Constrained Invariant Motions for Networked Multi-Agent Systems Mauro Franceschelli ⋆ , Magnus Egerstedt † , Alessandro Giua ⋆ and Cristian Mahulea ‡ ⋆ Electrical and Electronic Engineering † Electrical and Computer Engineering ‡ Computer Science and Systems Engineering University of Cagliari Georgia Institute of Technology University of Zaragoza 09123 Cagliari, Italy Atlanta, GA 30332, USA 50018 Zaragoza, Spain mauro.franceschelli@diee.unica.it magnus@ece.gatech.edu cmahulea@unizar.es giua@diee.unica.it Abstract— In this paper we propose a methodology to solve the constrained consensus problem, i.e., the consensus problem for multi-agent systems with constrained dynamics. We propose a decentralized one-step horizon optimization problem to be solved iteratively by the agents to achieve rendezvous at the centroid of the network while ensuring the connectivity of the network and the feasibility of the agents motion respect to their constrained kinematics. We also provide simulations of the algorithm behavior. I. I NTRODUCTION The consensus problem, i.e., the problem of having a collection of agents’ states reach a common value in the presence of network and information sharing constraints, has recently received considerable attention. For a representative sample, see [1], [2], [4], [6], [9], [8], [11], [16]. A common approach to this problem is to use a linear, nearest neighbor control strategy, resulting in a linear dynamic system driven by the graph Laplacian associated with the underlying network topology. In [14] was proposed a method to probe a network of agents executing a Laplacian-based control strategy with an application to fault detection. Furthermore a decentralized al- gorithm was proposed to recover the initial network centroid for a network of agents after a failure detection dragged the network away from the desired location. In this paper we propose a framework to address the Constrained Consensus Problem, i.e., the consensus problem with constrained agents’ trajectories and constrained mission objectives, based on the results of [14]. The proposed methodology follows decentralized model predictive control applied to multi-agent systems [12], [13]. Our approach differs form the ones in the literature in that it allows to perform consensus on the average in the presence of constraints. In particular we show that the iterative solution of a one step-horizon optimization problem, involving only information locally available by the agents without any communication, can efficiently solve complex constrained consensus problem. We focus on the application of our algorithm to rendezvous in multi-agent systems assuming single integrator agents with bounded speed. Such assump- tion is taken to decouple the motion coordination problem from the low level control of the single agent. The motion coordination algorithm basically specifies the set point that the low level controller need to track within a specified time horizon. The constraints that we consider are kinematic constraints of the agents, network constraints such as net- work connectivity preservation and mission constraints such rendezvous at the initial network centroid while ensuring all the other constraints along their motion. II. NETWORK MODEL We will be considering networks of agents, whose nominal state evolution is governed by a discrete time consensus equation that can be written quite generally as x(k + 1) = Px(k), (1) where P is a stochastic, indecomposable, aperiodic matrix, as discussed in [7]. Moreover, x ∈ R n is an aggregated state vector, with each component x i representing a scalar state associated with agent i =1,...,n. We model the network as an undirected graph G = V × E, with V being a set of vertices V = {1,...,n} that represent the agents, and where the edge set E ⊆ V × V encodes the network topology in that (i, j ) ∈ E if and only if agents i and j can share information. The graph can be encoded through its adjacency matrix A , i.e., a n × n matrix such that a i,j =1 if and only if (i, j ) ∈ E and is 0 elsewhere. Let N i ⊂ V be the set of vertices adjacent to vertex i, and let |N i | denote its cardinality. We can then define the degree matrix Δ as the diagonal matrix whose diagonal entries are Δ i,i = |N i |. Using these matrices, a standard, discrete time model of consensus networks is the one defined by x(k + 1) = (I −ǫL)x(k), where I is the identity matrix, L =Δ−A is the graph Laplacian of the graph G, and ǫ> 0 the sampling time. Under this dynamics, the matrix P in Equation (1) becomes P = I − ǫL. (2) Following the notation in [7], we will refer to P as a Perron matrix, and this is the particular choice of P -matrix that will be used throughout the paper. For the developments in this paper, we will not necessarily assume that the network is static, i.e. edges may be removed or added, thus resulting in a change in network topology. As such, we will in fact let the network be represented by a time varying, undirected graph G(t) = (V,E(t)), where the edge set is time dependent. This could be caused by communication failures, or by the movements of the 2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009 FrC19.4 978-1-4244-4524-0/09/$25.00 ©2009 AACC 5749