Formation Merging Control Of Leader-follower Networks In 3D Space

The paper studies two questions of formation control in 3D. The first question con-centrates on the fundamental coordination problem that requires a network of agents to achieve a specific but arbitrary formation shape. We show that all similar formation sub-ject to only shape constraints are those that lie in the equilibrium set of system whose sensor graph is the same as the formation graph. The relative Laplacian satisfies certain rank condition and that a formation shape can be realized almost surely if and only if the graph modeling the inter-agent specification of the formation shape is i-rooted (the rank of Laplacian which relative to the graph is n-i). The second question studies the forma-tion merging problem for a leader-follower network under directed and Markov switching topologies. We show that the followers can asymptotically merge themselves to the for-mation for arbitrarily initial configurations. Each follower selects its neighbors and also its control law according to the target formation they aim to achieve and thus it allows directed and Markov switching topologies. It is shown that a globally rigid formation can be merged asymptotically for the leader-follower network in a setup with directed and Markov switching graphs if and only if every follower has at least a path from the set of leaders or the rank of Laplacian which relatives to the graph is m (the number of followers is m).