Navigation Coordinate System

Abstract

Why does navigation study coordinate systems? We know that describing object motion is relative to a certain reference frame. The task of the navigation system is to determine the motion parameters of the carrier, that is, to determine the series of navigation parameters such as the carrier position, speed, attitude and its change rate in a certain coordinate system. In other words, the navigation system itself is a reference coordinate system for the carrier, which is often known as the spatiotemporal reference. Therefore, when studying the navigation problem, it comes first to determine the coordinate system, which is not only the mathematical basis for defining navigation parameters and navigation solutions but also the basic problem of all navigation technologies and navigation systems. It is very important to describe navigation parameters and solve navigation problems. This chapter will highlight the relevant knowledge about the navigation coordinate system.

Why does navigation study coordinate systems? We know that describing object motion is relative to a certain reference frame. The task of the navigation system is to determine the motion parameters of the carrier, that is, to determine the series of navigation parameters such as the carrier position, speed, attitude and its change rate in a certain coordinate system. In other words, the navigation system itself is a reference coordinate system for the carrier, which is often known as the spatiotemporal reference. Therefore, when studying the navigation problem, it comes first to determine the coordinate system, which is not only the mathematical basis for defining navigation parameters and navigation solutions but also the basic problem of all navigation technologies and navigation systems. It is very important to describe navigation parameters and solve navigation problems. This chapter will highlight the relevant knowledge about the navigation coordinate system.

2.1 Main Coordinate Systems

2.1.1 Types of Coordinate Systems

2.1.1.1 Cartesian Coordinate System in Three-Dimensional Space

The coordinate system was first put forward by Descartes in 1637, namely, the rectangular coordinate system, which mainly includes a two-dimensional plane rectangular coordinate system and a three-dimensional space rectangular coordinate system.

The coordinate system has three basic elements: origin, axis, and unit. To select and determine a coordinate system, it is usually necessary to determine (1) two poles, namely, the base pole and the second pole; (2) two planes, namely, the base plane and the second plane; and (3) three axes, namely, the base axis, the second axis and the third axis. The base pole (line) is the symmetrical axis of the coordinate system, such as the rotation axis of the earth. The base plane is a plane perpendicular to the base pole, such as the equatorial plane. The second plane is perpendicular to the base plane and contains the base pole (line), such as the Greenwich meridian plane. The second pole (line) is the intersection line between the base plane and the second plane. The third axis is orthogonal to the base axis and the second axis, forming a right-handed rectangular coordinate system. Once the above elements are determined, a coordinate system is basically determined.

All points in space can be projected to three coordinate axes, and their coordinates {x, y, z} can be obtained (Fig. 2.1). Three-dimensional space rectangular coordinate systems are widely used in the navigation field. The inertial coordinate system, geographic coordinate system, body coordinate system and navigation coordinate system are all three-dimensional space rectangular coordinate systems.

Fig. 2.1

Cartesian coordinate system

2.1.1.2 Spherical Coordinate System

Points in a two-dimensional plane can be described in Cartesian coordinates $\left( {X,Y} \right)$, which can also be expressed by polar coordinates $({\rho ,\theta })$ . Similarly, points in three-dimensional rectangular coordinate systems $\left( {x, y, z} \right)$ can also be described by the spherical coordinate system $(r, \alpha, \delta )$ (as shown in Fig. 2.2), where r is the length of the vector (OM),α is longitude, and δ is latitude $\left( {0 \le r < \infty ,0 < \alpha < 2\pi ,\;0 \le \delta \le \pi } \right)$.

Fig. 2.2

Spherical coordinate system

Spherical coordinate systems are widely used in navigation. Usually, the navigation positioning is on the surface of the earth, which is a sphere. Because the earth is not a regular sphere, ellipsoidal coordinates are often used in practice. The ellipsoid coordinate system can be regarded as a general form of the spherical coordinate system. Geodetic coordinates and celestial coordinates commonly used are spherical coordinates. Longitude, latitude and altitude are defined in a spherical coordinate system, and their common symbols are ${(\lambda, \varphi, L)}.$

2.1.1.3 Other Coordinate Systems

Cylindrical coordinate systems and conical coordinate systems are also used in navigation; for example, cylindrical coordinate systems are often used in navigation mapping, such as Mercator projections and Gauss projections.

2.1.2 Common Navigation Coordinate System

2.1.2.1 Inertial Coordinate System

An inertial coordinate system is a coordinate system describing inertial space. The so-called inertial space refers to the space in which an object can keep stationary or uniform linear motion when it is not subjected to force or the resultant force is zero.

The inertial system is a very important concept in mechanics. In the inertial coordinate system, the relationship between force and motion is described by Newton's law. With the development of physics, Einstein created a new space–time theory in the twentieth century, which developed the concept of Newton's inertial system.

The book follows Newton's idea of an inertial system. To establish an inertial coordinate system, we must find a reference object that is stationary or moving uniformly in inertial space. However, according to the principle of gravitation, there is no object absolutely under zero force in the universe, so the ideal inertial reference system cannot be found in reality. On the other hand, in practical applications, it is not necessary to find an absolutely accurate inertial reference system but only to find an approximate inertial reference system. Thus, the approximation depends on the needs of the requirement.

According to different approximations, many different inertial systems can be obtained. For objects other than the sun in the solar system, the Sun-center (heliocentric) coordinate system is very similar to the inertial system, and for objects other than the Earth in the Earth-Moon system, the centroid coordinate system of the Earth is very similar to the inertial system. The following are several inertial systems commonly used in navigation. The main difference between them lies in the selection of the origin and the definition of the axis.

• (1) Sun-center inertial coordinate system (SCI system)

The origin of the SCI system is at the center of the Sun's sphere. The orbital plane of the Z_s axis is vertical to the Earth's revolution, and the X_s and Y_s axes are in the right-hand coordinate system in the plane of the Earth's orbit of revolution (Fig. 2.3), which is ecliptic. The SCI is suitable for the navigation and positioning of interstellar vehicles. The angular velocity of the sun's rotation around the galactic center is approximately 0.001 angular seconds/year, and the centripetal acceleration of the sun's movement around the galactic center is approximately 2.4 × 10^–10 m/s². The angular velocity and acceleration of the sun's motion around the Galaxy are far less than the minimum that human instruments can measure at present. Therefore, the use of SCI ignores the movement of the sun in the Galaxy and has sufficient accuracy.

Fig. 2.3

Sun-center inertial coordinate system

• (2) Earth-center inertial coordinate system (ECI system)

The coordinate origin of the ECI system is taken at the geocentric center of the earth. The Z_i axis is along the Earth's axis of rotation, and the X_i axis and Y_i axis are in the equatorial plane of the Earth. The X_i axis, Y_i axis, and Z_i axis form a right-handed coordinate system. The X_i axis usually points to the intersection of the equatorial plane of the earth and the ecliptic plane of the sun, which is called the equinox (spring equinox and autumn equinox) and does not change with the rotation of the earth (Fig. 2.4). The ECI is suitable for studying the navigation and positioning of vehicles near the Earth’s surface. Near-Earth navigation mostly uses ECI as an inertial coordinate system. For example, the motion of an Earth satellite can be calculated in the ECI system.

Fig. 2.4

Geocentric inertial coordinate system

This approximate inertial reference system ignores the acceleration caused by the gravitational pull of the sun, moon and other stars on the earth. The acceleration of the Earth's common centripetal acceleration caused by the sun's gravity is approximately 6.05 × 10^–3 m/s². The centripetal acceleration of the earth caused by the moon's gravitation is approximately 3.4 × 10^–5 m/s². The acceleration caused by the gravitation of other stars is two orders of magnitude smaller than that of both above, so the acceleration of the origin of the geocentric inertial coordinate system is approximately 6 × 10^–6 m/s². This magnitude can be sensitized by current accelerometers, so in some applications of inertial technology, the ECI is not accurate enough. However, if we only care about the motion of objects relative to the earth, the acceleration caused by the gravitation of the sun and other stars to the earth and the near-earth objects is approximately 10^–6 m/s². The magnitude can be neglected.

• (3) Celestial coordinate system (C system)

The celestial coordinate system uses spherical coordinates to describe the position of celestial objects (stars, planets, etc.). The position of celestial objects in the celestial coordinate system only indicates the direction of celestial objects without considering the distance of celestial objects. The celestial sphere rotates around the Earth’s axis from east to west. The intersection of the axis of rotation and the celestial sphere in the northern (southern) hemisphere is called the northern (southern) celestial pole, namely, $P_{N} \left( {P_{S} } \right)$. The northern and southern ecliptic poles are represented by $\Pi \left( {\Pi^{\prime}} \right)$, respectively. The angle between the ecliptic plane and the celestial equatorial plane is called the ecliptic obliquity $\varepsilon$, with an average value of approximately 23.5° (Fig. 2.5).

Fig. 2.5

Points and circles on the celestial sphere

The coordinate origin of the Ecliptic coordinate system is taken at the geocentric center of the earth. The Z_i axis is along the line of $\Pi \left( {\Pi^{\prime}} \right)$, and the X_i axis and Y_i axis are in the ecliptic plane of the earth. The X_i axis usually points to the spring equinox, and the X_i axis, Y_i axis, and Z_i axis form a right-handed coordinate system (Fig. 2.6). Based on the coordination, the ecliptic longitude and latitude are defined.

Fig. 2.6

Ecliptic coordinate system

2.1.2.2 Earth-Center Earth-Fixed Coordinate System (ECEF System)

The ECEF system is also known as the Earth coordinate system (e system). The origin of the coordinate system in the center of the earth, because the earth is always rotating, the earth coordinate system and the earth are fixed, rotating with the earth polar axis Z_e. The X_e axis is at the intersection of the equatorial plane and the primary meridian plane, and the Ye axis is also in the equatorial plane. The three axes form right-hand rectangular coordinates (Fig. 2.7).

Fig. 2.7

ECEF system and geographic coordinate system (E-N-U)

Considering the autorotation and revolution of the earth, the rotation angular velocity of the e system relative to the inertial reference system is

$$ \omega_{ic} \approx 15.0411\left( {^\circ /h} \right) \approx 7.2921 \times 10^{ - 5} \,{\text{rad/s}} $$

(2.1)

2.1.2.3 Earth Surface Coordinate System

• (1) Geographic coordinate system (g system)

The g system is the most commonly used coordinate system (Fig. 2.8). The origin of the g system is the observer or the center of the vehicle. The axis X_g can point east in the local horizontal plane. The axis Y_g then points north along the local meridian circle. The axis Z_g points to the zenith along the normal of the local reference ellipsoid. Three axes form the right-hand rectangular coordinate system (E-N-U) (Fig. 2.7). The Chinese traditional conception of six directions (i.e., Up, Down, Fore, Behind, Left and Right—“六合 liu he”) is the geographical coordinate system.

Fig. 2.8

Geographic coordinate system (N-W-U)

The three axes of the g system can be selected in different ways. The right-hand rectangular coordinate system can be constructed in the order of “north, west, up” (N-W-U) (Fig. 2.8) or “north, east and down” (N-E-D).

When the vehicle moves on the surface of the earth, the position of the vehicle relative to the earth changes constantly, and the g system of different places has different angular positions relative to the earth. That is, the motion of the vehicle will cause the local g system to rotate relative to the e system.

• (2) Horizontal coordinate system (h system)

The so-called horizon is a plane with an observer on the Earth’s surface at its center, which is the dividing line between the visible and invisible parts of the celestial sphere. The celestial perpendicular line is consistent with the local gravity field through the observer. The two points above and below the observer’s head are the zenith Z_h and the bottom $Z^{\prime}_h$ . Z_h is a celestial vertical axis, and the horizon plane is the base plane, which forms the horizontal coordinate system (Fig. 2.5).

• (3) Tangent plane coordinate system (t system)

The t system is the fixed earth coordinate system. The tangent plane is the plane tangent to the geodetic reference ellipsoid, and the tangent point is the origin. This point is usually chosen as the landing site, navigation radar station, or some other convenient reference points. The axis X_t points east, the axis Y_t points north, and the axis Z_t points zenith and perpendicular to the reference ellipsoid, which form the right-hand rectangular coordinate system.

For stationary vehicles, the g system and the t system are identical. For the moving vehicle, the origin of the tangent plane is fixed, and the origin of the g system is the projection of the origin of the vehicle on the geoid. The t system is often used in local navigation, e.g., aircraft relative path navigation.

• (4) Wandering azimuth coordinate system (w system)

The w system is defined on the basis of the geographical coordinate system. The origin of the w system is at the center of the vehicle; the axis X_w is perpendicular to the axis Y_w and the axis Z_w. The angle $\alpha$, known as the drift azimuth, is between the axis Y_w in the local horizontal plane and the northern meridian circle (counterclockwise positive). The axis Z_w points to the zenith along the normal direction outside the ellipsoid $\alpha$ satisfy.

$$ \alpha = - \lambda \sin \varphi $$

(2.2)

In the formula, $\lambda$ and $\varphi$ are the longitude and latitude of the vehicle center, respectively.

The use of the w system avoids the singularity of the g system in the polar region, so it is widely used in the polar navigation algorithm (Fig. 2.9).

Fig. 2.9

Wandering azimuth coordinate system

• (5) Greenwich grid coordinate system

The origin of the G system is the location of the vehicle. The plane parallel to the Greenwich meridian at the point is taken as the grid plane, and the horizontal plane where the vehicle is located is taken as the tangent plane. The intersection line between the grid plane and the tangent plane is defined as the grid north, and the angle between the grid north and the true north is represented as the grid azimuth $\sigma$. The zenith direction of the grid coincides with the geographic zenith direction, and the grid east is in the tangent plane and perpendicular to the grid north to form a right-hand rectangular coordinate system (Fig. 2.10).

Fig. 2.10

Grid coordinate system

2.1.2.4 Body Coordinate System (b System)

The b system is a general term for the aircraft coordinate system (aircraft), ship coordinate system (ship), star coordinate system (satellite), etc. (Fig. 2.11). Taking the body coordinate system of a ship as an example, the body coordinate system is fixed on the ship, and its origin is the mass center of the ship. Axis X_b points to starboard along the transverse axis of the ship, axis Y_b points to the bow along the longitudinal axis of the ship, and axis Z_b is perpendicular to the deck plane and points upward.

Fig. 2.11

Body coordinate system (R-F-U)

2.1.2.5 Other Coordinate Systems

In addition to the above coordinate systems, some other important coordinate systems are used in the navigation list below.

• (1) Navigation coordinate system (n system)

  The n system is the coordinate system selected by different navigation systems according to the principles when solving the navigation parameters.

• (2) Platform coordinate system (p system)

  For the inertial navigation system based on the stabilized platform, the p system is fixed with the physical platform, whose origin is located at the mass center of the platform, and the axes are consistent with the three axes indicated by the platform. For a strap down inertial navigation system, the p system is realized by the direction cosine matrix stored in the computer, so it is also called a “mathematical platform”.

• (3) Computing coordinate system (c system)

  The c system refers to the navigation coordinate system determined by the computer navigation output. It has some error with the actual n system. The c system is often used in the analysis of navigation error.

• (4) Gyroscope coordinate system (gr system)

  The gr system is a coordinate system used to indicate the spatial orientation of the gyroscope rotor spin axis, which is fixed on the gyroscope internal frame. The origin is the center of the gyroscope bracket. The X_gr axis coincides with the rotor shaft but does not participate in the rotation of the rotor. The Y_gr axis coincides with the inner ring axis, and the Z_gr axis is always perpendicular to the X_gr and Y_gr planes. Three axes form the right-hand rectangular coordinate system.

2.2 The Coordinate System Transformation

In navigation research, an object is often abstracted as a coordinate system, and the motion of an object relative to another object is equivalent to the linear and angular motion of a coordinate system relative to another one. In the navigation system, one set of coordinate systems is connected to the studied object, and another set is connected to the selected reference space, the latter constituting the reference coordinate system of the former.

2.2.1 Coordinate Representation of the Vector

To define most motion parameters, including position, velocity, acceleration and angular velocity, three coordinate systems are involved:

• (1) Body coordinate system (object frame), indicating the carrier, b;

• (2) Reference coordinate system (reference frame), indicating the reference coordinate system of the movement, β;

• (3) Projection coordinate system (resolving frame), indicating the measurement coordinate system, γ.

The b system cannot be the same as the β system; otherwise, there will be no relative motion. The γ system may be the b system, β system, or other coordinate systems. Just define the axis direction of the γ system and do not need to define its origin. The selection of the γ system does not affect the amplitude of the vector.

To fully describe the motion parameters, the above three coordinate systems should be clearly defined. The following symbols are indicated by:

$$ x_{{{\beta b}}}^{\gamma } $$

(2.3)

Vector x can represent the Cartesian position, velocity, acceleration, angular velocity, etc., which describes the projected representation of the motion vector in frame b relative to frameβ.

There are two main differences between the two rectangular coordinate systems. (1) The origins are different, that is, the origin of one coordinate system is displaced from another; (2) one coordinate system rotates relative to another. For example, the relationship between the g system (OX_gY_gZ_g) at a point on the Earth’s surface and the E system (OX_eY_eZ_e).

Navigation involves many coordination transformation calculations, which include the transformation of points between different coordinate systems, such as transformations between rectangular coordinate systems and spherical coordinate systems, and various angle transformations between rectangular coordinate systems.

2.2.2 Position Transformation in the Coordinate System

The navigation system often transforms the position parameters between the spherical coordinate system and the rectangular coordinate system, which is as follows:

• (1) Transform the relationship between spherical coordinates and rectangular coordinates (Fig. 2.12):

  Fig. 2.12

  

  Schematic diagram of the spherical coordinates

  $$ \left[ \begin{gathered} x \hfill \\ y \hfill \\ z \hfill \\ \end{gathered} \right] = r\left[ \begin{gathered} \cos \delta \cos \alpha \\ \cos \delta \sin \alpha \\ \sin \delta \\ \end{gathered} \right] $$

  (2.4)

• (2) Transform the relationship between rectangular coordinates and spherical coordinates:

  $$ \begin{gathered} r = \sqrt {x^{2} + y^{2} + z^{2} } \hfill \\ \alpha = \arctan \frac{y}{x} \hfill \\ \delta = \arctan \frac{z}{{\sqrt {x^{2} + y^{2} } }} \hfill \\ \end{gathered} $$

  (2.5)

Position transformation can be commonly used in the e system. Let us suppose a point S on the Earth’s surface in the e system, obtain $\varvec{r}_{S} = (x_{S}, y_{S}, z_{S})$; the latitude and longitude coordinates are $P (\lambda, L)$; the distance from the point to the Earth’s center is called the center radius, namely, $\varvec{r}_{S} = | \varvec{r}_{S} |$, and the transformation formula between the Earth’s rectangular coordinates and spherical coordinates is as follows.

Transform the relationship between spherical coordinates to rectangular coordinates:

$$ \left\{ \begin{gathered} x_{s} = R_{N} \cos L\cos \lambda \hfill \\ y_{s} = R_{N} \cos L\sin \lambda \hfill \\ z_{s} = R_{N} (1 - e)^{2} \sin L \hfill \\ \end{gathered} \right. $$

(2.6)

where R_N is the radius of the prime vertical circle and e is the eccentricity of the reference ellipsoid.

Transform the relationship between rectangular coordinates to spherical coordinates:

$$ \left\{ \begin{gathered} {\lambda = arctan}\frac{y}{x} \hfill \\ L = \arctan \left[ {\frac{1}{(1 - e^2 )}\frac{z}{{\sqrt {x^2 + y^2 } }}} \right] \hfill \\ \end{gathered} \right. $$

(2.7)

2.2.3 Angle Relationship Representation Between Coordinate Systems

Determining the angular position and conversion relationship between two different coordinate systems is the fundamental problem of navigation. Navigation uses the g, b, p and c coordinate system, etc. In many cases, the origins of these frames are often approximately treated as the same, with no relative displacement, while mainly studying the relative angular variation. In the coordinate system transformation with different origins, the relative displacement between the origins can also be reflected by the angular position relationship between the coordinate systems. For example, the angular relationship between the e coordinate system and the g system can be determined by the angles λ and φ, which determine the position of the origin of the g coordinate system in the e coordinate system. Therefore, the angular relationship between the two coordinate systems is the key to analyzing the coordinate angular interconversion. The main methods to describe the angular position of the coordinate system are the directional cosine, Euler angle, quaternion and rotating vector methods.

2.2.3.1 Directional Cosine Matrix

The directional cosine method is a method of representing the coordinate transformation matrix by the directional cosine of the vector. The coordinate transformation matrix from coordinate system a to coordinate system b can be represented by a 3 × 3 azimuthal cosine matrix, i.e.,

$$ {\varvec{C}}_{{a}}^{b} = \left[ {\begin{array}{*{20}c} {{\varvec{C}}_{11} } & {{\varvec{C}}_{12} } & {{\varvec{C}}_{13} } \\ {{\varvec{C}}_{21} } & {{\varvec{C}}_{22} } & {{\varvec{C}}_{23} } \\ {{\varvec{C}}_{31} } & {{\varvec{C}}_{32} } & {{\varvec{C}}_{33} } \\ \end{array} } \right] $$

(2.8)

The various elements of the matrix are shown below. The unit vector of axes $x_{{\text{a}}}$,$y_{{\text{a}}}$,$z_{{\text{a}}}$ can be represented by i_a, j_a and k_a. The unit vector of axes $x_{{\text{b}}}$,$y_{{\text{b}}}$,$z_{{\text{b}}}$ can be represented by i_b, j_b, and k_b. Let any vector r be represented in i_a, j_a, k_a and i_b, j_b, k_b, respectively:

$$ \begin{gathered} {\varvec{r}}^{\text{a}} = x_{\text{a}} {\varvec{i}}_{\text{a}} + y_{\text{a}} {\varvec{j}}_{\text{a}} + z_{\text{a}} {\varvec{k}}_{\text{a}} \hfill \\ {\varvec{r}}^{\text{b}} = x_{\text{b}} {\varvec{i}}_{\text{b}} + y_{\text{b}} {\varvec{j}}_{\text{b}} + z_{\text{b}} {\varvec{k}}_{\text{b}} \hfill \\ \end{gathered} $$

(2.9)

The projection of vector r on the three axial vectors of coordinate system b is as follows:

$$ \begin{gathered} x_{\text{b}} = {\varvec{r}}^{\text{a}} \cdot {\varvec{i}}_{\text{b}} = ({\varvec{i}}_{\text{a}} \cdot {\varvec{i}}_{\text{b}} )x_{\text{a}} + ({\varvec{j}}_{\text{a}} \cdot {\varvec{i}}_{\text{b}} )y_{\text{a}} + ({\varvec{k}}_{\text{a}} \cdot {\varvec{i}}_{\text{b}} )z_{\text{a}} \hfill \\ y_{\text{b}} = {\varvec{r}}^{\text{a}} \cdot {\varvec{j}}_{\text{b}} = ({\varvec{i}}_{\text{a}} \cdot {\varvec{j}}_{\text{b}} )x_{\text{a}} + ({\varvec{j}}_{\text{a}} \cdot {\varvec{j}}_{\text{b}} )y_{\text{a}} + ({\varvec{k}}_{\text{a}} \cdot {\varvec{j}}_{\text{b}} )z_{\text{a}} \hfill \\ z_{\text{b}} = {\varvec{r}}^{\text{a}} \cdot {\varvec{k}}_{\text{b}} = ({\varvec{i}}_{\text{a}} \cdot {\varvec{k}}_{\text{b}} )x_{\text{a}} + ({\varvec{j}}_{\text{a}} \cdot {\varvec{k}}_{\text{b}} )y_{\text{a}} + ({\varvec{k}}_{\text{a}} \cdot {\varvec{k}}_{\text{b}} )z_{\text{a}} \hfill \\ \end{gathered} $$

(2.10)

In Eq. (2.10), the transformation coefficients of the dot-product form (i_a i_b) are the direction cosine, which can be obtained from the following form:

$$ {\varvec{i}}_{\text{a}} \cdot {\varvec{i}}_{\text{b}} = |{\varvec{i}}_{\text{a}} ||{\varvec{i}}_{\text{b}} |cos\theta_{i_{\text{b}} i_{\text{a}} } = \cos \theta_{i_{\text{b}} i_{\text{a}} } $$

(2.11)

where $\cos \theta_{{i_{{\text{b}}} i_{{\text{a}}} }}$ is the cosine between the two unit vectors i_a and i_b.

The above transformation equations can be further written in matrix form as follows:

$$ \left[ \begin{gathered} x_{\text{b}} \hfill \\ y_{\text{b}} \hfill \\ z_{\text{b}} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {{\varvec{i}}_{\text{a}} \cdot {\varvec{i}}_{\text{b}} } & {{\varvec{j}}_{\text{a}} \cdot {\varvec{i}}_{\text{b}} } & {{\varvec{k}}_{\text{a}} \cdot {\varvec{i}}_{\text{b}} } \\ {{\varvec{i}}_{\text{a}} \cdot {\varvec{j}}_{\text{b}} } & {{\varvec{j}}_{\text{a}} \cdot {\varvec{j}}_{\text{b}} } & {{\varvec{k}}_{\text{a}} \cdot {\varvec{j}}_{\text{b}} } \\ {{\varvec{i}}_{\text{a}} \cdot {\varvec{k}}_{\text{b}} } & {{\varvec{j}}_{\text{a}} \cdot {\varvec{k}}_{\text{b}} } & {{\varvec{k}}_{\text{a}} \cdot {\varvec{k}}_{\text{b}} } \\ \end{array} } \right]\left[ \begin{gathered} x_{\text{a}} \hfill \\ y_{\text{a}} \hfill \\ z_{\text{a}} \hfill \\ \end{gathered} \right] $$

(2.12)

let

$$ \varvec{C}_{\text{a}}^{\text{b}} = \left[ {\begin{array}{*{20}c} {{\varvec{i}}_{\text{b}} \cdot {\varvec{i}}_{\text{a}} } & {{\varvec{i}}_{\text{b}} \cdot {\varvec{j}}_{\text{a}} } & {{\varvec{i}}_{\text{b}} \cdot {\varvec{k}}_{\text{a}} } \\ {{\varvec{j}}_{\text{b}} \cdot {\varvec{i}}_{\text{a}} } & {{\varvec{j}}_{\text{b}} \cdot {\varvec{j}}_{\text{a}} } & {{\varvec{j}}_{\text{b}} \cdot {\varvec{k}}_{\text{a}} } \\ {{\varvec{k}}_{\text{b}} \cdot {\varvec{i}}_{\text{a}} } & {{\varvec{k}}_{\text{b}} \cdot {\varvec{j}}_{\text{a}} } & {{\varvec{k}}_{\text{b}} \cdot {\varvec{k}}_{\text{a}} } \\ \end{array} } \right]\; $$

(2.13)

Namely, formula (2.12) can be written as

$$ {\varvec{r}}^{{\text{b}}} = {\varvec{C}}_{{\text{a}}}^{{\text{b}}} {\varvec{r}}^{{\text{a}}} \; $$

(2.14)

Similarly, we can obtain

$$ {\varvec{r}}^{{\text{a}}} = {\varvec{C}}_{{\text{b}}}^{{\text{a}}} {\varvec{r}}^{{\text{b}}} \; $$

(2.15)

$$ \varvec{C}_{\text{b}}^{\text{a}} = \left[ {\begin{array}{*{20}c} {{\varvec{i}}_{\text{b}} \cdot {\varvec{i}}_{\text{a}} } &{{\varvec{j}}_{\text{b}} \cdot {\varvec{i}}_{\text{a}} } & {{\varvec{k}}_{\text{b}} \cdot {\varvec{i}}_{\text{a}} } \\ {{\varvec{i}}_{\text{b}} \cdot {\varvec{j}}_{\text{a}} } & {{\varvec{j}}_{\text{b}} \cdot {\varvec{j}}_{\text{a}} } & {{\varvec{k}}_{\text{b}} \cdot {\varvec{j}}_{\text{a}} } \\ {{\varvec{i}}_{\text{b}} \cdot {\varvec{k}}_{\text{a}} } & {{\varvec{j}}_{\text{b}} \cdot {\varvec{k}}_{\text{a}} } & {{\varvec{k}}_{\text{b}} \cdot {\varvec{k}}_{\text{a}} } \\ \end{array} } \right] $$

(2.16)

Nine elements in $C_{{\text{a}}}^{{\text{b}}}$ and $C_{{\text{b}}}^{{\text{a}}}$ are all directional cosines between the axes of two coordinate systems, which reflects the angular relationship between two coordinate systems, called the directional cosine matrix from coordinate system a to b, $C_{{\text{a}}}^{{\text{b}}}$, and called the directional cosine matrix from coordinate system b to a, $C_{{\text{b}}}^{{\text{a}}}$.

Available by formulas (2.13) and (2.16):

$$ (\varvec{C}_{{\text{a}}}^{{\text{b}}} )^{ - 1} = \varvec{C}_{{\text{b}}}^{{\text{a}}} = (\varvec{C}_{{\text{a}}}^{{\text{b}}} )^{{\text{T}}} $$

(2.17)

The directional cosine matrix is an orthogonal matrix. It is transitive, which enables the transformation between multiple coordinate systems easily. For example, in the aforementioned problem, if there is a third coordinate system $x_{{\text{c}}} y_{{\text{c}}} z_{{\text{c}}}$ and the directional cosine matrix of the coordinate system $x_{{\text{b}}} y_{{\text{b}}} z_{{\text{b}}}$ to the coordinate system $x_{{\text{c}}} y_{{\text{c}}} z_{{\text{c}}}$ is $C_{{\text{b}}}^{{\text{c}}}$, then the directional cosine matrix of the coordinate system $x_{{\text{a}}} y_{{\text{a}}} z_{{\text{a}}}$ to the coordinate system $x_{{\text{c}}} y_{{\text{c}}} z_{{\text{c}}}$ can be expressed as

$$ \varvec{C}_{{\text{a}}}^{{\text{c}}} = \varvec{C}_{{\text{b}}}^{{\text{c}}} \varvec{C}_{{\text{a}}}^{{\text{b}}} $$

(2.18)

2.2.3.2 Euler Angle

The directional cosine matrix between two three-dimensional rectangular frames has nine elements, depending on the properties of the orthogonal matrix, and the quadratic sum of the three elements of the orthogonal matrix is one. There are actually six constraints, and only three elements are independent. This shows that the angular relationship between any two three-dimensional rectangular coordinate systems can be completely described by three independent rotation angles, which are called Euler angles.

By referring to the rotation of the coordinate system, we can define the transformation matrix between the orthogonal coordinate systems based on the rotation angle (i.e., the Euler angle). In other words, the first reference frame x_ay_az_a rotates 3 times to produce the second reference frame x_by_bz_b: first rotate around the x_a-axis with the angle α₁; then rotate around the new y-axis with the angle α₂; finally rotate around the new z-axis with the angle α₃, and obtain the second reference frame x_by_bz_b (Fig. 2.13).

Fig. 2.13

Angular relation of the coordinate system

2.2.3.3 Quaternion

• (1) Quaternion Definition

A quaternion is a number composed of four elements:

$$ \varvec{Q}(q_{0} ,q_{1} ,q_{2} ,q_{3} ) = q_{0} + q_{1}\varvec{i} + q_{2}\varvec{j} + q_{3}\varvec{k} $$

(2.19)

where q₀, q₁, q₂ and q₃ are real numbers; i, j, k are mutual orthogonal unit vectors and the virtual units of $\sqrt { - 1}$, which are embodied in the following quaternionic multiplication relationship:

$$ \left. \begin{gathered} \varvec{i} \otimes \varvec{i} = - 1,\;\;\;\;\varvec{j} \otimes \varvec{j} = - 1,\;\;\;\;\varvec{k} \otimes \varvec{k} = - 1 \hfill \\ \varvec{i} \otimes \varvec{j} = \varvec{k},\;\;\;\;\;\;\;\varvec{j} \otimes \varvec{k} = \varvec{i},\;\;\;\;\;\;\varvec{k} \otimes \varvec{i} = \varvec{j} \hfill \\ \varvec{j} \otimes \varvec{i} = - \varvec{k},\;\;\;\;\varvec{k} \otimes \varvec{j} = - \varvec{i},\;\;\;\;\varvec{i} \otimes \varvec{k} = - \varvec{j} \hfill \\ \end{gathered} \right\} $$

(2.20)

where $\otimes$ indicates the quaternionic multiplication.

Quaternions, proposed by the Irish mathematician Hamilton, can be regarded as a super complex number consisting of the real part scalar q₀ and the virtual part three-dimensional vectors. The three axes i, j, and k are orthogonal to the virtual part and can actually be regarded as a vector in a four-dimensional space. Similar to the mathematical significance of complex number multiplication, quaternionic multiplication can easily solve the complex vector rotation transformation problem in three-dimensional space.

• (2) The Relationship Between the Quaternions and the Directional Cosine Matrix

The reference coordinate system β is provided, and the unit vectors of the x₀, y₀, andz₀axes are i₀, j₀, and k₀, respectively. The axes $x_{{\text{b}}}$$y_{{\text{b}}}$$z_{{\text{b}}}$ of the body coordinate system b rotate around the fixed point O. To explain the relative angular position and rotation relationship of the body, point A on the body and turning point O are taken to obtain the position vector OA (Fig. 2.14). The position vector can simplify the angular position of the body.

Fig. 2.14

Equivalent rotation of the rigid body

Let the b system rotate relative to the β system; the initial position vector OA = r. After time t, the position vector OA' = r′. According to Euler's theorem, and only considering the angular position of the b system at time 0 and t, the rotation of the b system from position A to position A' can be achieved directly by rotating around the axis u (unit vector) with the angle θ, where $\varvec{u} = l\varvec{i}_{0} + m\varvec{j}_{0} + n\varvec{k}_{0}$. Position vectors make a conical motion, with A and A' on the same circle and r and r' on the same conical plane.

Let

$$ \left\{ \begin{gathered} q_{0} = \cos \frac{\theta }{2} \hfill \\ q_{1} = l\sin \frac{\theta }{2} \hfill \\ q_{2} = m\sin \frac{\theta }{2} \hfill \\ q_{3} = n\sin \frac{\theta }{2} \hfill \\ \end{gathered} \right. $$

(2.21)

In addition, construct quaternions based on q₀, q₁, q₂, andq₃:

$$ \begin{gathered} \varvec{Q} = q_0 + q_1 {\varvec{i}}_0 + q_2 {\varvec{j}}_0 + q_3 {\varvec{k}}_0 = \cos \frac{\theta }{2} + (l{\varvec{i}}_0 + m{\varvec{j}}_0 + n{\varvec{k}}_0 )sin\frac{\theta }{2} \\ = \cos \frac{\theta }{2} + {\varvec{u}}^\beta \sin \frac{\theta }{2} \\ \end{gathered} $$

(2.22)

where ${\varvec{u}}^\beta$ is the rotation axis direction and θ is the turning angle. The following conclusions can be obtained:

• (a) Quaternion Q describes the fixed rotation of the body, that is, when only concerned about the angular position of the b system relative to β system, the b system can be considered to be formed by a disposable equivalent rotation of the system β directly; Q contains all the information of this equivalent rotation, and it can prove that the vectors r and r′ through rotation satisfy the following relationship:

  $$ \varvec{r}^{\prime} = \varvec{Q}\cdot \varvec{r}\cdot \varvec{Q}^* $$

  (2.23)

  where $\varvec{Q}^{*} = q_0 - q_1 {\varvec{i}}_0 - q_2 {\varvec{j}}_0 - q_3 {\varvec{k}}_0$and is the conjugate of the quaternions Q.

• (b) It can be deduced that there is the following correspondence between quaternion Q and the coordinate transformation matrix from system b to system β.

  $$ \varvec{C}_{{\text{b}}}^{\beta } = \left[ {\begin{array}{*{20}c} {1 - 2(q_{2}^{2} + q_{3}^{2} )} & {2(q_{1} q_{2} - q_{0} q_{3} )} & {2(q_{1} q_{3} + q_{0} q_{2} )} \\ {2(q_{1} q_{2} + q_{0} q_{3} )} & {1 - 2(q_{1}^{2} + q_{3}^{2} )} & {2(q_{2} q_{3} - q_{0} q_{1} )} \\ {2(q_{1} q_{3} - q_{0} q_{2} )} & {2(q_{2} q_{3} + q_{0} q_{1} )} & {1 - 2(q_{1}^{2} + q_{2}^{2} )} \\ \end{array} } \right] $$

  (2.24)

2.2.4 Common Coordinate System Transformation

The following lists the common coordinate transformations in navigation.

2.2.4.1 The Inertial Coordinate System—The Earth Coordinate System

From the ECI system X_iY_iZ_i to the e system X_eY_eZ_e, only one rotation is needed:

$$ \varvec{C}_{{\text{i}}}^{{\text{e}}} = \left[ {\begin{array}{*{20}c} {\cos (\omega_{{{\text{ie}}}} t - \lambda_0)} & {\sin (\omega_{{{\text{ie}}}} t - \lambda_0)} & 0 \\ { - \sin (\omega_{{{\text{ie}}}} t - \lambda_0)} & {\cos (\omega_{{{\text{ie}}}} t - \lambda_0)} & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] $$

(2.25)

Conversely,

$$ \varvec{C}_{{\text{i}}}^{{\text{e}}} = \left[ {\begin{array}{*{20}c} {\cos (\omega_{{{\text{ie}}}} t - \lambda_0)} & { - \sin (\omega_{{{\text{ie}}}} t - \lambda_0)} & 0 \\ {\sin (\omega_{{{\text{ie}}}} t - \lambda_0)} & {\cos (\omega_{{{\text{ie}}}} t - \lambda_0)} & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] $$

(2.26)

where $\omega_{{{\text{ie}}}}$ is the earth rotation angular velocity, t is the navigation time, and $\lambda_{0}$ is the angle between the X axes of the ECI system and the Earth coordinate system.

2.2.4.2 Inertial Coordinate System – Geographic Coordinate System (N-E-D)

The inertial coordinate system X_iY_iZ_i must pass through two consecutive rotations from the geographic coordinate system X_gY_gZ_g.

First,

$$ [\lambda^{\prime}]z_{{\text{i}}} = \left[ {\begin{array}{*{20}c} {\cos \lambda^{\prime}} & {sin\lambda^{\prime}} & 0 \\ { - sin\lambda^{\prime}} & {\cos \lambda^{\prime}} & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] $$

(2.27)

Second,

$$ \left[ { - \left( {\frac{{\uppi }}{2} + L} \right)} \right]_{{Y^{\prime}_{{\text{i}}} }} = \left[ {\begin{array}{*{20}c} {\cos \left( {\frac{{\uppi }}{2} + L} \right)} & 0 & {\sin \left( {\frac{{\uppi }}{2} + L} \right)} \\ 0 & 1 & 0 \\ { - \sin \left( {\frac{{\uppi }}{2} + L} \right)} & 0 & {\cos \left( {\frac{{\uppi }}{2} + L} \right)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - \sin L} & 0 & { - \cos L} \\ 0 & 1 & 0 \\ {\cos L} & 0 & { - \sin L} \\ \end{array} } \right] $$

(2.28)

Thus, the coordinate transformation matrix from the i system to the g system is

$$ \begin{gathered} \varvec{C}_{{\text{i}}}^{{\text{g}}} = \left[ { - \left( {\frac{{\uppi }}{2} + L} \right)} \right]_{{Y^{\prime}_{{\text{i}}} }} [\lambda^{\prime}]_{{Z_{{\text{i}}} }} \\ = \left[ {\begin{array}{*{20}c} { - \sin L} & 0 & { - \cos L} \\ 0 & 1 & 0 \\ {\cos L} & 0 & { - \sin L} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\cos \lambda^{\prime}} & {\sin \lambda^{\prime}} & 0 \\ { - \sin \lambda^{\prime}} & {\cos \lambda^{\prime}} & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] \\ = \left[ {\begin{array}{*{20}c} { - \cos \lambda^{\prime}\sin L} & { - \sin \lambda^{\prime}\sin L} & { - \cos L} \\ { - \sin \lambda^{\prime}} & {\cos \lambda^{\prime}} & 0 \\ {\cos \lambda^{\prime}\cos L} & {\sin \lambda^{\prime}\cos L} & { - \sin L} \\ \end{array} } \right] \\ \end{gathered} $$

(2.29)

where L is geographical latitude and λ′ = L − L₀ + ω_iet is geographical longitude.

Otherwise, the transformation matrix from the g system to the ECI system is

$$ \varvec{C}_{{\text{g}}}^{{\text{i}}} = \left[ {\begin{array}{*{20}c} { - \cos \lambda^{\prime}\sin L} & { - \sin \lambda^{\prime}} & {\cos \lambda^{\prime}\cos L} \\ { - \sin \lambda^{\prime}\sin L} & {\cos \lambda^{\prime}} & {\sin \lambda^{\prime}\cos L} \\ { - \cos L} & 0 & { - \sin L} \\ \end{array} } \right] $$

(2.30)

2.2.4.3 Earth Coordinate System—Geographic Coordinate System (N-E-D)

From the ECI system to the g system, it must go through two consecutive plane rotations, and we can obtain the “north, east and down” geographic coordinate system.

First,

$$ [\lambda ]_{{Z_{{\text{c}}} }} = \left[ {\begin{array}{*{20}c} {\cos \lambda } & {\sin \lambda } & 0 \\ { - \sin \lambda } & {\cos \lambda } & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] $$

(2.31)

Second,

$$ \begin{gathered} \left[ { - \left( {\frac{{\uppi }}{2} + L} \right)} \right]_{{Y^{\prime}_{{\text{e}}} }} = \left[ {\begin{array}{*{20}c} {\cos \left( {\frac{{\uppi }}{2} + L} \right)} & 0 & {\sin \left( {\frac{{\uppi }}{2} + L} \right)} \\ 0 & 1 & 0 \\ { - \sin \left( {\frac{{\uppi }}{2} + L} \right)} & 0 & {\cos \left( {\frac{{\uppi }}{2} + L} \right)} \\ \end{array} } \right] \\ = \left[ {\begin{array}{*{20}c} { - \sin L} & 0 & { - \cos L} \\ 0 & 1 & 0 \\ {\cos L} & 0 & { - \sin L} \\ \end{array} } \right] \\ \end{gathered} $$

(2.32)

Thus, the coordinate transformation matrix from the ECI system to the g system is

$$ \begin{gathered} \varvec{C}_{{\text{e}}}^{{\text{g}}} = \left[ {\begin{array}{*{20}c} { - \sin L} & 0 & { - \cos L} \\ 0 & 1 & 0 \\ {\cos L} & 0 & { - \sin L} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\cos \lambda } & {\sin \lambda } & 0 \\ { - \sin \lambda } & {\cos \lambda } & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] \\ = \left[ {\begin{array}{*{20}c} { - \cos \lambda \sin L} & { - \sin \lambda \sin L} & { - \cos L} \\ { - \sin \lambda } & {\cos \lambda } & 0 \\ {\cos \lambda \cos L} & {\sin \lambda \cos L} & { - \sin L} \\ \end{array} } \right] \\ \end{gathered} $$

(2.33)

where λ = L − L₀ = λ′ − ω_iet is the difference between the local and the origin geographical longitude (when L₀ = 0, λ is the geographical longitude).

Otherwise, the coordinate transformation matrix from the g system to the ECI system is

$$ {\varvec{C}}_{\text{g}}^{\text{e}} = \left[ {\begin{array}{*{20}c} { - \cos \lambda \sin L} & { - \sin \lambda } & {\cos \lambda \cos L} \\ { - \sin \lambda \sin L} & {\cos \lambda } & {\sin \lambda {\text{co}} {\text{s}} L} \\ { - \cos L} & 0 & { - \sin L} \\ \end{array} } \right] $$

(2.34)

2.2.4.4 The Geographic Coordinate System—The Body Coordinate System

The attitude angles are the three angles between the body coordinate system and the geographic coordinate system. Their definitions are defined as follows.

• (1) Direction angle ($\psi$): the angle between the longitudinal axis of body X_b and the north axis (N) in the horizontal plane, and clockwise is positive;

• (2) pitch angle ($\theta$): the angle between the longitudinal axis X_b of the body and the horizontal plane in the vertical plane, and elevation is positive;

• (3) Rolling Angle ($\phi$): the angle between the longitudinal axis of the body Y_b and the horizontal plane in the cross section, and the left lift is positive.

Note: According to the attitude angle definition, a geometric diagram between the b system X_bY_bZ_b and the g system (N-E-D) is drawn (Fig. 2.15). The transformation process from the g system (N-E-D) to the b system (F-R-D, “front, right and down”) is introduced. According to the geometry relation of the Euler angle, the g system can obtain the b system by successively rotating with heading angle ($\psi$), pitch angle (θ) and roll angle ($\phi$).

Fig. 2.15

Definition of attitude

The first rotation:

$$ [\psi ]_{D} = \left[ {\begin{array}{*{20}c} {\cos \psi } & {sin\psi } & 0 \\ { - sin\psi } & {\cos \psi } & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] $$

(2.35)

The second rotation:

$$ [\theta ]_{Yb1} = \left[ {\begin{array}{*{20}c} {\cos \theta } & 0 & { - sin\theta } \\ 0 & 1 & 0 \\ {sin\theta } & 0 & {\cos \theta } \\ \end{array} } \right] $$

(2.36)

The third rotation:

$$ [\phi ]_{Xb2} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & {\cos \phi } & {sin\phi } \\ 0 & { - \sin \phi } & {\cos \phi } \\ \end{array} } \right] $$

(2.37)

Therefore, the coordinate transformation matrix from the g system to the b system is

$$ \begin{gathered} {\varvec{C}}_{\text{g}}^{\text{b}} = [\phi ]_{Xb2} [\theta ]_{Yb1} [\psi ]_{\text{D}} \\ = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & {\cos \phi } & {\sin \phi } \\ 0 & { - \sin \phi } & {\cos \phi } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\cos \theta } & 0 & { - \sin \theta } \\ 0 & 1 & 0 \\ {\sin \theta } & 0 & {\cos \theta } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\cos \psi } & {\sin \psi } & 0 \\ { - \sin \psi } & {\cos \psi } & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] \\ = \left[ {\begin{array}{*{20}c} {\cos \psi \cos \theta } & {\sin \psi \cos \theta } & { - \sin \theta } \\ { - \sin \psi \cos \phi + \cos \psi \sin \theta \sin \phi } & {\cos \psi \cos \phi + \sin \psi \sin \theta \sin \phi } & {cos\theta \sin \phi } \\ {\sin \psi \sin \phi + \cos \psi \sin \theta \cos \phi } & { - \cos \psi \sin \phi + \sin \psi \sin \theta \cos \phi } & {cos\theta \cos \phi } \\ \end{array} } \right] \\ \end{gathered} $$

(2.38)

Otherwise, the coordinate transformation matrix from the b system to the g system is

$$ {\varvec{C}}_{\text{b}}^{\text{g}} = \left[ {\begin{array}{*{20}c} {\cos \psi \cos \theta } & { - \sin \psi \cos \phi + \cos \psi \sin \theta \sin \phi } & {sin\psi sin\phi + cos\psi \sin \theta \cos \phi } \\ {\sin \psi \cos \theta } & {\cos \psi \cos \phi + \sin \psi \sin \theta \sin \phi } & { - cos\psi sin\phi + \sin \psi sin\theta cos\phi } \\ { - \sin \theta } & {\cos \theta \sin \phi } & {cos\theta \cos \phi } \\ \end{array} } \right] $$

(2.39)

The Euler angle can be calculated based on the elements in the directional cosine matrix ${\varvec{C}}_g^b$:

$$ \tan \phi = \frac{{C_{23} }}{{C_{33} }} = \frac{\sin \phi \cos \theta }{{\cos \phi \cos \theta }} = \frac{\sin \phi }{{\cos \phi }},\;\;\;\;\phi = \tan^{ - 1} \left( {\frac{{C_{23} }}{{C_{33} }}} \right) $$

(2.40)

$$ \tan \psi = \frac{{C_{12} }}{{C_{11} }} = \frac{\sin \psi \cos \theta }{{\cos \psi \cos \theta }} = \frac{\sin \psi }{{\cos \psi }},\;\;\;\;\psi = \tan^{ - 1} \left( {\frac{{C_{12} }}{{C_{11} }}} \right) $$

(2.41)

$$ - \tan \theta = \frac{{C_{13} }}{{\sqrt {1 - C_{13}^{2} } }} = \frac{ - \sin \theta }{{\sqrt {1 - \sin^{2} \theta } }} = \frac{ - \sin \theta }{{\cos \theta }},\;\;\;\;\theta = \tan^{ - 1} \left( {\frac{{ - C_{13} }}{{\sqrt {1 - C_{13}^{2} } }}} \right) $$

(2.42)

The attitude information of the body can be calculated from the coordinate transformation matrix ${\varvec{C}}_b^g$.

The transformation relationship between the various coordinate systems is shown in Fig. 2.16.

Fig. 2.16

Transform relationships between the coordinate systems in the navigation system

2.3 Geodetic Coordinate System

2.3.1 Mathematical Description of the Earth’s Shape

At present, humans mainly focus on navigation near the surface of the earth. How can the Earth be described as accurately as possible in the coordinate system?

The earth is an approximate sphere with a complex shape, not a standard sphere. There are mountains, deep valleys and plains on land and reefs and trenches in the ocean. It is a very complex and irregular surface. Therefore, the natural surface of the earth is not a mathematical surface and cannot be directly calculated [3].

For the convenience of scientific research, the irregular natural form of the earth can be replaced by a form that can be expressed mathematically. Assuming that the sea water in the ocean is in a completely static and balanced state, i.e., Without the influence of ocean currents, tides, wind and waves, the calm sea surface is called a geoid. The geoid is a level that corresponds to the hypothetical and fully equilibrium sea level and is perpendicular to the plumb lines everywhere. If it extends to the mainland to form a continuous level, it will extend to the entire surface of the earth, which is called the geoid ellipsoid (Figs. 2.17 and 2.18).

Fig. 2.17

Geoid diagram

Fig. 2.18

Geoid ellipsoid

2.3.1.1 Earth Sphere

In general, applications, the shape of the earth is regarded as a sphere with radius R, that is, the first approximation of the earth, while the ellipsoid is regarded as its second approximation. The average radius of the Earth's sphere is R ≈ 6371.02 ± 0.05 km, and the earth rotation angular velocity is ω  ≈ 7.29 × 10^–5 rad/s.

2.3.1.2 Earth Ellipsoid

In more accurate navigational calculations, it is necessary to assume the earth as a slightly smooth ellipsoid (Fig. 2.19). The ellipsoid is composed of ellipses ${{P_{N} QP_{s} Q^{\prime}}}$ rotating around its minor semi axis ${{P_{N} P_{s} }}$.

Fig. 2.19

A schematic diagram of the earth’s ellipsoid

The important parameters of the Earth's ellipsoid are the major semi axis a, minor semi axis b, oblate c and eccentricity e. The interrelations between them are:

$$ c = \frac{a - b}{a};e = \frac{{\sqrt {a^{2} - b^{2} } }}{a}; $$

(2.43)

In different historical periods, the measured results are different, so the parameters of the ellipsoid solid are different. Table 2.1 lists the main parameters of various ellipsoids once commonly used in the world.

Table 2.1 Earth ellipsoid parameters

2.3.2 Common Geodetic Coordinate System

The geodetic coordinate system uses longitude, latitude and height to describe spatial position $(B,L,H)$. The definition of a geodetic coordinate system includes the origin of the coordinate system, the direction and scale of the three coordinate axes, and four basic parameters of the Earth’s ellipsoid (the major semi axis $a$, the second-order spherical harmonic coefficient of the Earth's gravity field $J_{2}$, the gravitational constant $GM$ and the angular velocity of the Earth's rotation $\omega$). Due to historical and technological reasons, China has established and used many different geodetic coordinate systems in different periods, and geodetic coordinate systems have also changed from the reference ellipsoid centric coordinate system to the geocentric coordinate system. The geodetic coordinate system commonly used in China is briefly introduced here.

2.3.2.1 1980 Xi’an Coordinate System

In 1980, a reference-ellipsoid-centric coordinate system was established through the adjustment of the National celestial geodetic network, also known as the Xi'an coordinate system. The origin of the Xi'an coordinate system was located in Yongle Town, Jinghe County, Shaanxi Province.

The four geometric and physical parameters of the ellipsoid parameters adopted in the coordinate system were recommended by the IAG in 1975. The minor semi axis of the ellipsoid is parallel to the rotation axis of the earth, and the starting meridian plane is parallel to the Greenwich mean astronomical meridian plane. The geoid of the ellipsoid is in good agreement in China.

The coordinate system uses four basic ellipsoidal parameters, including both geometric and physical parameters. The values are recommended by the 16th Congress of IUGG 1975.

Major semi axis	$a = 6378140\,{\text{m}}$
Gravitational constant	$GM = {3}{\text{.986005}} \times {10}^{{{14}}} \,{\text{m}}^{{3}}/{\text{s}}^{2}$
Dynamic shape factor	$J_{2} = 1.08263 \times 10^{ - 3}$
Earth rotation angular velocity	$\omega = 7.292115 \times 10^{ - 5} \,{\text{rad}}/\text{s}$

The coefficient of the second-order main sphere function of the J₂ earth gravity field is a function of oblateness. The elevation system is based on the average sea level of China's Yellow Sea in 1956. Other coordinate system constants can be derived from the above four basic constants.

Compared with the Beijing coordinate system in 1954, the Xi'an coordinate system in 1980 has the following advantages: (1) the origin of the coordinates is located in China; and (2) the reference ellipsoid is more suitable. The minor semi axis of the ellipsoid points to the polar origin JYD1968.0. (3) The ellipsoid is in good agreement with China's geoid, with an average difference of 10m across the country. (4)The accuracy of geocentric coordinates obtained from the Xi'an coordinate system in 1980 has been improved through different types of mathematical models and their conversion parameters.

However, the following problems still exist in the Xi'an coordinate system in 1980: (1) It is still a two-dimensional coordinate system and cannot provide high-precision three-dimensional coordinates. (2) The ellipsoid used is approximately 3 m larger than that used in the satellite positioning system, which may cause errors in the surface length with the magnitude $5 \times 10^{ - 7}$. (3) The ellipsoidal minor semi axis points to the polar origin JYD1968.0, which is different from the general international terrestrial coordinate system such as ITRS or the direction of the ellipsoidal short axis such as WGS84 (BIH1984.0) used in GPS (Fig. 2.20).

Fig. 2.20

WGS84 coordinate system definition

2.3.2.2 WGS84 International Coordinate System

The full name of the WGS-84 coordinate system is World Geodatic System-84. It is the coordinate system adopted by GPS. The ephemeris parameters of GPS are based on this coordinate system. It is the most widely used global geodetic reference system in the field of navigation and surveying.

The WGS-84 coordinate system is an ECEF system. The coordinate origin is located at the center of mass of the earth, the Z axis points to the agreed polar direction of the earth defined by BIH 1984.0, the X axis points to the intersection of the starting meridian plane and the equator of BIH 1984.0, and the Y axis forms a right-handed system with the X axis and the Z axis.

The four basic ellipsoidal parameters adopted by the rotating reference ellipsoid are as follows:

Major semi axis	$a = 6378137\,{\text{m}}$
Gravitational constant	$GM = 3.986004418 \times 10^{{{14}}} \,{\text{m}}^{{3}}/{\text{s}}^{2}$
Dynamic shape factor	$J_{2} = 1.08263 \times 10^{ - 3}$
Earth rotation angular velocity	$\omega = 7.292115 \times 10^{ - 5} \,{\text{rad}}/\text{s}$

The WGS84 coordinate system was developed by NIMA (National Image and Mapping Agency) and its predecessor, the Defense Mapping Agency (DMA) of the United States Department of Defense, from the initial world geodetic coordinate system WGS60 and developed on the basis of the subsequent WGS66 and WGS72. Through the use of GPS, the implementation of the WGS84 reference frame has made great progress. The WGS84 coordinate system is realized by calculating GPS tracking stations with absolute accurate coordinates.

2.3.2.3 PZ-90 Coordinate System

The PZ-90 coordinate system was adopted by GLONASS in 1993. It belongs to the ECEF system. The coordinate reference frame of the GLONASS system is realized by a series of coordinates of tracking stations. Before 1993, GLONASS adopted the Soviet Geodetic System (SGS-85) in 1985. After 1991, the GLONASS system control center accepted the suggestion of the Russian Map Bureau, improved the longitude orientation and Z-axis origin position of SGS-85, and established the PZ-90 (Parametry Zelmy 1990; English translation: Parameters of the Earth) coordinate system. PZ-90 is based on Doppler observations, satellite laser ranging (SLR), GEO-IK altimetry satellite and GLONASS satellite radar ranging and other large amounts of data, which are calculated through the combined adjustment of the ground network and space network. Since 1993, GLONASS has changed to the PZ-90 coordinate system [4].

The coordinate system PZ-90 adopted by GLONASS is defined as follows:

• (1) The origin of the coordinates is located at the mass center of the earth;

• (2) The Z axis points to the conventional terrestrial pole recommended by the International Earth Rotation Service (IERS);

• (3) The X-axis points to the zero meridian intersection defined by the Earth's equator and the International Bureau of Time (BIH);

• (4) The Y-axis satisfies the right-hand coordinate system.

The reference ellipsoid parameters and other parameters used in the PZ-90 coordinate system are listed as follows.

Major semi axis	a = 6378136 m
Gravitational constant	GM=3.9860044 × 1014 m3/s2
Dynamic shape factor	J2 = 1.0825257 × 10–3
Earth rotation angular velocity	$\omega = 7.292115 \times 10^{ - 5} {\text{rad}}/\text{s}$

Although PZ-90 is a geocentric geostationary coordinate system and its definition is the same as that of the International Terrestrial Reference Frame (ITRF), there are some differences between the defined coordinate system and the actual coordinate system due to the inevitable positioning and measurement errors of the tracking station. In fact, PZ-90 differs from WGS 84 and ITRF in origin, orientation and scale of coordinates. The difference in coordinates between PZ-90 and WGS-84 on the Earth's surface can reach 20 mm.

2.3.2.4 CGCS2000 China Geodetic Coordinate System

Since July 1, 2008, China has launched the National Geodetic Coordination System (CGCS 2000) and used 10 years to complete the conversion from the current geodetic coordinate system. The CGCS 2000 national geodetic coordinate system is adopted in the Beidou navigation satellite system in China.

The origin of the CGCS 2000 national geodetic coordinate system is the mass center of the Earth, including the ocean and atmosphere; the Z axis of the coordinate system points from the origin to the Earth reference pole of epoch 2000.0, and the direction of the epoch is calculated from the initial direction of epoch 1984.0 given by the International Bureau of Time. The directional time evolution ensures that no residual global rotation occurs relative to the crust, and the X axis points from the origin to the intersection between the reference meridian and the equatorial surface of the Earth (epoch 2000.0). The Y axis, Z axis and X axis constitute a right-hand orthogonal coordinate system, using the scale in the sense of GR.

The ellipsoid parameters adopted in CGCS2000 are as follows:

Major semi axis	a = 6378137 m
Gravitational constant	GM = 3.986004418 × 1014 m3 /s2
Dynamic shape factor	J2 = 1.082629832258 × 10–3
Earth rotation angular velocity	$\omega = 7.292115 \times 10^{ - 5} {\text{rad}}/\text{s}$
Oblate	f = 1/298.257222101

The CGCS2000 national geodetic coordinate system is basically consistent with the International Terrestrial Reference Coordination System (ITRF). ITRFis more accurate than WGS84. In addition, the GTRF is the geodetic coordinate system used by the GTRF, which is also fully used for reference in the ITRF.

2.3.2.5 Comparison Between Geocentric Coordinate System and Reference-Ellipsoid-Centric Coordinate System

WGS-84, PZ-90, CGCS2000, GTRF and ITRF are all geocentric coordinate systems, which are quite different from the 1980 Xi'an coordinate system in China. The differences between them are mainly reflected in the following aspects:

• (1) Different ellipsoidal positioning methods. The reference-ellipsoid-centric coordinate system is used to study the local spherical shape. Under the principle of minimizing the correction of the land survey data to the ellipsoid, the ellipsoid coordinate system, which is most consistent with the selected local geoid, is not conducive to the study of global shape and plate motion and is unable to establish a unified global geodetic coordinate system. WGS-84, PZ-90 and CGCS2000 are geocentric coordinate systems and the ellipsoid positioning is the closest to the global geoid.

• (2) Implementation technology is different. The reference-ellipsoid-centric coordinate system adopts traditional geodetic means, i.e., measuring the distance and direction between landmarks. The position of each point relative to the starting point is obtained by the adjustment method, and the coordinates of each point under the paracentric system are determined. The WGS-84, PZ-90 and CGCS 2000 frameworks are used to obtain the geocentric coordinates of stations under the ITRF framework through space geodetic observation technology.

• (3) Different origins. The origin of the reference-ellipsoid-centric coordinate system is quite different from the mass center of the earth, and the origin of the geocentric coordinate system is located in the mass center of the earth.

• (4) Different accuracies. Due to the limitation of objective conditions at that time, the reference-ellipsoid-centric coordinate system lacks high-precision external control, and its long-distance accuracy is low. Today, the reference-ellipsoid-centric coordinate system has difficulty meeting the needs of users. The precision of CGCS 2000 is 10 times higher than that of the reference-ellipsoid-centric coordinate system, and the relative precision can reach 10^–7~10^–8.

In other words, the geocentric coordinate system is conducive to the maintenance and rapid updating of the coordinate system by modern space technology, the coordinate determination of high-precision geodetic control stations, and the improvement of mapping efficiency. It can better clarify various geographic and physical phenomena on the earth, especially the movement of space objects. Thus, the geocentric coordinate system has been the general trend.

Appendices

Appendix 2.1: Questions

• 1. How do the rectangular coordinate system, spherical coordinate system and cylindrical coordinate system apply in navigation? What coordinate system is the latitude and longitude information and Mercator projection commonly used in navigation?

• 2. How can we understand the inertial coordinate system? Why are there many different definitions of inertial coordinate systems in navigation? Among the sun-center inertial coordinate system, earth-center inertial coordinate system, geographic coordinate system and body coordinate system, what are the coordinate systems not affected by the Earth's rotation?

• 3. Please list the definitions, English terms and abbreviations of the various coordinate systems introduced in the textbook.

• 4. What are the common coordinate transformation methods in navigation? What are the main mathematical methods by which to represent the angular relationship between coordinate systems in navigation?

• 5. What are the first and second approximations to the mathematical description of the earth shape? What is the geoid? How can the shape of the real earth surface be accurately understood?

• 6. What are the main elements of the definition of the geodetic coordinate system? The differences in the parameters in Xi'an 1980, WGS84, PZ 90, CGCS2000 and other geodetic coordinates are compared.

• 7. What are the main differences between geocentric coordinate systems and paracentric coordinate systems?

• 8. How can we understand the great significance of the concept of introducing the coordinate system in the development of modern science? Is there an ultimate inertial coordinate system?

Appendix 2.2: The Significance of Describing the World Based on the Coordinate System

Different coordinate systems show different human concerns, which to some extent reflects the relativity and generality of people’s cognitive process. A large number of coordinate systems produce the abstraction of the objective world with a particular way of professional thinking. The correlation between different coordinate systems constitutes a dynamic and complex world outlined in the “description” of the coordinate frame. All kinds of complex carrier movement relationships are reflected in the changing and complex linear and angular movements between different coordinate systems.

A deep understanding of coordinate systems can expand the understanding of navigation and of the key foundation of modern physics. In fact, describing the world in the space–time frame of the coordinate systems is a major advance in the development of modern science. In particular, the introduction of the mathematical analysis method based on coordinate systems by Descartes has transformed the ancient eastern concepts, such as “liuhe (“六合”, six directions of the world)” and “shifang (“十方”, ten directions of the world)”, into an abstract but accurate space–time framework, which changed the human understanding of the world and laid the foundation of modern scientific research paradigm. Knowing this can help readers to deepen their thinking and understanding of scientific thinking.

In addition, a profound and fundamental question of physics is whether there is an ultimate inertial coordinate system in the objective world. To answer this question involves the Newtonian system of physics and the Einstein system of physics. Seeking answers to this problem encourages readers to fully understand the inertial coordinate system, and more importantly, to expand their thinking dimension with the rational thinking in the space–time frame, and to further address the “ontological” problem.