# 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
\right[\begin{array}{ccc}1&0&0\\ 0&\cos(\theta)&\sin(\theta)\\ 0&-\sin(\theta)&\cos(\theta)\end{bmatrix}\right]\left[ \begin{array}{c}0\\ 0\\ \dot{\varphi}\end{array} \right]_{a}+\left[\begin{array}{c}\cos(\gamma)&0&- \sin(\gamma)\\ 0&1&0\\ \sin(\gamma)&0&\cos(\gamma)\end{bmatrix}\right]\left[ \begin{array}{c}\dot{ \theta}\\ 0\\ 0\\ 1\end{array} \right]_{1}+\left[ \begin{array}{c}0\\ \dot{\gamma}\\ 0\end{array} \right]_{2}\right\} \tag{4}\]  
Figure 1: The operational principle of SINS (strapdown inertial navigation systems). From Equation (4), the Euler angle velocities induced by the angular velocities of the body frame with respect to the navigation frame denoted in the body frame can be written as:  
\[\left[ \begin{array}{c}\dot{\varphi}\\ \dot{\theta}\\ \dot{\gamma}\end{array} \right] = \left[ \begin{array}{ccc}-\sin(\gamma)\cos(\theta)&\cos(\gamma)&0\\ \sin(\theta)&0&1\\ \cos(\gamma)\cos(\theta)&\sin(\gamma)&0\end{array} \right]^{-1}\left[ \begin{array} []{c}\omega^{b}_{bbx}\\ \omega^{b}_{nby}\\ \omega^{b}_{nbz}\end{array} \right]_{b}\] \[= \frac{1}{\cos(\theta)}\left[ \begin{array}{ccc}-\sin(\gamma)&0& \cos(\gamma)\\ \cos(\theta)\cos(\gamma)&0&\cos(\theta)\sin(\gamma)\\ \sin(\theta)\sin(\gamma)&\cos(\theta)&-\sin(\theta)\cos(\gamma)\end{array}  \right]\left[ \begin{array}{c}\omega^{b}_{nby}\\ \omega^{b}_{nbz}\end{array} \right]_{b}\] \[= \left[ \begin{array}{ccc}-\frac{\sin(\gamma)}{\cos(\theta)} \omega^{b}_{nbx}+\frac{\cos(\gamma)}{\cos(\theta)}\omega^{b}_{nbz}\\ \cos(\gamma)\omega^{b}_{nbx}+\sin(\gamma)\omega^{b}_{nbz}\\ \tan(\theta)\sin(\gamma)\omega^{b}_{nbx}+\omega^{b}_{nby}-\tan(\theta)\cos( \gamma)\omega^{b}_{nbz}\end{array} \right]\]  
According to Equation (5), the Euler angles can be calculated by introducing the integral operation:  
\[\varphi=\varphi_{0}+\int_{t0}^{t}\dot{\varphi}dt=\varphi_{0}+\int_{t0}^{t} \biggl{[}-\frac{\sin(\gamma)}{\cos(\theta)}\omega^{b}_{nbx}+\frac{\cos(\gamma )}{\cos(\theta)}\omega^{b}_{nbz}\biggr{]}dt \tag{6}\]  
\[\theta=\theta_{0}+\int_{t0}^{t}\dot{\theta}dt=\theta_{0}+\int_{t0}^{t}\left[ \cos(\gamma)\omega^{b}_{nbx}+\sin(\gamma)\omega^{b}_{nbz}\right]dt \tag{7}\]  
\[\gamma=\gamma_{0}+\int_{t0}^{t}\dot{\gamma}dt=\gamma_{0}+\int_{t0}^{t}\left[ \tan(\theta)\sin(\gamma)\omega^{b}_{nbx}+\omega^{b}_{nby}-\tan(\theta)\cos( \gamma)\omega^{b}_{nbz}\right]dt \tag{8}\]  
Figure 2: The Euler angles transformation relation for SINS. where \(\varphi_{0}\), \(\theta_{0}\), and \(\gamma_{0}\) respectively represent the initial head, pitch, and roll of the SINS. According to Equations (6)-(8), the body attitude matrix from the navigation frame to the body frame can be expressed by Euler angles \(\varphi\), \(\theta\), and \(\gamma\) in the head, pitch, and roll:  
\[ \begin{array}{ll}C_{n}^{b}&=\left[\begin{array}{ccc}\cos(\gamma)\cos( \varphi)-\sin(\gamma)\sin(\varphi)\sin(\theta)&\cos(\gamma)\sin(\varphi)+\sin( \gamma)\cos(\varphi)\sin(\theta)&-\sin(\gamma)\cos(\theta)\\ -\sin(\varphi)\cos(\theta)&\cos(\varphi)\cos(\theta)&\sin(\theta)\\ \sin(\gamma)\cos(\varphi)+\cos(\gamma)\sin(\varphi)\sin(\theta)&\sin(\gamma) \sin(\varphi)-\cos(\gamma)\cos(\varphi)\sin(\theta)&\cos(\gamma)\cos(\theta) \end{array}\right]\\ &=\left[\begin{array}{ccc}T_{11}&T_{12}&T_{13}\\ T_{21}&T_{22}&T_{23}\\ T_{31}&T_{32}&T_{33}\end{array}\right]\end{array}  \tag{9}\]  
where \(T_{ij}\) is the element of the transformation matrix \(C_{n}^{b}\) with \(i=1,2,3\); \(j=1,2,3\).