# Active Disturbance Rejection Strategy for Distance and Formation Angle Decentralized Control in Differential-Drive Mobile Robots

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Related Works

#### 1.3. Contribution

- It utilizes the robust ADRC scheme (with a custom error-based high-order ESO) that allows the follower agent to keep a desired distance and formation angle with respect to its own leader in spite the external disturbances, i.e., linear and lateral slipping parameters as well as unknown leader dynamics and velocities.
- It only depends on the distance and formation angle measurements.
- It is developed using solely a kinematic model based on the distance and the formation angle between a pair of robots, taking into account the front point of the differential-drive mobile robots.

## 2. Leader–Follower Problem

#### 2.1. Considered Class of Systems

**Assumption**

**1.**

**Remark**

**1.**

**Remark**

**2.**

#### 2.2. Problem Statement

- Leader tracks a prescribed trajectory, i.e.,$$\underset{t\to \infty}{lim}({\mathbf{\chi}}_{n}-{\mathbf{\chi}}^{\ast})=0,$$
- Agent ${R}_{i}$ maintains a desired distance ${d}_{ij}^{\ast}$ and a desired formation angle ${\alpha}_{ij}^{\ast}$ with respect to the agent ${R}_{j}$, i.e.,$$\underset{t\to \infty}{lim}({\mathbf{\eta}}_{ij}-{\mathbf{\eta}}_{ij}^{\ast})=0,$$

## 3. Proposed Control System

#### 3.1. Leader–Follower Scheme Based on Distance and Formation Angle between the Agents

**Proposition**

**1.**

**Proof.**

#### 3.2. Followers Control Strategy

#### 3.3. Leader Control Strategy

## 4. Experimental Validation

#### 4.1. Experimental Platform

^{©}with a precision of $0.5$ [mm] that measure the position and orientation of each robot in an area of $5\times 4$ [m${}^{2}$] with a sample time of $0.005$ s. Each robot has several reflective markers with different patterns to be detectable by the TRACKER

^{©}cameras’ software (see Figure 2b).

**Remark**

**3.**

#### 4.2. First Experiment

#### 4.3. Second Experiment

**Remark**

**4.**

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ADRC | Active Disturbance Rejection Control |

PID | Proportional Integral Derivative Control |

ESO | Extended State Observer |

GPIO | Generalized Proportional Integral Observer |

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**Figure 1.**Schematic diagram of two differential-drive wheeled mobile robots in the leader–follower configuration.

**Figure 2.**Overview of the experimental setup. (

**a**) Differential-drive wheeled robots used in the test. (

**b**) Communication flow chart.

**Figure 4.**Trajectory in the plane of the mobile robots for the first experiment. (

**a**) Trajectory in the plane with the GPIO approach. (

**b**) Trajectory in the plane with the PI approach.

**Figure 5.**Leader trajectory tracking performance. (

**a**) Leader tracking for the first experiment. (

**b**) Leader trajectory error.

**Figure 6.**Distances and formation angles between the robots for the first experiment. (

**a**) Distances between the robots. (

**b**) Formation angles between the robots.

**Figure 7.**Distances and orientation angles errors for the first experiment. (

**a**) Distance error. (

**b**) Orientation error.

**Figure 8.**Control inputs for the robots for the first experiment. (

**a**) Longitudinal velocities. (

**b**) Angular velocities.

**Figure 10.**Trajectory in the three dimensional space of the mobile robots for the second experiment. (

**a**) Trajectory in the three dimensional space with the GPIO approach. (

**b**) Trajectory in the three dimensional space with the PI approach.

**Figure 11.**Trajectory in the plane of the mobile robots for the second experiment. (

**a**) Trajectory in the plane with the GPIO approach. (

**b**) Trajectory in the plane with the PI approach.

**Figure 12.**Leader trajectory tracking performance for the second experiment. (

**a**) Leader tracking. (

**b**) Leader trajectory error.

**Figure 13.**Distances and formation angles between the robots for the second experiment. (

**a**) Distances between the robots. (

**b**) Formation angles between the robots.

**Figure 14.**Distances and orientation angles errors for the second experiment. (

**a**) Distance error. (

**b**) Orientation error.

**Figure 15.**Control inputs for the robots for the second experiment. (

**a**) Longitudinal velocities. (

**b**) Angular velocities.

**Figure 16.**On line total disturbance estimation. (

**a**) Total disturbance estimation of the longitudinal velocity. (

**b**) Total disturbance estimation of angular velocity.

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**MDPI and ACS Style**

Ramírez-Neria, M.; González-Sierra, J.; Luviano-Juárez, A.; Lozada-Castillo, N.; Madonski, R.
Active Disturbance Rejection Strategy for Distance and Formation Angle Decentralized Control in Differential-Drive Mobile Robots. *Mathematics* **2022**, *10*, 3865.
https://doi.org/10.3390/math10203865

**AMA Style**

Ramírez-Neria M, González-Sierra J, Luviano-Juárez A, Lozada-Castillo N, Madonski R.
Active Disturbance Rejection Strategy for Distance and Formation Angle Decentralized Control in Differential-Drive Mobile Robots. *Mathematics*. 2022; 10(20):3865.
https://doi.org/10.3390/math10203865

**Chicago/Turabian Style**

Ramírez-Neria, Mario, Jaime González-Sierra, Alberto Luviano-Juárez, Norma Lozada-Castillo, and Rafal Madonski.
2022. "Active Disturbance Rejection Strategy for Distance and Formation Angle Decentralized Control in Differential-Drive Mobile Robots" *Mathematics* 10, no. 20: 3865.
https://doi.org/10.3390/math10203865