# Coordinated Formation Design of Multi-Robot Systems via an Adaptive-Gain Super-Twisting Sliding Mode Method

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## Abstract

**:**

## Featured Application

**multi-robot systems, formation maneuvers, sliding mode control, adaptive control**.

## Abstract

## 1. Introduction

- The multiple-input-multiple-output dynamics of the formation problem were formulated.
- An adaptive-gain super-twisting sliding mode control method was developed by the formation maneuvers of uncertain multi-robot systems.
- The control method is with the guaranteed closed-loop stability in the sense of Lyapunov.
- The adaptive gains were theoretically bounded even if the boundaries of the uncertainties and disturbances were unknown.

## 2. Modelling

#### 2.1. Model of a Robot

#### 2.2. Model of a Leader-Follower Formation 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 dynamics of formation maneuvers of the multi-robot system can have the form of (8) in light of the leader-follower scheme [14].

**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 depicted by

_{1}, F

_{2}, P

_{1}, and P

_{2}are written by

## 3. Control Design

#### 3.1. Sliding Mode Design and Its Input-Output Dynamics

**Assumption 1.**

**Assumption 2.**

#### 3.2. Adaptive-Gain Super-Twisting Sliding Mode Design

**Assumption 3.**

**Assumption 4.**

#### 3.3. Stability Analysis of the Closed-Loop Control System

**Theorem 1.**

- a parameter$\mu >0$so that${\alpha}_{k}$satisfies (25) if$||{s}_{ik}|{|}_{2}>\mu $at t = 0;$${\alpha}_{k}>\frac{{\delta}_{1}(\lambda +4{\epsilon}^{2})-\epsilon (4{\delta}_{4}+1)}{\lambda (1-{\gamma}_{1})}+\frac{{[2\epsilon {\delta}_{1}-2{\delta}_{4}-\lambda -4{\epsilon}^{2}]}^{2}}{12\epsilon \lambda (1-{\gamma}_{1})}$$Here$\lambda $is an arbitrary positive constant and${\delta}_{4}$ is determined by (23).
- a finite time${t}_{F}>0$so that the sliding modes of${s}_{ik}$are reached in the finite time${t}_{F}$regarding to the adaptive-gain super-twisting sliding mode control method, that is,$\forall t>{t}_{F}$, $\exists $$||{s}_{ik}|{|}_{2}\le {\eta}_{1}$and$||{\dot{s}}_{ik}|{|}_{2}\le {\eta}_{2}$. Here,${\eta}_{1}>\mu $and${\eta}_{2}>0$.
- both${\alpha}_{k}$and${\beta}_{k}$are bounded.

**Proof.**

**Case 1.**

**Case 2.**

_{1}is the time instant when ${s}_{ik}$ enters the domain $||{s}_{ik}|{|}_{2}\le \mu $ and t

_{2}is the moment when ${s}_{ik}$ leaves the domain. Once $||{s}_{ik}|{|}_{2}$ becomes $\mu <||{s}_{ik}|{|}_{2}<{\eta}_{1}$, we have

_{2}is the time instant when ${s}_{ik}$ leaves the domain $||{s}_{ik}|{|}_{2}<\mu $ and t

_{3}is the moment when ${s}_{ik}$ enter the domain $||{s}_{ik}|{|}_{2}<\mu $. Subsequently, (53) can be drawn from (51) and (52).

## 4. Implementation

#### 4.1. Multi-Robot Simulation Platform

#### 4.2. Simulation Results

#### 4.2.1. String Formation When Moving along a Circular Trajectory

#### 4.2.2. String Formation When Moving along an S-Shape Trajectory

#### 4.2.3. String Formation When Moving Along a Straight Trajectory

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Comparisons of the state variables by different control methods. (

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

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

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

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

**Figure 5.**Comparisons of the control inputs from the follower 2. (

**a**) Acceleration by AST-SMC, (

**b**) angular acceleration by AST-SMC, (

**c**) acceleration by DI-TSMC, (

**d**) angular acceleration by DI-TSMC, (

**e**) acceleration by ST-SMC, (

**f**) angular acceleration by ST-SMC.

**Figure 6.**Comparisons of the control inputs from the follower 3. (

**a**) Acceleration by AST-SMC, (

**b**) angular acceleration by AST-SMC, (

**c**) acceleration by DI-TSMC, (

**d**) angular acceleration by DI-TSMC, (

**e**) acceleration by ST-SMC, (

**f**) angular acceleration by ST-SMC.

**Figure 7.**Sliding surfaces of the two followers. (

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

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

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

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

**Figure 8.**Adaptive gains of the two followers. (

**a**) ${\alpha}_{2}$, (

**b**) ${\beta}_{2}$, (

**c**) ${\alpha}_{3}$, (

**d**) ${\beta}_{3}$.

© 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/).

## Share and Cite

**MDPI and ACS Style**

Qian, D.; Zhang, G.; Chen, J.; Wang, J.; Wu, Z.
Coordinated Formation Design of Multi-Robot Systems via an Adaptive-Gain Super-Twisting Sliding Mode Method. *Appl. Sci.* **2019**, *9*, 4315.
https://doi.org/10.3390/app9204315

**AMA Style**

Qian D, Zhang G, Chen J, Wang J, Wu Z.
Coordinated Formation Design of Multi-Robot Systems via an Adaptive-Gain Super-Twisting Sliding Mode Method. *Applied Sciences*. 2019; 9(20):4315.
https://doi.org/10.3390/app9204315

**Chicago/Turabian Style**

Qian, Dianwei, Guigang Zhang, Jiarong Chen, Jian Wang, and Zhimin Wu.
2019. "Coordinated Formation Design of Multi-Robot Systems via an Adaptive-Gain Super-Twisting Sliding Mode Method" *Applied Sciences* 9, no. 20: 4315.
https://doi.org/10.3390/app9204315