# A General Euler Angle Error Model of Strapdown Inertial Navigation Systems
## 2 The Proposed General Euler Angle Error Model
### Attitude Solution by Euler Angle Differential Equations  
Replacing the inertial platform of GINS, the digital computational platform of SINS, the computational frame inside the navigation computer, is theoretically equivalent to the local level navigation frame when there are no navigation errors. Gyro output is used to maintain the digital computational platform, and the specific force measurements from the accelerometer triads are resolved, then the velocity and position are acquired by double integration as shown in Figure 1. However, no matter which reference frame is chosen, all quantities must be transformed to the computational frame prior to integration. How to determine the body attitude matrix is vital, because it represents the transformation of the body frame with respect to the navigation frame. The body attitude matrix can be calculated by Euler angles \(\varphi\), \(\theta\), and \(\gamma\) in turn along the head, pitch, and roll axes. These Euler angles can be resolved in real-time using the angular velocities of the body frame with respect to the navigation frame denoted in the body frame, and the angular velocity can be calculated by:  
\[\mathbf{\omega_{nb}^{b}}=\mathbf{\omega_{ib}^{b}}-C_{n}^{b}\mathbf{\omega_{in}^{n}}=\mathbf{ \omega_{ib}^{b}}-C_{n}^{b}(\mathbf{\omega_{ie}^{n}}+\mathbf{\omega_{en}^{n}}) \tag{1}\]  
where \(\mathbf{\omega_{ib}^{b}}\) is the angular velocity vector of the body frame with respect to the inertial frame denoted in the body frame, and can be directly measured by gyros, and \(\mathbf{\omega_{in}^{n}}\) is the angular velocity vector of the navigation frame with respect to the inertial frame denoted in the navigation frame \(\mathbf{\omega_{in}^{n}}=\mathbf{\omega_{ie}^{n}}+\mathbf{\omega_{en}^{n}}\). \(\mathbf{\omega_{ie}^{n}}\) is the angular velocity vector of the Earth frame with respect to the inertial frame denoted in the navigation frame. \(\mathbf{\omega_{en}^{n}}\) is the angular velocity vector of the navigation frame with respect to the Earth frame denoted in the navigation frame. They can be respectively calculated by:  
\[\mathbf{\omega_{in}^{n}}=\mathbf{\omega_{ie}^{n}}+\mathbf{\omega_{en}^{n}}=\left[ \begin{array} []{c}\omega_{iex}^{n}\\ \omega_{iey}^{n}\\ \omega_{enz}^{n}\end{array} \right]+\left[ \begin{array}{c}\omega_{enx}^{n}\\ \omega_{eny}^{n}\\ \omega_{enz}^{n}\end{array} \right]=\left[ \begin{array}{c}0\\ \omega_{ie}\cos(L)\\ \omega_{ie}\sin(L)\end{array} \right]+\left[ \begin{array}{c}-V_{N}/(R_{M}+H) \\ V_{E}/(R_{N}+H)\\ V_{E}\tan(L)/(R_{N}+H)\end{array} \right] \tag{2}\]  
As shown in Figure 2, in order to accurately derive the Euler angle error differential equations, the transformation relation between the angular velocity of the body frame with respect to the navigation frame denoted in the body frame and the Euler angle velocities in head, pitch, and roll with respect to the navigation frame, can be expressed as:  
\[\left[ \begin{array}{c}\omega_{nbx}^{b}\\ \omega_{nby}^{b}\\ \omega_{nbz}^{n}\end{array} \right]_{b}=R_{y}^{2}(\gamma)R_{x}^{1}(\theta)R_{ z}^{n}(\varphi)\left[ \begin{array}{c}0\\ 0\\ \dot{\varphi}^{n}\end{array} \right]_{n}+R_{y}^{2}(\gamma)R_{x}^{1}(\theta) \left[ \begin{array}{c}\dot{\theta}^{n}\\ 0\\ 0\end{array} \right]_{1}+R_{y}^{2}(\gamma)\left[ \begin{array}{c}0\\ \dot{\gamma}^{n}\\ 0\end{array} \right]_{2} \tag{3}\]  
where \(\omega_{nbi}^{b}\) is the angular velocity of the body frame with respect to the navigation frame denoted in the body frame along \(i\) axis, \(R_{y}^{2}(\gamma)\), \(R_{x}^{1}(\theta)\), and \(R_{z}^{n}(\varphi)\) respectively represent attitude transformation matrixes induced by Euler angles \(\gamma\), \(\theta\), and \(\varphi\) along the roll, pitch, and head. Equation (3) can be further rewritten by Euler angles:  
\[\left[ \begin{array}{c}\omega_{abx}^{b}\\ \omega_{abx}^{b}\\ \omega_{abx}^{b}\end{array} \right]_{b}=\left\{ \begin{bmatrix}\cos(\gamma)&0&- \sin(\gamma)\\ 0&1&0\\ \sin(\gamma)&0&\cos(\gamma)\end{bmatrix}