# LSOS: An FG Position Method Based on Group Phase Ranging Ambiguity Estimation of BeiDou Pseudolite
## 3 Algorithmic Principle
### FG Optimization Positioning Method Assisted by SFLA
#### 3.4.1 BDAPS FG Model
\[h(\mathbf{p}_{r,k})=\begin{bmatrix}\|\mathbf{p}_{r,k}-\mathbf{p}_{k}^{2}\|- \|\mathbf{p}_{r,k}-\mathbf{p  
\(\Gamma_{k}\) is the measurement noise covariance at k-time, and \(h(\mathbf{p}_{r,k})\) is the nonlinear measurement function of the measurement model at k-time. The square mahalanobis distance in the formula calculates the residuals of the single difference between the measured \(h(\mathbf{p}_{r,k})\) and the distance estimated by the group phase at the current K time.  
From the cost function of the factor node, it can be seen that the factor node \(f_{k}^{meas}(\mathbf{p}_{r,k})\) is only related to the navigation state quantity \(\mathbf{p}_{r,k}\) at the current moment.  
The process model is a prior estimation model, which represents the process of estimating the state of the system from the previous time to the current time. In the application process of BDAPS, there are mainly two stages involved in this process: (1) the estimation process of ranging measurement based on doppler; (2) Position estimation process based on doppler. The following two process factor graph models are modeled respectively.  
In the estimation of ranging measurements based on doppler, the pseudo-satellite signal doppler feature is caused by the receiver movement. Based on the doppler measurement \(\mathbf{d}_{r,k}^{s}=(d_{r,k}^{1},d_{r,k}^{2},\ldots,d_{r,k}^{N})\) of K at the current moment, the current doppler-based velocity relation.  
\[\mathbf{y}_{r,k}^{d}=(\lambda d_{r,k}^{1},\lambda d_{r,k}^{2},\ldots,\lambda d _{r,k}^{N})^{T} \tag{26}\]  
The current time estimation is obtained by smoothing the estimated range and doppler values or the carrier phase difference between the fore-and-aft epoch. After the predicted value of the system state is obtained, the residual value is calculated according to the measured value of the current time obtained in the process of group phase estimation. \[f_{p,k}^{pro}(\mathbf{ps}_{k})=d(err_{k}(\mathbf{ps}_{k^{\prime}},g(\mathbf{ps }_{k}))) \tag{27}\]  
Similarly, the probability estimation process is transformed from the above equation into  
\[ \begin{array}{l}P(\mathbf{ps}_{k}|\mathbf{ps}_{k-1})\propto f_{k}^{pro}( \mathbf{ps}_{k})\\ =\exp(-\frac{1}{2}\|err_{k}(\mathbf{ps}_{k^{\prime}},g(\mathbf{ps}_{k}))\|_{ \tau_{k}}^{2})\\ =\exp(-\frac{1}{2}\|\mathbf{ps}_{k}-g(\mathbf{ps}_{k})\|_{\tau_{k}}^{2})\end{array}  \tag{28}\]  
In the process of location estimation based on doppler, the system status \(\mathbf{p}_{r,k-1}\) at the last moment and the system status transition function are used to predict the system status value \(\mathbf{p}_{r,k}^{\wedge}\) at the current K moment. Affected by the receiver's doppler drift, we usually get the doppler value from the speed doppler \(dop_{v,k}^{N}\) and the doppler drift doppler \(dop_{cdf,k}^{N}\), the correlation equation can be expressed as  
\[d_{r,k}^{N}=dop_{v,k}^{N}+dop_{cdf,k}^{N} \tag{29}\]  
\[ \begin{array}{l}\mathbf{y}_{r,k}^{d}\\ =\lambda(\left[\begin{array}{c}dop_{v,k}^{1}\\ dop_{v,k}^{2}\\ \ldots\\ dop_{v,k}^{N}\end{array}\right]+\left[\begin{array}{c}dop_{cdf,k}^{1}\\ dop_{cdf,k}^{1}\\ \ldots\\ dop_{cdf,k}^{N}\end{array}\right])\end{array}  \tag{30}\]  
Therefore, the relation equation between velocity and position information can be written as  
\[\mathbf{y}_{r,k}^{d}=\mathbf{v}_{r,k}^{T}\mathbf{e}_{r,k}^{s}+\mathbf{\varepsilon }_{r,k}^{d} \tag{31}\]  
\(\mathbf{v}_{r,k}\) is the speed value of the user, \(\Delta t\) is the unit time interval of the data output, the interval time is not necessarily 1 s, under normal circumstances, the data rate of the user terminal output has 1 Hz, 5 Hz, 10 Hz, 20 Hz and so on. \(\mathbf{e}_{r,k}^{s}\) is the direction vector between the current position and the transmitting antenna position of BDAPS at the estimated position at the current K time. \[\mathbf{v}_{r,k}=\frac{\mathbf{p}_{r,k}-\mathbf{p}_{r,k-1}}{\Delta t}=