# 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  
To describe the motion of the proposed mobile buoy, three coordinate frames are defined according to the standard method [35], as shown in Figure 4, including the inertial frame \(\mathbf{E}\)**-XYZ**, the body frame \(\mathbf{e}\)**-xyz** and the velocity frame \(\mathbf{e}^{\prime}\)**-x\(\mathbf{y}^{\prime}\)**-z**. The inertial frame is anchored to a reference point on the water surface, which be used to describe the position and attitude of the buoy. The initial heading of the mobile buoy on the water surface is the \(\mathbf{X}\)-axis of the inertial frame, the \(\mathbf{Z}\)-axis is vertically downward along the direction of the gravity and the \(\mathbf{Y}\)-axis can be defined by the right-hand rule. The body frame is fixed to the wing, and its origin \(\mathbf{e}\) is fixed to the center of buoyancy of the mobile buoy. The \(\mathbf{x}\)-axis of the body frame points to the bow of the buoy along the axis of the wing, the \(\mathbf{z}\)-axis is perpendicular to the \(\mathbf{x}\)-axis and the \(\mathbf{y}\)-axis can be acquired by the right-hand rule. As for the velocity frame, the origin coincides with the body frame, and the \(\mathbf{x}^{\prime}\) axis points to the velocity direction. The \(\mathbf{z}^{\prime}\) axis is vertically downward to the \(\mathbf{x}^{\prime}\) axis in the plane \(\mathbf{e}\)**-xz**. The direction of the \(\mathbf{y}^{\prime}\) axis is obtained by the right-hand rule. With reference to the inertial frame, the position and the attitude of the mobile buoy are described by position vector \(\mathbf{b}=[x,\,y,\,z]\) and angle vector \(\mathbf{\eta}=[\phi,\,\theta,\,\psi]\), respectively. In the body frame, the velocity and the angular velocity of mobile buoy can be defined as \(\mathbf{V}=[u,\,v,w]\) and \(\mathbf{\Omega}=[p,\,q,\,r]\), respectively. Moreover, the assumptions about the motion of the mobile buoy are given as follows: (1) The mobile buoy has six degrees of freedom, including surge, sway, heave, roll, pitch and yaw, where the dynamic response of the surge, heave and pitch degrees of freedom are focused on to assess the performance of the vehicle in profiling motion and the other degrees of freedom are ignored. (2) In this study, external disturbances such as wind, waves and currents, which are influenced by non-ideal environmental conditions, are not taken into consideration. The angle which the body frame rotates around the \(\mathbf{Y}\)-axis of the inertial frame can be expressed by \(\varphi\). \[\varphi=\alpha_{w}+\theta \tag{3}\]  
where \(\alpha_{w}\) is the wing rotation angle and \(\theta\) represents the pitch angle. In the diving phase of the mobile buoy, both \(\alpha_{w}\) and \(\theta\) are positive, and in the climbing phase of the mobile buoy, both \(\alpha_{w}\) and \(\theta\) are negative. There is a specific mapping relationship between the variables in the inertial frame and the variables in the body frame, and the velocity vector of the mobile buoy in the inertial frame can be expressed as:  
\[\dot{\mathbf{b}}=\left( \begin{array}{c}\dot{X}\\ \dot{Y}\\ \dot{Z}\end{array} \right)=\left( \begin{array}{c}V_{x}\\ V_{y}\\ V_{z}\end{array} \right)=\mathbf{R}_{E}^{\text{B}}\left( \begin{array}{c}u\\ v\\ w\end{array} \right) \tag{4}\]  
Figure 4: Coordinate frames of the mobile buoy. where \(V_{x}\), \(V_{y}\) and \(V_{z}\) represent the velocities of the mobile buoy along the \(\mathbf{X}\)-axis, \(\mathbf{Y}\)-axis and \(\mathbf{Z}\)-axis, respectively, all within the inertial frame. \(\mathbf{R}_{E}^{B}\) is the rotation matrix that maps the variables in the body frame to the inertial frame, which can be expressed as:  
\[\mathbf{R}_{E}^{B}= \begin{pmatrix}\cos\psi\cos\varphi&\cos\psi\sin\varphi\sin\phi- \sin\psi\cos\phi&\cos\psi\sin\varphi\cos\phi+\sin\psi\sin\phi\\ \sin\psi\cos\varphi&\sin\psi\sin\varphi\sin\phi+\cos\psi\cos\phi&\sin\psi\sin \varphi\cos\phi-\cos\psi\sin\phi\\ -\sin\varphi&\cos\varphi\sin\phi&\cos\varphi\cos\phi\end{pmatrix}