Non-Collision Conditions in Multi-Agent Virtual Leader-Based Formation Control

Formation control is one of the most important issues of group coordination for multi-agent robots systems. Some schemes are based on the leader-followers approach where some robots are considered as group leaders which influence the group behaviour. In this work, we address a formation strategy using a virtual leader which has communication with the rest of the follower robots, considered as omnidirectional robots. The virtual leader approach presents advantages such as analysis simplification and fewer sensing requirements in the control law implementation. The formation control is based on attractive potential functions only. The control law guarantees the convergence to the desired formation but, in principle, does not avoid inter-agent collisions. A set of necessary and sufficient non-collision conditions based on the explicit solution of the closed-loop system is derived. The conditions allow concluding from the initial conditions whether or not the agents will collide. The results are extended to the case of unicycle-type robots.


Introduction
Multi-agent robots systems (MARS) are understood as groups of autonomous robots (called agents) coordinated to achieve cooperative tasks. The range of applications includes toxic residue cleaning, transportation and manipulation of large objects, exploration, search and rescue, and simulation of biological entities' behaviours [1]. The research areas of MARS encompass motion coordination, task assignment, communications, sensor networks, etc. [2]. Formation control is an important subarea of motion coordination. It refers both to omnidirectional and mobile wheeled robots. The goal is to guarantee the convergence to a particular formation pattern avoiding inter-agent collision at the same time [3,11]. The problem is complex because it is assumed that the agents have a limited sensing capacity and no robot possesses global information about the position of all robots. Therefore, the control strategies are decentralized and the main intention is to achieve the desired global behaviours through local interactions [13].
Different schemes of formation control exist [4,5]. Some schemes consider all agents with the same capacities and achieve formation patterns, without a specific position, following some simple behaviour rules such as neighbours-rules, swarm intelligence, physics-based techniques, etc. [5,6,7,8]. Other formation schemes are based on formation graphs which represent specific interagent communication channels to achieve a specific geometric pattern [9,10,29]. Some tools of graph theory and linear systems are used to prove the convergence to the desired formation for special formation graphs such as the complete formation graph (i.e., where every agent measures the position of the rest of the group [18]), cyclic pursuit [13], undirected graphs [21], leaders-followers graphs (i.e., where all robots have communication with one or more group leaders [11,12]), etc. In this paper, we utilize a formation graph where one agent is considered as a virtual leader. Some advantages of the use of virtual leaders are the simplification of the analysis and best performance of the closed-loop system. Some important works about virtual leaders are [14,15,16].
The non-collision problem is generally solved using the negative gradient of an artificial potential function [17]. In the standard methodology, this function is the sum of Attractive (APF) and Repulsive Potential Functions (RPF). The APF are designed according to the desired interagent distances of a particular formation, whereas RPF serve to create repulsive forces to avoid collisions. In decentralized strategies, these RPF appear only when the minimum allowed distance between agents is violated. A formation control law based on APF only, guarantees the convergence to the desired formation, but inter-robot collisions could occur for some initial conditions. A formation control law based on the sum of APF and RPF guarantees non-collision, but does not ensure convergence to the desired formation for all initial conditions. This occurs because the robots can be trapped in undesired equilibrium points. The analysis to calculate these equilibria and the trajectories which do not converge to the desired formation is very complex [18]. A complete survey of convergence and collision avoidance based on APF and RPF was presented in our previous work [28].
Some research on the analysis of non-collision based on repulsive forces also exists. In [18], it is shown that the undesired equilibria are unstable for a complete graph formation strategy, however, they are not explicitly calculated. In [19,20], some "navigation functions" are designed with attractive and repulsive behaviour. However, the RPF are centralized and high-order functions, i.e., every agent needs to be fed back with the position of the rest of agents. The main disadvantages are that the system is no longer decentralized and the approach requires a high computational cost in real-time implementations. In [21], a general form of decentralized RPF approach applied to undirected formation graphs is obtained, showing the complexity of the analysis for the general case. However, the analysis of convergence to the desired formation assumes that the basin of attraction of the undesired equilibria and the set of admissible initial conditions are disjointed. Other attempts to achieve global convergence without collisions in formation control include: hybrid architectures where a high-level supervisor switches momentarily to reactive non-collision strategy [30], the use of small disturbances in order to escape from undesired equilibria which exhibit a saddle point behaviour [31] and the use of discontinuous repulsive vector fields [32]. Despite successful implementations, the repulsive forces are heuristically designed and therefore, formal proof about global convergence are rarely found (for example [33]).
Instead of using the mix of APF and some type of repulsive forces, in this paper a simple formation strategy based on APF only is used. Then, our main result is to obtain necessary and sufficient non-collision conditions based on the explicit solution of the closed-loop system. The main idea is to predict, from the initial positions, whether or not the agents will collide. Doing this, the convergence and the non-collision requirements are satisfied in a subset of initial conditions within the workspace. Preliminary results for the case of another formation strategy for three agents only were reported in [22]. In this paper, the non-collision conditions and their geometric interpretation are generalized for the case of any number of agents formed with respect to a virtual leader. A related result reported in the literature is [13], where non-collision conditions are obtained for the case of point robots formed in a cyclic pursuit graph. Roughly speaking, if the arguments of the polar coordinates of the initial conditions of the robots define a strictly increasing sequence, then the robots will not collide for any time instant.
The paper is organized as follows: Section 2 introduces a formal problem statement. Section 3 describes the formation control strategy for agents, considered as points in plane, using APF; also, convergence to the desired formation is demonstrated. In Section 4, noncollision conditions are derived. The result is extended to the case of unicycle formations in Section 4.1. Section 5 presents numerical simulations. Finally, concluding remarks are offered in Section 6.
n as the desired relative position of every i R in the formation given by The desired relative position of the group leader can be considered as a combination of the desired positions of n z with respect to every follower in the group. The formation strategy using n R as virtual leader is shown in  defined with respect to a global frame. Other works consider either the desired positions of robots using local frames [24,25] or the relative distances and orientations between robots [11,26,27].

Problem Statement
The control goal is to design a control law that circumscribes each robot.

Remark 1.
It is important to observe that the collisions between the leader and every follower are not analysed because they do not occur in the physical world. Thus, we only analyse the collisions between followers.
Remark 2. The star-shaped topology shown in Fig. 1 has been chosen because it renders symmetric expressions for the distance between any pair of agents. Another topology rendering the same kind of symmetry is the socalled undirected complete graph [18], where every agent detects the position of the rest of the agents. We have chosen the former topology because it requires less communication channels between agents.
For completeness of the paper, let us introduce the following definition.
Definition 1. The desired relative position of n mobile agents given by (3)-(4) is said to be a closed-formation if

Control strategy
In this section, a formation strategy is presented which ensures the convergence to a desired formation only. The collision problem will be analysed in Section 4. For system (1), local potential functions are defined by The functions i V are positive definite and reach their global minimum ( Using these functions, we define a control law given by where  0 k is a design parameter. Theorem 1. Consider the system (1) and the control law (7). Suppose that  0 k and the desired formation is closed. Then, in the closed-loop system (1)-(7) the agents converge exponentially to the desired formation, i.e., Proof: The closed-loop system (1)-(7) has the form A change of coordinates for the closed-system (8) is defined by The dynamics of the coordinates (9) are given by where

Remark 3.
Note that M is almost identical to the socalled Laplacian Matrix [21] of the formation given by (3)-(4). The only difference is the  n th row, due to the factor  1 2( 1) n in the last expression of (7).

Non-collision conditions
The control law (7) guarantees that all agents converge exponentially to the desired formation but inter-agent collisions can occur starting from some initial positions. Collision-free trajectories between follower robots are defined by the condition where d is the diameter of a circle that circumscribes each robot.
In order to characterize collision-free trajectories, the first step is to obtain an analytical expression of the solution of the closed-loop system (8). Such expression is provided by the next proposition.
where   Then, the solution of (15) is given by where  0 (0) p p is the vector of initial conditions of coordinates p and After some algebra, the solution in original coordinates can be written as in (13). in the condition of collision-free trajectories (12), we obtain  0 t . Using this notation, we establish our main result. and suppose that Then, anyone of the following three conditions is sufficient to guarantee non-collision between the i-th and j-th agents, i.e.,

Remark 4.
Hypothesis 2) means that the desired distance between agent i R and j R must be greater than d .
Hypothesis 3) means that agents do not collide at  0 t .

Proof of Theorem 2: (sufficiency) Note that the function
By hypothesis 2) and 3) The derivative of ( ) ij f t is given by ( ) ij f t is decreasing, crosses by zero at time instant  t and after that, becomes increasing. This occurs only when condition iii) is satisfied.
 Remark 5. The distance between the leader and the i-th follower, i.e.,  The analysis of the non-collision conditions for the case of the leader and a follower is more complex, but it is not studied because these collisions do not occur in the physical world. However, it is an issue for further theoretical research.  Another choice of formation graph which allows a simple explicit solution of the closed-loop system is the so-called complete graph, where each agent detects the position of all the others agents. This solution produces symmetric expressions for the distances between agents and hence allows obtaining a result similar to Theorem 2. This result has been reported for the case of three robots only in [22].

5.
Agents' positions in space in case

Extension to formation control of unicycles
The analysis of Section 4 can be adapted to the case of unicycles. The kinematic model of each agent i R , as shown in Fig. 6, is given by  Fig. 6. The coordinates  i are given by The idea of controlling coordinates  i instead of the centre of the wheels axis is frequently found in the mobile robot literature in order to avoid singularities in the control law [11]. The kinematics of (26) is given by

Conclusions
An analysis of non-collision conditions of a simple formation control strategy for multi-agent robots is presented. The formation strategy is based on a virtual leader. This virtual leader can be emulated by a computer or another control device and serves for the communication between followers. The formation control is based on attractive potential functions which only ensure the convergence to the desired formation, however, agents might still collide. To ensure collisionfree trajectories, a set of necessary and sufficient conditions, based on the exact solution of the closed-loop system, is presented. The main idea is to predict, from the initial conditions, whether or not the agents will collide. The virtual leader serves to calculate and simplify the non-collision conditions. The main result has an application in experimental work where the analysis of these conditions can be achieved off-line and the robots can be protected from undesired shocks. The results are extended to the case of unicycle-type robots.

Acknowledgments
The first author acknowledges financial support from UIA and CONACyT, Mexico.