Research on an Autonomous Underwater Helicopter with Less Thrusters

Abstract

Autonomous underwater helicopter, referred to as AUH, has high maneuverability in the horizontal plane and stable movement in the vertical direction due to its disc shape. Thus, the AUH demonstrates great advantages when working in scenarios that require high accuracy of horizontal movement, fixed height and depth, operation near the seafloor, and so on. In this paper, we propose a new design for an autonomous underwater helicopter with three thrusters in the vertical direction (three-vertical-thrusters), so it is equipped with fewer thrusters while maintaining maneuverability and motion stability. The three-vertical-thruster AUH not only achieves stable attitude control, but also reduces the number of thrusters, enabling the AUH to save space, reduce drag, and decrease power consumption. The three-vertical-thruster structure is designed first and compared with the existing four-vertical-thrusters type to verify its advantages through dynamic analysis and hydrodynamic simulation. The three-vertical-thruster AUH is then modelled, and a compensation method is proposed for its more complex control. The three-vertical-thruster AUH’s controllability and stability are also verified by experiments on the basis of the experimental prototypes.

1. Introduction

An autonomous underwater vehicle (AUV) is one of the important pieces of equipment for underwater scientific research and engineering. The traditional torpedo-shaped AUV is suitable for long distance navigation because of its streamline profile and good linear performance [1,2]. However, most AUVs with the torpedo shape have poor maneuverability and vertical motion stability [3], which make them unsuitable for small-scale work near the seabed with complex terrain. Hereto, a new type of disc-shaped AUV was proposed by Zhejiang University in 2017 [4]. They fitted the disc-shaped fuselage with thrusters that allowed it to move with multiple degrees of freedom in the water, acting like the helicopter on land, and they called it the autonomous underwater helicopter (AUH). It has high maneuverability in the horizontal plane and stable movement in the vertical direction [5,6]. Thus, it has great advantages when working in scenarios that require high accuracy of horizontal movement, fixed height and depth, operation near the seabed, close-up inspection [7], cruising between base stations, seabed detection, etc.

Up to now, many studies related to AUH have been carried out, such as parametric design [8], wave impact [9], and hydrodynamic simulation [10,11]. The development of AUH has been constantly updated: in the original version, motion was achieved by only two thrusters in the vertical direction and four thrusters in the horizontal direction. The first-generation AUH [12] is shown in Figure 1. For the second-generation AUH, the number of vertical thrusters was increased to four for the consideration of the attitude stability when moving in the vertical direction, and the number of horizontal thrusters was changed to two due to size limitations [13], as shown in the Figure 1. In subsequent versions, however, the AUHs mostly followed a strictly disc-shaped design, and the thruster configuration remained the same.

Based on previous studies, it is concluded that: In the recent versions, AUHs are always equipped with two horizontal thrusters. More thrusters can compromise the good hydrodynamic properties of the disc shape and increase energy losses. Due to their zero turning radius, reducing to two thrusters does not affect their high maneuverability. It also saves space to some extent. Therefore, the AUH noted later in this paper defaults to two horizontal thrusters. In the vertical direction, the number of thrusters of the existing AUH is two or four, but mostly four. The AUH with two vertical thrusts (two-vertical-thruster AUH, the AUH with x thrusters in the vertical direction is referred to as the x-vertical-thruster AUH for short in this paper) cannot control the roll attitude in the water. In contrast, the four-vertical-thruster AUH can adjust and control any posture so as to maintain a stable motion. However, it adds two more thrusters, which not only take up more space but also increase energy losses and navigation resistance. In order to have more installation space for sensors and reduce energy consumption, we want to find a solution that equips the AUH with fewer thrusters while maintaining high maneuverability and motion stability.

Therefore, in this paper, we removed one vertical thruster from the four-vertical-thruster AUH, and we ensure it can still achieve roll attitude control to maintain a stable attitude in the water. Next, we propose the concept of a three-vertical-thruster AUH that is equipped with fewer thrusters (two in the horizontal direction and three in the vertical direction) while maintaining high maneuverability and motion stability. It can control the roll attitude compared with the two-vertical-thruster AUH and can save one propeller to maintain better hydrodynamic performance and save more space compared with the four-vertical-thruster AUH. These points will be proved by dynamic analysis and hydrodynamic simulation in Section 2. In addition, the shortcomings of the three-vertical-thruster AUH are also obvious. For example, the odd number of thrusters greatly increases the difficulty of motion control, which will also be summarized in Section 2. In Section 3, we will describe the control system of the three-vertical-thruster AUH and propose a control compensation method for it based on the existing PID control to give it stable motion. PID control is not the optimal control method, and there are many other control methods for underwater vehicles. For example, Vu et al. proposed the robust station-keeping (SK) control algorithm in 2020, which was based on sliding mode control (SMC) theory to guarantee stability and better performance of a hovering over-actuated autonomous underwater vehicle (HAUV) despite the existence of model uncertainties and ocean current disturbance in the horizontal plane (HP) [14]; in 2021, they also carried out research on nonlinear dynamics and robust positioning control of the over-actuated AUV based on the dynamic sliding mode control (DSMC) theory [15]. These are of great significance for the research in this paper. However, the control method is not the focus of this paper; we just want to distinguish the difference of the control between the three-vertical-thruster AUH and the four-vertical-thruster AUH and prove that the three-vertical-thruster AUH is controllable, which is verified by the experimental results in Section 4. Thus, we choose to use the simplest PID controller to achieve the AUHs’ control in this paper. In addition, in Section 4, the motion stability of the three-vertical-thruster AUH will also be verified by comparing the attitude angles with the four-vertical-thruster AUH when they are in motion. Finally, concluding remarks are given in Section 5.

2. Comparison of the Three-Vertical-Thruster AUH with Other AUHs

2.1. Dynamic Analysis

The AUH model can be simplified as shown in Figure 2, and the force analysis is also shown in the figure.

According to the principle of the reaction, the thruster rotates to produce an axial thrust force, $F=C_T{\omega }^2$, which is proportional to the square of its rotating speed. This force applied to the AUH will generate a torque, ${\tau }_{hor}=d\times F$, which causes the AUH to pitch and roll. In addition, a radial thrust is also generated, which is applied to the AUH to generate a torque that rotates the AUH around its central axis, ${\tau }_{ver}=K_M{\omega }^2$, called anti-torsional torque. The torque is proportional to the square of its rotating speed as well.

The expression $C_T$ is the thrust coefficient of the thruster; $K_M$ is the anti-torsion coefficient, which represents the growth rate of anti-torsional torque with the increase in rotating speed; $\omega$ is the rotating speed of the thruster; and $d$ is the vector from the fulcrum to the force acting point.

The force analysis of vertical motion is also shown in Figure 2. According to Newton’s second law of motion, the equation of linear motion can be written as:

$$F-F_f=m\frac{dV}{dt}$$

(1)

The equation of rotation motion is:

$$\tau -M_f=J\frac{d\Omega }{dt}$$

(2)

where $F$ is the thrust force produced by thrusters; $\tau$ is the thruster torque; $F_f$ is the fluid resistance; $M_f$ is the resistance moment of the z-axis; $V$ is the velocity; $\Omega$ is the angular velocity; $m$ is the mass; and $J$ is the inertia.

For the horizontal motion, $F=F_5+F_6$, $\tau =F_{5}a-F_{6}a+{\tau }_z$. Therefore, the above two equations can be written as:

$$\begin{gathered} F_5+F_6-F_f=m\frac{dV}{dt} \\ {F}_{5}a-F_{6}a+{\tau }_z-M_f=J\frac{d\Omega }{dt} \end{gathered}$$

where $F_5$ and $F_6$ are the thrust force produced by horizontal thrusters, and $a$ is the radius of the AUH. Therefore, it is not difficult to draw the conclusion that two horizontal thrusters are sufficient for sailing and turning.

Because there are only two thrusters in the horizontal direction, the AUH can only complete two actions in the horizontal direction: advancing and rotating. Therefore, reducing to two thrusters in the horizontal direction changes the AUH from the original 6-DOF motion to 5-DOF motion. Thus, it is unable to move in the x direction in the coordinate system in Figure 2. The thruster model can be represented by $T={\left[\begin{array}{cccccc}{F}_x& F_y& F_z& {\tau }_x& {\tau }_y& {\tau }_z\end{array}\right]}^T$; $F_n$ and ${\tau }_n$, respectively, correspond to thrust force and torque produced by thrusters of the $n$-axis.

For the vertical motion of the two-vertical-thruster AUH, assume that the thrust force of the horizontal thruster is zero. The thruster model can be written as:

$$\{\begin{array}{c} F_x=0\\ F_y=C_T\left({\omega }_5^2+{\omega }_6^2\right)\\ F_z=C_T\left({\omega }_1^2+{\omega }_2^2\right)\\ {\tau }_x=C_T{l}_2\left({\omega }_1^2-{\omega }_2^2\right)\\ {\tau }_y=0\\ {\tau }_z=K_M\left(-{\omega }_1^2+{\omega }_2^2\right)\end{array}$$

(3)

when the AUH maintains a stable attitude, and ${\omega }_1={\omega }_2$ is obtained according to the torque balance ${\tau }_x={\tau }_y={\tau }_z=0$. However, if it is disturbed by the flow at this moment, represented as

, shown in Figure 2b, the roll moment produced by the thrusters

. Therefore, the two-vertical-thruster AUH cannot actively balance the disturbance, but can only rely on its own recovery moment because the position of the buoyancy is higher than the center of gravity to eliminate the disturbance.

For the vertical motion of the three-vertical-thruster AUH, the thrust force of the horizontal thruster is also zero. The thruster model is:

$$\{\begin{array}{c} F_x=0\\ F_y=C_T\left({\omega }_5^2+{\omega }_6^2\right)\\ F_z=C_T\left({\omega }_1^2+{\omega }_2^2+{\omega }_3^2\right)\\ {\tau }_x=C_T\left({\omega }_1^2{l}_1+{\omega }_2^2{l}_1-{\omega }_3^2{l}_2\right)\\ {\tau }_y=C_T\left(-{\omega }_1^{2}l+{\omega }_2^{2}l\right)\\ {\tau }_z=K_M\left(-{\omega }_1^2-{\omega }_2^2+{\omega }_3^2\right)\end{array}$$

(4)

First, according to the torque balance${\tau }_x={\tau }_y=0$ when stable, we get the equation:

$${\omega }_1={\omega }_2=\sqrt{\frac{l_2}{2l_1}}{\omega }_3$$

(5)

Therefore, for the three-vertical-thruster AUH, the outputs of the three thrusters in the vertical direction should be different, otherwise the stable attitude cannot be maintained. Moreover, the outputs of the three thrusters should follow the determined numerical relationship in Equation (5). Therefore, for the three-vertical-thruster AUH, a certain compensation coefficient should be given to the single thruster for motion control in the vertical direction. This coefficient is calculated by Equation (4):

$$\begin{gathered} {\tau }_x=C_T{\omega }_1^2{l}_1+C_T{\omega }_2^2{l}_1-k{C}_T{\omega }_3^2{l}_2=0 \\ k=\frac{2l_1}{l_2} \end{gathered}$$

(6)

Then, ${\tau }_z=0$, the anti-torsional torque generated by the thrusters should also be zero. The calculation result is $l_2=2l_1$. However, in the actual mechanical design, considering the internal structure, general layout, etc., it is impossible to strictly meet the size requirement. Therefore, when the three-vertical-thruster AUH is actually in operation, there will always be an anti-torsional torque, which makes the AUH tend to rotate around its central axis. Therefore, the horizontal thrusters are still needed to balance this torque when the three-vertical-thruster AUH is diving.

For the vertical motion of the four-vertical-thruster AUH, the thrust force of the horizontal thruster is zero. The thruster model is:

$$\{\begin{array}{c}{F}_x=0\\ F_y=C_T\left({\omega }_5^2+{\omega }_6^2\right)\\ F_z=C_T\left({\omega }_1^2+{\omega }_2^2+{\omega }_3^2+{\omega }_4^2\right)\\ {\tau }_x=C_T{l}_2\left({\omega }_1^2+{\omega }_2^2-{\omega }_3^2-{\omega }_4^2\right)\\ {\tau }_y=C_T{l}_1\left(-{\omega }_1^2+{\omega }_2^2+{\omega }_3^2-{\omega }_4^2\right)\\ {\tau }_z=K_M\left({\omega }_1^2-{\omega }_2^2+{\omega }_3^2-{\omega }_4^2\right)\end{array}$$

(7)

Let the torque balance, ${\tau }_x={\tau }_y={\tau }_z=0$, as well. We can draw the result: ${\omega }_1={\omega }_2={\omega }_3={\omega }_4$. That is, for the vertical four push, as long as the output of their thrusters is the same, they can maintain a stable attitude.

From what has been discussed above, the following conclusions can be drawn: The two-vertical-thruster AUH cannot completely control the motion posture, whereas the other two AUHs can, which was the reason why we no longer chose the two-vertical-thruster structure and optimized the AUH on the basis of the four-vertical-thruster structure. For the three-vertical-thruster AUH, it is more difficult to control the motion, and the anti-torsional torque cannot be eliminated by connecting the propeller reversely due to the asymmetry of the number of thrusters. However, as countermeasures, we can give the single thruster a compensation coefficient, which is calculated from Equation (6), to balance the outputs of the three thrusters. In addition, we can use the thrust difference of the horizontal thrusters to offset the anti-torsional torque.

2.2. Hydrodynamic Simulation and Analysis

The hydrodynamic performance of different AUH models is evaluated with computational fluid dynamics (CFD) simulation method. Here, we mainly consider the number of thrusters in the vertical direction, so only the horizontal (surge) motion is studied. Thus, the AUH models (the three-vertical-thruster AUH and the four-vertical-thruster AUH, shown in the Figure 3) are simplified to keep only the vertical flow channels and neglect all the holes and accessories, which can greatly simplify the simulation calculation. The basic parameters of the AUH model are shown in

Table 1. Basic parameters of the AUH model used in simulation.

Parameter	Diameter (m)	Height (m)	Weight (kg)
Value	0.6	0.3	20.0

.

In the simulation, the AUH model is placed in a 6 × 6 × 9 m³ cube filled with water, as shown in Figure 4. The center of the AUH is 3 m away from the inflow boundary as well as the side wall, and 6 m away from the outflow boundary. The left wall is set as velocity inlet with a fixed horizontal velocity value, and the right wall is set as pressure outlet. The other four boundary surfaces are all set as symmetric boundaries. The AUH model edge is set as a stationary wall to ensure no sliding boundary conditions.

We adopt the commercial CFD software, ANSYS Fluent, and the geometry model of the AUH is composed with SOLIDWORKS. Next, we import the model to ANSYS ICEM, where structured meshes are generated from the basic geometric model of the AUH. The grid figure shown in Figure 4. The total cell number is 504,184. The Y+ value ranges from 18 to 50, concentrated around 35, which matches the suggested value for the $k-\epsilon$ turbulence model. The principal conditions employed in the CFD analysis are shown in

Table 2. Principal conditions employed in the CFD analysis.

Water tank size	6 × 6 × 9 m³
Turbulence model	$k-\epsilon$ model
Y+ value	18.0~50.0
Total no. of elements	504,184

.

The Navier–Stokes (NS) equations are:

$$\begin{gathered} \frac{\partial \overline{u}}{\partial x}+\frac{\partial \overline{v}}{\partial x}+\frac{\partial \overline{w}}{\partial x}=0 \\ \rho \frac{d\overline{V}}{dt}+\rho \left(\overline{V}·\nabla \right)\overline{V}=-\nabla \overline{p}+\mu {\nabla }^2\overline{V} \end{gathered}$$

where $\overline{V}={\left[ u v w \right]}^T$ is the velocities vector; $\rho$ is the fluid density; $\overline{p}$ is the fluid pressure; and $\mu$ is the dynamic viscosity.

The mass and momentum are considered with the Reynolds averaged method, which corresponds to the Reynolds Averaged Navier–Stokes (RANS) equations. In addition, the $k-\epsilon$ two-equation turbulence model is introduced to make it closed.

The simulation results are shown in Figure 5 and

Table 3. Statistical results of the simulation data at $v=0.5 \mathrm{m}/\mathrm{s}$.

Model	Average Value of Drag Coefficient	Average Value of Lift Coefficient	$$\mathbf{Variance} \mathbf{of} \mathbf{Lift} \mathbf{Coefficient} (\times {10}^{-4})$$
Three-vertical-thrusters with one thruster in front	0.4838	0.0316	3.4185
Three-vertical-thrusters with two thrusters in front	0.4886	0.0477	1.3933
Four-vertical-thrusters	0.5215	0.0178	0.7158

and

Table 4. Statistical results of the simulation data at $v=1 \mathrm{m}/\mathrm{s}$.

Model	Average Value of Drag Coefficient	Average Value of Lift Coefficient	$$\mathbf{Variance} \mathbf{of} \mathbf{Lift} \mathbf{Coefficient} (\times {10}^{-3})$$
Three-vertical-thrusters with one thruster in front	1.6231	0.1158	5.0000
Three-vertical-thrusters with two thrusters in front	1.6065	0.2015	3.0000
Four-vertical-thrusters	1.7546	0.0613	0.8698

.

It can be clearly seen that the resistance of the AUH increases with the increase in the number of vertical thrusters, and the change becomes more obvious when the AUH swims faster. When the moving speed $v=0.5 \mathrm{m}/\mathrm{s}$, the resistance of the four-vertical-thruster AUH is approximately 6.5% larger than the three-vertical-thruster type, and when $v=1 \mathrm{m}/\mathrm{s}$, the resistance of the four-vertical-thruster AUH is 9% larger than the three-vertical-thruster one. Therefore, reducing the number of vertical thrusters can indeed reduce the fluid drag during navigation and save power to a certain extent. For the three-vertical-thruster structure, the resistance of one thruster in front and two thrusters in front is almost the same. Therefore, it can be considered that the two structures have no influence on the navigation resistance of the AUH.

The four-vertical-thruster structure receives the minimum lift during navigation, and the change of lift is most stable. As the rotating object will generate a lift force when moving, the flat structure of the AUH is prone to a small rotation under the influence of inertia when moving, which generates the lift force that inclines the body. Whereas in the three-vertical-thruster structure, when the structure with one thruster is in the front, the thrust force provided by one thruster is not stable enough, resulting in a drastic change in its lift, which is not conducive to the motion stability of the AUH. Thus, under comprehensive consideration, we would better select the structure of three-vertical-thrusters with two thrusters in front, which has the best hydrodynamic performance.

2.3. Comparison Summary

Based on the analysis above, it can be concluded that the two-vertical-thruster AUH has the disadvantage of not being able to control the roll attitude in the water, which will lead to its inability to maintain stable motion in the vertical direction when big turbulences come. Therefore, we no longer consider the two-vertical-thruster AUH. The four-vertical-thruster AUH is the most stable and the motion control is also very simple, but the damage to the hydrodynamic profile is also largest, leading to the largest fluid resistance and the maximum energy loss. The three-vertical-thruster AUH combines all the advantages of being able to control roll attitude and maintain stable motion in the water while also saving one thruster to release more space and reduce fluid resistance losses as well.

For the three-vertical-thruster AUH, it also has its own disadvantages, such as the asymmetric structure, which make motion control difficult, and uneven weight distribution. Fortunately, we have corresponding solutions, which are proved to be effective in the following experiments.

All these advantages and disadvantages are summarized in

Table 5. Comparison of AUHs with different numbers of thrusters in the vertical direction.

Number of Thrusters	Advantages	Disadvantages
3	① Can control roll attitude ② Stable movement in vertical direction ③ Save one thruster	① The motion control is more difficult ② Unfavorable to weight balance ③ The anti-torsional torque cannot be offset
4	① Can control roll attitude ② Most stable movement in vertical direction ③ Easy motion control	① Maximum occupied space ② Maximum fluid resistance ③ Largest damage to the shell

.

3. Control Compensation of the Three-Vertical-Thruster AUH

3.1. Mathematical Model

To simplify the model before modeling, we made the following assumptions:

• The center of buoyancy of the AUH coincided with the center of mass.
• No environmental disturbances acted on the AUH.
• The AUH was considered as a rigid body.
• The hydrodynamic coefficients of the AUH were unchanged.
• All parts of the AUH were uniformly stressed.

The coordinate frame system is presented in Figure 6. The absolute linear position of the AUH is defined in the inertial frame with $\xi ={\left[ x y z \right]}^T$, and the angular position is defined in the inertial frame with$\eta ={\left[ \phi \theta \psi \right]}^T$. The origin of the body frame is considered in the center of mass of the vehicle. The rotation matrix from the body frame to the inertial frame is:

$$R=\left[\begin{array}{ccc}cos\Psi cos\theta & cos\Psi sin\theta sin\phi -sin\Psi cos\phi & cos\Psi sin\theta cos\phi +sin\Psi sin\phi \\ sin\Psi cos\theta & sin\Psi sin\theta sin\phi +cos\Psi cos\phi & sin\Psi sin\theta cos\phi -cos\Psi sin\phi \\ -sin\theta & cos\theta sin\phi & cos\theta cos\phi \end{array}\right]$$

(8)

In the inertial frame, Equation (1) in Section 2 can be written as:

$$R\times F_T-R\times F_f+G-F_B=m\frac{dV}{dt}$$

(9)

where $F_T$ is the force vector produced by thrusters; $F_f={\left[ f_x f_y f_z \right]}^T$ is the fluid resistance force, which can be calculated by $f_x\approx C_t\rho v^{2}S/2$, where $C_t$ is total fluid resistance coefficient; $G=mg$ is the gravity force, where $g$ is the gravity coefficient; $F_B=K_{b}mg$ is the buoyancy, where $K_b$ is the buoyancy coefficient equal to the ratio of gravity and buoyancy; $V={\left[ u v w \right]}^T$ is a linear velocities vector; and $R$ is the rotation matrix from the body frame to the inertial frame, shown as Equation (9).

Equation (2) in Section 2 can be written as the following equation:

$${\tau }_T-M_f-\Omega \times \left(J\Omega \right)=J\frac{d\Omega }{dt}$$

(10)

where ${\tau }_T$ is the thruster torque; $M_f={\left[ M_{fx} M_{fy} M_{fz}\right]}^T$is the fluid resistance moment; $M_{fx}=C_{w}av$; $C_w$ is the proportional coefficient of water resistance torque and rotational angular velocity; $a$ is the radius of the AUH; $\Omega ={\left[ p q r \right]}^T$ is the angular velocities vector; and $\Omega \times \left(J\Omega \right)$ represents the centripetal forces.

Additionally, because of the symmetries of the disc shape, we can view the inertia $J$ as a diagonal matrix:

$$J=\left[\begin{array}{ccc}{J}_x& 0& 0\\ 0& J_y& 0\\ 0& 0& J_z\end{array}\right]$$

(11)

The above five equations (Equations (4), (8)–(11)) can be solved to obtain the dynamic model of three-vertical-thruster AUH. Without considering the anti-torsional torque and with great simplifying, the mathematical model can be written as follows [13]:

$$\ddot{x}=\frac{C_T}{m}\left({\omega }_4{}^2+{\omega }_5{}^2\right)cos\Psi +\frac{1}{2m}{C}_{tx}\rho S_1\left({\dot{y}}^{2}sin\Psi -{\dot{x}}^{2}cos\Psi \right)$$

(12)

$$\ddot{y}=\frac{C_T}{m}\left({\omega }_4{}^2+{\omega }_5{}^2\right)sin\Psi -\frac{1}{2m}{C}_{ty}\rho S_1\left({\dot{y}}^{2}cos\Psi +{\dot{x}}^{2}sin\Psi \right)$$

(13)

$$\ddot{z}=-\frac{C_T}{m}\left({\omega }_1{}^2+{\omega }_2{}^2+{\omega }_3{}^2\right)-\frac{1}{2m}{C}_{tz}\rho {\dot{z}}^2{S}_2+\left(1-K_b\right)g$$

(14)

$$\ddot{\phi }=\frac{C_T}{J_x}l\left(-{\omega }_1{}^2+{\omega }_2{}^2\right)-\frac{C_w}{J_x}a\dot{\phi }$$

(15)

$$\ddot{\theta }=\frac{C_T}{J_y}\left({\omega }_1^2{l}_1+{\omega }_2^2{l}_1-{\omega }_3^2{l}_2\right)-\frac{C_w}{J_y}a\dot{\theta }$$

(16)

$$\ddot{\Psi }=\frac{C_T}{J_z}a\left({\omega }_4{}^2-{\omega }_5{}^2\right)-\frac{C_w}{J_z}a\dot{\Psi }$$

(17)

where ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$ are the rotating speed of the three vertical thrusters; ${\omega }_4$ and ${\omega }_5$ are the rotating speed of the two horizontal thrusters; $C_{tx}=C_{ty}$ is the resistance coefficient in the horizontal plane; $C_{tz}$ is the resistance coefficient in the vertical direction; and $S_1$ and $S_2$ are the horizontal and vertical wet surface area of the AUH. The expressions $K_b$, $C_T$, $C_w$, $a$, $l$, $l_1$, and $l_2$ were all noted earlier and are not repeated here.

3.2. Control Algorithm

The control system is responsible for the AUH depth control, navigation tracking, and attitude adjustment, etc. We have learned from previous work that a PID controller is sufficient to meet the control needs of the AUH, and a good control effect has been achieved through the PID controller [16]. Therefore, in this paper, we also choose a PID controller to realize the control method of the three-vertical-thruster AUH.

The control system is designed around a STM32H743 microprocessor based on an ARM-M7 kernel, which integrates the depth meter, the altimeter, and the inertial measurement unit (IMU) sensor and power output unit interface, positioning system interface, and energy unit interface.

For the PID algorithm, P, I, and D correspond to proportional, integral, and differential, respectively. The proportion link can speed up the dynamic response of the system. The integration link can reduce the steady state error. The differential link can reduce the overshoot of the system and make the system more stable. For the discrete PID control system, the time–domain equation is as follows:

$$u_i\left(t\right)=K_{P}e\left(i\right)+K_i{\displaystyle \sum }_{j=0}^{i}e\left(j\right)+K_d\left(e\left(t\right)-e\left(t-1\right)\right)$$

(18)

where $K_P$ represents the proportional gains; $K_I$ represents the integral gains; and $K_d$ represents the differential gains.

We build the control models in MATLAB according to the above obtained dynamic model (Equations (14)–(17)). The control system block diagram is given in Figure 7.

The heading control is relatively simple, and the adjustment of PID control parameters is relatively easy. However, the motion control of the AUH in the vertical direction is more difficult. Due to the special structure of the three-vertical-thrusters, the three thrusters in the vertical direction advance in the same direction during the depth control, whereas the two thrusters at the front end and one thruster at the end are required to advance in the reverse direction for the control of the pitch angle during the attitude control, and only the two thrusters at the front end are required to advance in the reverse direction for the control of the roll angle, so the control coupling is complicated.

Another problem is that when the three thrusters in the vertical direction work at the same time, due to the different number of thrusters at both ends, the same rotation speed of the thrusters will generate a moment that makes the AUH pitch. Therefore, we propose a solution to this problem: when we control the motion in the vertical direction, we can add a compensation coefficient to the two thrusters at the front end in the vertical direction, so that when the three thrusters in the vertical direction work at the same speed, the torque generated by the two thrusters at the front end is the same as that generated by the one thruster at the rear end. The calculation method of this compensation coefficient has already been given in the Section 2. It can be calculated by Equation (6).

Therefore, the problem of difficult vertical motion control of a three-vertical-thruster AUH is solved by adding a compensation coefficient, which is verified by subsequent experiments.

4. Experiments and Results

To verify the motion control feasibility and stability of the three-vertical-thruster AUH, several comparative experiments were carried out based on the experimental prototypes. It is noted that the experimental part of this paper is still focused on comparison. The feasibility and stability of the three-vertical-thruster AUH are verified by comparing the three-vertical-thruster AUH and the four-vertical-thruster AUH with their experiment results from the same experiments. The experimental prototypes are shown in the Figure 8, and its basic parameters are shown in

Table 6. Basic parameters of the experimental prototypes.

The Number of Thrusters	Diameter (m)	Height (m)	Weight (kg)
3	0.6	0.3	$\le$22
4	0.6	0.3	≤27

.

The control parameters in the PID controller, which were obtained by constant testing before the experiments, are shown in

Table 7. PID control parameters in experiments.

Properties	Symbols	Values
Yaw	$\left[\begin{array}{ccc}{K}_p& K_i& K_d\end{array}\right]$	$\left[\begin{array}{ccc}10& 0.01& 20\end{array}\right]$
Pitch	$\left[\begin{array}{ccc}{K}_p& K_i& K_d\end{array}\right]$	$\left[\begin{array}{ccc}10& 0.01& 12\end{array}\right]$
Roll	$\left[\begin{array}{ccc}{K}_p& K_i& K_d\end{array}\right]$	$\left[\begin{array}{ccc}5& 0.01& 6\end{array}\right]$
Depth	$\left[\begin{array}{ccc}{K}_p& K_i& K_d\end{array}\right]$	$\left[\begin{array}{ccc}20& 0.01& 24\end{array}\right]$

.

To verify the feasibility and stability of the three-vertical-thruster AUH, we conduct the diving experiment as follows: the AUH dives to a certain depth and maintains that depth while collecting attitude angle data and depth data during the descent. The results are shown in Figure 9 and Figure 10.

From the experiments, we make the following conclusion: we have realized the three-vertical-thruster AUH depth and attitude control, and it has a good control effect, which proves its feasibility. Moreover, the attitude angles of the AUH for the three-vertical-thruster AUH and the four-vertical-thruster AUH are both less than 5°, which shows that the attitude stability of the three-vertical-thruster AUH is not worse than that of the four-vertical-thruster AUH. Therefore, the three-vertical-thruster AUH can be considered to have good stability.

We also conducted a turning experiment to prove good maneuverability of the three-vertical-thruster AUH. We rotated the AUH while it was going straight by a fixed angle (90 degrees in the experiment) and the new heading was kept for the next move, while its yaw angle data were recorded. The yaw control step response curves are plotted in Figure 11.

Through the curves, we can draw the following conclusions: the three-vertical-thruster AUH also has a good control effect in turning motion. Due to the different experimental environment and prototype conditions, the two AUHs in the turning experiment rotate at different angular velocities. However, it can be confirmed that the angular velocity of the three-vertical-thruster AUH is not slower than that of the four-vertical-thruster AUH, which also proves that the three-vertical-thruster AUH has good maneuverability.

5. Conclusions

In this paper, a new type of AUH with three thrusters in the vertical direction is proposed. Under the premise of maintaining high maneuverability and motion stability, it has fewer thrusters compared with existing AUHs, which was proved by dynamic analysis. As a result, it retains the hydrodynamic performance of its disc shape to the maximum extent and reduces the fluid resistance when moving in the horizontal direction. Through CFD simulation, we know that it can reduce the fluid resistance by 6.5–10% compared with the four-vertical-thruster type, and the reduction depends on the moving speed of the AUH. In addition, although the weight distribution of the three-vertical-thruster AUH is not symmetrical, it can be compensated for by adding counterweight to the light parts. Although the motion control in the vertical direction is relatively complicated, the control compensation method we proposed in this paper has also achieved great control performance compared with the four-vertical-thruster AUH shown in experimental results. The attitude angles of the three-vertical-thruster AUH are all less than 5°.

The three-vertical-thruster structure seems to be a good design for an AUH because reducing the number of vertical thrusters can bring many benefits, such as less energy consumption and more installation space for sensors, but there is still much research not yet investigated, such as the appropriate control algorithm for its motion under the disturbance, the vehicle’s stability at high moving speed, motion sensitivity, and so on.