MC Rosenberg, JL Proctor, KM Steele (2024) “Quantifying changes in individual-specific template-based representations of center-of-mass dynamics during walking with ankle exoskeletons using Hybrid-SINDy”

Journal Article in Scientific Reports

Ankle exoskeletons alter whole-body walking mechanics, energetics, and stability by altering center-of-mass (CoM) motion. Controlling the dynamics governing CoM motion is, therefore, critical for maintaining efficient and stable gait. However, how CoM dynamics change with ankle exoskeletons is unknown, and how to optimally model individual-specific CoM dynamics, especially in individuals with neurological injuries, remains a challenge.

Depictions of walking conditions, phase variables, and example template state variables. (A) Two-dimensional depictions of template model applied to human walking without and with ankle exoskeletons (left). The phase portrait (right) defined a phase variable, , used to cluster kinematically similar measurements for model fitting. Colors denote gait phases corresponding to first and second double-limb support, single-limb support, and swing of the right leg. (B) Stride-averaged global CoM position, velocity, and acceleration for an exemplary unimpaired adult in the anterior–posterior, vertical, and mediolateral directions. The three exoskeleton conditions are shown in panels (B) and (C): shoes-only (solid lines), zero-stiffness exoskeletons (K0; dashed lines), and stiff exoskeletons (KH; dotted lines). (C) Template position and velocity states used to fit the template signatures were defined by sagittal- and frontal-plane leg angles, and leg length.Aim: Evaluate individual-specific changes in CoM dynamics in unimpaired adults and one individual with post-stroke hemiparesis while walking in shoes-only and with zero-stiffness and high-stiffness passive ankle exoskeletons.

Methods: To identify optimal sets of physically interpretable mechanisms describing CoM dynamics, termed template signatures, we leveraged hybrid sparse identification of nonlinear dynamics (Hybrid-SINDy), an equation-free data-driven method for inferring sparse hybrid dynamics from a library of candidate functional forms.

Results: In unimpaired adults, Hybrid-SINDy automatically identified spring-loaded inverted pendulum-like template signatures, which did not change with exoskeletons (p > 0.16), except for small changes in leg resting length (p < 0.001). Conversely, post-stroke paretic-leg rotary stiffness mechanisms increased by 37–50% with zero-stiffness exoskeletons.

Interpretation: While unimpaired CoM dynamics appear robust to passive ankle exoskeletons, how neurological injuries alter exoskeleton impacts on CoM dynamics merits further investigation. Our findings support Hybrid-SINDy’s potential to discover mechanisms describing individual-specific CoM dynamics with assistive devices.

UW Data Science Seminar with Megan Ebers

Title slide from the UW eScience Data Science seminar that says "Mobile sensing with shallow recurrent decoder networks. Megan R. Ebers"

Steele lab member and postdoctoral scholar, Megan Ebers, was featured in the Winter 2024 UW Data Science Seminar series. You can watch her full presentation on “Mobile sensing with shallow recurrent decoder networks” linked HERE on UW eScience Institute’s YouTube channel.

Abstract: Sensing is a fundamental task for the monitoring, forecasting, and control of complex systems. In many applications, a limited number of sensors are available and must move with the dynamics, such as with wearable technology or ocean monitoring buoys. In these dynamic systems, the sensors’ time history encodes a significant amount of information that can be extracted for critical tasks. We show that by leveraging the time-history of a sparse set of sensors, we can encode global information of the measured high-dimensional system using shallow recurrent decoder networks. This paradigm has important applications for technical challenges in climate modeling, natural disaster evaluation, and personalized health monitoring; we focus especially on how this paradigm has the potential to transform the way we monitor and manage movement-related health outcomes.

Bio: Megan Ebers is a postdoctoral scholar in applied mathematics with UW’s NSF AI Institute in Dynamic Systems. In her PhD research, she developed and applied machine learning methods for dynamics systems to understand and enable human mobility. Her postdoctoral research focuses on data-driven and reduced-order methods for complex systems, so as to continue her work in human-centered research challenges, as well as to extend her research to a broader set of technical challenges, including turbulent flow modeling, natural disaster monitoring, and acoustic object detection.

MR Ebers, KM Steele, JN Kutz (2024) “Discrepancy Modeling Framework: Learning missing physics, modeling systematic residuals, and disambiguating between deterministic and random effects”

Journal Article in SIAM Journal on Applied Dynamical Systems

Physics-based and first-principles models pervade the engineering and physical sciences, allowing for the ability to model the dynamics of complex systems with a prescribed accuracy. The approximations used in deriving governing equations often result in discrepancies between the model and sensor-based measurements of the system, revealing the approximate nature of the equations and/or the signal-to-noise ratio of the sensor itself. In modern dynamical systems, such discrepancies between model and measurement can lead to poor quantification, often undermining the ability to produce accurate and precise control algorithms.

Top panel: An approximate dynamical model f(·) provides estimates of system behavior used for both reconstruction and forecasting (shaded region), x(t). However, true behavior x0(t) (without observation noise) deviates from these estimates. The goal of discrepancy modeling is to learn a discrepancy model that recovers the missing physics and augments the approximate dynamics to improve system characterization, ˜x(t). Bottom panel: There are two approaches for building a discrepancy model to estimate missing physics: (i) modeling systematic state-space residual between the approximate state space, x(t), and true state space, x0(t), and (ii) learning the deterministic dynamical error between the true dynamics, x˙ 0(t) = f(x0(t)) + g(x0(t)), and the approximate dynamics, x˙ (t) = f(x(t)). In real-world systems, the true system behavior is noisily observed, yk = x0(tk) + N(μ, σ), model-measurement mismatch contains both deterministic and random effects; measurements yk = y(kΔt) denote a continuous dynamical system’s full state noisily observed at discrete time points.

Aim: Introduce a discrepancy modeling framework to identify the missing physics and resolve the model-measurement mismatch with two distinct approaches: (i) by learning a model for the evolution of systematic state-space residual, and (ii) by discovering a model for the deterministic dynamical error. Regardless of approach, a common suite of data-driven model discovery methods can be used.

Method: Specifically, we use four fundamentally different methods to demonstrate the mathematical implementations of discrepancy modeling: (i) the sparse identification of nonlinear dynamics (SINDy), (ii) dynamic mode decomposition (DMD), (iii) Gaussian process regression (GPR), and (iv) neural networks (NN). The choice of method depends on one’s intent (e.g., mechanistic interpretability) for discrepancy modeling, sensor measurement characteristics (e.g., quantity, quality, resolution), and constraints imposed by practical applications (e.g., state- or dynamical-space operability).

Results: We demonstrate the utility and suitability for both discrepancy modeling approaches using the suite of data-driven modeling methods on three continuous dynamical systems under varying signal-to-noise ratios. Finally, we emphasize structural shortcomings of each discrepancy modeling approach depending on error type.

Interpretation: In summary, if the true dynamics are unknown (i.e., an imperfect model), one should learn a discrepancy model of the missing physics in the dynamical space. Yet, if the true dynamics are known yet model-measurement mismatch still exists, one should learn a discrepancy model in the state space.

MR Ebers, MC Rosenberg, JN Kutz, KM Steele (2023) “A machine learning approach to quantify individual gait responses to ankle exoskeletons”

Journal Article in Journal of Biomechanics:

Physiological and biomechanical responses to mechanical assistance from wearable technology are highly variable, especially for clinical populations; tools to predict how users respond to different types of exoskeleton assistance may optimize the prescription process and uncover underlying mechanisms driving locomotor changes in the context of personalized wearable/assistive technology.

Aim: The purpose of this study was to determine if a discrepancy modeling framework could quantify individual-specific gait responses to ankle exoskeletons.

Method: We employ a machine learning technique — neural network based discrepancy modeling — on gait data from 12 non-disabled adults to capture within-participant differences in walking dynamics without vs. with a bilateral passive elastic ankle exoskeletons applying 5 N-m/deg of torque. We fit three models: Nominal gait (no exo), Exo, and Discrepancy. Then, post-fitting, we extend the Nominal by the Discrepancy Model (Augmented). We hypothesize that if Augmented (Nom+Discrep) can capture similar amount of variability as the Exo model, then it can be inferred that the discrepancy model accurately captures how a user will respond to an exoskeleton — without direct information about that user’s physiology or motor coordination.

Results:While joint kinematics during Exo gait were well predicted using the Nominal model (median 𝑅2 = 0.863 − 0.939), the Augmented model significantly increased variance accounted for (𝑝 < 0.042, median 𝑅2 = 0.928 − 0.963). For EMG, the Augmented model (median 𝑅2 = 0.665 −
0.788) accounted for significantly more variance than the Nominal model (median 𝑅2 = 0.516 − 0.664). Minimal kinematic variance was left unexplained by the Exo model (median 𝑅2 = 0.954 − 0.978), but only accounted for 72.4%–81.5% of the median variance in EMG during Exo gait across all individuals.

Interpretation:Discrepancy modeling successfully quantified individuals’ exoskeleton responses without requiring knowledge about physiological structure or motor control. However, additional measurement modalities and/or improved resolution are needed to characterize Exo gait, as the discrepancy may not comprehensively capture response due to unexplained variance in Exo gait.

MH Schwartz, KM Steele, AJ Ries, AG Georgiadis, BA MacWilliams (2022) “A model for understanding the causes and consequences of walking impairments”

Journal Article in PLOS ONE:

Causal inference is inherently ambiguous since we cannot observe multiple realizations of the same person with different characteristics. Causal models must be evaluated through indirect means and reasoning.

Aim: The main objectives in conducting this study were to (1) propose a comprehensive model for quantifying the causes and consequences of walking impairments and (2) demonstrate the potential utility of the model for supporting clinical care and addressing basic scientific questions related to walking.

Method: This paper introduced a model consisting of 10 nodes and 23 primary causal paths and demonstrated the model’s utility using a large sample of gait data.

Results: The model was plausible, captured some well-known cause-effect relationships, provided new insights into others, and generated novel hypotheses requiring further testing through simulation or experiment.

Interpretation: This model is a proposal that is meant to be critically evaluated, validated or refuted, altered, and improved over time. Such improvements might include the introduction of new nodes, variables, and paths.