\begin{array}{c}{\omega }_{en}^{n}=\left[\begin{array}{ccc}\mathrm{cos}\alpha & \mathrm{sin}\alpha & 0\\ -\mathrm{sin}\alpha & \mathrm{cos}\alpha & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}-\frac{{V}_{N}}{{R}_{M}}\\ \frac{{V}_{E}}{{R}_{N}}\\ 0\end{array}\right]=\left[\begin{array}{ccc}\mathrm{cos}\alpha & \mathrm{sin}\alpha & 0\\ -\mathrm{sin}\alpha & \mathrm{cos}\alpha & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}-\frac{{V}_{x}\mathrm{sin}\alpha +{V}_{y}\mathrm{cos}\alpha }{{R}_{M}}\\ \frac{{V}_{x}\mathrm{cos}\alpha -{V}_{y}\mathrm{sin}\alpha }{{R}_{N}}\\ 0\end{array}\right]\\ =\left[\begin{array}{c}-{V}_{x}\mathrm{sin}\alpha \mathrm{cos}\alpha \left(\frac{1}{{R}_{M}}-\frac{1}{{R}_{N}}\right)-{V}_{y}\left(\frac{{\mathrm{cos}}^{2}\alpha }{{R}_{M}}-\frac{{\mathrm{sin}}^{2}\alpha }{{R}_{N}}\right)\\ {V}_{x}\left(\frac{{\mathrm{cos}}^{2}\alpha }{{R}_{M}}+\frac{{\mathrm{sin}}^{2}\alpha }{{R}_{N}}\right)+{V}_{y}\mathrm{sin}\alpha \mathrm{cos}\alpha \left(\frac{1}{{R}_{M}}-\frac{1}{{R}_{N}}\right)\\ 0\end{array}\right]\end{array}$ (10)

${\omega }_{in}^{n}={\omega }_{ie}^{n}+{\omega }_{en}^{n}\text{ }=\left[\begin{array}{c}\Omega \mathrm{sin}\alpha \mathrm{cos}L-{V}_{x}\mathrm{sin}\alpha \mathrm{cos}\alpha \left(\frac{1}{{R}_{M}}-\frac{1}{{R}_{N}}\right)-{V}_{y}\left(\frac{{\mathrm{cos}}^{2}\alpha }{{R}_{M}}-\frac{{\mathrm{sin}}^{2}\alpha }{{R}_{N}}\right)\\ \Omega \mathrm{cos}\alpha \mathrm{cos}L+{V}_{x}\left(\frac{{\mathrm{cos}}^{2}\alpha }{{R}_{M}}+\frac{{\mathrm{sin}}^{2}\alpha }{{R}_{N}}\right)+{V}_{y}\mathrm{sin}\alpha \mathrm{cos}\alpha \left(\frac{1}{{R}_{M}}-\frac{1}{{R}_{N}}\right)\\ \Omega \mathrm{sin}L\end{array}\right]=\left[\begin{array}{c}{\omega }_{inx}^{n}\\ {\omega }_{iny}^{n}\\ {\omega }_{inz}^{n}\end{array}\right]$ (11)

$\left[\begin{array}{ccc}{\stackrel{˙}{C}}_{11}& {\stackrel{˙}{C}}_{12}& {\stackrel{˙}{C}}_{13}\\ {\stackrel{˙}{C}}_{21}& {\stackrel{˙}{C}}_{22}& {\stackrel{˙}{C}}_{23}\\ {\stackrel{˙}{C}}_{31}& {\stackrel{˙}{C}}_{32}& {\stackrel{˙}{C}}_{33}\end{array}\right]=\left[\begin{array}{ccc}0& 0& -{\omega }_{eny}^{n}\\ 0& 0& {\omega }_{enx}^{n}\\ {\omega }_{eny}^{n}& -{\omega }_{enx}^{n}& 0\end{array}\right]\left[\begin{array}{ccc}{C}_{11}& {C}_{12}& {C}_{13}\\ {C}_{21}& {C}_{22}& {C}_{23}\\ {C}_{31}& {C}_{32}& {C}_{33}\end{array}\right]$ (12)

$\stackrel{˙}{V}={f}^{n}-\left(2{\omega }_{ie}^{n}+{\omega }_{en}^{n}\right)×V+{g}^{n}$ (13)

${\psi }_{真}=\psi +\alpha$ (17)

$\stackrel{˙}{\alpha }=\Omega \mathrm{sin}L-\left(\Omega \mathrm{sin}L+\frac{{V}_{E}}{{R}_{N}}\mathrm{tan}L\right)=-\frac{{V}_{E}}{{R}_{N}}\mathrm{tan}L$ (18)

Figure 4. Mechanical layout block diagram of walking azimuth inertial navigation system

4. 基于惯性平台调制的高纬度工作方案

Figure 5. Semi-analytical high latitude mechanical arrangement of inertial navigation with rotation modulation of inertial platform

$k\left(t\right)={k}_{0}-{\int }_{0}^{t}\left(\Omega \mathrm{sin}\phi +{\omega }_{TZ}\mathrm{sin}\phi +\frac{{V}_{x}}{R}\mathrm{tan}\phi \right)\text{d}t$ (19)

$\left[\begin{array}{c}{\omega }_{kpx}\\ {\omega }_{kpy}\end{array}\right]=\left[\begin{array}{cc}\mathrm{cos}k& \mathrm{sin}k\\ -\mathrm{sin}k& \mathrm{cos}k\end{array}\right]\left[\begin{array}{c}{\omega }_{cx}\\ {\omega }_{cy}\end{array}\right]$ (20)

$\delta {\stackrel{˙}{C}}_{e}^{p}={\Omega }_{pe}\cdot \delta {C}_{e}^{p}$ (21)

${\Omega }_{pe}=-{\Omega }_{ep}=\left[\begin{array}{ccc}0& 0& -{\omega }_{epy}\\ 0& 0& {\omega }_{epx}\\ {\omega }_{epy}& -{\omega }_{epx}& 0\end{array}\right]$ (22)

${\omega }_{ep}$ 有如下关系：

$\left[\begin{array}{c}{\omega }_{epx}\\ {\omega }_{epy}\end{array}\right]=\left[\begin{array}{cc}-\frac{1}{{R}_{px}}& \frac{1}{{\xi }_{k}}\\ -\frac{1}{{\xi }_{k}}& \frac{1}{{R}_{py}}\end{array}\right]\left[\begin{array}{c}{V}_{px}\\ {V}_{py}\end{array}\right]$ (23)

$\left\{\begin{array}{l}\frac{1}{{R}_{px}}=\frac{{\mathrm{cos}}^{2}k}{{R}_{M}}+\frac{{\mathrm{sin}}^{2}k}{{R}_{N}}\\ \frac{1}{{R}_{py}}=\frac{{\mathrm{sin}}^{2}k}{{R}_{M}}+\frac{{\mathrm{cos}}^{2}k}{{R}_{N}}\\ \frac{1}{{\xi }_{k}}=\left(\frac{1}{{R}_{N}}-\frac{1}{{R}_{M}}\right)\mathrm{sin}k\mathrm{cos}k\end{array}$ (24)

$\left\{\begin{array}{l}{\omega }_{epx\left(t\right)}={C}_{13}\Omega +{\omega }_{epx\left(t-1\right)}\\ {\omega }_{epy\left(t\right)}={C}_{23}\Omega +{\omega }_{eyx\left(t-1\right)}\end{array}$ (25)

$\left\{\begin{array}{l}{\stackrel{˙}{V}}_{px}={A}_{px}+\Omega {C}_{33}{V}_{py}\\ {\stackrel{˙}{V}}_{py}={A}_{py}-\Omega {C}_{33}{V}_{px}\end{array}$ (26)

5. 格网导航

${C}_{g}^{G}=\left[\begin{array}{ccc}\mathrm{cos}\sigma & -\mathrm{sin}\sigma & 0\\ \mathrm{sin}\sigma & \mathrm{cos}\sigma & 0\\ 0& 0& 1\end{array}\right]$ (27)

$\sigma$ ：格网角。

${C}_{b}^{G}={C}_{g}^{G}{C}_{b}^{n}=\left[\begin{array}{ccc}×& \mathrm{sin}\left(\psi -\sigma \right)\mathrm{cos}\theta & ×\\ ×& \mathrm{cos}\left(\psi -\sigma \right)\mathrm{cos}\theta & ×\\ -\mathrm{sin}\gamma \mathrm{cos}\theta & \mathrm{sin}\theta & \mathrm{cos}\gamma \mathrm{cos}\theta \end{array}\right]$ (28)

$\psi$ ：地理航向；

$\theta$ ：纵摇角；

$\gamma$ ：横摇角；

${\psi }_{格}=\psi -\sigma$ (29)

6. 极区导航输出

Figure 6. Polar navigation mechanical layout

7. 结论

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