This article continues from the [Part 1] Reference Frames in Gyroscope Motion: Global Definitions
1. Body Coordinate System / Body Frame
The Body coordinate system is represented as \(Ox_by_bz_b\) (referred to as the \(b\) system). The Body Coordinate System (or Body Frame) is rigidly attached to the Body, as shown in the figure below.
Its origin coincides with the aircraft’s center of gravity (CG). The \(x_b\) axis aligns with the lateral axis (to the right), the \(y_b\) axis aligns with the longitudinal axis (forward), and the \(z_b\) axis aligns with the vertical axis (upward). The three coordinate axes form a right-hand Cartesian coordinate system. Among them, the \(Oy_bz_b\) plane is the longitudinal symmetry plane of the aircraft. The reference coordinate system is attached to the aircraft and defined within the aircraft carrying the navigation system. This coordinate system is fixed on the aircraft and moves along with the movement of the aircraft at all times. For strapdown inertial navigation systems, the three measurement axes of the inertial sensor of the navigation system are usually consistent with the three coordinate axes of the reference coordinate system. The positional relationship between the aircraft coordinate system and the horizon coordinate system is as shown in the figure below.
The angular positions of an aircraft relative to the ground plane, namely the heading angle, pitch angle and roll angle, can be represented by the angular positions of the aircraft coordinate system relative to the geographic coordinate system. The relative positions of the Body coordinate system and the geographic coordinate system are shown in the following figure, and the yaw angle \(\boldsymbol{\psi}\), pitch angle \(\boldsymbol{\theta}\) and roll angle \(\boldsymbol{\gamma}\) are defined.
Yaw angle \(\boldsymbol{\psi}\) : The angle formed by the projection of the longitudinal axis \(y_b\) onto the horizontal plane and the geographic coordinate system \(y_g\). The rotation of \(y_g\) in the negative direction of \(z_g\) is considered positive.
Pitch angle \(\boldsymbol{\theta}\) : The angle formed by the longitudinal axis of the aircraft \(y_b\) and the horizontal plane. It is defined as positive when rotated counter-clockwise from the horizontal plane upwards to the \(y_b\) axis.
Roll Angle \(\boldsymbol{\gamma}\) : The angle between the body’s vertical axis and the vertical plane containing the longitudinal axis. It is defined as positive when rolling to the right from the vertical plane, as viewed from the tail. It can be considered that the aricraft coordinate system \(Ox_by_bz_b\) is obtained by rotating the geographic coordinate system \(Ox_gy_gz_g\) three times.
Assume that the aircraft is flying horizontally at a heading angle \(\boldsymbol{\psi}\). At this time, the aircraft coordinate system and the geographic coordinate system only differ by this heading angle \(\boldsymbol{\psi}\). When the body coordinate system rotates in the positive direction of the \(x_h\) axis at an angular velocity \( \boldsymbol{\dot{\theta}} \) by a pitch angle of \(\boldsymbol{\theta}\) to reach a new position \( \boldsymbol{O x_h’ y_h’ z_h’} \), and then rotates in the positive direction of the \( \boldsymbol{y_h’} \) axis at an angular velocity \(\boldsymbol{\dot{\gamma}}\) by a roll angle of \(\boldsymbol{\gamma}\) to reach position \(Ox_by_bz_b\), there is a difference of a pitch angle of \(\boldsymbol{\theta}\) and a roll angle of \(\boldsymbol{\gamma}\) between the body coordinate system \( \boldsymbol{b} \) and the horizontal coordinate system \(h\). The pitch angle of \(\boldsymbol{\theta}\) and the roll angle of \(\boldsymbol{\gamma}\) are called the attitude angles of the aircraft.
From this, it can be seen that if an artificial geographic coordinate system is established using a gyroscope on the aircraft and compared with the body coordinate system, the aircraft’s heading angle, pitch angle and roll angle can be measured. Similarly, if a horizontal coordinate system is established using a gyroscope on the aircraft and compared with the horizontal coordinate system, the aircraft’s pitch angle and roll angle can be measured. When the aircraft is in level flight, the axes of the aircraft coordinate system and the horizontal coordinate system are coincident.
The navigation coordinate system is represented as \(Ox_ny_nz_n\) (referred to as the \(n\) system). The coordinate system used by the inertial navigation system when solving navigation parameters is called the navigation coordinate system. For a platform inertial navigation system, the ideal platform coordinate system is the navigation coordinate system. For a strapdown inertial navigation system, since its navigation parameter is not within the body coordinate system, the signals of the accelerometer must be decomposed in a coordinate system that is more convenient for calculating navigation parameters, and then the navigation calculation is carried out, and this coordinate system is the navigation coordinate system.
3. Platform Coordinate System / Platform Frame
The platform coordinate system is represented as \(Ox_py_pz_p\) (referred to as the \(p\) system). In the platform inertial navigation system, This coordinate system describes the frame to which the actual platform (physical platform) points. In the strapdown inertial navigation system, since there is no actual physical platform, this coordinate system describes the mathematical platform. The platform coordinate system is an important coordinate system for navigation calculations and attitude reference. In an ideal situation, assuming the platform has no errors, such a platform coordinate system is called the ideal platform coordinate system, and it can be a geographic coordinate system or some other coordinate systems.
4. Computational Coordinate System (or Computer Frame)
The computational coordinate system is represented as \(Ox_cy_cz_c\) (referred to as the \(c\) system) and is a virtual coordinate system for facilitating navigation computation. Usually, it is a geographic coordinate system established based on the latitude \(\boldsymbol{\phi}\) and longitude \(\boldsymbol{\lambda}\) as the origin. It is different from the geographic coordinate system \(Ox_gy_gz_g\) established based on the actual position of the vehicle. The angle between these two coordinate systems is the positioning error of the inertial navigation system, and it is commonly used when describing inertial navigation errors and deriving inertial navigation error equations.
The angle between the platform coordinate system \(Ox_py_pz_p\) and the geographic coordinate system \(Ox_gy_gz_g\) is called the platform’s attitude angle \(\boldsymbol{\phi}\), while the angle between the platform coordinate system \(Ox_py_pz_p\) and the computational coordinate system \(Ox_cy_cz_c\) is called the platform drift angle \(\boldsymbol{\psi}\).