This paper treats the problem of the merging of formations, where the underlying model of a formation is graphical. We first analyze the rigidity and persistence of meta-formations, which are formations obtained by connecting several rigid or persistent formations. Persistence is a generalization to directed graphs of the undirected notion of rigidity. In the context of moving autonomous agent formations, persistence characterizes the efficacy of a directed structure of unilateral distance constraints seeking to preserve a formation shape. We derive then, for agents evolving in a two- or three-dimensional space, the conditions under which a set of persistent formations can be merged into a persistent meta-formation, and give the minimal number of interconnections needed for such a merging. We also give conditions for a meta-formation obtained by merging several persistent formations to be persistent.
Deep Dive into Rigidity and persistence for ensuring shape maintenance of multiagent meta formations (extd version).
This paper treats the problem of the merging of formations, where the underlying model of a formation is graphical. We first analyze the rigidity and persistence of meta-formations, which are formations obtained by connecting several rigid or persistent formations. Persistence is a generalization to directed graphs of the undirected notion of rigidity. In the context of moving autonomous agent formations, persistence characterizes the efficacy of a directed structure of unilateral distance constraints seeking to preserve a formation shape. We derive then, for agents evolving in a two- or three-dimensional space, the conditions under which a set of persistent formations can be merged into a persistent meta-formation, and give the minimal number of interconnections needed for such a merging. We also give conditions for a meta-formation obtained by merging several persistent formations to be persistent.
Recently, significant interest has been shown on the behavior of autonomous agent formations (groups of autonomous agents interacting which each other) [2,4,7,9,19], and more recently on meta-formations, which is the name ascribed to an interconnection of formations, generally with the individual formations being separate [1,25]. By autonomous agent, we mean here any human-controlled or unmanned vehicle moving by itself and having a local intelligence or computing capacity, such as ground robots, air vehicles or underwater vehicles. Many reasons such as obstacle avoidance and dealing with a predator can indeed lead a (meta-)formation to be split into smaller formations which are later re-merged. Those smaller formations need to be organized in such a way that they can behave autonomously when the formation is split. Conversely, some formations may need to be temporarily merged into a meta-formation to accomplish a certain task, this meta-formation being split afterwards.
The particular property of formations and meta-formations which we analyze here is persistence. This graph-theoretical notion which generalizes the notion of rigidity to directed graphs was introduced in [9] to analyze the behavior of autonomous agent formations governed by unilateral distance constraints: Many applications require the shape of a multi-agent formation to be preserved during a continuous move. For example, target localization by a group of unmanned airborne vehicles (UAVs) using either angle of arrival data or time difference of arrival information appears to be best achieved (in the sense of minimizing localization error) when the UAVs are located at the vertices of a regular polygon [5]. Other examples of optimal placements for groups of moving sensors can be found in [17]. This objective can be achieved by explicitly keeping some inter-agent distances constant. In other words, some inter-agent distances are explicitly maintained constant so that all the inter-agent distances remain constant. The information structure arising from such a system can be efficiently modelled by a graph, where agents are abstracted by vertices and actively constrained inter-agent distances by edges.
We assume here that those constraints are unilateral, i.e., that the responsibility for maintaining a distance is not shared by the two concerned agents but relies on only one of them. This unilateral character can be a consequence of the technological limitations of the autonomous agents. Some UAV’s can for example not efficiently sense objects that are behind them or have an angular sensing range smaller than 360 • [3,8,20]. Also, some of the authors of this paper are working with agents in which optical sensors have blind three dimensional cones. It can also be desired to ease the trajectory control of the formation, as it allows so-called leader-follower formations [2,6,21]. In such a formation, one agent (leader) is free of inter-agent distance constraints and is only constrained by the desired trajectory of the formation, and a second agent (first follower) is responsible for only one distance constraint and can set the relative orientation of the formation. The other agents have no decision power and are forced by their distance constraints to follow the two first agents. This asymmetry is modelled using directed edges in the graph. Intuitively, an information structure is persistent if, provided that each agent is trying to satisfy all the distance constraints for which it is responsible, it can do so, with all the inter-agent distances then remaining constant, and as a result the formation shape is preserved. A necessary but not sufficient condition for persistence is rigidity [9], which intuitively means that, provided that all the prescribed distance constraints are satisfied during a continuous displacement, all the inter-agent distances remain constant (These concepts of persistence and rigidity are more formally reviewed in the next section). The above notion of rigidity can also be applied to structural frameworks where the vertices correspond to joints and the edges to bars. The main difference between rigidity and persistence is that rigidity assumes all the constraints to be satisfied, as if they were enforced by an external agency or through some mechanical properties, while persistence considers each constraint to be the responsibility of a single agent. As explained in [9], persistence implies rigidity, but it also implies that the responsibilities imposed on each agent are not inconsistent, for there can indeed be situations where this is so, and they must be avoided. Rigidity is thus an undirected notion (not depending on the edge directions), while persistence is a directed one. Both rigidity and persistence can be analyzed from a graph-theoretical point of view, and it can be proved [9,22,28] that if a formation is rigid (resp. persistent), then almost all formations represented by the same graph are rigid (resp. persiste
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