# Global Fixed-Time Sliding Mode Trajectory Tracking Control Design for the Saturated Uncertain Rigid Manipulator

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## Abstract

**:**

## 1. Introduction

- (1)
- For the sake of eliminating the effect of external environment disturbance and uncertainty on the system, the novel adaptive fixed-time sliding mode disturbance observer (AFSMDO) is presented while guaranteeing the fixed-time convergence of the observer. This method can not only break the limitation that the disturbance observer depends on a constant upper bound of the disturbance’s change rate but also improve the disturbance estimation accuracy of the observer by adjusting the parameters adaptively.
- (2)
- The nonlinear functions considering the tracking error of the manipulator are introduced into the nonlinear fixed-time sliding mode control (NFSMC) to solve the singularity problem of FTSMC. Based on the FTSMC, the tracking error of the manipulator converges to the arbitrarily small zero region in the fixed time by effective combination with the designed AFSMDO.
- (3)
- In view of the situation that the input torque of the manipulator trajectory tracking control system is too large, leading to the supersaturation of the manipulator, the fixed-time saturation compensator (FTSC) is designed in this paper. When the manipulator’s actuator torque occurs saturation, the FTSC can efficiently adjust the control input by introducing a saturation function so that the designed control strategy can maintain the operation of the manipulator normally and protect the manipulator from damage. Additionally, compared to the other saturation compensators in previous literature, the NFSMC-AFSMDO combined with the proposed saturation compensator can effectively improve the manipulator trajectory tracking speed and reduce joint chattering.

## 2. Preliminaries and Notions

#### 2.1. Preliminaries

**Lemma 1.**

**Lemma 2.**

**Lemma 3.**

**Lemma 4.**

#### 2.2. Notions

#### 2.3. Dynamic Model of the Manipulator

## 3. System Control Scheme Design

#### 3.1. The Design of the Adaptive Fixed-Time Sliding Mode Disturbance Observer

**Remark 1.**

**Theorem 1.**

**Proof.**

**Remark 2.**

**Remark 3.**

#### 3.2. Nonsingular Fixed-Time Sliding Mode Controller Design

**Remark 4.**

#### 3.3. Fixed-Time Saturation Compensator Design

**Remark 5.**

**Theorem 2.**

**Proof.**

## 4. Stability Analysis

**Theorem 3.**

#### 4.1. Analysis of Stability in the Reaching Phase

**Remark 6.**

#### 4.2. Analysis of Stability in the Sliding Phase

**Case 1.**

**Case 2.**

**Remark 7.**

## 5. Simulation Comparisons

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Position tracking comparison. (

**a**) Position tracking curve at the initial values [1 rad, 2 rad]. (

**b**) Position tracking curve at the initial values [80 rad, 80 rad].

**Figure 4.**Position tracking error. (

**a**) Position tracking error curve at the initial values [1 rad, 2 rad]. (

**b**) Position tracking error curve at the initial values [80 rad, 80 rad].

**Figure 5.**Joint velocity tracking comparison. (

**a**) Joint velocity tracking curve at the initial values [1 rad, 2 rad]. (

**b**) Joint velocity tracking curve at the initial values [80 rad, 80 rad].

**Figure 6.**Joint velocity tracking error. (

**a**) Joint velocity tracking error curve at the initial values [1 rad, 2 rad]. (

**b**) Joint velocity tracking error curve at the initial values [80 rad, 80 rad].

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**MDPI and ACS Style**

Nie, J.; Hao, L.; Lu, X.; Wang, H.; Sheng, C.
Global Fixed-Time Sliding Mode Trajectory Tracking Control Design for the Saturated Uncertain Rigid Manipulator. *Axioms* **2023**, *12*, 883.
https://doi.org/10.3390/axioms12090883

**AMA Style**

Nie J, Hao L, Lu X, Wang H, Sheng C.
Global Fixed-Time Sliding Mode Trajectory Tracking Control Design for the Saturated Uncertain Rigid Manipulator. *Axioms*. 2023; 12(9):883.
https://doi.org/10.3390/axioms12090883

**Chicago/Turabian Style**

Nie, Jun, Lichao Hao, Xiao Lu, Haixia Wang, and Chunyang Sheng.
2023. "Global Fixed-Time Sliding Mode Trajectory Tracking Control Design for the Saturated Uncertain Rigid Manipulator" *Axioms* 12, no. 9: 883.
https://doi.org/10.3390/axioms12090883