Distributed multiple-bipartite consensus in networked Lagrangian systems with cooperative–competitive interactions

In combination with the collective behavior evolution of bipartite consensus and cluster/group consensus, this paper proposes the notion of multiple-bipartite consensus in networked Lagrangian systems (NLSs), effectively integrating the above emergent collective behaviors in cooperative–competitive networks. The problems of leaderless multiple-bipartite consensus and leader-following multiple-bipartite tracking consensus are solved via deploying the designed distributed adaptive torque controllers for uncertain NLSs. Compared with the traditional bipartite consensus framework, antagonistic interactions can exist in the same subnetwork. By introducing an acyclic partition and adding the integral item in the distributed adaptive torque control protocols, the explicit expressions of the final states are eventually obtained in the leaderless case. Moreover, the leader-following case can be realized in finite time resorting to two predefined estimators embedded in the scheme. The effectiveness of two scenarios has been illustrated through numerical simulations with ten heterogeneous mechanical manipulators.