Each leg’s COM work appears to be performed at a metabolic cost. For variations of both step length and step width, the measured metabolic rate is approximately proportional to the rate of COM work (Fig. 5). We report here the net metabolic rate, measured from total oxygen consumption rate during walking, subtracting the rate for quiet standing. As with work, net metabolic cost increases approximately with Ė ∝ l4 (R2 = 0.95) when varying step length, and Ė ∝ w2 (R2 = 0.91) for step width. Over the step lengths shown, metabolic cost ranges from 78 to 276% of the nominal rate at 1.25 m·s−1. With changing step width, metabolic cost increases up to 43%, ranging from 2.4 to 3.4 W·kg−1. For reference, the nominal net metabolic rate at 1.25 m·s−1 is 2.3 ± 0.3 W·kg−1, for a dimensionless net metabolic cost of transport (energy per weight and distance traveled) of 0.19 ± 0.02.
These combined results are interlinked. The mechanical work curves appear individually (Fig. 5), but should be considered to be alternate views of a single surface (Fig. 4c). It may not be surprising for any single measure of gait to change when an independent variable is manipulated. However, the results shown here vary in different ways with step length and step width and, more importantly, with trends predicted from a single model of step-to-step transitions. The metabolic rates vary in a similar manner, approximately proportional to the mechanical work rates.
We use these results to estimate the contribution of step-to-step transitions to the net rate of metabolic energy expenditure during walking at 1.25 m·s−1. We assume that double support is a suitable interval for evaluating step-to-step transitions, and that all of the positive work is performed actively with an efficiency of 25% (6). The negative work could be performed passively at no cost, or actively at an efficiency of −120%. Adding these contributions yields a crude estimate for the cost of step-to-step transitions, of approximately 60 to 70% of the net metabolic cost of walking at 1.25 m·s−1.
It may also cost energy to move the legs back and forth relative to the body. We previously hypothesized that such a cost might act as a tradeoff against step-to-step transitions (10,11). The high cost of step-to-step transitions alone would favor walking at very short steps and high step frequencies. Periodic actuation of the opposing hips for this purpose could thereby reduce step-to-step transition costs. Walking could then resemble the rolling of a wheel (13), except for the difficulty of moving the legs quickly. In humans, the metabolically optimum step length does not occur at very short steps, indicating a cost that increases sharply with step frequency (11) and that trades off against step-to-step transitions. We hypothesized that forcing the legs to move quickly would exact a substantial metabolic cost increasing sharply with frequency (specifically, a rate proportional to f4 × l). The activity of the hips can be interpreted as having two components. One is analogous to springlike actuation caused by a combination of muscle and tendon, most likely exacting a metabolic cost for moving the legs relative to the body. The second is nonspringlike, net positive work performed by the hips over a stride, which may ultimately contribute to pushoff, and would therefore be included in step-to-step transition costs.
There are surely other metabolic costs for walking not considered here. Lateral leg motion and step variability associated with stabilizing balance might also exact a metabolic cost (12). They appear to act in concert with the step width dependency (Fig. 5) to explain why the preferred step width is energetically optimal (6). Other possible costs might be for supporting body weight, balancing the trunk, or actively moving the arms during walking. These contributions are by no means exhaustive, nor will they necessarily sum linearly. However, they also appear not to change significantly under the experimental conditions considered here.
Refined Conceptual Model
The measures used thus far show when mechanical work is performed on the COM, but not which joints or muscles perform this work, nor when or whether this work is performed actively by muscle fibers. In human walking, COM redirection occurs over a period longer than double support and with nonzero displacement, and the single-support phase never behaves exactly as an inverted pendulum. Insight into these behaviors, as well as the possible muscular sources of work, can be gained by comparing our results against joint-power data published by Winter (15).
Here, we examine joint power (resultant joint moment multiplied by joint angular velocity) for normal walking, in the context of the four intervals of the stance phase (Figs. 3a and 6), demarcated by sign changes in the instantaneous rate of individual-limb COM work (6). We propose interpretations of a few aspects of joint power that can be related directly to our simple model, cautioning that these interpretations are necessarily descriptive rather than prescriptive. For convenience, we take “muscle” and “tendon” to mean the active contractile component of muscle and series elastic components, respectively. We also define “midstance” to mean the instance between rebound and preload, when the stance leg is approximately vertical and the knee is near its maximum extension.
The collision phase refers to the interval after heel strike when substantial negative COM work is performed by the leading leg. Negative work is actively performed first at the ankle joint, and then at the knee, but in summed amount insufficient to account for work performed on the COM. Much of the negative COM work is therefore likely not attributable to joints or muscles of the leading leg.
Some of the negative work might be performed elsewhere, most obviously by the shoe, heel pad, and plantar ligament, but also in the damped motion of fat, viscera, and muscle. Nonrigid or wobbling mass accounts for the majority of body mass. It plays a major role in passively dissipating energy during running and jumping, one that might also apply to walking. Unfortunately, this dissipation is difficult to quantify theoretically or empirically. Rigid body inverse dynamics methods can only assign dissipation to modeled degrees of freedom, and the addition of more degrees of freedom, especially for soft tissues, demands parameters and data that are exceedingly difficult to determine or verify. Here, the nonparametric nature of COM work measurements comes to advantage, because they require only knowledge of external forces, and the integrated forces can capture soft tissue contributions to COM motion. Measured COM work does not quantify total energy change, but it may quantify the negative work associated with collisions better than the present alternatives.
Rebound is characterized by positive COM work, as the stance leg extends before midstance. Some of this work can be attributed to extension of the stance knee. Quadriceps muscles are active during this interval, indicating extensor force and quite possibly work. But the loading conditions and timing also admit the possibility of some elastic rebound at the knee, to an unknown degree. Rebound of the knee, whether elastic or not, can be considered a direct consequence of the collision, because the amount of extension will largely be dependent on the amount of flexion occurring during collision. This may explain why the rebound work rate increases with the collision work rate (Fig. 3).
There may be tradeoffs in the amount of rebound extension desirable. A fully extended knee minimizes the force needed to support body weight at midstance, but reaching that state likely requires work. This tradeoff means that metabolic cost is likely minimized with less than full extension at midstance.
Not all of the COM work observed during rebound occurs at the knee. Some work may instead be attributable to the hips (6). They perform net positive work that moves the swing leg with respect to the body and accelerates the inverted pendulum. The term “rebound” therefore roughly refers to a time interval during which the knee extends after being flexed, where the amount of COM work performed during this interval is not all performed at the knee, and not necessarily elastically.
After midstance, preload is characterized by negative COM work, which can largely be attributed to the ankle joint. Substantial work is likely performed on the Achilles tendon (8), such that the muscle fibers may actually be isometric, or even perform positive work. Elastic energy storage provides three potential advantages. First, it may allow the work for pushoff to be performed over a long duration including both rebound and preload, rather than during pushoff alone. Second, it allows pushoff energy to be derived not only from ankle muscles but also from the inverted pendulum motion. Temporal and spatial distribution of pushoff work might allow muscle to perform at optimum efficiency, avoiding the need to produce high forces for short durations and low efficiency. Finally, slowing of the inverted pendulum not only stores energy, but also reduces the COM velocity so that less energy is lost at collision. Preloading may ultimately allow the net positive work generated by the hips over a stride to contribute substantially to pushoff. These various mechanisms may explain why the preload work rate increases with the pushoff work rate (Fig. 3).
Positive COM work during pushoff is almost entirely attributable to the ankle joint. The knee and hip joints perform little net work over this interval, whereas the stance-leg ankle produces the largest single burst of positive work in the entire stride. As stated, some or even all of this positive work may result from elastic energy stored in tendon. But even if the tendon performs most of pushoff, there are several reasons why muscles might actively perform work to store that energy elastically. First, the energy lost at collision cannot be regenerated by muscle, and only a fraction is likely stored and returned elastically, meaning that active work must restore that energy. Second, the proportionality between step-to-step transition work and metabolic energy also indicates that much of pushoff is actively powered. Elastic energy storage can obscure the timing and even the source, but it is nevertheless likely that some muscles, not necessarily the ankle extensors alone, perform the work for pushoff, and at substantial metabolic cost.
We briefly consider the swing phase, which is dominated by pendulum dynamics (14), but with significant muscle activity at both the beginning and end of swing. Some of the active hip torque may be springlike in the sense of speeding the pendulum motion (11) without performing much net work over a stride. Elastic tendon may contribute to this motion, reducing the muscle work (but not force) needed to speed and slow the limb. But whether work or force dominates, actively moving the legs back and forth must cost metabolic energy. Examination of hip power also reveals that the hip performs net positive work over a stride, possibly contributing to COM motion, and ultimately to pushoff through the energy storage described.
Reexamining these interpretations, it is apparent that work performed at a joint does not necessarily indicate which muscles perform work, or when. Pushoff might even be partially powered by the positive work performed by hip flexor muscles during collision and rebound, directly accelerating the inverted pendulum. The COM energy might then be stored in Achilles tendon as the ankle extensors activate during preload phase, culminating with release of that energy at pushoff. Performing positive work by multiple muscles, and for relatively long durations, might reduce the demands for peak muscle force and power, perhaps allowing the muscles to operate at higher efficiency and to avoid fatigue.
Given these refinements, the reality of human walking might differ in many details from inverted pendulum arcs. Almost the entire gait cycle is spent in some combination of either redirecting the COM velocity, or recovering from or preparing for it. This may reflect competing mechanical and metabolic demands. Step-to-step transitions are mechanically the least costly if pushoff is performed impulsively at high forces and short durations, whereas muscle is most metabolically efficient at moderate forces exerted over moderate durations. Such tradeoffs imply that the simple pendulum is not quite a biological ideal. Energy is expended, not in the vain attempt to emulate the ideal, but rather to reconcile the unavoidable step-to-step transitions associated with a pendulum against presumably higher force and work requirements for nonpendular motion.
The simple pendulum model predicts energy expenditure not for pendulum motion itself, but rather for the transition between steps. Work is required to redirect the COM between pendular arcs, with positive work performed by the trailing leg just before or simultaneous with negative work by the leading leg. Experimental results indicate that most of this work occurs during double support, but with pushoff beginning before this interval and collision continuing beyond it, all with proportional metabolic cost. COM work is also performed before and after midstance, some of it a consequence of collision and pushoff. Both the rates of this work and of metabolic energy expenditure increase approximately with the fourth power of step length and the second power of step width. Step-to-step transitions appear to be significant not only in their direct contributions to the energetics of walking, but also in explaining why humans prefer certain gait parameters for step length, width, and frequency.
Despite its simplistic nature, this model provides useful insight into human walking. Step-to-step transitions explain why mechanical energy must be dissipated in the periodic motion of the limbs, and this dissipation requires that positive work must then be performed to restore the energy lost. This hardly constitutes a full explanation of the metabolic cost of walking, but it offers quantitative predictions supported by simple accessible models and experimental tests made through relatively simple measurements. These measurements suggest a substantial metabolic cost associated with step-to-step transitions as a major consequence of walking like an inverted pendulum.
The authors thank R. Kram for his collaboration. This work supported in part by NIH R21 DC006466 and NSF 0320308.
1. Alexander, R.M. Simple models of human motion. Appl. Mech. Rev
. 48:461–469, 1995.
2. Bertram, J.E., and A. Ruina. Multiple walking speed-frequency relations are predicted by constrained optimization. J. Theor. Biol
. 209:445–453, 2001.
3. Cavagna, G.A. Force platforms as ergometers. J. Appl. Physiol
. 39:174–179, 1975.
4. Collins, S.H., M. Wisse, and A. Ruina. A three-dimensional passive-dynamic walking robot with two legs and knees. Int. J. Robot. Res
. 20:607–615, 2001.
5. Donelan, J.M., R. Kram, and A.D. Kuo. Mechanical and metabolic determinants of the preferred step width in human walking. Proc. R. Soc. Lond. B
. 268:1985–1992, 2001.
6. Donelan, J.M., R. Kram, and A.D. Kuo. Mechanical work
for step-to-step transitions is a major determinant of the metabolic cost
of human walking. J. Exp. Biol
. 205:3717–3727, 2002.
7. Donelan, J.M., R. Kram, and A.D. Kuo. Simultaneous positive and negative external mechanical work
in human walking. J. Biomech
. 35:117–124, 2002.
8. Fukunaga, T., K. Kubo, Y. Kawakami, S. Fukashiro, H. Kanehisa, and C.N. Maganaris. In vivo
behaviour of human muscle tendon during walking. Proc. R. Soc. Lond. B
. 268:229–233, 2001.
9. Garcia, M., A. Chatterjee, and A. Ruina. Efficiency, speed, and scaling of passive dynamic walking. Dyn. Stabil. Syst
. 15:75–100, 2000.
10. Kuo, A.D. Energetics of actively powered locomotion using the simplest walking model. J. Biomech. Eng
. 124:113–120, 2002.
11. Kuo, A.D. A simple model of bipedal walking
predicts the preferred speed-step length relationship. J. Biomech. Eng
. 123:264–269, 2001.
12. Kuo, A.D. Stabilization of lateral motion in passive dynamic walking. Int. J. Robot. Res
. 18:917–930, 1999.
13. McGeer, T. Passive dynamic walking. Int. J. Robot. Res
. 9:62–82, 1990.
14. Mochon, S., and T.A. McMahon. Ballistic walking. J. Biomech
. 13:49–57, 1980.
15. Winter, D.A. The Biomechanics and Motor Control of Human Gait: Normal, Elderly and Pathological
. 2nd ed. Waterloo, Ontario: Waterloo Biomechanics, 1991, 143.
Keywords:©2005 The American College of Sports Medicine
human locomotion; bipedal walking; center of mass; mechanical work; oxygen consumption; metabolic cost