# Autonomous Underwater Robot Fuzzy Motion Control System with Parametric Uncertainties

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## Abstract

**:**

## 1. Introduction

## 2. Description of the Problem

## 3. Mathematical Model

_{2}. In this layer, a sea current is observed with a speed of w

_{2}. The density of water in this layer is taken equal to ${\rho}_{2}$.

_{1}. The density of water in this layer is taken to be equal to ${\rho}_{1}$. The initial submersion depth of the underwater vehicle is H

_{1}.

_{kp}is the square of the wings’ surface, c

_{0_spsh}is the coefficient of the body’s form, S

_{w}is the square of the body’s surface and c

_{0_w}is the coefficient of added mass of the water.

## 4. Synthesis of a Fuzzy Controller to Stabilize the Depth of an Underwater Robot

## 5. Simulation Results

## 6. The Study of the Efficiency of the Fuzzy Control System with a Stepwise Change in the Depth of Immersion AUV

_{1}and k

_{2}is ineffective. For example, let the new value of the given depth $h=1$ m. Figure 8 shows the operation of the FC, the coefficients k

_{1}and k

_{2}of which correspond to a depth of 10 m, and in Figure 9 the operation of the FC with the coefficients k

_{1}= 1.01 and k

_{2}= 12, specially tuned to change the depth by 1 m, is shown.

## 7. Study of the Efficiency of Fuzzy Control Systems with a Harmonic Control Law

_{2}. The results of determining the coefficient k

_{1}for various values of the coefficient k

_{2}under the condition of minimizing the error ${\epsilon}_{1}$ are given in Table 1, where ${\epsilon}_{1}$, ${\epsilon}_{2}$ are the total error for a stepwise and sinusoidal input signal, respectively; ${n}_{1}$ is the number of oscillations of the control action with a stepwise input signal, ${n}_{2}$ is the number of oscillations of the control effect when crossing through zero with a sinusoidal input signal (see Figure 10), and ${h}_{e\hspace{0.17em}\hspace{0.17em}\mathrm{max}}$ the maximum absolute value of the error when moving along a sinusoid.

_{2}, the error ${\epsilon}_{2}$ decreases, which allows one to achieve high-quality control with a sinusoidal input signal. Figure 10 shows the simulation results for k

_{2}= 100. It was also established that in this case for the step input signal, the error ${\epsilon}_{1}$ almost does not change, but the overshoot increases significantly, and the control signal becomes unacceptable for a real system since it has a large number of oscillations (see Table 1).

_{1}and k

_{2}of FC was also found in the automatic control of vertical movement for stepwise and harmonic input signals. This means that each of the above modes requires a separate synthesis of FC coefficients, which limits the application of such a regulator in practice [11,12,13].

## 8. Comparative Analysis of the Performance of a Fuzzy Control System and a System with PD Controller

_{1}and k

_{2}were determined experimentally. The experimental results are shown in Table 2, and the best characteristics that were obtained with coefficients k

_{1}= 60, k

_{2}= 420 are shown in Figure 11.

## 9. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 6.**Simulation of an underwater robot using typical blocks of the Simulink package and proposed mathematical model.

**Figure 7.**Graph of the given and the current depth of the AUV versus time without noise or external disturbances.

**Figure 8.**Graph of the given and the current depth of the AUV versus time with harmonic exogenous disturbance.

**Figure 9.**Control of the vertical movement of the AUV with a harmonic input signal: h

_{ref}is a graph of the depth; h

_{fuzz}—control signal.

**Figure 10.**Control of the vertical movement of the MPR with a regulator tuned to a harmonic input signal: h

_{ref}is a graph of the depth; h

_{fuzz}—control signal.

**Figure 11.**The transition process of stepwise changes in the depth of the AUV with PD-regulator: a graph of changes in depth.

Indicator | Arguments | |||||
---|---|---|---|---|---|---|

k_{1} | 0.64 | 0.61 | 0.61 | 0.62 | 0.61 | 0.62 |

k_{2} | 10 | 20 | 30 | 40 | 50 | 60 |

${n}_{1}$ | 3 | 4 | 5 | 6 | 6 | 7 |

${\epsilon}_{1}$, m∙s | 130.0 | 129.7 | 129.7 | 129.6 | 129.7 | 129.7 |

${n}_{2}$ | 1 | 1 | 2 | 2 | 2 | 2 |

${h}_{e\mathrm{max}}$, m | 1.00 | 0.60 | 0.42 | 0.33 | 0.27 | 0.23 |

${\epsilon}_{2}$, m∙s | 56.0 | 29.1 | 19.6 | 14.92 | 12.10 | 10.22 |

Indicator | Arguments | ||||||
---|---|---|---|---|---|---|---|

k_{1}, V/m | 200 | 150 | 100 | 90 | 80 | 70 | 60 |

k_{2}, V/m | 2000 | 1500 | 1000 | 900 | 700 | 500 | 420 |

${\epsilon}_{1}$, m∙s | 153.8 | 153.9 | 154.0 | 154.0 | 148.1 | 141.6 | 141.2 |

k_{1}, V/m | 40 | 35 | 30 | 25 | 20 | 15 | 10 |

k_{2}, V/m | 280 | 250 | 210 | 180 | 150 | 120 | 80 |

${\epsilon}_{1}$, m∙s | 141.3 | 141.9 | 141.4 | 142.3 | 143.7 | 146.3 | 147.4 |

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**MDPI and ACS Style**

Zhilenkov, A.; Chernyi, S.; Firsov, A.
Autonomous Underwater Robot Fuzzy Motion Control System with Parametric Uncertainties. *Designs* **2021**, *5*, 24.
https://doi.org/10.3390/designs5010024

**AMA Style**

Zhilenkov A, Chernyi S, Firsov A.
Autonomous Underwater Robot Fuzzy Motion Control System with Parametric Uncertainties. *Designs*. 2021; 5(1):24.
https://doi.org/10.3390/designs5010024

**Chicago/Turabian Style**

Zhilenkov, Anton, Sergei Chernyi, and Andrey Firsov.
2021. "Autonomous Underwater Robot Fuzzy Motion Control System with Parametric Uncertainties" *Designs* 5, no. 1: 24.
https://doi.org/10.3390/designs5010024