Entrainment to Natural Oscillations via Uncoupled Central Pattern Generators

Mechanical systems can often be controlled efficiently by exploiting a resonance. An optimal trajectory minimizing an energy cost function is found at (or near) a natural mode of oscillation. Motivated by this fact, we consider the natural entrainment problem: the design of nonlinear feedback controllers for linear mechanical systems to achieve a prescribed mode of natural oscillation for the closed-loop system. We adopt a set of distributed central pattern generators (CPGs) as the basic control architecture, inspired by biological observations. The method of multivariable harmonic balance (MHB) is employed to characterize the condition, approximately, for the closed-loop system to have a natural oscillation as its trajectory. Necessary and sufficient conditions for satisfaction of the MHB equation are derived in the forms useful for control design. It is shown that the essential design freedom can be captured by two parameters, and the design parameter plane can be partitioned into regions, in each of which approximate entrainment to one of the natural modes, with an error bound, is predicted by the MHB analysis. Control mechanisms underlying natural entrainment, as well as limitations and extensions of our results, are discussed.