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 (tce) and flight (tfe) times, instead of ground contact (tc) and flight (tf) times, are usually collected outside the laboratory using inertial sensors. Unfortunately, tce and tfe cannot be related to tc and tf because the exact shape of vertical ground reaction force is unknown. However, using a sine wave approximation for vertical force, tce and tc as well as tfe and tf could be related. Indeed, under this approximation, a transcendental equation was obtained and solved numerically over a tce x tfe 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 tce values. Then, reconstructed tc (using the model) was compared to corresponding experimental ground truth. A systematic bias of 35 ms was depicted, demonstrating that ground truth tc values were larger than reconstructed ones. Nonetheless, error in the reconstruction of tc from tce 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 tc and tf in real time, which would allow coaches and practitioners to better evaluate running performance and to prevent running-related injuries.