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ResearchCite this article: Seethapathi N, Srinivasan M.2015 The metabolic cost of changing walkingspeeds is significant, implies lower optimalspeeds for shorter distances, and increasesdaily energy estimates. Biol. Lett. 11:20150486.http://dx.doi.org/10.1098/rsbl.2015.0486

Received: 8 June 2015Accepted: 21 August 2015

Subject Areas:biomechanics, behaviour, neuroscience

Keywords:legged locomotion, walking, acceleration,preferred speeds, metabolic cost,energy optimality

Authors for correspondence:Nidhi Seethapathie-mail: seethapathi.1@osu.eduManoj Srinivasane-mail: srinivasan.88@osu.edu

Electronic supplementary material is availableat http://dx.doi.org/10.1098/rsbl.2015.0486 orvia http://rsbl.royalsocietypublishing.org.

Biomechanics

The metabolic cost of changing walkingspeeds is significant, implies loweroptimal speeds for shorter distances,and increases daily energy estimatesNidhi Seethapathi and Manoj Srinivasan

Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA

NS, 0000-0002-5159-9717; MS, 0000-0002-7811-3617

Humans do not generally walk at constant speed, except perhaps on a treadmill.Normal walking involves starting, stopping and changing speeds, in addition toroughlysteady locomotion. Here, we measure the metabolic energy cost of walk-ing when changing speed. Subjects (healthy adults) walked with oscillatingspeeds on a constant-speed treadmill, alternating between walking slower andfaster than the treadmill belt, moving back and forth in the laboratory frame.The metabolic rate for oscillating-speed walking was significantly higher thanthat for constant-speed walking (6–20% cost increase for +0.13–0.27 m s21

speed fluctuations). The metabolic rate increase was correlated with twomodels: a model based on kinetic energy fluctuations and an inverted pendulumwalking model, optimized for oscillating-speed constraints. The cost of changingspeeds may have behavioural implications: we predicted that the energy-opti-mal walking speed is lower for shorter distances. We measured preferredhuman walking speeds for different walking distances and found people pre-ferred lower walking speeds for shorter distances as predicted. Further,analysing published daily walking-bout distributions, we estimate that thecost of changing speeds is 4–8% of daily walking energy budget.

1. IntroductionWalking in typical human life requires changing speeds. Most daily walkingappears to happen in short bouts [1], starting and ending at rest. Here, to betterunderstand such behaviour, we measure the metabolic cost of changing walkingspeeds. Although much is known about constant-speed walking [2,3], the cost ofchanging speeds has not been measured without non-inertial treadmill speedchanges or step-frequency control [4]. Here, we show that the cost of changingspeed is significant and an appreciable fraction of daily walking energy budget.This cost may have behavioural implications: we predict lower optimal walkingspeeds for short distances; we then measured and found that our subjectsprefer lower speeds for shorter distances.

2. Material and methods(a) Experiment: metabolic cost of oscillating-speed walkingSubjects (N ¼ 16, 12 males and 4 females, 23.25+2.1 years, height 177.08+7.4 cm,mass 75.99+12.94 kg, mean+ s.d.) performed both ‘steady’ (constant-speed) and‘oscillating-speed’ walking trials. Oscillating walking speeds were achieved on a con-stant-speed treadmill by alternately walking faster and slower than the belt (figure 1a).Two distinct audible tones of durations Tfwd and Tbck alternated in a loop indicatingwhether the subjects should move towards the treadmill front or rear. We usedthree (Tfwd,Tbck) combinations, (1.9,1.9) s, (2.8,2.8) s and (1.9,2.8) s, obtaining different

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speed fluctuations. We instructed subjects to walk between fixedpositions on the treadmill (0.48 m apart) giving mean excursionlength L ¼ 0.41+0.08 m (figure 1a). The subjects obeyed theimposed back-and-forth time period constraints: mean periodsdiffered from prescribed periods by 0.97+0.24%. While humansdo not usually walk with oscillating speeds, this protocol wasdesigned to isolate the cost of changing speed.

Oscillating-speed trials were at one or both constanttreadmill speeds 1.12 and 1.56 m s21 (equal to the meanspeeds): 10 subjects at both speeds, four subjects at 1.12 m s21

only and two at 1.56 m s21 only, with random speed order.Steady walking trials were performed at speeds ranging from0.89 to 1.78 m s21, including 1.12 and 1.56 m s21.

Metabolic rate per unit mass (W kg21) was estimated usingrespirometry (Oxycon Mobile), approximated as _E ¼ 16:58 _VO2þ4:51 _VCO2 ( _V in ml kg21 s21), denoted _Esteady and _Eosc for steady-and oscillating-speed trials, respectively. Trials lasted 7 min:4 min to reach metabolic steady state and 3 min to estimate themean metabolic rate. The speed oscillation periods (3.8–5.6 s) aremuch smaller than typical metabolic time-constants (30 s), so ourmetabolic steady state is nominally constant. A sacral marker’smotion was measured with marker-based motion capture.

(b) Experiment: preferred walking speedSubjects (N ¼ 10) were asked to walk ten distances (D ¼ 0.5, 1, 2,4, 6, 8, 10, 12, 14 and 89 m) at a comfortable speed, starting and

ending at rest (figure 1d ). We had three trials per distance, alltrials in random order, but performed 0.5–1 m trials separately.

(c) Model 1: kinetic energy fluctuationsIn this model, we attribute the metabolic cost increase for oscillat-ing-speed walking over steady walking to fore–aft kinetic energyfluctuations beyond what happens within each step in constant-speed walking. Figure 1b shows fore–aft velocity vx(t) of thesacral marker for oscillating-speed and steady walking, approxi-mating centre of mass motion. Smoothing vx(t) with an averagingwindow equal to step period gives !vxðtÞ, removing within-stepspeed fluctuations (figure 1b). The mass-normalized metaboliccost increase for oscillating-speed walking over steady walkingdue to the kinetic energy fluctuations for each cycle is modelledas DEke ¼ ð!v 2

max % !v 2minÞðh%1

pos þ h%1negÞ=2, where !vmax and !vmin are

maximum and minimum smoothed fore–aft speeds for thatcycle and hpos ¼ 0.25 and hneg ¼ 1.2 are typical positive andnegative muscle work efficiencies [5]. The model-predicted meta-bolic rate increase D _Eke for each oscillating-speed trial was themedian DEke/Tperiod over all cycles.

(d) Model 2: inverted pendulum walkingWe consider inverted pendulum walking of a point-mass biped, forwhich the total walking metabolic cost is the sum of (i) a step-to-steptransition cost (described below) and (ii) a leg-swing cost [6]. Using

1.010 2 3 1 2 30

1.5

2.0

2.5

fore

–aft

spee

d v x (

ms–1

)

un-smoothed smoothed treadmill speed

subjects walk with changing speeds,moving between these positions in the lab frame

treadmillfront

treadmillrear

treadmill beltat constant

speed

movingto front

movingto rear

(a)

(d )

(c)

(e)(b) steady walking

bungee cord prescribesmaximum walk excursion

oscillating-speed walking

natural within-stepspeed fluctuations in

normal walking

L

preferred walking speedsfor a finite distance

inverted pendulum walking, alternating between fast and slow

START STOPSTOP

instantaneousdeceleration

to rest

bout distance D

idealized walking bout

speed

time tSTARTinstantaneousacceleration

to vopt

walk at constantspeed vopt

fast (again)fast

one complete period

slow

heel-strikepush-off

centreof mass

trajectory

vafter velocityjust afterheel-strike

vbefore velocityjust beforepush-off

push-offimpulse

heel-strikeimpulse

mechanics of the step-to-step transition

t t

trailingleg

leadingleg

vafter

vbefore

umax–

–

–uavgumin

Figure 1. Experimental protocols and theoretical models. (a) Subject walking with oscillating speeds on a constant-speed treadmill, walking faster and slower thanthe belt, moving between two prescribed positions. A longitudinal bungee cord (never to be made taut) constrains the rear-most position and a bungee cordperpendicular to sagittal plane (not shown, never to be touched) constrains the forward-most position. (b) Sacral marker fore – aft velocities, original and smoothed.(c) A nine-step periodic inverted pendulum walking motion, with five steps faster and four steps slower than the mean speed. Initial and final stance-leg directionsare shown for each step (red and blue). Details of one step-to-step transition are shown; downward velocity at the end of one step is redirected by push-off andheel-strike impulses. (d ) Measuring preferred walking speeds as a function of bout distance D; subjects start and stop at rest. (e) An idealized bout: human travelsthe whole distance D at single speed vopt, starting and stopping instantaneously.

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numerical optimization, we found the multi-step-periodic invertedpendulum walking motion (e.g. figure 1c) satisfying our oscillat-ing-speed experimental constraints and minimizing this metaboliccost. Here, we derived and used expressions for step-to-steptransition cost for non-constant-speed walking, generalizing pre-vious constant-speed expressions (see electronic supplementaryinformation). This step-to-step cost accounts for the push-off andheel-strike work to redirect the centre of mass velocity during thestep-to-step transition (figure 1c), and depends on leg-angles andcentre of mass velocities. The model prediction D _Eip is the differencebetween the optimal oscillating-speed and constant-speed costs atthe same mean speed. A metabolic cost term proportional to the inte-gral of leg forces contributed almost equally to oscillating-speedand constant-speed walking costs and did not contribute to theirdifference (see the electronic supplementary material for details).

3. Results(a) Metabolic rate of oscillating-speed walkingMetabolic rates for all six oscillating-speed trials (P1–P6,figure 2a) were significantly higher than the correspondingsteady-state costs. Metabolic rate increment over constant-

speed walking D _Eexpt was significantly greater than zero forall trials (one-sample t-test, for all p , 2 & 1023, figure 2a).Oscillating-speed trials with higher speed fluctuations hadhigher metabolic rates with one exception (P1 . P3 . P2,P4 . P6 and P4 . P5, all p , 0.02).

(b) Model predictionsBoth the kinetic energy fluctuation model D _Eke and theinverted pendulum model D _Eip were correlated with measur-ed metabolic rate increments D _Eexpt (figure 2b). The kineticenergy model and experimental costs are best-fitted by theline: D _Eexpt ¼ lke ðD _EkeÞ % 0:04, with lke ¼ 0.67 whether weuse trial means (R2 ¼ 0.96, 95% CI of lke ¼ 0.48–0.86) or alldata (R2 ¼ 0.24, 95% CI of lke ¼ 0.39–0.95). Similarly, theinverted pendulum model and experimental costs are best-fitted by D _Eexpt ¼ lip ðD _EipÞ þ 0:05 with lip ¼ 0.79 (R2 ¼ 0.88,95% CI of lip ¼ 0.39–1.19).

(c) Daily energy budget for starting and stoppingHumans mostly walk in short bouts [1]. For simplicity,we idealize a bout of distance D and mean speed v as

0.5

“average” walkingspeed over wholedistance

“steady-state”walking speed

optimal walking speedwithin 1% of optimum energywithin 2% of optimum energy

experiment (preferred speeds)

theoretical predictions (optimal speeds)

comparing preferred (expt) and energy-optimal walking speeds (model) as a function of distance

0 2 4 6 8 10 12 14 890

1.2

0.8

0.4

1.6

2.0 speeds, m s–1

distance walked (m)

long-distance

P4(1.9,1.9)

P5(2.8,2.8) P6

(1.9,2.8)

0

P1(1.9,1.9)

P2(2.8,2.8)

P3(1.9,2.8)

DE· ex

pt (W

kg–1

)

prescribed (Tfwd,Tbck) in seconds

0.6

1.2

00 55550 1.0 1.5

−0.5

0

0.5

1.0

1.5mean speed

vavg = 1.12 m s–1

mean speed

vavg = 1.56 m s–1

incremental cost of unsteady walking over steady walking for 6 unsteady trials: P1–P6

3 × 10–3 1 × 10–4 1 × 10–6 1 × 10–3 4 × 10–5

00

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1.0

slope0.79 slope

0.67

kinetic energy model inverted pendulum model

experiment versus model-predicted metabolic increments

expe

rim

enta

l DE· ex

pt (W

kg–1

)

expe

rim

enta

l DE· ex

pt (W

kg–1

)

model-predicted DE·ke (W kg–1) model-predicted DE

·ip (W kg–1)

p-values = 1 × 10–4

(a)

(c)

(b)

Figure 2. (a) Difference D _Eexpt between oscillating- and constant-speed walking metabolic rates for six oscillating-speed trials (P1 – P6): three (Tfwd,Tbck) combinationsand two mean speeds. Box plot shows median (red bar), 25 – 75th percentile (box), and 10 – 90th percentile (whiskers); p-values use one-sided t-tests for the alternativehypothesis that metabolic rate differences are from a distribution with greater-than-zero mean. (b) D _Eexpt compared with kinetic energy model D _Eke and invertedpendulum model D _E ip; we show experimental and model means (black filled circles), the best-fit line (red, solid) and all subjects’ trials (scatter plot, grey dots);no scatter plot for inverted pendulum model as it produces only one prediction per trial. (c) Distance-dependence of the model-based energy-optimal walkingspeeds (blue, solid) and experimentally measured preferred speeds (red and black error bars). Ranges of model-based energy-optimal speeds within 1% (blueline, thin) and 2% (blue band) of optimal energy cost are shown. We show whole-bout ‘average’ speeds (red) and ‘steady-state’ speeds over middle 1.42 m(black, thick), indistinguishable from over middle 0.75 m (grey, thin). Average preferred speeds for 0.5 – 14 m trials were significantly lower than that for the89 m trial ( paired t-test, p , 0.01); similarly, the ‘steady-state’ speeds for 2 – 6 m were significantly lower than that for 89 m ( paired t-test, p , 0.04).

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instantaneously accelerating from rest to speed v, walkingat constant speed v and stopping instantaneously at timeD/v (figure 1e). The total metabolic energy per unit massEbout(D,v) for this idealized bout has two components:(i) a starting and stopping cost, extrapolating from the kin-etic energy model, lkeðh%1

pos þ h%1negÞv2=2 and (ii) a cost for

steady-state walking at speed v, given by _Esteady ¼ aþ bv2

with a ¼ 2.22 W kg21 and b ¼ 1.15 W kg21 (m s– 1)22 [2], sothat Ebout ¼ lkeðh%1

pos þ h%1negÞv2=2þ ðaþ bv2ÞD=v: Applying

this model to data in [1] with vopt ¼ffiffiffiffiffiffiffia=b

p¼ 1:39 m s%1 and

the 95% CI of lke suggests that starting–stopping costs are4–8% of daily walking energy expenditure (electronic sup-plementary material); this cost fraction (4–8%) may applyprimarily to the subject population of [1], adults working inoffices, but could be estimated for other populations giventhe distribution of their daily walking bout lengths.

(d) Optimal and preferred walking speeds are lowerfor shorter distances

For the idealized bout of distance D (figure 1e), the energy-opti-mal walking speed vopt that minimizes Ebout(D,v) is given bythe implicit function: lkev3

optðh%1neg þ h%1

posÞ=ða% bv2optÞ ¼ D:

This metabolically optimal speed increases with distance D,approaching vopt ¼

ffiffiffiffiffiffiffia=b

pfor large distances (figure 2c).

As predicted by the distance-dependence of optimalwalking speeds, preferred human walking speeds in our exper-iment, both ‘average’ and ‘steady-state’ speeds, increased withdistance (figure 2c). ‘Average’ preferred speed is the meanspeed over the whole bout; a proxy for the ‘steady-state’ pre-ferred speed is the mean over the bout’s middle 0.75 m(indistinguishable from averaging the middle 1.4 m). Model-predicted optimal speeds have a 0.96 correlation coefficient(Pearson’s) with experimental steady-state preferred speeds,which were within 1–2% optimal cost. Our subjects couldaccelerate to higher mean or steady-state speeds, but they pre-ferred not to. Therefore, the time taken to accelerate–deceleratecannot explain lower speeds for shorter distances.

4. DiscussionWe have shown that oscillating-speed walking costs more thanconstant-speed walking. These cost-increments are correlated

with kinetic energy fluctuation and inverted pendulummodel predictions; inverted pendulum model predictionswere closer to experimental values (regression slope closer to1), perhaps because the kinetic energy model ignores walkingmechanics. The cost of changing speeds implies lower energy-optimal speeds for shorter distances, reflected in our preferredspeed experiments here and previous amputee data [7].

Preferred walking speeds are used to quantify mobilityand rehabilitation [8], so bout distances should be chosen toavoid artificially lowering speeds. Using the cost of changingspeeds may improve daily activity tracking, energy balanceestimations for obesity, and metabolic estimations duringsports (e.g. soccer [9]).

A previous experiment [4] considered walking with greaterspeed fluctuations (+0.15 to +0.56 m s21) than our study(+0.13 to +0.27 m s21) and similar kinetic energy fluctuations(electronic supplementary material), and found significant costincrease over steady walking for their highest speedfluctuations. However, this study [4] required walking on an oscil-lating-speed treadmill belt or controlling step durations inoverground walking (derived from oscillating-speed treadmilltrials). An oscillating-speed treadmill, being a non-inertial frame(in contrast to a constant-speed treadmill), can perform mechan-ical work, and is not mechanically equivalent to overgroundoscillating-speed walking (as noted in [4]). Further, prescribingstep durations to control overground speed fluctuations isdifferent from prescribing speed fluctuations directly [10].

Future work could involve overground experiments (sayby having subjects follow a laser projection [11]), detailedbiped and metabolic cost models (including muscle force andhistory dependence), using different speed fluctuations andmeasuring metabolic cost while subjects alternate betweenwalking, stopping and starting (being directly applicable towalking bouts, relying less on extrapolation).

Ethics. The Ohio State University’s IRB approved the experiments.Subjects gave informed consent.Data accessibility. Data available through Dryad (http://dx.doi.org/10.5061/dryad.15v26).Authors’ contributions. N.S. performed all experiments, analysis and mod-elling, partly in discussion with M.S. M.S. performed some analyses.N.S. and M.S. wrote the paper.Competing interests. We declare we have no competing interest.Funding. This work was supported by NSF grant no. 1254842.

References

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2. Bobbert AC. 1960 Energy expenditure in leveland grade walking. J. Appl. Physiol. 15, 1015 – 1021.

3. Srinivasan M. 2009 Optimal speeds for walking andrunning, and walking on a moving walkway. Chaos19, 026112. (doi:10.1063/1.3141428)

4. Minetti AE, Ardigo LP, Capodaglio EM, Saibene F.2001 Energetics and mechanics of human walkingat oscillating speeds. Am. Zool. 41, 205 – 210.(doi:10.1093/icb/41.2.205)

5. Margaria R. 1976 Biomechanics and energetics ofmuscular exercise. Oxford, UK: Clarendon Press.

6. Srinivasan M. 2011 Fifteen observations on thestructure of energy-minimizing gaits in manysimple biped models. J. R. Soc. Interface 8, 74 – 98.(doi:10.1098/rsif.2009.0544)

7. Klute GK, Berge JS, Orendurff MS, Williams RM,Czerniecki JM. 2006 Prosthetic intervention effects onactivity of lower-extremity amputees. Arch. Phys. Med.Rehabil. 87, 717–722. (doi:10.1016/j.apmr.2006.02.007)

8. Lamontagne A, Fung J. 2004 Faster is better:implications for speed-intensive gait training afterstroke. Stroke 35, 2543 – 2548. (doi:10.1161/01.STR.0000144685.88760.d7)

9. Osgnach C, Poser S, Bernardini R, Rinaldo R, DiPrampero PE. 2010 Energy cost and metabolicpower in elite soccer: a new match analysisapproach. Med. Sci. Sports Exerc. 42, 170 – 178.(doi:10.1249/MSS.0b013e3181ae5cfd)

10. Bertram JEA, Ruina A. 2001 Multiple walking speed-frequency relations are predicted by constrainedoptimization. J. Theor. Biol. 209, 445 – 453. (doi:10.1006/jtbi.2001.2279)

11. Minetti AE, Gaudino P, Seminati E, Cazzola D. 2012The cost of transport of human running is notaffected, as in walking, by wide acceleration/deceleration cycles. J. Appl. Physiol. 114, 498 – 503.(doi:10.1152/japplphysiol.00959.2012)

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Supplementary Information Appendix for

The metabolic cost of changing walking speeds is significant,

implies lower optimal speeds for shorter distances,

and increases daily energy estimates

Nidhi Seethapathi and Manoj SrinivasanMechanical and Aerospace Engineering, The Ohio State University, Columbus OH 43210 USA

S1 Deriving step-to-step transition work for changing speeds

The mechanical work done to redirect the velocity of the center of mass from downward to upward whentransitioning from one step to the next is thought to be a major determinant of the metabolic cost of walkingat a constant speed [1, 2, 3].

Mathematical expressions for this step-to-step transition cost have previously been derived for such steady statewalking, with the simplifying assumption that humans walk with an inverted pendulum gait [4, 1, 2, 3]. Whena person walks at continuously varying speeds, as in our experiments, the step-to-step transition will includea change in both the magnitude and direction of the COM velocity. Here, we derive an expression for thework done in the step-to-step transition when walking at changing (non-constant) speeds, thereby generalizingprevious work [4, 1, 2, 3]. We allow that the length of the leg to change during the step-to-step transition(unequal ✓

before

and ✓

after

) while having constant leg length during the stance phase.

Push-o↵ before heel-strike. Figure S1 describes the transition from one inverted pendulum to the nextusing push-o↵ and heel-strike impulses; panels a-c describe situations in which the push-o↵ happens entirelybefore heel-strike, which we consider first. In particular, Figure S1a-b shows a finite reduction in speed beingaccomplished during the step-to-step transition, with push-o↵ before heel-strike. We focus on Figure S1b in the

following derivation. Here, vector�!OA with magnitude OA = V

before

and making angle ✓before

with horizontal, is

the body velocity just before push-o↵ at the end of one inverted pendulum phase. Vector�!OC, with magnitude

OC = V

after

and making angle ✓

after

, is the body velocity just after heel-strike at the beginning of the nextinverted pendulum phase.

A push-o↵ impulse is applied along the trailing leg to change velocity�!OA to

�!OB, along

�!AB. Then, a heel-strike

impulse is applied along the leading leg to change velocity�!OB to

�!OC, along

�!BC. The push-o↵ positive work

W

pos

is the kinetic energy change from�!OA and

�!OB given by 1

2

mOB2 � 1

2

mOA2 and the heel-strike negative

work is the kinetic energy change from�!OB to

�!OC given by W

neg

given by 1

2

mOC2� 1

2

mOB2, which simplify to:

W

pos

=1

2m (AB)2 and W

neg

=1

2m (BC)2 respectively. (1)

As opposed to the steady walking situation [4, 1, 2, 3], when changing speeds, the step-to-step push-o↵ positivework W

pos

and the step-to-step heel-strike negative work W

neg

will be unequal.

First, we note that angle that in triangle EBD, \DEB = ⇡/2 � ✓

after

, \BDE = ⇡/2 � ✓

before

, and \EBD =✓

after

+ ✓

before

. We use the geometric relations that AB = AD-BD, BC = EB+CE, AD = OA tan ✓before

, CE =

1

Ο

A

B

C

D

velocity just after heel-strike

Velocity change due topush-off impulse along trailing leg

Velocity change due to heel-strike impulse along leading leg

velocity just before push-off

a) Hodograph for unsteady walking -- speed decrease (push-off before heel-strike)

E

θbθ

a

A

Step to step transition with push-off before heel-strike (details of panel-a)

d) Step to step transition with heel-strike before push-off

θa

θa

AAθ

b

θb

Ο

CQ

G

F

P

θa

θa

θb

θb

Ο

A

B

C

D E

c) Speeding upb) Slowing down

Slowing down

θa

Aθ

b

θb

θa

Figure S1: Step-to-step transition to change speed. a) The walking motion is assumed to be inverted pendulum-like with the transitions from one inverted pendulum step to the next accomplished using push-o↵ and heel-strikeimpulses. Overlaid is the ‘hodograph’ (a depiction of velocity changes) during the step-to-step transition, when push-o↵happens entirely before heel-strike. b) Details of the velocity changes during step-to-step transition, with push-o↵ beforeheel-strike and slowing down. c) Analogous to panel-c, except the walking speeds up during the transition. d) Velocitychanges and impulses when the heel-strike precedes push-o↵ entirely.

2

OC tan ✓after

,

ED =OA

cos ✓before

� OC

cos ✓after

,

BD

ED=

cos ✓after

sin (✓after

+ ✓

before

), and

EB

ED=

cos ✓before

sin (✓after

+ ✓

before

)(2)

in Eq. 1 to obtain:

W

pos

=1

2m

V

before

tan ✓before

� cos ✓after

sin (✓after

+ ✓

before

)

✓V

before

cos ✓before

� V

after

cos ✓after

◆�2

and (3)

W

neg

=1

2m

V

after

tan ✓after

+cos ✓

before

sin (✓after

+ ✓

before

)

✓V

before

cos ✓before

� V

after

cos ✓after

◆�2

, (4)

with OA = V

before

and OC = V

after

. We obtain exactly the same expressions for speeding up during step-to-steptransition as shown in figure S1c (though the figure looks superficially di↵erent, a couple of negative signs cancel,thereby giving the same answers). Note that the main qualitative di↵erence between panels b and c of figure

S1 is that the velocity�!OB is above or below the horizontal.

An implicit assumption in the above derivation is that the push-o↵ and heel-strike impulses do not requiretensional leg forces (the leg cannot pull on the ground) and are in the directions shown. This requirement issatisfied when the ratio of the two speeds obey the following condition:

cos (✓after

+ ✓

before

) V

after

V

before

1

cos (✓after

+ ✓

before

).

When V

after

/V

before

= cos (✓after

+ ✓

before

), the necessary push-o↵ impulse becomes zero and when V

before

/V

after

=cos (✓

after

+ ✓

before

), the necessary heel-strike impulse becomes zero.

Heel-strike before push-o↵. When the heel-strike impulse precedes the push-o↵ impulse, the negative workW

neg

by the heel-strike and the positive work by the push-o↵ W

pos

are given by the respective kinetic energychanges:

W

neg

=1

2m (OA2 �OG2) and W

pos

=1

2m (OC2 �OG2), (5)

where OA = V

before

, OC = V

after

, OG2 = OQ2 + QG2, OQ = OAcos (✓after

+ ✓

before

), angle \QGC = ✓

after

+✓

before

, QG = CGcos (✓after

+ ✓

before

), and CG = AB in figure S1b. i.e.,

CG = AB = V

before

tan ✓before

� cos ✓after

sin (✓after

+ ✓

before

)

✓V

before

cos ✓before

� V

after

cos ✓after

◆

using the derivation for push-o↵ before heel-strike; the final expression is easily obtained by substituting theserelations into equation 5. As has been shown before for steady state walking [4, 2], we find that a transitionwith heel-strike before push-o↵ requires more metabolic cost than push-o↵ before heel-strike even when changingspeeds with the simplifying assumption that ✓

before

= ✓

after

.

S2 Optimal multi-step inverted pendulum gaits satisfying experi-mental protocol

We use a metabolic cost that is a sum of two terms: (1) the step-to-step transition cost and (2) a swing cost.

Step-to-step transition cost. The step-to-step transition cost E

s2s

is a weighted sum of the push-o↵ workW

pos

and heel-strike work W

neg

, summed over all steps, and scaled by the approximate e�ciencies of positiveand negative work respectively:

E

s2s

=X

steps

⌘

�1

pos

W

pos

+ ⌘

�1

neg

W

neg

3

using the equations 3 and 4 for the work expressions.

Swing cost. The step-to-step transition cost does not account for the work required to swing the legs. We usea simple model of the metabolic cost required to swing the legs forward [5] equal to

E

swing

= µD

stride

/T

3

step

where T

step

is the step duration, Dstride

is the distance travelled by the swing foot during the step (distancebetween previous and successive foot contact points), and the proportionality constant µ = 0.06 when all otherquantities are non-dimensional, chosen so as to best fit steady walking metabolic costs [5].

Representing a multi-step inverted pendulum walking motion. Each step of the inverted pendulumwalking motion was represented using five variables: the initial leg angle ✓

0

, the initial (post-heel-strike) angularvelocity ✓

0

, step duration T

step

, the constant leg-length over the step `

leg

, and the foot-ground contact positionin the forward direction x

contact

. Nonlinear equality constraints make sure that the body position at the end ofone step is equal to that at the beginning of the next step.

Numerical optimization. We used numerical optimization to determine the multi-step walking motion thatsatisfies the oscillating-speed experimental protocol and minimizes the model metabolic cost as described above.The biped model alternates between a higher speed v

avg

+L/T

fwd

and a lower speed v

avg

�L/T

bck

, each lastinga few steps, so that the net average speed is v

avg

, and the forward and backward movement in lab frame haveperiods equal to T

fwd

and T

bck

. The number of steps for the forward and backward movements are chosen basedon the number of steady walking steps in the durations T

fwd

and T

bck

. Other constraints included an upperbound on the leg length (< `

max

) and a periodicity constraint on the body height over one period of back andforth walking. The optimization problem was solved in MATLAB using the optimization software SNOPT, whichemploys the sequential quadratic programming technique [6]. At each average speed, we also computed theoptimal constant-speed inverted pendulum walking gait (repeating calculations in [3, 7]), so as to subtract fromthe optimal oscillating-speed walking cost.

Leg force cost. We repeated the calculations above with a cost for leg force, proportional to the integral ofthe leg force, with a proportionality constant as in [8]. We found that this leg force cost did not change ouroverall predictions for the di↵erence between oscillating-speed and constant-speed metabolic costs, as both thesecosts increase by the almost same amount due to the leg force cost. This result can be explained intuitively asfollows: because the legs make relatively small angle with the vertical, as explained in [7], the average leg forceis approximately equal to the average vertical force, which has to be equal to the total body weight for periodicmotion – be it constant-speed walking or oscillating-speed walking.

S3 Daily energy budget for starting and stopping

Subjects in [9] performed a majority of the walking over a day in short bouts; they walked in 43914 boutsand took a total of 1717730 steps. Assuming a typical step length of 0.6 m [10], the subjects walked a totaldistance of 1030638 m. Assuming the subjects walked the whole distance at a constant speed of 1.4 m/s, wecan predict a total constant-speed energy expenditure to be 2262600 J, based on a parabolic relationship givenby E

steady

= a + bv

2 with a = 2.22 W/kg and b = 1.15 W/kg/(ms�1)2 [11]. But, such a cost would ignorethe cost of accelerating from and to rest at the start and the end of the bout. We can approximate this dailyunsteady cost for the 43914 bouts to be 137380 J by extrapolating our results from the kinetic energy-basedmodel with the unsteady cost for one bout given by, �ke(⌘�1

pos

+ ⌘

�1

neg

)v2/2, where �ke = 0.67. The ratio of thecost of changing speed to cost of walking is thus found to be 0.06, in other words, the unsteady cost of walkingper day is 6% of the steady cost on average. Using the 95% C.I. for �ke gives us 4-8% as reported in the mainmanuscript. This approximate calculation shows that the cost of changing speeds is a significant fraction of theenergy humans consume in daily walking.

4

S4 Comparison with a previous study

Our oscillating speed protocols had speed fluctuations between ±0.13 and ±0.27 m/s. As noted in the mainmanuscript, one previous article [12] attempted to measure the cost of changing speeds, with greater speedfluctuations (±0.15 to ±0.56 m/s) and higher kinetic energy fluctuations per unit time than our study. The rateof kinetic energy fluctuations for both experiments can be compared by comparing v�v/T , for a fluctuationbetween speeds v��v and v+�v in T seconds. In [12] v�v/T ranges between to 0.0226 and 0.127 m2s�3 andin our protocol, v�v/T ranges between 0.026 and 0.1108 m2s�3. Thus, the kinetic energy fluctuation rates weresimilar in the two studies. Nevertheless, the study [12] found significant increase only for their highest speedfluctuation but not lower. As noted in the main manuscript, this study [12] required walking on oscillating-speedtreadmill belts or controlling step durations in overground walking (derived from oscillating-speed treadmill).An oscillating-speed treadmill, being a non-inertial reference frame, can perform mechanical work on the subject,and is not mechanically equivalent to oscillating-speed walking overground. Further, controlling step durations[12] will produce incorrect speed fluctuations that do not obey the speed-step-duration relation for directlycontrolling walking speed, as established by Bertram and Ruina [13] in the case of steady walking.

References

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