# 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 differential equation of attitude transformation matrix for SINS can be given by:  
\[\dot{C}_{b}^{n}=C_{b}^{n}(\mathbf{\omega_{nb}^{b}}\times)=C_{b}^{n}\Big{[}\Big{(} \mathbf{\omega_{ib}^{b}}-C_{n}^{b}\mathbf{\omega_{in}^{n}}\Big{)}\times\Big{]}=C_{b}^ {n}\Big{[}\Big{(}\mathbf{\omega_{ib}^{b}}-C_{n}^{b}(\mathbf{\omega_{ie}^{n}}+\mathbf{ \omega_{en}^{n}})\Big{)}\times\Big{]} \tag{15}\]  
The SINS output includes not only sensing the angular velocity \(\mathbf{\omega_{ib}^{b}}\), but also harmful measurement error \(\delta\mathbf{\omega_{ib}^{b}}\) and random drift of gyros \(\varepsilon^{b}\), and can be expressed as:  
\[\mathbf{\hat{\omega}_{ib}^{b}}=\mathbf{\omega_{ib}^{b}}+\mathbf{\delta\omega_{ib}^{b}}+\bm {\varepsilon^{b}} \tag{16}\]  
The computational angular velocity of the computational frame with respect to the inertial frame denoted in the computational frame has errors, and can be expressed as:  
\[\mathbf{\omega_{ic}^{c}}=\mathbf{\omega_{in}^{n}}+\delta\mathbf{\omega_{in}^{n}}=\mathbf{ \omega_{ie}^{n}}+\mathbf{\omega_{en}^{n}}+\delta\mathbf{\omega_{ie}^{n}}+\delta\mathbf{ \omega_{en}^{n}} \tag{17}\]  
where \(\delta\mathbf{\omega_{in}^{n}}\) is the angular velocity error of the navigation frame with respect to the inertial frame denoted in the navigation frame \(\delta\mathbf{\omega_{in}^{n}}=\delta\mathbf{\omega_{ie}^{n}}+\delta\mathbf{\omega_{en}^{n}}\), and \(\delta\mathbf{\omega_{ie}^{n}}\) is the angular velocity error of the Earth frame with respect to the inertial frame denoted in the navigation frame. \(\delta\mathbf{\omega_{en}^{n}}\) is the angular velocity error of the navigation frame with respect to the earth frame denoted in the navigation frame. According to Equation (15), the differential equation of attitude transformation matrix from the body frame \(o_{b}x_{b}y_{b}z_{b}\) to the computational frame \(o_{c}x_{c}y_{c}z_{c}\) can be given by:  
\[\dot{C}_{b}^{c}=C_{b}^{c}\Big{[}\Big{(}\mathbf{\hat{\omega}_{ib}^{b}}-C_{c}^{b}\bm {\omega_{ie}^{c}}\Big{)}\times\Big{]} \tag{18}\]  
Figure 3: The coordinate systems and transformation relation for SINS. According to Equation (13), the differential equation of attitude transformation matrix \(\dot{C}_{b}^{c}\) can also be given by:  
\[\dot{C}_{b}^{c}=\dot{C}_{b}^{n}+\delta\dot{C}_{b}^{n} \tag{19}\]  
Substituting Equations (16), (17) and (19) into Equation (18), the differential equation of the attitude transformation matrix can be rewritten as:  
\[\dot{C}_{b}^{n}+\delta\dot{C}_{b}^{n}=[I-(\mathbf{\Phi}\times)]C_{b}^{n}\Big{\{} \Big{[}\mathbf{\omega}_{\mathbf{ib}}^{\mathbf{b}}+\mathbf{\delta}\mathbf{\omega}_{\mathbf{ib}}^{\mathbf{b}} +\mathbf{\varepsilon}^{\mathbf{b}}-C_{n}^{b}(I+\mathbf{\Phi}\times)\big{(}\mathbf{\omega}_\mathbf{{in}}^{\mathbf{n}}+\mathbf{\delta}\mathbf{\omega}_{\mathbf{in}}^{\mathbf{n}}\big{)}\Big{]}\times \Big{\}} \tag{20}\]  
Substituting Equation (15) into Equation (20), supposing that the attitude error is slight and ignoring the term as small as the second order and the high-order terms of error products, it is easy to obtain the differential equation of attitude transformation matrix errors:  
\[\delta C_{b}^{n}=C_{b}^{c}-C_{b}^{n}=-(\mathbf{\Phi}\times)C_{b}^{n} \tag{13}\]0  
Substituting Equation (14) into Equation (21), the differential equation of attitude transformation matrix errors can be rewritten as:  
\[\delta C_{b}^{n}=C_{b}^{c}-C_{b}^{n}=-(\mathbf{\Phi}\times)C_{b}^{n} \tag{13}\]1  
The differential equation of misalignment of SINS can be further rewritten as:  
\[\delta C_{b}^{n}=C_{b}^{c}-C_{b}^{n}=-(\mathbf{\Phi}\times)C_{b}^{n} \tag{13}\]2  
\[\delta C_{b}^{n}=C_{b}^{c}-C_{b}^{n}=-(\mathbf{\Phi}\times)C_{b}^{n} \tag{13}\]3  
The differential equation of misalignment errors can be expanded as:  
\[\left[ \begin{array}{c}\dot{\mathbf{\phi}}_{E}\\ \dot{\mathbf{\phi}}_{N}\\ \dot{\mathbf{\phi}}_{V}\end{array} \right] =\] \[+\left[