# 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}{c}-\frac{\delta V_{N}}{(R_{M}+H)}+\frac{V_ {N}}{(R_{M}+H)^{2}}\delta H+\delta\omega_{\mathbf{ib}}^{\mathbf{n}}+\varepsilon_{E} \\ \frac{\delta V_{F}}{(R_{N}+H)}-\omega_{\mathbf{ie}}\sin(L)\delta L-\frac{V_{F}}{(R_ {N}+H)^{2}}\delta H+\delta\omega_{\mathbf{ib}}^{\mathbf{n}}+\varepsilon_{N}\\ \frac{\tan(L)\delta V_{F}}{(R_{N}+H)}+\Big{(}\omega_{\mathbf{ie}}\cos(L)+\frac{V_ {F}\sec^{2}(L)}{(R_{N}+H)}\Big{)}\delta L-\frac{\tan(L)V_{F}}{(R_{N}+H)^{2}} \delta H+\delta\omega_{\mathbf{ib}}^{\mathbf{n}}+\epsilon_{V}\end{array} \right] \tag{25}\]  
According to Equations (16) and (17), the angular velocity error vector of the body frame \(o_{b}x_{b}y_{b}z_{b}\) with respect to the navigation frame \(o_{n}x_{n}y_{n}z_{n}\) denoted in the body frame \(o_{b}x_{b}y_{b}z_{b}\) is a main factor leading to the Euler angle error, and can be expressed by:  
\[\delta\mathbf{\omega}_{\mathbf{nb}}^{\mathbf{b}}=\delta\mathbf{\omega}_{\mathbf{ib}}^{\mathbf{b}}+ \mathbf{\varepsilon}^{\mathbf{b}}-\delta\mathbf{\omega}_{\mathbf{in}}^{\mathbf{b}}=\left[\begin{array} []{c}\delta\omega_{\mathbf{ib}}^{\mathbf{b}}+\varepsilon_{x}^{\mathbf{b}}-\delta\omega _{\mathbf{in}}^{\mathbf{b}}\\ \delta\omega_{\mathbf{ib}}^{\mathbf{b}}+\varepsilon_{y}^{\mathbf{b}}-\delta\omega_{\mathbf{in}} ^{\mathbf{b}}\\ \delta\omega_{\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i} \mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{ i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}\mathbf{i}  
According to Equations (5) and (31), the differential equations of Euler angle errors in head, pitch, and roll can be obtained. The differential equation of the pitch angle error is written by:  
\[ \begin{array}{ll}\delta\dot{\theta}&=\dot{\theta}^{c}-\dot{\theta}=\cos(\gamma ^{c})\omega^{b}_{nbx}+\sin(\gamma^{c})\omega^{b}_{nbx}-\left[\cos(\gamma^{c}- \delta\gamma)\omega^{b}_{nbx}+\sin(\gamma^{c}-\delta\gamma)\omega^{b}_{nbz} \right]\\ &=[1-\cos(\delta\gamma)]\left[\cos(\gamma^{c})\omega^{b}_{nbx}+\sin(\gamma^{c} )\omega^{b}_{nbz}\right]+\left[\cos(\gamma^{c})\omega^{b}_{nbz}-\sin(\gamma^{ c})\omega^{b}_{nbx}\right]\sin(\delta\gamma)\end{array}  \tag{32}\]  
By Taylor expansion, \(\sin(\delta\gamma)\approx\delta\gamma\), \(\cos(\delta\gamma)\approx 1-(\delta\gamma)^{2}/2\), and the differential equation of pitch angle error can be rewritten as:  
\[\delta\dot{\theta}=\left[\cos(\gamma^{c})\omega^{b}_{nbz}-\sin(\gamma^{c}) \omega^{b}_{nbx}\right]\delta\gamma+(\delta\gamma)^{2}\left[\cos(\gamma^{c} )\omega^{b}_{nbx}+\sin(\gamma^{c})\omega^{b}_{nbz}\right]/2 \tag{33}\]  
Ignoring the term as small as the second order, the differential equation of pitch angle errors can be written as:  
\[\delta\dot{\theta}\approx\left[\cos(\gamma^{c})\omega^{b}_{ntz}-\sin(\gamma^{ c})\omega^{b}_{nbx}\right]\delta\gamma \tag{34}\]  
Similarly, the differential equation of the roll angle error can be obtained as:  
\[\begin{array}{ll}\delta\dot{\gamma}=\dot{\gamma}^{c}-\dot{\gamma}=\tan( \theta^{c})\left[\sin(\gamma^{c})\omega^{b}_{nbx}-\cos(\gamma^{c})\omega^{b}_ {nbz}\right]+\omega^{b}_{nby}-\tan(\theta^{c}-\delta\theta)\left[\sin(\gamma^ {c}-\delta\gamma)\omega^{b}_{nbx}-\cos(\gamma^{c}-\delta\gamma)\omega^{b}_{ nbz}\right]-\omega^{b}_{nby} \tag{35}\]  
Based on Taylor expansion, \(\tan(\theta^{c}-\delta\theta)=\tan(\theta^{c})-\sec^{2}(\theta^{c})\delta\theta\), \(\sin(\delta\theta)\approx\delta\theta\), and the differential equation of roll angle error can be rewritten as:  
\[