Dynamics of Human Walking at Steady Speeds

Biped locomotion is discussed through a Lagrangian formulation for velocity-dependent, body driving forces. An analysis of level walking in humans is given through the known experimental data on the ground-reaction force and the external work without recourse to inverted-pendulum modeling. At a certain speed, rectilinear motion of the center of mass with its backward rotation along a shortened hypocycloid is ensured by double-frequency nonlinear oscillations, whose energy cost is 1% of the external work. With increasing speed, a peculiarity and an instability of the trajectory indicate, respectively, a slow-to-normal gait crossover and the maximal fast walking speed. Key words: integrative biology, biped locomotion, human gaits, muscles.

Figure 1

Qualitative (a) and quantitative (b) analyses of human walking on a level ground in the sagittal plane. (a) The dotted lines indicate schematically the two-step trajectory $Y(X)$ of the center of mass in the ground system and the one-step cycles for the libration velocity $\Delta \mathbf{V}$ and the driving force $\Delta \mathbf{F}$, in the inertial system. The coordinate systems are described in the text. (b) The force-platform data on slow walking gait are reproduced from Fig. 1 in Ref. 15 (the human case of speed $3.9\mathrm{k}\mathrm{m}{\mathrm{h}}^{-1}$). Solid lines describe the force-time linear harmonics through Eqs. (2) with the fitting amplitudes $\Delta l_0=0.012\mathrm{m}$ and $\Delta h_0=0.016\mathrm{m}$. The symmetric lines depicted by hatched areas correspond to the predicted force given in Eqs. (4) and fitted with $\Delta l_1=\Delta l_0/2$, $\Delta h_1=\Delta h_0/2$, and the cyclic frequency ${\omega }_0=9.37{\mathrm{s}}^{-1}$. Asymmetric deviations of the recorded $\mathbf{F}$ are associated with (i) the naturally distinct exertions by left and right feet and (ii) in the difference between the time intervals for the single-foot ($T_{\mathrm{s}\mathrm{c}}$) and double-foot ($T_{\mathrm{d}\mathrm{c}}$) ground contacts, with $T_{\mathrm{s}\mathrm{c}}+T_{\mathrm{d}\mathrm{c}}=T_{\mathrm{c}}$.

Figure 2

The eccentricity of the c.m.-trajectory characteristic ellipses in different walking gaits. Points are predictions for the shrunken ($\sqrt{1-W_f^{+(\exp )}/W_v^{+(\exp )}}$) and the flatter ($\sqrt{1-W_v^{+(\exp )}/W_f^{+(\exp )}}$) ellipses estimated from the experimental data [3] ($◯$) and through the analysis ($\square$) given in Fig. 1b. Experimental uncertainty is indicated by error bars. Solid curve results from third-order polynomial fits to the data [3]. Inserts show characteristic closed orbits in the inertial coordinate system (length unit: $0.025\mathrm{m}$) moving with different speeds, indicated by arrows.

Figure 3

External work predicted in Eq. (8) is estimated per unit mass and unit length ($◯$) with the help of $r^{(\exp )}$, $W_f^{+(\exp )}$, and $W_v^{+(\exp )}$ (taken from Fig. 2 in Ref. 3); ${\omega }_0^{(\exp )}=4.94+4.02V$ that fits the human data [12], and ${\omega }_1=6.37-6.15V+2.38V^2$ deduced from Eq. (9). The lines are least-square linear fits. Insert: experimental data [3] on $r^{(\exp )}$ ($●$) with third-order polynomial fitting.