# Leader–Follower Formation and Disturbance Rejection Control for Omnidirectional Mobile Robots

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## Abstract

**:**

## 1. Introduction

- An inter-robot dynamical model, dependent on the distance, heading angle, and orientation angles is proposed using dynamical models of leader and follower robots. The resulting equations are rewritten as an inter-robot perturbed dynamical model, where the conveniently aggregated single perturbation contains viscous and Coulomb frictions, centripetal forces, and other unmodeled dynamics.
- A general proportional integral observer (GPIO), as seen in [43], is proposed to estimate the aggregated perturbation. A formation control law, based on the active disturbance rejection control (ADRC) approach, is then defined for the leader and follower robots using the position, distance, formation angle, and estimated perturbation. The approach becomes a robust setup ready to overcome unmodeled dynamics in real time.
- Experimental work utilizing laboratory-scale omnidirectional mobile robots and supported with VICON
^{©}motion capture system verifies the accuracy of the parameters of the assumed dynamical models. It validates the efficacy of the proposed control.

## 2. Problem Statement

- The leader robot follows a desired trajectory, that is, $\underset{t\to \infty}{lim}({\mathbf{q}}_{L}-{\mathbf{q}}^{*})=0$, where ${\mathbf{q}}^{*}={\left[\begin{array}{ccc}{x}^{*}& {y}^{*}& {\theta}^{*}\end{array}\right]}^{\top}$, with ${x}^{*}$, ${y}^{*}$, and ${\theta}^{*}$ being the leader’s desired position in X, desired position in Y, and desired orientation, respectively;
- The follower agent keeps a desired distance ${d}^{*}$ and a formation angle ${\alpha}^{*}$ concerning the leader robot, and a desired orientation ${\theta}_{F}^{*}$, that is, $\underset{t\to \infty}{lim}({\mathit{\eta}}_{LF}-{\mathit{\eta}}^{*})=0$, where ${\mathit{\eta}}^{*}={\left[\begin{array}{ccc}{d}^{*}& {\alpha}^{*}& {\theta}_{F}^{*}\end{array}\right]}^{\top}$.

## 3. Modeling Based on Distance and Formation Angle

**Remark**

**1**

**.**

## 4. Control Strategy

#### 4.1. Leader Controller Design

**Assumption**

**1.**

**Theorem**

**1.**

**Proof.**

#### 4.2. Follower Controller Design

**Theorem**

**2.**

**Proof.**

**Assumption**

**2.**

## 5. Numerical Simulations and Real-Time Experiments

#### 5.1. Numerical Simulation

^{©}with a sample time of $0.01$ s. The parameters of the used robots are ${m}_{s}=1.82$ kg, ${r}_{s}=0.03$ m, $\delta =\frac{\pi}{6}$ rad, $D=0.11$ m, ${I}_{{w}_{s}}=3.06\times {10}^{-5}$ kg·m${}^{2}$, and ${I}_{{0}_{s}}=0.0071$ kg·m${}^{2}$. The initial conditions are ${\mathbf{q}}_{L}\left(0\right)={\left[\begin{array}{ccc}-0.6& -0.4& 0\end{array}\right]}^{\top}$ and ${\mathbf{q}}_{F}\left(0\right)={\left[\begin{array}{ccc}-0.85& 0.032& 0\end{array}\right]}^{\top}$, while the perturbations are

#### 5.2. Real-Time Experiments

^{©}library, using Bluetooth communication protocol which is programmed on ESP32 micro-controller with ARDUINO-ESPRESSIF (https://docs.espressif.com/projects/arduino-esp32/en/latest/api/bluetooth.html, (accessed on 14 August 2023)) as it is shown in Figure 9. An STM32F4 Discovery board received the torque reference of each motor, and using a nonlinear function approximate the conversion of torque to PWM, as follows:

#### 5.2.1. First Case Study

#### 5.2.2. Second Case Study

#### 5.3. Discussion

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

GPIO | General Proportional Integral Observer |

SM | Sliding Mode |

ADRC | Active Disturbance Rejection Control |

RMS | Root Mean Square |

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**Figure 3.**Simulation: comparison of the distance error, the formation angle error, and the orientation error.

**Figure 4.**Simulation: comparison of the leader’s control inputs. (

**a**) Control inputs for the leader with the GPIO. (

**b**) Control inputs for the leader with the SM.

**Figure 5.**Simulation: Comparison of the follower’s control inputs. (

**a**) Control inputs for the follower with the GPIO. (

**b**) Control inputs for the follower with the SM.

**Figure 8.**The omnidirectional mobile robots (leader ${R}_{L}$ and follower ${R}_{F}$) used in the experiments.

**Figure 12.**Exp1: control inputs for the two robots. (

**a**) Control inputs for the leader. (

**b**) Control inputs for the follower.

**Figure 13.**Exp1: linear and angular velocities for both robots. (

**a**) Linear velocities for both robots. (

**b**) Angular velocities for both robots.

**Figure 14.**Exp1: disturbance estimation using GPIO. (

**a**) Linear disturbance estimation. (

**b**) Angular disturbance estimation.

**Figure 17.**Exp2: control inputs for the two robots. (

**a**) Control inputs for the leader. (

**b**) Control inputs for the follower.

**Figure 18.**Exp2: linear and angular velocities for both robots. (

**a**) Linear velocities for both robots. (

**b**) Angular velocities for both robots.

**Figure 19.**Exp2: disturbance estimation using GPIO. (

**a**) Linear disturbance estimation. (

**b**) Angular disturbance estimation.

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## Share and Cite

**MDPI and ACS Style**

Ramírez-Neria, M.; González-Sierra, J.; Madonski, R.; Ramírez-Juárez, R.; Hernandez-Martinez, E.G.; Fernández-Anaya, G.
Leader–Follower Formation and Disturbance Rejection Control for Omnidirectional Mobile Robots. *Robotics* **2023**, *12*, 122.
https://doi.org/10.3390/robotics12050122

**AMA Style**

Ramírez-Neria M, González-Sierra J, Madonski R, Ramírez-Juárez R, Hernandez-Martinez EG, Fernández-Anaya G.
Leader–Follower Formation and Disturbance Rejection Control for Omnidirectional Mobile Robots. *Robotics*. 2023; 12(5):122.
https://doi.org/10.3390/robotics12050122

**Chicago/Turabian Style**

Ramírez-Neria, Mario, Jaime González-Sierra, Rafal Madonski, Rodrigo Ramírez-Juárez, Eduardo Gamaliel Hernandez-Martinez, and Guillermo Fernández-Anaya.
2023. "Leader–Follower Formation and Disturbance Rejection Control for Omnidirectional Mobile Robots" *Robotics* 12, no. 5: 122.
https://doi.org/10.3390/robotics12050122