Oscillators are ubiquitous in nature and are usually associated with the existence of an asymptotic phase that governs the long-term dynamics of the oscillator. We show that the asymptotic phase can be estimated using a carefully chosen series expansion that directly computes the phase response curve (PRC) and provides an algorithm for estimating the coefficients of this series. Unlike previously available data-driven phase estimation methods, our algorithm can use observations that are much shorter than a cycle; has proven convergence rate bounds as a function of the properties of measurement noise and system noise; will recover phase within any forward invariant region for which sufficient data are available; recovers the PRCs that govern weak oscillator coupling; and recovers isochron curvature and recovers nonlinear features of isochron geometry. Our method may find application wherever models of oscillator dynamics need to be constructed from measured or simulated time-series.

Mechanical systems naturally evolve on principal bundles describing their inherent symmetries. The ensuing factorization of the configuration manifold into a symmetry group and an internal shape space has provided deep insights into the locomotion of many robotic and biological systems. On the other hand, the property of differential flatness has enabled efficient, effective planning and control algorithms for various robotic systems. Yet, a practical means of finding a flat output for an arbitrary robotic system remains an open question. In this work, we demonstrate surprising new connections between these two domains, for the first time employing symmetry directly to construct a flat output. We provide sufficient conditions for the existence of a trivialization of the bundle in which the group variables themselves are a flat output. We call this a geometric flat output, since it is equivariant (i.e. it preserves the symmetry) and often global or almost global, properties not typically enjoyed by other flat outputs. In such a trivialization, the motion planning problem is easily solved, since a given trajectory for the group variables will fully determine the trajectory for the shape variables that exactly achieves this motion. We provide a partial catalog of robotic systems with geometric flat outputs and worked examples for the planar rocket, planar aerial manipulator, and quadrotor.