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There are limitations on the extent to which manually constructedmathematical models can capture relevant aspects of legged locomotion. Evensimple models for basic behaviors such as running involve non-integrabledynamics, requiring the use of possibly inaccurate approximations in the designof model-based controllers. In this study, we show how data-driven frequencydomain system identification methods can be used to obtain input--outputcharacteristics for a class of dynamical systems around their limit cycles,with hybrid structural properties similar to those observed in leggedlocomotion systems. Under certain assumptions, we can approximate hybriddynamics of such systems around their limit cycle as a piecewise smooth lineartime periodic system (LTP), further approximated as a time-periodic, piecewiseLTI system to reduce parametric degrees of freedom in the identificationprocess. In this paper, we use a simple one-dimensional hybrid model in which alimit-cycle is induced through the actions of a linear actuator to illustratethe details of our method. We first derive theoretical harmonic transferfunctions of our example model. We then excite the model with small chirpsignals to introduce perturbations around its limit-cycle and presentsystematic identification results to estimate the harmonic transfer functionsfor this model. Comparison between the data-driven HTF model and itstheoretical prediction illustrates the potential effectiveness of suchempirical identification methods in legged locomotion.

Identification of a Hybrid Spring Mass Damper via Harmonic Transfer Functions as a Step Towards Data-Driven Models for Legged Locomotion