Many aspects of human walking are determined by minimization of metabolic energy expenditure. For example, the preferred step length (Atzler and Herbst, 1927) and step width (Donelan et al., 2001) minimize energy expenditure for a given steady speed, and the preferred steady speed approximately coincides with minimum energy expenditure per distance traveled (Figure 1A, Ralston, 1958). This speed, as well as the economy of walking, both decline with age, disability, or poor health. As such, preferred speed is widely employed as a clinically useful indicator of overall mobility (Afilalo et al., 2010; Studenski et al., 2011). However, there are naturally many other factors that also influence walking. All walking tasks have a beginning and end, and some may spend little or no time at steady speed. Some tasks may also occur with a degree of urgency, and some individuals may habitually walk faster than others, for reasons not obviously explained by economy. Energy economy is a powerful and objective explanation for steady walking speed, but it does not readily accommodate these everyday observations. Realistic walking tasks must therefore be governed by more than energy economy alone.

Figure 1

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The specific energy measure thought to govern steady walking speed is the gross metabolic cost of transport (COT). Defined as energy expended per distance travelled and body weight (or mass), it has a convex dependency on speed. Its minimum (termed min-COT here) seems to predict the steady preferred speed of about 1.25 ms^-1, as widely reported for laboratory settings (Ralston, 1958; Martin et al., 1992; Willis et al., 2005; Browning and Kram, 2005; Browning et al., 2006; Rose et al., 2006; Entin et al., 2010), and observed of some other animals (Hoyt and Taylor, 1981; Langman et al., 2012; Watson et al., 2011). However, much of actual daily living also involves relatively short bouts of walking (Figure 1B), with about half of daily bouts taking less than 16 steps as reported by Orendurff et al., 2008. Such bouts, say of distance $D$ (Figure 1C), may spend substantial time and energy on starting from and stopping at rest, and relatively little time at steady speed. For example, in short bouts of walking up to about a dozen steps, peak speed is slower than the steady optimum, and only attains that value with more steps (Seethapathi and Srinivasan, 2015). There is a substantial energetic cost to changing speeds that could account for 4–8% of daily walking energy budget (Seethapathi and Srinivasan, 2015). If energy economy is important for walking, it should apply to an entire walking bout or task, and not only to steady speed.

Another important factor for walking is time. Time is valuable in part because energy is always expended even when one is at rest (Jetté et al., 1990), and because walking faster can save time to reach destination, but at greater energy cost (Ralston, 1958). Time is also subjectively valuable, because the urgency of a task, or even of an individual’s personality, surroundings, or culture, could influence their speed. It has long been observed that people walk faster in big cities than in small towns, by a factor of more than twofold (about 0.75–1.75 ms^-1), or about ±40% of 1.25 ms^-1 (Bornstein and Bornstein, 1976). Perhaps population density affects a person’s valuation of time (Bornstein, 1979; Levine and Bartlett, 1984; Li and Cao, 2019). Time is certainly a factor in deciding whether to walk or run (Summerside et al., 2018), and is considered an important factor in the general vigor of movements, beyond walking alone (Labaune et al., 2020). It is clearly worthwhile to expend more energy if time is of essence.

There are, however, challenges to incorporating time into walking. One method is to factor time into the equivalent of temporally discounted reward (Shadmehr et al., 2010), which refers to offering a reduced reward for longer durations, typically employed in fields such as movement vigor, foraging theory (Green and Myerson, 1996), and reinforcement learning (Sutton and Barto, 2018). Another is to express time as a cost that increases for longer movement durations, trading off against greater energy cost for shorter durations. Both the energy cost for an entire walking bout, plus a cost for time duration, could thus be combined into a single objective function to be minimized (Wong et al., 2021). This presents a second challenge, which is how to determine the optimum. Unlike the case of steady walking at a single speed (Figure 1A), an entire walking bout requires a time-varying trajectory of walking speed. This cannot be determined from the cost of transport curve, but can potentially be predicted by a quantitative, dynamical model. Simple models of walking (Figure 7), based on the pendulum-like dynamics of walking, can predict aspects such as optimal step length and step width (Kuo et al., 2005) for a steady speed, and optimal speed fluctuations for uneven steps (Darici and Kuo, 2022a; Darici and Kuo, 2022b). It remains to be determined whether they can predict the energetics and timing of walking bouts with transient conditions.

The purpose of the present study was to test whether the combined costs of energy and time can predict dynamic variation in walking speed. We propose a basic quantitative objective function called the Energy-Time hypothesis, which includes a cost for total energy expenditure or mechanical work for a walking bout, plus a penalty increasing with the bout’s time duration. We apply this objective to a computational walking model, using dynamic optimization to predict dynamic speed profiles for walking bouts of varying distance (Figure 1D). For relatively short walking bouts, this hypothesis predicts speeds that vary within a bout, and speed profiles that vary across bout distances. For longer distances, it predicts a steady walking speed, not as an explicit outcome but rather as an emergent behavior. To test these predictions, we performed a human subjects experiment, comparing empirical speed profiles against model predictions. If the model is able to predict human speed profiles, it may suggest that a valuation of time and energy can influence walking, and thus be compatible with walking bouts of any distance and any degree of urgency.

We had sought to test whether humans optimize not only metabolic energy but also a valuation of time spent walking. Although the prevailing theory of minimizing the energetic COT explains steady walking, it does not explain shorter walks that lack a steady speed, nor does it readily accommodate individual tendencies toward faster or slower speeds. We found that humans walk bouts of finite distance with a trajectory of speeds varying with distance. These bouts fall within a consistent family of trajectories across subjects, despite individual differences in overall speed or duration. These results are in agreement with a simple mechanistic model of walking, governed by optimization. The findings suggest that humans optimize a combined objective that trades off the energy to arrive at destination against the time it takes to get there.

Each human walking bout consisted of a dynamically varying trajectory of speed with an inverted U shape. Many of these bouts included a period of steady walking, at speed similar to (but usually faster than) the supposed min-COT (Ralston, 1958), but mainly for the longer distances (Figure 4). Shorter bouts of say 10 m or less exhibited a relatively brief peak slower than the typical min-COT speed (Figure 5B). All such bouts also spent substantial time and energy in acceleration and deceleration (Figure 5B). Moreover, short distances such as this are quite ecological, accounting for about half of the daily living walking bouts reported by Orendurff et al., 2008, with acceleration and deceleration potentially accounting for 4–8% of daily walking energy budget (Seethapathi and Srinivasan, 2015). This contrasts with the min-COT hypothesis, which predicts only a single steady speed, and cannot explain how to start or end a bout. People usually walk to a destination of known and fixed distance, for which it is more sensible to minimize the total energy for that distance, rather than energy per distance (min-COT). This is not to dismiss the energy spent for steady walking, which has been well-characterized in laboratory settings where the vast majority of published studies have been performed. But in daily living, humans walk a variety of speeds and distances, many too short to be steady. To our knowledge, this is the first study to consider the time and energy spent in brief walking bouts.

Even though there were considerable differences between individuals, each subject was quite consistent within their own walking bouts. Those with a slower or faster peak speed during longer bouts were also consistently so during shorter, non-steady bouts (Figure 5), as evidenced by the 54% reduced variability after normalizing peak speeds by the longest bout. Moreover, the bouts across all subjects were scalable to a single, self-similar family of trajectories (Figure 4). These trajectories were not consistent with a fixed acceleration or deceleration profile (Figure 2A), and instead exhibited a greater peak speed and longer time to that peak with greater distance (Figure 4B). This pattern suggests that there are systematic criteria or principles that govern walking bouts of finite distance. Even though some individuals are faster than others (Figure 4A), they all seem to follow similar principles.

These observations agree with the primary hypothesis that humans optimize for energy and time. The trade-off between the two, described by the valuation of time $C_T$ (for human metabolic cost, or $c_T$ in terms of model work), is readily explained for steady walking. A valuation of zero corresponds to minimizing the gross metabolic cost of transport (Figure 6B), yielding the min-COT speed. (Or equivalently, resting metabolic rate could be treated as part of a running time cost, trading off against the net metabolic rate for walking, as in Figure 6C.) Of course, humans do not always walk the same speed, and faster speeds save time but cost more energy to cover a given distance. One’s actual steady speed must therefore be observed empirically, from which their valuation of time may also be estimated (ranging roughly 0–2 MET, Figure 6B). But what sets time valuation apart is its predictive value for non-steady walking. The speed trajectory for a fixed bout distance need not even contain a steady portion, and is often dominated by acceleration and deceleration. The Energy-Time model can nevertheless predict an entire family of trajectories across a variety of distances (Figure 3) despite individual-specific step lengths and time valuations. The model also suggests how peak speeds, and time to peak, and durations should increase with distance (Figure 3), similar to human data. All this is based on one free parameter, the individual-specific valuation of time. That valuation may depend on complex physiological and socio-psychological traits, but it nonetheless appears to have predictive value for a given context. Not tested here is the presumption that different contexts, for example changing the saliency of a task or adding time pressure, will also lead to systematic changes in walking bouts. If an individual’s valuation of time can be estimated empirically, our hypothesis provides an operational means of integrating it into a quantitative model.

These predictions are produced by a mechanistic model governed almost entirely by dynamics. The timing comes from the dynamics of pendulum-like walking, and the energetics from the step-to-step transition between pendulum-like steps. The step-to-step transition requires mechanical work to accelerate and to restore collision losses, such that for short walks it is uneconomical to accelerate quickly to min-COT speed (Figure 2, steady min-COT). An alternative is to accelerate more gradually, but that is costly because of the high peak speed attained (Figure 2, steady accel). The model thus favors an intermediate and smooth acceleration to a slower and continuously varying speed with an inverted U shape. Separate studies have found step-to-step transition work to predict human metabolic energy expenditure as a function of step length (Donelan et al., 2002) and changing speed (Seethapathi and Srinivasan, 2015). Here, we have constrained the pendulum-like dynamics so that there is only one physical parameter, step length, which in any case has very little effect on the characteristic shape of speed trajectories (Figure 3). As such, this was set to nominal value and not fitted to data, making the model as predictive as possible. Indeed, the very same model also predicts human compensation strategies for walking on surfaces of uneven height (Darici and Kuo, 2022a; Darici and Kuo, 2022b). Of course, the human body has many degrees of freedom capable of far different motions, but model analysis suggests that pendulum-like walking is the most economical means to move the COM at slow to moderate speeds (Srinivasan and Ruina, 2006), and that push-off during the step-to-step transition is the most economical means of powering such pendulum-like walking (Kuo, 2001). These models are predicated on mechanical work as the major cost, and the COM as the major inertia in the system.

It is instructive to consider what other models might explain or predict our experimental results. We did not explore more complex models here, but would expect similar predictions to result from any model based on pendulum-like walking and step-to-step transitions. This includes those cited in the previous paragraph, as well as a family of such models including three-dimensional motion (Donelan et al., 2001), knees and bi-articular actuators (Dean and Kuo, 2009), and plantarflexing ankles (Zelik et al., 2014). It is also possible that more complex, musculoskeletal models also perform substantial work and expend energy for step-to-step transitions, and might therefore agree with the present model. But here, relatively simple principles account for a fairly wide array of predictions (Figure 3), which are unlikely to result from happenstance. We are also unaware of any current hypothesis that could plausibly substitute for the present one. We therefore doubt if alternative models not based on pendulum-like principles could predict or reproduce these results, except with numerous fitted parameters.

This model is optimized with an additional control parameter, for the valuation of time. Time has long been recognized as a factor in the pace of life (Levine and Bartlett, 1984), and in reward and vigor in motor control (Shadmehr et al., 2010). It is typically expressed as a temporal discounting of reward, which appears key to human decision making and the theory of reinforcement learning. Here we expressed it as a trade-off equivalency between energy and time. This was mainly due to the need for compatibility with our energetics model, but also because neither model nor experiment included an explicit reward to be discounted. We used a simple, linear valuation of time in terms of energy, rather than a nonlinear, exponential or hyperbolic temporal discounting factor (Green and Myerson, 1996). Energy is a physiological cost endemic to life, that is not obviously more or less valuable at different points in time. It is sufficient to predict and explain the present results, and there is currently insufficient evidence to favor a nonlinear cost over our linear valuation. But regardless of the particular formulation, a default valuation of time may be an individualistic trait, generalizable to other tasks such as hand and eye movements (Labaune et al., 2020). Indeed, we have found a similar valuation of time to explain how reaching durations and speed trajectories vary with reaching distance (Wong et al., 2021). Another implication of our model is that humans may incorporate prediction of time within central nervous system internal models. Such models have long been proposed to explain humans predict and adapt their movement trajectories, for example to novel dynamics (Todorov, 2004). If movement duration is also part of human planning, it suggests the ability to predict not only movement trajectories and energetics, but also time. Here such prediction is made operational within a quantitative model.

Valuation of time offers another perspective on minimizing the gross energetic cost of transport. Actual walking tasks are not purely steady, and are probably planned with consideration of what happens at the destination. Long and Srinivasan, 2013 proposed a task to minimize the total energy expended to walk to destination within a more than ample allocation of time. They showed that total energy should be optimized by mixing resting and walking (and running if necessary). Suppose the task is extended to an indefinite duration, where a considerable amount of time is spent resting. The optimal total energy and walking duration may be found by applying our Energy-Time objective with time valuation ($C_T$) equal to zero (Figure 6). Walking faster than optimal would yield more time to rest, but at a greater total energy cost for walking. Walking slower would cost less energy for the walking motion alone, but at a greater total cost due to less time available to rest. After all, $C_T$ is the energy one is willing to expend to save a unit of time, and the resting rate is the energy expended to rest for a unit of time. This may seem like a trivial restatement of the min-COT hypothesis, but it differs in two important ways. First, it can predict both the duration of walking and the entire speed trajectory, even for short bouts where there is no steady portion. Second, it considers how valuable time is at the destination. Minimizing the gross cost of transport is most sensible for maximizing the survivable range distance (Srinivasan, 2009), which may not be a concern in modern life where survival rates are high, walks short, and calories plentiful. Rather, it may be a sensible default to value time at close to the resting rate (particularly for long walks), and then to vary the valuation dependingon context. One might thus rush toward a long-lost friend or hurry in a big city, if the time spent at destination is far more valuable than resting (Bornstein, 1979; Levine and Bartlett, 1984; Li and Cao, 2019). Similarly, we do not consider walking slowly to be a waste of energy per distance, but rather a waste of time. Even then, there are cases when humans might wish to waste time, for example to avoid an odious task, according to the expression ‘the slow march to the gallows’.

The consistency of individual walking trajectories may have practical implications. Although walking speed is used as a clinical indicator of mobility, it is difficult to standardize (Middleton et al., 2015), because evaluations may be confined to the length of the available walkway, which may be too short (e.g. less than 10 m) for a steady speed to be reached. But given the time to walk a fixed distance, it may be possible to predict the duration and steady speed for another distance, referenced from a universal family of walking trajectories. We have identified one such family that applies to healthy young adults with pendulum-like gait. We do not know whether that family also applies to older adults, who prefer slower steady speeds and expend more energy to walk the same speed (Malatesta et al., 2003). Perhaps an age-related valuation of time might explain some of the differences in speed. Of course, some clinical conditions might be manifested by a deviance from that family, perhaps in the acceleration or deceleration phases, or in how the trajectories vary with distance. If quantified, such deviance might prove clinically useful. The methodology employed here does not require specialized equipment beyond inertial measurement units, and the characterization of distance-dependent speed trajectories can potentially provide more information than available from steady speed alone.

The Energy-Time hypothesis could be tested by further inquiries. We have thus far regarded the valuation of time as a relatively fixed parameter for each subject. That valuation is likely influenced, and therefore testable, by many contextual factors, including physiological and socio-psychological variables and task constraints. For example, caffeine intake, feeding status (e.g. Taylor and Faisal, 2018), or monetary reward could be used make time more valuable as a trade-off against energy. Conversely, energy may be helpful for assessing the valuation of time (or temporally discounted reward), which is not easy to measure other than indirectly. (Energy is also a universal currency, because all animals use energy, whereas only some use money.) Walking has a well-characterized physiological energy cost, and could serve as a useful trade-off against time or reward. The hypothesized optimal gait is the point at which the costs of energy and time have equal and opposite slopes (i.e., partial derivatives) with respect to an independent variable such as speed (e.g. Figure 6C), carried load, or incline. There are thus a variety of opportunities to manipulate the energetic cost of walking, as a means to assess the proposed valuation of time.

There are a number of limitations to this study. Although we tested model predictions in terms of speed trajectories, we did not measure mechanical work or metabolic energy expenditure in human subjects, which would provide greater insight regarding the proposed trade-offs against time. We also did not evaluate each individual’s metabolic cost of transport vs. speed, which would reveal more precise differences between the min-COT speed and the actual self-selected speed. Nor did we evaluate gait kinematics or kinetics, which may be helpful for detailing other ways that walking bouts vary with distance. The simple walking model also only includes a crude representation of step-to-step transitions, which we have crudely estimated to account for as much as 70% of net metabolic cost in steady walking (Donelan et al., 2002). We did not include other factors such as forced leg motion and step length modulation (Doke et al., 2005, up to say 33% of cost) that likely also affect energetic cost, and could therefore be used to test the valuation of time. Nor did we include factors such stability (Bauby and Kuo, 2000; Donelan et al., 2004; Rebula et al., 2017) and three-dimensional motion (Donelan et al., 2001), despite being part of our previous models, because we believe them to contribute little to the present task. In fact, because the optimal control model successfully completes the walking task, its feedforward motion attains implicit stability, which could reduce (but not eliminate) the need for feedback stabilization (Darici and Kuo, 2022a; Darici and Kuo, 2022b). We also did not include an explicit reward, which could facilitate assessment of energy and time in terms of other trade-offs such as money or food. In fact, the Energy-Time hypothesis should be regarded as a subset of the many factors that govern human actions, rendered here in a simple but quantitative form.

We experimentally tested how human walking speed varies with walking distance. The speed trajectories observed from human subjects were compared against predictions from the Energy-Time hypothesis and against the minimum-COT speed. To formulate the hypothesis and make quantitative predictions, we expressed it as an optimal control problem including both energy and time. We first state the hypothesis for human walking, and describe how it is adapted for a simple walking model to yield predicted speed trajectories. This is then followed by description of the experiment regarding human walking speed, and finally an analysis of steady speed as a property of the model.

Walking model

We use the ‘simplest walking model’ (Kuo, 2002) to operationalize this optimization problem (Figure 7A). The model treats the stance leg as an inverted pendulum and requires mechanical work to power the gait. The body center of mass (COM) is modeled as a point mass supported by the stance leg, so that each pendulum-like step follows an arc, which itself requires no energy input. Work is performed during the step-to-step transition (Figure 7B), to redirect the COM velocity from forward-and-downward the end of one arc, and forward-and-upward at the beginning of the next. This is accomplished most economically with an active, impulsive push-off along the axis of the trailing leg, immediately followed by an impulsive, dissipative collision between the rigid leading leg and ground. In steady gait, the optimal push-off restores the collision losses, with mutually canceling impulses of equal magnitude. Speeding up is a matter of a greater push-off than collision, and a net increase in COM velocity during the step-to-step transition (Figure 7C). Positive and negative work are proportional to the square of the push-off and collision impulses, respectively (Kuo, 2002), so that speeding up also dissipates less collision energy than steady gait. Slowing down is the same in reverse, with collisions exceeding push-offs. This model predicts how step-to-step transition work for steady walking should increase as a function of step length and step width (Donelan et al., 2002). The model mainly predicts mechanical work for push-off, which appears to be a proportional predictor of the majority of human metabolic energy during steady walking (Donelan et al., 2002). That work also yields a mechanical cost of transport that varies curvilinearly with steady speed, similar to the empirical metabolic curve (Figure 1a; Ralston, 1958). There are of course other contributions to the metabolic cost of walking such as to move the swing leg (Kuo, 2001), but of smaller magnitude than step-to-step transitions, which are to be tested alone for their predictive value. Details of this model have been described in greater detail previously (Darici et al., 2020; Kuo, 2002), and are recounted only briefly here.

Figure 7

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A walking bout consists of a sequence of $N$ steps, starting and ending at rest. It may be described by the discrete sequence of body speeds v_i ($i=1,2,\mathrm{\dots },N$), each equal to the distance traveled for step $i$ divided by that step’s time duration ${\tau }_i$. The model begins at rest in an upright position (Figure 7D), and is set into motion by a forward initiation impulse acting on the pelvis. In humans, the torso can serve as an inertia that the hip muscles can act against, but for simplicity this action is represented as a translational impulse at the pelvis, summarized by the associated positive work u₀. The total positive work performed by the model consists of the work from initiation and the successive push-offs, a sequence u_i ($i=0,1,\mathrm{\dots },N$). There is also a corresponding sequence of dissipative collision impulses by the leading leg, and a dissipative gait termination to end at upright.

The step-to-step transition starts just before leading leg ground contact contact, and consists of a perfectly impulsive push-off from the trailing leg, followed in immediate succession by a perfectly inelastic and impulsive collision of the leading leg with ground. The COM velocity at the end of one stance phase is $v_i^{-}$, directed forward and downward according to the pendulum arc. Mechanical work is performed only during the step-to-step transition, with a succession of ideal impulses. First is positive push-off work from the trailing leg, directed from its foot to the COM, and second is a perfectly inelastic heel-strike collision of the leading leg with ground, directed from the leading foot to the COM. For brevity, the equations presented here use dimensionless versions of quantities, with body mass $M$, gravitational acceleration $g$, and leg length $L$ as base units. The push-off work is denoted u_i (in units of mass-normalized work), and the push-off and collision sequence act to redirect the COM velocity to $v_i^{+}$ at the beginning of the next stance phase, directed forward and upward according to the next pendulum arc. Using impulse-momentum, the step-to-step transition is described by

(3) ${\displaystyle v_i^{+}=v_i^{-}\cos 2\alpha +\sqrt{2u_i}\sin 2\alpha }$

where $2\alpha$ is the inter-leg angle (Figure 7A). There is no work performed during the passive, inverted pendulum phases, and so the step-to-step transition is responsible for all energy inputs (u_i) and energy losses (from collisions).

The dynamics of an inverted pendulum describe all of the other motion in the system, consisting of the falling of one inverted pendulum toward the step-to-step transition, and the rising of the next inverted pendulum toward mid-stance. These dynamics determine the respective velocities and timing of these respective instances. The velocities may be found through conservation of energy:

(4) ${\displaystyle v_i^{-}=\sqrt{2(1-\cos \alpha )+v_i^2}}$

(5) ${\displaystyle v_{i+1}=\sqrt{(v_i^{+}{)}^2+2(\cos \alpha -1)}}$

The step time ${\tau }_i$ is defined as the time for the stance leg angle $\theta$ to move between successive mid-stance instants, and the corresponding velocities from v_i to $v_{i+1}$. It may be regarded as the sum of a time ${\tau }_i^{-}$ from mid-stance to the step-to-step transition, and then the time ${\tau }_i^{+}$ from the step-to-step transition until next mid-stance. Using the linearized dynamics, the dimensionless time ${\tau }_i^{-}$ of step $i$ is

(6) ${\displaystyle {\tau }_i^{-}=\log \frac{\alpha +\sqrt{v_i^2+{\alpha }^2}}{v_i}.}$

The other time ${\tau }_i^{+}$ is

(7) ${\displaystyle {\tau }_i^{+}=\log \frac{\sqrt{v_i^{+}+\alpha }}{\sqrt{v_i^{+}-\alpha }}}$

For comparison with experiment, we also defined an average (as opposed to mid-stance) speed for each step $i$ as the step length divided by the step time between mid-stance instances,

(8) ${\displaystyle {\mathrm{B}\mathrm{o}\mathrm{d}\mathrm{y}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{e}\mathrm{d}}_i=\frac{2L\sin \alpha }{{\tau }_i^{-}+{\tau }_i^{+}}}$

The trajectory of this body speed is plotted for different walking bouts, for both model and experiment. The equations for body speed and step time are summarized as constraints $f$ and $g$ below.

We chose nominal parameters to correspond to typical human walking. A person with body mass $M$ 70 kg and leg length $L$ of 1 m may typically walk at 1.25 ms^–1, with step length of 0.68 m and step time of 0.58 s, and corresponding fixed constant value $\alpha =0.35$. The step length was also varied in parameter sensitivity studies. Using dynamic similarity, parameters and results are reported here either in SI units, or in normalized units with body mass $M$, gravitational acceleration $g$, and $L$ as base units.

Experimental methods

We tested the model predictions by experimentally measuring the speed profiles of healthy adults walking a series of short distances, ranging about 2–20 steps. Subjects ($N=10$, 6 male and 4 female, 24–38 yr) were instructed to walk at a comfortable speed in ten distance conditions, starting from standing at one numbered marker on the ground, and ending at another as requested by the experimenter. After each trial, there was a brief waiting interval of about 15–30 s, to reduce interference between successive trials and to avoid any incentive to rush through trials. The walking surface was a level sidewalk. The numbered markers were separated by distances of 1.1, 1.7, 2.2, 2.8, 3.3, 3.8, 5.1, 7, 9.1, and 12.7 m. Subjects were provided with a simple task upon reaching the target: They were provided a pointer stick and instructed to walk to and touch the pointer to the target marker. This was intended to provide a context for the task, reflecting the fact that humans often walk to a particular destination to accomplish a task. Each distance condition was conducted a total of four times in two pairs of out-and-back trials, with the distances in random order. There were therefore a total of 400 trials, from 10 subjects walking 10 distances, each four times.

Walking speeds were measured from foot-mounted inertial measurement units (IMUs). These were used to compute the spatiotemporal trajectory of each foot in 3D, which was then processed to yield forward walking speed for the body per step. Each IMU (Opal sensors, APDM Inc, Portland, Oregon) was placed on the top of each foot, taped to the outside of the shoe. The recorded data of linear acceleration and angular velocity data were integrated using a previously described algorithm (Rebula et al., 2013) to yield foot trajectories. Briefly, the algorithm detects footfalls as instances in time when the foot is momentarily at rest on the ground, as defined by thresholds for acceleration and angular velocity. The footfall instance was defined as the mid-point of the below-threshold interval, and used to correct the integrated foot velocity (from gravity-corrected inertial accelerations) to zero, thus reducing IMU integration drift. The footfalls were also used to segment data into discrete strides, from which speed and length of each stride was calculated. (Subjects also wore another IMU on a waist belt, the data from which was used to demarcate the trials, but not for any further quantitative analysis.)

There were a few other analysis adjustments required to produce forward walking data. The absolute position and compass heading of the IMUs were unknown, yielding independent foot trajectories with no relation to each other. However, the experimental conditions called for forward walking for a known distance, so we rotated each foot path to align them into a single forward direction. We also assumed that both feet travelled approximately the same distance for each walk, and translated and rescaled the start and end points to match each other, to yield a processed position-time graph of the two feet (see representative data in Figure 8). We also devised a definition for the starting and ending times for each trial based on IMU data. Humans initiate their gait by shifting their weight before moving the feet (Mann et al., 1979), so that the footfall threshold defined above may not detect the actual gait initiation. We therefore defined a rough approximation to gait initiation and termination, starting before and ending after threshold crossing, by an amount equal to half the average below-threshold time during walking. This adjustment may be incorrect compared to actual weight shift by several tens of milliseconds. The experiment is mainly concerned with speed profiles over time on the order of several seconds. The accuracy of the experiment can thus tolerate small errors in detecting gait initiation or termination.

Figure 8

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Finally, the body’s walking speed and length of each step were calculated as follows (Figure 8). The trajectory of each foot’s strides were found to cross each other, approximating the time in mid-stance when one foot passes by the other. These points of intersection were used to define step length as the spatial distance between intersections, and step time as the temporal difference between intersections. These were used to determine the average speed for each step (‘body speed’), defined as the step length divided by step time ending at each intersection. This assumes that the body moves as much as the feet between mid-stance instances. These discrete, step-averaged data were used to produce trajectories of body speed for each bout (Figure 8, inset), without regard to continuous-time undulations in velocity for the body center of mass. For comparison with these data, similar discrete body speeds and times were computed from model predictions.

We used these data to test the model predictions. We examined how human speed profiles varied with bout distance, and exhibited more rounded peaks for shorter bouts and flatter ones for longer ones. We tested for self-similarity by scaling the profiles by speed and time and performing statistical tests regarding peak speeds and walking durations. We tested whether a saturating exponential describes the increase in peak speed with bout distance ($R^2$; test 95% confidence interval of parameters not including zero). Expecting a self-similar shape for the peak speed vs. distance relationship, we scaled the curves by peak speed and tested for a single exponential. We tested self-similarity in terms of a reduction of variability in peak speed (standard deviation across subjects) for each condition, comparing non-normalized to normalized peak speeds (rescaled to mean overall peak speed) with paired t-test. We examined the walking durations as a function of bout distance, and also tested self-similarity by significant reduction in standard deviations across subjects, comparing non-normalized to normalized data (rescaled to mean longest duration) with paired t-test. We also described walking durations in terms of rise and fall times (between 10% and 90% of peak speed).

Prior to the experiment, subjects provided informed consent as approved by the University of Calgary Conjoint Health Research Ethics Board (REB21-1497). Pre-established exclusion criteria included significant health or other conditions that preclude ability to walk on uneven terrain or moderate hiking trails; no prospective participants were excluded. Subjects were recruited from the community surrounding the University of Calgary; the city has a moderately affluent population of about 1.4  M, with a developed Western culture. The experiment was performed once.

Data (Carlisle and Kuo, 2023, https://github.com/kuo-lab/short_walk_experiment) and code (https://github.com/kuo-lab/simplelocomotionmodel, copy archived at Kuo, 2023) for this study are in publicly-accessible archives.

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Author details

1. Rebecca Elizabeth Carlisle

   Biomedical Engineering Program, University of Calgary, Calgary, Canada

   Contribution

   Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Writing – original draft, Writing – review and editing

   Competing interests

   No competing interests declared

2. Arthur D Kuo

   1. Biomedical Engineering Program, University of Calgary, Calgary, Canada
   2. Faculty of Kinesiology, University of Calgary, Calgary, Canada

   Contribution

   Conceptualization, Formal analysis, Funding acquisition, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review and editing

   For correspondence

   arthurdkuo@gmail.com

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-5233-9709

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