On a Methodological Error of the Inertial Vertical

O. A. Artyukhovsky, A. I. Studenikin

(Presented by the scientific-technical seminar of the Department of Gyroscopic Instruments and Devices)

One of the principal errors of precision inertial navigation systems is the error that arises as a result of neglecting the nonsphericity of the Earth in the formulas of the onboard algorithm. In this connection, it is of interest to investigate the equations of inertial navigation obtained with allowance for the nonsphericity of the Earth, and to estimate the methodological errors of the inertial vertical.

We shall assume that the Earth is an ellipsoid of revolution with a small flattening in the direction of the poles. In the appropriate places, by the parameters of the terrestrial spheroid we shall mean the parameters of Krasovsky’s ellipsoid. Let us introduce the following orthogonal coordinate system—the system \(\xi_0\eta_0\zeta_0\), connected with the Earth. We place the origin \(O\) of this coordinate system at the center of the Earth. We direct the axis \(O\eta_0\) along the vector of the angular velocity of the Earth’s rotation \(\omega\). For the axis \(O\xi_0\) we take the line of intersection of the equator with the plane of the Greenwich meridian. The axis \(O\zeta_0\) is perpendicular to the first two, so that the system of axes is right-handed.

At the launch point we place the coordinate system \(\xi^1\eta^1\zeta^1\), whose axis is directed upward along the radius vector \(\bar R\), and the axis \(O\eta_1\)—in the plane of the orthodromic meridian in the direction toward the orthodromic pole.

The geocentric coordinate system \(\xi^1\eta^1\zeta^1\) thus determines the orthodromy as a central section of the terrestrial spheroid passing either through two given points or through one point specified by geocentric coordinates and an azimuth. The selected orthodromy will be characterized by the following table of direction cosines between the axes \(\xi_0\eta_0\zeta_0\) and \(\xi^1\eta^1\zeta^1\):

\[ \left. \begin{aligned} n_{11} &= \cos \lambda_0 \cdot \sin N_0 + \sin \lambda_0 \cdot \sin \varphi_0 \cdot \sin N_0,\\ n_{12} &= \cos \varphi_0 \cdot \sin N_0,\\ n_{13} &= \sin \varphi_0 \cdot \cos \lambda_0 \cdot \sin N_0 - \sin \lambda_0 \cdot \cos N_0,\\ n_{21} &= \sin \lambda_0 \cdot \sin \varphi_0 \cdot \cos N_0 - \cos \lambda_0 \cdot \sin N_0,\\ n_{22} &= \cos \varphi_0 \cdot \sin N_0,\\ n_{23} &= \cos \lambda_0 \cdot \sin \varphi_0 \cdot \cos N_0 + \sin \lambda_0 \cdot \sin N_0,\\ n_{31} &= \sin \lambda_0 \cdot \cos \varphi_0,\quad n_{32} = -\sin \varphi_0,\quad n_{33} = \cos \lambda_0 \cdot \cos \varphi_0. \end{aligned} \right\} \tag{1} \]

\(\lambda_0\) is the longitude of the launch point, \(\varphi_0\) is the geocentric latitude of the launch point, \(N_0\) is the orthodromic angle at the launch point. Let us associate with the current coordinates of the aircraft the geocentric reference frame \(\xi_1\eta_1\zeta_1\). The relative position of the systems \(\xi\eta\zeta\) and \(\xi^1\eta^1\zeta^1\) will be determined by a table of direction cosines compiled from geocentric orthodromic coordinates.

Multiplying the matrices written in Tables 1 and 2, we obtain the direction cosines between the axes: \(\xi_0\eta_0\zeta_0\) and \(\xi_1\eta_1\zeta_1\).

Table 1

Table 2

Table 3

Table 4

\[ \begin{aligned} m_{11}&=n_{11}\cos \Lambda_1-n_{31}\sin \Lambda_1,\\ m_{12}&=n_{12}\cos \Lambda_1-n_{32}\sin \Lambda_1,\\ m_{13}&=n_{13}\cos \Lambda_1-n_{33}\sin \Lambda_1,\\ m_{21}&=n_{11}\sin \Lambda_1\sin \Phi_1+n_{21}\cos \Phi_1+n_{31}\sin \Phi_1\cos \Lambda_1,\\ m_{22}&=n_{12}\sin \Lambda_1\sin \Phi_1+n_{22}\cos \Phi_1+n_{32}\sin \Phi_1\cos \Lambda_1,\\ m_{23}&=n_{13}\sin \Lambda_1\sin \Phi_1+n_{23}\cos \Phi_1+n_{33}\sin \Phi_1\cos \Lambda_1,\\ m_{31}&=n_{11}\sin \Lambda_1\cos \Phi_1-n_{21}\sin \Phi_1+n_{31}\cos \Lambda_1\cos \Phi_1,\\ m_{32}&=n_{12}\sin \Lambda_1\cos \Phi_1-n_{22}\sin \Phi_1+n_{32}\cos \Lambda_1\cos \Phi_1,\\ m_{33}&=n_{13}\sin \Lambda_1\cos \Phi_1-n_{23}\sin \Phi_1+n_{33}\cos \Lambda_1\cos \Phi_1. \end{aligned} \tag{2} \]

Let us assume that on board the aircraft the geodetic vertical is materialized, with which one of the axes of the coordinate system \(\xi\eta\zeta\) coincides. Denoting the angles of deviation of the geodetic vertical from the direction of the radius vector in the plane of the orthodromic meridian by \(\nu_1\), and in the direction perpendicular to the plane of the meridian by \(\nu_2\), we obtain the relation between the coordinate systems \(\xi_1\eta_1\zeta_1\) and \(\xi\eta\zeta\) in the form of Table 4, written taking into account the smallness of the angles \(\nu_1\) and \(\nu_2\). The angles \(\nu_1\) and \(\nu_2\) are found from the simple relations

\[ \nu_1=\mu\cdot\cos N,\quad \nu_2=-\mu\cdot\sin N, \tag{3} \]

which relate the values of these angles to the geometric shape of the Earth.

The function \(\mu(B)\) is given by means of a trigonometric series [1], and for us it is sufficient to retain the first two terms:

\[ \mu = c \cdot \sin 2B - \frac{1}{2}c^2 \cdot \sin 4B, \tag{4} \]

where

\[ c=(a^2-b^2)(a^2+b^2)^{-1}, \]

\(a,\ b\) are the semiaxes of the Krasovsky ellipsoid, and \(B\) is the geodetic latitude.

In this case it is expedient to take the following values as the coordinates of the aircraft:

\[ \begin{aligned} \Phi &= \Phi_1 + \nu_1,\\ \Lambda &= \Lambda_1 + \frac{\nu_2}{\cos \Phi}. \end{aligned} \tag{5} \]

The choice of orthodromic coordinates in the form (5) is justified by the fact that the conversion of coordinates from the geographic system to the orthodromic one and vice versa can be carried out by using matrix transformations of the form (1) and (2). This can be shown. Let the aircraft be located at a point with geodetic coordinates \(B\) and \(L\), and orthodromic coordinates \(\Phi\) and \(\Lambda\). They correspond to geocentric coordinates \(\varphi = B-\mu\), \(\lambda=L\); \(\Phi_1=\Phi-\nu_1\); \(\Lambda_1=\Lambda-\nu_2(\cos\Phi)^{-1}\). Substituting the latter into the right-hand sides of formulas (1) and (2), we obtain two systems of identities. For example, for \(m_{32}\) we shall have

\[ -\left(n_{12}\cdot \sin\Lambda_1\cos\Phi_1 - n_{22}\sin\Phi_1 + n_{32}\cdot \cos\Lambda_1\cdot \cos\Phi_1\right)=\sin\varphi. \]

Substituting in the last expression \(\Phi\) and \(\Lambda\) instead of \(\Phi_1\) and \(\Lambda_1\), and taking account of (1), (2), (3), we obtain, with accuracy up to terms \(C^2\),

\[ -\left[ n_{12}\cdot \sin\left(\Lambda_1+\frac{\nu_2}{\cos\Phi}\right)\cdot \cos(\Phi_1+\nu_1) - n_{22}\sin(\Phi_1+\nu_1) + \right. \]

\[ \left. + n_{32}\cos\left(\Lambda_1+\frac{\nu_2}{\cos\Phi}\right)\cdot \cos(\Phi_1+\nu_1) \right] = \sin\varphi + \mu\cdot \cos\varphi = \sin B. \]

The basic equation of inertial navigation, written with respect to the coordinate system \(\xi\eta\zeta\), has the form [2, 3]

\[ \left. \begin{aligned} j_\xi &= V'_\xi + V_\zeta \Omega_\eta - V_\eta \Omega_\zeta - f_\xi,\\ j_\eta &= V'_\eta + V_\xi \Omega_\zeta - V_\zeta \Omega_\xi - f_\eta,\\ j_\zeta &= V'_\zeta + V_\eta \Omega_\xi - V_\xi \Omega_\eta - f_\zeta, \end{aligned} \right\} \tag{6} \]

where \(\Omega_\xi,\ \Omega_\eta,\ \Omega_\zeta,\ V_\xi,\ V_\eta,\ V_\zeta\) are the components of the absolute angular and linear velocities of the radius vector. Introducing into consideration the relative linear velocities of the object and using Table 4, it can be shown that the projections of the absolute angular and linear velocities on the axes \(\xi\eta\zeta\) will be equal to:

\[ \Omega_\xi = -\frac{W_{\eta 1}}{R} + \omega\cdot \cos\varphi\sin N -\left(-\frac{W_{\xi 1}}{R}\operatorname{tg}\Phi_1-\omega\sin\varphi\right)\nu_2; \]

\[ \Omega_\eta = \frac{W_{\xi 1}}{R} + \omega\cdot \cos\varphi\cdot \cos N +\left(-\frac{W_{\xi 1}}{R}\operatorname{tg}\zeta_1-\omega\cdot \sin\varphi\right)\nu_1; \]

\[ \Omega_\zeta = -\frac{W_{\xi 1}}{R}\operatorname{tg}\Phi_1 -\omega\sin\varphi +\left(\frac{W_{\eta 1}}{R}+\omega\cdot \cos\varphi\cdot \sin N\right)\nu_2 - \]

\[ -\left(\frac{W_{\xi 1}}{R}+\omega\cdot \cos\varphi\cdot \cos N\right)\nu_1; \tag{7} \]

\[ \begin{aligned} V_{\xi} &= W_{\xi 1} - V_{\zeta}\nu_2 + R\omega \cos \varphi \cdot \cos N;\\ V_{\eta} &= W_{\eta 1} + V_{\zeta}\nu_1 - R\omega \cdot \cos \varphi \cdot \sin N;\\ V_{\zeta} &= W_{\nu 1} + W_{\xi 1}\nu_2 - W_{\eta 1}\nu_1 + R\omega \cos \varphi \cdot \cos N \cdot \nu_2 + R\omega \cdot \cos \varphi \cdot \sin N \cdot \nu_1, \end{aligned} \]

and for system (6) we shall have

\[ \begin{aligned} j_{\xi}={}& W'_{\xi 1} - (V_{\zeta 1}\nu_2)' + \frac{W_{\xi 1}V_{\zeta 1}}{R} + \frac{W_{\eta 1}W_{\xi 1}}{R}\operatorname{tg}\Phi_1 +{}\\ &+ 2V_{\zeta 1}\omega \cdot \cos \varphi \cdot \cos N + 2W_{\eta 1}\omega \cdot \sin \varphi - R\omega^2 \cdot \sin \varphi \cdot \cos \varphi \cdot \sin N +{}\\ &+\left( \frac{W_{\xi 1}^2}{R} +\frac{W_{\eta 1}^2}{R} +2W_{\xi 1}\omega \cdot \cos \varphi \cdot \cos N -\right.\\ &\left.\qquad -2W_{\eta 1}\omega \cdot \cos \varphi \cdot \sin N + R\omega^2\cos^2\varphi \right)\nu_2 - f_{\xi}. \end{aligned} \tag{8} \]

\[ \begin{aligned} j_{\eta}={}& W'_{\eta 1} + (V_{\zeta}\nu_1)' - \frac{W_{\xi 1}^{2}}{R}\operatorname{tg}\Phi_1 + \frac{V_{\zeta 1}W_{\eta 1}}{R} -{}\\ &- 2W_{\xi 1}\omega \cdot \sin \varphi - 2V_{\zeta 1}\omega \cdot \cos \varphi \cdot \sin N - R\omega^2 \sin \varphi \cdot \cos \varphi \cdot \cos N -{}\\ &-\left( \frac{W_{\xi 1}^{2}}{R} +\frac{W_{\eta 1}^{2}}{R} +2W_{\xi 1}\omega \cdot \cos \varphi \cdot \cos N -2W_{\eta 1}\omega \cdot \cos \varphi \cdot \sin N +\right.\\ &\left.\qquad + R\omega^2\cos^2\varphi \right)\nu_1 - f_{\eta}. \end{aligned} \]

\[ \begin{aligned} j_{\zeta}={}& V'_{\zeta} + (W_{\xi 1}\nu_2)' - (W_{\eta 1}\nu_1)' - \frac{W_{\eta 1}^{2}}{R} - \frac{W_{\xi 1}^{2}}{R} +{}\\ &+ 2W_{\eta 1}\omega \cdot \cos \varphi \cdot \sin N - 2W_{\xi 1}\omega \cdot \cos \varphi \cdot \cos N - R\omega^2\cos^2\varphi +{}\\ &+\left( \frac{W_{\xi 1}W_{\zeta 1}}{R} +\frac{W_{\xi 1}W_{\eta 1}}{R}\operatorname{tg}\Phi_2 +2W_{\eta 1}\omega \cdot \sin \varphi +2V_{\zeta 1}\omega \cos \varphi \cos N -\right.\\ &\left.\qquad - R\omega^2 \cos \varphi \cdot \sin \varphi \cdot \sin \varphi \cdot \sin N \right)\nu_2 -\left( \frac{W_{\eta 1}W_{\zeta 1}}{R} -\frac{W_{\xi 1}^{2}}{R}\operatorname{tg}\Phi_1 -\right.\\ &\left.\qquad -2W_{\xi 1}\omega \sin \varphi -2V_{\zeta 1}\omega \cos \varphi \cdot \sin N - R\omega^2 \cdot \cos \varphi \cdot \sin \varphi \cdot \cos N \right)\nu_1 - f_{\zeta}. \end{aligned} \]

Let us pass in (7) and (8) to the geodetic coordinates \(B\) and \(\Phi\), and also to the components of the relative velocity in the system \(\xi\eta\zeta\), for which purpose in (8) we substitute the following correspondences:

\[ \begin{gathered} \Phi_1=\Phi-\nu_1;\qquad \varphi=B-\mu;\qquad W_{\xi 1}=W_{\xi}+V_{\zeta}\nu_2;\\ W_{\eta 1}=W_{\eta}-V_{\zeta}\nu_1;\qquad V_{\zeta 1}=V_{\zeta}-W_{\xi}\nu_2+W_{\eta}\nu_1, \end{gathered} \]

of which the last three are justified by the natural assumption \(V_{\zeta}\nu_{1,2}=V_{\zeta 1}\nu_{1,2}\), \(W_{\eta}\nu_{1,2}=W_{\eta 1}\nu_{1,2}\). It is also taken into account that, for a gyroplatform oriented along the gravity-force vector, the relations

\[ - R\omega^2 \sin \varphi \cdot \cos \varphi \cdot \sin N + R\omega^2\cos^2\varphi\,\nu_2 - f_{\xi}=0, \]

\[ - R\omega^2 \sin \varphi \cdot \cos \varphi \cdot \cos N - R\omega^2\cos^2\varphi \cdot \nu_1 - f_{\eta}=0. \]

are valid. After simple transformations we obtain, for the readings of the horizontal accelerometers and the platform-correction signals,

\[ j_{\xi}=W'_{\xi}+\frac{W_{\xi}W_{\eta}}{R}\operatorname{tg}\Phi +2W_{\eta}\omega \cdot \sin B +2V_{\zeta}\omega \cdot \cos B \cdot \cos N, \]

\[ \begin{gathered} j_{\eta}=W'_{\eta}-\frac{W_{\xi}^{2}}{R}\tan\Phi-2W_{\xi}\omega\sin B-2V_{\xi}\omega\cos B\cdot\sin N,\\ \Omega_{\xi}=-\frac{W_{\eta}}{R}+\omega\cdot\cos B\cdot\sin N+\frac{W_{\xi}}{R}\tan\Phi\,\nu_{2},\\ \Omega_{\eta}=\frac{W_{\xi}}{R}+\omega\cdot\cos B\cdot\cos N-\frac{W_{\xi}}{R}\tan\Phi\,\nu_{1},\\ \Omega_{\zeta}=-\frac{W_{\xi}}{R}\tan\Phi-\omega\sin B-\frac{W_{\eta}}{R}\nu_{2}, \end{gathered} \tag{9} \]

where \(R=a(1-c\cdot\sin^{2}B)\).

If the third term in the expressions for \(\Omega_{\xi}\), \(\Omega_{\eta}\), \(\Omega_{\zeta}\) is neglected, which is valid in flight along an orthodrome (\(\Phi=\nu_{1};\ W_{\eta}=0\)), then it may be assumed that relations (9) are written for a sphere of geodetic latitudes \(B\). The spherical structure of formulas (9) and the possibility noted above of using spherical formulas for transforming the coordinates \(\Phi\) and \(\Lambda\) into \(B\) and \(L\) give grounds for considering the operation of an inertial vertical on a sphere of variable radius, obtained as the spherical mapping of an ellipsoid onto a sphere. Therefore, for analyzing the methodological errors of simpler algorithms (absence of rotary accelerations containing \(V_{\zeta}\), constant value of \(R\), etc.), one may use the general differential equations of error of an inertial system (2) for a spherical Earth. In this case, in the right-hand sides of the equations for the error in constructing the vertical it is necessary to substitute the differences between the methodologically exact formulas (9) and those adopted in the computer, interpreting these differences as errors equivalent to accelerometer errors and drifts of stabilizing gyroscopes. When the complete formulas (9) with variable \(R\) are implemented in the computer, the right-hand sides of the error equations are equal to zero. Consequently, in flight the platform will be oriented along the geodetic vertical.

Continuous computation of the coordinates from the equations:

\[ \Lambda_{\mathrm{c}}=\Lambda_{0}+\int_{0}^{t}\frac{1}{R\cos\Phi}\left\{\int_{0}^{t}W'_{\xi}\,dt\right\}dt;\qquad \Phi_{\mathrm{c}}=\Phi_{0}-\int_{0}^{t}\frac{1}{R}\left\{\int_{0}^{t}W'_{\eta}\,dt\right\}dt; \]

\[ W'_{\xi}=-j_{\xi}-\frac{W_{\xi}W_{\eta}}{R}\tan\Phi-2W_{\eta}\omega\sin B-2V_{\zeta}\omega\cos B\cos N; \]

\[ W'_{\eta}=j_{\eta}+\frac{W_{\xi}^{2}}{R}\tan\Phi+2W_{\xi}\omega\sin B+2V_{\zeta}\omega\cos B\cdot\sin N, \]

would seem to have to yield the values \(\Phi\) and \(\Lambda\). However, this does not occur. Indeed, the expressions for the relative accelerations \(W'_{\xi}\) and \(W'_{\eta}\) from (8) for the case of horizontal flight will have the form

\[ W'_{\xi}=W'_{\xi1}+2C^{2}W_{\eta0}\cos B\cdot\sin 2B\cdot\sin N\cdot B' -C^{2}W_{\eta0}\sin 2B\,(\sin 2B\cdot\sin N)'; \]

\[ W'_{\eta}=W'_{\eta1}+2C^{2}W_{\eta0}\cdot\cos B\cdot\sin 2B\cdot\cos N\cdot B' -C^{2}W_{\eta0}\sin 2B\,(\sin 2B\cdot\cos N)'. \]

As before, with accuracy up to terms \(C^{2}\), we have

\[ W'_{\xi}=W'_{\xi1};\qquad W'_{\eta}=W'_{\eta1}. \]

Consequently, the inertial system will compute the geocentric coordinates \(\Lambda_{1}\) and \(\Phi_{1}\). Taking into account that at the starting point the exact coordinates are entered into the computer, the total error will be

\[ \Delta\Phi=c\sin 2B\cdot\cos N-\nu_{1}^{0}, \]

\[ \Delta\Lambda=-\frac{c\cdot\sin 2B\cdot\sin N}{\cos\Phi}. \tag{10} \]

The methodological error is purely geometric. Its maximum value reaches 22 km.

Conclusions

If, for a spheroidal Earth, the plane of the orthodrome coincides with the central section, then the conversion of orthodromic coordinates in the form (5) into geodetic coordinates is performed by the usual formulas.

The methodologically exact equations of the inertial orienter admit a spherical representation on a sphere of variable radius; that is, they can be obtained from the “spherical formulas” by replacing the geocentric coordinates $\Phi_1$ and $\varphi$ with the geodetic coordinates $\Phi$ and $B$. In this case the inertial system is the generator of the geodetic vertical, but has an error in the coordinate. Simple corrections may be provided by the expressions (10), taken with the opposite sign.

Literature

1. G. V. Bagratuni. Course of Spherical Geodesy. Publishing House of Geographical Literature, Moscow, 1962.

2. V. D. Andreev. On the theory of inertial systems for autonomous determination of coordinates of a moving object. PMM, vol. 28, issue 1, 1964.

3. K. L. Mac-Clure. Theory of Inertial Navigation. “Nauka,” Moscow, 1964.