## 6 Results and Discussion
### Observer Simulation under Conditions of Cable Disturbing Force with Compensation  
To observe the effects of the cable disturbing force compensation on the performance of the observer, the cable force is considered in the ideal ROV model. The cable disturbing force mentioned above is generated by using the lumped parameter method, regarded as the true value in practical environments. The initial cable shape in the vertical plane is shown in Figure 17. The observer simulation is conducted under the conditions of a cable disturbing force with disturbance compensation. The parameters of the observer remain unchanged, \(L=diag\{5,5,5,5,5\}\), \(\rho=diag\{0.5,0.5,0.5,5,5\}\). The effects of the measurement noise are ignored. The current velocity averages (\(-0.5\) m/s, \(0\) m/s, \(0\) m/s). The simulation time is \(100\) s, and the simulation step size is \(0.001\) s.  
When the propeller thrust acting on the ROV is \(F_{T}=(2000N,0N,0N)\), the following simulation results are obtained.  
The simulation results show that when the longitudinal thrust acts on the ROV system, the observation accuracy of state variables \(u_{r},w_{r},u_{c}\) is improved considerably by the addition of the cable disturbing force compensation to the observer model compared with the observations without compensation (Figures 18 and 19). The absolute value of the velocity observation error decreases from \(0.05\) m/s to \(0.01\) m/s, and the vertical displacement observation error becomes \(0.12\) m after \(100\) s (Figure 20), which is acceptable. The cable disturbing force error between the observed value and the true value is small, and the observed result can accurately reflect the variation in the disturbance force (Figure 21).  
Figure 16: Observed error of velocity and position: (**a**) velocity error; (**b**) position error.  
Figure 17: Initial cable shape in the vertical plane.  
Figure 19: Observed result of ROV position: (**a**) position in the X direction; (**b**) position in the Z direction.  
Figure 18: Observed result of velocity: (**a**) current velocity in the X direction; (**b**) relative velocity of ROV in the x direction.  
Figure 20: Observed error of velocity and position: (**a**) velocity error; (**b**) position error.  
It is concluded that as the existence of the disturbing force deteriorates the observer performance, modeling the disturbance force and compensating for the observer is an effective method to increase the accuracy of the observer without modifying its parameters, although disturbing force modeling cannot fully characterize the cable disturbing force.  
When the ROV's resultant thrust is \(F_{T}=(3000N,0N,0N)\), the simulation results are as follows.  
When the ROV's vertical thrust increases from 2000 \(N\) to 3000 \(N\), the observation results of the state variables \(u_{r},w_{r},u_{c}\) are still effective and have a high observation precision (Figures 22 and 23). The absolute value of the velocity observation error converges within 0.01 m/s (Figure 24). The vertical displacement observation error is relatively large, mainly because the vertical component error between the observed value and the true value of the cable disturbing force is relatively large, of which the relative error is around 10% (Figure 25). It is illustrated that the established simplified model of the cable disturbing force can accurately reflect the change in the disturbing force at different velocities, which is effective and universal.  
Figure 21: Cable disturbing force (LPM): (a) cable force in the X direction; (b) cable force in the Z direction.  
Figure 22: Observed result of velocity: (a) current velocity in the X direction; (b) relative velocity of ROV in the x direction.