Maintenance-free strap-down ship's gyro compass

Abstract

Method for operation of a strap-down ship's gyro compass for optimal calculation of position and course angle on a ship, with three rotational rate sensors each mutually aligned to each other at a right angle, and two nominally horizontally aligned orthogonal acceleration sensors, without required specification of the geographic latitude and/or of the speed of the ship, characterized by the steps: a. Preparation of a set of dynamic equations based on the angular velocity components detected by the rotational rate sensor, b. Preparation of a measured data equation based on the force components detected by the acceleration sensors, c. Determination of the properties of the lever arm between the strap-down ship's gyro compass and the point of rotation of the ship, and d. Determination of the properties of the earth's rotation and of the ship's angular velocity, wherein their properties are determined on the basis of the set of dynamic equations and the measured data equation, and are used in each case as parameters for calculation of the geographic latitude and/or speed of the ship.

Claims

1. A method for operation of a strap-down ship's gyro compass for optimal calculation of position and course angle on a ship, with three rotational rate sensors each mutually aligned to each other at a right angle, and two nominally horizontally aligned orthogonal acceleration sensors, without required specification of the geographic latitude and/or of the speed of the ship, said method comprising: a. preparing a set of dynamic equations based on the angular velocity components detected by the rotational rate sensors, b. preparing of a measured data equation based on the force components detected by the acceleration sensors, c. determining the properties of a lever arm between the strap-down ship's gyro compass and the point of rotation of the ship, and d. determining the properties of the earth's rotation and of the ship's angular velocity, wherein their properties are determined on the basis of the set of dynamic equations and the measured data equation and are used in each case as parameters for calculation of the geographic latitude and/or speed of the ship.

2. The method according to claim 1, wherein the error corrections of the acceleration sensors are determined by an optimal filter on the basis of an expanded state-space representation.

3. The method according to claim 2, wherein the error corrections of the rotational rate sensors are determined by the optimal filter on the basis of an expanded state-space representation.

4. A method for operation of a strap-down ship's gyro compass for optimal calculation of position and course angle on a ship, with three rotational rate sensors each mutually aligned to each other at a right angle, and two nominally horizontally aligned orthogonal acceleration sensors, without required specification of the geographic latitude and/or of the speed of the ship, said method comprising: a. preparing a set of dynamic equations based on the angular velocity components detected by the rotational rate sensors, b. preparing of a measured data equation based on the force components detected by the acceleration sensors, c. determining the properties of a lever arm between the strap-down ship's gyro compass and the point of rotation of the ship, and d. determining the properties of the earth's rotation and of the ship's angular velocity, wherein their properties are determined on the basis of the set of dynamic equations and the measured data equation and are used in each case as parameters for calculation of the geographic latitude and/or speed of the ship, wherein the set of dynamic equations includes one equation that reads as follows:
.sub.nb=C.sub.nb.Math..sub.nb.sup.b=C.sub.nb.Math.(.sub.ib.sup.b+.sub.ib.sup.b)(.sub.en.sup.n+.sub.ie.sup.n).Math.C.sub.nb.

5. The method according to claim 4, wherein the measured data equation reads as follows:
f.sup.b+f.sup.b={dot over (v)}.sup.bC.sub.nb.sup.T.Math.g.sup.n with $\begin{array}{c}{\underset{\_}{\underset{\_}{C}}}_{\mathrm{nb}}\left(,,\right)={\underset{\_}{\underset{\_}{R}}}_3\left(-\right).Math.{\underset{\_}{\underset{\_}{R}}}_2\left(-\right).Math.{\underset{\_}{\underset{\_}{R}}}_1\left(-\right)\\ =\left(\begin{array}{ccc}c.Math.c& s.Math.s.Math.c-c.Math.s& c.Math.s.Math.c+s.Math.s\\ c.Math.s& s.Math.s.Math.s+c.Math.c& c.Math.s.Math.s-s.Math.c\\ -s& s.Math.c& c.Math.c\end{array}\right)\end{array}$ wherein: C.sub.nb corresponds to the transformation matrix with the roll, pitch and course angle of the compass, .sub.ib.sup.b corresponds to the components of angular velocity measured by the three rotational rate sensors, .sub.ie.sup.n corresponds to the earth's rotational rate components in the North-East-lower reference system as functions of the geographic latitude, .sub.en.sup.n corresponds to the ship's angular velocity components in the North-East lower reference system as functions of the compass velocity vector on the earth's curved surface, .sub.ib.sup.b,f.sup.b corresponds to the error corrections of the rotational rate sensor and acceleration sensors, f.sup.b corresponds to the force components measured by the acceleration sensors arranged nominally horizontal and orthogonal, and g.sup.n corresponds to the earth's gravity as g.sup.n=(0 0 g).sup.T.

6. The method according to claim 4, wherein the error corrections of the acceleration sensors are determined by an optimal filter on the basis of an expanded state-space representation.

7. The method according to claim 6, wherein the error corrections of the rotational rate sensors are determined by the optimal filter on the basis of an expanded state-space representation.

8. A method for operation of a strap-down ship's gyro compass for optimal calculation of position and course angle on a ship, with three rotational rate sensors each mutually aligned to each other at a right angle, and two nominally horizontally aligned orthogonal acceleration sensors, without required specification of the geographic latitude and/or of the speed of the ship, said method comprising: a. preparing a set of dynamic equations based on the angular velocity components detected by the rotational rate sensors, b. preparing of a measured data equation based on the force components detected by the acceleration sensors, c. determining the properties of a lever arm between the strap-down ship's gyro compass and the point of rotation of the ship, and d. determining the properties of the earth's rotation and of the ship's angular velocity, wherein their properties are determined on the basis of the set of dynamic equations and the measured data equation and are used in each case as parameters for calculation of the geographic latitude and/or speed of the ship, wherein the measured data equation reads as follows:
f.sup.b+f.sup.b={dot over (v)}.sup.bC.sub.nb.sup.T.Math.g.sup.n with $\begin{array}{c}{\underset{\_}{\underset{\_}{C}}}_{\mathrm{nb}}\left(,,\right)={\underset{\_}{\underset{\_}{R}}}_3\left(-\right).Math.{\underset{\_}{\underset{\_}{R}}}_2\left(-\right).Math.{\underset{\_}{\underset{\_}{R}}}_1\left(-\right)\\ =\left(\begin{array}{ccc}c.Math.c& s.Math.s.Math.c-c.Math.s& c.Math.s.Math.c+s.Math.s\\ c.Math.s& s.Math.s.Math.s+c.Math.c& c.Math.s.Math.s-s.Math.c\\ -s& s.Math.c& c.Math.c\end{array}\right)\end{array}$ wherein: C.sub.nb corresponds to the transformation matrix with the roll, pitch and course angle of the compass, .sub.ib.sup.b corresponds to the components of angular velocity measured by the three rotational rate sensors, .sub.ie.sup.n corresponds to the earth's rotational rate components in the North-East-lower reference system as functions of the geographic latitude, .sub.en.sup.n corresponds to the ship's angular velocity components in the North-East lower reference system as functions of the compass velocity vector on the earth's curved surface, .sub.ib.sup.b,f.sup.b corresponds to the error corrections of the rotational rate sensor and acceleration sensors, f.sup.b corresponds to the force components measured by the acceleration sensors arranged nominally horizontal and orthogonal, and g.sup.n corresponds to the earth's gravity as g.sup.n=(0 0 g).sup.T.

9. The method according to claim 8, wherein the error corrections of the acceleration sensors are determined by an optimal filter on the basis of an expanded state-space representation.

10. The method according to claim 9, wherein the error corrections of the rotational rate sensors are determined by the optimal filter on the basis of an expanded state-space representation.

Description

BRIEF DESCRIPTION OF THE DRAWING

(1) FIG. 1 shows a schematic overview of the components of a strap-down ship's gyro compass designed according to the invention.

DETAILED DESCRIPTION OF THE PREFERED EMBODIMENTS

(2) FIG. 1 shows a schematic overview of the components of a strap-down ship's gyro compass designed according to the invention, with a compass measuring unit, a compass signal processing unit and the parameters obtained and/or processed by the two units.

(3) The invention will be explained in greater detail below based on particularly preferred design embodiments:

(4) Estimation of the Lever Arm

(5) The non-linear system dynamics are reflected by the following dynamic equation:

(6) ${\stackrel{.}{\underset{\_}{\underset{\_}{C}}}}_{\mathrm{nb}}={\underset{\_}{\underset{\_}{C}}}_{\mathrm{nb}}.Math.{\underset{\_}{\underset{\_}{}}}_{\mathrm{nb}}^b={\underset{\_}{\underset{\_}{C}}}_{\mathrm{nb}}.Math.\left({\underset{\_}{\underset{\_}{}}}_{\mathrm{ib}}^b+{\underset{\_}{\underset{\_}{}}}_{\mathrm{ib}}^b\right)-\left({\underset{\_}{\underset{\_}{}}}_{\mathrm{en}}^n+{\underset{\_}{\underset{\_}{}}}_{\mathrm{ie}}^n\right).Math.{\underset{\_}{\underset{\_}{C}}}_{\mathrm{nb}}$

(7) The non-linear measured data equation is as follows:
f.sup.b+f.sup.b={dot over (v)}.sup.bC.sub.nb.sup.T.Math.g.sup.n
with

(8) $\begin{array}{c}{\underset{\_}{\underset{\_}{C}}}_{\mathrm{nb}}\left(,,\right)={\underset{\_}{\underset{\_}{R}}}_3\left(-\right).Math.{\underset{\_}{\underset{\_}{R}}}_2\left(-\right).Math.{\underset{\_}{\underset{\_}{R}}}_1\left(-\right)\\ =\left(\begin{array}{ccc}c.Math.c& s.Math.s.Math.c-c.Math.s& c.Math.s.Math.c+s.Math.s\\ c.Math.s& s.Math.s.Math.s+c.Math.c& c.Math.s.Math.s-s.Math.c\\ -s& s.Math.c& c.Math.c\end{array}\right)\end{array}$
wherein: C.sub.nb corresponds to the transformation matrix with the roll, pitch and course angle of the compass, .sub.ib.sup.b corresponds to the components of angular velocity measured by the three rotational rate sensors, .sub.ie.sup.n corresponds to the earth's rotational rate components in the North-East-lower reference system as functions of the geographic latitude, .sub.en.sup.n corresponds to the ship's angular velocity components in the North-East lower reference system as functions of the compass velocity vector on the earth's curved surface, .sub.ib.sup.b,f.sup.b corresponds to the error corrections of the rotational rate sensor and acceleration sensors, f.sup.b corresponds to the force components measured by the acceleration sensors arranged nominally horizontal and orthogonal, and g.sup.n corresponds to the earth's gravity as g.sup.n=(0 0 g).sup.T.

(9) It should be noted that .sub.ib.sup.b, .sub.ie.sup.n, .sub.en.sup.n and .sub.ib.sup.b are skew-symmetrical matrices which are each formed from the vectors .sub.ib.sup.b, .sub.ie.sup.n, .sub.en.sup.n and .sub.ib.sup.b.

(10) Also applying are .sub.ie.sup.n(.sub.N .sub.E .sub.D).sup.T=.Math.(cos 0sin ).sup.T, wherein represents the angular velocity of the earth, y represents the geographic latitude and

(11) ${}_{\mathrm{en}}^n{\left(\begin{array}{ccc}{}_N& {}_E& {}_D\end{array}\right)}^T=\frac{1}{R}.Math.{\left(\begin{array}{ccc}{v}_E& -v_N& -v_E.Math.\tan \end{array}\right)}^T,$
wherein R, .sub.N and .sub.E represent the radius of the earth and the North and East velocity components.

(12) Now in order to increase the availability, accuracy and operating reliability of the strap-down ship's gyro compass, in particular in rough seas, an additional estimation of the lever arm components is performed, wherein the specific force measurements f.sub.X.sup.b and f.sub.Y.sup.b are corrected by: deviations in the acceleration sensors b.sub.X.sup.b and b.sub.Y.sup.b estimated by means of optimal filter; lever arm components l.sub.X.sup.v, l.sub.Y.sup.v and l.sub.Z.sup.v, estimated by means of optimal filter, that is, their resulting centrifugal and rotational accelerations ship longitudinal acceleration effects .sub.X.sup.v, estimated by means of velocity filter, and ship longitudinal velocity {circumflex over (v)}.sub.X.sup.v, and/or Coriolis acceleration effects resulting therefrom, as estimated by means of velocity filter.

(13) The difference between the corrected specific force measurements and the scaled gravity vector, that is, ((C.sub.n/b).sub.3,1 (C.sub.n/b).sub.3,2).sup.T, is called the residual vector and is formed as a function of the corrections of the deviations in the acceleration sensor and the lever arm components estimated by means of optimal filter, and also of the roll and pitch angle error corrections and , which are expressed as corrections of the pitch angle errors .sub.N/ and .sub.E/{tilde over (E)}, in conformance with the definitions E.sup.nC.sub.nb.Math.C.sub.{circumflex over (n)}b.sup.T and

(14) ${\underset{\_}{\underset{\_}{E}}}^n\left(\begin{array}{ccc}0& -{\mathrm{.Math.}}_D& {\mathrm{.Math.}}_{E/\tilde{E}}\\ {\mathrm{.Math.}}_D& 0& -{\mathrm{.Math.}}_{N/\tilde{N}}\\ -{\mathrm{.Math.}}_{E/\tilde{E}}& {\mathrm{.Math.}}_{N/\tilde{N}}& 0\end{array}\right)$
N, E and D denote the North, East and vertically downward direction, wherein and {tilde over (E)} denote the unknown, initial orientation of the strap-down ship's gyro compass during the rough alignment phase.
(C.sub.vb).sub.i,j denotes the element in line i and column j of the transformation matrix from the compass coordinate system b into the ship coordinate system v, with axes of the v-coordinate system pointing: toward the bow, along the longitudinal axis of the ship to starboard along the transverse axis of the ship, and toward the hull along the normal axis of the ship.

(15) The constancy matrix C.sub.vb is determined during compass installation on the ship by referencing the azimuth of the pier. Note that C.sub.nv=C.sub.nb.Math.C.sub.vb.sup.T with the identity C.sub.nv(.sub.v,.sub.v,.sub.v)=R.sub.3(.sub.v).Math.R.sub.2(.sub.v).Math.R.sub.1(.sub.v), contains the ship roll, pitch and course angles. ({circumflex over ()}.sub.ib.sup.b).sub.X, ({circumflex over ()}.sub.ib.sup.b).sub.Y, ({circumflex over ()}.sub.ib.sup.b).sub.Z and {circumflex over (f)}.sub.X.sup.b, {circumflex over (f)}.sub.Y.sup.b are determined from the usually measured increases in angle and velocity .sub.X.sup.b, .sub.Y.sup.b, .sub.Z.sup.b and v.sub.X.sup.b, v.sub.Y.sup.b by means of suitable deep pass filtering. The initial values of the lever arm components l.sub.X.sup.v, l.sub.Y.sup.v and l.sub.Z.sup.v, are set to 0.

(16) $\frac{1}{g}.Math.\left(\begin{array}{c}{\hat{f}}_X^b\\ {\hat{f}}_Y^b\end{array}\right)+\frac{1}{g}.Math.\left(\begin{array}{c}{b}_X^b\\ b_Y^b\end{array}\right)-\frac{1}{g}\left(\begin{array}{c}{\left(C_{\mathrm{vb}}\right)}_{1,1}\\ {\left(C_{\mathrm{vb}}\right)}_{1,2}\end{array}\right).Math.{\hat{a}}_X^v+\frac{1}{g}.Math.\left(\begin{array}{c}{\left(C_{\mathrm{vb}}\right)}_{1,2}.Math.{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Z-{\left(C_{\mathrm{vb}}\right)}_{1,3}.Math.{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Y\\ {\left(C_{\mathrm{vb}}\right)}_{1,3}.Math.{\left({\hat{}}_{\mathrm{ib}}^b\right)}_X-{\left(C_{\mathrm{vb}}\right)}_{1,1}.Math.{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Z\end{array}\right).Math.v_X^v+\frac{1}{g}.Math.\left(\begin{array}{c}{\left(\begin{array}{c}-{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Y^2-{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Z^2\\ {\left({\hat{}}_{\mathrm{ib}}^b\right)}_X.Math.{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Y-{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Z\\ {\left({\hat{}}_{\mathrm{ib}}^b\right)}_X.Math.{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Z+{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Y\end{array}\right)}^T\\ {\left(\begin{array}{c}{\left({\hat{}}_{\mathrm{ib}}^b\right)}_X.Math.{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Y-{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Z\\ -{\left({\hat{}}_{\mathrm{ib}}^b\right)}_X^2-{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Z^2\\ {\left({\hat{}}_{\mathrm{ib}}^b\right)}_Y.Math.{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Z-{\left({\hat{}}_{\mathrm{ib}}^b\right)}_X\end{array}\right)}^T\end{array}\right).Math.{\underset{\_}{\underset{\_}{C}}}_{\mathrm{vb}}^T\left(\begin{array}{c}{l}_X^v\\ l_Y^v\\ l_Z^v\end{array}\right)+\left(\begin{array}{c}{\left(C_{n/\tilde{n}b}\right)}_{3,1}\\ {\left(C_{n/\tilde{n}b}\right)}_{3,2}\end{array}\right){\left(\begin{array}{cc}{r}_X^b& r_Y^b\end{array}\right)}^T=\left(\left(\begin{array}{cccc}-{\left(C_{n/\tilde{n}b}\right)}_{2,1}& -{\left(C_{n/\tilde{n}b}\right)}_{1,1}& -\frac{1}{g}& 0\\ -{\left(C_{n/\tilde{n}b}\right)}_{2,2}& -{\left(C_{n/\tilde{n}b}\right)}_{1,2}& 0& -\frac{1}{g}\end{array}\right)-\frac{1}{g}\left(\begin{array}{c}{\left(\begin{array}{c}-{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Y^2-{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Z^2\\ {\left({\hat{}}_{\mathrm{ib}}^b\right)}_X.Math.{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Y-{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Z\\ {\left({\hat{}}_{\mathrm{ib}}^b\right)}_X.Math.{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Z-{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Y\end{array}\right)}^T\\ {\left(\begin{array}{c}{\left({\hat{}}_{\mathrm{ib}}^b\right)}_X.Math.{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Y-{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Z\\ -{\left({\hat{}}_{\mathrm{ib}}^b\right)}_X^2-{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Z^2\\ {\left({\hat{}}_{\mathrm{ib}}^b\right)}_Y.Math.{\left({\hat{}}_{\mathrm{ib}}^b\right)}_Z-{\left({\hat{}}_{\mathrm{ib}}^b\right)}_X\end{array}\right)}^T\end{array}\right).Math.{\underset{\_}{\underset{\_}{C}}}_{\mathrm{vb}}^T\right).Math.\left(\begin{array}{c}{\mathrm{.Math.}}_{N/\tilde{N}}\\ {\mathrm{.Math.}}_{E/\tilde{E}}\\ b_X^b\\ b_Y^b\\ l_X^v\\ l_Y^v\\ l_Z^v\end{array}\right)$
Estimation of the Earth's and Ship's Angular Velocity Components

(17) If the geographic latitude and ship velocity are not available, then the state vector is reproduced as follows:
x=(.sub..sub.{tilde over (E)}.sub.Dd.sub.d.sub.{tilde over (E)}d.sub.Db.sub.X.sup.bb.sub.Y.sup.bl.sub.X.sup.vl.sub.Y.sup.vl.sub.Z.sup.v).sup.T

(18) The dynamic matrix appears as follows:

(19) $\underset{\_}{\underset{\_}{F}}=\left(\begin{array}{ccc}\left(\begin{array}{cccccc}0& {}_D& -{}_{\tilde{E}}& 1& 0& 0\\ -{}_D& 0& {}_{\tilde{N}}& 0& 1& 0\\ {}_{\tilde{E}}& -{}_{\tilde{N}}& 0& 0& 0& 1\\ 0& 0& 0& -{}_{d^{\tilde{n}}}& 0& 0\\ 0& 0& 0& 0& -{}_{d^{\tilde{n}}}& 0\\ 0& 0& 0& 0& 0& -{}_{d^{\tilde{n}}}\end{array}\right)& {\underset{\_}{\underset{\_}{0}}}_{62}& {\underset{\_}{\underset{\_}{0}}}_{63}\\ {\underset{\_}{\underset{\_}{0}}}_{26}& \mathrm{diag}\left(\begin{array}{c}-{}_{b_X^b}\\ -{}_{b_Y^b}\end{array}\right)& {\underset{\_}{\underset{\_}{0}}}_{23}\\ {\underset{\_}{\underset{\_}{0}}}_{36}& {\underset{\_}{\underset{\_}{0}}}_{32}& \mathrm{diag}\left(\begin{array}{c}-{}_{l_X^v}\\ -{}_{l_Y^v}\\ -{}_{l_Z^v}\end{array}\right)\end{array}\right)$

(20) In this case we have .sub..Math.cos .Math.cos =d.sub., .sub.{tilde over (E)}.Math.cos .Math.sin =d.sub.{tilde over (E)} and .sub.D.Math.sin =d.sub.D.

(21) Therefore, a tan 2 (d.sub.{tilde over (E)},d.sub.),

(22) After has been calculated to a significant figure, the compass will operate with the actual N-E-D reference coordinate system, that is, the compass has entered into the operating mode and the correction model for the state vector is:
x=(.sub.N.sub.E.sub.Dd.sub.Nd.sub.Cd.sub.Db.sub.X.sup.bb.sub.Y.sup.bl.sub.X.sup.vl.sub.Y.sup.vl.sub.Z.sup.v).sup.T
and the dynamic matrix is:

(23) $\underset{\_}{\underset{\_}{F}}=\left(\begin{array}{cccc}\left(\begin{array}{ccc}0& {}_D& -{}_E\\ -{}_D& 0& {}_N\\ {}_E& -{}_N& 0\end{array}\right)& \left(\begin{array}{ccc}1& \cos {}_v.Math.\sin {}_v& 0\\ 0& -\cos {}_v.Math.\cos {}_v& 0\\ -\frac{d_C.Math.d_D}{d_N^2}.Math.\cos {}_v.Math.\sin {}_v& \frac{d_D}{d_N}.Math.\cos {}_v.Math.\sin {}_v& 1+\frac{d_C}{d_N}.Math.\cos {}_v.Math.\sin {}_v\end{array}\right)& {\underset{\_}{\underset{\_}{0}}}_{32}& {\underset{\_}{\underset{\_}{0}}}_{33}\\ {\underset{\_}{\underset{\_}{0}}}_{33}& \mathrm{diag}\left(\begin{array}{c}-{}_{d_X^b}\\ -{}_{d_Y^b}\\ -{}_{d_Z^b}\end{array}\right)& {\underset{\_}{\underset{\_}{0}}}_{32}& {\underset{\_}{\underset{\_}{0}}}_{33}\\ {\underset{\_}{\underset{\_}{0}}}_{23}& {\underset{\_}{\underset{\_}{0}}}_{23}& \mathrm{diag}\left(\begin{array}{c}-{}_{b_X^b}\\ -{}_{b_Y^b}\end{array}\right)& {\underset{\_}{\underset{\_}{0}}}_{23}\\ {\underset{\_}{\underset{\_}{0}}}_{33}& {\underset{\_}{\underset{\_}{0}}}_{33}& {\underset{\_}{\underset{\_}{0}}}_{32}& \mathrm{diag}\left(\begin{array}{c}-{}_{l_X^v}\\ -{}_{l_Y^v}\\ -{}_{l_Z^v}\end{array}\right)\end{array}\right)$
wherein

(24) 0${}_N=-d_N-d_C.Math.\cos {}_v.Math.\sin {}_v$${}_E=d_C.Math.\cos {}_v.Math.\cos {}_v$${}_D=-d_D-\frac{d_C.Math.d_D}{d_N}.Math.\cos {}_v.Math.\sin {}_v$
and

(25) $d_C=-\frac{v_X^v}{R}.$
Calibration of Deviation of Angular Velocity Sensor

(26) If the geographic latitude and the ship's velocity are provided, then the components of the earth's and ship's angular velocity can be calculated, and thus the measured angular velocity can be corrected; also their sensor-induced errors can be estimated and/or calibrated. The much smaller deviations in the rotational rate sensors during compass operation are corrected by application of the following state vector:
x=(.sub.N.sub.E.sub.Dd.sub.X.sup.bd.sub.Y.sup.bd.sub.Z.sup.bb.sub.X.sup.bb.sub.Y.sup.bl.sub.X.sup.vl.sub.Y.sup.vl.sub.Z.sup.v).sup.T
and the dynamic matrix reproduced below:

(27) $\underset{\_}{\underset{\_}{F}}=\left(\begin{array}{cccc}\left(\begin{array}{ccc}0& {}_D& -{}_E\\ -{}_D& 0& {}_N\\ {}_E& -{}_N& 0\end{array}\right)& {\underset{\_}{\underset{\_}{C}}}_{\mathrm{nb}}& {\underset{\_}{\underset{\_}{0}}}_{32}& {\underset{\_}{\underset{\_}{0}}}_{33}\\ {\underset{\_}{\underset{\_}{0}}}_{33}& \mathrm{diag}\left(\begin{array}{c}-{}_{d_X^b}\\ -{}_{d_Y^b}\\ -{}_{d_Z^b}\end{array}\right)& {\underset{\_}{\underset{\_}{0}}}_{32}& {\underset{\_}{\underset{\_}{0}}}_{33}\\ {\underset{\_}{\underset{\_}{0}}}_{23}& {\underset{\_}{\underset{\_}{0}}}_{23}& \mathrm{diag}\left(\begin{array}{c}-{}_{b_X^b}\\ -{}_{b_Y^b}\end{array}\right)& {\underset{\_}{\underset{\_}{0}}}_{23}\\ {\underset{\_}{\underset{\_}{0}}}_{33}& {\underset{\_}{\underset{\_}{0}}}_{33}& {\underset{\_}{\underset{\_}{0}}}_{32}& \mathrm{diag}\left(\begin{array}{c}-{}_{l_X^v}\\ -{}_{l_Y^v}\\ -{}_{l_Z^v}\end{array}\right)\end{array}\right)$
Fast and Rough Tuning

(28) If the geographic latitude is known, then a faster and more reliable calculation of can be performed from the estimated horizontal components of the earth's angular velocity.

(29) The state vector is as follows:
x=(.sub..sub.{tilde over (E)}.sub.Dd.sub.d.sub.{tilde over (E)}b.sub.X.sup.bb.sub.Y.sup.bl.sub.X.sup.vl.sub.Y.sup.vl.sub.Z.sup.v).sup.T

(30) The dynamic matrix is as follows:

(31) $\underset{\_}{\underset{\_}{F}}=\left(\begin{array}{ccc}\left(\begin{array}{ccccc}0& {}_D& -{}_{\tilde{E}}& 1& 0\\ -{}_D& 0& {}_{\tilde{N}}& 0& 1\\ {}_{\tilde{E}}& -{}_{\tilde{N}}& 0& 0& 0\\ 0& 0& 0& -{}_{d^{\tilde{n}}}& 0\\ 0& 0& 0& 0& -{}_{d^{\tilde{n}}}\end{array}\right)& {\underset{\_}{\underset{\_}{0}}}_{52}& {\underset{\_}{\underset{\_}{0}}}_{53}\\ {\underset{\_}{\underset{\_}{0}}}_{25}& \mathrm{diag}\left(\begin{array}{c}-{}_{b_X^b}\\ -{}_{b_Y^b}\end{array}\right)& {\underset{\_}{\underset{\_}{0}}}_{23}\\ {\underset{\_}{\underset{\_}{0}}}_{35}& {\underset{\_}{\underset{\_}{0}}}_{32}& \mathrm{diag}\left(\begin{array}{c}-{}_{l_X^v}\\ -{}_{l_Y^v}\\ -{}_{l_Z^v}\end{array}\right)\end{array}\right)$

(32) with .sub..Math.cos .Math.cos =d.sub., .sub.{tilde over (E)}.Math.cos .Math.sin =d.sub.{tilde over (E)} and .sub.D.Math.sin , so that a tan 2 (d.sub.{tilde over (E)},d.sub.).

(33) To ensure that the compass measuring unit is suitable only for the strap-down ship's gyro compass, its measured increases in angle and velocity are scaled to a lesser accuracy than can be used for a gyro compass. When the scaled increment is received by the compass signal processing unit, the increments are descaled before their subsequent use within the strap-down compass signal processing path and corrected as described above.