Article
Time-Optimal Velocity Tracking Control for Consensus
Formation of Multiple Nonholonomic Mobile Robots
Hamidreza Fahham
1,
*, Abolfazl Zaraki
2
, Gareth Tucker
1
and Mark W. Spong
3
Citation: Fahham, H.; Zaraki, A.;
Tucker, G.; Spong, M.W.
Time-Optimal Velocity Tracking
Control for Consensus Formation of
Multiple Nonholonomic Mobile
Robots. Sensors 2021, 21, 7997.
https://doi.org/10.3390/s21237997
Academic Editor: Felipe Jiménez
Received: 12 November 2021
Accepted: 27 November 2021
Published: 30 November 2021
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4.0/).
1
Institute of Railway Research, School of Computing and Engineering, University of Huddersfield,
Huddersfield HD1 3DH, UK; G.J.Tucker@hud.ac.uk
2
Centre for Artificial Intelligence, Robotics and Human-Machine Systems (IROHMS), Cardiff University,
Cardiff CF24 3AA, UK; zarakia@cardiff.ac.uk
3
Erik Jonsson School of Engineering & Computer Science, University of Texas at Dallas, 800 W Campbell Rd.,
Richardson, TX 75080, USA; mspong@utdallas.edu
* Correspondence: h.faham@hud.ac.uk
Abstract:
The problem of velocity tracking is considered essential in the consensus of multi-wheeled
mobile robot systems to minimise the total operating time and enhance the system’s energy efficiency.
This study presents a novel switched-system approach, consisting of bang-bang control and consensus
formation algorithms, to address the problem of time-optimal velocity tracking of multiple wheeled
mobile robots with nonholonomic constraints. This effort aims to achieve the desired velocity
formation in the least time for any initial velocity conditions in a multiple mobile robot system. The
main findings of this study are as follows: (i) by deriving the equation of motion along the specified
path, the motor’s extremal conditions for a time-optimal trajectory are introduced; (ii) utilising a
general consensus formation algorithm, the desired velocity formation is achieved; (iii) applying the
Pontryagin Maximum Principle, the new switching formation matrix of weights is obtained. Using
this new switching matrix of weights guarantees that at least one of the system’s motors, of either the
followers or the leader, reaches its maximum or minimum value by using extremals, which enables
the multi-robot system to reach the velocity formation in the least time. The proposed approach
is verified in a theoretical analysis along with the numerical simulation process. The simulation
results demonstrated that using the proposed switched system, the time-optimal consensus algorithm
behaved very well in the networks with different numbers of robots and different topology conditions.
The required time for the consensus formation is dramatically reduced, which is very promising. The
findings of this work could be extended to and beneficial for any multi-wheeled mobile robot system.
Keywords:
time-optimal; velocity tracking; consensus formation; switching control; multi-robot systems
1. Introduction
Thanks to the greater functionality and performance of cooperative, mobile robot sys-
tems versus single mobile robots, they are highly beneficial and used in many applications
such as hazardous material handling, surveillance, environment exploration, transportation
of large objects, etc. Although the cooperative nature of these systems may result in greater
efficiency and operational capability compared to a single mobile robot, it introduces a
challenging control problem that must deal with consensus formation. For instance, in
the works presented in [
1
,
2
] there are two examples in the domain of formations control
for multi-agent systems in surveillance applications, and the control of spacecraft using
formations control is presented in [3–6].
The study of consensus formation control of multi-agent systems (Figure 1) can be
classified into three categories: a leader–follower approach [
7
–
9
], the virtual structure
approach [
10
,
11
], and decentralised control [
12
,
13
]. The leader–follower approach consists
of a leader agent, a robot/vehicle or human operator who is entrusted to track a specified
trajectory, and the follower agents are designed to follow the leader agent while achieving
Sensors 2021, 21, 7997. https://doi.org/10.3390/s21237997 https://www.mdpi.com/journal/sensors
