Distributed Network of Coupled Oscillators With Multiple Limit Cycles

When designing feedback controllers to achieve periodic movements, a reference trajectory generator for oscillations is an important component. Using autonomous oscillators to this effect, rather than directly crafting periodic signals, may allow for systematic coordination in a distributed manner and storage of multiple motion patterns within the nonlinear dynamics, with potential extensions to adaptive mode switching through sensory feedback. This article proposes a method for designing a distributed network that possesses multiple stable limit cycles from which various output patterns are generated with prescribed frequency, amplitude, temporal shapes, and phase coordination. In particular, we adopt, as the basic dynamical unit, a simple nonlinear oscillator with a scalar complex state variable, and derive conditions for their distributed interconnections to result in a network that embeds desired periodic solutions with orbital stability. We show that the frequencies and phases of target oscillations are encoded into the network connectivity matrix as its eigenvalues and eigenvectors, respectively. Various design examples will illustrate the proposed method, including generation of human gaits for walking and running.