# Tracking Control of Robot Manipulator with Friction Compensation Using Time-Delay Control and an Adaptive Fuzzy Logic System

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## Abstract

**:**

## 1. Introduction

## 2. Review and Problem of TDC

#### 2.1. Review of TDC

**Assumption 1.**

**Assumption 2.**

**Assumption 3.**

#### 2.2. TDE Error Due to Friction

**Remark 1.**

## 3. Time-Delay Control with Adaptive Fuzzy Logic System

#### 3.1. Derivation of the Proposed Controller

**Remark 2.**

#### 3.2. Fuzzy Logic Term

#### 3.3. Adaptive Scheme

#### 3.4. Stability Analysis

**Theorem 1.**

**Proof.**

**Remark 3.**

## 4. Numerical Experiments

_{1}denotes the mass of first link; ${l}_{cl}$ is the distance between the mass center of the first link and the first joint; ${I}_{l}$ is the moment of inertia of the first link; ${m}_{e}$ is the mass of second link with payload; ${l}_{ce}$ is the distance between the mass center of second link and the second joint; ${I}_{e}$ is the moment of inertia of the second link; ${\delta}_{e}$ is the angle relative to the original second link. The physical parameters of the robot manipulator are shown in Table 1.

- 1.
- FCC: This is a time-delay estimation controller equipped with AFLS (as given in Equation (20)), which is thoroughly described in Section 3 of this paper. The control gains for this controller are set as follows: ${\overline{M}}_{1}=0.15$, ${\overline{M}}_{2}=0.15$, ${k}_{11}=10$, ${k}_{12}=10$, ${k}_{21}=70$, and ${k}_{22}=70$.
- 2.
- NTSM: This technique is founded on the principles of TDE and sliding-mode control. It achieves the high-precision control of nonlinear dynamic systems by incorporating supplementary terms on the sliding surface. These terms help to mitigate the impacts that traditional sliding-mode control may generate, resulting in a superior level of control.
- 3.
- TDC: A widely used control method for systems with a delay that effectively solves delay problems and improves the stability and precision of the control system.

#### 4.1. Comparative Experiment of Three Controllers under No-Friction Disturbance

**Remark 4.**

#### 4.2. Comparative Experiment of Three Controllers under Normal Friction Disturbance

#### 4.3. Comparative Experiment of Three Controllers under Significant Frictional Disturbance

#### 4.4. Further Analysis of Error for Three Algorithms

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Performance indices evaluation during the last three cycles under no-friction disturbance.

**Figure 8.**The control input voltage curves of the three controllers under normal friction disturbance.

**Figure 9.**Performance indices evaluation during the last three cycles under normal friction disturbance.

**Figure 10.**Position-tracking curves for the three controllers under significant frictional disturbance.

**Figure 12.**The control input voltage curves of the three controllers under significant frictional disturbance.

**Figure 13.**Performance indices evaluation during the last three cycles under significant frictional disturbance.

Indices | ${\mathit{l}}_{1}$ | ${\mathit{l}}_{2}$ | ${\mathit{l}}_{\mathit{c}1}$ | ${\mathit{l}}_{\mathit{c}\mathit{e}}$ | ${\mathit{I}}_{1}$ | ${\mathit{I}}_{2}$ | ${\mathit{m}}_{1}$ | ${\mathit{m}}_{\mathit{e}}$ | ${\mathit{\delta}}_{\mathit{e}}$ |
---|---|---|---|---|---|---|---|---|---|

Value | 1 | 1.2 | 0.5 | 1 | 0.083 | 0.4 | 1 | 3 | 0 |

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**MDPI and ACS Style**

Sun, Y.; Liang, X.; Wan, Y.
Tracking Control of Robot Manipulator with Friction Compensation Using Time-Delay Control and an Adaptive Fuzzy Logic System. *Actuators* **2023**, *12*, 184.
https://doi.org/10.3390/act12050184

**AMA Style**

Sun Y, Liang X, Wan Y.
Tracking Control of Robot Manipulator with Friction Compensation Using Time-Delay Control and an Adaptive Fuzzy Logic System. *Actuators*. 2023; 12(5):184.
https://doi.org/10.3390/act12050184

**Chicago/Turabian Style**

Sun, Yao, Xichang Liang, and Yi Wan.
2023. "Tracking Control of Robot Manipulator with Friction Compensation Using Time-Delay Control and an Adaptive Fuzzy Logic System" *Actuators* 12, no. 5: 184.
https://doi.org/10.3390/act12050184