On a hierarchical control strategy for multi-agent formation without reflection

This paper considers a formation shape control problem for point agents in a two-dimensional ambient space, where the control is distributed, is based on achieving desired distances between nominated agent pairs, and avoids the possibility of reflection ambiguities. This has potential applications for large-scale multi-agent systems having simple information exchange structure. One solution to this type of problem, applicable to formations with just three or four agents, was recently given by considering a potential function which consists of both distance error and signed triangle area terms. However, it seems to be challenging to apply it to formations with more than four agents. This paper shows a hierarchical control strategy which can be applicable to any number of agents based on the above type of potential function and a formation shaping incorporating a grouping of equilateral triangles, so that all controlled distances are in fact the same. A key analytical result and some numerical results are shown to demonstrate the effectiveness of the proposed method.

I. Introduction

Formation shape control for multi-agent systems is one of the most actively studied topics due to its potential in various applications and theoretical depth. Surveys of formation shape control are found in [1] and [5]. According to the sensing capability and agent-interaction topology, most of the existing methods can be classified as (a) Position-based control, (b) Displacement-based control, and (c) Distance-based control (see [5]). In the case of (a), we need the absolute position of each agent, often to a high accuracy, so sensors (like GPS) could be expensive. In the case of (b), most of the existing works require that all the different local coordinate systems associated with each agent should be aligned with a global coordinate system. This might be difficult from the implementation viewpoint. In contrast, in the case of (c), each agent only requires the relative position information in its own local coordinate system. Hence, at least in part because of the implied saving in sensor requirements relative to (a) and (b), distance-based formation control has attracted a considerable attention recently (see references in [2]–[5], [8], [9]). One major drawback is that there can be many undesirable equilibria when using the gradient control laws that are typically suggested. Because of the overall system's nonlinear control nature, it is not trivial to guarantee the convergence to the desired formation shape from all or almost all initial conditions. Also, the approach may require more inter-agent interactions to achieve a desired rigid formation compared to the case of (b). For example, if the target system is described by four agents with five edges (i.e., inter-agent interactions) and their lengths in a two dimensional space, the system is not uniquely specified up to congruence, due to the possibility of reflection ambiguities, either of individual triangles in the formation or the whole formation. Six edges are necessary, if individual triangle reflection ambiguities are to be avoided, while a reflection ambiguity remains for the whole formation.