# In-Motion Coarse Alignment Method Based on Position Loci and Optimal-REQUEST for SINS
## 3 The Principle of Vector Observation Based on Position Loci
### Discrete Recursive Formulae
\begin{array}{l}\int_{0}^{t}\int_{0}^{\tau}C_{n(\sigma)}^{n0}\mathsf{g}^{n}d \sigma d\tau=\sum\limits_{k=0}^{M-1}\int_{t_{k}}^{t_{k+1}}\int_{0}^{\tau}C_{n( \sigma)}^{n0}\mathsf{g}^{n}d\sigma d\tau=\sum\limits_{k=0}^{M-1}\int_{t_{k}}^{t _{k+1}}\left(\int_{0}^{t_{k}}C_{n(\sigma)}^{n0}\mathsf{g}^{n}d\sigma+\int_{t_{k} }^{\tau}C_{n(\sigma)}^{n0}\mathsf{g}^{n}d\sigma\right)d\tau\\ =\sum\limits_{k=0}^{M-1}\Delta t_{G}\int_{0}^{t_{k}}C_{n(\sigma)}^{n0}\mathsf{ g}^{n}d\sigma+\sum\limits_{k=0}^{M-1}\int_{t_{k}}^{t_{k+1}}\int_{t_{k}}^{\tau}C_{n( \sigma)}^{n0}\mathsf{g}^{n}d\sigma d\tau\\ =\sum\limits_{k=0}^{M-1}\Delta t_{G}\sum\limits_{m=0}^{k-1}C_{n(t_{m})}^{n0} \int_{t_{m}}^{t_{m+1}}C_{n(\sigma)}^{n(t_{m})}\mathsf{g}^{n}d\sigma+\sum \limits_{k=0}^{M-1}C_{n(t_{k})}^{n0}\mathsf{f}_{t_{k}}^{\tau}\int_{t_{k}}^{ \tau}C_{n(\sigma)}^{n(t_{k})}\mathsf{g}^{n}d\sigma d\tau\end{array}  \tag{21}\]  
Substituting Equations (19)-(21) into Equation (14), the complete formula for the measurement vector is shown below:  
\[ \begin{array}{l}\beta_{p}=C_{n(t_{M})}^{n0}\mathsf{r}^{n}-\sum\limits_{k=0}^ {M-1}C_{n(t_{k})}^{n0}\int_{t_{k}}^{t_{k+1}}C_{n(\tau)}^{n(t_{k})}(\omega_{in }^{n}\times)\mathsf{r}^{n}d\tau-\mathsf{t}^{n}(0)+\sum\limits_{k=0}^{M-1}C_{n( t_{k})}^{n0}\int_{t_{k}}^{t_{k+1}}C_{n(\tau)}^{n(t_{k})}(\omega_{ie}^{n} \times)\mathsf{r}^{n}d\tau\\ -\sum\limits_{k=0}^{M-1}\Delta t_{G}\sum\limits_{m=0}^{k-1}C_{n(t_{m})}^{n0} \int_{t_{m}}^{t_{m+1}}C_{n(\tau)}^{n(t_{m})}(\omega_{in}^{n}\times)\mathsf{r} ^{n}d\tau-\sum\limits_{k=0}^{M-1}C_{n(t_{k})}^{n0}\int_{t_{k}}^{t_{k+1}}\int_{ t_{k}}^{\tau}C_{n(\sigma)}^{n(t_{k})}(\omega_{in}^{n}\times)\mathsf{r}^{n}d \sigma d\tau\\ -\sum\limits_{k=0}^{M-1}\Delta t_{G}\sum\limits_{m=0}^{k-1}C_{n(t_{m})}^{0} \int_{t_{m}}^{t_{m+1}}C_{n(\sigma)}^{n(t_{m})}\mathsf{g}^{n}d\sigma-\sum \limits_{k=0}^{M-1}C_{n(t_{k})}^{0}\int_{t_{k}}^{t_{k+1}}\int_{t_{k}}^{\tau}C_{ n(\sigma)}^{n(t_{k})}\mathsf{g}^{n}d\sigma d\tau\end{array}  \tag{22}\]  
Assuming that the terms related to \((\omega_{ie}^{n}\times)\) and \((\omega_{in}^{n}\times)\) are so small compared with other terms in Equation (22) that they can be ignored during calculation, then the measurement vectors can be approximately determined as:  
\[ \begin{array}{l}\beta_{p}=C_{n(t_{M})}^{n0}\mathsf{r}^{n}-\mathsf{t}^{n}(0)- \sum\limits_{k=0}^{M-1}\Delta t_{G}\sum\limits_{m=0}^{k-1}C_{n(t_{m})}^{n0} \int_{t_{m}}^{t_{m+1}}C_{n(\sigma)}^{n(t_{m})}\mathsf{g}^{n}d\sigma-\sum\limits _{k=0}^{M-1}C_{n(t_{k})}^{n0}\int_{t_{k}}^{t_{k+1}}\int_{t_{k}}^{\tau}C_{n(\sigma )}^{n(t_{k})}\mathsf{g}^{n}d\sigma d\tau\end{array}  \tag{23}\]  
Meanwhile, the attitude matrix \(C_{n(t)}^{n(t_{k})}\) can be determined approximately as follows:  
\[ \begin{array}{l}C_{n(t)}^{n(t_{k})}=\mathrm{I}+\frac{\sin\lVert\varphi_{n} \rVert}{\lVert\varphi_{n}\rVert}(\varphi_{n}\times)+\frac{1-\cos\lVert\varphi_ {n}\rVert}{\lVert\varphi_{n}\rVert^{2}}(\varphi_{n}\times)^{2}\approx\mathrm{I}+ (\varphi_{n}\times)\end{array}  \tag{24}\]  
where \(\varphi_{n}\approx\int_{t_{k}}^{t}\omega_{ie}^{n}d\tau\approx(t-t_{k})\omega_{ in}^{n}\). The reason for this approximation is that \(\omega_{in}^{n}\) slowly changes during the coarse alignment. Further discretizing the Equation (23), we obtain the following formulae:  
\[