A simple model of mechanical effects to estimate metabolic cost of human walking

Abstract

Since the advent of energy measurement devices, gait experiments have shown that energetic economy has a large influence on human walking behavior. However, few cost models have attempted to capture the major energy components under comprehensive walking conditions. Here we present a simple but unified model that uses walking mechanics to estimate metabolic cost at different speeds and step lengths and for six other biomechanically-relevant gait experiments in literature. This includes at various gait postures (e.g. extra foot lift), anthropometric dimensions (e.g. added mass), and reduced gravity conditions, without the need for parameter tuning to design new gait trajectories. Our results suggest that the metabolic cost of walking can largely be explained by the linear combination of four costs-swing and torso dynamics, center of mass velocity redirection, ground clearance, and body weight support. The overall energetic cost is a tradeoff among these separable components, shaped by how they manifest under different walking conditions.

Figure 1

The metabolic cost model and its four components, shown as cost of transport, at different walking speeds and step frequencies with experimental data reported in Bertram for comparison. The overall cost of transport is composed of the swing and torso cost from sagittal and frontal dynamics (3LP dynamics), CoM velocity redirection, ground clearance, and weight support costs. Each component is dominant at different speed-step frequency combinations. CoM redirection is costly at long step lengths, foot lift at slow speeds and high frequencies, and weight support at slow speeds. These components can be combined with constant muscle efficiency (red crosses) or variable efficiency (blue crosses) to yield costs more similar to experimental data (mean represented by black circles, standard deviation by vertical lines).

Figure 2

Comparison of model metabolic rate (red lines) with data from six walking experiments from literature (black circles, solid lines). Model predictions include with variable efficiency (solid blue line) and with constant efficiency (dashed red line). The six comparisons were on (A) step width, (B) added mass to the leg, (C) extra foot lift, (D) simulated reduced gravity, (E) CoM flat-trajectory walking, and (F) walking with obesity. Fitting equations, from the original experiments when possible, were used to investigate trends (see Table S1). Patch layers represent the contribution of each cost component (yellow: ground clearance, orange: 3LP dynamics, green: CoM redirection, purple: weight support).

Figure 3

The four energy components of the metabolic cost model and their formulations. The cost components are from (A) 3LP dynamics, (B) CoM velocity redirection, (C) ground clearance, and (D) weight support. (A) 3LP is composed of three linear pendulums (blue), two represent the legs and one for the trunk. Translational and rotational kinetic energies are calculated from the linear and angular velocities of each segment. The 3LP cost is the integral of the positive component of the kinetic energy change rate. (B) CoM velocity redirection cost accounts for the vertical work to change CoM velocity at the step-to-step transitions, which is not accounted for by the 3LP model. Similar to Kuo, the magnitude of the velocity redirection, and thus kinetic energy, depends on geometry (i.e. the angle α between the legs). This angle comes from 3LP geometry (represented in blue). (C) Ground clearance cost is the potential energy to lift the leg. We used a constant c of 16.5% of leg length for lift height. Since the vertical CoM displacement must be constant, there is a corresponding penalty to move the ‘rest of body’ mass down. (D) Supporting the body during stance requires extensor muscular force to keep the leg from collapsing. The metabolic cost of the vasti muscles performing leg extension is calculated from the Alexander-Minetti curve, following the work of Srinivasan. The leg angle β(t) is derived from 3LP geometry (blue), and we used a constant knee angle θ of 8.4 degrees.