# 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  
To estimate velocity as the initial step from the established PVM, an inverse matrix is constructed using the dynamic pressures of four sensors and its regression coefficients obtained from the previous step. Based on the last analysis, the regression outcomes select pressure sensors 4 to 7, and the second subscript indices of the PVM coefficients in Equation (9) represent the sensor number. The estimated velocities for straight, turning, and gliding motions are presented in Tables 6-8, demonstrating accurate predictions of the AUV's velocity. \[ \begin{bmatrix}U^{2}\\ \alpha^{2}+\beta^{2}\\ U^{2}\big{(}\alpha^{2}+\beta^{2}\big{)}\\ U(q+r)\end{bmatrix} = \begin{bmatrix}C_{14}&C_{24}&C_{34}&C_{54}\\ C_{15}&C_{25}&C_{35}&C_{55}\\ C_{16}&C_{26}&C_{36}&C_{56}\\ C_{17}&C_{27}&C_{37}&C_{57}\end{bmatrix} ^{-1} \begin{bmatrix}P_{I,A}-C_{41} \big{(}q^{2}+r^{2}\big{)}-C_{64}\\ P_{I,5}-C_{42}\big{(}q^{2}+r^{2}\big{)}-C_{65}\\ P_{I,6}-C_{43}\big{(}q^{2}+r^{2}\big{)}-C_{66}\\ P_{I,7}-C_{43}\big{(}q^{2}+r^{2}\big{)}-C_{67}\end{bmatrix}  \tag{9}\]  
\[relative\ err.\ (\%)=(U_{est}-U)/U\times 100 \tag{10}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\mathbf{U}\) & **1.0** & **1.5** & **2.0** & **2.5** & **3.0** & **3.5** & **4.0** \\ \hline Predicted (knots) & \(0.991\) & \(1.498\) & \(2.002\) & \(2.504\) & \(3.004\) & \(3.498\) & \(3.999\) \\ Relative Err. (\%) & \(-0.944\) & \(-0.109\) & \(0.098\) & \(0.163\) & \(0.118\) & \(-0.048\) & \(-0.016\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Predicted velocities and relative errors for straight motion. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Sensor** & \(\mathbf{C_{1}}\) & \(\mathbf{C_{2}}\) & \(\mathbf{C_{3}}\) & \(\mathbf{C_{4}}\) & \(\mathbf{C_{5}}\) & \(\mathbf{C_{6}}\) & \(\mathbf{D_{1}}\) & \(\mathbf{D_{2}}\) & \(\mathbf{D_{3}}\) & \(\mathbf{D_{4}}\) \\ \hline
1 & \(-305.1\) & \(-396.1\) & \(-206.1\) & \(-3803.3\) & \(1976.2\) & \(-174.5\) & \(-1533.5\) & \(-2328.1\) & \(1947.2\) & \(145.28\) \\
2 & \(-64.5\) & \(-364.6\) & \(-2336.0\) & \(-52.7\) & \(-543.6\) & \(-177.0\) & \(-1350.0\) & \(97.21\) & \(829.76\) & \(46.93\) \\
3 & \(-288.4\) & \(-328.4\) & \(-1701.4\) & \(-510.0\) & \(-34.3\) & \(-179.5\) & \(-1098.3\) & \(264.52\) & \(480.09\) & \(-100.02\) \\
4 & \(-433.6\) & \(-295.8\) & \(-1311.6\) & \(-307.3\) & \(31.0\) & \(-180.4\) & \(-588.9\) & \(5.64\) & \(358.47\) & \(-18.15\) \\
5 & \(-515.2\) & \(-257.3\) & \(-1119.0\) & \(-209.4\) & \(36.3\) & \(-180.5\) & \(-656.3\) & \(-79.53\) & \(248.60\) & \(-21.85\) \\
6 & \(-535.6\) & \(-232.4\) & \(-1075.2\) & \(-209.8\) & \(42.2\) & \(-180.1\) & \(-488.2\) & \(-52.10\) & \(128.54\) & \(-21.81\) \\
7 & \(-488.6\) & \(-230.2\) & \(-1192.8\) & \(-252.3\) & \(42.4\) & \(-179.7\) & \(-365.5\) & \(11.14\) & \(26.08\) & \(-22.23\) \\
8 & \(-323.7\) & \(-228.5\) & \(-1522.9\) & \(-254.4\) & \(-6.0\) & \(-183.5\) & \(-325.8\) & \(-36.75\) & \(29.30\) & \(-13.58\) \\
9 & \(-212.7\) & \(-236.8\) & \(-1743.7\) & \(-261.3\) & \(-25.1\) & \(-181.9\) & \(-297.3\) & \(-31.41\) & \(11.47\) & \(58.43\) \\
10 & \(-158.8\) & \(-244.4\) & \(-1843.9\) & \(-252.2\) & \(-23.4\) & \(-181.0\) & \(-265.1\) & \(-23.95\) & \(-18.61\) & \(104.23\) \\
11 & \(-126.1\) & \(-248.1\) & \(-1900.1\) & \(-20.1\) & \(-229.9\) & \(-180.4\) & \(-240.3\) & \(-24.64\) & \(-41.54\) & \(144.74\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: PVM coefficients. For straight, turning, and gliding motions, it is confirmed that the maximum error is within 1%, 2.5%, and 7%, respectively, demonstrating accurate velocity predictions. Furthermore, utilizing the predicted velocity and dynamic pressure, the drift angles are estimated through Equation (11) to Equation (12), and the second subscript indices of the PVM coefficients in Equations (10) and (11) represent the sensor number.