A leader follower model has recently been proposed for homogeneous deformation of a multi-agent system (MAS) in ℝn. Researchers have shown how a desired homogeneous transformation can be designed by choosing proper trajectories for n + 1 leader agents and can be learned by every follower through local communication. However, existing work requires every follower to communicate with n + 1 adjacent agents, where communication between every two adjacent followers is constrained to be bidirectional. These requirements limit the total allowable number of agents, so an arbitrary number of agents may not be able to acquire a desired homogeneous mapping by local interaction. Additionally, if followers are not allowed to communicate with more than n + 1 neighboring agents, the convergence rate of actual positions to the desired positions (defined by a homogeneous transformation) may not be sufficiently high. The system may then considerably deviate from the desired configuration during transition. The main contribution of this article is to address these two issues, where each follower is considered to be a general linear system. It will be proven that followers can acquire desired positions prescribed by a homogeneous mapping in the presence of disturbance and measurement noise by applying a new finite-time reachability model under either fixed or switching topologies, if: (i) communication among followers is defined by a directed and strongly connected subgraph, (ii) each follower applies a consensus protocol with communication weights that are consistent with the positions of the agents in the initial configuration, and (iii) every follower i is allowed to communicate with mi≥n+1 local agents. With this strategy, an MAS with an arbitrary number of agents with linear dynamics can acquire a desired homogeneous mapping in the presence of disturbance and measurement noise, where convergence rate can be enhanced by increasing the number of communication links.
In this paper, a new framework for evolution of multi-agent systems (MAS) based on principles of continuum mechanics is developed. Agents are treated as mass particles of a continuum whose evolution (both translation and deformation) is modeled as a homeomorphism from a reference to the current configuration. Such a mapping assures that no two mass particles of the continuum occupy the same location at any given time, thus guaranteeing that inter-agent collision is avoided during motion. We show that a special class of mappings whose Jacobian is only time varying and not spatially varying has some desirable properties that are advantageous in studying swarms. Two specific scenarios are studied where the evolution of a swarm from one configuration to another occurs with no inter-agent collisions while avoiding obstacles, under (i) zero inter-agent communication and (ii) local inter-agent communication. In the first case, a desired map is computed by each agent all knowing the positions of a few leader agents in a reference and the desired configurations. In the second case, paths of n + 1 leader agents evolving in an n-D space are known only to the leaders, while positions of follower agents evolve through updates that are based on positions of n + 1 adjacent agent through local communication with them. The latter is based on a set of weights of communication of follower agents that are predicated on certain distance ratios assigned on the basis of the initial formation of the MAS. Properties of homogeneous maps are exploited to characterize the necessary communication protocol.
The continuum deformation leader-follower cooperative control strategy models vehicles in a multi-agent system as particles of a deformable body. A desired continuum deformation is defined based on leaders' trajectories and acquired by followers in real-time through local communication. The existing continuum deformation theory requires followers to be placed inside the convex simplex defined by leaders. This constraint is relaxed in this paper. We prove that under suitable assumptions any n + 1 (n = 1, 2, 3) vehicles forming an n-D simplex can be selected as leaders while followers, arbitrarily positioned inside or outside the leading simplex, can acquire a desired continuum deformation in a decentralized fashion. The paper's second contribution is to assign a one-to-one mapping between leaders' smooth trajectories and homogeneous deformation features obtained by continuum deformation eigen-decomposition. Therefore, a safe and smooth continuum deformation coordination can be planned either by shaping homogeneous transformation features or by choosing appropriate leader trajectories. This is beneficial to efficiently plan and guarantee collision avoidance in a large-scale group. A simulation case study is reported in which a virtual convex simplex contains a quadcopter vehicle team at any time t; A* search is applied to optimize quadcopter team continuum deformation in an obstacle-laden environment.
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