Feedback Stabilization and Trajectory Tracking of a Drifting Vortex Near a Robotic Fish Using Shape Control

Abstract

This article presents a feedback control design that adjusts the trajectory of a drifting vortex in a freestream near a deformable Joukowski foil using camber shape control. We derive the dynamics of a point vortex in flow around a Joukowski foil using a potential flow model and provide numerical analysis of the number, stability, and controllability of open-loop equilibrium points of the vortex-foil system. We show that the position of a point vortex can effectively be regulated using a full-state feedback camber control law based on a linear–quadratic regulator design while maintaining the validity of the dynamics model. We then provide a vortex trajectory tracking method that adjusts sinusoidal camber changes using full-state feedback applied to the phase offset of the sinusoid, implemented through an optimization loop and backstepping. We present an illustrative example of effective trajectory tracking achieved through closed-loop control, along with a visualization of the region of convergence for several different system parameters and initial conditions. The results demonstrate the ability to track a desired feasible trajectory with minimal persistent error across a wide range of system configurations.