# Inertial-Navigation-Aided Single-Satellite Highly Dynamic Positioning Algorithm
## 2 Single-Satellite System Model  
Under the conditions of GNSS denial, a single-satellite positioning system could be an effective complement [11]. Most targets, such as vehicles, personnel, ships and aircraft are dynamic, that is to say, their track, direction and movement time are uncertain. Therefore, in this paper, inertial navigation is the motion of the target that must be compensated by the INS. The principle of the inertial-navigation-aided single-satellite positioning method is shown in Figure 1.  
Figure 1: Single-satellite positioning based on inertial navigation system (INS). Here, \(\{X_{i},Y_{i},Z_{i}\}\) denotes the satellite position at time \(i\), \(R_{i}\) denotes the pesudorange between satellite and target at time \(i\) and \(\{x_{i},y_{i},z_{i}\}\) denotes the position of the target at time \(i\). Then, the pesudorange at the three successive moments, as shown in Figure 1, can be described as  
\[R_{i}=\sqrt{(X_{i}-x_{i})^{2}+(Y_{i}-y_{i})^{2}+(Z_{i}-z_{i})^{2}}+\delta t_{u} \tag{1}\]  
\[R_{i+1}=\sqrt{(X_{i+1}-x_{i+1})^{2}+(Y_{i+1}-y_{i+1})^{2}+(Z_{i+1}-z_{i+1})^{2} }+\delta t_{u} \tag{2}\]  
\[R_{i+2}=\sqrt{(X_{i+2}-x_{i+2})^{2}+(Y_{i+2}-y_{i+2})^{2}+(Z_{i+2}-z_{i+2})^{2} }+\delta t_{u} \tag{3}\]  
where \(\delta t_{u}\) represents the error caused by the receiver-clock offset. With the aid of inertial navigation [12], we can obtain the position of the target at these three successive moments as  
\[x_{i+1} = x_{i}+\Delta x_{i}\] \[y_{i+1} = y_{i}+\Delta y_{i} \tag{4}\] \[z_{i+1} = z_{i}+\Delta z_{i}\] \[x_{i+2} = x_{i+1}+\Delta x_{i+1}\] \[y_{i+2} = y_{i+1}+\Delta y_{i+1}\] (5) Figure 1:0  
where \(\Delta x_{i},\Delta y_{i},\Delta z_{i}\) represent the displacement of the target by the INS from the \(i\)th time point to the \(i+1\)th time point, respectively. we substituted (4) and (5) into Equations (2) and (4) and the pseudorange between the dynamic targets and the satellites is given in simplified form:  
Figure 1:1 Figure 1:2 Figure 1:3 Figure 1:4 Figure 1:5 Figure 1:6 Figure 1:7 Figure 1:8  
There are four unknowns in the three above equations, so we could not settle the position [13]. In addition, we included the height measurement that was directly obtained from the altimeter at time \(i\) to determine the target position, since it satisfies  
Figure 1:9  
The initial estimation of the target position at time point \(i\)th can be obtained by a least-mean-square algorithm. Based on the Taylor expansion, the estimation problem can be described as  
\[R_{i}=\sqrt{(X_{i}-x_{i})^{2}+(Y_{i}-y_{i})^{2}+(Z_{i}-z_{i})^{2}}+\delta t_{u} \tag{1}\]0  
where  
\[R_{i}=\sqrt{(X_{i}-x_{i})^{2}+(Y_{i}-y_{i})^{2}+(Z_{i}-z_{i})^{2}}+\delta t_{u} \tag{1}\]1\[\mathbf{x}=\left[ \begin{array}{c}x_{i}\\ y_{i}\\ z_{i}\end{array} \right] \tag{11}\]  
and  
\[\mathbf{G}_{m}=\left[ \begin{array}{c}-\frac{x_{i+m,k-1}}{R_{i+m,k-1}}\\ -\frac{y_{i+m,k-1}}{R_{i+m,k-1}}\\ -\frac{z_{i+m,k-1}}{R_{i+m,k-1}}\\ 1\end{array} \right]\,m=0,1,2 \tag{12}\]  
\[R_{i+m,k-1}=\sqrt{(X_{i+m}-x_{i+m,k-1})^{2}+(Y_{i+m}-y_{i+m,k-1})^{2}+(Z_{i+m}- z_{i+m,k-1})^{2}} \tag{13}\]  
\[\mathbf{G}_{4}=\left[ \begin{array}{c}-\frac{x_{i+1,k-1}}{\sqrt{x_{i+1,k-1}^{ 2}+y_{i+1,k-1}^{2}+z_{i+1,k-1}^{2}}}\\ -\frac{y_{i+1,k-1}^{2}-y_{i+1,k-1}^{2}+z_{i+1,k-1}^{2}}{\sqrt{x_{i+1,k-1}^{2}+ y_{i+1,k-1}^{2}+z_{i+1,k-1}^{2}}}\\ 0\end{array} \right] \tag{14}\]  
\[\mathbf{b}=\left[ \begin{array}{c}R_{i}-R_{i,k-1}-\delta t_{u,k-1}\\ R_{i+1}-R_{i+1,k-1}-\delta t_{u,k-1}\\ R_{i+2}-R_{i+1,k-1}-\delta t_{u,k-1}\\ h_{i+1}-\sqrt{x_{i+1,k-1}^{2}+y_{i+1,k-1}^{2}+z_{i+1,k-1}^{2}}\end{array} \right] \tag{15}\]  
Equation (10) contains four unknown parameters. The least-squares method was utilized to jointly solve the dynamic results at each iteration [14], \(\Delta\mathbf{x}\), as follows:  
\[\Delta\mathbf{x}=(\mathbf{G}^{T}\mathbf{G})^{-1}\mathbf{G}^{T}\mathbf{b} \tag{16}\]  
\[\left[ \begin{array}{c}x_{i+1,k}\\ y_{i+1,k}\\ z_{i+1,k}\end{array} \right]=\left[ \begin{array}{c}x_{i+1,k-1}\\ y_{i+1,k-1}\\ z_{i+1,k-1}\end{array}