# Design and Modelling of an Amphibious Spherical Robot Attached with Assistant Fins
## Simulation and Experiments  
### Simulation  
Simulations were carried out in MATLAB in order to reveal characteristics of the dynamic model. The zero-state response under a prescribed input of the rolling motion motor is provided. Parameters used in the simulation are \(m_{s}=1.6\) kg, \(m_{p}=2\) kg, \(m_{r}=2.2\) kg, \(\zeta=0.02\), \(l_{OA}=0.08\) m, \(I_{sYY}=0.0308\) kg\(\cdot\)m\({}^{2}\), \(I_{pYY}=0.00053\) kg\(\cdot\)m\({}^{2}\), \(I_{rYY}=0.0045\) kg\(\cdot\)m\({}^{2}\), \(\tau_{l}=0.4\) Nm, \(\tau_{r}=0\) Nm, \(C_{\rm{Lmax}}=0.92\), \(C_{\rm{Dmax}}=1.12\), respectively. The frictional resistance exerted on the spherical shell is \(R_{f}=0.04\) N when the equivalent velocity is set as \(V_{equ}=0.4\) m/s.  
The time-varying states of those three generalized coordinates are shown in Figure 7.  
The simulation results show that the linear velocity of the robot \(\dot{x}\), the angle of the IDU platform \(\theta\) and the angular velocity of the spherical shell \(\dot{\alpha}_{s}\) proceed to a prescribed value in an oscillation manner as time increases, while the angular velocity of the IDU platform \(\dot{\theta}\) oscillates and then attenuates to zero; the travel distance \(x\) increases in a nearly linear manner, when \(\zeta=0.02\). If \(\zeta=0\), oscillation will be magnified for \(\dot{x}\), \(\theta\), \(\dot{\theta}\) and \(\dot{\alpha}_{s}\), and \(\theta,\alpha_{s}\), \(\dot{\alpha}_{s}\) will have a large amplitude while \(x\) keeps the same.  
It can be concluded from the simulation results that a steady linear speed can be achieved by increasing the viscous damping between the spherical shell and the IDU platform; moreover, a less oscillating IDU platform can be achieved at the same time, which is favorable in terms of the performance of sensors and cameras mounted on the IDU.  
Simulations were carried out to evaluate the effects of two parameters on the robot's steady-state velocity when operating on the water, i.e., \(m_{r}\) and \(\Delta C_{f}\), the adjustable rotator mass and the roughness coefficient of the spherical shell. Figure 8a shows the velocity curve when the rotator mass is set as \(0.8m_{r}\), \(m_{r}\) and \(1.2m_{r}\), respectively, while Figure 8b shows the velocity curve when the roughness coefficient is set as \(0R_{f}\), \(0.5R_{f}\) and \(1.5R_{f}\)respectively, with \(m_{r}\), representing the nominal mass of the rotator and \(R_{f}\) the nominal frictional resistance.  
The results indicate that, the reduction of the rotator mass to \(0.8m_{r}\) results in lower steady-state velocity of the robot, while the increase of the rotator mass has no apparent effect. It is observed that, the spherical shell floats upward when the rotator mass is reduced, leading to lower area of fins immersed under water, thereby reducing the forces exerted on the fins, affecting the steady-state velocity.  
On the other hand, the spherical shell dives downward when the rotator mass is increased, which leads to larger area of fins immersed under water; however, this effect is found to be less significant, since the water line is approaching the great circle of the spherical shell that parallels to the surface of water. The wetted surface area increases at the same time, leading to increased hydrodynamic forces because it is positively correlated to the wetted surface area.  
As a matter of fact, the current value of \(m_{r}\) was chosen such that it roughly maximizes the average area of the fin under water, under the condition that the mass of the robot is kept reasonably low, which is the cause of the behavior observed here.  
As for the roughness coefficient, it is found that it has no apparent effect, because the proportion of roughness in the hydrodynamic forces is low.  
Figure 7: Time-varying states of the generalized coordinates. (**a**) Travel distance of the robot; (**b**) Speed of the robot; (**c**) Angle of the IDU platform; (**d**) Angular velocity of the IDU platform; (**e**) Angle of the spherical shell; (**f**) Angular velocity of the spherical shell.