Fig 1.
Comparison of work-minimizing model of locomotion with human walking and running gaits, in terms of body center of mass (COM) trajectories and vertical ground reaction forces vs. time.
The model’s COM trajectories feature a sharp, instantaneous redirection due to ideal impulsive vertical ground reaction forces (asterisks denote infinitely high peaks, shown truncated) in both walking and running. In contrast, human gait has much smoother COM trajectories and more rounded vertical leg forces with finite peaks.
Fig 2.
Optimization model for bipedal locomotion.
(A) The model has a point mass body and massless, telescoping legs driven by axial force actuators. (B) Force magnitudes (f and fc for the two legs) are represented as piecewise linear functions of time within a step (defined by control points, magnified in inset). (C) The state vector includes body positions and velocities (x, y, ẋ, ẏ), which result from the leg forces and equations of motion.
Fig 3.
Vertical ground reaction force trajectories for various formulations of force cost Jf (Equation 3).
Examples show the effect of varying the derivative order n and norm exponent p. Values for α were chosen to qualitatively match human forces where possible. The values for α in the cost function are (left to right), for amplitude force costs: [4.5 ⋅ 10−1, 5.0 ⋅ 10−1, 5.5 ⋅ 10−1], 5.0 ⋅ 10−2, 4.0 ⋅ 10−2, 4.5 ⋅ 10−2, and 2.0 ⋅ 10−2, and for fluctuation force costs: 2.7 ⋅ 10−3, 1.5 ⋅ 10−2, 2.3 ⋅ 10−2, 1.1 ⋅ 10−4, and 1.6 ⋅ 10−3.
Fig 4.
Vertical ground reaction forces for gaits optimized only for force cost.
For amplitude costs (n = 0, left), the optimized forces resemble square waves or an instantaneous spike. For fluctuation costs (n > 0, right), forces smoothly rise from zero to a peak, then return to zero smoothly. All gaits (except a linear amplitude cost) exert leg forces for both legs throughout the stride except for instantaneous swing phases. In general, forces that optimize costs for force amplitude exhibit force discontinuities, while forces optimal for fluctuation costs have no discontinuities.
Fig 5.
Optimized walking and running gaits as a function of weighting α.
Shown are COM trajectories, vertical leg forces vs. time, fore-aft leg forces vs. time, and ground contact periods. Work-minimizing gaits (left side, α = 0) approach impulsive walking (top) and running (bottom), while force fluctuation minimizing gaits (right side, α = 1) have very smooth leg forces. Intermediate combinations of the two costs result in gaits with more human-like attributes, both in leg forces and in COM trajectories (shown in gray for human). Trajectories shown here are for cost parameters n = 2 and p = 2. Values for α for walking are (left to right): 0, 2.68 ⋅ 10−4, 1.61 ⋅ 10−3, 2.68 ⋅ 10−3, and 1. Values for running are: 0, 2.15 ⋅ 10−3, 8.05 ⋅ 10−3, 5.37 ⋅ 10−2, and 1.
Fig 6.
Summary measures of optimized gaits as a function of weighting α: Work cost, force fluctuation (fl.) cost, double support time (for walking) or leg duty factor (or ground contact time, for running) as fraction of stride, and peak vertical force.
As the optimization cost function is changed from work only (α = 0) to force fluctuation only (α = 1), the work cost increases and force fluctuation cost decreases, both monotonically. The resulting double support time and leg duty factor (ground contact time) increase, and peak force decreases. Values for human (shown as dotted lines) are between the extremes found at work-minimal and force fluctuation minimal gaits.
Fig 7.
Force vs. displacement (left) and vertical stiffness (right) for optimized gaits.
The force vs. displacement relationship shows multiple regions of effective stiffnesses, defined as the slope of the force-displacement curve. Optimization yields walking gaits with two stiffnesses, one relatively low during single support, and one relatively high during double support (with stiffnesses typical of human shown with dotted lines). Fitting line segments to these periods yields estimates of effective vertical stiffness (right) for the model, as a function of weighting coefficient α. Model results for intermediate α weights are similar to those of human.
Fig 8.
Optimized leg forces vs leg displacements.
Work optimized gaits approach infinite leg stiffness (slope of a linear fit to forces) during upward velocity redirection (roughly corresponding to double support in walking, contact phase in running). As the optimization cost includes a larger weighting α for force fluctuation, the optimal leg stiffness decreases for both walking and running.