# Second-Order Sliding Mode Formation Control of Multiple Robots by Extreme Learning Machine

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- An architecture that combines second-order sliding mode control and the extreme learning machine technique is investigated.
- The closed-loop stability of this combination is presented in the sense of Lyapunov.
- Some numerical results for different formation patterns are demonstrated to support the combination.

## 2. Formation Model

#### 2.1. A Single Robot

#### 2.2. A Leader–Follower Pair

_{ik}means the distance between the leader’s center and the follower’s front castor, formulated by

**x**

_{ik}= [x

_{1}x

_{2}x

_{3}x

_{4}]

^{T}. Let ${x}_{1}={l}_{ik}$, ${x}_{2}={\dot{l}}_{ik}$, ${x}_{3}={\psi}_{ik}$, and ${x}_{4}={\dot{\psi}}_{ik}$. According to the formation objective, the relative distance ${l}_{ik}$ and the relative bearing angle ${\psi}_{ik}$ are determined as the formation control output. Then, the formation dynamics of this leader–follower pair among the multiple robots can have the form of Equation (8) in light of the leader–follower scheme.

**x**

_{ik}is the system state vector and

**y**

_{ik}is the system output vector. Further,

**A**

_{ik},

**B**

_{ik,}

_{1},

**B**

_{ik,}

_{2}and

**h**(

**x**

_{ik}) are defined as

_{1}, F

_{2}, P

_{1}, and P

_{2}are written as

## 3. Formation Control Design

#### 3.1. Sliding Surfaces and Input–Output Dynamics

**Assumption**

**1.**

**Assumption**

**2.**

#### 3.2. Super-Twisting Sliding Mode Control Design

#### 3.3. Super-Twisting Sliding Mode Control Design via ELM

**Theorem**

**1.**

**Proof.**

## 4. Simulation Results

#### 4.1. Multirobot Platform

#### 4.2. Formation Tasks

#### 4.2.1. String Formation Moving Along a Circular Trajectory

#### 4.2.2. Triangular Formation Moving in a Circular Trajectory

#### 4.2.3. Maneuvers from a String Formation to a Triangular One

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 5.**Comparisons of the state variables by different methods: (

**a**) ${l}_{12}$, (

**b**) ${\psi}_{12}$, (

**c**) ${l}_{13}$, (

**d**) ${\psi}_{13}$.

**Figure 6.**Comparisons of the control inputs from Follower 2: (

**a**) acceleration by STW (sliding mode control) with ELM (extreme learning machine), (

**b**) angular acceleration by STW with ELM, (

**c**) acceleration by STW, (

**d**) angular acceleration by STW, (

**e**) acceleration by SMC with NDOBC, and (

**f**) angular acceleration by SMC (sliding mode control) with NDOBC.

**Figure 7.**Comparisons of the control inputs from Follower 3: (

**a**) acceleration by STW with ELM, (

**b**) angular acceleration by STW with ELM, (

**c**) acceleration by STW, (

**d**) angular acceleration by STW, (

**e**) acceleration by SMC with NDOBC, and (

**f**) angular acceleration by SMC with NDOBC.

**Figure 8.**Sliding surfaces of the two followers: (

**a**) ${s}_{12,1}$, (

**b**) ${s}_{12,2}$, (

**c**) ${s}_{13,1}$, (

**d**) ${s}_{13,2}$.

**Figure 9.**ELM outputs of the two followers: (

**a**) estimation of ${d}_{12,1}$, (

**b**) estimation of ${d}_{12,2}$, (

**c**) estimation of ${d}_{13,1}$, (

**d**) estimation of ${d}_{13,2}$.

**Figure 10.**Estimation errors of the ELM for the two followers: (

**a**) ${e}_{12,1}$, (

**b**) ${e}_{12,2}$, (

**c**) ${e}_{13,1}$, (

**d**) ${e}_{13,2}$.

**Figure 11.**Triangular formation of this multirobot platform when moving along a circular trajectory.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Qian, D.; Zhang, G.; Wang, J.; Wu, Z.
Second-Order Sliding Mode Formation Control of Multiple Robots by Extreme Learning Machine. *Symmetry* **2019**, *11*, 1444.
https://doi.org/10.3390/sym11121444

**AMA Style**

Qian D, Zhang G, Wang J, Wu Z.
Second-Order Sliding Mode Formation Control of Multiple Robots by Extreme Learning Machine. *Symmetry*. 2019; 11(12):1444.
https://doi.org/10.3390/sym11121444

**Chicago/Turabian Style**

Qian, Dianwei, Guigang Zhang, Jian Wang, and Zhimin Wu.
2019. "Second-Order Sliding Mode Formation Control of Multiple Robots by Extreme Learning Machine" *Symmetry* 11, no. 12: 1444.
https://doi.org/10.3390/sym11121444