# Analysis of Adaptive Algorithms Based on Least Mean Square Applied to Hand Tremor Suppression Control

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## Abstract

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## 1. Introduction

#### Related Publications

## 2. Mathematical Models of Tremors in Upper Limbs

#### 2.1. Physiological Tremor Model by Time Series

#### 2.2. Mathematical Model of Parkinsonian Tremor

## 3. Vibration Active Control

#### 3.1. Adaptive Algorithms

#### 3.1.1. Least-Mean Square (LMS) Algorithm

#### 3.1.2. Normalized Least Mean Square (NLMS) Algorithm

#### 3.1.3. Filtered-X Least Mean Square (Fx-LMS) Algorithm

## 4. Simulation Configurations

#### 4.1. Simulation Setup for Physiological Tremors Control

#### 4.2. Simulation Setup for Pathological Tremors Control

## 5. Results

#### 5.1. Simulation Result of Physiological Tremors Control

- Scenario 1 (Fx-LMS): The behavior of the Fx-LMS algorithm (Scenario 1) in the active control of physiological tremor vibration is shown in Figure 6. The steady state is reached after approximately 4000 samples. Figure 7 presents the power spectral density (PSD) which quantifies the error distribution over the frequency spectrum.

- Scenario 2 (Fx-NLMS): The Fx-NLMS algorithm has normalized data in both the main and secondary loops, and its result for controlling the physiological tremor signal is illustrated by Figure 8. In this case, approximately 4500 samples are needed for the control signal to converge, following the reference values, and it is enough to minimize the tremors. In Figure 9, we present the power spectral density (PSD) to quantify the error distribution over the frequency spectrum.

- Scenario 3 (Fx-LMS&NLMS): In Scenario 3, the hybrid, the control structure uses the Fx-LMS–NLMS algorithms, which have normalized data in the secondary loop error. This arrangement of algorithms is a suggestion of this work regarding a change in the normalization of the secondary path only. This combination enables a gain in performance. Figure 10 shows the control result of the algorithm, where the steady state is reached with approximately 4000 samples. Figure 11 provides a clear and quantitative analysis of the frequency content of the error signal.

#### 5.2. Simulation Result of Pathological Tremors Control

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Active Vibration Control | AVC |

Analog-to-digital converters | ADC |

Digital Signal Processor | DSP |

Digital-to-analog converters | DAC |

Essential Tremors | ET |

Filtered-x Least Mean Square | Fx-LMS |

Filtered-x Normalized Least Mean Square | Fx-NLMS |

Finite Impulse Response | FIR |

Internal Model Control | IMC |

Least Mean Square | LMS |

Mean Square Error | MSE |

Normalized Least Mean Square | NLMS |

Parkinson’s Disease | PD |

Proportional, Integrative, and Derivative | PID |

Recursive Least Square | RSL |

World Health Organization | WHO |

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**Figure 2.**Weight diagram of the LMS algorithm [26].

**Figure 6.**Control evaluation results produced by the Fx-LMS algorithm for 30 runs of a Monte Carlo simulation.

**Figure 8.**Control evaluation results produced by the Fx-NLMS algorithm for 30 runs of a Monte Carlo simulation.

**Figure 12.**Comparison of the normalized MSE behavior averaged over 30 runs. Ragged curves (yellow): Monte Carlo simulation of Fx-LMS. Ragged curves (blue): Monte Carlo simulation of Fx-NLMS. Ragged curves (red): Monte Carlo simulation of Fx-LMS–NLMS.

**Figure 13.**Control evaluation results produced by the FsinX-LMS algorithm for 30 runs of a Monte Carlo simulation.

**Figure 14.**Normalized-MSE results produced by the FsinX-LMS algorithm for 30 runs of a Monte Carlo simulation.

**Figure 15.**Behavior of the power spectral density (PSD) of the error signal by Fx-LMS in the pathological tremor control scenario.

Author | Technique | Control |
---|---|---|

[13] | Motors’ arrangement | Passive |

[14] | PID Control/Low-Pass Filter | Active |

[15] | PID Control | Active |

[16] | Neural Network | Active |

[17] | Adaptive Filter/Intern Control | Active |

This paper | Adaptive Filters | Active |

Algorithm | Samples | MSE |
---|---|---|

Fx-LMS | 4000 | −22 dB |

Fx-NLMS | 4500 | −22 dB |

Fx-LMS–NLMS | 4000 | −28 dB |

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**MDPI and ACS Style**

Araújo, R.S.A.; Tironi, J.C.; Parreira, W.D.; Borges, R.C.; De Paz Santana, J.F.; Leithardt, V.R.Q.
Analysis of Adaptive Algorithms Based on Least Mean Square Applied to Hand Tremor Suppression Control. *Appl. Sci.* **2023**, *13*, 3199.
https://doi.org/10.3390/app13053199

**AMA Style**

Araújo RSA, Tironi JC, Parreira WD, Borges RC, De Paz Santana JF, Leithardt VRQ.
Analysis of Adaptive Algorithms Based on Least Mean Square Applied to Hand Tremor Suppression Control. *Applied Sciences*. 2023; 13(5):3199.
https://doi.org/10.3390/app13053199

**Chicago/Turabian Style**

Araújo, Rafael Silfarney Alves, Jéssica Cristina Tironi, Wemerson Delcio Parreira, Renata Coelho Borges, Juan Francisco De Paz Santana, and Valderi Reis Quietinho Leithardt.
2023. "Analysis of Adaptive Algorithms Based on Least Mean Square Applied to Hand Tremor Suppression Control" *Applied Sciences* 13, no. 5: 3199.
https://doi.org/10.3390/app13053199