# Design and Depth Control of a Buoyancy-Driven Profiling Float
## 3 Depth Control System
### Motion Modeling  
In the process of modeling, vertical down is defined as the positive direction of the coordinate. The motion equation of the float can be established by Equation (1) through the vertical gravity, buoyancy, and resistance, as shown in Figure 2.  
\[M\ddot{h}=G-F_{b}-F_{r} \tag{1}\]  
where \(M\) is the mass of the float, \(h\) is the depth of the water where the float is located, \(G\) is gravity, \(G=Mg\), \(g=9.8\) m/s\({}^{2}\), \(F_{b}\) is buoyancy, and \(F_{r}\) is resistance.  
Figure 2: Main forces acting on a moving float.  
The resistance can be calculated by Equation (2); by replacing \(v^{2}\) with \(v\cdot|v|\), the direction of resistance is obtained.  
\[F_{r}=\frac{1}{2}C_{r}v|v|\rho A \tag{2}\]  
where \(A\) Is the cross-sectional area of the float, \(v\) is the speed of the float in the water, and \(C_{r}\) is the drag coefficient, which is related to the shape and speed of the float [13]. The expected speed of the float in the process of diving was not more than 0.2 m/s. Through ANSYS, the resistance \(F_{r}\) of the float under the condition of 0.2 m/s can be calculated. The drag coefficient can be obtained by further transforming (2) into (3). The final \(C_{r}\) value was 0.66.  
\[C_{r}=\frac{2F_{r}}{\rho Av^{2}} \tag{3}\]  
The density of seawater is influenced by temperature, salinity, and pressure. In the upper layer of water, the change of density is more drastic, so the influence caused by the change of seawater density could not be ignored in the simulation. Equation (4) indicates the density change of the upper layer of seawater fitted with a group of Argo floats.  
\[\rho=0.02073h+1023 \tag{4}\]  
The buoyancy of a float (5) is affected by the density of the liquid at its location and the volume of water pumped out. The volume change of the float has two components. One is the active volume change caused by the float pushing the linear actuator, and the other is the passive volume change caused by the deformation of the body with the change of depth. In Equation (6) \(V_{d}\) consists of the initial volume of the float \(V_{0}\), the volume of active change \(\Delta V_{r}\) and the volume of deformation \(\Delta V_{P}\).  
\[F_{b}=\rho V_{d}g \tag{5}\]  
\[V_{d}=V_{0}+\Delta V_{r}+\Delta V_{P} \tag{6}\]  
The deformation of the float depends on the material selected and the pressure applied. For an acrylic cylinder, the compression deformation can be calculated by Equation (7)  
\[\Delta V_{P}=2\pi R_{o}\cdot\Delta R_{o}\cdot L+\pi R_{o}^{2}\Delta L \tag{7}\]  
where \(R_{o}\) is the external radius of the cylinder, and \(L\) is the length of the cylinder, and its variation can be calculated by Equations (8) and (9) [24].  
\[\Delta R_{o}=\frac{R_{o}\cdot P}{E\cdot\left(R_{i}^{2}-R_{o}^{2}\right)}\times \left[(1-2\mu)R_{i}^{2}+(1+\mu)R_{o}^{2}\right] \tag{8}\]  
\[\Delta L=\frac{R_{o}^{2}\cdot P}{E\cdot\left(R_{i}^{2}-R_{o}^{2}\right)}\times \left[(1-2\mu)L\right] \tag{9}\]  
where \(P\) is the pressure, \(E\) is the elastic modulus of the material, \(R_{i}\) is the internal radius of the cylinder, and \(\mu\) is the Poisson's ratio.  
The specific values and units of each parameter are reported in Table 1.