Formation-Preserving Properties of Cooperative Kinematic Agents With or Without External Influence of Target Attraction

In this paper, we investigate formation-preserving properties of a system of cooperative kinematic agents in response to external influence of (virtual) target attraction. When the flocking agents track a target, their group size will generally shrink, and the geometry of the group will deform (i.e., in a shape-variant manner) unless certain conditions are satisfied. In this paper, we focus on the formation-preserving properties between the two stable formations of the system without and with target attraction. To investigate such properties, we introduce a new concept of formation-preserving stiffness (FPS) to characterize the degree of variation between the two formations, and whether the variation is shape-invariant (i.e., with a uniform scaling between each pair of the agents) or shape-variant. The perspective in this paper is unique, with the main contributions as follows. First, we express the FPS for coupled agents with a general topology. Second, we provide the necessary and sufficient topological conditions that guarantee shape invariance of the two formations with or without target attraction. Finally, we characterize the shape-variant FPS with respect to some commonly used topologies, and interpret the characteristics of the FPS for each topology, particularly from the perspective of relative topological status of agents in each topology.