Markov Chain Approach to Probabilistic Guidance for Swarms of Autonomous Agents

Abstract

AbstractMotivated by biological swarms occurring in nature, there is recent interest in developing swarms comprised completely of engineered agents. The main challenge for developing swarm guidance laws compared to earlier formation flying and multi‐vehicle coordination approaches is the sheer number of agents involved. While formation flying applications might involve up to 10 to 20 agents, swarms are desired to contain hundreds to many thousands of agents. In order to deal with the sheer size, the present paper makes a break with past deterministic methods, and considers the swarm as a statistical ensemble for which guidance can be performed from a probabilistic point of view. The probability‐based approach takes advantage of the law of large numbers, and leads to computationally tractable and implementable swarm guidance laws. Agents following a probabilistic guidance algorithm make statistically independent probabilistic decisions based solely on their own state, which ultimately guides the swarm to the desired density distribution in the configuration space. Two different synthesis methods are introduced for designing probabilistic guidance laws. The first is based on the Metropolis‐Hastings (M‐H) algorithm, and the second is based on using linear matrix inequalities (LMIs). The M‐H approach ensures convergent swarm behavior subject to enforced desired motion constraints, while the LMI approach additionally ensures exponential convergence with a prescribed decay rate, and allows minimization of a cost function that reflects fuel expenditure. In addition, both algorithms endow the swarm with the property of self‐repair, and the capability to strictly enforce zero‐probability keep‐out regions. This last property requires a slight generalization of the Perron‐Frobenius theory, and can be very useful in swarm applications that contain regions where no agents are allowed to go. Simulation examples are given to illustrate the methods and demonstrate desired properties of the guided swarm.