## 3 Dynamic Model of ROV
### Umbilical Cable Force  
The umbilical cable plays an important role in facilitating the power supply and the communication function between the ROV and the support vessel. However, the attachment of the cable and the drag relative to the current places some restrictions on the maneuverability of the ROV. Therefore, the estimation of the corresponding effect caused by the umbilical cable and the current will be helpful for the controller design of the ROV. However, thus far, most of the researchers have neglected the effect of the umbilical cable because of the complexity involved, particularly in including the current effect. There are two types of modeling techniques available to predict the response of tethered systems: continuous analytical methods and discrete numerical models [26]. Discrete numerical models are valid for some nonlinear properties such as the quadratic drag and the spatially varying properties of cables. The nonlinear coupling motion principle between the tether and the vehicle can be included in these models. The most prevalent numerical approaches used nowadays for determining the hydrodynamic performance of an underwater tethered system are the lumped mass method [27], the finite difference method [28; 29], and the finite element method [30; 31]. The lumped mass method is adopted in this study for cable simulation because of its simplicity and effectiveness [26]. In the present study, for simplifying the problem, the following assumptions are made to solve the configuration and tension of the umbilical cable attached to the ROV:  
1. The umbilical cable is incompressible. 2. The cable surface is relatively smooth, ignoring the attachments on the cable. 3. The bending stiffness of the cable is ignored. The umbilical cable can only resist the tension force, but not the bending moment and the compression force. 4. The torsional rigidity and the quality of cable point rotation effect, which do not consider the torque, are ignored. The umbilical cable force is one of the most important nonlinear disturbance forces on the ROV when the current is strong and the cable length is sufficiently long. The movement of a flexible cable moving in a fluid can be described by the following lumped parameter equations. The cable model is described using the lumped parameters in the earth-fixed frame of O-XYZ. The coordinate system for analyzing the umbilical cable is shown in Figure 4. The origin O coincides with the end point of the umbilical cable. As shown in the figure, the cable is divided into \(n\) segments by \((n+1)\) nodes. The node number is sorted from the bottom to the top. The first node is on the ROV, and the last node is on the tether management system (TMS) or the surface ship. The \(j\)th segment refers to the segment between the \(j\)th node and the \((j+1)\)th node. Moreover, the segment is considered a linear elastic unit with the same tension directing along the tangent of the segment as that shown in Figure 4.  
According to the lumped mass model, the mass of each segment is distributed on the two boundary nodes equally and the nodes are connected with a light spring. Therefore, the first node bears half of the mass of the first segment as well as the last node in particular. For the nodes except the first and the last, the kinetics equations of each node can be obtained by using [PERSON]'s laws of motion as follows:  
\[m_{i}\frac{du_{i}}{dt}=W_{i}-B_{i}+F_{Ai}+D_{i}+T_{i}-T_{i-1} \tag{19}\]  
\[\frac{dr_{i}}{dt}=u_{i} \tag{20}\]  
where \(m_{i}\) is the \(i\)th lumped mass of the cable, \(u_{i}\) and \(r_{i}\) are the velocity and position vectors of the \(i\)th lumped mass with respect to the fixed inertial reference, and \(W_{i}\) and \(B_{i}\) are the gravity and buoyancy of the \(i\)th lumped mass. \(D_{i}\) represents the drag force, and \(T_{i}\) represents the tension force of the \(i\)th segment unit.