# Energetic Consequences of Walking Like an Inverted Pendulum: Step-to-Step Transitions

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1 ARTICLE Energetic Consequences of Walking Like an Inverted Pendulum: Step-to-Step Transitions Arthur D. Kuo, 1 J. Maxwell Donelan, 2 and Andy Ruina 3 1 Departments of Mechanical Engineering & Biomedical Engineering, University of Michigan, Ann Arbor, MI; 2 School of Kinesiology, Simon Fraser University, Burnaby, BC, Canada; and 3 Department of Theoretical & Applied Mechanics, Cornell University, Ithaca, NY KUO, A.D, J.M. DONELAN, and A. RUINA. Energetic consequences of walking like an inverted pendulum: Step-to-step transitions. Exerc. Sport Sci. Rev., Vol. 33, No. 2, pp , Walking like an inverted pendulum reduces muscle-force and work demands during single support, but it also unavoidably requires mechanical work to redirect the body s center of mass in the transition between steps, when one pendular motion is substituted by the next. Production of this work exacts a proportional metabolic cost that is a major determinant of the overall cost of walking. Key Words: human locomotion, bipedal walking, center of mass, mechanical work, oxygen consumption, metabolic cost INTRODUCTION It costs several times the metabolic energy for a human to walk as it does to bicycle the same distance. Walking is often likened to the motion of two coupled pendula, because the stance leg behaves like an inverted pendulum moving about the stance foot, and the swing leg like a regular pendulum swinging about the hip. This analogy is used to explain conservation of energy during walking. In the absence of a dissipative load, the sustained periodic motion of a pendulum (or any jointed linkage of body segments) requires no net mechanical work. The pendulum analogy is attractive for its simplicity, but it also presents a paradox, because there is no reason why friction or other dissipation within the joints should be greater for walking than for bicycling. Why, then, is walking so much more costly? It is not that humans have little concern for energy. Walking appears to be highly optimized with respect to metabolic cost. Humans prefer to walk at the particular combination of step length, step frequency, and even step width that is energetically optimal (2,5,11). Walking is less costly than other human gaits, even though it still falls far short of wheeled transport. Address for correspondence: Arthur D. Kuo, Depts. of Mechanical Engineering & Biomedical Engineering, University of Michigan, 3200 G G Brown Building, 2350 Hayward St., Ann Arbor, MI ( Accepted for publication: December 29, /3302/88 97 Exercise and Sport Sciences Reviews Copyright 2005 by the American College of Sports Medicine The relatively high cost of walking would surely be better understood if there were a more complete physiological model of the energetics of work and force production by muscle than the one that presently exists. But because the same physiology is shared between walking and bicycling, part of the answer may lie in physics. The mechanics of walking require a transition between pendulum-like phases. In contrast to a wheel, a single inverted pendulum can only transport the body s center of mass (COM) a limited distance. Humans continue movement by transferring from one pendulum-like stance leg to the next. We propose that significant metabolic energy is expended as a consequence of this transition. Positive and negative mechanical work must be performed on the COM to accomplish these step-to-step transitions, and this work exacts a proportional metabolic cost. This cost is an unavoidable consequence of inverted pendulum behavior, and comprises a substantial fraction of the overall cost of walking. The concept of step-to-step transitions in terms of the mechanics of an ideal, simple inverted pendulum is reviewed. We formulate a power law describing the major factors that contribute to transition costs, yielding predicted trends that can be tested independently from the model. We present experimental tests of the mechanical work predicted by stepto-step transitions, as well as tests of a metabolic cost proportional to this work. We then consider additional factors that might contribute to the overall metabolic cost of walking, followed by a refined interpretation of our results in the context of other data from the literature. 88

6 during double support increases approximately with Ẇ l4 when speed increases proportional to step length and step frequency is kept fixed (R ; R 2 indicates degree of variability about each curve fit), and approximately with Ẇ w 2 when only step width is varied (R ). The rate of work over an entire step also increases with similar relationships for Ẇ l4 (R ) and Ẇ w2 (R ), but with different proportionalities. This confirms the observation that step-to-step transition work is not confined to double support alone. And, as discussed, it also might not be confined to pushoff and collision intervals. 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 l 4 (R ) when varying step length, and Ė w2 (R ) 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 W kg 1, for a dimensionless net metabolic cost of transport (energy per weight and distance traveled) of 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 stepto-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 Figure 4. Predictions of step-to-step transitions for changing step length and width. (a) Walking with increasing step length and fixed step frequency theoretically requires pushoff work to increase sharply (6), because the body s center-of-mass (COM) velocity increases in magnitude, and the angle through which it must be redirected also increases. (b) Walking with increasing step width and fixed step length and frequency also requires increasing pushoff (5), but to a lesser degree because the magnitude of collision velocity is dominated by the (fixed) walking speed, and therefore changes little. Step width does, however, contribute to the change in direction of COM velocity, so that the rate of work performed on the COM is predicted to increase approximately with the square of step width. (c) Predicted work rate is actually a function of both step width and step length, represented as a surface. Variations of step width alone and step length alone yield separate curves along the surface equations 6 and 8, with a larger predicted cost for step length (increasing with the fourth power). Step length and step width are shown as a fraction of leg length L. Volume 33 Number 2 April 2005 Energetic Consequences of Inverted Pendulum Walking 93

7 Figure 5. Experimental results comparing measured mechanics (top) and metabolics of walking (bottom) vs increasing step lengths (left) and step widths (right), against predictions for step-to-step transitions. Upper left shows that average rate of mechanical work performed on the body s center of mass (COM), Ẇ, increases approximately with fourth power of step length (accounting for leg motion in an additional squared term (6)), keeping step frequency fixed. Upper right shows that work rate also increases approximately with the square of step width as predicted (Fig. 4), keeping step length and frequency fixed. Both total work measured over a step (filled symbols) and negative work measured over double support (DS, unfilled circles) change with condition, indicating that work related to step-to-step transitions is not limited to double support alone. Bottom row shows that net metabolic rates (subtracting the cost of quiet standing) also increase approximately proportional to mechanical work rates, indicating a proportional cost for step-to-step transitions. All data are plotted in dimensionless units (right and top axes), but scaled to more familiar dimensional units (left and bottom axes, scaled by the mean nondimensionalizing factor) for convenience. The curve fits are derived from the model predictions of equations 6 9. Increasing step lengths were applied as fractions of each subject s preferred step length l* at 1.25 m s 1, and increasing step widths as fractions of each subject s leg length, L. The parameters C and D are coefficients, different for each curve, whose values were derived from curve fit rather than from models. Data are from two separate studies (5,6) with 9 and 10 subjects, respectively, recomputed here using a single consistent procedure. (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 f 4 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 94 Exercise and Sport Sciences Reviews

10 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. Acknowledgments The authors thank R. Kram for his collaboration. This work supported in part by NIH R21 DC and NSF References 1. Alexander, R.M. Simple models of human motion. Appl. Mech. Rev. 48: , Bertram, J.E., and A. Ruina. Multiple walking speed-frequency relations are predicted by constrained optimization. J. Theor. Biol. 209: , Cavagna, G.A. Force platforms as ergometers. J. Appl. Physiol. 39: , Collins, S.H., M. Wisse, and A. Ruina. A three-dimensional passivedynamic walking robot with two legs and knees. Int. J. Robot. Res. 20: , 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: , Donelan, J.M., R. Kram, and A.D. Kuo. Mechanical work for step-tostep transitions is a major determinant of the metabolic cost of human walking. J. Exp. Biol. 205: , Donelan, J.M., R. Kram, and A.D. Kuo. Simultaneous positive and negative external mechanical work in human walking. J. Biomech. 35: , 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: , Garcia, M., A. Chatterjee, and A. Ruina. Efficiency, speed, and scaling of passive dynamic walking. Dyn. Stabil. Syst. 15:75 100, Kuo, A.D. Energetics of actively powered locomotion using the simplest walking model. J. Biomech. Eng. 124: , Kuo, A.D. A simple model of bipedal walking predicts the preferred speed-step length relationship. J. Biomech. Eng. 123: , Kuo, A.D. Stabilization of lateral motion in passive dynamic walking. Int. J. Robot. Res. 18: , McGeer, T. Passive dynamic walking. Int. J. Robot. Res. 9:62 82, Mochon, S., and T.A. McMahon. Ballistic walking. J. Biomech. 13:49 57, Winter, D.A. The Biomechanics and Motor Control of Human Gait: Normal, Elderly and Pathological. 2nd ed. Waterloo, Ontario: Waterloo Biomechanics, 1991, 143. Volume 33 Number 2 April 2005 Energetic Consequences of Inverted Pendulum Walking 97

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