The Cost of Leg Forces in Bipedal Locomotion: A Simple Optimization Study

<div><p>Simple optimization models show that bipedal locomotion may largely be governed by the mechanical work performed by the legs, minimization of which can automatically discover walking and running gaits. Work minimization can reproduce broad aspects of human ground reaction forces, such as a double-peaked profile for walking and a single peak for running, but the predicted peaks are unrealistically high and impulsive compared to the much smoother forces produced by humans. The smoothness might be explained better by a cost for the force rather than work produced by the legs, but it is unclear what features of force might be most relevant. We therefore tested a generalized force cost that can penalize force amplitude or its <i>n</i>-th time derivative, raised to the <i>p</i>-th power (or <i>p</i>-norm), across a variety of combinations for <i>n</i> and <i>p</i>. A simple model shows that this generalized force cost only produces smoother, human-like forces if it penalizes the rate rather than amplitude of force production, and only in combination with a work cost. Such a combined objective reproduces the characteristic profiles of human walking (<i>R</i>² = 0.96) and running (<i>R</i>² = 0.92), more so than minimization of either work or force amplitude alone (<i>R</i>² = −0.79 and <i>R</i>² = 0.22, respectively, for walking). Humans might find it preferable to avoid rapid force production, which may be mechanically and physiologically costly.</p></div

Optimization model for bipedal locomotion.

<p>(A) The model has a point mass body and massless, telescoping legs driven by axial force actuators. (B) Force magnitudes (<i>f</i> and <i>f</i>_<i>c</i> 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 (<i>x</i>, <i>y</i>, <i>ẋ</i>, <i>ẏ</i>), which result from the leg forces and equations of motion.</p

Vertical ground reaction force trajectories for various formulations of force cost <i>J</i>_<i>f</i> (Equation 3).

<p>Examples show the effect of varying the derivative order <i>n</i> and norm exponent <i>p</i>. Values for <i>α</i> were chosen to qualitatively match human forces where possible. The values for <i>α</i> 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.</p

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.

<p>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.</p

Optimized walking and running gaits as a function of weighting <i>α</i>.

<p>Shown are COM trajectories, vertical leg forces vs. time, fore-aft leg forces vs. time, and ground contact periods. Work-minimizing gaits (left side, <i>α</i> = 0) approach impulsive walking (top) and running (bottom), while force fluctuation minimizing gaits (right side, <i>α</i> = 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 <i>n</i> = 2 and <i>p</i> = 2. Values for <i>α</i> 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.</p

Optimized leg forces vs leg displacements.

<p>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 <i>α</i> for force fluctuation, the optimal leg stiffness decreases for both walking and running.</p

Summary measures of optimized gaits as a function of weighting <i>α</i>: 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.

<p>As the optimization cost function is changed from work only (<i>α</i> = 0) to force fluctuation only (<i>α</i> = 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.</p

Force vs. displacement (left) and vertical stiffness (right) for optimized gaits.

<p>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 <i>α</i>. Model results for intermediate <i>α</i> weights are similar to those of human.</p