# Development of a Mobile Buoy with Controllable Wings: Design, Dynamics Analysis and Experiments
## 3 Fluid-Multibody Coupling Dynamics Model
### Multibody Dynamics Model of the Mobile Buoy
 \tag{5}\]  
The velocity of the mobile buoy in the inertial frame can be expressed as:  
\[V=\sqrt{V_{\rm x}^{2}+V_{y}^{2}+V_{z}^{2}} \tag{6}\]  
The geometric structure of the mobile buoy can be regarded as a multibody system, as shown in Figure 5, which can be divided into three parts: the hull mass \(m_{h}\) (assuming a uniform distribution), the nonuniformly distributed mass \(m_{w}\) and the variable ballast mass \(m_{b}\). The body frame, designated \(e\)-\(\mathbf{x}y\)z, originates \(e\) at the geometric center (GC), which is the application point for the buoyancy and moments. The point mass \(m_{h}\) is simplified and positioned at GC, which are regarded as rigid mass points during the modeling. \(m_{w}\) is located at \(\mathbf{r_{w}}\) with respect to the GC, providing a restoring moment that allows the proposed mobile buoy to return to the original equilibrium position after an external interference. The variable mass \(m_{b}\) (water in the tank) is the adjustable mass, considered as a mass point with constant position (located in GC) and variable mass, which can control the diving and climbing of the proposed mobile buoy by adjusting the buoyancy. Thus, the total mass \(m_{t}\) can be expressed as:  
\[m_{t}=m_{h}+m_{w}+m_{b} \tag{7}\]  
Normally, the effect of seawater density variation on buoyancy control is negligible. In this case, the mobile buoy substitutes a fixed water volume, and its mass can be represented as \(m\). Thus, the net buoyancy mass \(m_{0}\) in the water can be represented as follows:  
\[m_{0}=m_{t}-m \tag{8}\]  
The motion of the mobile buoy in the vertical plane is an important way to obtain the oceanographic information. Figure 6 shows the forces acting on the mobile buoy as it moves in the vertical plane. \(F_{d}\) and \(F_{l}\) are the drag force and the lift force exerted on the mobile buoy in the flow field, respectively, which can be expressed as:  
\[F_{l}=F_{l-fin}+F_{l-fuselage} \tag{9}\]  
\[F_{d}=F_{d-fin}+F_{d-fuselage} \tag{10}\]  
where \(F_{l\_fin}\) is the lift force acting on the wings, \(F_{l\_fin}\) is the lift force acting on the fuselage. \(F_{d\_fin}\) is the drag force of the wings. \(F_{d\_fin}\) is the drag force of the fuselage. The horizontal  
Figure 5: The mass distribution of the mobile buoy. hydrodynamic forces acting on the proposed mobile buoy can be obtained by projecting \(F_{d}\) and \(F_{l}\), which can be calculated as:  
\[F_{x-fluid}=F_{l}\sin\gamma+F_{d}\cos\gamma \tag{11}\]  
\[F_{z-fluid}=F_{l}\cos\gamma+F_{d}\sin\gamma \tag{12}\]  
where \(\gamma\) represents the glide angle. \(F_{x-fluid}\) represents the hydrodynamic force in the \(\mathbf{X}\)-axis direction and \(F_{z-fluid}\) is the hydrodynamic force in the \(\mathbf{Z}\)-axis direction. The forward motion is generated by the combined effect of lift and drag, and the propulsive force in \(\mathbf{X}\)-axis direction is mainly provided by the horizontal component of the wing lift. During the Argo mode, when the wings of the mobile buoy are erected, the angle of attack is approximately \(90^{\circ}\), causing the lift generated by the wing to be small or even negligible according to [PERSON]'s lift formula [36]. Therefore, the mobile buoy lacks horizontal displacement capability due to the horizontal component of lift being ignored, and can only perform vertical diving and climbing in an attitude of near vertical motion. During the Glider mode, the wing rotation angle \(\alpha_{w}\) can be controlled to ensure the horizontal component of the lift is sufficient to generate the forward propulsive force, thus driving the mobile buoy forward.