Voronoi-Based Deployment of Multi-Agent Systems

Abstract

This thesis investigates convergence in the framework of Voronoi-based deployment of a multi-agent system to a convex polytopic multi-dimensional environment. The deployment objective is to drive the system into a stable static configuration which exhibits optimal coverage of the target environment. To this end, the system is subjected to a collection of decentralized control laws steering each agent towards a Chebyshev center of its associated time-varying polytopic Voronoi-neighborhood. In non-degenerate cases, socalled Chebyshev configurations of the multi-agent system achieves the above objective. In these configurations, all agents are at a Chebyshev center of their Voronoi-neighborhood. Proving convergence to the set of Chebyshev configurations is an open research question. This is the most pertinent issue with regards to the framework viability. While such a property is supported by simulations, neither complete formal convergence proofs nor formal characterizations of the equilibria to be achieved exist. This thesis is oriented towards strengthening the theoretical convergence results. For the special case of deployment to one-dimensional environments, we prove convergence to an unique static Chebyshev configuration. Moreover, we highlight connections to discrete time averaging systems and show how the system converges to consensus on the Chebyshev radii. The remaining results apply in the general case of multi-dimensional environments. We introduce a novel undirected interaction graph as a theoretical tool for a deeper understanding of the multi-agent system's functioning. Exploiting this graph, we prove that the set of static configurations are Chebyshev configurations in which all subsets of agents within the same connected component of the interaction graph are in consensus on their Chebyshev radii. Finally we prove convergence to a Chebyshev configuration, with consensus on the Chebyshev radii, provided the interaction graph is connected along the trajectories of the multi-agent system. Throughout the presentation, the theoretical results are motivated and supported by simulations.

Decentralized MPC for UAVs Formation Deployment and Reconfiguration with Multiple Outgoing Agents

“…α coincides with its associated Chebyshev center c c α . When Assumption 7 is verified and the agents are steered with a full state-feedback control law, it has been proven in [12,24] that the formation converges towards a static configuration if and only if it is a Chebyshev configuration.…”

“…The control algorithm for deployment and reconfiguration of a multi-agent system presented in this paper is tested in simulation with MATLAB. The optimization solvers for the problems (12) and (14) are generated with CVXGEN [17]. This tool generates a custom solver for a given optimization problem with a quadratic cost function and linear constraints.…”

Guaranteed Voronoi-based Deployment for Multi-Agent Systems under Uncertain Measurements

“…The constraint (19e) ensures that X λ is a controlled λ-contractive set [18] for system (2). Then, the nominal system converges to a static Chebyshev configuration [8], [10], where x α (k + 1) = x α (k) = x obj α . Finally, it was shown in [16] that, with the control law given in (11), the real position x α and speed v α of the agent α ∈ Σ satisfy the constraints x α ∈ X and v α ∈ U provided that x α (0) ∈ S and e α (0) ∈ S. Then, with the chosen constraints and with the robust tube-based MPC control algorithm, the real position of each agent is guaranteed to stay inside its GV cell which is a safety region.…”

“…Several existing results focus on driving the MAS into a centroidal Voronoi configuration where the target point is the center of mass of the Voronoi cell [5], [6], [7] but this point can be difficult to compute. A simpler objective is to consider the Chebyshev center of the cell [8], [9], [10], i.e. the center of the largest euclidean ball inscribed in the cell.…”

“…Several control algorithms have been studied for the deployment of a MAS such as consensus based algorithms [13], [1], potential field [2], gradient ascent [12] or model predictive control (MPC) [14], [9], [15]. Moreover, in the case of a Chebyshev center based control algorithm, a statefeedback control law ensures that a static optimized coverage configuration is attained [8], [10]. In order to take into account possible disturbances on the system, robust MPC techniques have been proposed in [16], [17] for a single system, based on tubes of robust positively invariant sets [18], [19] centered on the system's nominal trajectory.…”