# Development of an Autonomous Underwater Helicopter with High Maneuverability
## 3 Dynamic Analysis of the AUH
### Analysis of Motion in the Vertical Direction  
The landing and take-off functions have different requirements compared to forward and lateral propulsion. Stable landing process requires AUH to be subjected to stable downward force, but the design of underwater vehicles generally requires a slight positive buoyancy. It is undesirable that the power consumption is greatly increased by having to use the propellers to maintain contact  
Figure 4: Force analysis of horizontal linear motion. (**a**) Initial state; (**b**) motion state. with the seab when landing for a long time under the slight positive buoyancy, or conversely, by operating under the condition of the negative buoyancy. Therefore, in addition to the two groups of propellers, a buoyancy adjustment system is added to adjust the buoyancy to complete the stable landing and take-off. In the case of the driving forces mentioned above, four DOFs are limited, and two DOFs are still free. The following analysis shows that the remaining two degrees of freedom needn't be limited. The two DOFs are the y-direction translation and the \(x\)-axis rotation. Because the y-direction translation can be limited, as described above, primarily the \(x\)-axis rotation is discussed. \(X\)-axis rotation, i.e., rolling, is not a motion function that needs to be actively controlled, but an interference that needs to be suppressed in the motion of the AUH. There are two reasons why rolling does not need active force: (1) The source of rolling interference is less, and the intensity is small. (2) The damping of the rolling motion is great. Because the AUH mainly works near the seabed, its interference sources are mainly ocean current and internal waves. The deep-sea current is relatively stable. AUH's spindle-like structure in the horizontal plane causes it to be less resistant, therefore making it difficult for rolling moment to be produced in the current. The general scale of internal wave is larger, and the occurrence region is only in the stable density decomposition layer; therefore, it is difficult for a torque effect to be produced on the AUH, with its relatively small size. Therefore, most of the rolling interference is turbulence or collision. When AUH is subjected to the interference moment \(M_{t}\), the swing motion is performed as shown in Figure 5. Vibration can be described by Equation (13). \[-GR_{BG}\sin\psi-M_{fh}=J\frac{d^{2}\psi}{dt^{2}} \tag{13}\]  
where \(\psi\) is the angle of roll. \(M_{fh}=\rho C_{d}R^{1.5}\left(d\psi/dt\right)^{2}\) is a moment generated by fluid resistance \(f_{h}\). Hence, Equation (9) can be adjusted to (14). \[-GR_{BG}\sin\psi-\rho C_{d}R^{1.5}\frac{d\psi}{dt}=J\frac{d^{2}\psi}{dt^{2}} \tag{14}\]  
Solving Equation (13) with reference to Equations (8)-(12) gives Equation (15):  
\[\left\{ \begin{array}{c}\psi=\frac{M_{t}J}{EK}\Big{(}e^{t\left(E+K\right)/ \left(2I\right)}-e^{t\left(E-K\right)/\left(2I\right)}\Big{)}\\ A=-GR_{BG}\\ E=-\rho C_{d}R^{1.5}\\ K=\sqrt{E^{2}+4 AH}\end{array} \right. \tag{15}\]  
From Equation (15), similar to Equation (13), since \(E<0\), when \(K>0\), the roll motion is overdamped, and when \(K<0\), the pitch motion is underdamped. In addition, a zero roll angle will eventually be  
Figure 5: \(X\)-axis disturbing vibration. (**a**) Initial state; (**b**) motion state. maintained. As shown in Figure 5, because of the special structure of disc shape, the \(M_{fh}\) is greater when the body is rolling, which means the damping term becomes greater. Therefore, the vibration decays rapidly and returns to a steady horizontal state without active force. As mentioned above, using four thrusters in combination with the unique disc shape gives the AUH the characteristics of high maneuverability in the horizontal plane and a certain stability in the vertical direction. Based on the analysis, it is found that if the attack angle of horizontal navigation and the moment of inertia can be further decreased, the response of horizontal linear motion and rotation motion will be faster, which means having higher maneuverability.