# Inertial-Navigation-Aided Single-Satellite Highly Dynamic Positioning Algorithm
## 3 Navigation-Data Fusion
\[\hat{\eta}_{i,k}^{-}=\sqrt{(\hat{\xi}_{i+1,x,k}^{-}-\Delta x_{i}-p_{i}^{x})^{2 }+(\hat{\xi}_{i+1,y,k}^{-}-\Delta y_{i}-p_{i}^{y})^{2}+(\hat{\xi}_{i+1,z,k}^{ -}-\Delta z_{i}-p_{i}^{z})^{2}} \tag{33}\]  
\[\hat{\eta}_{i+1,k}^{-}=\sqrt{(\hat{\xi}_{i+1,x,k}^{-}-p_{i+1}^{x})^{2}+(\hat {\xi}_{i+1,y,k}^{-}-p_{i+1}^{y})^{2}+(\hat{\xi}_{i+1,z,k}^{-}-p_{i+1}^{z})^{2}} \tag{34}\]  
\[\hat{\eta}_{i+2,k}^{-}=\sqrt{(\hat{\xi}_{i+1,x,k}^{-}+\Delta x_{i+1}-p_{i+2}^{ x})^{2}+(\hat{\xi}_{i+1,y,k}^{-}-\Delta y_{i+1}-p_{i+2}^{y})^{2}+(\hat{\xi}_{i+1,z,k}^ {-}-\Delta z_{i+1}-p_{i+2}^{z})^{2}} \tag{35}\]  
\[\hat{\eta}_{hj+1,k}^{-} = \sqrt{(\hat{\xi}_{i+1,x,k}^{-})^{2}+(\hat{\xi}_{i+1,y,k}^{-})^{2} +(\hat{\xi}_{i+1,z,k}^{-})^{2}} \tag{36}\] \[k = 1,2,\cdots,2l+1\]  
All symbols in Equations (36-39) are scalars. For example, \(\hat{\eta}_{i+1,k}^{-}\) represents the \(k\)th column of the solution of vector \(\hat{\boldsymbol{\eta}}_{t}^{-}\) at time \(i\); \(\hat{\xi}_{i+1,x,k}^{-}\), \(\hat{\xi}_{i+1,y,k}^{-}\) and \(\hat{\xi}_{i+1,z,k}^{-}\) represent the \(x,y,z\) value of the \(k\)th column of solution of vector \(\boldsymbol{\eta}_{t}^{-}\) at time \(i+1\). Then, predictive value \(\hat{\mathbf{z}}_{t}^{-}\) of \(\mathbf{z}_{t}\) is as follows:  
\[\hat{\mathbf{z}}_{t}^{-}=\sum_{i=0}^{2l}w_{i}(\boldsymbol{\eta}_{t}^{-})_{i} \tag{37}\]  
UT transformation was utilized to obtain a priori covariance matrix \(\mathbf{P}_{t}^{-}\)  
\[\mathbf{P}_{t}^{-}=\sum_{i=0}^{2l}w_{i}(\boldsymbol{\xi}_{t}^{-}-\hat{ \mathbf{X}}_{t}^{-})(\boldsymbol{\xi}_{t}^{-}-\hat{\mathbf{X}}_{t}^{-})_{t}^{T} \tag{38}\]  
In order to make the dynamic positioning prediction accurate, initialization was only performed once and the measurement-update part was as follows:  
\[\mathbf{P}_{t}^{-}=\sum_{i=0}^{2l}w_{i}(\boldsymbol{\xi}_{t}^{-}-\hat{ \mathbf{X}}_{t}^{-})(\boldsymbol{\xi}_{t}^{-}-\hat{\mathbf{X}}_{t}^{-})_{i}^{T} \tag{39}\]In order to make the dynamic positioning prediction accurate, initialization was only performed once and the measurement-update part was as follows:  
\[\mathbf{P}_{t}^{yy}=\sum_{i=0}^{2l}w_{i}(\boldsymbol{\eta}_{t}^{-}-\hat{z}_{t}^{ -})(\boldsymbol{\eta}_{t}^{-}-\hat{z}_{t}^{-})_{i}^{T} \tag{40}\]  
\[\mathbf{P}_{t}^{xy}=\sum_{i=0}^{2l}w_{i}(\xi_{t}^{-}-\hat{\mathbf{X}}_{t}^{-}) (\boldsymbol{\eta}_{t}^{-}-\hat{z}_{t}^{-})_{i}^{T} \tag{41}\]  
\[\hat{\eta}_{i+1,k}^{-}=\sqrt{(\hat{\xi}_{i+1,x,k}^{-}-p_{i+1}^{x})^{2}+(\hat {\xi}_{i+1,y,k}^{-}-p_{i+1}^{y})^{2}+(\hat{\xi}_{i+1,z,k}^{-}-p_{i+1}^{z})^{2}} \tag{34}\]0  
\[\hat{\eta}_{i+1,k}^{-}=\sqrt{(\hat{\xi}_{i+1,x,k}^{-}-p_{i+1}^{x})^{2}+(\hat {\xi}_{i+1,y,k}^{-}-p_{i+1}^{y})^{2}+(\hat{\xi}_{i+1,z,k}^{-}-p_{i+1}^{z})^{2}} \tag{34}\]1  
\[\hat{\eta}_{i+1,k}^{-}=\sqrt{(\hat{\xi}_{i+1,x,k}^{-}-p_{i+1}^{x})^{2}+(\hat {\xi}_{i+1,y,k}^{-}-p_{i+1}^{y})^{2}+(\hat{\xi}_{i+1,z,k}^{-}-p_{i+1}^{z})^{2}} \tag{34}\]2  
The Kalman filter gives the estimated value at the new time point according to the a priori estimate. The \(\hat{\cdot}\) of the state variable indicates the prediction estimate, right label-indicates the a priori value, \(\mathbf{k}_{t}\) indicates the Kalman filter coefficient matrix at time \(t\) and \(\mathbf{P}_{t}^{xy}\) and \(\mathbf{P}_{t}^{yy}\) are the prediction covariance error matrix of the single-satellite navigation system. The computational complexity of the UKF filter was smaller than that of the EKF with the same accuracy and tracking performance was also better than EKF. It is very suitable for dynamic-target location and has good robustness. Combining INS input compensation and the initial least-squares calculation value, the UKF filter can provide stable rough positioning for a considerable period of time and can meet the requirements of single-satellite navigation.