Complexity

Volume 2017 (2017), Article ID 9404193, 13 pages

https://doi.org/10.1155/2017/9404193

## Formation Tracking of Heterogeneous Mobile Agents Using Distance and Area Constraints

^{1}Engineering Department, Universidad Iberoamericana, 01219 Mexico City, Mexico^{2}Electrical Engineering Department, Universidad Católica del Uruguay, 11600 Montevideo, Uruguay^{3}Physics and Mathematics Department, Universidad Iberoamericana, 01219 Mexico City, Mexico^{4}Mathematics Department, Universidad Católica del Uruguay, 11600 Montevideo, Uruguay

Correspondence should be addressed to E. D. Ferreira-Vazquez; yu.ude.ucu@ierrefne

Received 26 May 2017; Accepted 9 August 2017; Published 20 September 2017

Academic Editor: Michele Scarpiniti

Copyright © 2017 E. G. Hernandez-Martinez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents two formation tracking control strategies for a combined set of single and double integrator agents with an arbitrary undirected communication topology. The first approach is based on the design of distance-based potential functions with interagent collision avoidance using local information about the distance and orientation between agents and the desired trajectory. The second approach adds signed area constraints to the desired formation specification and a control strategy that uses distance as well as area terms is designed to achieve tracking convergence. Numerical simulations show the performance from both control laws.

#### 1. Introduction

The consensus problem of multiagent systems has gained considerable interest in the research community recently. For instance, Shang and Ye [1] proposed a leader-follower nonlinear distributed control algorithm for single integrator agents, based on local information such that the following agents track their corresponding leaders. Higher order multiagent systems have been studied by Wang et al. [2] using multiple Lyapunov functions, showing that consensus can be reached under a switching topology within a finite set of digraphs with an average delay time between changes.

An extension to the consensus problem, called formation control, studies strategies to distribute agents in geometric patterns avoiding interagent collisions [3]. In distance-based formation control (DFC) the agents achieve a formation pattern defined by interagent distances according to a predefined communication graph, for example, in Ramazani et al. [4]. The DFC allows a more decentralized and scalable setup than position based control strategies because the control laws can be implemented using local sensors on the agents. Representative works are of Dimarogonas and Johansson [5] for the case of undirected communication graphs or Oh and Ahn [6] that used an Euclidean distance matrix. The DFC has been addressed for the case of single or double integrators, separately in Zavlanos et al. [7] and leader-follower schemes in Anderson et al. [8]. Control of formations specified only by interagent angles has been studied by Basiri et al. [9]. Formation and communication changes have also been studied to find suitable control laws that maintain rigidity in Zelazo et al. [10]. Experimental work is found in Fidan et al. [11] using local video cameras and Antonelli et al. [12] with laser range finder devices. Our previous contributions in DFC are in Lopez-Gonzalez et al. [13] for a distance-orientation scheme, Ferreira-Vazquez et al. [14] adding desired internal angles in the formation pattern to compensate for the flip ambiguity problem, and Ferreira-Vazquez et al. [15] using planar as well as volume constraints for DFC in 3D. A recent paper by Anderson et al. [16] makes a detailed convergence analysis of DFC with signed area constraints using standard potential functions. However, their results are restricted to formations of three and four agents without any collision avoidance techniques.

In some applications, the agents must follow a prescribed trajectory maintaining the DFC simultaneously. This problem is known as* formation tracking* or* marching control*. The models for the agents depending on the approximation used, kinematic or dynamical, could be of first or second order which involves different physical robots with an appropriate linearization. Therefore, the combination of these simplified models is the reason to consider this setup as a heterogeneous multiagent system (e.g., Zheng et al. [17], Hernandez-Martinez et al. [18]).

For first-order agents, the leader-follower schemes add the reference velocity for the leader, whereas the follower agents estimate this velocity using an adaptive method in Kang et al. [19]. In Rozenheck et al. [20], a subset of leader agents follows the marching path and the followers combine a proportional-integral control with the standard gradient method. The velocity and position of a target are communicated to the leader agent in Cai and De Queiroz [21]. The orientation of the formations and the enclosing and tracking of a moving target is studied in Garcia De Marina et al. [22] cancelling the phenomena of distance mismatches due to the local measurement of distance. Alternative works are of Xiao et al. [23] for a leader-follower distance maintenance and obstacle avoidance for unicycle-type agents using nonlinear model predictive control with neurodynamic optimization.

For the case of double integrator models, in Dimarogonas and Johansson [24] formation tracking is achieved transmitting a common velocity. The case of unmanned aerial vehicles (UAVs) is studied in Zhang et al. [25], where a linearized elastic distance vector adapts to the changes of the follower velocities. A set of underwater mobile agents in a vertical plane formation is studied in Do [26], where an optimal assignment algorithm selects the appropriated reference trajectories to the agents. Switched routines of tracking and encircling of a moving target in a V-shape DFC are addressed in Dang and Horn [27] using attractive or repulsive and rotational force fields.

The analysis of a combined set of first-second-order agents becomes a useful result in the sense that heterogeneous agents can be formed within the same setup. Our previous work in Hernandez-Martinez et al. [18] studies the DFC for terrestrial (first-order) and aerial (second-order) agents. Also, the potential of collaborative work of aerial and ground agents is shown in Harik et al. [28], where an aerial agent provides way points to the ground agents to carry objects in unsafe industrial areas using a predictive vision to keep a distance and bearing to the leader.

This paper extends our previous work in Hernandez-Martinez et al. [18], focusing on static DFC, for the case of formation tracking of heterogeneous agents communicated by an arbitrary undirected graph. The main contributions and originality of this work are given by the next points:(i)The approach applies to a combined set of first- and second-order agents, where a second-order leader is chosen to follow a desired marching path whilst all the agents maintain a DFC.(ii)Two control laws are designed using Lyapunov techniques. The first strategy depends on distance and orientation measurements only applied to rigid undirected formation topologies of distance. Convergence to the formation setup is shown for an arbitrary number of agents.(iii)In order to avoid symmetric solutions, a second strategy is designed to add desired area constraints related to the standard triple product of a subset of robots. It defines a new area based or planar topology based on triplets of robots separately of the previous distance formation topology. The combination of distance and area restrictions helps to avoid undesired final patterns of the agents and the convergence to the desired formation eliminating some distance links.(iv)Both approaches are based on a general class of artificial potential functions with attractive and repulsive behavior designed to ensure convergence and collision avoidance. The performance of the strategies is shown by numerical simulations.The paper is organized in the following sections. Section 2 presents the problem definition for the mixed group of agents and the distance-based formation topology. Section 3 addresses the first strategy of DFC. The definition of the planar topology related to the area constraints is given in Section 4 and the addition of this planar topology to the previous DFC is studied in Section 5, completed with the analysis of the appendix section. Numerical simulations for both strategies are shown in Section 6. Finally, some conclusion remarks are given in Section 7.

#### 2. Problem Definition

Let be the set of mobile agents with positions in the plane . Consider the first agents as single-integrators and the rest as double integrators; that is,where is the input vector of the agent , and is the velocity of the second-order agents. As shown below, the single and double integrators are related to kinematic or dynamic posture models of mobile agents, respectively. This allows a feasible combination of distinct modeling complexities into the same motion coordination scheme. Consider the last agent to be the leader agent.

Assume that each agent is communicated to agents that belong to its adjacent subset (note that ). Thus, the possible interagent communication defines a distance-based formation graph (DFG) given bywhere is the set of vertices related to the agents; is the set of edges that represent the possible interagent communications; therefore if . The set , , contains the desired distances between and ; that is, , , within a desired formation pattern [13].

A well-defined DFG must be connected, that is, there are no isolated nodes (), and rigid, where at least communication edges are defined for the agents. It is assumed that for all . This paper addresses the case of undirected graphs, that is, bidirectional communication between agents, where if , then , for all .

On the other hand, define the distance between and as and as the angle of with respect to a reference axis (e.g., the magnetic pole of the Earth), as shown in Figure 1. It is assumed that the values of distance and orientation can be measured by the combination of local sensors as laser range finders or lidar and magnetometers.