Energy optimality predicts curvilinear locomotion
aa r X i v : . [ q - b i o . N C ] A ug Energy optimality predicts curvilinear locomotion
Geoffrey L. Brown, , , † Nidhi Seethapathi, , , † Manoj Srinivasan , ∗ Mechanical and Aerospace Engineering, the Ohio State University, Columbus, OH 43210 Feinberg School of Medicine, Northwestern University, Chicago, IL 60611 Department of Bioengineering, University of Pennsylvania, Philadelphia, PA 19104 † Equal co-authorship. ∗ To whom correspondence should be addressed; E-mail: [email protected].
Everyday human locomotion requires changing directions and turning. How-ever, while straight-line walking behavior is approximately explained by en-ergy minimization, we do not yet have a unified theoretical account of non-straight-line (i.e., curvilinear) locomotion, despite its ecological importance.Here, we show that many non-straight-line walking phenomena are predictedby including an energy cost for turning. We quantified the cost of turning inhumans, showing that the metabolic rate of walking increases with decreasingradius for fixed speed. We then used this metabolic cost to mathematicallypredict energy-optimal movement patterns for five tasks of varying complex-ity: walking in circles, turning in place, walking through an angled corridor,walking freely from point to point while having to turn, and walking throughdoors while maneuvering around obstacles. In these tasks, humans moved atspeeds and paths approximately predicted by energy optima. Thus, we pro-vide a unified theoretical account that predicts diverse curvilinear locomotorphenomena. ne sentence summary. Using new measurements of human energy expenditure while turningand using optimization-based theoretical models, we predict how and explain why people movethe way they do in many ecological tasks involving walking, turning, and path planning.
Introduction
Most real-world walking tasks requires changing direction and turning. In one previous studythat tracked walking behavior over many days, 35-45% of steps within a home or office environ-ment required turns ( ). Despite the importance of turning in ecological locomotion, we do nothave a coherent theoretical account of the various human behavior observed in such walking.Human subject experiments ( ) and mathematical models (
5, 8–15 ) have suggested that en-ergy optimality explains many aspects of straight line locomotion, at least approximately. How-ever, we do not know if such energy optimality generalizes to walking with turning. Here, weobtain a better understanding of walking with turning, first quantifying its increased energeticdemands and then showing that accounting for these increased energetic demands correctlypredicts a variety of curvilinear human walking behavior.Over the years, researchers have measured human behavior in a few different tasks involvingturning, for instance, walking through variously angled corridors ( ), walking from point-to-point while having to turn ( ), and walking through doors and avoiding obstacles (
18, 19 ).Previous theoretical attempts at explaining human behaviors in these tasks often posited thathumans minimized smoothness-related terms (e.g., acceleration, jerk, the derivative of accel-eration). However, these theoretical attempts were not physiologically-based, required fittingmodel parameters to behavioral data, were generally fit to only one experiment, and/or did notgeneralize to multiple experiments ( ). As we argue later on, these models could not simul-taneously explain the paths and speeds observed in human curvilinear walking, often predictingzero speeds for simple walking tasks. In contrast, we produce a theoretical account that does2ot have these limitations and is broadly predictive.Here, we perform human subject experiments, first quantifying the metabolic cost of hu-mans walking in circles and showing that walking with turning costs substantially more thanwalking in a straight line. We then use this empirically-derived metabolic cost model with anoptimization-based framework to make a number of behavioral predictions about humans walk-ing in tasks of different complexity. We compared our predictions to five different experiments,each containing a qualitatively different walking or turning task. These five experiments con-sist of two new behavioral experiments we performed here and data from three prior studies.Specifically, we predict that humans would walk slower when turning in smaller circles or withgreater curvature, which we compare with our own behavioral experiments, correctly predictingthe lowered walking speeds. We show that the speed at which humans turn in place is approxi-mately predicted by minimizing the cost of turning, again comparing with our own experiments.Finally, minimizing the same metabolic model, we predict more complex walking behavior ob-served in three previous studies: walking freely from point to point ( ), walking through doorsand avoiding obstacles (
18, 19 ), and walking and turning along an angled corridor ( ). Weshow that energy optimality explains many qualitative and quantitative features of human walk-ing including not taking sharp turns, path shapes adopted while walking with turning, and speedreductions during turns, as observed in these prior studies ( ), but heretofore not predictedby a single model. Results
Turning increases the energy cost of walking substantially.
We measured the metabolicenergy expenditure of seventeen human subjects while walking with turning. To measure thecost of turning, we instructed the subjects to walk in circles of different radii and at differenttangential speeds along the circle. We provided feedback to them ensure that they were able3o perform the required task (see Figure 1A and the Methods section). Each subject performedat least 16 trials of 4 radii and at least 4 speeds. We measured metabolic energy expenditureusing indirect calorimetry, that is, by tracking respiratory oxygen and carbon dioxide flux ( ).Figure 1B shows the resulting mass-normalized metabolic rate, that is, metabolic energy perunit time per unit subject mass ˙ E , as a function of prescribed speed v and path radius R . Thesemeasurements show that for a given prescribed walking speed, the metabolic energy expenditurewas higher for smaller radii, or equivalently, for higher path curvature (Figure 1). For instance,directly comparing the measured metabolic rates at equal speeds but different radii, we findthat walking at 1 m radius was more expensive than walking at 4 m radius by 0.59 W/kg onaverage ( p = 7 × − using a right-tailed t-test), 1 m radius was more expensive than 2 mradius by 0.46 W/kg on average ( p = 1 . × − ), and 2 m radius was more expensive than4 m radius by 0.15 W/kg on average ( p = 0 . ). These differences constitute large energypenalties relative to the resting metabolic rate: 40%, 20%, and 10% respectively. Because thesedifferences are computed at matched walking speeds, the differences are the same whether weconsider metabolic expenditure per unit time or per unit distance. Walking metabolic rate is well-modeled by a quadratic function of linear and angularspeeds.
The total metabolic rate was described well by the following quadratic function ofthe linear speed v (tangent to the walking path) and the angular speed ω = v/R as follows(Figure 1B): ˙ E = α + α v + α v R = α + α v + α ω , (1)with α = 2 . ± . W/kg, α = 1 . ± . W/(kg.ms − ), α = 0 . ± . W/(kg.rad.s − ) , giving the metabolic rate ˙ E in W/kg (normalized by body mass), where v isin ms − , radius R is in meters, and ω is in rad.s − . The p values for the three coefficients allsatisfied p < − , compared to a null constant model of zero coefficients. Equation 1 explains4 M e t a b o l i c r a t e p e r u n i t m a ss ( W / k g ) R (m) Walking speed v (m/s)Speed is constrainedby constraining lap durationMetabolic ratemeasurementSubjects walk in circles of different radii (curvature) and speedsR = 1, 2, 3, 4 mHumansubject( A ) Metabolic experiments with humans to measure cost of turning How much energy does it take to turn while walking? ( B ) Metabolic energy cost of walking while turningCircles drawn on the floor Prescribed R-v combinations Figure 1:
Energy cost of turning from humans walking in circles. ( A ) We estimated the metabolic rateas a function of walking speed and turning radius by having subjects walk in circles of a few differentradii and at a few different speeds at each radius. Speeds were constrained by having them complete lapsat prescribed durations. Metabolic rate was estimated using respiratory gas analysis. ( B ) The metabolicrate data per unit body mass ( ˙ E ) is higher for higher speeds and lower radii. The prescribed speeds andradii ( v, R ) at which the data was collected is shown as blue dots on the horizontal plane ( ˙ E = 0 plane);raw metabolic data points from four representative subjects are shown as green dots (see SI for all data).The wireframe surface shows the best-fit model ˙ E = α + α v + α ω , capturing the nonlinear increasewith both linear velocity v and angular velocity ω ; the model captures 88.1% of the data variance. R value, not to be confused with radius-squared). Metabolic model captures straight-line walking as a less expensive special case.
Settingangular speed ω to zero or radius R to infinity in equation 1 gives ˙ E for straight line walking: α + α v . Thus, the quadratic expression (Eq. 1) generalizes the classic quadratic expressionused for the metabolic rate of walking in a straight-line, namely, α + α v ( ). Previous studiesof overground or treadmill straight line walking (
2, 4 ) have estimated α ≈ − . W/kg and α ≈ . − . W/(kg.ms − ), and our estimates are squarely in this same range. Because the5oefficient α > with p = 10 − , the model (Eq. 1) confirms that the estimated metabolic rateto be higher for lower radii R for a given tangential speed v . This radius dependence implies,for instance, that at speed v = 1 . m/s, reducing the radius R from infinity to 1 m inducesan additional cost ( α v /R ) of about 43% of the total straight-line walking metabolic rate( α + α v ). This turning cost is about 60% of the net straight-line walking metabolic rate, thatis, over and above the resting metabolic rate ( α + α v − e rest ). In the rest of this article, weshow that this additional cost term for turning has a variety of behavioral implications. We thencompare these theoretical behavioral predictions with experiment. Prediction: optimal walking speeds are lower for smaller circles.
When walking a long-enough distance in a straight line in the absence of time constraints, humans usually walk closeto the speed that minimizes the total metabolic cost per unit distance E ′ = ˙ E/v (see (
2, 4, 5, 21 )and see ( ) for generalization to shorter distances). This energy optimal speed is sometimescalled the maximum range speed ( ) as it also maximizes the straight-line distance traveledwith a fixed energy budget. Analogously, for walking in circles (Figure 2A), we hypothesizethat humans will use speeds that minimize the total cost per unit distance E ′ = ˙ E/v = α /v +( α + α /R ) v . This cost per unit distance is minimized when the slope ∂E ′ /∂v = 0 , thatis, at the speed v opt = q α / ( α + α /R ) (see Figure 2A). This optimal speed v opt is lowerfor lower radius R , thus predicting that humans would prefer to walk slower in smaller circles(Figure 2B). Figures 2C-D provide intuition for how the turning cost lowers the optimal speedfor circle-walking. Note that the mass-normalized cost per unit distance E ′ is a scaled versionof the cost of transport ( ); the cost of transport is a non-dimensional quantity given by E ′ /g . Experiment: Human preferred speeds are lower in smaller circles as predicted by energyoptimality.
We asked people to walk naturally on circular paths of four different radii and ina straight line (Figure 2A). As predicted by energy optimality, we found that humans preferred6 E n e r g y p e r u n i t d i s t a n c e ( J / k g / m ) ( E ) Optimal per-distance cost is higher for lower radii1 2 3 4Radius of circle (m) Prediction vs Experiment: Humans walk slower in smaller circles, as predicted by energy optimality B ) Preferred walking speeds vs energy optimal speeds W i t h i n % o f op t i m a l E W i t h i n % o f op t i m a l EE n e r g y op t i m a l s p ee d Predictions:
StraightLine
Experiment:Human preferred walking speeds E n e r g y c o s t p e r u n i t d i s t a n c e ( J / k g / m ) ( C ) Energy cost landscape vs speed & radiusLocus of minimaat every radius( A ) Subjects walk in circles of different radii (curvature) at whatever speeds feel naturalHumansubjectwalkingCircles drawn on the floor R = 2 mspeed (m/s)2 0 1 2 0 1 2 0 1 2468 E n e r g y p e r d i s t a n c e speed (m/s)2468speed (m/s)012Straight line walking cost Turning cost+ = Walking with turningR = inf R = 1 m R = 1 mR = InfR = 2 mR = Inf( D ) How a turning cost shifts the optimal speed to lower valuesHigher for smaller radii Lowerspeed for lower R Straight-line walkingWalking in circles Figure 2:
Prediction vs behavior: Preferred walking in circles. ( A ) To test behavioral predictionsof energy optimal walking, we asked subjects to walk on circles of different radii at whatever speedsthey found natural. ( B ) Human preferred walking speeds and model-predicted optimal walking speeds.Humans walk slower for smaller radii, as also predicted by energy optimality. The yellow box-plot showshuman preferred walking speed along with individual data points (box indicates 25 th , median, and 75 th percentile and whiskers indicate the range). The solid dark blue line is the optimal tangential speed v opt for every radius. Also shown are two bands denoting speeds for which the metabolic cost per distanceis within 1% (lighter blue) and 2% (darker blue) of the optimum cost. Most humans seem to be within2% of their energy optima. ( C ) Metabolic cost per unit distance per unit mass. Minimizing this functionat each radius produces model predictions in panel-b, identical to the dark blue line panel-c. ( D ) Theturning cost per unit distance is linear in velocity and modifies the walking cost in a manner that theoptimal speed is lower for smaller radii or higher curvatures. ( E ) The optimal metabolic cost per unitdistance as a function of the radius. v opt = q α /α = 1 . m/s, which agreeswith typical human preferred walking speeds in previous studies (
5, 21, 22 ) as well as the trialshere.
Corollary: A straight line path is optimal if there are no other constraints or obstacles.
Conventional wisdom dictates that (in the absence of other constraints), a straight line pathwould be optimal to travel a given distance. This conventional wisdom relies on implicit as-sumptions about the metabolic energy landscape unavailable before our measurements. Assumethat the distance to be traveled is long-enough ( ) so that we minimize the metabolic cost perunit distance: E ′ = ˙ E/v = α /v + ( α + α /R ) v , ignoring any small initial or final transientcosts. Then, at any speed, including the optimal speed, this cost is minimized when the turningradius R goes to infinity: that is, walking in a straight line is optimal. This result is reflectedin Figure 2E, which shows that the minimum cost per unit distance at any given turning radius,and the minimum is achieved when radius R goes to infinity. The optimality of the straight linepath are not generally true in the presence of obstacles or constraints such as those on initialand final body orientation, as considered later in this results section. Prediction: minimizing energy cost of turning-in-place predicts an optimal turning rate.
Humans often need to turn in place while standing, to re-orient their body — to face a newdirection. Turning in place or spinning in place (Figure 3A) is a special case of walking incircles with radius R → and speed v → , while the angular velocity ω = v/R remains non-8ero. Extrapolating using these limits, we obtain the metabolic rate of turning-in-place to be: ˙ E = α + α ω . The metabolic cost of turning in place per unit angle (analogous to metaboliccost per unit distance) is ˙ E/ω = α /ω + α ω . This cost per unit angle is optimized by steadyoptimal turning speed ω opt = q α /α = 1.46 rad/s = 83.6 degrees/s (Figure 3B-C). M e t a b o l i c r a t e ( W / k g ) angular velocity (rad/s)04812 E n e r g y p e r u n i t a n g l e ( J / k g / r a d ) Energy optimal turning speedangular velocity (rad/s)( B ) Metabolic cost of turning in place: per time and per angle ( C ) Energy optimal speed and human preferred speed( A ) Turning in place total turn angle (degrees)Within 5% of optimal EWithin 1% of optimal E Model prediction:
Energy optimal speed
Experiment:
Human preferred walking speeds012 a n g u l a r s p ee d ( r a d / s ) α Prediction vs. Experiment: Turning in place
Figure 3:
Prediction vs behavior: Turning in place. ( A ) To further test behavioral predictions of energyoptimality, we asked subjects to turn by a given angle α , starting and ending at rest. ( B ) Metabolic rateand the cost per unit turning angle, obtained by extrapolating the model to turning-in-place. The bluelines and bands shown denote optimal turning speeds and the set of speeds within 1% or 5% of optimalenergy cost. ( C ) Human preferred turning speeds (yellow box plot and individual data points) largelyoverlap with the turning speeds within 5% of the optimal cost. Experiment: humans turn in place at close to the energy optimal turning rate.
We per-formed behavioral experiments in which humans turned in place by a fixed turn angle α (Figure3A), starting and ending at rest. The average human turning speeds for large turns of 270 de-grees and 360 degrees largely overlap with each other and almost entirely overlap with the setof steady turning speeds that are within 5% of the optimal turning cost. Two ways of generalizing to complex paths: face the movement direction or not.
Wenow generalize the metabolic cost of walking in circles to walking on arbitrary paths, but first,we discuss what assumptions such generalizations make. When walking, we can conceptually9istinguish between the direction in which our body moves (velocity direction, angle β , Figure4A) and the direction in which the body faces (body torso orientation, angle θ ). As a simpleexample, in normal straight-line walking, these two directions are aligned ( θ = β ), but in“sideways walking” ( ), the body moves perpendicular to how the body faces (Figure 4B). Thus,it is not essential that we walk in a manner that we always face the movement direction. So, weconsider two ways of walking: walking while not always facing the movement direction (Figure4C) and walking while always facing the movement direction (Figure 4D). We call these kindsof walking “holonomic” and “non-holonomic” respectively, borrowing this terminology fromclassical mechanics, control theory, and other prior work (
23, 24 ). The term ‘non-holonomic’simply implies that the system obeys a velocity constraint – here, the constraint is that thebody velocity direction is always along body orientation. Holonomic walking has no suchvelocity constraint and thus non-holonomic walking is a special case of holonomic walking. Wegeneralize the metabolic cost model of equation 1 to both these types of walking (see Methods).The rest of this article uses this cost model to make predictions about how people walk in morecomplex situations involving turning. Because holonomic walking is more general, we use thistype of walking to make predictions in the rest of this main manuscript, but we also show resultsfrom non-holonomic walking in supplemental figures.
Prediction vs. experiment: Going from A to B with constraints on initial and final direc-tion.
Mombaur et al ( ) performed human subject trials in which the human started from restat point A and ended at rest at point B, starting and ending with different body orientations (Fig-ure 5A). The required body orientations were provided as arrows drawn on the ground. Subjectswere not constrained in any other way, say by obstacles or time limits. Having different requiredbody orientations at A and B requires the subjects to turn. For seven different end-point andbody orientation combinations, we computed the metabolically optimal turning trajectory with10 D ) “Non-holonomic” walkingWalking with body alwaysfacing movement direction( C ) “Holonomic” walkingWalking with body not always facing movement direction( B ) “Holonomic” walkingallows both forward (normal)and sideways walkingSideways walkingForward (normal) walkingFrontBackLeft Right Direction in whichperson is facing θβHuman bodytop viewCenter of massvelocity directionCenter of masstrajectory (top view)Tangent to trajectory( A ) Defining body orientation and velocity direction (top view) (arrows always aligned)(arrows not always aligned) Two kinds of walking: always facing or not always facing movement direction
Figure 4:
Two kinds of walking: face where you are going or not. ( A ) Visual representation of thebody from a top view, introducing different notations for the direction in which the body is moving (redarrow) and the body orientation (blue arrow). ( B ) While forward walking has body orientation alignedwith movement direction, it is possible to walk sideways, so that the body orientation is perpendicular tothe movement direction. ( C ) Walking in a manner that that the body orientation need not be aligned withthe movement direction (“holonomic”); that is, the blue and red arrows need not be aligned. ( D ) Walkingin a manner that the body orientation is the same as the movement direction (“non-holonomic”); that is,the blue and red arrows are aligned. x vs t and y vs t ) and body orientation ( θ vs t ) without any fitting parameters (Figure 5B-D). In theseresulting optimal paths as well as the human paths, humans walk in a manner that the body ori-entation is not identical to movement direction. Constraining the body orientation to be alwaysaligned with the movement direction (non-holonomic walking) produces less good predictionsof both body position and orientation (Supplementary figure S1). Prediction vs. experiment: Optimal path planning between two doorways.
In another setof previous studies (
18, 23, 25 ), researchers instructed human subjects to walk through two setsof doors A and B facing in different directions and separated by a few meters (Figure 6A). Thesubjects started 2 m before A and ended 2 m beyond B. In all these trials, as in ( ), humanschose smooth paths that gradually turn rather than, say, achieve the same task using sharpturns or too many direction changes (Figure 6B-C). Again, we used trajectory optimization tocompute the energy-optimal way of performing this task. The resulting optimal trajectories aresimilar to the human trajectories in data (Figure 6), which are within 2% of the optimal cost(Supplementary Figure S3). The predictions from the holonomic and non-holonomic modelsare almost the same, with the non-holonomic model taking a slightly wider turn near the door.For these longer distance bouts (compared to those in the previous paragraph ( )) with widerturns, even when the walker is not constrained to be non-holonomic, it is energy optimal to benearly non-holonomic – that is, walk in a manner that the body nearly faces movement direction,as also observed in experiment ( ).An important constraint for the optimal path calculation here, in contrast to the compari-12 ( r a d ) -404 y ( m ) -4 0 4 -4 0 4 -4 0 4 -4 0 4 -4 0 4 -4 0 4 -4 0 4x (m) x (m) x (m) x (m) x (m) x (m) x (m) -303 x ( m ) -303-303 y ( m ) A AB B
Prediction vs. Experiment: Walking from A to B, starting and ending with different body orientations
Energy optimality-based holonomic model predicts data from Mombaur et al
Initial body orientation Final body orientation Energy-optimality-based model prediction +/- 5% of optimal cost Human data, Mombar et al
Mombaur et al best fit
Top-view Top-viewTarget 1 Target 2 Target 3 Target 4 Target 5 Target 6 Target 70.4 m walk( A )( B )( C )( D ) Figure 5:
Prediction vs behavior: Path planning, starting and ending at rest. ( A ) Mombaur et al ( )asked subjects to walk short distances, starting at rest at point A and ending at rest at point B. The subjectshad to start facing one direction (light green arrow) and end facing possibly another direction (orangearrow). ( B, C, D ) The body position ( x, y ) and body orientation θ as a function of time. Non-holonomicmodel predictions are solid dark blue with a light blue band indicating trajectories withing 5% of theoptimum cost; experimental data are dashed dark green, and the best-fit model in Mombaur et al ( ) isindicated in dashed red line. We see that our energy optimization-based model predictions mostly passthrough the center of the experimental data. Just for targets 3 and 7, subjects started and ended withslightly different body orientations than prescribed, so these were used in the optimizations presented.See Supplementary Figures S1 and S2 for variants of this figure with alternate assumptions. ), is that the body path does not intersect with the doors and has aminimum clearance from the doors. The clearance used is consistent with typical human dimen-sions ( ) and is also consistent with behavioral data ( ). In the absence of such a clearanceconstraint, the optimal path ignores the walls of the doors and shows a sharper turn near the end-point B. Thus, it is important to note that explaining human behavior may require consideringconstraints such as avoiding obstacles in addition to minimizing energy-like cost functions. Prediction vs. experiment: Humans slow down turning a corner in an angled corridor.
A common task that most walking humans undertake every day is turning a corner in an angledcorridor (Figure 6D). A previous study by Dias et al ( ) had subjects walk around angled cor-ridors and measured the walking speeds during the turn. Again, using trajectory optimization,we computed the metabolically optimal path in such angled corridors, specifically computingthe walking speed during the turn. We find that the optimal walking speed is lower during theturn and that this turning speed is lower for turning by a greater angle (Figure 6E-F). The exper-imentally observed human speeds from ( ) are almost identical to the model-predicted turningspeed; specifically, the distribution of human turning speeds overlaps with the model-predictedband of speeds within 2% of the optimal cost (Figure 6F). Speed reductions during turns weresimilarly observed by Sreenivasa et al ( ), who performed turning by different turn-angles ina cyclical task that alternated between turning and straight line walking. In all these turningtasks, as predicted by the model, humans do not usually use ‘sharp turns’ when smooth turnsare possible. Prediction vs. experiment: To go sideways, walk sideways or turn and walk forward.
Humans do not usually walk sideways. In a previous article, Handford and Srinivasan ( )showed that sideways walking costs three times more energy than walking forward at theirrespective optimal speeds. Now, consider a situation in which someone wants to go from A14o B, but is initially (and finally) facing perpendicular to the line AB (Supplementary FigureS4). That is, they want to move sideways. For instance, they are working at a kitchen counterand want to move sideways. Should they walk sideways or turn by 90 degrees and walk facingforward? To make a prediction, we can compare the cost of walking sideways and the costof turning and walking forward and turning again for distance D . Using this comparison, wepredict that humans should walk sideways for less than a critical distance D crit = 0 . m; forlarger distances, walking sideways is more expensive than turning by 90 degrees and walkingfacing forward. Indeed, target 4 in Figure 5 asks subjects to travel sideways by 0.4 m, startingand ending facing forward ( ). Subjects, as predicted by the model, did not turn and walkforward, but instead stepped sideways while mostly facing forward. Minimizing energy is more parsimonious and more broadly predictive than other opti-mality hypotheses.
We have predicted a wide variety of experimental data, with no fittingparameters, directly by minimizing an empirically based metabolic model of walking. We nowbriefly compare the energy-cost-based hypothesis with some other hypotheses that have previ-ously been explored. Smooth body trajectories can be predicted by cost functions that maximizesmoothness, such as acceleration, ‘jerk’ (derivative of acceleration) or ‘snap’ (second deriva-tive of acceleration). Such cost functions have been most successful in arm reaching ( ), butthey have also been employed to predict walking trajectories (
17, 18 ). However, such smooth-ness maximizing cost functions, taken seriously, produces two un-ecological predictions. First,they cannot predict the velocity at which people move – more specifically, the optimal wayto minimize jerk or snap is to perform a task with infinitesimal speed over an arbitrarily longperiod of time. Thus, such smoothness maximizing cost functions require a constraint on theaverage velocity of the task to produce meaningful results. In contrast, our metabolic energyapproach naturally produces an optimal velocity for any task, without having to constrain it.15 second consequence of the jerk minimization hypothesis, even with a velocity constraint, isthat it produces ‘scale-invariant solutions’. That is, for the tasks in Figure 6A, it produces pathsthat ‘look the same’ for a 10 meter walk versus a 10 km walk, producing many-km long pathexcursions that humans would never use. Arachavaleta et al ( ) used an objective functionequal to the integral of ( v + ω ) over the path. Again, minimizing this objective without avelocity constraint produces the result that humans should move at infinitesimal velocity. Soin their calculations, Arachavaleta et al ( ) simply constrained the instantaneous walking ve-locity to be constant, without allowing it to change through the path (as it does in humans).Thus, these hypotheses are not truly predictive – they require further assumptions about humanbehavior which need not be made with the more parsimonious energetics-based approach. Fi-nally, Mombaur et al ( ) used a model-fitting procedure to select an objective function that bestpredicts human walking trajectories in their experiments: their objective function terms relatedto linear and angular velocity acceleration, work, jerk, and task time duration. However, theirbest-fit cost function, because it is dominated by linear and angular acceleration terms,s cannotexplain observed speeds for walking in circles, turning in place, or even walking in a straightline for longer distances. Thus, each of the cost functions considered previously by researcherscould be considered as ‘overfit’ to one or two experimental conditions and cannot predict all thediverse phenomena predicted by our model with no task-specific fitting. Discussion
Here, we first measured the energetics of humans walking in circles and showed that the energycost has a substantial dependence on path curvature. That is, turning increases the cost of walk-ing and this turning cost increases quadratically with the turning angular velocity. We then usedthe experimentally-derived metabolic energy model to predict energy optimal walking behav-ior in various locomotor scenarios, explaining many experimentally-measured and real world16 D ) Model predictions from energy optimality−2 0 2048( C ) Potential path possibilities not chosen by humans and not optimal( B ) Human paths in experiment Data from Arachavaleta et alDoors to walk throughIntial directionFinal directionPathway to walk through −2 0 2048Avoid sharpturnsAvoid ramblingpaths−2 0 2048 (A ) Task: Go via A to B, changing direction, and avoiding obstaclesDoors to walk throughFinal directionDoors to walk through (Top-view) (Top-view)B BA ABBBB Prediction vs. Experiment: Going from A to B, walking through doors and avoiding obstacles
Energy optimality-based model predicts data from Arachavaleta et al
Othertargetpoints Holonomic -2 0 2048 Non-holonomic -5 0 5 -5 0 5 -5 0 5-5 0 5 -5 0 5-505 -505 I n i t i a l p o i n t & d i r e c t i on C o rr i d o r ( . m w i d e ) Wall F i n a l p o i n t & d i r e c t i on Turn angle β (D ) Task: Walking and turning in an angled corridor (E ) Model-derived paths for various turn angles t u r n i n g s p ee d s ( m / s ) W i t h i n % o f op t i m a l E W i t h i n % o f op t i m a l E M od e l - d e r i v e d op t i m a l s p ee d H u m a n t u r n i ngd a t a
45 deg 135 deg 180 deg60 deg 90 deg
Prediction vs. Experiment: Walking speeds while turning an angled corridor
Energy optimality-based model predicts data from Dias et al (F ) Larger turn angle imples more speed reduction in model and in humansHumansModel Data from Dias et al
Figure 6: (A, B, C) Prediction vs behavior: Path planning through doors. ( A ) Humans were asked( ) to walk through a doorway at point A (pink parallel lines) to another doorway at point B, stopping2 m beyond the second door. The second door B had five different locations as shown (red circles). Thewalls of the doorways serve as obstacles to be avoided. ( B ) Human data redrawn from ( ) are for headpaths. ( C ) A human or the model are capable of sharp turns and otherwise complex paths to achieve thetask. c) But model predictions for the body path from energy optimality are qualitatively similar to humanpaths, despite not having to constrain the average velocity as in ( ). See Supplementary Figure S3 forbands containing trajectories within 2% of the optimal cost. (D, E, F) Prediction vs behavior: Turningin an angled corridor. ( D ) A turning task, involving walking along a straight corridor and turning intoanother corridor angled with respect to the first. ( E ) Energy optimal paths for the task. Subject enters thefirst corridor and leaves the second corridor in a straight line path at their energy optimal speed and clearsthe wall. ( F ) Experimental data from Dias et al ( ) shows that the human speeds during the turn arelower for a larger turn angles, as also predicted by the model-derived energy optimal paths. The humanspeed distribution is enturely captured by the set of speeds within 2% of the optimal cost. uman walking behavior, some via new experiments and some by comparison with data fromprior experimental studies. Thus, we have presented a unified theoretical account of curvilinearhuman locomotion, while includingWe have found that humans slow down when turning and slow down more when turningin a tighter curve. For instance, we saw this behavior when we asked our subjects to walk incircles of different radii (Figures 2). One alternate hypothesis for slowing down while turningis to avoid slipping. While fast-moving cars and bicycles slow down on tight curves to avoidslipping, humans in our experiments were far from any danger of slipping. We estimated thefoot-ground friction coefficient as µ = 0 . to . , corresponding to friction cone angles of 30to 50 degrees ( = tan − µ ). But the maximum leg angle in our circle walking (1 m at 1.5 m/s)was 12 degrees, much less than the friction cone angles, giving a safety factor of at least 2 fromslipping.One could conjecture that there are greater stability issues while turning and that these sta-bility issues contribute to the subjects being cautious and lowering their speed. However, suchconjecture seems unnecessary and not parsimonious as energy minimization seems to largelyexplain the speed reductions. A standard open question in movement control is to what extenthumans prioritize energy or effort on the one hand and stability or robustness to uncertainty onthe other ( ). Addressing this question is beyond the scope of this study, as we have basedour behavioral predictions on empirically derived energy costs. The measured energy costsin our experiment already include any trade-offs that humans had to make to walk stably andefficiently while turning. Thus, we should distinguish our approach from the theoretical limitof true energy optimality in the limit of perfect control and the absence of perturbations oruncertainty (neither of which is physically feasible).In the real world, there may be other additional concerns that may make a human walkfaster than normal. For instance, there could be a cost for time or constraints on time taken to18omplete the movement (
4, 30 ). Here, we have considered conditions where such explicit time-pressure does not exist. Some locomotor behavior may have been adapted to typical conditionsin a Bayesian or probabilistic sense. For instance, while walking along a corridor, it may begood to not walk too close to the wall for a few reasons: lower the likelihood of bumping intothe wall inadvertently or avoiding bumping into an oncoming human who is also walking closeto the wall. It may be interesting to examine such human-human interaction without collisionas a cooperative or non-cooperative game theoretic scenario, with payoffs related to minimizingenergy costs.It is sometimes argued that energy optimality or energy economy may be useful only in‘steady state tasks’ as opposed to ‘transient tasks’ or may only be useful when the energy savedis substantial. Here we have shown broad agreement of behavior with empirical energy opti-mality in short transient tasks (e.g., the turning part a corner) that consume a very small amountof energy. For instance, we estimate the total energy cost of a turn in the angled corridor (Figure6D-F) to be about 10 J/kg and the savings relative to a non-optimal turn (e.g., not slowing downor using a sharper turn) are a small fraction of this cost, equivalent to just over a second of rest-ing energy expenditure. Here, as in many of the situations we considered, the experimentallyobserved behavior were within 2% of the optimal energy costs from the model, corresponding tosimilarly small amounts of energy differences (0.2 J/kg, equivalent to resting energy expenditurefor one seventh of a second). That energy optimality provides an account of diverse transientbehavior with low energy requirements is perhaps an indication of how much the nervous sys-tem values energy savings, all else being equal. Our results are agnostic to how the near-energyoptimality is achieved, whether it is hard-wired evolutionarily, acquired while learning to walkduring childhood, or if the energy optimal trajectories are computed in real time by the nervoussystem. It is likely a mixture of all such mechanisms: energy optimality correctly predictedlowered walking speeds for shorter distances (a task with low total energy ( )), walking speeds19n sideways walking (an uncommon task ( )), and stride frequency in the presence of a externalexoskeleton (an uncommon task with dynamic changes in the energy landscape ( )).We could better understand the walking mechanics here by simultaneously measuring thebody segment motions (motion capture) and the ground reaction forces e.g., ( ). With typi-cal laboratory settings, these measurements are likely feasible only over two steps, a fractionof the whole circle. Nevertheless, such measurements could enable inverse dynamic analyseswith 3D multi-body models (e.g., ( )), estimating joint torques, joint work, which, with furtheroptimality assumptions, give muscle forces and work. Such analyses could help pinpoint themechanistic reason for the increased metabolic cost for turning. They may also further illu-minate the asymmetric role of the two legs for circle walking ( ). Simple biped modelsseem unable to explain the increased cost of walking in circles (see Supplementary Informa-tion). Thus, once such 3D biped models are able to explain the energetics of walking in circles,we can then use them to directly compute the energy optimal solutions for any given task.Our results suggest numerous further experiments to test model predictions and inform im-provements to the model: for instance, walking through many via points with freedom to choosethe intervening path, comparing walking sideways versus turning, walking around obstacles,walking around moving obstacles (such as other people), etc. We obtained a cost of spinningin place by extrapolation, whose accuracy can be improved by using smaller radii. Mecha-nisms for the turning cost could be probed by predicting and testing experimentally how thecost varies after various manipulations of the system: adding mass or moment of inertia to thetrunk or the legs ( ). Our empirical metabolic cost model could be tested further by energeticmeasurements of humans walking on sinusoidal paths, by moving side to side on treadmills.Models for curvilinear locomotion, especially running, might help estimate the metaboliccost during sports (e.g., soccer), which involve extensive speed and direction changes. Reducedtop running speeds around a track or while cornering have been partly attributed to the cen-20ripetal forces ( ), but corresponding sub-maximal running studies have not been performed.Similarly, it may be worth considering whether energetic or power considerations constrain fastmaneuvers ( ). About 20% of steps in household settings ( ) and 35-45% of steps in commonwalking tasks in home and office environments involve turns ( ). Further ambulatory studiesof human walking (lasting many days and using wearable sensors ( )), combined with ourempirical metabolic model, could estimate the relative cost of turning over and above steadystraight-line walking in daily life.In conclusion, through experiments and mathematical models, we have provided a unifiedtheoretical predictive account of human walking in non-straight-line paths and with turningfrom the perspective of energy optimality. Further experiments to test model predictions are in-dicated, which will also inform improvements in the mathematical models. Better understand-ing of the mechanics and energetics of human locomotion while turning would be a useful toolin computer animation, robotics, and especially to design robotic legs, prostheses, and assistivedevices that aid walking in real world scenarios where curvilinear locomotion is essential. Methods
We performed three different experimental studies, one for measuring the metabolic cost ofturning and two for measuring human behavior while turning. We performed multiple model-based optimization calculations to predict energy optimal trajectories and speeds under differenttask constraints to compare with a number of different behavioral experiments, including ourown.
Experiment: Metabolic cost of humans walking in circles.
Subjects were instructed towalk along circles drawn on the ground. The subjects were instructed to keep the circle directlybeneath their feet or between their two feet, but never entirely to one side of their feet. All sub-21ects walked with the circle between their feet with non-zero step width. We used four differentcircle radii ( R = 1 , , , m, N radii = 4 ). At each radius, subjects performed four walkingtrials, each with a different constant tangential speed v in the range 0.8-1.58 m/s, resulting in N trials = 16 trials per subject; one subject performed fewer trials ( N trials = 13 ). Tangentialspeeds were enforced by specifying a duration for each lap around the circle. See Supplemen-tary Information for the list of specific lap durations and tangential speeds. A timer providedauditory feedback at the end of every half lap duration (for R = 3 , m) or full lap duration(for R = 1 , m), so that subjects could speed up or slow down as necessary. Within a fewlaps of such auditory-feedback-driven training, subjects walked at the desired average speed,completing each lap almost coincident with the desired lap time. Subjects maintained the speedwith continued auditory feedback for 6-7 minutes: 4 minutes for achieving metabolic steadystate and 2-3 minutes for obtaining an average metabolic rate ˙ E . Subjects used clockwise orcounter-clockwise circles as preferred. Subjects were instructed to walk and never jog or run,and all subjects always walked.The trial order was randomized over speed and radius for seven subjects (mass . ± kg,height . ± . m, mean ± s.d. and age range 22-27). For ten other subjects (mass ± kg,height . ± . , mean ± s.d. and age range 21-27), the trial order increased monotonicallyin speed and radius. Nevertheless, the overall regression relations were the same when both setsof data were analyzed in the same manner, suggesting the lack of any order effect. Metabolicrate per unit mass ˙ E was estimated during resting and circular walking using respiratory gasanalysis (Oxycon Mobile with wind shield, < ˙ E = 16 .
58 ˙V O + 4 .
51 ˙V CO W / kg withvolume rates ˙V in mls − kg − ( ). To obtain accurate velocity and angular velocity dependenceof the metabolic cost, small systematic differences between observed and actual lap times werecorrected by using measured trial-specific lap times and a linear model. To improve estimates ofthe steady state metabolic rate, we fit an exponential to the metabolic data to the last 3 minutes22f data and determined the extrapolated steady state. This resulted in less than 2% changes inany of the coefficients in equation 1, compared to just taking the mean metabolic rate over thelast 2 minutes or using a weighted least squares procedure. Experiment: Preferred walking speeds in circles.
Subjects’ preferred walking speeds weremeasured by asking them to walk in a straight-line and along circles of radius 1 m, 2 m, 3 m,and 4 m at whatever speed they found comfortable; the subjects walked for about 100 m in eachof these trials (four 4 m laps, eight 2 m laps, etc.) and the second half of the walk was timed toestimate average tangential speed ( ). Two trials were performed for each radius and all trialswere in random order of radii. The subject population for these trials was distinct from thosefor characterizing the energy cost of walking in circles (age 22.6 ± ± ± Experiment: Preferred turning-in-place speeds.
Subjects’ (age 26 ± ±
11 kg, height 1.75 ± Model: Metabolic cost of arbitrary walking paths.
Our walking-in-circles metabolic exper-iments constrained the circular paths that the feet travel on rather than the paths that the bodytravels in. If the foot travels in a circle of radius R and has an effective tangential velocity v , thebody center of mass travels in a circle of smaller radius R b and slightly lower tangential velocity23 b . This is because the legs slope into the circle ( ). We first obtain a body-based descriptionof the empirical metabolic cost: ˙ E = α ′ + α ′ v b + α ′ ω b , where ω b = v/R = v b /R b for circlewalking. We find α ′ = 2 . W/kg, α ′ = 1 . W/kg/ ( ms − ) , and α ′ = 1 . W/kg/ ( rad.s − ) .These coefficients explain the metabolic data roughly as well as the original model (explainingabout 88% variance).Any non-circular or non-straight-line walking path can be described as a curve with con-stantly changing curvature. That is, each point on the curve has a distinct radius of curvature.If we assume non-holonomic walking, in which the body always faces the movement direction,we can directly apply a metabolic rate of the form ˙ E = α ′ + α ′ v + α ′ ω b , where v b is theinstantaneous body velocity, ω b = v b /R b is the angular velocity, and R b is the instantaneous ra-dius of curvature. To generalize to holonomic walking, that is, allowing the body to not alwaysface the velocity direction, we distinguish between the body velocity component along the bodyorientation v f (forward) and the body velocity component perpendicular to the body orientation v s (sideways). We then use a metabolic cost model of the form: ˙ E = α + α v f + α ω b + α s v s .Here, the new term α s v s is the incremental cost of sideways walking. In a previous article,Handford and Srinivasan ( ) characterized this quantity and derived the estimate α s ≈ . W/kg/(ms − ) .Because walking along arbitrary paths may involve or require changing walking speeds, weinclude the additive metabolic cost of changing speeds, previously characterized by Seethapathiand Srinivasan ( ). This study showed that accounting for this cost predicts lower speeds forshort distance walking bouts using energy optimality, as seen in humans (
5, 40 ). See Supple-mentary Appendix for more details about the metabolic cost model.
Model: energy-optimality-based behavioral predictions.
We compare measured experi-mental human behavior in a number of different walking tasks to the energy optimal walking24ehavior predictions. The energy optimal walking behavior is obtained by minimizing the totalmetabolic cost of the walking task. For the simplest two tasks, namely, walking in circle andturning in place (Figures 2-3) – the optimization assumes steady state and requires only basiccalculus. So the complete analytical reasoning for the prediction is provided entirely in theResults section. For more complex walking tasks where the walking path is not pre-determined(Figures 5,6), we solve for the total time duration, body position, body orientation, and theirderivatives as functions of time using numerical trajectory optimization methods ( ). For thistrajectory optimization, we use the metabolic cost function described in the previous paragraph.We perform two versions of the optimization, holonomic and non-holonomic, with the latterobeying the constraint that the body always faces the velocity direction. We solve additionaloptimization problems to obtain trajectories within a certain percent of the optimal cost. See Supplementary Information for more mathematical details of the numerical optimization.
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Journal of Rehabilitation Research & Development (2005). Acknowledgments
We thank Carmen Swain and Blake Holderman for advice and help regarding metabolic equip-ment during early pilot testing, Alison Sheets for comments on an early draft, and V. Joshi forhelp with some experimental setups.
Funding:
This work was supported in part by NSF grantCMMI-1254842 and its writing in part by an NIH grant. The authors declare that they have nocompeting interests.
Author contributions:
Conceptualization, GB, NS, and MS; Methodol-ogy, GB, NS, and MS; Software and Formal Analysis, GB and MS; Writing – Original Draft,GB and MS; Writing – Review & Editing, NS and MS; Supervision, MS; Funding Acquisi-tion, MS.
Data and materials availability.
All data will be available in public domain with norestrictions through the Dryad database upon acceptance.