# Engineering Applications of Adaptive Kalman Filtering Based on Singular Value Decomposition (SVD)
## 5 Numerical Experiments: Applications in Navigation
\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}{*}{7} & \(\left[ \begin{array}{cc}\frac{-1}{i\tau_{var}}\\ &\frac{-1}{i\tau_{var}}\end{array} \right]\) & \(\left[ \begin{array}{cc}0&0&0\\ 0&0&0\\ 0&0&\sigma_{jerk}^{2}\Delta t^{2}\end{array} \right]\) & \(\sigma_{pos}^{2}\) & \(\sigma_{pos}^{2}\) \\ \hline \multirow{2}{*}{7} & \(\left[ \begin{array}{cc}\frac{-1}{i\tau_{var}}\\ &\frac{-1}{i\tau_{var}}\end{array} \right]\) & \(\left[ \begin{array}{cc}0&0&0\\ 0&0&0\\ 0&0&0\\ 0&0&\sigma_{jerk}^{2}\Delta t^{2}\end{array} \right]\) & \(\sigma_{pos}^{2}\) & \(\sigma_{pos}^{2}\) \\ \hline \multirow{2}{*}{8} & \(\left[ \begin{array}{cc}\frac{-1}{i\tau_{var}}\\ &\frac{-1}{i\tau_{var}}\end{array} \right]\) & \(\left[ \begin{array}{cc}0&0&0\\ 0&0&0\\ 0&0&\sigma_{jerk}^{2}\Delta t^{2}\end{array} \right]\) & \(\sigma_{pos}^{2}\) & \(\sigma_{pos}^{2}\) \\ \hline \multirow{2}{*}{8} & \(\left[ \begin{array}{cc}\frac{-1}{i\tau_{var}}\\ &\frac{-1}{i\tau_{var}}\end{array} \right]\) & \(\left[ \begin{array}{cc}0&0&0\\ 0&0&0\\ 0&0&0&\sigma_{jerk}^{2}\Delta t^{2}\end{array} \right]\) & \(\sigma_{pos}^{2}\) & \(\sigma_{pos}^{2}\) \\ \hline \multirow{2}{*}{9} & \(\left[ \begin{array}{cc}\frac{-1}{i\tau_{var}}\\ &\frac{-1}{i\tau_{var}}\end{array} \right]\) & \(\left[ \begin{array}{cc}0&0&0\\ 0&0&0\\ 0&0&\sigma_{jerk}^{2}\Delta t^{2}\end{array} \right]\) & \(\sigma_{pos}^{2}\) & \(\sigma_{pos}^{2}\) \\ \hline \multirow{2}{*}{10} & \(\left[\  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c} \hline \hline
**Velocity** & **Filter** & \multicolumn{3}{c}{**RMS N Pos. Err (no)**} & \multicolumn{3}{c}{**RMS E Pos. Err (no)**} & \multicolumn{3}{c}{**RMS D Err Pos. (no)**} & \multicolumn{3}{c}{**RMS N Vol. Err (no)**} & \multicolumn{3}{c}{**RMS S Vel. Err (no)**} & \multicolumn{3}{c}{**RMS D Vel. Err (no)**} \\ \cline{3-14}
**(dpp)** & **Algorithm** & \multicolumn{3}{c}{**Model**} & \multicolumn{3}{c}{**Model**} & \multicolumn{3}{c}{**Model**} & \multicolumn{3}{c}{**Model**} & \multicolumn{3}{c}{**Model**} & \multicolumn{3}{c}{**Model**} \\ \cline{3-14}  & **3** & **5** & **6** & **3** & **5** & **6** & **3** & **5** & **6** & **3** & **5** & **6** & **3** & **5** & **6** & **3** & **5** & **6** & **3** & **5** & **6** \\ \hline \multirow{9}{*}{**90**} & K-SVD & 18.99\% & 13.1344 & 8.6985 & 13.2594 & 9.5004 & 7.696 & 4.9553 & 3.5034 & 3.3165 & 13.4455 & 18.1012 & 21.3424 & 12.1186 & 13.3198 & 15. Figure 4: Plan view of the estimated and true locations at 90 kph using the conventional Kalman filter and singular value decomposition based (K-SVD) adaptive algorithm. Figure 5: Plan view of the estimated and true locations at 140 kph using the conventional Kalman filter algorithm. Figure 6: Plan view of the estimated and true locations at 140 kph using the K-SVD filter algorithm. Figure 8: Plan view of the estimated and true locations at 240 kph using the conventional Kalman filter algorithm. Figure 7: Plan view of the estimated and true locations at 140 kph using the K-SVD adaptive algorithm. Figure 9: Plan view of the estimated and true locations at 240 kph using the K-SVD filter algorithm. Figure 11: Plan view of the estimated and true locations at 360 kph using the conventional Kalman filter algorithm. Figure 12: Plan view of the estimated and true locations at 360 kph using the K-SVD filter algorithm. Figure 10: Plan view of the estimated and true locations at 240 kph using the K-SVD adaptive algorithm. Comparing the values for each model in Table 4 at different velocities, it is clear that when the velocity is increased (e.g., from 90 to 360 kph), the conventional KF always increases the RMS error absolute value well above the K-SVD by about 172%. At the model level, the best performer in all cases is model 5.