Tracking a Diffusing Three-Dimensional Source via Nonholonomic Extremum Seeking

摘要

One method for autonomously exploring unknown environments is to treat the environment as a scalar potential field. In this scenario, one or more robots acquire measurements of the field (e.g., radiation, temperature, or carbon dioxide concentrations) and use those measurements to guide their path as they explore. An important behavior robots may need is that of extremum seeking-that is, the ability to autonomously drive toward a region containing a maximum or a minimum of the field and reliably stay with this extremum even if its location is moving. In this paper, we describe an extremum seeking control law for three-dimensional scalar potential fields built upon a previous two-dimensional law. We derive an equilibrium trajectory, prove its local stability for the case of a radial scalar field, and characterize the stability as a function of parameters by numerically evaluating its Floquet multipliers. Additionally, we consider the case where the field is constant except for a small range around the source and allow the source to be diffusing with an unknown diffusion coefficient. We derive an approximate expected first passage time and use numerical simulations to show how to optimally select parameters so that the expected tracking time is maximized.