# Establishment of a Pressure Variation Model for the State Estimation of an Underwater Vehicle
## 2 Methodology and Results
### Pressure Variation Model (PVM)
#### 2.2.2 State Estimation
\begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{Motion}\) & **Motion Variable** & **Note** \\ \hline \multirow{2}{*}{Turning w/drift} & \(U=3.0\) [knots] & \\  & \(\beta=10\), \(20\) [deg] & \\  & \(r=10\), \(20\), \(30\) [deg/s] & \\ \hline \multirow{2}{*}{Gliding w/drift} & \(U=1.5\), \(3.0\) [knots] & \\  & \(\alpha=-10\), \(-20\) [deg] & \\  & \(\beta=10\), \(20\) [deg] & \\ \hline \multirow{2}{*}{Spiral} & \(U=3.0\) [knots] & \\  & \(\alpha=-10\), \(-20\), \(-30\) [deg] & \\  & \(r=10\), \(20\), \(30\) [deg/s] & \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Predicted drift angle and relative errors for gliding motion (\(\beta=0\)). For the gliding with drift motion, velocity predictions were made for angle of attacks between \(-10\) and \(-20\) deg and drift angles between 0 and 20 deg, at speeds of 1.5 and 3 knots. Then, the predicted velocity was within approximately 3% across all drift angle conditions (Table 14). Velocity predictions were made for spiral motion at an angle of attack between \(-10\) and \(-30\) degrees and angular velocities between 10 and 30 rad/s, at a design speed of 3 knots. The predicted velocity was accurate within 12.5% for all angles and angular velocities (Table 15). Then, the model estimates the drift angles using the previously predicted velocity and pressure difference data. At the design speed of 3 knots, the angle of attack for the turning with drift motion is expected to be near zero. For the drift angle, except for the condition at 20 deg when the angular velocity is 10 deg/s, the maximum error is approximately 8%. For the 20 deg drift angle with an angular velocity of 10 rad/s, the prediction showed a higher error rate of about 20%, due to the slightly lower accuracy in the earlier velocity prediction (Table 16). \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(\mathbf{r}\) & \multicolumn{4}{c}{**10**} & \multicolumn{4}{c}{**20**} & \multicolumn{4}{c}{**30**} \\ \hline \(\mathbf{\beta}\) & **0** & **10** & **20** & **0** & **10** & **20** & **0** & **10** & **20** \\ \hline Predicted (knots) & 3.004 & 2.960 & 2.681 & 2.994 & 3.059 & 2.934 & 2.996 & 3.182 & 3.116 \\ Relative Err. (\%) & 0.119 & 1.354 & 10.64 & 0.196 & 1.947 & 2.204 & 0.134 & 6.071 & 3.865 \\ \hline \end{tabular}
\end{table} 
Table 13: Predicted velocities and relative errors for turning with drift motion (\(U\) = 3.0 knots, \(\alpha=0\)). Figure 5: AUV motion patterns (multiple). \[relative\ err.\ (\%)=(\beta_{est}-\beta)/\beta\times 100 \tag{14}\]  
Drift angles practice for speeds of 1.5 and 3 knots, with the angle of attacks between \(-10\) and \(-20\) deg and drift angles between 0 and 30 deg for gliding with drift motion. The drift angles predict within approximately 10% across all drift angle conditions (Table 17). In addition, for spiral motion, drift angles predict angles of attack between \(-10\) and \(-30\) deg and angular velocities between 10 and 30 deg, at a design speed of 3 knots. The angle of attack was predicted within 15% across all drift angle conditions, except for the condition at \(-20\) deg when the angular velocity is 10 deg/s. For the \(-20\) deg angle of attack with angular velocity 10 rad/s, the prediction showed a higher error rate of about 40% due to the relatively lower accuracy in the earlier velocity prediction (Table 18). \[relative\ err.\ (\%)=(\beta_{est}-\beta)/\beta\times 100 \tag{15}\]  
Drift angles practice for speeds of 1.5 and 3 knots, with the angle of attacks between \(-10\) and \(-20\) deg and drift angles between 0 and 30 deg for gliding with drift motion. The drift angles predict within approximately 10% across all drift angle conditions (Table 17). In addition, for spiral motion, drift angles predict angles of attack between \(-10\) and \(-30\) deg and angular velocities between 10 and 30 deg, at a design speed of 3 knots. The angle of attack was predicted within 15% across all drift angle conditions, except for the condition at \(-20\) deg when the angular velocity is 10 deg/s.