# Engineering Applications of Adaptive Kalman Filtering Based on Singular Value Decomposition (SVD)
## 5 Numerical Experiments: Applications in Navigation
Models 3 and 5 do not include the three acceleration states (north, east, vertical)._  
Table 2 lists the number of state variables and the minimum and maximum number of visible satellites. Table 3 lists the parametric models used in this research to model a single component of the motion of a host vehicle. The model parameters mentioned in Table 3 include the standard deviation \(\sigma\) and the correlation time constant \(\tau\) of position, velocity, acceleration, and jerk (derivative of acceleration) [2]. The example 3 makes use of the Algorithm 1 adaptive computation from Equations (29) and (30), and the resulting algorithm is called K-SVD adaptive. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{6}{c}{\(\|\mathbf{RMSE}_{ij}\|_{2}\), \(i=1,\ldots,19\)**while**Growing**Ill-Conditioned \(\delta\)\(\rightarrow\)\(\leftarrow\)\(\leftarrow\)\(\leftarrow\)**Round-Off**} & \multicolumn{6}{c}{**Execution**Time (ms)**} \\ \cline{2-19}  & \(10^{-1}\) & \(10^{-2}\) & \(10^{-3}\) & \(10^{-4}\) & \(10^{-5}\) & \(10^{-6}\) & \(10^{-7}\) & \(10^{-8}\) & \(10^{-9}\) & \(10^{-10}\) & \(10^{-11}\) & \(10^{-12}\) & \(10^{-13}\) & \(10^{-14}\) & \(10^{-15}\) & \(10^{-16}\) & \\ \hline Kalman filter (KF) & 0.318 & 0.32 & 3.2 & 3.185 & 3.186 & 3.186 & 3.186 & 3.186 & 3.846 & NaN & NaN & NaN & NaN & NaN & NaN & NaN & 1.75 \\ \hline K-SVD & 0.318 & 0.32 & 3.2 & 3.186 & 3.186 & 3.186 & 3.186 & 3.186 & 3.186 & 3.186 & 3.186 & 3.186 & 3.186 & 3.186 & 3.186 & 3.186 & 3.186 & 1.2095 & 1.87 \\ \hline SQR-KF & 0.317 & 0.32 & 3.2 & 3.185 & 3.186 & 3.186 & 3.186 & 3.189 & 3.380 & NaN & NaN & NaN & NaN & NaN & NaN & NaN & 0.123 \\ \hline UD-KF & 0.318 & 0.32 & 3.2 & 3.185 & 3.186 & 3.186 & 3.201 & 2.607 & NaN & NaN & NaN & NaN & NaN & NaN & NaN & 0.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Vector norm of absolute errors for each component of the dynamic state—i.e., \(\|\mathbf{RMSE}_{ij}\|_{2}\) (example 2). RMSE, root mean square error; KF, Kalman filter; SVD, singular value decomposition; SQRK, square root; UD, unit triangular matrix diagonal; NaN, not a number. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model No. **} & \multicolumn{2}{c}{**Model Parameters (Each Axis)**} & \multirow{2}{*}{**Independent**} & \multirow{2}{*}{**Dependent**} \\ \cline{2-2} \cline{4-4}  & \multicolumn{2}{c}{**F**} & \multicolumn{2}{c}{**Q**} & & & \\ \hline \multirow{2}{*}{3} & \(\left[ \begin{array}{cc}0&1\\ 0&0\end{array} \right]\) & \(\left[ \begin{array}{cc}0&0\\ 0&\sigma_{acc}^{2}\Delta t^{2}\end{array} \right]\) & \(\sigma_{acc}^{2}\) & \(\sigma_{pos}^{2}\rightarrow\infty\) \\ \hline \multirow{2}{*}{5} & \(\left[ \begin{array}{cc}0&1\\ 0&\frac{-1}{i\tau_{val}}\\ &\frac{-1}{i\tau_{var}}\end{array} \right]\) & \(\left[ \begin{array}{cc}0&0&0\\ 0&0&0\\ 0&0&\sigma_{jerk}^{2}\Delta t^{2}\end{array} \right]\) & \(\sigma_{pos}^{2}\) & \(\sigma_{pos}^{2}\rightarrow\infty\) \\ \hline \multirow{2}{*}{6} & \(\left[ \begin{array}{cc}\frac{-1}{i\tau_{var}}\\ &\frac{-1}{i\tau_{var}}\end{array} \right]\) & \(\left[ \begin{array}{cc}0&0&0\\ 0&0&0\\ 0&0&\sigma_{jerk}^{2}\Delta t^{2}\end{array} \right]\) & \(\sigma_{pos}^{2}\) & \(\sigma_{pos}^{2}\) & \(\sigma_{pos}^{2}\) \\ \hline \multirow{2}{*}{6} & \(\left[ \begin{array}{cc}\frac{-1}{i\tau_{var}}\\ &\frac{-1}{i\tau_{var}}\end{array} \right]\) & \(\left[ \begin{array}{cc}0&0&0\\ 0&0&0\\ 0&0&\sigma_{jerk}^{2}\Delta t^{2}\end{array} \right]\) & \(\sigma_{pos}^{2}\) & \(\sigma_{pos}^{2}\) & \(\sigma_{pos}^{2}\) \\ \hline \multirow{2}{*}{6} & \(\left[ \begin{array}{cc}\frac{-1}{i\tau_{var}}\\ &\frac{-1}{i\tau_{var}}\end{array} \right]\) & \(\left[ \begin{array}{cc}0&0&0\\ 0&0&0\\ 0&0&\sigma_{jerk}^{2}\Delta t^{2}\end{array} \right]\) & \(\sigma_{pos}^{2}\) & \(\sigma_{pos}^{2}\) & \(\sigma_{pos}^{2}\) \\ \hline \multirow{2}{*}{6} & \(\left[ \begin{array}{cc}\frac{-1}{i\tau_{var}}\\ &\frac{-1}{i\tau_{var}}\end{array} \right]\) & \(\left[