## 2 Tri-Stable Stochastic Resonance Theory  
### AUV Actuator Dynamics Model  
The omnidirectional actuator spindle speed control part controls the permanent magnet synchronous motor to rotate according to the speed of the main control unit, and then drives the omnidirectional actuator main shaft to rotate, thereby bringing the actuator to rotate. The spindle speed control part is composed of a power amplifier, a system controller, a permanent magnet synchronous motor, a signal detecting device, and a speed reducer. The spindle speed control composition diagram is shown in Figure 1. The motor driver, permanent magnet synchronous motor, drive board, power supply, etc., are built up to form the system. When the omnidirectional actuator pitch angle is used for periodic motion, it will generate six directions of motion, which will change the speed and actuator of the actuator load of the submersible. If the omnidirectional actuator is moving at a uniform speed, the forces in all directions will balance forces to keep the speed stable. However, when the force of the omnidirectional actuator changes, the force of each fire blade will be uneven, causing the change of the spindle speed and direction. The mathematical model of the spindle speed control system is established by considering various factors. The equation of motion of the spindle system is obtained by the momentum moment theorem  
\[2\pi I\frac{dn}{dt}=M_{E}-M_{P}-M_{f}, \tag{1}\]  
\[m\frac{dV_{s}}{dt}=T-R, \tag{2}\]  
In the Formula (1): \(M_{E}\) is the driving torque of the motor, \(M_{P}\) is the resistance torque of the blade, \(T\) is the effective thrust of the actuator, \(M_{f}\) is the friction torque of the propeller, \(I\) is the moment of inertia of the propulsion motor, and \(\frac{dn}{dt}\) is the rotation speed change rate. In Equation (2): \(m\) is the mass of the submersible, \(\frac{dV_{s}}{dt}\) is the rate of change of the speed of the omnidirectional actuator, \(R\) is the hull resistance. Obtained by the momentum moment theorem  
\[M_{E}=9.55\frac{P_{e}}{n_{e}}, \tag{3}\]  
\[M_{P}=K_{Q}\rho n^{2}d_{p}^{5}, \tag{4}\]  
Among them, \(d_{p}\) is the actuator diameter, \(\rho\) is the seawater density, \(P_{e}\) is the power of the drive motor, \(n\) is the propeller speed, and \(n_{e}\) is the motor speed. In Formula (1), the moment of inertia \(I\) of the main propulsion motor is equal to the sum of the motor, the actuator shaft, and the blade to the moment of inertia of the submerged main propulsion system. \[I=I_{motor}+I_{shaft}+I_{blade}, \tag{5}\]  
Among them, \(I_{shaft}\) is the moment of inertia of the actuator shaft, \(I_{motor}\) is the moment of inertia of the internal rotating part of the propulsion motor, \(I_{blade}\) is the moment of inertia of the actuator. In the  
Figure 1: Spindle speed control composition diagram. actuator design, if there is no detailed design information, it can be utilized. The empirical formula is calculated  
\[I_{blade}=\frac{1}{4}k_{1}k_{2}k_{3}k_{4}\gamma\theta\frac{t_{0}d_{p}^{4}}{g}, \tag{6}\]  
where \(k_{2}\) is the blade section parameter, \(k_{3}\) is the blade shape coefficient, and these two coefficients can be selected according to the actual situation, \(k_{4}\) is the blade clamp ratio coefficient, \(k_{1}\) is the paddle coefficient, which can be taken as 1.025-1.06, \(\gamma\) is the paddle Density, \(\theta\) is the width ratio of the paddle, \(t_{0}\) is the blade root thickness at the axis, and \(d_{p}\) is the paddle tip thickness, \(g\) is the gravitational acceleration. Finding the \(I_{blade}\) from Equation (6) requires a detailed understanding of the geometry of the actuator. If the detailed set parameters of the actuator are not available, an empirical formula can be used to estimate  
\[I_{blade}=\frac{W}{g}\left(\frac{kd_{p}}{2}\right)^{2}, \tag{7}\]  
where \(k\) is the coefficient, \(k=0.4\), and \(W\) is the weight of the full paddle. \(P_{e}\) is the power of the drive motor, \(n\) actuator speed, \(n_{e}\) is the motor speed.