# Design and Modelling of an Amphibious Spherical Robot Attached with Assistant Fins
## 3 Dynamic Model of the Robot
### Model Simplification  
The dynamic equations indicate that the amphibious robot is a highly coupled nonlinear system, which makes it difficult to analyze and control. To make the dynamic model practicable, a method for decoupling is applied, by separating the model into two basic motion modes, i.e., the rolling motion mode and the spinning motion mode.  
The dynamic equations, in its original form, represent a highly underactuated system with two inputs and eight state variables. The system has one input \(\tau_{l}\) and three state variables \((x,\theta,\alpha_{s})\) in the basic rolling motion mode, while it bears one input \(\tau_{r}\) and two state variables \((\psi,\alpha_{r})\) in the spinning motion mode; in a combined motion mode of these two, three extra variables \((\psi,z,\phi)\) exist. The focus here is on the linear motion of the robot when navigating on the water. In this case, the dynamic model can be described by just three generalized coordinates \(\mathbf{\eta}^{{}^{\prime}}=(x,\theta,\alpha_{s})\) when the others are set to zero. The Lagrangian then can be written as  
\[\begin{array}{c}L=\frac{1}{2}\big{(}m_{s}+m_{p}+m_{r}\big{)}\dot{x}^{2}+ \frac{1}{2}\big{(}I_{pYY}+I_{rYY}+m_{r}l_{OA}^{2})\dot{\theta}^{2}+m_{r}gl_{ OA}\epsilon\theta\\ -m_{r}l_{OA}\epsilon\dot{\theta}\dot{x}\dot{\theta}+\frac{1}{2}I_{sYY}\big{(} \dot{\theta}+\dot{\alpha}_{s}\big{)}^{2}+\frac{1}{12}\rho_{w}g\pi(4R-H_{0})H_ {0}^{3}\end{array}. \tag{40}\]  
Substituting the above Lagrangian into the [PERSON]'s equations, and let \(\Psi=\dot{\zeta}\dot{\alpha}_{s}^{2}/2\) be the viscous damping effect between the spherical shell and the IDU platform where \(\zeta\) denotes viscous damping coefficient, we obtain the simplified equations of motion of linear motion,  
\[M(\eta)\ddot{\eta}+C(\eta)\dot{\eta}+K(\mathbf{\eta})=\mathbf{\tau}, \tag{41}\]where  
\[\mathbf{M}(\mathbf{\eta})=\left[\begin{array}{ccc}m_{s}+m_{p}+m_{r}&-m_{r}gl_{OA}c\theta &0\\ -m_{r}gl_{OA}c\theta&I_{pYY}+I_{rYY}+I_{sYY}+m_{r}l_{OA}^{2}&I_{sYY}\\ 0&I_{sYY}&I_{sYY}\end{array}\right],\mathbf{C}(\mathbf{\eta})=\left[\begin{array}{ccc} 0&m_{r}l_{OA}s\theta\dot{\theta}&0\\ 0&0&0\\ 0&0&\zeta\end{array}\right],\]  
\[\mathbf{K}(\mathbf{\eta})=\left[\begin{array}{ccc}0\\ m_{r}l_{OA}gs\theta\\ 0\end{array}\right],\mathbf{\tau}=\left[\begin{array}{ccc}F_{hX}+F_{fX}\\ \tau_{hY}+\tau_{fY}\\ \tau_{hY}+\tau_{fY}+\tau_{l}\end{array}\right].\]  
The generalized forces \(\mathbf{\tau}\) are determined by Equation (31). The elements of \(\mathbf{\tau}\) are corresponding to the generalized coordinates \(x\), \(\theta\) and \(\alpha_{s}\), respectively, where \(F_{hX}\), consisting of the frictional resistance and the pressure resistance, is the hydrodynamic force exerted on the spherical shell, \(F_{fX}\) is the resultant force exerted on the fins.  
It can be concluded from Equation (41) that attaching assistant fins to the outer spherical shell will increase the propulsion force of the amphibious spherical robot. As a result, it will increase the acceleration of the robot, especially in the accelerating process when the robot starts from stationary. This can be validated by experiments presented in the following section. Since the magnitude of the propulsion force induced by one assistant fin correlate positively to the cross-section area of the fin, and the overall effect of all the fins are the summation of propulsion forces exerted on those fins immersed into the water, it will be helpful to use larger fins or more fins in order to improve the performance of the amphibious spherical robot.  
It is noteworthy that the forces exerted on the spherical shell by the water are estimated by empirical equations which significantly depends on their corresponding coefficients. Determining the accurate values of these coefficients is important in order to obtain a precise model for the amphibious spherical robot. However, this is not the emphasis here so it is put off for future research.