## 2 Modeling of The Compliant Buoy-Ballast Sensing System
### Distance Versus Speed in Steady-State Mode  
Figure 2 presents the scheme of the buoy-ballast V-shape model, showing the forces exerted on the buoys and ballast. It is assumed that the entire system moves at a constant velocity \(v\) driven by the ROV in the left-right direction. Figure 2: Model of the buoy–ballast V-system. The buoys and ballast move at the same constant speed. The total number of independent equations that govern the force compensation of the elements of the V-shape system at constant speed is 7 (detailed in Appendix A), which are  
\[T_{\text{B1}}^{x}=F_{d}^{buoy}+T_{1}\] (8) \[T_{\text{B1}}^{y}=-W_{buoy}\] (9) \[T_{\text{B2}}^{x}=-F_{d}^{ballast}-F_{d}^{CB_{1}}-F_{d}^{CB_{2}} -F_{d}^{buoy}T_{1}\] (10) \[T_{\text{B2}}^{y}=-W_{buoy}\] (11) \[T_{2}=2.F_{d}^{buoy}+F_{d}^{ballast}\] (12) \[(F_{d}^{buoy}+T_{1}+\frac{1}{2}.F_{d}^{CB_{1}}).\ell.\cos\theta_ {1}-W_{buoy}.\ell.\sin\theta_{1}=-C_{s}\Big{[}2(\frac{\pi}{2}-\theta_{1})+( \frac{\pi}{2}+\theta_{2})\Big{]}\] (13) \[-(F_{d}^{ballast}+F_{d}^{buoy}+T_{1}+F_{d}^{CB_{1}}+\frac{1}{2}.F_{d}^{CB_{2}}).\ell.\cos\theta_{2}-W_{buoy}.\ell.\sin\theta_{2}\] \[=C_{s}\Big{[}2(\frac{\pi}{2}+\theta_{2})+(\frac{\pi}{2}-\theta_{ 1})\Big{]}\] (14) where \(\bullet\): \(T_{\text{B1}}^{x}\) and \(T_{\text{B1}}^{y}\) are the components of the force \(\mathbf{T_{B1}}\) that is exerted by the cable portion \(CB_{1}\) of length \(\ell\) onto the left buoy. \(\bullet\): \(T_{\text{B2}}^{x}\) and \(T_{\text{B2}}^{y}\) are the components of the force \(\mathbf{T_{B2}}\) that is exerted by the cable portion \(CB_{1}\) of length \(\ell\) onto the right buoy. \(\bullet\): \(F_{d}^{buoy}\) is the intensity of the drag force exerted by the fluid on the buoy. \(\bullet\): \(F_{d}^{ballast}\) is the intensity of the drag force exerted by the fluid on the ballast. \(\bullet\): \(F_{d}^{CB_{1}}\) and \(F_{d}^{CB_{2}}\) are, respectively, the intensity of the drag force exerted by the fluid on the cable portion \(CB_{1}\) and \(CB_{2}\). \(\bullet\): \(C_{s}\) is the bending stiffness coefficient of the cable. \(\bullet\): \(\theta_{1}\) and \(\theta_{2}\) are, resp., the vertical angle of the \(CB_{1}\) cable portion and the \(CB_{2}\) cable portion. \(\bullet\): \(T_{1}\) is the magnitude of the horizontal scalar force exerted by the cable that links the left buoy to the remote station. \(\bullet\): \(T_{2}\) is the horizontal scalar force exerted by the cable that links the right buoy to the ROV. In the case of the figure, the intensity \(T_{2}\) is positive since the ROV pulls on the cable to move from left to right. The drag force magnitudes \(F_{d}^{CB_{1}}\) and \(F_{d}^{CB_{2}}\) are calculated as follows, based on the diameter \(d_{c}\) of the cable  
\[F_{d}^{CB_{1}} = \frac{1}{2}.C_{D}^{cable}.d_{c}.\ell.\cos\theta_{1}.\varphi_{w}. \
u^{2}\]  
where \(C_{D}^{cable}\) is dependent on the ratio of length to diameter. The forces exerted on the portion of cable from the station to \(B_{1}\) are the drag force, namely \(T_{1}^{d}\), and the resisting force exerted by the winch on the station side, namely \(T_{1c}\), which depends on the design of the feeder rotary joint. Since the cable is neutrally buoyant, there is no horizontal tension caused by apparent gravity (which would exist in the case of a weighing cable that takes the shape of a catenary in resting mode). \[T_{\text{B1}}^{x}=F_{d}^{buoy}+T_{1}\]0  
where \(T_{1}c\) is considered as a constant in the steady-state mode, whereas \(T_{1}^{d}\) can be expressed as \[T_{\text{B1}}^{x}=F_{d}^{buoy}+T_{1}\]1  
where \(dh\) is the depth offset between \(B_{1}\) and the other extremity of the cable on the station side. Equations (13) and (14) can be numerically solved for \(\theta_{1}\) and \(\theta_{2}\) to obtain the evolution of the inter-buoy distance \(d=\ell(\sin\theta_{1}-\sin\theta_{2})\) as a function of the velocity \(v\) (see Section 3.3).