## 3 Modeling and Error Analysis of the Navigation Process in RINS
### The Navigation Algorithm Calculation Process
In one calculation period T, the attitude updating equation in n- and b-frame could be, respectively, expressed as follows:  
\[C^{b(m-1)}_{b(m)}=M_{RV}(\phi^{b}_{ib(m)}) \tag{3}\]  
\[C^{n(m)}_{n(m-1)}=\left(C^{n(m-1)}_{n(m)}\right)^{\rm T}=M^{\rm T}_{RV}(\phi^ {n}_{in(m)})\approx M^{\rm T}_{RV}(T\omega^{n}_{in(m)}) \tag{4}\]  
where the \(\phi^{b}_{ib(m)}\) and \(T\omega^{n}_{in(m)}\) denote the angular increment of the carrier and earth during one calculation period T. \(M^{\rm T}_{RV}\) is the operator which would change the rotation vector into the equivalent transformation matrix with the following equation. \[M^{\rm T}_{RV}(\phi)=I+\frac{\sin\phi}{\phi}(\phi\times)+\frac{1-\cos\phi}{ \phi^{2}}(\phi\times)^{2} \tag{5}\]where \(\phi\) is the equipment rotational vector and \(\phi\times\) is the skew symmetric matrix of the rotation vector \(\phi\). Velocity-updating equation:  
Once the attitude of the carrier is determined, we could obtain the dynamic movement of the carrier through the following equation. \[\dot{v}^{n}(t)=C_{b}^{n}(t)f_{sf}^{b}(t)-\left[2\omega_{ie}^{n}(t)+\omega_{en}^ {n}(t)\right]\times v^{n}(t)+g^{n}(t) \tag{6}\]  
where the \(f_{sf}^{b}(t)\) is the specific force measured by the accelerometer at time t, \(\omega_{ie}^{n}\) is the angular velocity of earth in i- and e-frame and \(\omega_{en}^{n}\) is the angular velocity in e- and n-frame. In this equation, \(2\omega_{ie}^{n}\times v_{en}^{n}\) is the Gothic acceleration caused by the carrier's relative movement with earth, and \(\omega_{en}^{n}\times v_{en}^{n}\) is the carrier's movement induced by centripetal acceleration. By integrating the above equation over a time period of t\({}_{m-1}\) to t\({}_{m}\), we could obtain the velocity-updating equation. \[ \begin{array}{ll}v_{m}^{n(m)}-v_{m-1}^{n(m-1)}&=\int_{t_{m-1}}^{t_{m}}C_{b} ^{n}(t)f_{sf}^{b}(t)\mathrm{d}t+\int_{t_{m-1}}^{t_{m}}-\left[2\omega_{ie}^{n} (t)+\omega_{en}^{n}(t)\right]\times v^{n}(t)+g^{n}(t)\mathrm{d}t\\ &=\Delta v_{sf(m)}^{n}+\Delta v_{cor/g(m)}^{n}\end{array}  \tag{7}\]  
Position-updating equation:  
Compared with attitude and velocity, the updating of position can be obtained in an easier and intuitive process. The position updating equation can be confirmed once the attitude and velocity of the carrier is given. The updating of the longitude, latitude and altitude of the carrier could be expressed as follows:  
\[\dot{L}=\frac{1}{R_{M}+h}v_{N}^{n},\dot{\lambda}=\frac{\sec L}{R_{N}+h}v_{E}^{ n},\dot{h}=v_{U}^{n} \tag{8}\]  
Which can be expressed in a concise matrix form, which is  
\[\dot{p}=M_{pv}v^{n} \tag{9}\]  
where the character is defined as  
\[p= \begin{bmatrix}L\\ \lambda\\ h\end{bmatrix} ,M_{pv}= \begin{bmatrix}0&1/R_{Mh}&0\\ \sec L/R_{Nh}&0&0\\ 0&0&1\end{bmatrix}  \tag{10}\]  
\[R_{Mh}=R_{M}+h,\ R_{Nh}=R_{N}+h \tag{11}\]  
\[R_{M}=\frac{R_{N}(1-e^{2})}{\left(1-e^{2}\sin^{2}L\right)},\ R_{N}=\frac{R_{e }}{\left(1-e^{2}\sin^{2}L\right)^{1/2}},\ e=\sqrt{2f-f^{2}} \tag{12}\]  
Through the above equations, we could obtain the attitude-, velocity- and position-updating processes of a navigation algorithm's calculation process.