Optimization of energy and time predicts dynamic speeds for human walking

Abstract

Humans make a number of choices when they walk, such as how fast and for how long. The preferred steady walking speed seems chosen to minimize energy expenditure per distance traveled. But the speed of actual walking bouts is not only steady, but rather a time-varying trajectory, which can also be modulated by task urgency or an individual’s movement vigor. Here we show that speed trajectories and durations of human walking bouts are explained better by an objective to minimize Energy and Time, meaning the total work or energy to reach destination, plus a cost proportional to bout duration. Applied to a computational model of walking dynamics, this objective predicts dynamic speed vs. time trajectories with inverted U shapes. Model and human experiment (𝑁 = 10) show that shorter bouts are unsteady and dominated by the time and effort of accelerating, and longer ones are steadier and faster and dominated by steady-state time and effort. Individual-dependent vigor may be characterized by the energy one is willing to spend to save a unit of time, which explains why some may walk faster than others, but everyone may have similar-shaped trajectories due to similar walking dynamics. Tradeoffs between energy and time costs can predict transient, steady, and vigor-related aspects of walking.

The value of time in the invigoration of human movements when interacting with a robotic exoskeleton

“…An alternative formulation considers vigor as the outcome of the minimization of a subjective weighting between a cost of time (CoT) and a cost of movement, modulated by the expected reward (20,21), which is convenient to model vigor in reaching tasks (28,29). When reward is not explicit (e.g., pointing to a light spot), movement vigor could then be determined by a common trade-off between time and effort, which could represent a trait-like feature of individuality (30)(31)(32)(33)(34). Empirical evidence of such a subjective CoT was recently reported in an isometric reaching task without explicit reward (35).…”

“…Empirical evidence of such a subjective CoT was recently reported in an isometric reaching task without explicit reward (35). On the basis of this premise, several computational models were developed to account for the vigor of individuals during walking (34) and reaching (20,29,31,35,36), from a similar minimum time-effort (MTE) principle. Estimation of the underlying CoT in reaching was obtained from point-to-point movements of various amplitudes, using effort costs traditionally represented in motor control (29,31), although other factors such as accuracy or comfort may also modulate vigor in general (37)(38)(39)(40)(41).…”

Walking speeds are lower for short distance and turning locomotion: Experiments and modeling in low-cost prosthesis users

“…This average speed includes the acceleration and deceleration periods as characterized by Miff et al [ 82 ], who found that the acceleration and deceleration periods were about 1.5 to 1.7 seconds with a weak dependence on speed. So, the reduction in average speed is partly due to a greater portion of the bout being spent in acceleration-deceleration and partly due to reduced steady walking speed [ 17 , 82 , 91 ]. We did not measure the velocities during the acceleration-deceleration phases, their durations, or the steady walking speed.…”

The value of time in the invigoration of human movements when interacting with a robotic exoskeleton

“…An alternative formulation considers vigor as the outcome of the minimization of a subjective weighting between a cost of time (CoT) and a cost of movement, modulated by the expected reward [20,21], which is convenient to model vigor in reaching tasks [28,29]. When reward is not explicit (e.g., pointing to a light spot), movement vigor could then be determined by a common tradeoff between time and effort, which could represent a trait-like feature of individuality [30][31][32][33][34]. Empirical evidence of such a subjective CoT was recently reported in an isometric reaching task without explicit reward [35].…”

“…Empirical evidence of such a subjective CoT was recently reported in an isometric reaching task without explicit reward [35]. Based on this premise, several computational models were developed to account for the vigor of individuals during walking [34] and reaching [20,29,31,35,36], from a similar minimum time-effort (MTE) principle. Estimation of the underlying CoT in reaching was obtained from point-to-point movements of various amplitudes, using effort costs traditionally represented in motor control [29,31], even though other factors such as accuracy or comfort may also modulate vigor in general [37][38][39][40][41].…”