# Design and Modelling of an Amphibious Spherical Robot Attached with Assistant Fins
## 3 Dynamic Model of the Robot
### Equations of Motion  
The equations of the spherical robot's motion when it navigates on the surface of water are deduced by [PERSON]'s methods. When evaluating the forces exerted on the robot's components, forces that can be obtained from a potential function are conservative, and should be formulated into the potential energy. Forces associated with energy dissipation are nonconservative and should be formulated into the generalized forces. First, we need to construct the Lagrangian by calculating the kinetic energy and the potential energy of those three components of the robot. The kinetic energy of the spherical shell is the sum of the energy due to the translational motion of its center of mass and the energy due to rotation about the center of mass, that is,  
\[T_{s}=\frac{1}{2}m_{s}\Big{(}\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\Big{)}+\frac{ 1}{2}\Big{(}I_{sXX}\ ^{S}\omega_{sx}{}^{2}+I_{sYY}\ ^{S}\omega_{sy}{}^{2}+I_{sZZ}\ ^{S}\omega_{sz}{}^{2}\Big{)}, \tag{32}\]  
where \(I_{sXX}\), \(I_{sYY}\) and \(I_{sZZ}\) are moments of inertia of the spherical shell about its three axes of body frame \(O_{S}\)-\(X_{S}Y_{S}Z_{S}\), respectively. As the spherical shell is symmetrical, those three axes are its principle axes. The total number of fins attached to the spherical shell is \(2n\) with \(n\) fins on each hemisphere. Each of those attached fins can be reckoned as with mass \(m_{f}\) and radius of gyration \(R/2\) approximately, so \(I_{sXX}\), \(I_{sYY}\) and \(I_{sZZ}\) can be written as  
\[I_{sXX}=\frac{2}{3}m_{s}R^{2}-\frac{5}{6}nm_{f}R^{2},\ I_{sYY}=\frac{2}{3}m_{s }R^{2}-\frac{5}{6}nm_{f}R^{2},\ I_{sZZ}=\frac{2}{3}m_{s}R^{2}-\frac{5}{6}nm_{f }R^{2}.\]  
The gravity and the buoyancy on the spherical shell are related only to the generalized coordinates so that they are conservative, then the potential energy of the spherical shell is  
\[V_{s}=m_{s}gz+\rho_{w}g\pi\Big{(}-R(H_{0}-z)^{3}/3+(H_{0}-z)^{4}/12\Big{)}. \tag{33}\]  
The first term in the right-hand side of the above equation is potential energy corresponding to the gravity, and the second term is the potential energy arising from the buoyancy. Similarly, the kinetic energy of the IDU platform is  
\[T_{p}=\frac{1}{2}m_{p}\Big{(}\dot{x}^{2}+\dot{y}^{2}_{p}+\dot{z}^{2}_{p}\Big{)} +\frac{1}{2}\Big{(}I_{pXX}\ ^{D}\omega_{px}{}^{2}+I_{pYY}\ ^{D}\omega_{py}{}^{2}+I_{pZZ}\ ^{D}\omega_{pz}{}^{2}\Big{)}, \tag{34}\]  
where \(I_{pXX}\), \(I_{pYY}\) and \(I_{pZZ}\) are moments of inertia of the IDU platform about its three axes of body frame \(O_{D}\)-\(X_{D}Y_{D}Z_{D}\), respectively. They can be obtained by referring to the cuboid model  
\[I_{pXX}=\frac{1}{12}m_{p}\Big{(}4R^{2}+l_{p}^{2}\Big{)},\ I_{pYY}=\frac{1}{6}m _{p}l_{p^{\prime}}^{2},\ I_{pZZ}=\frac{1}{12}m_{p}\Big{(}4R^{2}+l_{p}^{2}\Big{)}.\]  
The potential energy of the IDU platform is  
\[V_{p}=m_{p}gz_{p}. \tag{35}\]  
The kinetic energy of the rotator is  
\[T_{r}=\frac{1}{2}m_{r}\Big{(}\dot{x}^{2}_{A}+\dot{y}^{2}_{A}+\dot{z}^{2}_{A} \Big{)}+\frac{1}{2}\Big{(}I_{rXX}\ ^{A}\omega_{rx}{}^{2}+I_{rYY}\ ^{A}\omega_{ry}{}^{2}+I_{rZZ}\ ^{A}\omega_{r \pi}{}^{2}\Big{)}, \tag{36}\]where \(I_{rXX}\), \(I_{rYY}\) and \(I_{rZZ}\) are moments of inertia of the rotator about its three axes of body frame \(A\)-\(X_{A}Y_{A}Z_{A}\), respectively.