## ABSTRACT

During each step of human walking, the swing foot passes close to the ground with a small but (usually) non-zero clearance. The foot can occasionally scuff against the ground, with some risk of stumbling or tripping. The risk might be mitigated simply by lifting the foot higher, but presumably at increased effort, of unknown amount. Perhaps the normally preferred ground clearance is a trade-off between competing costs, one for lifting the foot higher and one for scuffing it. We tested this by measuring the metabolic energy cost of lifting and scuffing the foot, treating these apparently dissimilar behaviors as part of a single continuum, where scuffing is a form of negative foot lift. We measured young, healthy adults (*N*=9) lifting or scuffing the foot by various amounts mid-swing during treadmill walking, and observed substantial costs, each well capable of doubling the net metabolic rate for normal walking (gross cost minus that for standing). In relative terms, the cost for scuffing increased over twice as steeply as that for lifting. That relative difference means that the expected value of cost, which takes into account movement variability, occurs at a non-zero mean clearance, approximately matching the preferred clearance we observed. Energy cost alone is only a lower bound on the overall disadvantages of inadvertent ground contact, but it is sufficient to show how human behavior may be determined not only by the separate costs of different trade-offs but also by movement variability, which can influence the average cost actually experienced in practice.

## INTRODUCTION

The foot momentarily passes close to the ground about mid-way through each swing phase of walking. It does so with a peak speed over a stride, about three times walking speed (Winter, 1992), thus presenting a risk of unexpected ground contact and therefore susceptibility to tripping or stumbling, which are leading causes of falls among older adults (Barrett et al., 2010; Begg and Sparrow, 2006). Inadequate foot–ground clearance may also be an issue for persons with clinical conditions such as drop-foot or weak hamstrings (Cruz and Dhaher, 2009), leading to compensations such as hip hiking and swing leg circumduction (Cruz and Dhaher, 2009; Kerrigan et al., 2000), which also have negative consequences. Even healthy individuals avoid unwanted ground contact, for example on uneven terrain, by lifting the foot higher mid-swing (Gates et al., 2012). Greater ground clearance may, however, also come with a cost, such as greater energy expenditure (Voloshina et al., 2013). There may thus be two competing trade-offs: a cost for making inadvertent ground contact (‘scuffing’) and one for lifting the foot mid-swing (‘lifting’). Together, these could explain the (non-zero) clearance humans normally prefer. We therefore sought to test the trade-offs between lifting the foot higher and allowing it to scuff against ground.

There are several reasons why lifting the foot could be costly (Fig. 1). For an otherwise normal gait, lifting the foot higher entails greater potential energy mid-swing, as well as a longer travel path and thus greater kinetic energy, both entailing more muscular effort. The effort of swinging the leg could potentially be reduced with the help of elastic tendons (Kuo, 2002; Doke and Kuo, 2007; Dean and Kuo, 2009), but simple walking models indicate that the elastic energy for improved ground clearance would nevertheless require more muscular effort (Dean and Kuo, 2011). In fact, even if the model is driven largely by passive dynamics and elastic tendons, it would attain greatest economy if the swing foot could pass through the ground without stumbling (Dean and Kuo, 2009, fig. 7). Of course, any model can only suggest how humans might hypothetically behave. It is therefore important to test whether lifting the foot higher actually requires more mechanical work performed by the body, and whether that also increases the metabolic cost.

There are also likely costs to scuffing the ground mid-swing. These go beyond energy expenditure to include less quantifiable costs such as the consequences of stumbling – for example, recovery actions needed to maintain balance – and consequences of falling, such as pain and injury. Stumbling and lifting the foot therefore have categorically different costs, and categorically different motions, making it difficult to compare their respective trade-offs. We therefore propose the simplification of treating the motions along a single continuum of vertical ground clearance, where positive values signify lifting the foot higher and negative values signify striking the ground harder (as if the foot could drop through the floor unimpeded). As for the costs, a typical optimization approach is to consider multiple contributions (e.g. Winters and Helm, 2000), each weighted and summed to yield a single objective function with arbitrary units. A second simplification is that, rather than contend with the many costs of scuffing, we measure only the metabolic cost of repeated scuffing, and treat it as a lower bound on the overall cost of unwanted ground contact. We therefore do not include stumbling and falling, which would only add to the incentive for greater ground clearance.

Another factor in the foot's motion is the notion of risk or variability (see probability distribution in Fig. 1). If the foot were controlled with perfect precision, a ground clearance of zero might be optimal, because it would entail no excess effort to lift the foot, while also avoiding scuffing. However, the actual foot motion is variable from step to step, perhaps due to imperfect motor control, and behaves according to a slightly skewed normal distribution (e.g. mean about 1.56 cm, standard deviation about 0.25 cm; Begg et al., 2007). On some terrains, non-smoothness of the ground surface will also present variability and occasionally cause insufficient ground clearance, and even contribute to a risk of falling (Best and Begg, 2008). This can be accounted for with the expected value function (Papoulis and Pillai, 2002), which considers both the probability distribution of the foot's motion and the costs for various mean clearances, to yield an overall probabilistic cost for ground clearance.

Here, we investigated the energetic cost and biomechanical consequences of foot-to-ground clearance during swing. We propose that the preferred foot-to-ground clearance is governed by interactions between the two hypothesized cost trade-offs of lifting and scuffing the foot, mediated by some probability distribution for the step-by-step variability of foot motion. As much of gait appears to be energetically optimal (e.g. Donelan et al., 2001; Doke et al., 2005; Elftman, 1966; Ralston, 1958), the preferred clearance height may closely match with the lowest metabolic cost. We expected that lifting the foot higher should require more muscular effort and come at greater metabolic cost. We also expected that negative foot lift, meaning greater amounts of scuffing, should require more effort and lead to greater metabolic cost as well. These costs, along with the probability distribution of foot motion, may explain the preferred ground clearance during normal walking.

## MATERIALS AND METHODS

### Ground clearance cost model

We propose two simple models for the cost of lifting the foot and of scuffing it against ground (Fig. 1). The cost of lifting the foot *C*_{LH}(*z*) is modeled as increasing with the maximum height *z* above the ground, and the cost of scuffing *C*_{SI}(*z*) is modeled as increasing with the opposite, a virtual negative lift where the foot would pass below ground if unimpeded. Of course, with the ground as an obstacle, the actual result is a normal ground reaction force (GRF), and hence scuffing, modeled as a proportional drag force. Because foot motion is imperfect and exhibits variability, an additional element is added to represent the probability distribution *p _{z}*(

*w*) for deviations

*w*about a mean ground clearance

*z*. The total, expected cost for a mean lift height is the probability-weighted sum of the two individual costs, (1)

where represents the average metabolic energy expenditure rate, evaluated as the expected value of total cost.

A higher energetic cost is expected for either alternative, lifting the foot higher or scuffing the foot harder as a function of vertical foot lift (Eqn 1). The cost of increasingly positive lift *z* of the swing foot could minimally be due to the work needed to lift the swing leg, but also include costs for the coordinative responses throughout the body to produce that motion while maintaining balance and forward gait. Regardless of the mechanism, the effort of lifting the foot might be such that least effort would be achieved with zero, or even negative, foot lift, if not for the obstacle posed by the ground. However, in reality, negative lift is not usually achievable because of scuffing, which is also expected to be costly. There may be multiple contributors to that cost, which is nonetheless expected to increase with more negative lift as a consequence of the drag force produced by the ground against the foot, which the human must counteract to avoid stumbling. Indeed, any drag impulse must be counteracted with an equal and opposite positive impulse from the rest of the body to recover the original walking speed.

We expect that both lifting and scuffing costs should be minimized near zero foot lift *z*. This alone would imply that humans should prefer zero foot lift if not for variability of foot motion. Human motions are not perfectly repeatable, as a result of imperfect motor control, noisy sensors, actuators and neurons, as well as a number of variations in the surrounding environment. With some two-sided probability distribution *p _{z}*(

*w*) of deviations about the nominal

*z*, the minimum cost should typically be biased away from the steeper of lifting and scuffing costs. Here, we posit that the cost of scuffing is steeper than the cost of lifting, therefore favoring slightly positive nominal foot lift.

The present study represents an extreme simplification of foot scuffing. In real life, scuffing is often an unexpected and singular occurrence, followed by multiple steps for recovery. Instead, we employed purposeful, continuous scuffing during steady gait, which simplifies experimental control over the amount of scuffing, as well as measurement of the associated biomechanics and energetics. Continuous scuffing might adequately reproduce the consequences of relatively light scuffing, which does not necessarily induce stumbling and recovery. The model (Eqn 1) therefore serves mainly as a conceptual basis for explaining the trade-offs between lifting and scuffing the foot, rather than an accurate representation of the complexities of the experimental measurements. The main purpose here was to test and quantify the hypothesized costs of lifting and scuffing.

### Experiment

We measured the costs of walking at different foot clearance heights performed by young, healthy adults. We used real-time visual feedback to enforce varying amounts of foot lift and scuffing during treadmill walking. We measured metabolic energy expenditure as the cost, and also characterized the associated gait kinematics and kinetics. Foot lift was measured relative to the ground, and additionally characterized by the overall mechanical work performed on the body center of mass (COM) and by the joints, while scuffing was characterized by the horizontal impulse (time-integral of fore–aft force) produced by the foot against the ground surface mid-swing (75–100% of stride, defined as starting from heel-strike). Eight subjects (*N*=8, 1 female, 7 male) walked at varying levels of ground clearance during leg swing at a constant speed of 1.25 m s^{−1} on an instrumented treadmill. Subjects walked normally and with three levels of foot lifting and three levels of scuffing during the swing phase of walking (Fig. 2). The thresholds, termed low, medium and high, to achieve during swing were 0.1, 0.2 and 0.3 m for foot lift and 100, 200 and 300 N for scuff force. Lift thresholds were measured from the treadmill surface, and scuff force was defined as horizontal force produced by the foot, measured as the drag GRF mid-swing. One subject's scuffing data were excluded because the scuff continued smoothly into heel-strike, making the impulse indistinguishable from normal heel contact. Subjects received visual feedback of toe marker height during foot lift conditions and fore–aft GRF during scuffing conditions. In each case, these were displayed in real time along with a target threshold for either scuff force or lift height. Trials were performed in randomized order, and were 6 min in duration. Subjects' age ranged from 18 to 29 years, and their body mass *M* was 73.8±11.1 kg (mean±s.d.) and leg length *L* was 0.93±0.05 m. All subjects provided written informed consent prior to the study, according to Institutional Review Board procedures.

We measured metabolic power and gait mechanics using standard procedures. Net metabolic rate (in W) was estimated from the rate of oxygen consumption and carbon dioxide production using standard conversion factors (Brockway, 1987). Steady-state metabolic power was averaged over the last 2 min of each 6 min trial, and the rate for quiet standing (99.2±17.1 W, 0.0467±0.0123 dimensionless) was subtracted from the gross rate to yield net metabolic power. Kinematic and kinetic data were also recorded over the same time with motion capture (PhaseSpace, San Leandro, CA, USA), using a standard 24-marker set (Zelik and Kuo, 2010). Foot lift was measured from the vertical height of the toe marker (fifth metatarsal) during swing, relative to its height during quiet standing. The marker can exhibit one or two peaks – in particular, one peak when the foot is lifted – and so lift height *z*_{LH} was defined as the first peak height after toe-off (Fig. 3D). For normal walking, we also measured the distribution of the minimum clearance (between the two peaks), as an indicator of *p _{z}*(

*w*). Scuffing was measured and characterized by the scuff impulse , defined as the integral of drag force (aft GRF) during the swing phase (Fig. 3C), normally zero when there is no scuffing. As a point of comparison, we calculated the average total horizontal impulse generated per stride for normal walking (26.2±4.32 N s), which was many times greater than the scuff impulses induced experimentally. We also measured the rate of work performed on the COM, termed instantaneous COM work rate (Donelan et al., 2002), computed as the inner product of the GRFs of each leg and the COM velocity. Standard kinematics and inverse dynamics procedures (Visual3D, C-Motion, Germantown, MD, USA) yielded ankle, knee and hip angles, and joint moments and powers. As a simple summary of joint actions, we defined summed joint power as the net power from ankle, knee and hip of one leg at each point in the stride. The positive intervals of COM work rate and summed joint power during a stride were integrated to yield positive COM work and summed joint work per stride. Average positive COM and summed joint work rates for both sides of the body were calculated from the corresponding positive work per stride by dividing by stride time and multiplying by 2. The same computation was performed for negative work rates. Finally, step lengths and widths were also computed, along with root-mean-square variabilities, for each trial.

We expected that the increasing levels of each condition would affect the corresponding measure and metabolic cost. Thus, scuff impulse was expected to increase with the amount of scuffing, and lift height with the amount of foot lifting. We also tested for simple relationships between these measures and metabolic rate. For scuffing, we expected its metabolic cost to be an odd function of scuff impulse , meaning the cost should decrease as scuffing approaches zero and would continue to decrease past the origin if a negative (i.e. assistive) drag force were possible. We therefore performed a linear fit between scuff impulse and net metabolic rate. For foot lift, it was less clear how it should determine energetic cost, and so we tested for net metabolic rate increasing both linearly and quadratically with lift height *z*_{LH}. Fits were performed for all of the data from each condition simultaneously, allowing each subject to have an individual constant offset. Statistical tests were performed on the coefficients, with a significance level of α=0.05. The effect of each condition on corresponding measures ( and *z*_{LH}) was also tested (ANOVA followed by *post hoc t*-tests with Holm–Sidak correction for multiple comparisons; Glantz, 2005).

To facilitate comparison between the relative costs of scuffing and lifting, we devised a scaling factor to translate between the two. Scaling foot lift *z*_{LH} to scuff impulse requires a transformation from units of distance to units of momentum. We defined the scaling factor *m*_{f}** g**/

*v*

_{f}, where

**is acceleration due to gravity, and**

*g**m*

_{f}is the mass of the lower leg and

*v*

_{f}is its speed, approximated as 16.1% of body mass

*M*(Winter, 2004) and 3.75 m s

^{−1}(three times walking speed; Winter, 1992), respectively. This may be interpreted as transforming the gravitational potential energy of the lifted foot and lower leg into work as the foot is scuffed against the ground. Such scaling is used to examine the relative metabolic costs of scuffing and lifting, where scuffing and lifting are plotted opposite each other on a common, scaled axis. This allows the costs to be compared in terms of the slope of metabolic rate versus either scuffing or lifting.

Dimensionless measurements are reported here, using the base units of body mass *M*, standing leg length *L* and gravitational acceleration ** g**. Force was non-dimensionalized by

*M*(mean 723.6 N), moment and work by

**g***M*(mean 670.8 N m), power by

**g**L*M*

**g**^{1.5}

*L*

^{0.5}(mean 2182 W), step length and width by

*L*(mean 0.9265 m), and step time by (mean 0.3072 s).

## RESULTS

We found each of the experimental conditions to yield varying levels of scuff impulse or lift height (Fig. 3). The conditions also resulted in substantial changes in metabolic energy expenditure, as well as alterations to gait biomechanics. Net metabolic rate increased substantially within the range of conditions, to about 1.9 and 2.3 times the normal expenditure rate for scuff and foot lift, respectively. The increase was less steep for increasing lift height (treated as a continuous variable), than for increasing scuff impulse. Positive and negative joint and COM work rate also increased with greater foot lift. In contrast, foot scuffing had less obvious effects on biomechanical measures despite its relatively high energetic cost.

Significant changes resulted from each of the discrete walking conditions for foot scuffing and lifting (Fig. 3, Table 1). These were observable in the form of aft-directed GRFs for scuffing and higher foot trajectories for lifting, summarized by significant changes in scuff impulse and lift height *z*_{LH}, respectively (all *P*<0.05). The scuffing conditions resulted in aft-directed impulses up to 2.13±0.46 N s for the high condition, equivalent to about 8.6% the aft-directed impulse for an entire normal stride. The lifting conditions resulted in heights about 1.62 to 3.41 times greater than the normal lift of 0.0886 m, referring to the first peaks in toe clearance.

As expected, the preferred, normal lift height approximately coincided with minimum metabolic cost (Fig. 4). Net metabolic rate increased with a greater magnitude of lift height or scuff impulse, treating the two measures as continuous variables (see Table 2). There was an approximately linear trend for scuff impulse, with slope −7.03±1.99 (metabolic rate per unit impulse, mean±95% confidence interval, CI) and offset 0.088±0.0316 (mean±s.d., equivalent to about −69.0 W N^{−1} s^{−1} and 183 W, respectively), with *R*^{2}=0.747 (*P*=3.95e−7). With the aforementioned scaling factor to convert scuff impulse to negative height, this slope is equivalent to −0.9097±0.2571. There was also a significant trend for lift height with quadratic coefficient 0.9876±0.2363 and offset 0.0877±0.0409 (equivalent to about 2517 W m^{−2} and 185 W, respectively), with *R*^{2}=0.77 (*P*=1.10e−8). Alternatively, applying a linear rather than quadratic fit for lift height, the trend remained significant, with linear coefficient 0.4395±0.0928 (metabolic rate per unit lift height) and offset 0.0472±0.0410, with *R*^{2}=0.8116 (*P*=1.12e−9), with approximately half the steepness of the analogous slope for scuffing (0.9097 after scaling to metabolic rate per negative lift height). Thus, the cost of scuffing was far steeper than that for lifting, regardless of the fitting method. As a simple indicator of magnitude, dragging the foot with 11% of the normal stance phase ground reaction impulse caused an approximate doubling of metabolic cost. A similar doubling of cost resulted from lifting the foot about 23 cm higher than normal. Minimum clearance during normal walking, as measured from the trough of the marker trajectory, was 0.0157±0.0046 m (mean±s.d.).

The major costs above were associated with relatively minor changes in step parameters for foot lift, and even smaller changes for scuffing (Table 2). Subjects exhibited slightly longer step length, step width, and step period and shorter double support duration with increasing lift height (for each 1 m additional lift: 0.53 m longer, 0.17 m wider, and 0.43 s more time, and −0.11 s more time; all *P*<0.05). There were also small increases in step length and width RMS variability (0.0504 m and 0.0588 m per 1 m lift, respectively, *P*<0.05). However, the only changes in step parameters for scuffing were slightly increased step width and step length variability (−0.0093 m and −0.0036 m per 1 N s additional scuff impulse, respectively, *P*<0.05).

The effects of foot lifting and scuffing were somewhat more evident in the mechanics of walking, particularly for lifting. Qualitatively examining force and power trajectories (Fig. 5), greater foot lift appeared to magnify the first peak of the vertical GRF, the positive and negative peaks of the fore–aft GRF, and the magnitudes of COM and summed joint power. In contrast, scuffing seemed to have much less effect, perhaps slightly reducing the second peak of the vertical GRF and push-off power (see Fig. 3 for representative data, and Fig. 5 for across-subject average data). In terms of joint kinematics (Fig. 6), lifting the foot appeared to require more flexion in the knee and hip during swing, while scuffing produced relatively minor changes. Lift height also produced changes in joint power while only ankle push-off seemed to reduce for scuffing.

These observations are supported by quantitative differences in the work performed by the body (see Fig. 7). Scuffing resulted in only minor or non-significant changes in average work rates on the COM and by the summed joints, whereas foot lifting resulted in much greater and significant increases in work (see Table 2 for details and individual joint results). In particular, greater foot lift entailed more positive and negative work on the COM and by the joints, particularly the ankle and knee for positive work, and hip for negative work.

## DISCUSSION

We hypothesized that humans compromise between the additional effort to lift the foot higher during swing, and that associated with scuffing of the foot on the ground. We found both types of deviations to be energetically more costly than the normal swing, with scuffing particularly sensitive to even small amounts of contact. These costs could in part explain the preferred ground clearance, if movement variability is also taken into account. This is because the average cost of clearing the ground by a nominal and positive amount can also depend on relatively infrequent but costly scuffing. Another aspect of this study is the treatment of two dissimilar options, lifting versus scuffing the foot, in a continuous fashion. We next consider the implications of this approach, for examining not only limb swing but also more general movements.

One notable finding was that the costs of scuffing or lifting the foot can be quite high (Fig. 4). Within the range of conditions considered, both were well capable of doubling the normal net metabolic rate. In the case of foot lifting, the cost is partially attributable to mechanical work, as indicated by two measures: work performed on the body COM (which the swing leg contributes to) and work performed by the lower extremity joints (Fig. 7 and Table 2). Work is needed to lift the leg, direct it on a longer path, and slow its descent as it nears the ground (Fig. 3) with each swing phase. It also appears that other adjustments take place throughout the stride, including a greater overall amplitude of COM power and summed joint power (Fig. 5). The apparently simple act of raising the swing foot therefore entails a coordinated action affecting the entire stride. The same is true for scuffing, albeit more subtly, with force imparted on the ground through the swing knee and hip (Fig. 6, Table 2). However, this had only a modest effect on overall work rates (small or non-significant effects, Fig. 7), suggesting that scuffing is primarily an action peripheral to the COM, perhaps accompanied by additional coordination throughout the body and potentially observable at the muscle level. In fact, both lifting and scuffing appear to entail a complex series of compensations that are difficult to predict and subtle to measure. It would be challenging to predict a more than doubling of metabolic cost from gait analysis measurements alone. The gait analysis presented here therefore serves mainly to illustrate possible indicators of the increased energetic cost, rather than being a comprehensive explanation.

During normal walking, the foot does not undergo the extremes of lifting or scuffing, but rather clears the ground with a small and usually positive amount. Although the experiment entailed large extremes, these were intended to reveal continuous trends in metabolic cost from discrete experimental conditions. The individual costs for lifting and scuffing are difficult to separate from one another near zero ground clearance, and so the surrounding trends were used to extrapolate the costs close to that boundary. Here, it is evident that, even though the energetic costs were only slightly elevated above normal, they increased more sharply with scuffing than with lifting. These relative sensitivities are proposed to determine the optimum clearance, mediated by the variability of foot motion. The typical variability of foot motion admits the possibility of occasional scuffing with only a 2 deg change in swing ankle angle (Winter, 1992). Scuffing may therefore be costly enough to make it preferable to avoid it by lifting the foot higher on average. Most humans probably scuff infrequently on flat, smooth floors, but the incidence likely increases on uneven surfaces. That scuffing does occasionally happen may be surmised by the wear pattern on shoe soles, or the marks left on tiled floors. We speculate that scuffing happens occasionally because it would be costlier to completely avoid it.

To address movement variability, we treated scuffing as if it could lie along a continuum with lifting the foot. We thus introduced scuffing as a consequence of negative virtual clearance of the foot, along with a scaling factor between lift height and scuff impulse. It is only with such a conversion that the relative sensitivities of metabolic cost with respect to lifting and scuffing can be compared (see opposing sides of Fig. 4). While similar results could have been achieved with summed joint work, this provides a means to compare otherwise dissimilar options, which are not easily incorporated into an optimization framework. Here, they are treated as convertible into a single independent variable. However, the actual cost of scuffing in the real world is likely considerably greater than the purely energetic cost measured here (on a flat, constant-speed treadmill), making its steepness (left side of Fig. 4) greater than measured here. The steeper that cost, the greater the incentive for higher average ground clearance, despite its higher metabolic cost. It is also possible that many other human behaviors depend in part on movement variability when determining an optimum or preference. For example, Begg et al. (2007) speculated that less effort may be needed to increase the skewness of the distribution (i.e. only tightly control the variability closer to the ground) than to decrease its variability.

There are a number of limitations to this study. We attempted to induce lifting and scuffing of the foot as deviations from normal gait, but this does not necessarily reproduce the adjustments that occur in real-world situations. For example, actual scuffing is usually unintentional, and could induce fear, injury avoidance and recovery actions not examined here. We also did not address the potential cost of controlling movement variability, or consider the effect of treadmill speed on the cost of ground clearance. Previous studies have shown that preferred ground clearance increases with walking speed (De Asha and Buckley, 2015; Schulz et al., 2010). At high walking speeds, scuffing would be expected to induce greater drag force, and therefore become costlier, thus favoring more clearance. Conversely, it is possible that scuffing becomes less critical at very low walking speeds, when it might even be helpful to scuff at the end of the swing to bring the foot to rest. We have also assumed that scuffing is adequately characterized by the drag force it produces against the ground and have not explored other possible effects. For example, scuffing may also entail some collision work against the ground, assumed here to be small as a result of the relatively low vertical GRF (Fig. 5). Another consideration is the possibility of unintended effects from experimental controls; for example, cognitive effort from visual feedback conditions. The present study therefore only represents a simple characterization of possible contributing factors that determine the average foot–ground clearance.

There are also several possible implications from our findings. One is a potential explanation for the increased clearance observed on uneven terrain (Gates et al., 2012; Voloshina et al., 2013). Uneven ground would be expected to increase the cost of scuffing, and perhaps shift it in the direction of positive foot lift (Fig. 1). It may also affect balance and gait steadiness, and cause a wider, and perhaps skewed, probability distribution for foot motion. Both of these effects would cause the optimum average foot clearance to increase. There may also be other factors in the opposite direction. For example, walking on surfaces with low friction could be more forgiving of scuffing, as humans anecdotally appear more willing to scuff when wearing slippers or walking on sand, grass or light snow, conditions where the drag force appears low. This also suggests a potential opportunity to lift the swing foot less when traversing stepping stones raised above ground. A reduced amount or probability of drag would be expected to favor reduced foot lift. The approach applied here may also affect other scenarios, such as the effects of fatigue or age. Older adults tend to walk at slower speeds and with increased clearance variability compared with younger adults (Begg et al., 2007), which may be due to differing biomechanics or control, and result in differing costs for scuffing or lifting the foot, and perhaps contribute to increased risk for stumbling or falling. Although we have explored a relatively innocuous situation here, the trade-offs between scuffing and lifting the foot may affect the overall energetic cost of walking as well as risks for injury.

## Acknowledgements

The authors thank Daniel J. Bertoni for assistance in data collection.

## FOOTNOTES

**Competing interests**The authors declare no competing or financial interests.

**Author contributions**A.R.W. and A.D.K. conceived the study and drafted the manuscript. A.R.W. designed and carried out the experiments and data analyses. Both authors read and approved the final manuscript.

**Funding**This work was supported in part by the Office of Naval Research (ETOWL program), National Institutes of Health (AG030815), and US Department of Defense (W81XWH-09-2-0142, National Defense Science & Engineering Graduate Fellowship Program). Deposited in PMC for release after 12 months.

- Received January 15, 2016.
- Accepted July 22, 2016.

- © 2016. Published by The Company of Biologists Ltd