Application of Oscillator Dynamics for Deviated Pursuit Formations: Preliminary Results

In this article, the application of Hopf oscillator for a group of agents connected cyclically is studied in regards to improve the robustness of the linear deviated cyclic pursuit protocol. The linear cyclic pursuit protocol suffers from the robustness problem due to the rotation angle of the line of sight between the agents, as a small perturbation in the rotation angle can cause the system to become structurally unstable, which makes the overall system either stable or unstable. To maintain a geometric pattern the linear system should maintain its marginal stability. To address this issue, individual agents connected via cyclic protocol are modeled to behave like Hopf oscillators. Consequently, two control laws are introduced based on Hopf oscillators, one with a static target at the origin and other with a virtual circular moving target. With a static target we show that it is possible to utilize the coupling gain to render geometric patterns and with a virtual target we can render geometric patterns inherently with the amplitude and frequency parameters of the virtual target along with rotation angle flexibility. For both the cases we utilize Poincaré map to show the existence of trochoidal like quasi-periodic trajectories.