# 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
Based on the previous regression analysis, pressure sensors 2 to 5 were used to estimate the drift angles (\(\alpha\), \(\beta\)), and the relative error for straight, turning, and gliding motions is provided in Tables 9-11. \[\left[\mathcal{U}^{2}\alpha\right]= \begin{bmatrix}D_{12}\\ D_{13}\\ D_{14}\\ D_{15}\end{bmatrix} +\begin{bmatrix}\Delta P_{d(T-B  
For straight and turning motions with zero drift angles, the state estimation results indicate proximity to zero across all velocity and angular velocity conditions, affirming accurate predictions. Moreover, during the gliding motion, when the angle of attack varied from 0 to \(-30\) degrees in intervals of 5 degrees at speeds ranging from 1 to 4 knots, the error stayed within 10%, excluding the low-speed 1 knot condition. The numerical results (Section 2.4) indicate that as the speed increases, the pressure variations corresponding to the drift angle become more pronounced, leading to higher prediction accuracy. Conversely, under low-speed conditions, the pressure variations are relatively small, resulting in lower prediction accuracy. Additionally, this confirms that the drift angle was well-predicted, closely approximating zero. Previously, regression coefficients were derived from single-motion data, enabling the prediction of the velocity and drift angles of the AUV using an array of pressure sensors. As the objective of this study was to apply this PVM to estimate the state during multiple motions, the aim was to validate the PVM and state estimation methods based on the numerous motion conditions listed in Table 12. Figure 5 depicts the definitions for each motion. For the turning with drift motion, velocity predictions were made for angular velocities between 10 and 30 rad/s and drift angles between 0 and 20 deg at a designed speed of 3 knots. With an angle of attack of zero, the model predicts velocities within approximately 10% across all angular velocity and drift angle conditions (Table 13). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\mathbf{U}\) & \multicolumn{2}{c}{**1.0**} & \multicolumn{2}{c}{**2.0**} & \multicolumn{2}{c}{**3.0**} & \multicolumn{2}{c}{**4.0**} \\ \hline \multirow{2}{*}{\(\mathbf{\alpha}\)} & **Predicted** & **Relative** & **Predicted** & **Relative** & **Predicted** & **Relative** & **Predicted** & **Relative** \\  & **\(\mathbf{\alpha}\)** & **Er. (\%)** & **\(\mathbf{\alpha}\)** & **Er. (\%)** & **\(\mathbf{\alpha}\)** & **Er. (\%)** & **\(\mathbf{\alpha}\)** & **Er. (\%)** \\ \hline
0 & 1.32 & - & 0.32 & - & 0.14 & - & 0.08 & - \\ \(-5\) & \(-3.75\) & \(-25.07\) & \(-4.84\) & \(-3.23\) & \(-5.09\) & 1.86 & \(-5.19\) & 3.78 \\ \(-10\) & \(-8.36\) & \(-16.43\) & \(-9.91\) & \(-0.93\) & \(-10.31\) & 3.12 & \(-10.50\) & 5.04 \\ \(-15\) & \(-12.57\) & \(-16.22\) & \(-14.99\) & \(-0.04\) & \(-15.46\) & 3.08 & \(-15.82\) & 5.48 \\ \(-20\) & \(-16.90\) & \(-15.48\) & \(-19.80\) & \(-0.99\) & \(-20.82\) & 4.11 & \(-21.27\) & 6.34 \\ \(-25\) & \(-24.10\) & \(-3.61\) & \(-25.74\) & \(2.95\) & \(-26.81\) & 7.23 & \(-27.27\) & 9.07 \\ \(-30\) & \(-30.18\) & \(0.60\) & \(-27.66\) & \(-7.81\) & \(-27.58\) & \(-8.08\) & \(-27.28\) & \(-9.08\) \\ \hline \(\mathbf{U}\) & \multicolumn{2}{c}{**1.0**} & \multicolumn{2}{c}{**2.0**} & \multicolumn{2}{c}{**3.0**} & \multicolumn{2}{c}{**4.0**} \\ \hline \(\mathbf{\alpha}\) & \multicolumn{2}{c}{**Predicted \(\mathbf{\beta}\)**} & \multicolumn{2}{c}{} \\ \hline
0 & 1.17 & 0.29 & 0.13 & 0.07 \\ \(-5\) & 1.12 & 0.28 & 0.13 & 0.07 \\ \(-10\) & 1.08 & 0.29 & 0.13 & 0.07 \\ \(-15\) & 1.00 & 0.29 & 0.13 & 0.08 \\ \(-20\) & 1.06 & 0.29 & 0.13 & 0.08 \\ \(-25\) & 1.13 & 0.30 & 0.14 & 0.08 \\ \(-30\) & 1.16 & 0.27 & 0.12 & 0.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Validation conditions.