Rigidity Maintenance Control for Multi-Robot Systems

Abstract

Abstract-Rigidity of formations in multi-robot systems is important for formation control, localization, and sensor fusion. This work proposes a rigidity maintenance gradient controller for a multi-agent robot team. To develop such a controller, we first provide an alternative characterization of the rigidity matrix and use that to introduce the novel concept of the rigidity eigenvalue. We provide a necessary and sufficient condition relating the positivity of the rigidity eigenvalue to the rigidity of the formation. The rigidity maintenance controller is based on the gradient of the rigidity eigenvalue with respect to each robot position. This gradient has a naturally distributed structure, and is thus amenable to a distributed implementation. Additional requirements such as obstacle and inter-agent collision avoidance, as well as typical constraints such as limited sensing/communication ranges and line-of-sight occlusions, are also explicitly considered. Finally, we present a simulation with a group of seven quadrotor UAVs to demonstrate and validate the theoretical results.

Decentralized rigidity maintenance control with range measurements for multi-robot systems

“…In our previous work (Zelazo et al, 2012), we introduced a related matrix termed the symmetric rigidity matrix. A main result of (Zelazo et al, 2012) was to provide a necessary and sufficient conditions for rigidity in the plane in terms of the positivity of a particular eigenvalue of the symmetric rigidity matrix; this eigenvalue we term the rigidity eigenvalue. This result is in the same spirit as the celebrated Fiedler eigenvalue 1 and its relation to the connectivity of a graph (Godsil and Royle, 2001).…”

Distributed combinatorial rigidity control in multi-agent networks

“…In determining and controlling the rigidity of the framework F x over time, one could continuously check infinitesimal rigidity and take suitable action towards preservation, as is explored in [19]. However, in [20] it is shown that almost all realizations of G are either infinitesimally rigid or flexible, indicating that rigidity can be approached generically, by examining the underlying graph G. Such a combinatorial characterization of graph rigidity in the plane was first described by Laman in [23], and is summarized as follows 1 : Theorem 2.1 (Graph rigidity, [23]): A graph G = (V, E) over realizations in R 2 having n ≥ 2 nodes is rigid if and only if there exists a subsetĒ ⊆ E consisting of |Ē| = 2n − 3 edges satisfying the property that for any non-empty subsetÊ ⊆Ē, we have |Ê| ≤ 2k − 3, where k is the number of nodes in V that are endpoints of (i, j) ∈Ê.…”

“…Notice that Laman's theorem places restrictions on the network edges, thus we must only evaluate the Laman conditions during transitions in G. When compared to continuum methods such as [19], exploiting Theorem 2.1 in rigidity control yields a fundamentally more efficient and robust solution, as continuous computation and communication resources are not required for implementation. Such insights motivate the problem investigated in this work:…”

“…static framework perspective. Finally, the most closely related work to this paper is [19], where a continuous gradient control for maintaining infinitesimal rigidity (as opposed to combinatorial rigidity) is proposed that applies the notion of a rigidity eigenvalue, although without decentralization.…”

Decentralized control of parallel rigid formations with direction constraints and bearing measurements

“…Rigidity is a well-known and fundamental tool in the context of formation control based on distances [5], [6], [7]. Parallel rigidity [8], [9], that is, the bearing counterpart of 'distance rigidity', has also been recently introduced for controlling bearing-constrained formations.…”