In this paper a continuum approach for controlling the motion of a swarm of particles (“agents”) is presented. The control objective is to move the swarm from an initial reference configuration to a final configuration possibly of a different shape and size, avoid obstacles and inter-agent collisions while satisfying hard constraints on agent kinematics. The agents are considered to be inside a rectangle and it is assumed that the task is to move the swarm so that at the final time the agents are confined to a rectangle of possibly different size and orientation. It is shown that the agents can locally control their motions so that a collision free transfer respecting all agent constraints can be achieved with minimal inter-agent communication. At the nucleus of this approach is the “deformation of the group shape from a given reference configuration to a desired configuration”. The key idea is to find an appropriate homeomorphism between the initial and final configurations that respect all agent constraints. We show that a class of homogeneous transformations has very beneficial attributes. In particular, each particle or agent has a well-defined path that is based solely on its reference position. It necessarily means that an agent does not have to know the location of any other agent once the motion map is made available to an agent. We emphasize: (1) that minimum to no communication between agents is required for its implementation, and (2) it is independent of the number of agents, meaning that the approach is completely scalable. These two attributes are major advantages that are not present in most currently known path planners for swarms. Presented will be simulation results to illustrate the key ideas of the proposed approach.

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