The position of an object in space can only be determined relative to another object. Thus, the latter object constitutes the reference frame when describing the motion of the former object. When the position of an object with respect to the reference frame changes, we say that the object has moved; conversely, if an object has no change in position with respect to the reference frame, we say that the object is stationary. The concepts of motion or rest are relative to the position of one object with respect to another. Therefore, the motion and rest of an object are only relative in nature. When describing the motion of an object, it is necessary to specify the reference frame being used, so that the described motion can have correct meaning. The reference frame is usually represented by a Cartesian coordinate system and is called the reference coordinate system or simply referred to as the reference frame. When studying gyroscopes or the motion of moving vehicles, a reference frame is also required.
1. Inertial Coordinate System / Inertial Reference Frame
The inertial coordinate system is represented as \(O x_i y_i z_i y_i\) (referred to as the \(i\) system). When studying the motion of an object, Newton’s laws of motion and various theorems derived from them are generally applied. Usually, the reference coordinate system that makes Newton’s laws of motion valid is called an inertial coordinate system or simply an inertial system.
The so-called inertial coordinate system refers to a coordinate system where the origin is a stationary point or a point in uniform linear motion, and there is no rotation. However, the inertial coordinate system defined in this way actually does not exist because all matter in the universe, including space itself, is in absolute motion, and rest is only relative. The Earth rotates around its own axis and together with other planets orbits the Sun. The entire solar system also rotates around the center of the Milky Way, and the entire Milky Way itself is also rotating. It is impossible to find an object in uniform linear motion, so the inertial space does not actually exist. Therefore, the choice of the inertial coordinate system is approximate and depends on the required or achieved accuracy. The so-called “inertial space” here is merely an artificial definition, and it is used as a reference benchmark to study the motion of gyroscopes.
Practice has shown that when studying the motion of general objects on Earth, using a coordinate system connected to the Earth is sufficiently accurate. Although this coordinate system has a rotational angular velocity due to the Earth’s rotation and its revolution around the Sun, and its origin also has a centripetal acceleration, these factors do not affect the accuracy.
However, when studying the movement of gyroscopes on Earth, the influence of the Earth’s rotation must be taken into account. At this time, a space associated with the solar system should be selected as the inertial space. Although this space is still rotating, it does not affect the accuracy. The space formed by the three coordinate axes of the inertial coordinate system actually represents the inertial space. The ones we commonly use are the heliocentric inertial coordinate system and the geocentric inertial coordinate system. For aviation navigation, the geocentric inertial coordinate system is sufficient and accurate enough. Without special indication, the inertial coordinate system refers to the geocentric inertial coordinate system. The geocentric inertial coordinate system is fixed relative to the inertial space and is independent of the Earth’s rotation, not participating in the Earth’s rotation. That is, the spatial directions of the coordinate axes do not rotate with the Earth’s rotation. The coordinate origin of the geocentric inertial coordinate system is the center of the Earth, the \(x_i\) axis and the \(y_i\) axis are within the Earth’s equatorial plane, \(x_i\) points to the vernal equinox (the vernal equinox is the starting point for astronomical \(z_i\) axis points to the Earth’s rotation axis (the Earth’s polar axis), and \(y_i\) axis is determined according to the right-hand rule.
In order to facilitate the study of the gyroscope’s movement, the origin of the inertial coordinate system is usually set at the support center of the gyroscope, and the three coordinate axes point towards specific stars. The origin of this inertial coordinate system moves along with the gyroscope, but it is still a coordinate system that does not rotate relative to the stars.
2. Earth Coordinate System / Earth-Centered Earth-Fixed Reference Frame
The Earth coordinate system is represented as \(O x_e y_e z_e\) (referred to as the e-system). The Earth coordinate system is also known as the geocentric fixed coordinate system. It is fixed to the Earth, and the spatial directions of the coordinate axes change along with the rotation of the Earth. Its coordinate origin is located at the center of the Earth. There are various specific definitions for the currently used Earth coordinate system. The commonly used Earth coordinate system takes the Earth center as the origin, \(Ox_ey_e\) is in the equatorial plane, the \(z_e\) axis points to the North Pole, coinciding with the Earth’s rotation axis, \(x_e\) axis points to the zero longitude line (the intersection line of the equatorial plane and the prime meridian plane), and \(y_e\) axis is determined according to the right-hand rule (pointing to the 90° east longitude line), as shown in the figure below.
3. Geographic Coordinate System
The geographic coordinate system is represented as \(Ox_gy_gz_g\) (referred to as the g system). The geographic coordinate system is shown in the above figure. Its origin coincides with the center of gravity of the vehicle. The \(x_g\) axis is horizontal and points eastward, the \(y_g\) axis is horizontal and points northward, and the \(z_g\) axis is perpendicular to the local vertical line and points to the zenith. Obviously, the three coordinate axes of the geographic coordinate system form a right-handed Cartesian coordinate system in the order of east-north-sky, where the \(Ox_gy_g\) plane is the local horizontal plane, the \(Oy_gz_g\) plane is the local meridian plane, and thus the geographic coordinate system is the reference coordinate system for measuring the attitude angles and heading angles of the vehicle. Of course, the north-west-sky system or north-east-earth system is also often used as the geographic coordinate system. The geographic coordinate system marked in the above figure is the north-east-sky system. The determination of the axial directions of the geographic coordinate system is related to the usage habits, convenience of use, and whether it is located in the eastern or western hemisphere. From the perspective of facilitating navigation and calculation, the difference is not significant.
4. Horizon Coordinate System / Local Horizontal Reference Frame
The horizontal coordinate system is represented as \(Ox_hy_hz_h\) (referred to as the h system). The horizontal coordinate system is shown in the following figure.
The coordinate origin is taken at the center of gravity of the vehicle, the \(y_h\) axis is horizontal and points in the direction of heading, the \(z_h\) axis is perpendicular to the local ground and points towards the zenith, and the \(x_h\) axis is also horizontal and forms a right-handed rectangular coordinate system with the \(y_h\) and \(z_h\) axes. The \(Ox_hy_h\) plane is the local horizontal plane, and the \(Oy_hz_h\) plane is the longitudinal vertical plane of the vehicle. Therefore, when determining the attitude angles of the vehicle, using the horizontal coordinate system is more direct and convenient.
The horizontal coordinate system also moves along with the vehicle, so it is called the local horizontal coordinate system. The horizontal coordinate system also follows the rotation of the Earth. The horizontal coordinate system is related to the aircraft’s flight path, so it is also called the track coordinate system. The horizontal coordinate system \(h\) and the geographic coordinate system \(g\) differ only by a heading angle of \( \boldsymbol{\psi} \), and their relationship is as shown in the figure above. When studying the motion of the gyroscope relative to the horizontal coordinate system, the coordinate origin can be taken at the pivot point of the ring frame, and the orientations of the coordinate axes remain the same as above.