# A General Euler Angle Error Model of Strapdown Inertial Navigation Systems
## 2 The Proposed General Euler Angle Error Model
### Modeling of Relative Euler Angle Errors  
The coordinate systems and transformation relation for SINS are shown in Figure 3. The misalignment are computational platform angle errors with respect to the navigation frame in the navigation system [33]. Generally, it is a small angle from the navigation frame \(o_{n}x_{n}y_{n}z_{n}\) to the SINS computational frame \(o_{c}x_{c}y_{c}z_{c}\), and can be expressed by a variable \(\boldsymbol{\Phi}=\left[ \begin{array}{cc}\phi_{E}&\phi_{N}&\phi_{V}\end{array}  \right]^{T}\) with \(\phi_{i}\) being misalignment along the east, north, and vertical directions. The coordinate transformation matrix from the navigation frame \(o_{n}x_{n}y_{n}z_{n}\) to the computational frame \(o_{c}x_{c}y_{c}z_{c}\) is referred to as an attitude matrix, and can be written as:  
\[ \begin{array}{ll}C_{n}^{c}&=\left[\begin{array}{ccc}\cos(\phi_{N})&0&-\sin (\phi_{N})\\ 0&1&0\\ \sin(\phi_{N})&0&\cos(\phi_{N})\end{array}\right]\left[\begin{array}{ccc}1&0 &0\\ 0&\cos(\phi_{E})&\sin(\phi_{E})\\ 0&-\sin(\phi_{E})&\cos(\phi_{E})\end{array}\right]\left[\begin{array}{ccc} \cos(\phi_{V})&\sin(\phi_{V})&0\\ -\sin(\phi_{V})&\cos(\phi_{V})&0\\ 0&0&1\end{array}\right]\\ &\approx\boldsymbol{I}-\left[\begin{array}{ccc}(\phi_{N}^{c}+\phi_{V}^{c})/ 2&-(\phi_{V}+\phi_{E}\phi_{N})&\phi_{N}\\ \phi_{N}&(\phi_{E}^{c}+\phi_{V}^{c})/2&-\phi_{E}\\ -(\phi_{N}+\phi_{E}\phi_{V})&(\phi_{E}+\phi_{N}\phi_{V})&(\phi_{E}^{c}+\phi_{N }^{c})/2\end{array}\right]\\ &\approx\boldsymbol{I}-\left[\begin{array}{ccc}0&-\phi_{V}&\phi_{N}\\ \phi_{V}&0&-\phi_{E}\\ -\phi_{N}&\phi_{E}&0\end{array}\right]=\boldsymbol{I}-(\boldsymbol{\Phi}\times) \end{array}  \tag{10}\]  
where \(\phi_{E}\), \(\phi_{N}\), \(\phi_{V}\) are small angles, and \(\cos(\phi_{E})=\cos(\phi_{N})=\cos(\phi_{V})\approx 1\), \(\sin(\phi_{E})\approx\phi_{E}\), \(\sin(\phi_{N})\approx\phi_{N}\), \(\sin(\phi_{V})\approx\phi_{V}\). According to Equations (9) and (10), the body attitude matrix from the computational frame to the body frame can be calculated as:  
\[ \begin{array}{ll}C_{c}^{b}&=C_{n}^{b}C_{c}^{n}=\left[\begin{array}{ccc}T_{ 11}&T_{12}&T_{13}\\ T_{21}&T_{22}&T_{23}\\ T_{31}&T_{32}&T_{33}\end{array}\right]\left[\begin{array}{ccc}1&-\phi_{V}& \phi_{N}\\ \phi_{V}&1&-\phi_{E}\\ -\phi_{N}&\phi_{E}&1\end{array}\right]\\ &=\left[\begin{array}{ccc}\cos(\gamma^{c})\cos(\varphi^{c})-\sin(\gamma^{c}) \sin(\varphi^{c})\sin(\theta^{c})&\cos(\gamma^{c})\sin(\varphi^{c})+\sin(\gamma^ {c})\cos(\varphi^{c})\sin(\theta^{c})&-\sin(\gamma^{c})\cos(\theta^{c})\\ -\sin(\varphi^{c})\cos(\theta^{c})&\cos(\varphi^{c})\cos(\theta^{c})\\ \sin(\gamma^{c})\sin(\varphi^{c})+\cos(\gamma^{c})\sin(\varphi^{c})\sin(\theta^ {c})&\sin(\gamma^{c})\sin(\varphi^{c})-\cos(\gamma^{c})\cos(\varphi^{c})\sin( \theta^{c})&\cos(\gamma^{c})\cos(\theta^{c})\end{array}\right]\\ &=\left[\begin{array}{ccc}C_{11}&C_{12}&C_{13}\\ C_{21}&C_{22}&C_{23}\\ C_{31}&C_{32}&C_{33}\end{array}\right]\end{array}  \tag{11}\]  
where \(\varphi^{c}\), \(\theta^{c}\), and \(\gamma^{c}\) are the head, pitch, and roll with respect to the computational frame \(o_{c}x_{c}y_{c}z_{c}\), and \(C_{ij}\) is an element of the transformation matrix \(C_{c}^{b}\) with \(i=1,2,3\); \(j=1,2,3\). Since the misalignment \(\mathbf{\Phi}\) from the navigation frame \(o_{n}x_{n}y_{n}z_{n}\) to the computational frame \(o_{c}x_{c}y_{c}z_{c}\), according to Equation (11), the transformation relation among attitude matrixes can be expressed as:  
\[C_{b}^{c}=C_{n}^{c}C_{b}^{n}=[I-(\mathbf{\Phi}\times)]C_{b}^{n} \tag{12}\]  
\[\delta C_{b}^{n}=C_{b}^{c}-C_{b}^{n}=-(\mathbf{\Phi}\times)C_{b}^{n} \tag{13}\]  
\[\delta\tilde{C}_{b}^{n}=-\Big{(}\mathbf{\Phi}\times\Big{)}C_{b}^{n}-(\mathbf{\Phi} \times)C_{b}^{n}\mathbf{\omega_{nb}^{b}}\times \tag{14}\]  
where \(\delta C_{b}^{n}\) represents the error of attitude transformation matrix from the body frame \(o_{b}x_{b}y_{b}z_{b}\) to the navigation frame \(o_{n}x_{n}y_{n}z_{n}\).