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This paper discusses a formation control problem in which a target formation is defined with both distance and signed area constraints. The control objective is to drive spatially distributed agents to reach a unique target rigid formation shape (up to rotation and translation) with desired inter-agent distances. We define a new potential function by incorporating both distance terms and signed area terms and derive the formation system as a gradient system from the potential function. We start with a triangle formation system with detailed analysis on the equilibrium and convergence property with respect to a weighting gain parameter. For an equilateral triangle example, analytic solutions describing agents' trajectories are also given. We then examine the four-agent double-triangle formation and provide conditions to guarantee that both triangles converge to the desired side distances and signed areas.

This paper provides a unified solution for a general distributed control problem of multi-agent systems based on the gradient-flow approach. First, a generalized coordination is presented as a control objective which represents a wide range of coordination tasks (e.g., consensus, formation and pattern decision) in a unified manner. Second, a necessary and sufficient condition for the gradient-based controllers to be distributed is derived. It turns out that the notion of clique (i.e., complete subgraph) plays a crucial role to obtain any distributed controllers. Furthermore, all such controllers are explicitly characterized with free design parameters. Third, it is shown how to choose an optimal controller in a systematic way among all distributed ones, where an optimality measure is introduced for the generalized coordination. Finally, the effectiveness of the proposed method is demonstrated through simulations, where a distributed pattern decision is discussed as an example of the generalized coordination.

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