# Three-Dimensional Modeling of a Robotic Fish Based on Real Carp Locomotion

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Three Dimensional Dynamic Model and Motion Equations of the Robotic Fish

#### 2.1. Carp Locomotion

#### 2.2. Swimming Motion of the Robotic Fish

#### 2.3. Hydrodynamic Forces Acting on the Tail

#### 2.4. Modeling of the Up-Down Motion Mechanism

#### 2.5. Three Dimensional Model Equations of Motion

## 3. Implementations of the Fish-Like Motion

- Proportional optimum link lengths according to actual size,
- Flapping frequency,
- Swimming speed performance,
- Proportional physical parameters according to the carp,
- Trajectory tracking.

#### 3.1. Ability of the Fish-Like Motion

#### 3.2. Guidance and Trajectory Tracking

_{1}) reference and the second path is a 3D CCW circular path (P

_{2}) reference. Trajectory tracking responses indicate the time-dependent performances of the robotic fish model. Figure 19 shows the closed loop P

_{1}tracking performance during a 60 s simulation time.

_{1}reference are ±0.18 m and 0.2 Hz, respectively. The depth of P

_{1}is 3 m. The robotic fish tracks the P

_{1}reference with a speed of 0.42 m/s. Figure 20 presents the P

_{1}tracking errors of speed and orientation angles.

_{1}reference during closed loop tracking.

_{2}tracking performance during a 64 s simulation time. Diameter of the P

_{2}is 8 m and depth is 1.28 m. Initial positions of the robotic fish and P

_{2}reference begin the origin at X = 0 m, Y = 0 m, Z = 0 m.

_{2}reference with a speed of 0.4 m/s. Figure 23 also presents the P

_{2}tracking errors of speed and orientation angles.

_{1}and P

_{2}references on the x, y and z axis are evaluated by Root Mean Square Error (RMSE) performance index and given as:

_{2}tracking underwater. Figure 25 shows the top view above the sea, and the isometric view of the underwater environment is also shown in Figure 26.

## 4. Conclusions

_{1}and P

_{2}) are tracked to evaluate sufficient swimming performances in Figure 19 and Figure 22. The proposed closed loop control system is also adapted to VRML in order to validate the P

_{2}tracking in virtual marine environment. These simulation and experimental analyses show that the derived model achieves real carp motions to implement robotic fish prototypes for biomimetic design.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Kinematic scheme of the robotic fish: (

**a**) Equivalent model; (

**b**) Description of the one link.

**Figure 2.**Trajectory study of two swimming patterns of carp. The fish is simplified as a 5-joint linkage represented by different colors. The light color curves indicate a trajectory of the corresponding joint within a period of swimming pattern: (

**a**) Forward swimming pattern, period 350 ms; (

**b**) Turning swimming pattern, period 550 ms.

**Figure 3.**Path tracking of the all links: (

**a**) Forward swimming pattern; (

**b**) Turning swimming pattern.

**Figure 4.**One period of the tail: (

**a**) Trace of the travelling wave for 350 ms; (

**b**) Caudal fin angles for forward swimming pattern: ${\theta}_{5}=[\pi -0.6\pi ,\pi -1.15\pi ]$.

**Figure 5.**Real fish angles according to the active links: (

**a**) Forward swimming mode; (

**b**) Turning swimming mode.

**Figure 11.**Free-swimming motion of the robotic fish. It is noted that there are three swimming modes: Forward, Counter Clock Wise (CCW) and Clock Wise (CW) turning.

**Figure 17.**Closed loop responses of the 6-DoF system during 40 s simulation time: (

**a**) Forward speed control; (

**b**) Yaw angles tracking control.

**Table 1.**The closed loop control performance: Maximum overshoot (%) and settling time (s) of forward speed for (0.1 m/s)–(0.4 m/s) and yaw angles for (−40°)–(+40°).

Interval | Maximum Overshoot (%) | Settling Time (s) | |
---|---|---|---|

Forward Speed (m/s) | 0.1 | 6.80 | 16.30 |

0.2 | 4.40 | 11.80 | |

0.3 | 2.23 | 15.40 | |

0.4 | 1.65 | 12.20 | |

Yaw Angles (°) | −40 | 37.50 | 18.30 |

−30 | 16.66 | 8.80 | |

−20 | 15 | 16.65 | |

−10 | 10 | 8.20 | |

0 | 36 | 14.50 | |

10 | 27.50 | 10.80 | |

20 | 6.50 | 21.50 | |

30 | 15 | 17.60 | |

40 | 31.25 | 18.90 |

**Table 2.**The simultaneously closed loop control performance of the roll and pitch motions for step and stairs references.

Criteria | Step | Stairs | ||||||
---|---|---|---|---|---|---|---|---|

Value (°) | 20 | −15 | −20 | 0 | 20 | |||

Motion | Roll | Pitch | Roll | Pitch | Roll | Pitch | Roll | Pitch |

Maximum Overshoot (%) | - | - | - | - | 27 | - | 27.50 | - |

Settling Time (s) | 0.38 | 0.37 | 0.20 | 0.24 | 0.18 | 0.26 | 0.19 | 0.24 |

**Table 3.**The tracking errors for proposed closed loop control system: P

_{1}and P

_{2}indicate the generated 3D path trajectories.

Trajectory | e_{x} (m) | e_{y} (m) | e_{z} (m) |
---|---|---|---|

P_{1} | 0.0968 | 0.0131 | 0.1058 |

P_{2} | 0.0827 | 0.0107 | 0.0920 |

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**MDPI and ACS Style**

Ozmen Koca, G.; Bal, C.; Korkmaz, D.; Bingol, M.C.; Ay, M.; Akpolat, Z.H.; Yetkin, S.
Three-Dimensional Modeling of a Robotic Fish Based on Real Carp Locomotion. *Appl. Sci.* **2018**, *8*, 180.
https://doi.org/10.3390/app8020180

**AMA Style**

Ozmen Koca G, Bal C, Korkmaz D, Bingol MC, Ay M, Akpolat ZH, Yetkin S.
Three-Dimensional Modeling of a Robotic Fish Based on Real Carp Locomotion. *Applied Sciences*. 2018; 8(2):180.
https://doi.org/10.3390/app8020180

**Chicago/Turabian Style**

Ozmen Koca, Gonca, Cafer Bal, Deniz Korkmaz, Mustafa Can Bingol, Mustafa Ay, Zuhtu Hakan Akpolat, and Seda Yetkin.
2018. "Three-Dimensional Modeling of a Robotic Fish Based on Real Carp Locomotion" *Applied Sciences* 8, no. 2: 180.
https://doi.org/10.3390/app8020180