# Development of a Mobile Buoy with Controllable Wings: Design, Dynamics Analysis and Experiments

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## Abstract

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## 1. Introduction

## 2. Mobile Buoy with Controllable Wings

_{w}is introduced in Figure 2b. The longitudinal axis of the fuselage (L-axis) coincides with the direction of antenna installation, and the horizontal axis of the fuselage (H-axis) that along the central axis of the spherical pressure shell is perpendicular to the L-axis. α

_{w}is the wing rotation angle (the angle where the wing axis intersects with the H-axis). During underwater navigation, the mobile buoy achieves up and down motion driven by the net buoyancy and is propelled forward by the horizontal lift component acting on wings. In this paper, two operation modes are divided by the range of wing rotation angles: Argo mode (−80° ≤ α

_{w}≤ −90° or 80° ≤ α

_{w}≤ 90°) and Glider mode (0° < α

_{w}≤ −80° in climbing, 0° < α

_{w}≤ 80° in diving) by regulating α

_{w}to control the ratio of horizontal displacement to diving depth to change the glide angle, as shown in Figure 2b. To ensure that the hydrodynamic forces acting on the fuselage are consistent in all directions, the shape of the mobile buoy is designed as a circular sphere consisting of two hemispherical pressure shells, thus contributing to the stability of the mobile buoy in both two motion modes. Stability is a critical aspect in the design process, which needs a reasonable arrangement of the internal configuration. For the proposed mobile buoy, the center of gravity is under the center of buoyancy and both are located on the same vertical line. In addition, the batteries and associated hardware of the buoyancy regulating system can be arranged at the bottom of the mobile buoy, which can provide the restoring moment to improve stability.

## 3. Fluid–Multibody Coupling Dynamics Model

#### 3.1. Fluid–Multibody Coupling Analysis Method

_{i}is the velocity component of the fluid, u

_{i}′ is the pulsation velocity, f

_{i}is the external force component acting on the fluid, μ is the dynamic viscosity and p is the pressure.

#### 3.2. Multibody Dynamics Model of the Mobile Buoy

**E-XYZ**, the body frame

**e-xyz**and the velocity frame

**e’-x’y’z’**. The inertial frame is anchored to a reference point on the water surface, which be used to describe the position and attitude of the buoy. The initial heading of the mobile buoy on the water surface is the

**X**-axis of the inertial frame, the

**Z**-axis is vertically downward along the direction of the gravity and the

**Y**-axis can be defined by the right-hand rule. The body frame is fixed to the wing, and its origin

**e**is fixed to the center of buoyancy of the mobile buoy. The

**x**-axis of the body frame points to the bow of the buoy along the axis of the wing, the

**z**-axis is perpendicular to the

**x**-axis and the

**y**-axis can be acquired by the right-hand rule. As for the velocity frame, the origin coincides with the body frame, and the

**x’**axis points to the velocity direction. The

**z’**axis is vertically downward to the

**x’**axis in the plane

**e-xz**. The direction of the

**y’**axis is obtained by the right-hand rule. With reference to the inertial frame, the position and the attitude of the mobile buoy are described by position vector

**b**= [x, y, z] and angle vector

**η**= [ϕ, θ, ψ], respectively. In the body frame, the velocity and the angular velocity of mobile buoy can be defined as

**V**= [u, v, w] and

**Ω**= [p, q, r], respectively. Moreover, the assumptions about the motion of the mobile buoy are given as follows: (1) The mobile buoy has six degrees of freedom, including surge, sway, heave, roll, pitch and yaw, where the dynamic response of the surge, heave and pitch degrees of freedom are focused on to assess the performance of the vehicle in profiling motion and the other degrees of freedom are ignored. (2) In this study, external disturbances such as wind, waves and currents, which are influenced by non-ideal environmental conditions, are not taken into consideration.

**Y**-axis of the inertial frame can be expressed by φ.

_{w}is the wing rotation angle and θ represents the pitch angle. In the diving phase of the mobile buoy, both α

_{w}and θ are positive, and in the climbing phase of the mobile buoy, both α

_{w}and θ are negative.

_{x}, V

_{y}and V

_{z}represent the velocities of the mobile buoy along the

**X**-axis,

**Y**-axis and

**Z**-axis, respectively, all within the inertial frame. ${\mathit{R}}_{E}^{B}$ is the rotation matrix that maps the variables in the body frame to the inertial frame, which can be expressed as:

_{h}(assuming a uniform distribution), the nonuniformly distributed mass m

_{w}and the variable ballast mass m

_{b}. The body frame, designated

**e-xyz**, originates

**e**at the geometric center (GC), which is the application point for the buoyancy and moments. The point mass m

_{h}is simplified and positioned at GC, which are regarded as rigid mass points during the modeling. m

_{w}is located at

**r**with respect to the GC, providing a restoring moment that allows the proposed mobile buoy to return to the original equilibrium position after an external interference. The variable mass m

_{w}_{b}(water in the tank) is the adjustable mass, considered as a mass point with constant position (located in GC) and variable mass, which can control the diving and climbing of the proposed mobile buoy by adjusting the buoyancy. Thus, the total mass m

_{t}can be expressed as:

_{0}in the water can be represented as follows:

_{d}and F

_{l}are the drag force and the lift force exerted on the mobile buoy in the flow field, respectively, which can be expressed as:

_{l-fin}is the lift force acting on the wings, F

_{l-fuselage}is the lift force acting on the fuselage. F

_{d-fin}is the drag force of the wings. F

_{d-fuselage}is the drag force of the fuselage. The horizontal hydrodynamic forces acting on the proposed mobile buoy can be obtained by projecting F

_{d}and F

_{l}, which can be calculated as:

_{x-fluid}represents the hydrodynamic force in the

**X**-axis direction and F

_{z-fluid}is the hydrodynamic force in the

**Z**-axis direction.

**X**-axis direction is mainly provided by the horizontal component of the wing lift. During the Argo mode, when the wings of the mobile buoy are erected, the angle of attack is approximately 90°, causing the lift generated by the wing to be small or even negligible according to Rayleigh’s lift formula [36]. Therefore, the mobile buoy lacks horizontal displacement capability due to the horizontal component of lift being ignored, and can only perform vertical diving and climbing in an attitude of near vertical motion. During the Glider mode, the wing rotation angle α

_{w}can be controlled to ensure the horizontal component of the lift is sufficient to generate the forward propulsive force, thus driving the mobile buoy forward. The equations of motion in both horizontal and vertical directions can be formulated as:

_{x-total}represents the resultant force in the direction of the

**X**-axis, F

_{z-total}represents the resultant force in the direction of the

**Z**-axis. The horizontal displacement of the mobile buoy is represented by x, while its vertical displacement is represented by z.

_{total}is the resultant moment. M

_{fluid}is the hydrodynamic moment. I represents the inertia of the mobile buoy concerning the

**y**-axis, affecting the pitching motion of the buoy.

#### 3.3. Hydrodynamics Model of the Mobile Buoy

## 4. Dynamics Simulation Results

#### 4.1. Dynamic Responses in Argo Mode

_{0}∈ [−0.1 kg, 0.1 kg] and a wing rotation angle α

_{w}of 90°. At the beginning of the simulation, the attitude and net buoyancy of the mobile buoy are controlled by the controllable wings and buoyancy/gravity to dive vertically. After reaching the target depth, the buoyancy system is adjusted again to drive the mobile buoy climbing vertically. Figure 9a,b illustrate the fluctuations in forward and heave velocities during the diving and climbing phases. It can be found that the heave velocity changes periodically with the net buoyancy mass, but the forward velocity fluctuates around zero. This is mainly due to the fact that the wing is subjected to less lift in its upright position, resulting in the forward velocity being far away from the heave velocity. From Figure 9c–e, it can be found that although the buoy has pitching motion, its angular velocity, pitch moment and pitch angle are small and can be negligible, thus ensuring the stability. The motion trajectory during an operation cycle in Argo mode is illustrated in Figure 9f, and it can be clearly seen that the horizontal displacement is almost negligible compared with the diving depth; thus, the diving and climbing motion can be assumed to be performed in the vertical direction.

#### 4.2. Dynamic Responses in Glider Mode

_{w}, as the key to attitude adjustment and mode switching, needs more attention regarding its effects on the motion performance of the proposed mobile buoy. Hence, the net buoyancy mass variation range stays constant, and the five variation ranges of the wing rotation angle, α

_{w}∈ [−10°, 10°], [−20°, 20°], [−30°, 30°], [−40°, 40°] and [−50°, 50°], are given to analyze the dynamic responses. The relationship between wing rotation angle and velocity is shown Figure 10a,b, where the forward velocity shows a slight increase followed by a decrease as α

_{w}increases, while the heave velocity increases with the increase in α

_{w}. Figure 10c shows the time domain curve of the angular velocity with different α

_{w}. The smaller the angle of rotation of the wing, the more violent the fluctuations in pitching motion, and the less stable the buoy. Due to the periodic change of the wing rotation angle, the pitch angle of the buoy also undergoes a periodic variation, as shown in Figure 10d. In Figure 10e,f, notice that the F

_{x-total}and M

_{total}fluctuate greatly at 23 s. The main reason for this phenomenon is the simultaneous rotation of the wings during the buoyancy adjustment process, which puts the buoy into the climb phase and prepares it for the gliding motion that follows. In addition, it should be pointed out the buoy experienced the unstable and stable phases of the sawtooth glide successively, and its velocity, angular velocity and forces can be stabilized after a brief unstable state. Figure 10g illustrates the impact of the wing rotation angle on the motion velocity, clearly indicating that the motion velocity increases within a certain range of the wing rotation angle. In addition, it can be seen from Figure 10h that the motion trajectories of the buoy are different under different wing rotation angle. With the increase in the wing rotation angle, the horizontal displacement first increases and then decreases, and the horizontal displacement is maximum when the wing rotation angle is about 30°.

_{0}determines the buoyancy, which serves as the primary power source for the motion of the mobile buoy. In order to explore the dynamic response of the mobile buoy under different m

_{0}, five adjusting ranges of the net buoyancy mass, m

_{0}∈ [−0.1 kg, 0.1 kg], [−0.2 kg, 0.2 kg], [−0.3 kg, 0.3 kg], [−0.4 kg, 0.4 kg] and [−0.5 kg, 0.5 kg], are analyzed, respectively. As shown in Figure 11a,b, the forward velocity and heave velocity are increased as net buoyancy mass increases. Figure 11c,d shows the time domain curve of the angular velocity and pitch angle with different net buoyancy mass. It can be found that the larger the net buoyancy mass, the more violent the pitching motion fluctuation, and the worse the stability of the mobile buoy. Meanwhile, as shown in Figure 11e,f, there are relatively large fluctuations in F

_{x-total}and M

_{total}at the initial moment and about 23 s due to the coupling of buoyancy, wing rotation and flow field. Figure 11g shows the impact of net buoyancy mass on the motion velocity, and it can be observed that the velocity increases with the increase in net buoyancy mass. Thus, reasonable control of m

_{0}can regulate the period of the sawtooth motion. In addition, the effect of net buoyancy mass on motion trajectories, as shown in Figure 11h. It can be clearly seen that the diving depth and horizontal displacement gradually increase with the increases in net buoyancy mass, but the gliding attitude is almost constant, and the glide angle maintains at about 42°. Finally, it can be seen from the simulation result that while both the control input, α

_{w}and m

_{0}, influence the dynamic responses of the buoy, net buoyancy mass has a larger impact on the buoy velocity, while wing rotation angle has a larger impact on mobile buoy’s motion attitude.

## 5. Experimental Verification

#### 5.1. Experiment Results of Argo Mode

#### 5.2. Experiment Results of Glider Mode

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Mobile buoy with controllable wings, showing (

**a**) shape and structural design, (

**b**) switching motion mode by controlling wing rotation angle.

**Figure 7.**The hydrodynamics model, showing (

**a**) the 3D computational domain, (

**b**) a zoom-in view of the flow field mesh in X-Y slice plane, (

**c**) the surface mesh of the mobile buoy.

**Figure 9.**Dynamic responses in Argo mode, showing (

**a**) forward velocity, (

**b**) heave velocity, (

**c**) pitch angle, (

**d**) pitch moment, (

**e**) angular velocity around the

**Y**-axis direction and (

**f**) motion trajectory.

**Figure 10.**Dynamic responses under different wing rotation angles during the Glider mode, showing (

**a**) forward velocity, (

**b**) heave velocity, (

**c**) angular velocity around the

**Y**-axis direction (

**d**) pitch angle, (

**e**) force, (

**f**) moment, (

**g**) motion velocity and (

**h**) motion trajectory.

**Figure 11.**Dynamic responses under different net buoyancy masses during the Glider mode, showing (

**a**) forward velocity, (

**b**) heave velocity, (

**c**) angular velocity around the

**Y**-axis direction (

**d**) pitch angle, (

**e**) force, (

**f**) moment, (

**g**) motion velocity and (

**h**) motion trajectory.

**Figure 12.**Photographic sequences of motion process in Argo mode, showing (

**a**) diving phase, (

**b**) climbing phase.

**Figure 13.**Photographic sequences of motion process in Glider mode with different wing rotation angles during diving phase, showing (

**a**) α

_{w}= 30°, (

**b**) α

_{w}= 40° and (

**c**) α

_{w}= 50°.

**Figure 14.**Photographic sequences of motion process in Glier mode with different wing rotation angles during climbing phase, showing (

**a**) α

_{w}= 30°, (

**b**) α

_{w}= 40° and (

**c**) α

_{w}= 50°.

**Figure 15.**Photographic sequences of motion process in Glider mode with different net buoyancy mass during diving phase, showing (

**a**) m

_{0}= 0.3, (

**b**) m

_{0}= 0.4 and (

**c**) m

_{0}= 0.5.

**Figure 16.**Photographic sequences of motion process in Glider mode with different net buoyancy mass during climbing phase, showing (

**a**) m

_{0}= 0.3, (

**b**) m

_{0}= 0.4 and (

**c**) m

_{0}= 0.5.

Parameter | Value |
---|---|

Mass of mobile buoy m_{t} (kg) | 14.7 |

Inertia of the buoy with respect to y-axis I (kg·m^{2}) | 2.1 |

Span of the buoy W (m) | 1.4 |

Buoy fuselage length L (m) | 0.35 |

Water depth H (m) | 10 |

Fluid density ρ (g/cm^{3}) | 1.025 |

Buoy fuselage diameter d (m) | 0.35 |

Nonuniformly distributed mass m_{w} (kg) | 3 |

Net buoyancy mass m_{0} (kg) during the diving phase | 0.1~0.5 |

Net buoyancy mass m_{0} (kg) during the climbing phase | −0.5~−0.1 |

Displacement of the nonuniform hull-distributed point mass m_{w} with respect to GC r (m)_{w} | 0.14 |

Variation range of the wing rotation angle α_{w} (°) | −90°~90° |

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## Share and Cite

**MDPI and ACS Style**

Wang, H.; Chen, J.; Feng, Z.; Du, G.; Li, Y.; Tang, C.; Zhang, Y.; He, C.; Chang, Z.
Development of a Mobile Buoy with Controllable Wings: Design, Dynamics Analysis and Experiments. *J. Mar. Sci. Eng.* **2024**, *12*, 150.
https://doi.org/10.3390/jmse12010150

**AMA Style**

Wang H, Chen J, Feng Z, Du G, Li Y, Tang C, Zhang Y, He C, Chang Z.
Development of a Mobile Buoy with Controllable Wings: Design, Dynamics Analysis and Experiments. *Journal of Marine Science and Engineering*. 2024; 12(1):150.
https://doi.org/10.3390/jmse12010150

**Chicago/Turabian Style**

Wang, Haibo, Junsi Chen, Zhanxia Feng, Guangchao Du, Yuze Li, Chao Tang, Yang Zhang, Changhong He, and Zongyu Chang.
2024. "Development of a Mobile Buoy with Controllable Wings: Design, Dynamics Analysis and Experiments" *Journal of Marine Science and Engineering* 12, no. 1: 150.
https://doi.org/10.3390/jmse12010150