# 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
\begin{array}{ll}\delta\dot{\gamma}=\tan(\theta^{c})\left\{\left[\cos(\gamma ^{c})\omega^{b}_{nbx}+\sin(\gamma^{c})\omega^{b}_{nbz}\right]\delta\gamma+ \left[\sin(\gamma^{c})\omega^{b}_{nbx}-\cos(\gamma^{c})\omega^{b}_{nbz}( \delta\gamma)^{2}/2\right]+\right.\\ \left.\sec^{2}(\theta^{c})\delta\theta\left\{\left[\sin(\gamma^{c})\omega^{b}_ {nbx}-\cos(\gamma^{c})\omega^{b}_{nbz}\right]-\left[\cos(\gamma^{c})\omega^{b} _{nbx}+\sin(\gamma^{c})\omega^{b}_{nbz}\right]\delta\gamma-\left[\sin(\gamma^{ c})\omega^{b}_{nbx}-\cos(\gamma^{c})\omega^{b}_{nbx}\right](\delta\gamma)^{2}/2 \right\}\end{array}  \tag{36}\]  
Ignoring the term as small as second order, the differential equation of the roll angle error can be rewritten as:  
\[ \begin{array}{ll}\delta\dot{\gamma}=\tan(\theta^{c})\left[\cos(\gamma^{c}) \omega^{b}_{nbx}+\sin(\gamma^{c})\omega^{b}_{nbz}\right]\delta\gamma+\sec^{2 }(\theta^{c})\left[\sin(\gamma^{c})\omega^{b}_{nbx}-\cos(\gamma^{c})\omega^{b }_{nbz}\right]\delta\theta\end{array}  \tag{37}\]  
Assuming \(\cos(\delta\theta)\approx 1-(\delta\theta)^{2  
When ignoring the term as small as second order, the differential equation of head angle error can be rewritten as:  
\[\delta\dot{\varphi}\approx\sec(\mathcal{\theta}^{c})\Big{\{}\Big{[}\cos(\gamma^ {c})\omega^{b}_{nbz}-\sin(\gamma^{c})\omega^{b}_{nbz}\Big{]}\tan(\theta^{c}) \delta\theta-\Big{[}\cos(\gamma^{c})\omega^{b}_{nbz}+\sin(\gamma^{c})\omega^{b} _{nbz}\Big{]}\delta\gamma\Big{\}} \tag{40}\]  
According to Equations (34), (37) and (40), the differential equation of the convected Euler angle errors can be obtained as:  
\[\left[ \begin{array}{c}\delta\theta^{*}\\ \delta\gamma^{*}\\ \delta\dot{\varphi}^{*}\end{array} \right]=\left[ \begin{array}{ccc}0&-\sin( \gamma^{c})\omega^{b}_{nbz}+\cos(\gamma^{c})\omega^{b}_{nbz}&0\\ \sec^{2}(\theta^{c})\big{(}\sin(\gamma^{c})\omega^{b}_{nbz}-\cos(\gamma^{c}) \omega^{b}_{nbz}\big{)}&\tan(\theta^{c})\cos(\gamma^{c})\omega^{b}_{nbz}+\tan (\theta^{c})\sin(\gamma^{c})\omega^{b}_{nbz}&0\\ \sec^{2}(\theta^{c})sin(\theta^{c})\big{(}-\sin(\gamma^{c})\omega^{b}_{nbz}+ \cos(\gamma^{c})\omega^{b}_{nbz}\big{)}&-\sec(\theta^{c})\big{(}\cos(\gamma^{c })\omega^{b}_{nbz}+\sin(\gamma^{c})\omega^{b}_{nbz}\big{)}&0\end{array} \right] \tag{41}\]  
According to Equation (41), the convected Euler angle errors can be written as:  
\[\left[ \begin{array}{c}\delta\theta^{*}\\ \delta\gamma^{*}\\ \delta\varphi^{*}\end{array} \right]=\left[ \begin{array}{c}\delta\theta_{0}+ \int_{\eta\phi}^{t}\delta\dot{\theta}^{*}dt\\ \delta\gamma_{0}+\int_{\eta\phi}^{t}\delta\dot{\gamma}^{*}dt\\ \delta\varphi_{0}+\int_{\eta\phi}^{t}\delta\dot{\varphi}^{*}dt\end{array} \right] \tag{42}\]  
From Equations (41) and (42), some conclusions can be obtained. Firstly, the convected Euler angle errors of SINS are affected by \(\omega^{b}_{nbz}\) and \(\omega^{b}_{nbz}\), the angular velocity of the body frame with respect to the navigation frame is denoted in the body frame along the \(X\) and \(Z\) axes, not \(\omega^{b}_{nby}\) along the \(Y\) axis. Secondly, the convected Euler angle errors are affected by the initial pitch error \(\delta\theta_{0}\) and roll error \(\delta\gamma_{0}\), rather than the initial head error \(\delta\varphi_{0}\). Finally, the pitch magnitude greatly affects the propagation rules of convected Euler angle errors in the head and roll, but not pitch. Since the state is an error vector, Equation (43) is an approximate expression which ignores the term as small as second order. According to Equations (29) and (41), the differential equation of the general Euler angle errors, including relative Euler angle errors and convected Euler angle errors, can be modeled as:  
\[\left[ \begin{array}{c}\delta\theta\\ \delta\gamma\\ \delta\varphi\end{array} \right] = \left[ \begin{array}{c}\dot{\phi}^{*}\\ \dot{\varphi}^{*}_{\gamma}\\ \delta\varphi\end{array} \right]+\left[ \begin{array}{c}\dot{\varphi}^{\phi ^{0}}\\ \dot{\varphi}^{*}_{\phi}\end{array} \right]\] \[= \left[