## 3 Simulations
### Generation of ROV Trajectories for Further Analysis  
**Generation of trajectories**. Trajectories are defined as a sequence of depth, orientation, and speed commands of the ROV. The ROV is speed-controlled with a PID control loop and with a proportional control loop in heave, yaw, roll, and pitch. Figure 7 presents the curves for the velocity, depth, and angle commands as well as the values measured during the simulation for an untethered ROV. Keyframes are indicated on this trajectory to provide reference points. This trajectory starts with forward movement of the ROV along its longitudinal axis at different successive speeds between points 0 and 1 (0.1 m/s, 0.2 m/s, and then 0.3 m/s) in parallel with a depth control of 0.5 m. Then, a command for a forward speed of 0.3 m/s is sent again (point 1), followed by a depth command at 2 m (point 2). Then, while moving forward at a constant speed, the ROV receives successive yaw commands: 45 deg to the left (point 3), 45 deg to the right (point 4), 0 deg, 90 deg to the right (point 5), and 180 deg (point 6). Finally, the forward speed command is set to zero, while the depth command is kept to 2 m (point 7). The trajectory projected in the horizontal plane, deduced from the longitudinal speed and the orientation of the ROV, is also presented at the bottom right. The difference between the set point and the actual displacements of the ROV is due to hydrodynamic phenomena, which slow down the ROV in water. Moreover, when there are changes of set point in depth and yaw, a significant acceleration is required, resulting in overloading of the thrusters and affecting the speed along the \(x\)-axis (Figure 7). **Control schemes**. The behavior of the system is compared for five different modes listed below. * Cable not controlled and without compliant system: the cable length is fixed to 15 m along the simulation. * Cable with passive compliant system (not controlled): the cable length is fixed to 15 m along the simulation. * Cable controlled with ROV speed command as input: the delivery speed of the cable is set to the ROV speed. * Cable controlled with distance between the buoys: a closed-loop PD controller maintains a target distance of 0.8 m between the buoys. * Hybrid control of the cable based on ROV speed and distance between buoys, which combines buoy distance PD controller and P controller using an ROV speed command. Furthermore, the target distance between the buoys is set to vary according to the ROV speed. The proportional part of the PD controller on the buoys' distance works  
Figure 6: Comparison of cable impact on ROV power with and without passive compliant system. only if the distance is at least 5 cm longer than the target (taut cable) or if it is less than the target for more than 1 s (slack cable). **Distance between buoys**. Figure 8 depicts the evolution of the distance between buoys for passive configuration and the three control modes. Unsurprisingly, the best control is obtained with the buoy distance-based control, which keeps the distance between buoys around 0.8 m, despite oscillations that appear when the ROV is turning, from keyframe 3 to shortly after keyframe 6. The hybrid controller allows the distance to be kept within reasonable values, between 0.4 m and 0.9 m. The distance between the buoys reaches more extreme limits (0.16 m and 1.1 m) when the cable is not controlled or ROV speed-controlled. **Cable delivery speed**. Figure 9 shows the delivery speed for the three control modes. When the ROV moves forward at low speed (before keyframe 1), the feeder's speed is almost the same for the three control modes. As only the linear speed is considered for the ROV speed-based control, the cable delivery speed remains constant between keyframes 1 and 7, not taking into account the orientation of the ROV. Oscillations are more  
Figure 8: Comparison of the evolution of the distance between the buoys in passive mode and in three control modes. Figure 7: Simulated trajectories of the ROV.