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

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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.

AM Spomer, BC Conner, MH Schwartz, ZF Lerner, KM Steele (2023) “Audiovisual biofeedback amplifies plantarflexor adaptation during walking among children with cerebral palsy”

Katie Landwehr JournalArticle, MotorControl, News, Publications , , , , , ,

Journal Article in Journal of NeuroEngineering and Rehabilitation

Biofeedback is a promising noninvasive strategy to enhance gait training among individuals with cerebral palsy (CP). Commonly, biofeedback systems are designed to guide movement correction using audio, visual, or sensorimotor (i.e., tactile or proprioceptive) cues, each of which has demonstrated measurable success in CP.

Figure 1. Experimental Protocol. Audiovisual (AV) biofeedback on soleus activity was provided for the more-affected limb alongside an auto-adjusting target score. Sensorimotor (SM) biofeedback was provided for the more-affected limb using an untethered ankle exoskeleton designed to impart a resistive ankle torque through stance, proportional to baseline values. Participants completed three data collection visits (pre-acclimation, post-acclimation, and follow-up), during which they walked with both biofeedback systems independently and in combination. Trials were pseudo-randomized within and between visits to ensure that feedback modalities were presented to each participant in a different order and control for fatigue and learning effects. Each trial was 10 min long and separated into baseline, feedback, and washout phases. All data analysis was performed for early (strides 1–30), mid (strides 91–110), and late (strides 181–210) feedback phases and washout (strides 1–30). Mean soleus activity for individual strides (purple dots) was normalized to baseline activity. Between the pre-acclimation and post-acclimation visits, participants completed four, 20-min acclimation sessions where they received additional practice with both systems

Aim: The aim of this study is to evaluate how the modality of biofeedback may influence user response which has significant implications if systems are to be consistently adopted into clinical care.

Method: In this study, we evaluated the extent to which adolescents with CP (7M/1F; 14 [12.5,15.5] years) adapted their gait patterns during treadmill walking (6 min/modality) with audiovisual (AV), sensorimotor (SM), and combined AV + SM biofeedback before and after four acclimation sessions (20 min/session) and at a two-week follow-up. Both biofeedback systems were designed to target plantarflexor activity on the more-affected limb, as these muscles are commonly impaired in CP and impact walking function. SM biofeedback was administered using a resistive ankle exoskeleton and AV biofeedback displayed soleus activity from electromyography recordings during gait. At every visit, we measured the time-course response to each biofeedback modality to understand how the rate and magnitude of gait adaptation differed between modalities and following acclimation.

Results: Participants significantly increased soleus activity from baseline using AV + SM (42.8% [15.1, 59.6]), AV (28.5% [19.2, 58.5]), and SM (10.3% [3.2, 15.2]) biofeedback, but the rate of soleus adaptation was faster using AV + SM biofeedback than either modality alone. Further, SM-only biofeedback produced small initial increases in plantarflexor activity, but these responses were transient within and across sessions (p > 0.11). Following multi-session acclimation and at the two-week follow-up, responses to AV and AV + SM biofeedback were maintained.

Interpretation: This study demonstrated that AV biofeedback was critical to increase plantarflexor engagement during walking, but that combining AV and SM modalities further amplified the rate of gait adaptation. Beyond improving our understanding of how individuals may differentially prioritize distinct forms of afferent information, outcomes from this study may inform the design and selection of biofeedback systems for use in clinical care.

ME Hoffman, KM Steele, JE Froehlich, KN Winfree, HA Feldner (2023) “Off to the park: a geospatial investigation of adapted ride-on car usage”

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Journal Article in Disability & Rehabilitation: Assistive Technology:

The accessibility of the built environment is an important factor to consider when providing a mobility device to a young child and their family to use in the community.

Figure 8. The accessibility scores for the sidewalks near each Participant’s (P5, P10, P17) home on the left and the drive path of the participant on the right. Participants generally avoided driving on streets that were not accessible.

Aim: To quantify the driving patterns of children using an adapted ride-on car in their home and community environment over the course of a year using an integrated datalogger.

Method: Fourteen children (2.5 ± 1.45 years old, 8 male: 6 female) used adapted ride-on cars outside and inside of their homes over the course of a year. We tracked their device use metrics with a custom datalogger and geospatial data. To measure environmental accessibility, we used the AccessScore from Project Sidewalk, an open-source accessibility mapping initiative, and the Walk Score, a measure of neighborhood pedestrian-friendliness.

Results: More play sessions took place indoors, within the participants’ homes. However, when the adapted ride-on cars were used outside the home, children engaged in longer play sessions, actively drove for a larger portion of the session, and covered greater distances. Most children tended to drive their ROCs in close proximity to their homes. Most notably, we found that children drove more in pedestrian-friendly neighborhoods and when in proximity to accessible paths.

Interpretation: The accessibility of the built environment is paramount when providing any form of mobility device to a child. Providing an accessible place for a child to move, play, and explore is critical in helping a child and family adopt the mobility device into their daily life.

“Gears of Progress” Podcast Launch!

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Gears of Progress Episode One featuring Elijah Kuska on computational biomechanics, synergies debates, and importance of education accessibility

Lab member, Sasha Portnova, launched a new podcast on research in rehabilitation and assistive technologies. The first episode features Steele Lab Alumni, Elijah Kuska, with a conversation on computational biomechanics, synergies debates, and the importance of education accessibility.

Gears of Progress Logo with three gears featuring assistive devices

Name: Gears of Progress

Platforms: Spotify, Apple Podcasts, Amazon Music, Castbox

Podcast launch date: Dec 1

Release frequency: bi-weekly on Fridays

Theme: Podcast about research and innovations in rehabilitation engineering and assistive technologies aimed to improve accessibility for people with disabilities. Every episode will feature engineers, medical professionals, end-users, and organizations who focus on improving the health and well-being of individuals with disabilities. We will be covering topics such as emerging tech, outcome measures, medical practice, public policy, accessibility education, and so much more!

Twitter: https://twitter.com/GearsOfProgress