Closing ranks in rigid multi-agent formations using edge contraction

Abstract

SUMMARYThis paper proposes a systematic approach to solve the closing rank problem for a rigid multi-agent formation, viz. restoring rigidity after loss of an agent. The approach is based on a particular graph operation, the edge contraction operation. It is proven that when an agent is lost in an arbitrary two-dimensional rigid formation, rigidity can always be restored by transferring all links to which this agent was incident on to one of its neighbors, though not in general any arbitrary one of them. From a graph theoretical point of view, this corresponds to contraction of a certain edge incident to the vertex representing the agent being lost. It is established, for any two-dimensional rigid formation (graph), that there exists at least two such edges that can be contracted to solve the closing ranks problem. Later, it is demonstrated that any potential decentralized algorithm to check if an arbitrary edge is contractible would need to use information on vertices and edges that can be at arbitrarily large distance from the edge considered; and a set of rigid graph theoretical results are established for several general settings, which can be used in selection of the edge to contract in these settings in order to solve the corresponding closing ranks problems. Partial results are also obtained for three-dimensional formations, and it is shown that the two-dimensional results do not generalize as such to higher dimension.

Edge contraction based maintenance of rigidity in multi-agent formations during agent loss

“…Despite the demonstrations of Section IV-A about potential requirements of massive information by closing ranks algorithms based on edge contraction, there exist classes of generic cases where one can easily determine that an edge is or is not 2-contractible, as implied by the following two propositions and their corollary, whose proofs are omitted here due to space limitations and can be found in [14]:…”

“…In the paper we focus on formations in 2 (2-dimensional Euclidean space), noting that discussions on extension of the results to be presented in the following sections to 3 can be found in [14], extended version of this paper. Hence, the rigidity and closing ranks notions and characteristics are introduced in 2 in this section, although most of them can be generalized for arbitrary dimensional space d (d ∈ {2, 3, .…”

“…First, we establish a sufficient condition in the following lemma, whose proof is omitted here and can be found in [14].…”

“…In proving Theorem 4, we use the following lemmas, whose proofs are omitted here due to space limitations and can be found in [14]:…”

Rigid formation construction from non-rigid components

“…The work of C. Yu are removed from a minimally rigid formation. Closing ranks problem was studied in [2] and [3] and different self-repair approaches were proposed to regain rigidity for non-rigid formations.…”