# Identification and Control of Rehabilitation Robots with Unknown Dynamics: A New Probabilistic Algorithm Based on a Finite-Time Estimator

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

**Remark**

**1.**

## 3. Gaussian Processes

#### 3.1. Gaussian Processes Formulation

#### 3.2. Comparison with Other Techniques

## 4. The Proposed Identification Algorithm

#### 4.1. Finite-Time Estimator

**Lemma**

**1.**

**Lemma**

**2.**

**Theorem**

**1.**

**Proof.**

**Lemma**

**3.**

#### 4.2. The Algorithm

#### 4.3. Feedback Linearization Augmented by the Proposed Algorithm

#### 4.4. Details for Implementing the Proposed Algorithm

## 5. Numerical Results

#### 5.1. Identifying Time-Varying Continuous Dynamic

#### 5.2. Identifying Time-Varying Discontinuous Dynamic

#### 5.3. Using Identified Dynamic for Control of the Rehabilitation Robot

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The applied torque to the system for identifying unknown dynamics presented in Equation (28).

**Figure 5.**The applied torque to the system for identifying unknown dynamics presented in Equation (29).

**Figure 6.**Values of ${\phi}_{1}$ and ${\dot{\phi}}_{1}$ obtained by applying the feedback linearization controller based on the information obtained using the proposed algorithm.

**Figure 7.**Values of ${\phi}_{2}$ and ${\dot{\phi}}_{2}$ obtained by applying the feedback linearization controller based on the information obtained using the proposed algorithm.

**Figure 8.**The applied torque to the system using the feedback linearization controller based on the proposed algorithm.

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**MDPI and ACS Style**

Alotaibi, N.D.; Jahanshahi, H.; Yao, Q.; Mou, J.; Bekiros, S.
Identification and Control of Rehabilitation Robots with Unknown Dynamics: A New Probabilistic Algorithm Based on a Finite-Time Estimator. *Mathematics* **2023**, *11*, 3699.
https://doi.org/10.3390/math11173699

**AMA Style**

Alotaibi ND, Jahanshahi H, Yao Q, Mou J, Bekiros S.
Identification and Control of Rehabilitation Robots with Unknown Dynamics: A New Probabilistic Algorithm Based on a Finite-Time Estimator. *Mathematics*. 2023; 11(17):3699.
https://doi.org/10.3390/math11173699

**Chicago/Turabian Style**

Alotaibi, Naif D., Hadi Jahanshahi, Qijia Yao, Jun Mou, and Stelios Bekiros.
2023. "Identification and Control of Rehabilitation Robots with Unknown Dynamics: A New Probabilistic Algorithm Based on a Finite-Time Estimator" *Mathematics* 11, no. 17: 3699.
https://doi.org/10.3390/math11173699