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Richard GonzalezRichard Gonzalez

Center Director, Research Center for Group Dynamics, Institute for Social Research
Director, BioSocial Methods Collaborative, RCGD
Amos N Tversky Collegiate Professor, Psychology and Statistics, LSA
Professor of Marketing, Stephen M Ross School of Business
Professor of Integrative Systems and Design, College of Engineering


E-mail: Email Richard Gonzalez
Address: Research Center for Group Dynamics
Institute for Social Research
University of Michigan
426 Thompson Street
Ann Arbor, Michigan 48106
Phone: 734-647-6785

Developing a new approach to modeling the ground reaction forces in elite human runners

Oct 6, 2023 | Announcements, Psychology, Statistics/Methods

Burns, G., Gonzalez, R., & Zernicke, R. (2021). Improving spring-mass parameter estimation in running with nonlinear regression methods. Journal of Experimental Biology, 224 . doi:10.1242/jeb.232850 PDF


Runners are commonly modeled as spring–mass systems, but the traditional calculations of these models rely on discrete observations during the gait cycle (e.g. maximal vertical force) and simplifying assumptions (e.g. leg length), challenging the predictive capacity and generalizability of observations. We present a method to model runners as spring–mass systems using nonlinear regression (NLR) and the full vertical ground reaction force (vGRF) time series without additional inputs and fewer traditional parameter assumptions. We derived and validated a time-dependent vGRF function characterized by four spring–mass parameters – stiffness, touchdown angle, leg length and contact time – using a sinusoidal approximation. Next, we compared the NLR-estimated spring–mass parameters with traditional calculations in runners. The mixed-effect NLR method (ME NLR) modeled the observed vGRF best (RMSE:155 N) compared with a conventional sinusoid approximation (RMSE: 230 N). Against the conventional methods, its estimations provided similar stiffness approximations (−0.2±0.6 kN m−1 ) with moderately steeper angles (1.2±0.7 deg), longer legs (+4.2±2.3 cm) and shorter effective contact times (−12±4 ms). Together, these vGRF-driven system parameters more closely approximated the observed vertical impulses (observed: 214.8 N s; ME NLR: 209.0 N s; traditional: 223.6 N s). Finally, we generated spring–mass simulations from traditional and ME NLR parameter estimates to assess the predictive capacity of each method to model stable running systems. In 6/7 subjects, ME NLR parameters generated models that ran with equal or greater stability than traditional estimates. ME NLR modeling of the vGRF in running is a useful tool for assessing runners holistically as spring–mass systems with fewer measurement sources or anthropometric assumptions. Furthermore, its utility as a statistical framework lends itself to more complex mixed-effects modeling to explore research questions in running.