## 5 Simulation Studies
### Results and Discussion  
The results of numerical simulations are shown in Figures 6-13. Figures 6-8 present the desired and actual spatial trajectories and their tracking errors. From these results it is observed that the proposed controller drives the UVMS to track the desired spatial linear and circular trajectories quite satisfactorily in both the proposed redundancy resolution technique (c1) and the comparative redundancy resolution technique (c2). Moreover, the proposed control scheme outperforms the H\(\infty\)-EKF method, and has smaller tracking errors in both positions and orientations under the conditions of model uncertainties, time-varying ocean currents, payload and sensory noises. Even though the H\(\infty\)-EKF method adopted a H\(\infty\) robust controller to compensate the estimated bias from the EKF, the residual tracking errors can not be fully eliminated, as shown in Figure 6b,c and Figure 6e,f. The proposed controller performs better than the H\(\infty\)-EKF method in terms of robustness, which is dedicated to the IDC and fuzzy compensator for reducing the perturbation effects. Figure 9 plots the norm of the vector \(\mathbf{F}_{td}\) (i.e., thruster forces and actuator torques) and energy consumption of the UVMS. It can be noted that the comparative redundancy resolution technique (c2) is consuming more energy in generating trajectories for the vehicle and manipulator during both the linear and circular trajectories tracking. However in the proposed redundancy resolution technique (c1), the UVMS states are adjusted by self-motion to minimize interaction effects between the vehicle and the manipulator. This is because of the introduction of the secondary task vector \(\dot{\xi}\) and the nonlinear function. For better understanding, the generated trajectories for the vehicle positions/attitudes and manipulator positions are presented in Figures 10 and 11. It can be seen from the results that the generated trajectories have larger differences on vehicle attitudes and joint angles than vehicle positions. This is because the adjustment of the vehicle position has little effect on reducing the interactive forces between the vehicle and the manipulator without affecting the primary task. Consequently, the energy consumption can be reduced by changing the vehicle attitude and joint angles. In addition, it is observed from Figures 10 and 11 that the small roll and pitch angles of the vehicle are kept in the proposed control scheme, which contributes to properly working of the vehicle's onboard sensors. Figure 6: Spatial linear and circular trajectories and their tracking errors. (_a_) Desired linear trajectory and tracking control results; (_b,c_) position tracking errors in positions and orientations; (_d_) desired circular trajectory and tracking control results; (_e,f_) tracking errors in positions and orientations. Figure 7: End-effector errors when tracking the linear trajectory. (_a_) \(x_{ee}\) error, (_b_) \(y_{ee}\) error, (_c_) \(z_{ee}\) error, (_d_) tracking errors in the end-effector roll direction, (_e_) tracking errors in the end-effector pitch direction, (_f_) tracking errors in the end-effector yaw direction. Figures 12 and 13 show the required thruster forces for the vehicle and actuator torques for the manipulator during the linear and circular trajectory tracking. It is observed that the thruster forces for the two trajectories are less in the proposed redundancy resolution technique (c1), which results in the reduced energy consumption. In addition, the thruster forces and actuator torques for both trajectories in the proposed control\({}_{c1}\) are within their constraints (\(\pm 60\) N for the thrusters and \(\pm 3\) N\(\cdot\)m for the actuators).