Nonlinear control by coupled oscillator: from nonholonomic systems to quasi-passive dynamic walker

In this paper, we propose a novel control method for nonholonomic constrained systems using coupled oscillators composed through the Kuramoto model. The Kuramoto model is a mathematical model used to describe synchronization phenomenona. We focus on the Kuramoto model because it is able to generate stable rhythmic signals that are modulable for engineering, and because it is sufficiently simple for mathematical analysis. The contributions of this paper are twofold. First, we propose the use of a pair of Kuramoto oscillators as dynamic controllers for a nonholonomic Brockett integrator system; next, we define a feedback control scheme by adjusting its angular velocities based on the system state. We then derive stability criteria for the entire feedback system and examine the effectiveness of the system through several numerical simulations. We also show that the proposed idea is applicable to a two-wheeled vehicle system, which is locally equivalent to the Brockett integrator under proper coordinate and input transformations. Second, we apply the proposed method to control the walking distance of a quadrupedal quasi-passive dynamic walker and examine the effectiveness of the proposed method through walking experiments.