A Multivariate Polynomial Regression to Reconstruct Ground Contact and Flight Times Based on a Sine Wave Model for Vertical Ground Reaction Force and Measured Effective Timings

Abstract

Effective contact ((Formula presented.)) and flight ((Formula presented.)) times, instead of ground contact ((Formula presented.)) and flight ((Formula presented.)) times, are usually collected outside the laboratory using inertial sensors. Unfortunately, (Formula presented.) and (Formula presented.) cannot be related to (Formula presented.) and (Formula presented.) because the exact shape of vertical ground reaction force is unknown. However, using a sine wave approximation for vertical force, (Formula presented.) and (Formula presented.) as well as (Formula presented.) and (Formula presented.) could be related. Indeed, under this approximation, a transcendental equation was obtained and solved numerically over a (Formula presented.) grid. Then, a multivariate polynomial regression was applied to the numerical outcome. In order to reach a root-mean-square error of 0.5 ms, the final model was given by an eighth-order polynomial. As a direct application, this model was applied to experimentally measured (Formula presented.) values. Then, reconstructed (Formula presented.) (using the model) was compared to corresponding experimental ground truth. A systematic bias of 35 ms was depicted, demonstrating that ground truth (Formula presented.) values were larger than reconstructed ones. Nonetheless, error in the reconstruction of (Formula presented.) from (Formula presented.) was coming from the sine wave approximation, while the polynomial regression did not introduce further error. The presented model could be added to algorithms within sports watches to provide robust estimations of (Formula presented.) and (Formula presented.) in real time, which would allow coaches and practitioners to better evaluate running performance and to prevent running-related injuries.