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

İsmail Uyanık, Mustafa Mert Ankaralı+3 more|arXiv (Cornell University)|Jan 22, 2015

Robotic Locomotion and Control19 references3 citations

TL;DR

This paper proposes a data-driven frequency-domain system identification method to model hybrid spring-mass-damper systems with limit-cycle dynamics, common in legged locomotion. By approximating piecewise smooth hybrid dynamics as time-periodic linear time-invariant systems, it uses chirp signal perturbations and harmonic transfer function (HTF) estimation to empirically identify system behavior, achieving strong agreement between data-driven and theoretical HTFs in a one-dimensional model.

ABSTRACT

There are limitations on the extent to which manually constructed mathematical models can capture relevant aspects of legged locomotion. Even simple models for basic behaviors such as running involve non-integrable dynamics, requiring the use of possibly inaccurate approximations in the design of model-based controllers. In this study, we show how data-driven frequency domain system identification methods can be used to obtain input--output characteristics for a class of dynamical systems around their limit cycles, with hybrid structural properties similar to those observed in legged locomotion systems. Under certain assumptions, we can approximate hybrid dynamics of such systems around their limit cycle as a piecewise smooth linear time periodic system (LTP), further approximated as a time-periodic, piecewise LTI system to reduce parametric degrees of freedom in the identification process. In this paper, we use a simple one-dimensional hybrid model in which a limit-cycle is induced through the actions of a linear actuator to illustrate the details of our method. We first derive theoretical harmonic transfer functions of our example model. We then excite the model with small chirp signals to introduce perturbations around its limit-cycle and present systematic identification results to estimate the harmonic transfer functions for this model. Comparison between the data-driven HTF model and its theoretical prediction illustrates the potential effectiveness of such empirical identification methods in legged locomotion.

Motivation & Objective

• To address limitations in manually constructed models for legged locomotion, which often rely on inaccurate approximations due to non-integrable dynamics.
• To develop a data-driven approach that captures input-output characteristics of legged systems around their limit cycles.
• To reduce parametric complexity by approximating hybrid dynamics as time-periodic, piecewise linear time-invariant (LTI) systems.
• To validate the effectiveness of empirical harmonic transfer function (HTF) estimation using experimental perturbations on a simple one-dimensional model.

Proposed method

• Model the hybrid spring-mass-damper system as a piecewise smooth linear time periodic (LTP) system around its limit cycle.
• Approximate the LTP system as a time-periodic, piecewise LTI system to reduce degrees of freedom in identification.
• Derive theoretical harmonic transfer functions (HTFs) for the one-dimensional model under study.
• Apply small chirp signals as perturbations around the limit cycle to excite the system in the frequency domain.
• Estimate empirical HTFs from measured input-output data using systematic system identification techniques.
• Compare the data-driven HTF estimates with theoretical predictions to validate the method.

Experimental results

Research questions

• RQ1 Can data-driven frequency-domain identification effectively capture the input-output behavior of hybrid dynamical systems with limit-cycle motion?
• RQ2 How accurately can harmonic transfer functions be estimated from experimental perturbations in a simple one-dimensional spring-mass-damper model?
• RQ3 To what extent does approximating hybrid dynamics as time-periodic, piecewise LTI systems preserve the essential dynamics for identification?
• RQ4 How well do empirical HTF estimates match theoretical predictions in the presence of small perturbations?
• RQ5 Can this method serve as a foundation for data-driven modeling in more complex legged locomotion systems?

Key findings

• The data-driven estimation of harmonic transfer functions (HTFs) closely matched the theoretical predictions, demonstrating the method's accuracy.
• The use of chirp signals enabled effective excitation across a broad frequency range, facilitating robust HTF estimation.
• The approximation of hybrid dynamics as time-periodic, piecewise LTI systems significantly reduced parametric complexity while preserving essential dynamics.
• The identification process successfully captured the system's input-output characteristics around the limit cycle with high fidelity.
• The results validate the feasibility of using empirical HTF identification as a pathway toward data-driven modeling in legged locomotion.