# Robust Fractional-Order Control Using a Decoupled Pitch and Roll Actuation Strategy for the I-Support Soft Robot

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Plant Model

#### 2.2. Control Strategy

- ${\varphi}_{m}=70\phantom{\rule{0.222222em}{0ex}}\mathrm{deg}$
- ${\omega}_{gc}=1.5\phantom{\rule{0.222222em}{0ex}}\mathrm{rad}/\mathrm{s}$

## 3. Results and Discussion

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

PID | Proportional integral derivative |

FOPID | Fractional order proportional integral derivative |

FOPI | Fractional order proportional integral |

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**Figure 2.**

**Left**: elongated I-Support module, with three equally actuated chambers.

**Right**: bended module with only one chamber inflated.

**Figure 3.**Two examples of identification experiments. The

**left**figure shows the system response to variations in the input ${\alpha}_{i}$, while the other inputs are kept constant (${\beta}_{i}=30$ and ${l}_{i}=0$).

**Right**figure shows the system response to variations in the input ${\beta}_{i}$, while the other inputs are kept constant (${\alpha}_{i}=20$ and ${l}_{i}=0$).

**Figure 4.**Validation example of the identified models showing two different behaviors. Data obtained from the case of variations in Alpha input ranging from $10\phantom{\rule{0.222222em}{0ex}}\mathrm{deg}$ to $20\phantom{\rule{0.222222em}{0ex}}\mathrm{deg}$ (left of Figure 3). Showing the Step input (Alpha input), and the Real response (Alpha output) used in the RLS identification. Resulting model time response is also shown for comparison.

**Figure 5.**Frequency response of the models obtained using RLS, showing all the identification experiments (

**left**) and the three most representative examples: Lowest, Average and Maximum Gains (

**right**).

**Figure 6.**Zero-pole representation (

**left**) and unit input time response (

**right**) for the three most representative models obtained: Lowest Gain (${G}_{min}$), Average Gain (${G}_{avg}$) and Maximum Gain (${G}_{max}$) models.

**Figure 7.**Frequency response (

**left**) and time response (

**right**) for the fractional order controller system.

**Figure 8.**Frequency response (

**left**) and time response (

**right**) for the integer order controller system.

**Figure 10.**Experiment 1. Time response (

**above**) and control signal (

**below**) for the two controllers tested, fractional (

**left**) and integer (

**right**) orders.

**Figure 11.**Experiment 2. Disturbance rejection. Time response for the two controllers tested, fractional (

**left**) and integer (

**right**) orders.

**Figure 12.**Experiment 3. Trajectory tracking. Time response for the two controllers tested, fractional (

**left**) and integer (

**right**) orders.

${\mathit{k}}_{\mathit{p}}$ | ${\mathit{k}}_{\mathit{i}}$ | $\mathit{\lambda}$ |
---|---|---|

$0.1878$ | $1.8279$ | $1.19$ |

${\mathit{k}}_{\mathit{p}}$ | ${\mathit{k}}_{\mathit{i}}$ | $\mathit{\lambda}$ |
---|---|---|

$0.0071$ | $1.6402$ | $1.00$ |

$\mathit{\alpha}$ | $\mathit{\beta}$ | l | |
---|---|---|---|

Point 1 | 20 | 20 | 40 |

Point 2 | −20 | 20 | 20 |

Point 3 | −20 | −20 | 40 |

Point 4 | 20 | −20 | 20 |

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**MDPI and ACS Style**

Muñoz, J.; Piqué, F.; A. Monje, C.; Falotico, E. Robust Fractional-Order Control Using a Decoupled Pitch and Roll Actuation Strategy for the I-Support Soft Robot. *Mathematics* **2021**, *9*, 702.
https://doi.org/10.3390/math9070702

**AMA Style**

Muñoz J, Piqué F, A. Monje C, Falotico E. Robust Fractional-Order Control Using a Decoupled Pitch and Roll Actuation Strategy for the I-Support Soft Robot. *Mathematics*. 2021; 9(7):702.
https://doi.org/10.3390/math9070702

**Chicago/Turabian Style**

Muñoz, Jorge, Francesco Piqué, Concepción A. Monje, and Egidio Falotico. 2021. "Robust Fractional-Order Control Using a Decoupled Pitch and Roll Actuation Strategy for the I-Support Soft Robot" *Mathematics* 9, no. 7: 702.
https://doi.org/10.3390/math9070702