## 4 Simulation Results
### Free Space Parking and Docking Tests  
This section highlights the capability of the system in performing free space parking maneuver (also used for homing) using the velocity field control technique and the capability to perform the docking maneuver in various challenging scenarios. Various trajectories of the vehicle while the parking maneuver is performed in free space with different parameters are presented in Figure 8a. Here, the control parameters--linear gain \(\lambda\), angular gain \(\gamma\), and modulation gain \(\sigma\)--of the task are varied. Increasing the value of \(\lambda\) \begin{table}
\begin{tabular}{r l r l} \hline \hline
**System Parameter** & **Value** & **Model Parameter** & **Value** \\ \hline DOF & 5 & Type & Torpedo \\ Length & 2 m & G & 9.810 \\ Width & 0.15 m & \(\rho\) & 1023.6 \\ Weight & 31 kg & CS & [0.000, 0.000, 0.070] \\ Main Propulsion & Central thruster & CB & [0.000, 0.000, 0.000] \\ Propellers & 5 & \
\begin{tabular}{l} Added Mass \\ Linear Drag \\ Quadratic Drag \\ Thruster distance \\ \end{tabular} &
\begin{tabular}{l} diag [\(-0.47\), \(-22.7\), \(-22.7\), \(-0.1\), \(-3.64\), \(-3.64\)] \\ diag [1.079, 10.21, 10.21, 0.5, 1.061, 1.061] \\ diag [0.794, 102.1, 102.1, 20.48, 15.191, 15.191] \\ \([-1.0\), 0.45, \(-0.45\), \(0.6\), \(-0.6\)] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: X300: System Parameters and dynamic model parameters. Thruster distance represents the longitudinal distance of each thruster from the center of body frame. allows the vehicle to reach the goal faster, and increasing the value of \(\gamma\) allows the vehicle to achieve the desired orientation sooner. Varying \(\sigma\) allows the vehicle to reach the final configuration faster or slower, while following the same path. The result of the free space docking simulations very well portrays how the vehicle perform the parking maneuver for various parameters to obtain the desired target position and orientation irrespective of its initial starting conditions. Action \(\mathcal{A}_{p}\) is used for this test. Figure 8b depicts the behavior of the vehicle while performing a docking maneuver at the bow side of a docking station traveling towards the vehicle. The vehicle follows a curvilinear velocity field trajectory towards the pre-docking position defined at a point in the near-vicinity of the station, and translates to the final dock position for completing the docking maneuver. In comparison to the work in [33], the translation is used only for the terminal docking maneuver, thereby also reducing drag and energy consumption. The \(\mathcal{A}_{p}\) and \(\mathcal{A}_{d}\) actions are exploited in these tests. The spheres represent the envelope obstacle placed around the docking station to avoid collision as the vehicle moves closer during homing and approach maneuver. Figure 9a depicts the behavior of the vehicle while performing a docking maneuver around the docking station at close proximity replicating the scenario presented in [30]. Here, the vehicle starts at close proximity from one side of the docking station and travels to the other side of the docking station without colliding with the station. Thanks to the tasks _Obstacle avoidance_ and _Field Velocity_ that are integrated to the task priority framework, the vehicle is able to move away from the close proximity of the station to go around and reach the pre-docking position present on the other side of the station by following a velocity field trajectory before translating to complete the docking maneuver. Figures 9a and 10a,b show the behavior of the vehicle when it needs to dock into a tight space, such as space-limited docks or piers. The use of pre-docking position allows the vehicle to home into the required orientation for docking without colliding with the surrounding dock structure. In a very extreme situation where the vehicle comes in close proximity with the dock structure, the obstacles and the _Obstacle Avoidance_ task allow the vehicle to safely move away from the dock structure while approaching.