# Fractional-Order Sensing and Control: Embedding the Nonlinear Dynamics of Robot Manipulators into the Multidimensional Scaling Method

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Fractional Integrodifferential Operators

#### 2.2. Variable Structure Control

#### 2.3. Fractional-Order Sensor

#### 2.4. The Multidimensional Scaling Technique

## 3. Multidimensional Analysis and Visualization of Variable Structure Control

#### 3.1. Integer Variable Structure Control and Fractional Sliding Mode

#### 3.2. Fractional Variable Structure Control and Integer Sliding Mode

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 3.**The Bode diagrams of a 6-stage fractional sensor, yielding the fractional orders: (

**top**) $\alpha =0.16$; (

**bottom**) $\alpha =0.40$.

**Figure 4.**The sensor time responses ${y}_{1}(t$) to the input signals $u\left(t\right)=\left\{\delta \right(t),h(t\left)\right\}$.

**Figure 5.**The 3-dim MDS locus for the IVSC-FSM, assessing the time-domain behavior, with the distances: (

**top**) Arccosine ${d}_{1}({\mathbf{v}}_{i}^{\left(t\right)},{\mathbf{v}}_{j}^{\left(t\right)})$; and (

**bottom**) Jaccard ${d}_{2}({\mathbf{v}}_{i}^{\left(t\right)},{\mathbf{v}}_{j}^{\left(t\right)})$. The points represent the test cases. On the left, lines connect points of constant $\delta $, and each color corresponds to points of constant $\alpha $. On the right, lines connect points of constant $\alpha $, and each color corresponds to points of constant $\delta $. The fractional order ${\alpha}_{q}\in [-0.5,1]$, and the width of the band ${\delta}_{q}\in [{10}^{-4},{10}^{1}]$.

**Figure 6.**The 3-dim MDS locus for the IVSC-FSM, assessing the frequency-domain behavior, with the distances: (

**top**) Arccosine ${d}_{1}({\mathbf{v}}_{i}^{\left(f\right)},{\mathbf{v}}_{j}^{\left(f\right)})$; and (

**bottom**) Jaccard ${d}_{2}({\mathbf{v}}_{i}^{\left(f\right)},{\mathbf{v}}_{j}^{\left(f\right)})$. The points represent the test cases. On the left, lines connect points of constant $\delta $, and each color corresponds to points of constant $\alpha $. On the right, lines connect points of constant $\alpha $, and each color corresponds to points of constant $\delta $. The fractional order ${\alpha}_{q}\in [-0.5,1]$, the width of the band ${\delta}_{q}\in [{10}^{-4},{10}^{1}]$, and ${f}_{r}\in [{10}^{-2},{10}^{2}]$ Hz.

**Figure 7.**The 3-dim MDS locus for the FVSC-ISM, assessing the time domain-behavior, with the distances: (

**top**) Arccosine ${d}_{1}({\mathbf{v}}_{i}^{\left(t\right)},{\mathbf{v}}_{j}^{\left(t\right)})$; and (

**bottom**) Jaccard ${d}_{2}({\mathbf{v}}_{i}^{\left(t\right)},{\mathbf{v}}_{j}^{\left(t\right)})$. The points represent the test cases. On the left, lines connect points of constant $\delta $, and each color corresponds to points of constant $\alpha $. On the right, lines connect points of constant $\alpha $, and each color corresponds to points of constant $\delta $. The fractional order ${\alpha}_{q}\in [-0.2,6]$, and the width of the band ${\delta}_{q}\in [{10}^{-4},{10}^{1}]$.

**Figure 8.**The 3-dim MDS locus for the FVSC-ISM, assessing the frequency-domain behavior, with the distances: (

**top**) Arccosine ${d}_{1}({\mathbf{v}}_{i}^{\left(f\right)},{\mathbf{v}}_{j}^{\left(f\right)})$; and (

**bottom**) Jaccard ${d}_{2}({\mathbf{v}}_{i}^{\left(f\right)},{\mathbf{v}}_{j}^{\left(f\right)})$. The points represent the test cases. On the left, lines connect points of constant $\delta $, and each color corresponds to points of constant $\alpha $. On the right, lines connect points of constant $\alpha $, and each color corresponds to points of constant $\delta $. The fractional order ${\alpha}_{q}\in [-0.2,0.6]$, the width of the band ${\delta}_{q}\in [{10}^{-4},{10}^{1}]$, and ${f}_{r}\in [{10}^{-2},{10}^{2}]$ Hz.

**Table 1.**Two optimal solutions determined by the PSO for $N=6$, yielding the fractional orders $\alpha =\{0.16,0.40\}$.

Order | Stage Elements | Parameters | |||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}$ | {${\mathit{M}}_{\mathit{i}}$,${\mathit{B}}_{\mathit{i}}$,${\mathit{K}}_{\mathit{i}}$},$\mathit{k}=1,\cdots ,6$ | I | $\mathit{\sigma}$ | $\Omega $ | |||||

1 | 2 | 3 | 4 | 5 | 6 | ||||

0.16 | 0.30, 0.09, 0.34 | 0.63, 0.33, 0.59 | 0.87, 0.22, 0.38 | 0.22, 0.98, 1.38 | 0.17, 0.44, 0.49 | 0.76, 0.77, 0.15 | 1.08 | 6.89 | 4.25 |

0.40 | 0.12, 0.78, 0.93 | 0.57, 0.22, 0.58 | 0.84, 0.75, 0.12 | 0.54, 0.08, 0.84 | 0.67, 0.36, 0.85 | 1.12, 1.15, 1.61 | 0.97 | 9.29 | 4.73 |

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**MDPI and ACS Style**

Lopes, A.M.; Tenreiro Machado, J.A.
Fractional-Order Sensing and Control: Embedding the Nonlinear Dynamics of Robot Manipulators into the Multidimensional Scaling Method. *Sensors* **2021**, *21*, 7736.
https://doi.org/10.3390/s21227736

**AMA Style**

Lopes AM, Tenreiro Machado JA.
Fractional-Order Sensing and Control: Embedding the Nonlinear Dynamics of Robot Manipulators into the Multidimensional Scaling Method. *Sensors*. 2021; 21(22):7736.
https://doi.org/10.3390/s21227736

**Chicago/Turabian Style**

Lopes, António M., and José A. Tenreiro Machado.
2021. "Fractional-Order Sensing and Control: Embedding the Nonlinear Dynamics of Robot Manipulators into the Multidimensional Scaling Method" *Sensors* 21, no. 22: 7736.
https://doi.org/10.3390/s21227736