# Design and Control of an Underwater Robot Based on Hybrid Propulsion of Quadrotor and Bionic Undulating Fin

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## Abstract

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## 1. Introduction

**Figure 1.**Swimming modes associated with (

**a**) BCF propulsion and (

**b**) MPF propulsion. Shaded areas contribute to thrust generation [28]. Reproduced with permission from J.E. Colgate, IEEE Journal of Oceanic Engineering; published by IEEE, 2004.

## 2. Mechanical Structure

## 3. Control Model

#### 3.1. Kinematic Model

#### 3.2. Dynamic Model

#### 3.2.1. Dynamic Model of Propellers

#### 3.2.2. Dynamic Model of Undulating Fin

#### 3.2.3. Model of Resilience

#### 3.2.4. Model of Resistance

#### 3.3. Equation of Motion Control

- 1.
- Ignore the additional mass force [32], i.e., $\mathit{M}(\mathit{v})=\mathrm{diag}\left(m,m,m,{J}_{xx},{J}_{yy},{J}_{zz}\right)$;
- 2.
- Assume that Coriolis force and centrifugal force have negligible influence on the motion [32], i.e., $\mathit{C}(\mathit{v})=0$;
- 3.
- According to the dynamic analysis, the control component of the robot along the Y-axis direction is 0, so the sway motion along Y-axis is ignored.

#### 3.4. Control Model and Strategy

## 4. Simulation and Experiment

#### 4.1. Prototype and Experimental Environment

#### 4.2. Simulation Parameters and Models

#### 4.3. Simulation and Experiment Results

#### 4.3.1. Heave Motion

- The robot can effectively realize the closed-loop control of depth, and the sinking and floating speed can reach 0.27 m/s;
- The heave motion of the robot is independent, the robot has no surge motion, sway motion, and tilt of attitude angle, which indicates that the robot has achieved independent motion along the Z-axis;
- The robot has good motion stability, the roll, pitch, and yaw angle of the robot remain stable during the process of descent, hovering, and floating.

#### 4.3.2. Yaw Motion

#### 4.3.3. Steering Motion

#### 4.3.4. Surge Motion

- The robot can achieve attitude stability in surge movement, and the angle fluctuation range is less than ±5°;
- The depth closed-loop controller and attitude angle closed-loop controller play an effective role in regulation;
- The surge motion of the robot is independent and does not accompany the motion of the other five degrees of freedom.

#### 4.3.5. Velocity Experiment

## 5. Discussion

#### 5.1. Performance of Maneuverability

#### 5.2. Performance of Stability

#### 5.3. Control Strategy

#### 5.4. Applicability

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Massm | $15.02\mathrm{kg}$ | |

Y-axis inertia${J}_{xx}$ | $0.119\mathrm{kg}\cdot {\mathrm{m}}^{2}$ | |

Mass parameter | Y-axis inertia${J}_{yy}$ | $0.592\mathrm{kg}\cdot {\mathrm{m}}^{2}$ |

Z-axis inertia${J}_{zz}$ | $0.563\mathrm{kg}\cdot {\mathrm{m}}^{2}$ | |

No-load buoyancyB | $160$ | |

Buoyant center${z}_{B}$ | −0.04 m | |

Distance between the propellersL | 0.281 m | |

Angle between the diago-nal$\alpha $ | 23.6° | |

Propeller parameter | Maximum rotating speed${w}_{\mathrm{max}}$ | $10,000\mathrm{rpm}$ |

Thrust coefficient${c}_{T}$ | $2.188\times {10}^{-7}$ | |

Reverse torque coefficient${c}_{M}$ | $1.944\times {10}^{-9}$ | |

Fin width$d$ | 0.143 m | |

Flexible arc angle$\alpha $ | $\pi /3$ | |

Arc inner diameterR | 0.8 m | |

Wave length$\mathsf{\lambda}$ | 0.419 m | |

undulating amplitude$\mathsf{\lambda}$ | 0.06 m | |

Fin coefficient${c}_{f}$ | $6.276\times {10}^{-4}$ | |

Fin parameter | Fin center${z}_{F}$ | 0.16 m |

Maximum undulating fre-quency${f}_{\mathrm{max}}$ | 6 Hz |

- 1.
- The model of surge motion: $\dot{u}=0.067{\tau}_{X}-(0.048+0.832|u|)u+0.653{s}_{\theta}$;
- 2.
- The model of heave motion: $\dot{w}=0.067{\tau}_{Z}-(0.733+5.256|w|)w-0.653{c}_{\theta}{c}_{\varphi}$;
- 3.
- The model of roll motion: $\dot{p}=8.403{\tau}_{K}-(0.639+2.823|p|)p-52.7{c}_{\theta}{s}_{\varphi}$;
- 4.
- The model of pitch motion: $\dot{q}=1.689{\tau}_{M}-(0.045+1.533|q|)q-52.7{s}_{\theta}$;The model of yaw motion: $\dot{r}=1.776{\tau}_{N}-(0.024+5.291|r|)r$.

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**Figure 2.**The overall design of the underwater robot. (

**a**) Design model of the underwater robot; (

**b**) Undulating fin tilts left side, ${\theta}_{left}$ is tilting angel; (

**c**) Undulating fin tilts right side, ${\theta}_{right}$ is tilting angel.

**Figure 3.**Design of the mechanical fin. (

**a**) Modular mechanical fin design; (

**b**) Phase adjustment mechanism; Four sides of the rectangular shaft correspond to four phases, that is, 0, π/2, π, and 3π/2; (

**c**) Cam mechanism; (

**d**) Self-adaptive oscillation fin ray; (

**e**) Initial geometry of the undulating fin; (

**f**) Straightened undulating fin; (

**g**) Driving and tilting transmission principle of the undulating fin.

**Figure 4.**Kinematics and mechanical model of the robot. (

**a**) Inertial coordinate system and robot coordinate system; (

**b**) Dynamic model of the robot.

**Figure 7.**Numerical simulation analysis of damping coefficient. (

**a**,

**b**) FEM of the robot; (

**c**) Resistance–velocity curve of uniaxial linear motion by CFD simulation; (

**d**) Moment of the resistance–velocity curve of uniaxial rotational motion by CFD simulation.

**Figure 9.**Results of heaving motion. (

**a**) Simulation result of heaving motion; (

**b**) Experiment result of heaving motion; (

**c**) Time sequence of heave motion.

**Figure 10.**Results of yaw motion. (

**a**) Simulation result of yaw motion; (

**b**) Experiment result of yaw motion; (

**c**) Time sequence of yaw motion.

**Figure 11.**Results of steering motion. (

**a**) Time sequence in steering-left motion experiment, in which case the undulating fin was tilted to the right side; (

**b**) Time sequence in steering-right motion experiment, in which case the undulating fin was tilted to the left side.

**Figure 12.**Results of surge motion. (

**a**) Simulation result of surge motion; (

**b**) Experiment result of surge motion; (

**c**) Time sequence of surge motion.

**Figure 13.**The (

**a**) Simulation and (

**b**) Experiment results of propulsive velocity (V

_{x}) versus undulating fre-quency (f).

Primary Damping Coefficient | Secondary Damping Coefficient | ||
---|---|---|---|

${X}_{u}$ | 0.7164 | ${X}_{u|u\mid}$ | 12.4744 |

${Y}_{v}$ | - | ${Y}_{v\left|v\right|}$ | - |

${Z}_{w}$ | 10.9893 | ${Z}_{w\left|w\right|}$ | 78.8397 |

${K}_{p}$ | 0.0760 | ${K}_{p\left|p\right|}$ | 0.3359 |

${M}_{q}$ | 0.0264 | ${M}_{q\left|q\right|}$ | 0.9075 |

${N}_{r}$ | 0.0134 | ${N}_{r\left|r\right|}$ | 2.9787 |

Name | Type | Velocity (m/s) | Velocity (BL/s) | Frequency (Hz) |
---|---|---|---|---|

Carangiform swimming robot [9] | BCF | 1.76 | 2.2 | 3.5 |

Ostraciiform underwater robot [12] | BCF | 0.3 | 2 | 14 |

Single motor actuated robotic fish [33] | BCF | 1.14 | 3.08 | 9 |

Wire-driven robotic fish [34] | BCF | 0.08 | 0.25 | 1 |

Four link robotic fish [35] | BCF | 0.32 | 0.8 | 2 |

EMA robotic fish [36] | BCF | 0.03 | 0.33 | 3.7 |

Robo-ray II [8] | MPF | 0.16 | 0.5 | 1.2 |

Robotic stingray [37] | MPF | 0.04 | 0.18 | 0.5 |

Undulatory Fin Prototype [38] | MPF | 0.67 | 1.45 | 1.5 |

Robot in this paper | MPF | 1.2 | 1.5 | 6 |

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## Share and Cite

**MDPI and ACS Style**

Zeng, X.; Xia, M.; Luo, Z.; Shang, J.; Xu, Y.; Yin, Q.
Design and Control of an Underwater Robot Based on Hybrid Propulsion of Quadrotor and Bionic Undulating Fin. *J. Mar. Sci. Eng.* **2022**, *10*, 1327.
https://doi.org/10.3390/jmse10091327

**AMA Style**

Zeng X, Xia M, Luo Z, Shang J, Xu Y, Yin Q.
Design and Control of an Underwater Robot Based on Hybrid Propulsion of Quadrotor and Bionic Undulating Fin. *Journal of Marine Science and Engineering*. 2022; 10(9):1327.
https://doi.org/10.3390/jmse10091327

**Chicago/Turabian Style**

Zeng, Xiaofeng, Minghai Xia, Zirong Luo, Jianzhong Shang, Yuze Xu, and Qian Yin.
2022. "Design and Control of an Underwater Robot Based on Hybrid Propulsion of Quadrotor and Bionic Undulating Fin" *Journal of Marine Science and Engineering* 10, no. 9: 1327.
https://doi.org/10.3390/jmse10091327