Tracking expanding star curves using guidance vector fields

This paper describes a constructive method for creating provably stable vector fields for a class of star shaped, three-dimensional curves. Using the approach, vector fields can be designed that provide essentially global exponential convergence to a desired curve expressed in cylindrical coordinates. Additional terms are added to the vector fields to account for time-varying curves and to enable normalization of the vector field in order to keep the resulting speed bounded. Simulation results demonstrate the method for tracking a variety of different closed curves and for following a family of patterns when time-varying elements are included.