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

Effective contact (t_ce) and flight (t_fe) time instead of ground contact (t_c) and flight (t_f) times are usually collected outside the laboratory using inertial sensors. Unfortunately, t_ce and t_fe cannot be related to t_c and t_f because the exact shape of vertical ground reaction force is unknown. However, using a sine-wave approximation for vertical force, t_ce and t_c as well as t_fe and t_f could be related. Indeed, under this approximation, a transcendental equation was obtained and solved numerically over a t_ce x t_fe grid. Then, a multivariate polynomial regression was applied to the numerical outcome. In order to reach a root mean square error of 0.5ms, the final model was given by an 8th order polynomial. As a direct application, this model was applied to experimentally measured t_ce values. Then, reconstructed t_c (using the model) were compared to corresponding experimental ground truth. A systematic bias of 35ms was depicted, demonstrating that ground truth t_c values were larger than reconstructed ones. Nonetheless, error in the reconstruction of t_c from t_ce 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 t_c and t_f in real time, which would allow coaches and practitioners to better evaluate running performance and to prevent running related injuries.