# Design and Modelling of an Amphibious Spherical Robot Attached with Assistant Fins
## 3 Dynamic Model of the Robot
### Generalized Forces  
The equations of motion of the amphibious spherical robot are derived based on force analyses when the robot is navigating on the water, as shown in Figure 6. The coordinate frame \(O\)-\(XYZ\) is set with its origin located at the center of the spherical shell \(O\) and its \(Y\) axis pointing outward the paper. Forces exerted on the spherical shell are mainly due to interactions between the fluid and the spherical shell. It includes the gravity of the spherical shell \(m_{s}\)\(g\), the buoyancy \(F_{b}\), the hydrodynamic force \(F_{hyd}\), the resultant force exerted on all fins \(\mathbf{F}_{tf}\) as well as their corresponding torques. \(F_{hyd}\) consists of two main components, i.e., the frictional resistance \(R_{f}\) and the pressure resistance \(R_{pv}\). The forces exerted on the IDU platform and the rotator are simply their gravities, \(m_{p}\)\(g\) and \(m_{t}\)\(g\), respectively. Two input torques \(\tau_{l}\) and \(\tau_{r}\) are generated from the rolling motion motor and the spinning motion motor. The buoyancy \(\mathbf{F}_{b}\), the frictional resistance \(R_{f}\), the pressure resistance \(R_{pv}\) and the resultant force exerted on all fins \(\mathbf{F}_{tf}\) are described in detail in the following. Figure 6: Force analysis of the amphibious spherical robot. Buoyancy  
The buoyancy exerted on the spherical shell equals the gravity of the water which has the same volume as the immerged part of the spherical shell. If we set the generalized coordinate \(z=0\) and let \(H_{0}\) represents the height from the bottom of the spherical shell to the surface of water when the robot is in static equilibrium on the water, then the buoyancy is  
\[F_{b}=\rho_{w}g\pi\Big{(}R(H_{0}-z)^{2}-(H_{0}-z)^{3}/3\Big{)}, \tag{19}\]  
where \(\rho_{w}\) is the density of water, \(g\) is the gravitational acceleration. The center of buoyancy lies vertically below the center of sphere, its \(z\) coordinate is  
\[z_{b}=z-\frac{3(H_{0}-z-2R)^{2}}{4(H_{0}-z-3R)}. \tag{20}\]  
(b)Frictional Resistance  
In contrast to the pure rolling motion on the ground, the amphibious spherical robot slides relative to the fluid surface when rolling on the water, as a consequence of poor frictional condition at the contact region. The frictional resistance between the spherical shell and the fluid can be derived from the theory adapted to a ship [28], assuming that it is equal to the frictional resistance on a flat plate that is equivalent to the immerged spherical shell. It should be noted that the direction of the frictional resistance on the spherical shell is the same as the robot's moving direction, instead of being in the opposite direction in the case of a ship. As a result, the frictional resistance acts as propulsion for the amphibious spherical robot. The frictional resistance exerted on the spherical shell \(R_{f}\) can be written as  
\[R_{f}=\Big{(}C_{f}+\Delta C_{f}\Big{)}.\frac{1}{2}\rho_{w}V_{equ}^{2}S_{w}, \tag{21}\]  
where \(C_{f}\) is the frictional resistance coefficient of an equivalent flat plate corresponding to the spherical shell; \(\Delta C_{f}\) is the roughness coefficient, determined by the roughness of the spherical shell in contact with the water; \(V_{equ}\) is the equivalent velocity of the robot relative to the water.