# 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  
In the previous subsection, we obtained the vector construction method based on position information. However, all the formulae are continuous and cannot be calculated in practical experiments. Thus, the discrete recursive formulae are studied in this subsection [21]. The reference vector \(\alpha_{p}\) can be approximated as below:  
\[\alpha_{p}(M)=\sum\limits_{k=0}^{M-1}\Delta t_{G}\sum\limits_{m=0}^{k-1}C^{b0 }_{b(t_{m})}\int_{t_{m}}^{t_{m+1}}C^{b(t_{m})}_{p(\sigma)}\hat{e}^{b}d\sigma+ \sum\limits_{k=0}^{M-1}C^{b0}_{b(t_{k})}\int_{t_{k}}^{t_{k+1}}\int_{t_{k}}^{ \tau}C^{b(t_{k})}_{b(\sigma)}\hat{e}^{b}d\sigma d\tau \tag{15}\]  
where \(\Delta t_{G}\) and \(M\) represent the sampling time of GPS and the current time, respectively. Specifically, \(t=t_{M}=M\Delta t_{G}\), \(t_{k}=k\Delta t_{G}\) and \(t_{m}=m\Delta t_{G}\). The integral term can be calculated by the two-sample method, which is as follows:  
\[\int_{t_{m}}^{t_{m+1}}C^{b(t_{m})}_{b(\sigma)}\hat{e}^{b}d\sigma=\Delta \mathbf{v}_{1}+\Delta\mathbf{v}_{2}+\frac{1}{2}(\Delta\theta_{1}+\Delta \theta_{2})\times(\Delta\mathbf{v}_{1}+\Delta\mathbf{v}_{2})+\frac{2}{3}( \Delta\theta_{1}\times\Delta\mathbf{v}_{2}+\Delta\mathbf{v}_{1}\times\Delta \theta_{2}) \tag{16}\]  
\[\int_{t_{k}}^{t_{k+1}}\int_{t_{k}}^{\tau}C^{b(t_{k})}_{b(\sigma)}\hat{e}^{b}d \sigma d\tau=\frac{\Delta t_{G}}{30}(25\Delta\mathbf{v}_{1}+5\Delta\mathbf{v} _{2}+12\Delta\theta_{1}\times\Delta\mathbf{v}_{1}+8\Delta\theta_{1}\times \Delta\mathbf{v}_{2}+2\Delta\mathbf{v}_{1}\times\Delta\theta_{1}+2\Delta \theta_{2}\times\Delta\mathbf{v}_{2}) \tag{17}\]  
where  
\[\left\{ \begin{array}{rcl}\Delta\mathbf{v}_{1}=\sum\limits_{l=1}^{\frac{D}{2 }}\hat{e}_{l}^{b}\Delta t_{S}&\Delta\mathbf{v}_{2}=\sum\limits_{l=\frac{D}{2}+ 1}^{D}\hat{e}_{l}^{b}\Delta t_{S}\\ \Delta\theta_{1}=\sum\limits_{l=1}^{\frac{D}{2}}\omega^{b}_{ib,l}\Delta t_{S}& \Delta\theta_{2}=\sum\limits_{l=\frac{D}{2}+1}^{D}\omega^{b}_{ib,l}\Delta t_{S }\end{array} \right. \tag{18}\]  
where \(D\) denotes the half sampling time of GPS, and \(\Delta t_{s}\) denotes the sampling time of the accelerometers and gyroscopes. After acquiring the recursive formulae of the reference vector, the measurement vector is derived in detail. The first integration term of Equation (14) is shown below:  
\[\int_{0}^{t}C^{n0}_{n(\tau)}\mathbf{r}^{n}d\tau=C^{n0}_{n(t)}\mathbf{r}^{n}- \int_{0}^{t}C^{n0}_{n(\tau)}(\omega^{n}_{in}\times)\mathbf{r}^{n}d\tau=C^{n0} _{n(t_{M})}\mathbf{r}^{n}-\sum\limits_{k=0}^{M-1}C^{n0}_{n(t_{k})}\int_{t_{k}}^ {t_{k+1}}C^{n(t_{k})}_{n(\tau)}(\omega^{n}_{in}\times)\mathbf{r}^{n}d\tau \tag{19}\]  
The second integration term can be given by:  
\[ \begin{array}{l}\int_{0}^{t}\int_{0}^{\tau}C^{n0}_{n(\sigma)}(\omega^{n}_{ie }\times)\mathbf{r}^{n}d\sigma d\tau=\int_{0}^{t}C^{n0}_{n(\tau)}(\omega^{n}_{ ie}\times)\mathbf{r}^{n}d\tau-\int_{0}^{t}\int_{0}^{\tau}C^{n0}_{n(\sigma)}( \omega^{n}_{in}\times)(\omega^{n}_{ie}\times)\mathbf{r}^{n}d\sigma d\tau\\ =\sum\limits_{k=0}^{M-1}C^{n0}_{n(t_{k})}\int_{t_{k}}^{t_{k+1}}C^{n(t_{k})}_{n( \tau)}(\omega^{n}_{ie}\times)\mathbf{r}^{n}d\tau-\sum\limits_{k=0}^{M-1} \Delta t_{G}\sum\limits_{m=0}^{k-1}C^{n0}_{n(t_{m})}\int_{t_{m}}^{t_{m+1}}C^{n( t_{n})}_{n(\tau)}(\omega^{n}_{in}\times)(\omega^{n}_{ie}\times)\mathbf{r}^{n}d \tau\\ -\sum\limits_{k=0}^{M-1}C^{n0}_{n(t_{k})}\int_{t_{k}}^{t_{k+1}}\int_{t_{k}}^{ \tau}C^{n(t_{k})}_{n(\sigma)}(\omega^{n}_{in}\times)(\omega^{n}_{ie}\times) \mathbf{r}^{n}d\sigma d\tau\end{array}  \tag{20}\]  
The last integration term is calculated as follows:\[