# Inertial-Navigation-Aided Single-Satellite Highly Dynamic Positioning Algorithm
## 3 Navigation-Data Fusion  
Considering the complex and changeable motion of the target and the inherent error of the INS, the initial estimation of the target position from the previous section needs to further be improved[15]. The traditional Kalman filter is based on the linear model, shows high positioning fluctuations and suffers divergence in sudden changes. Therefore, in this section, we present an unscented Kalman filter (UKF) -based inertial-navigation-aided single-satellite highly dynamic positioning algorithm. Suppose \(\mathbf{x}\) is a \(l\) dimensional random variable that follows Gaussian distribution, that is, \(\mathbf{x}\ \sim\ N(\mathbf{\overline{x}},\mathbf{P})\), where \(\mathbf{\overline{x}}\) is an \(l\) dimensional vector and \(\mathbf{P}\) is a \(l\times l\) symmetric semipositive definite square matrix. \(\mathbf{P}\) is decomposed by Cholesky decomposition as \(\mathbf{P}=\mathbf{\dot{P}}\cdot\mathbf{\dot{P}}^{T}\), where \(\mathbf{\dot{P}}\) is the lower triangular matrix. A new posterior random variable \(\mathbf{y}=f(\mathbf{x})\) is obtained through nonlinear transformation. In this case, unscented transformation (UT) is utilized to estimate the mean value and the covariance matrix of \(\mathbf{y}\). Choosing a set of \(2l+1\) sampling points that meet the selection scheme as \[\xi_{0} = \overline{\mathbf{x}}\] \[\xi_{i} = \overline{\mathbf{x}}+r\dot{\mathbf{p}}_{i}\] \[\xi_{i+1} = \overline{\mathbf{x}}-r\dot{\mathbf{p}}_{i} \tag{18}\] \[i = 1,3,\cdots,2l-1\]  
where \(r=\sqrt{l+\gamma}\) and \(\gamma=a^{2}(l+\kappa)-l\) represent the main scale parameter and \(\kappa\) represents a minor scale parameter; \(a\) indicates the extent to which the sampling point deviates from the expected value, which is usually a very small positive number. It can be seen that all \(\xi_{i}\) and \(\xi_{i+1}\) are symmetrical about \(\xi_{0}=\overline{\mathbf{x}}\), so weight factors can be defined as  
\[w_{0} = \frac{\gamma}{l+\gamma}\] \[v_{0} = \frac{\gamma}{l+\gamma}+1-a^{2}\] \[w_{i} = v_{i}=\frac{1}{2(l+\gamma)} \tag{19}\] \[i = 1,2,\cdots,2l\]  
The predicted state mean and covariance matrix can be obtained through the weighted sum. \[\overline{\mathbf{x}}=\sum_{i=0}^{2l}w_{i}\xi_{i} \tag{20}\]  
\[\mathbf{P}=\sum_{i=0}^{2l}v_{i}(\xi_{i}-\overline{\mathbf{x}})(\xi_{i}- \overline{\mathbf{x}})^{T} \tag{21}\]  
Propagating each sampling point through nonlinear function,  
\[\xi_{i} = \overline{\mathbf{x}}+r\dot{\mathbf{p}}_{i}\]0 \[\xi_{i} = \overline{\mathbf{x}}+r\dot{\mathbf{p}}_{i}\]1 \[\xi_{i} = \overline{\mathbf{x}}+r\dot{\mathbf{p}}_{i}\]2 \[\xi_{i} = \overline{\mathbf{x}}+r\dot{\mathbf{p}}_{i}\]3  
Then, the predicted measurement mean and covariance matrix can be obtained through the weighted sum. \[\xi_{i} = \overline{\mathbf{x}}+r\dot{\mathbf{p}}_{i}\]4  
\[\xi_{i} = \overline{\mathbf{x}}+r\dot{\mathbf{p}}_{i}\]5  
The UKF replaced the traditional EKF to solve nonlinear problems through unscented transformation. This transformation approximates the probability of distribution rather than the nonlinear function and avoids calculation of the Jacobi matrix. It provides estimation accuracy approximately to the third-order EKF, equally computationally intensive with the first-order EKF. In a single-satellite system, the decremental of computation load has great significance for performance improvement in highly dynamic positioning. In the inertial-navigation-aided single-satellite system, the augmented state is denoted as  
\[\mathbf{X}_{t}=\left[ \begin{array}{ccccccccc}x&y&z&v_{x}&v_{y}&v_{z}&a_{x}& a_{y}&a_{z}\\ \end{array} \right]_{t} \tag{25}\]where \(\left[x,y,z\right]_{t}\), \(\left[v_{x},v_{y},v_{z}\right]_{t}\) and \(\left[a_{x},a_{y},a_{z}\right]_{t}\) represent the position, velocity and acceleration of the positioning target at time \(t\), respectively.