Decentralized probabilistic density control of autonomous swarms with safety constraints

This paper presents a Markov chain based approach for the probabilistic density control of a large number, swarm, of autonomous agents. The proposed approach specifies the time evolution of the probabilistic density distribution by using a Markov chain, which guides the swarm to a desired steady-state distribution, while satisfying the prescribed ergodicity, motion, and safety constraints. This paper generalizes our previous results on density upper bound constraints and captures a general class of linear safety constraints that bound the flow of agents. The safety constraints are formulated as equivalent linear inequality conditions on the Markov chain matrices by using the duality theory of convex optimization which is our first contribution. With the safety constraints, we can facilitate proper low-level conflict avoidance policies to compute and execute the detailed agent state trajectories. Our second contribution is to develop (i) linear matrix inequality based offline methods, and (ii) quadratic programming based online methods that can incorporate these constraints into the Markov chain synthesis. The offline method provides a feasible solution for Markov matrix when there is no density feedback. The online method utilizes realtime estimates of the swarm density distribution to continuously update the Markov matrices to maximize the convergence rates within the problem constraints. The paper also introduces a decentralized method to compute the density estimates needed for the online synthesis method.

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