Figure 1.
A schematic of the walking model.
A point mass is restrained by rigid massless legs. The trailing ankle is actuated as a cocked (pre-loaded) spring released at the beginning of double stance. The hip joint and the leading ankle do not exert any torque.
Figure 2.
One step cycle of the walking model.
The end and beginning of a step is the moment when the leading foot collides with ground. During double stance the model moves as four linked bars. During single stance the model moves as an inverted pendulum.
Figure 3.
A free body diagram of the model during double stance.
Points A, B, C, and D denote the toe of the trailing leg, the heel of the leading leg, the point mass, and the trailing ankle respectively. The angle between the horizontal line and the line AC is defined as φ.
Table 1.
Parameter values for the ankle actuated model.
Figure 4.
Asymptotic stability of period-one gait.
Errors in initial conditions of angular velocity converge to zero as the number of steps increases.
Figure 5.
Entrainment to mechanical perturbations with a finite basin.
Stride period is plotted as a function of stride number; (a) shows entrained gaits, and (b) shows gaits that failed to entrain. For entrained gaits, the stride period converged to the perturbation period, τp, whereas stride period continued to fluctuate when gait was not entrained. Note that the model shows a narrow basin of entrainment. Any perturbation with τp>2τ0 or τp≤2τ0−80 (ms) did not entrain the model.
Figure 6.
Phase-locking at terminal stance of normal human walking and the mathematical model.
In (a), the experimental data of normal subjects from [24] are shown. The estimated phase difference between toe-off (initiation of swing) and the initiation of the perturbation pulse is plotted as a function of stride number. In (b), the phase difference between toe-off of the model and the initiation of the perturbation pulse is plotted for entrained gaits with (τp = 2τ0−50 ms) and various initial phases of the perturbation pulse. In both (a) and (b), regardless of the initial phase, the perturbation pulse converged to a phase close to toe-off; the model successfully reproduced the phase-locking at the end of double stance which was observed in the experiment.
Figure 7.
Torque profiles at the trailing ankle during successive cycles.
Time profile of torque to the ankle is plotted per stride when a perturbation with period of 1.8841 (s) (τp = 2τ0−50 ms) is applied. The perturbation pulses, which are superimposed on the intrinsic ankle actuation, drift along the gait cycle, but eventually phase lock at the end of double stance where the intrinsic ankle actuation torque approaches zero. The evolution of stride durations (displayed in red numbers) shows that the stride duration converges to the perturbation period; entrainment is achieved.
Figure 8.
The average speed of the model (step length/step duration), v, vs. the phase of a perturbation pulse, φ for a pulse of constant amplitude and duration.
When a pulse is located in a swing phase, it cannot accelerate the model, and v is lower than when the entire pulse is inside double stance. Average speed, v, increases as φ approaches 0, the onset of double stance, and decreases as φ approaches the end of double stance.