Variational Principles for Optimal Control of Left-Invariant Multi-Agent Systems with Asymmetric Formation Constraints
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We study an optimal control problem for a multi-agent system modeled by an undirected formation graph with nodes describing the kinematics of each agent, given by a left invariant control system on a Lie group. The agents should avoid collision between them in the workspace. This is accomplished by introducing appropriate potential functions into the cost functional for the optimal control problem, corresponding to fictitious forces, induced by the formation constraint among agents, that break the symmetry of the individual agents and the cost functions, and render the optimal control problem partially invariant by a Lie group of symmetries. Reduced necessary conditions for the existence of normal extrema are obtained using techniques from variational calculus on manifolds. The Hamiltonian formalism associated with the optimal control problem is explored through an application of Pontryagin's maximum principle for left-invariant systems where necessary conditions for the existence of normal extrema are obtained as integral curves of a Hamiltonian vector field associated to a reduced Hamiltonian function. By means of the Legendre transformation we show the equivalence of both frameworks. The discrete-time version of optimal control for multi-agent systems is studied in order to develop a variational integrator based on the discretization of an augmented cost functional in analogy with the Hamiltonian picture of the problem through the Hamilton-Pontryagin variational principle. Such integrator defines a well defined (local) flow to integrate the necessary conditions for local extrema in the optimal control problem. As an application we study an optimal control problem for multiple unicycles.
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