Finding resonance: Adaptive frequency oscillators for dynamic legged locomotion

There is much to gain from providing walking machines with passive dynamics, e.g. by including compliant elements in the structure. These elements can offer interesting properties such as self-stabilization, energy efficiency and simplified control. However, there is still no general design strategy for such robots and their controllers. In particular, the calibration of control parameters is often complicated because of the highly nonlinear behavior of the interactions between passive components and the environment. In this article, we propose an approach in which the calibration of a key parameter of a walking controller, namely its intrinsic frequency, is done automatically. The approach uses adaptive frequency oscillators to automatically tune the intrinsic frequency of the oscillators to the resonant frequency of a compliant quadruped robot. The tuning goes beyond simple synchronization and the learned frequency stays in the controller when the robot is put to halt. The controller is model free, robust and simple. Results are presented illustrating how the controller can robustly tune itself to the robot, as well as readapt when the mass of the robot is changed. We also provide an analysis of the convergence of the frequency adaptation for a linearized plant, and show how that analysis is useful for determining which type of sensory feedback must be used for stable convergence. This approach is expected to explain some aspects of developmental processes in biological and artificial adaptive systems that "develop" through the embodied system-environment interactions.