Uncertainty analysis for stride-time-derived modelling of lower limb stiffness: applying Taylor series expansion for error propagation on Monte-Carlo simulated data.

Abstract

Knowledge of uncertainty is valuable mainly in correctly appraising measured effects. In lower limb stiffness, which affects injury risk and athletic performance, uncertainty is often related to vertical (K_vert) and leg (K_leg) stiffness. Imprecisions in measurements of body mass (M), leg length (L), contact (t_c) and flight (t_f) time propagate through the calculations, augment stiffness uncertainty and inflate relevant effects. This study estimated the limits of this uncertainty as probable (E_prob) and upper bound (E_upper) errors by applying Taylor series expansion on Monte-Carlo simulated data. E_prob and E_upper were 1285 ± 221 N/m (3.9 ± 0.2%) and 1441 ± 248 N/m (4.4 ± 0.3%) in K_vert, and 222 ± 61 N/m (2.1 ± 0.1%) and 375 ± 109 N/m (3.6 ± 0.3%) in K_leg, respectively. To avoid the complexities of full Taylor series expansion, E_prob was predicted (R² ≈ 1) more simply as 0.89E_upper in K_vert and 11 + 0.56E_upper in K_leg. These uncertainties reflect mostly errors in t_c and t_f, and uncertainty in F_max, at kinematic sampling of 300 Hz and running at 4-5 m/s. With slower sampling or faster running these uncertainties rise, and their impact on similar lower limb stiffness effects could be substantial. Applying Taylor series expansion for error propagation on Monte-Carlo simulated data is valid for uncertainty analysis in any multivariable functional relationship.