# Establishment of a Pressure Variation Model for the State Estimation of an Underwater Vehicle
## 2 Methodology and Results
### Pressure Variation Model (PVM)
#### 2.2.1 Velocity Estimation  
First, to estimate the velocity \(U\) of the AUV, a single equation (Equation (2)) is used, similar to [PERSON] et al. (2020), but the same coefficients of PVM are set for all motion conditions. Representative captive model tests called PMM (planar motion mechanism) determine non-dimensional hydrodynamic maneuvering coefficients. Following the model's path and acquiring forces and moments under specific yaw and sway conditions, it measures hydrodynamic forces and estimates the hydrodynamic coefficient. \[P_{d\ (T+B+L+R)}=C_{1}U^{2}+C_{2}(\alpha^{2}+\beta^{2})+C_{3}U^{2}(\alpha^{2}+ \beta^{2})+C_{4}(\alpha^{2}+r^{2})+C_{5}U(q+r)+C_{6} \tag{2}\]  
\[P_{d,straight}=C_{1}U^{2}+C_{6} \tag{3}\]  
\[P_{d,gliding}=C_{1}U^{2}+C_{2}(\alpha^{2}+\beta^{2})+C_{3}U^{2}(\alpha^{2}+ \beta^{2})+C_{6} \tag{4}\]  
Figure 1: AUV shape and pressure sensor position. \[P_{d,turning}=C_{1}U^{2}+C_{4}(q^{2}+r^{2})+C_{5}U^{2}(q+r)+C_{6} \tag{5}\]  
where \(\alpha\) and \(\beta\) represent the drift angle on the vertical and horizontal plane, and \(q\) and \(r\) denote the angular velocities on the vertical and horizontal plane. Figure 2 depicts the definitions of these variables and Figure 3 depicts the specific definitions of each founded motion. We employed numerical analysis to predict the dynamic pressure in three motion conditions (straight, gliding, and turning). Based on these results, we derived PVM coefficients (\(C_{1}\)-\(C_{6}\)) through linear regression. At this point, the velocity and drift angles are known, and the angular velocity is assumed to be measured from the inertial sensor. Table 2 depicts the test conditions for obtaining PVM coefficients, and the performance of the acquired PVM evaluates the coefficient of determination \(R^{2}\) and the mean absolute error (MAE) between the estimated value and the measured one. \[MAE=\frac{1}{n}{\sum_{i=1}^{n}}\big{|}P_{d,est,i}-P_{d}\big{|} \tag{6}\]  
Here, \(n\) is the number of test cases, \(P_{d,est}\) is the estimated hydrodynamic pressure using Equation (2), and \(P_{d}\) is the calculated hydrodynamic pressure using CFD. Table 3 shows the coefficients of determination obtained from regression analysis for each motion condition concerning sensor 1 through 11. Figure 4 shows the overall process of AUV state estimation in this paper, and the details about the numerical analysis are described in Section 2.3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Motion** & \multicolumn{2}{c}{**Motion Variable**} & \multicolumn{2}{c}{**Note**} \\ \hline Straight & \(U=1.0\)–\(4.0\) [knots] *Interval \(0.5\) knots & \(\alpha=0\), \(\beta=0\), \(r=0\), \(q=0\) \\ \hline Turning & \(U=2.0\)–\(4.0\) [knots] *Interval \(1.0\) knots & \(\alpha=0\), \(\beta=0\), \(r\
eq 0\), \(q=0\) \\ \cline{2-4}  & \(r=0\)–\(30\) [deg/s] *Interval \(5\) deg/s & \(\alpha=0\), \(\beta=0\), \(r\
eq 0\), \(q=0\) \\ \hline Gliding & \(U=1.0\)–\(4.0\) [knots] *Interval \(0.5\) knots & \(\alpha\
eq 0\), \(\beta=0\), \(r=0\), \(q=0\) \\ \cline{2-4}  & \(\alpha=0\)–\(30\) [deg] *Interval \(5\) deg & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Test conditions. Figure 3: AUV motion patterns (single). Figure 2: Coordinate system and motion variables. Equation (2) enables us to predict the sum of the square of the drift angle. Still, it requires a second step to overcome the limitation of being able to separate individual components. Here, the difference in dynamic pressure between sensors located at the top and bottom, right and left sides, was formulated with the following equations (Equations (7) and (8)) and coefficients (\(D_{1}\)-\(D_{4}\)) obtained through linear regression using the velocity predicted at the previous step. Furthermore, as the AUV is axisymmetric, the coefficients for the pressure difference between sensors located at the top, bottom, right, and left sides can be equally applied.