## 5 Simulation Studies
### Simulation Conditions  
In the simulations, the UVMS's end-effector is commanded to follow a spatial circle with diameter 2.24 m and a straight line of length 7.0 m. The simulation time of the circular trajectory is 50 s, where the initial 10 s is used for acceleration, 30 s is used to follow the circle and the final 10 s is used for deceleration. The simulation time of the straight-line trajectory is 30 s, where in the initial 10 s the acceleration is a half-period sine function, and then it maintains zeros, and in the final 10 s it is a half-period negative sine function. The UVMS's end-effector maintains orientation during the two trajectory tracking tasks. The initial desired positions and orientations are the same as the initial actual positions and orientations. The initial desired and actual velocities and accelerations are zeros. The average speeds of the two trajectories both are 0.23 m/s. The sampling time for the simulation is \(t=20\) ms. In this case, the model uncertainties, external disturbances, payload and sensory noises in position and orientation measurements are introduced for simulating the real working environment. To reflect the uncertainties, it is assumed that the modeling inaccuracy for each parameter is 10%. The vector of time-varying ocean currents in the inertial frame is assumed to be governed by (38). It is supposed that the end-effector of the manipulator is attached with a payload of 1 kg (in water). The following sensory noises are introduced: gaussian noise of 0.01 m mean and 0.01 m standard deviation for the vehicle position measurements; 0.5\({}^{\circ}\) mean and 0.5\({}^{\circ}\) standard deviation for the vehicle attitude measurements; and 0.05\({}^{\circ}\) mean and 0.05\({}^{\circ}\) standard deviation for the manipulator's joint position measurements. In addition, the thruster dynamic characteristics are inserted into the simulation. Suppose that the thruster response delay time is 50 ms, and its efficiency is 95%. \[\
u_{c}=\left[0.15+0.1\text{cos}(0.3t),\ 0.05\text{cos}(0.1\pi t),\ 0.1\text{ cos}(0.2t),\ 0,\ 0,\ 0\right]^{T}\text{m/s} \tag{38}\]  
To implement solution (7), the primary task vector is \(\mathbf{x}_{Ed}=[\mathbf{\mu}_{ELd}^{T},\mathbf{\mu}_{EDd}^{T}]^{T}\). \(\mathbf{k}_{f}=\text{diag}(4,4,4,6,6,6)\) and \(\mathbf{k}_{s}=25\mathbf{I}\) for the two trajectories. Other parameters are shown in Table 5. A secondary task is designed to align the vehicle orientation and the joint position with the primary task in terms of reducing the coupling effects, as shown in (39). \[\dot{\mathbf{\eta}}_{sd}= \begin{bmatrix}\dot{\phi}\\ \dot{\theta}\\ \dot{\psi}\\ q_{1}\\ q_{2}\end{bmatrix} = \begin{bmatrix}0&0&0\\ 0&0&-\alpha_{1}\\ 0&\alpha_{2}&0\\ 0&-\alpha_{3}&0\\ 0&0&-\alpha_{4}\end{bmatrix} \dot{\mathbf{\eta}}_{ELd} \tag{39}\]  
where \(\dot{\mathbf{\eta}}_{ELd}\) is the linear part of \(\mathbf{x}_{Ed}\). \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & **Length** & **Diameter** & **Mass** & **Buoyancy** & **Weight** \\  & **(m)** & **(m)** & **(kg)** & **(N)** & **(N)** \\ \hline AUV & 1.78 & 0.26 & 78.2 & 771.59 & 767.14 \\ Link1 & 0.1 & 0.06 & 1.28 & 11.43 & 12.54 \\ Link2 & 0.3 & 0.0425 & 1.92 & 17.2 & 18.87 \\ Link3 & 0.3 & 0.0425 & 1.92 & 17.2 & 18.87 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The list of UVMS parameters. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Trajectory** & \(\mathbf{\pi_{1}}\) & \(\mathbf{\pi_{2}}\) & \(\mathbf{\pi_{3}}\) & \(\mathbf{\pi_{4}}\) & \(\mathbf{\lambda}\) & \(\mathbf{t_{s}}\) (\(\mathbf{s}\)) & \(\mathbf{K_{\xi}}\) \\ \hline Straight line & \(-0.02\) & 0.13 & 0 & 0.05 & 0.3 & 15 & \(\text{diag}(0,2,0.2,0,1,1,1,1)\) \\ Circle & \(-0.06\) & 0.2 & \(-0.2\) & 0.15 & 0.3 & 45 & \(\text{diag}(0.1,0.3,0.1,0,1,1,1,1,1)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters for the proposed redundancy resolution technique. The distribution matrix is defined as  
\[\mathbf{W}^{-1}=