In this paper, we address the decentralized and parallel construction of rigid graphs in the plane that optimize an edge-weighted objective function under cardinality constraints. Two auction-based algorithms to solve this problem in a decentralized fashion are first proposed. Centered around the notion of leader election, the first approach finds an optimal solution through a greedy bidding, while the second approach provides a sub-optimal solution which reduces complexity according to a sliding mode parameter. Then, by exploiting certain local structural properties of graph rigidity, a parallelization to build a portion of the optimal solution in constant time is derived. A theoretical characterization of algorithm performance is provided together with complexity analysis. Finally, simulation results are presented to corroborate the theoretical findings.

In this paper, we address the construction of optimally rigid networks that minimize an edge-weighted objective function over a planar graph. We propose two auction-based algorithms to solve this problem in a fully decentralized way. The first approach finds an optimal solution at the cost of high communication complexity; the second approach provides a sub-optimal solution while reducing the computational burden according to a sliding mode parameter ζ, yielding a tradeoff between complexity and optimality. A theoretical characterization of the optimality of the first algorithm is provided, and a closed form for the maximum gap between the optimal solution and the sub-optimal solution is also given. Simulation results are presented to corroborate the theoretical findings.

In this paper, we consider the problem of evaluating the rigidity of a planar network, while satisfying common objectives of real-world systems: decentralization, asynchronicity, and parallelization. The implications that rigidity has in fundamental multirobot problems, e.g., guaranteed formation stability and relative localizability, motivates this study. We propose the decentralization of the pebble game algorithm of Jacobs et al. , which is an O(n2) method that determines the generic rigidity of a planar network. Our decentralization is based on asynchronous messaging and distributed memory, coupled with auctions for electing leaders to arbitrate rigidity evaluation. Further, we provide a parallelization that takes inspiration from gossip algorithms to yield significantly reduced execution time and messaging. An analysis of the correctness, finite termination, and complexity is given, along with a simulated application in decentralized rigidity control. Finally, we provide Monte Carlo analysis in a Contiki networking environment, illustrating the real-world applicability of our methods, and yielding a bridge between rigidity theory and realistic interacting systems.

In this paper, a novel decentralized swarm aggregation algorithm for multi-robot systems with an integrated obstacle avoidance is proposed. In this framework, the interaction among robots is limited to their visibility neighborhood, i.e., robots that are within the visibility range of each other. Furthermore, to better comply with the hardware/software limitations of mobile robotic platforms, robots actuators are assumed to be saturated. A theoretical characterization of the main properties of the proposed swarm aggregation algorithm is provided. Simulations have been carried out to validate the theoretical results and experiments have been performed with a team of low-cost mobile robots to demonstrate the effectiveness of the proposed approach in real scenario.

In this paper, we consider the problem of evaluating the generic rigidity of an interconnected system in the plane, without a priori knowledge of the network's topological properties. We propose the decentralization of the pebble game algorithm of Jacobs et. al., an O(n2) method that determines the generic rigidity of a planar network. Our decentralization is based on asynchronous inter-agent message-passing and a distributed memory architecture, coupled with consensus-based auctions for electing leaders in the system. We provide analysis of the asynchronous messaging structure and its interaction with leader election, and Monte Carlo simulations demonstrating complexity and correctness. Finally, a novel rigidity evaluation and control scenario in the accompanying media illustrates the applicability of our proposed algorithm.

We propose a swarm aggregation algorithm based on local interactions in the presence of saturations on the robot actuators. This assumption allows to better model the physical limitations of actual mobile robotic platforms. In our framework, robot-to-robot interactions are limited to the visibility neighborhood, i.e., to robots that are within the range of visibility of each other. A theoretical analysis of the convergence properties is presented for the proposed swarm aggregation algorithm. Extensive simulations have been performed to corroborate the theoretical results. In addition, experiments with a team of low-cost mobile robots have been carried out to show the effectiveness of the proposed approach.