# Dynamic Walking of a Legged Robot in Underwater Environments

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Underwater Inverted Pendulum (UIP)

#### 2.1.1. Simplification of Buoyant Force

#### 2.1.2. Hydrodynamic Damping

#### 2.2. Underwater Zero Moment Point

#### Walking Pattern Generation

#### 2.3. Reaction Step for Balance Recovery

#### 2.4. Dynamic and Control Position

#### 2.4.1. Inverse Dynamic

#### 2.4.2. Control Position

#### 2.5. Description of Prototype

#### 2.5.1. Hydrodynamic Mass

#### 2.5.2. Damping Coefficients

#### 2.6. Experiment

## 3. Results

#### 3.1. Walking

#### 3.2. Reaction Step for Balance Recovery

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Development of underwater locomotion, where the locomotion types are represented on the vertical axis, and the year in which they were developed on the horizontal axis.

**Figure 2.**Inverted pendulum model in an underwater environment. Here, m’ is the hydrodynamic mass of the water, m is the mass of the robot trunk, D is the hydraulic damping force, mg is the weight, B is the buoyancy of the body and τ is the resulting torque.

**Figure 3.**Model Predictive control for the calculation of the position and velocity of the centre of mass.

**Figure 4.**Displacement of the Uzmp when the inverted pendulum is disturbed by the movement of the fluid (V).

**Figure 7.**(

**a**) A biped robot used by MIT’s leg laboratory, (

**b**) the Mabel robot of the University of Michigan, (

**c**) Atrias of the Oregon State University’s Dynamic Robotics Laboratory and (

**d**) Rabbit of Laboratoire de Grenoble Automatique.

**Figure 10.**Leg rotation mechanism, (

**a**) cylinder bar introduced, this rotates the leg clockwise, (

**b**) cylinder bar outside, this rotates the leg counterclockwise, (

**c**) top view of each actuator.

**Figure 11.**CAD design of the sensor. Inertial measurement unit 1 (IMU1) is in the sphere of the sensor, and IMU2 is in the base of the sensor. Also, the 2-UPS prismatic links and 1-RU rotational link are shown.

**Figure 18.**Hydrodynamic forces that are obtained by each fluid velocity to determine the damping constants.

**Figure 24.**Fluid velocities that are measured by the mechanical sensor at the time of water disturbance.

body | m (kg) | m’ (kg) | m + m’ (kg) |
---|---|---|---|

Hip | 4 | 4.2 | 8.2 |

Leg1 | 0.7 | 0.186 | 0.886 |

Leg2 | 0.7 | 0.186 | 0.886 |

axes | 0.5 | 0.079 | 0.579 |

sum | 10.551 |

Constants | |
---|---|

z | 0.7 m |

${X}_{u\left|u\right|}$ | $134.4{\text{}\mathrm{Ns}}^{2}/{\mathrm{m}}^{2}$ |

${X}_{u}$ | $18.6\text{}\mathrm{Ns}/\mathrm{m}$ |

$m$ | 4 kg |

${m}^{\prime}$ | 4.2 kg |

g | 9.8 m/s^{2} |

$\lambda $ | 0.7 |

MPC | Prediction control = 90, Control horizon = 20 |

${\left|\dot{x}\right|}_{0}$ | 0.02 m/s |

$\mu $ | 0.8 |

PI | Q1 (0.04, 300); Q2 (0.04, 300); L1 (0.05, 150); L2 (0.05, 150) |

kp | Q1 (−0.15); Q2 (−0.15); L1 (−0.1); L2 (−0.1) |

K | 120 N/m |

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**MDPI and ACS Style**

Portilla, G.; Saltarén, R.; Montero de Espinosa, F.; R. Barroso, A.; Cely, J.; Yakrangi, O.
Dynamic Walking of a Legged Robot in Underwater Environments. *Sensors* **2019**, *19*, 3588.
https://doi.org/10.3390/s19163588

**AMA Style**

Portilla G, Saltarén R, Montero de Espinosa F, R. Barroso A, Cely J, Yakrangi O.
Dynamic Walking of a Legged Robot in Underwater Environments. *Sensors*. 2019; 19(16):3588.
https://doi.org/10.3390/s19163588

**Chicago/Turabian Style**

Portilla, Gerardo, Roque Saltarén, Francisco Montero de Espinosa, Alejandro R. Barroso, Juan Cely, and Oz Yakrangi.
2019. "Dynamic Walking of a Legged Robot in Underwater Environments" *Sensors* 19, no. 16: 3588.
https://doi.org/10.3390/s19163588