## 5 Results And Discussion
### Curvilinear Trajectory  
Figure 18 shows the snapshots at keyframes 1, 3, and 4 in cable control mode. Figure 19 presents the results in terms of buoy gap, feeder speed, and delivered cable length. Figure 20 presents the yaw angle. Figure 21 shows the executed trajectories in the cases of passive slack cable, passive taut cable, and control mode. Figure 17: ROV depth and vertical depth control thrust in the three modes. Figure 18: Views of the overall system at different points along the curvilinear trajectory when the cable is controlled. The red arrows indicate the direction of motion of the ROV (video at [[https://youtu.be/fv-JTc8](https://youtu.be/fv-JTc8) PvIY]([https://youtu.be/fv-JTc8](https://youtu.be/fv-JTc8) PvIY) (accessed on 1 December 2023)). In control mode, the compliant system keeps its V-shape and a buoy gap average close to the target distance (0.8 m), except for during turns, where the behavior is highly dynamic because of the increased forward speed and the opposing directions of the successive turns, which causes the buoy gap to reach its bounds, namely 0.2 m and 1.2 m, during a short period of time. The buoy gap resulting from passive taut and passive slack modes demonstrates the added value of automatic cable length management in maintaining an average buoy gap, which allows the cable shape between the station and the buoy-ballast system to be controlled. For a flawless system without any disturbance, the yaw angle (Figure 20) must be constant during the ROV straight line commands (before point 2 and after point 5). It must also be linear during rotating commands (between points 2 and 3, then 3 and 4). There is a slight deviation of about 20 deg to the right during the first straight line (2.6 m) command of the ROV for the passive slack cable. This deviation is oriented to the left and its absolute value is doubled with the controlled cable and doubled again with the passive taut cable. In fact, the cable is fixed on the left side of the back of the ROV, which induces a slight deviation to the left for the controlled cable, both in a straight line and during a right rotation. This deviation is much larger for the passive taut cable. The left rotation of the ROV appears to be less affected in the control and passive taut modes. The impact of these deviations is observed in Figure 21, in which the ROV paths in top view are shown for the three modes. The distances traveled are quite similar for the controlled cable and the passive slack cable, whereas the turns are slightly different. The path of the ROV with the passive taut cable is completely distorted. The first rotation to the right (between points 2 and 3) is tightly confined, and the rotation to the left (between points 3 and 4) is quite irregular (much wider curvature in the middle than at the beginning and the end). Figure 19: Distance between buoys (flex sensor), feeder speed, and unwound length of cable in control mode, passive slack mode, and passive taut mode for the curvilinear trajectory (videos at [[https://youtu.be/LR4](https://youtu.be/LR4) BKefRSnM]([https://youtu.be/LR4](https://youtu.be/LR4) BKefRSnM) for passive slack mode and at [[https://youtu.be/74p5](https://youtu.be/74p5) Bzee9 KY]([https://youtu.be/74p5](https://youtu.be/74p5) Bzee9 KY) for passive taut mode (accessed on 1 December 2023)). The experiments also showed there is effective control of the cable using the compliant buoy-ballast system, which keeps the cable in a semi-stretched configuration, and the delivered length is appropriately managed, preventing the creation of cable loops that can occur in passive slack mode and reducing the risks of snagging and tangles, considering that turns in passive taut mode can cause the cable to become entangled with the robot. The control mode generates slight tension in the cable, which is transmitted to the ROV and results in a minor deviation in the trajectory of the system, which can be avoided by fixing the cable closer to the ROV's center of gravity.