Redirection of center-of-mass velocity during the step-to-step transition of human walking

2668 The Journal of Experimental Biology 212, 2668-2678 Publihed by The Company of Biologit 29 doi:1.1242/jeb.27581 Redirection of center-of-ma elocity during the tep-to-tep tranition of human walking Peter G. Adamczyk* and Arthur D. Kuo Department of Mechanical Engineering, Unierity of Michigan, Ann Arbor, MI 4819, USA *Author for correpondence (e-mail: padamczy@umich.edu) Accepted 25 May 29 SUMMARY Simple dynamic walking model baed on the inerted pendulum predict that the human body center of ma (COM) moe along an arc during each tep, with ubtantial work performed to redirect the COM elocity in the tep-to-tep tranition between arc. But human do not keep the tance leg perfectly traight and need not redirect their COM elocity preciely a predicted. We therefore teted a pendulum-baed model againt a wide range of human walking data. We examined COM elocity and work data from normal human ubject (N=1) walking at 24 combination of peed (.75 to 2. m 1 ) and tep length. Thee were compared againt model prediction for the angular redirection of COM elocity and the work performed on the COM during redirection. We found that the COM i redirected through angular change increaing approximately linearly with tep length (R 2 =.68), with COM work increaing with the quared product of walking peed and tep length (R 2 =.82), roughly in accordance with a imple dynamic walking model. Thi model cannot, howeer, predict the duration of COM redirection, which we quantified with two empirical meaure, one baed on angular COM redirection and the other on work. Both indicate that the tep-to-tep tranition begin before and end after double upport and lat about twice a long approximately 2 27% of a tride. Although a rigid leg model can predict trend in COM elocity and work, the non-rigid human leg perform the tep-to-tep tranition oer a duration coniderably exceeding that of double upport. Key word: locomotion, biomechanic, center of ma, elocity, redirection, tep-to-tep tranition, walking, work, inerted pendulum, leg compliance, hodograph, dynamic walking, tep length. INTRODUCTION The body center of ma (COM) moe like an inerted pendulum during human walking. A pendulum conere mechanical energy and need no work to moe along an arc (Alexander, 1991). By keeping the knee relatiely traight, the human tance leg alo upport body weight with relatiely little mucle force (Kuo, 27). Thee mechanical aing appear to be phyiologically releant becaue work production and force deelopment in mucle both require metabolic energy expenditure. The pendulum offer trong inight regarding the mechanim of walking, but it alo ha limitation. The human tance leg i not perfectly rigid, and the actual COM motion deiate omewhat from a pendulum arc. Here we examine the pendulum model prediction for how the COM i redirected between arc, to quantify how well it predict human COM elocity and to determine it limitation a a predictie model of human walking. The pendulum analogy i demontrated well by fluctuation of COM energy, a the kinetic energy fluctuate out of phae with graitational potential energy during walking (Caagna and Kaneko, 1977). By contrat, thee fluctuation occur in phae during running (Caagna et al., 1977), where a pring-ma analogy better decribe the compreion and extenion of the tance leg (Blickhan, 1989). The phaing and amplitude of energy fluctuation alo change during walking, a a function of gait parameter uch a peed, tep length and tep frequency (Willem et al., 1995). Some of thee change may be attributed to work performed on the wing leg, but ome may alo be aociated with the tance leg when it doe not behae a a perfectly rigid pendulum. Thi make it difficult to predict how gait parameter will affect energy fluctuation and, conerely, to relate obered fluctuation to actual COM motion. The imperfect rigidity of the tance leg ha preiouly been noted in eeral way. Alexander pointed out that the ground reaction force under each leg are explained much better by a model with axially compliant leg (Alexander, 1992). Human produce a characteritic ertical ground reaction force profile with two peak that are not produced by a rigid inerted pendulum model. The addition of compliance, imilar to the pring-ma running analogy, can predict uch a profile in forward dynamic imulation (Geyer et al., 26). Thee finding are corroborated by inere dynamic analye, which how that a telecoping pendulum (referring to axial lengthening and hortening) allow for much better matching of ground reaction force than a rigid one (Buczek et al., 26). The leg joint, mot notably the knee, hae long been obered to flex and extend during ingle upport (Winter, 1991), and the telecoping action i a imple mean of ummarizing the effect of multiple joint motion on COM motion. None of thee effect, howeer, are eaily quantified through energy fluctuation. Another limitation of the inerted pendulum analogy i that it only applie to the ingle upport phae of walking. Double upport i not pendular and intead function a a tranition between ingle upport phae (Donelan et al., 22a). The tep-to-tep tranition redirect the COM elocity from the downward portion of an arc precribed by the tance leg for one tep, to the upward portion of another arc precribed by the ucceeding tance leg for the next tep. Auming rigid tance leg and impulie colliion, imple model of dynamic walking predict that thee elocity change determine the mechanical work performed during the tranition

Redirection of center of ma elocity 2669 (Adamczyk et al., 26; Donelan et al., 22b; Kuo, 22). Empirical data ugget that thi work exact an approximately proportional metabolic cot in human (Donelan et al., 21; Donelan et al., 22a). Thee model do not, howeer, capture how human leg are imperfectly rigid and unable to produce ideal, intantaneou impule. Thi can potentially lead to incorrect prediction depending on the degree and nature of axial leg motion. Preiou tudie hae focued on the work of the tep-to-tep tranition primarily during double upport but hae largely oerlooked the poible dependency of COM elocity change on axial leg motion during ingle upport. The purpoe of thi tudy wa to examine how COM elocitie ary a a function of gait parameter uch a peed, tep length and tep frequency. We meaured COM elocitie between pendulumlike phae, acro a wide range of walking gait parameter. We analyzed the relationhip between elocity magnitude and direction, the impule proided by the two leg, and the mechanical work performed on the COM during the tranition between tep. Thee were then compared againt the prediction of imple model auming rigid leg. It i poible that human leg deiate from rigid leg model, and the conitency of that deiation may determine how ueful imple model are for predicting general trend in COM elocity redirection and the aociated work. MATERIALS AND METHODS We compared experimental meaurement of COM motion during human walking againt imple mathematical prediction. Experimental meaurement were made for a range of low to fat walking peed, hort to long tep, and low to high tep frequencie, the extreme of which might be mot expected to induce tance leg behaior not reembling a pendulum. Predicted quantitie included the magnitude and direction of COM elocity, the impule produced by each leg againt the COM, and the work performed on the COM through each leg. The prediction were baed on a imple model of dynamic walking that relie entirely on paie dynamic except for actie puh-off to produce gait. Thi model predict trend in the meaured quantitie a a function of walking peed and tep length. Thi ection begin with a brief ummary of the model, followed by decription of the experimental condition, aociated meaurement and quantitatie analye. Model We ued a preiouly deeloped dynamic walking model (Kuo, 22) to derie the dependency of COM motion on gait parameter. The model i a implification of human gait that diide the COM approximately inuoidal motion into two portion: an upper one that correpond roughly with ingle upport and a lower one for double upport. The upper portion i that which mot reemble a pendulum arc (ee Fig. 1A), with the COM elocity directed upward at the beginning and downward at the end. The forward peed reache minimum at the top of the arc, when the pendulum ha maximum graitational potential energy. We treat the lower portion of the path a a redirection phae (Fig. 1B) or tep-to-tep tranition (Donelan et al., 22b) that end with the beginning of the next pendulum phae. The model aume perfectly rigid leg of negligible ma upporting body ma concentrated at the peli, o that the COM moe atop a imple inerted pendulum (ee Fig.1C). Each foot ha ufficient ma to allow the wing leg to behae like a pendulum, but with negligible effect on the ret of the body. The model tep-to-tep tranition (Fig. 1D) conit of puh-off and colliion, treated a ucceie, intantaneou impule applied along the trailing and leading leg, repectiely. Thee impule perform all of the work in the model, determined entirely by the COM elocitie at beginning and end of the pendulum phae. Human do not produce uch force, but the work performed by the model obey imilar trend for more realitic force, a long a the actual duration and diplacement of the tep-to-tep tranition are relatiely mall. The principal prediction are for fluctuation in COM elocity and work performed on the COM by the indiidual leg. Thee come a a erie of linear relationhip, all of which may be decribed intuitiely with the pendulum model. The model contrain the COM trajectory along a erie of pendular arc, each with an angular excurion determined by tep length (Fig. 2A). Thi angular excurion alo dictate the directional change,, that the COM elocity mut undergo in the tep-to-tep tranition. Walking fater at a gien tep length (by increaing tep frequency) produce a higher elocity along the ame trajectory. The magnitude of COM elocity increae approximately linearly with the model walking peed (Fig. 2B), for any tep length. The angular redirection increae approximately linearly with the model tep length, for any walking peed. The work performed on the COM during the tep-to-tep tranition i proportional to the change in kinetic energy due to the puh-off and colliion. Thi energy change i, in turn, proportional to the quare of COM elocity change and therefore to the quared product of COM elocity magnitude and the angular redirection. Combining all of thee relationhip, the COM work per tep i predicted to be proportional to the quared product of walking peed and tep length. The mathematical detail of the work prediction are a follow. We refer to aerage walking peed a, and the COM elocitie at the beginning and end of the tep-to-tep tranition a and, repectiely (ee Fig.1D; Fig.2A). Model imulation (Kuo, 22) how that:, (1) and that bet economy i achieed if the trailing leg applie a puhoff impule, ufficient to reduce the ertical component of COM elocity to zero, immediately before heel-trike. The heel-trike colliion then produce an impule along the leading leg, uch that i directed along a new arc-like trajectory precribed by that leg. The angle between the leg increae with tep length, :, (2) for mall angle. The work, W, performed by uch a puh-off (and the negatie work performed by the colliion) i equal to the change in kinetic energie before and after puh-off: W = GM ( 2 mid 2 pre) = GM 2 pre tan 2, (3) where mid refer to the mid-tranition COM elocity between the puh-off and heel-trike impule, and M refer to body ma. Again, auming mall angle: W ( ) 2. (4) Combining Eqn1,2 and 4 yield work per tep in term of peed and tep length: W ( ) 2. (5) Although thee relationhip are deried with the ery imple model preented here, the addition of human-like ma ditribution and arc-haped feet ha preiouly been hown to hae little effect on the oerall linear form of the prediction except for an added contant offet term to the proportionalitie (Adamczyk et al., 26; Kuo, 21).

267 P. G. Adamczyk and A. D. Kuo A Human inerted pendulum and tep-to-tep tranition Pendulum Step-to-tep Pendulum tranition Ground reaction force (GRF) PO imp CO COM path B Human COM elocity change el Vertical horizontal elocity Up Pendulum.2 m 1 Fwd S-to-S C Model inerted pendulum and tep-to-tep tranition D Model COM elocity change Pendulum Pendulum Step-to-tep tranition Up Vertical horizontal elocity Pendulum S-to-S Fwd PO CO.2 m 1 Fig. 1. Body center of ma (COM) motion in human and in a imple walking model. (A) The human COM moe in a roughly inuoidal path, which may be diided into inerted pendulum and tep-to-tep tranition phae, correponding approximately to ingle and double upport, repectiely. Ground reaction force (GRF) from each leg produce puh-off and colliion force (highlighted with thicker line during double upport) that um to the impule ector (labeled PO for puh-off and CO for colliion) eparated by an angle, imp. (B) The COM elocity change during the tep-to-tep tranition, from a pre-tranition elocity ( ) to a pot-tranition elocity ( ), eparated by an angle, el. Thee change may alo be obered in a plot of ertical horizontal component of COM elocity (inet), which trace a counter-clockwie path, a the COM elocity change from upward to downward during the pendulum phae, and then from downward to upward due to puh-off and colliion force during the tep-to-tep tranition (S-to-S). (C) In the model, COM elocity i precribed by a imple pendulum, with an impulie tep-to-tep tranition. (D) The angular difference between the model impule i equal to the angular difference between and. The plot of COM elocitie, termed the COM hodograph (inet), alo trace a counter-clockwie path. Experiment We teted model prediction with meaurement of COM elocity and work from a wide range of human walking. We impoed 24 different combination of walking peed and tep length on 1 human ubject (fie male, fie female), with body ma, M, aeraging 68.9±12.2 kg (mean ±.d.) and leg length, L, aeraging.93±.5 m, and obered the impact of change to thee gait parameter on pre-tranition COM elocity, COM redirection, colliion work performed on the COM (ee Eqn 1, 2, 4, and 5) and timing of redirection. We meaured ground reaction force (GRF) while ubject walked oer ground and ued thee to compute the COM trajectory and colliion work, W, oer the coure of a tep, defined a heel-trike to oppoite heel-trike. We ued four different et of condition to map each ubject performance acro a range of peed and tep length urrounding normal walking (ee Fig. 3). Prior to thee condition, we ealuated each ubject preferred tep frequency, f*, and tep length, *, at a deignated nominal peed, *, of 1.25m 1 (where *=* f*), with peed meaured by photogate. The firt et of condition conited of natural walking (circle in Fig. 3), in which ubject walked oer ground at peed of.75, 1., 1.25, 1.5, 1.75 and 2.m 1 (.6 to 1.6 *), all at their own preferred tep length and frequency for each peed (ee Donelan et al., 22b). In the econd et of condition, ubject walked at the ame peed but with a contant tep frequency f* (contant frequency, CF; quare in Fig.3) et by a metronome (Donelan et al., 22a). Becaue peed equal tep length multiplied by tep frequency, thi protocol reulted in tep length ranging from.6 to 1.6 *. The third et of condition wa complementary to the econd; ubject maintained their preferred tep length, *, acro the ame range of peed by tepping to a metronome at frequencie from.6 to 1.6 f* (contant tep length, CS; diamond in Fig. 3). In the final et of condition, ubject aried both tep frequency and tep length in inere proportion to maintain the pecified peed, *, matching their tep frequency to a metronome beat ranging from.7 to 1.3 f* (contant peed, CV; triangle in Fig.3). All the data we analyzed were collected in conjunction with earlier tudie (Donelan et al., 22a; Donelan et al., 22b), in which ubject completed three trial per condition, with condition applied in random order acro all four et. All human ubject proided their informed conent, a approed by the Unierity of California Intitutional Reiew Board. COM elocity and work were etimated from GRF data. COM elocity wa determined by integrating three-dimenional GRF data (Caagna, 1975; Donelan et al., 22b), with integration contant for each tep baed on an aumption of periodic gait. Velocity and force data were then ued to calculate the intantaneou rate of work performed by each leg on the COM, defined a the dot product of each leg GRF againt COM elocity (Donelan et al., 22b). The

Redirection of center of ma elocity 2671 A Effect of gait ariation Normal walking Longer tep, ame peed B Model prediction Fater peed, ame tep length Fater peed with longer tep Fig. 2. (A) Effect of gait ariation and (B) model prediction baed on a imple dynamic walking model. (A) The magnitude of COM elocity increae with the model walking peed, and it directional change,, increae with the model tep length,. Fater walking peed and longer tep together require a greater change in COM elocity than either factor alone. (B) Model imulation predict that pretranition elocity ( ) will increae approximately linearly with walking peed (Eqn 1), angular redirection () will increae approximately linearly with tep length () (Eqn 2), and the aociated work (W) performed on the COM will increae (Eqn 4 and 5) with a predicted quantity ( ) 2, which i alo proportional to the quared product of walking peed and tep length ( ) 2. The model predict linear relationhip with unknown lope and offet to be determined from experimental data. pre W Model predictor, (. ) 2 work rate for each leg wa then integrated to yield the poitie and negatie COM work performed by each leg during the tep-to-tep tranition. To aid our analyi, we alo plotted the ertical and horizontal component of COM elocity againt each other oer the coure of a tep (ee Fig. 1B,D). The term hodograph (Greenwood, 1988) refer to a plot of elocity component, and o we refer to our plot a a COM hodograph. A practical iue in the comparion of experimental data with model i that human do not produce purely impulie force. The production of finite GRF for a finite duration mean that human need not redirect their COM elocity by the ame amount a the angle between the leg. We therefore defined eparate quantitie: el for the angular change in COM elocity, and imp for the angular difference in ground reaction impule. Both of thee quantitie are expected to increae with the model, but not necearily with equal proportion. Human alo need not produce equal amount of work during puh-off and colliion. We therefore computed eparate puh-off poitie work, W PO, and (magnitude of) colliion negatie work, W CO ; we expected both to increae with the model work, W. Another iue i the duration of the human tep-to-tep tranition. In the model, double upport occur in an intant, with puh-off and colliion impule coinciding with redirection of the COM. In human, double upport occur oer a finite duration that only approximately matche when puh-off and colliion work are performed, which in turn only approximately coincide with the extreme of COM elocity redirection. We defined the duration of the tep-to-tep tranition a the period, τ el, between extreme of direction for the COM elocity, referred to a and (ee Fig. 1B), locally urrounding double upport (Adamczyk et al., 26). Thi duration wa then ued to compute the elocity redirection and work meaure. We alo conidered two additional definition for the duration, one (τ DS ) baed on the double upport period a determined from ground reaction force, and one (τ work ) baed on the interal of COM work performed by the two leg (Doke et al., 25; Donelan et al., 22a). It will be hown that all three definition ere well in experimental comparion; for breity, only reult for τ el are reported here, with other reult reported in the Appendix. Data analyi We teted the model uing leat-quare fit to the prediction. The model predict a erie of trend with unknown coefficient C and D to be determined by each fit, with a different ubcript for each prediction. Although the implet walking model do not require an offet D, other model that include human-like ma or arc-haped feet (Adamczyk et al., 26) do predict an offet. Pre-tranition COM elocity wa teted with a model-baed fit to Eqn1: = C + D, (6) applied to all walking condition except the et in which walking elocity wa held contant (CV; ee Fig. 3). Redirection of COM elocity wa teted with fit baed on the actual angular change in elocity and on the impule produced by the indiidual leg, from Eqn2: el = C el + D el, (7) imp = C imp + D imp, (8) applied to all walking condition except the et in which tep length wa held contant (CS; ee Fig.3). Step-to-tep tranition work wa teted with eeral model-baed fit. Thee included model-baed fit to elocity-baed prediction of Eqn4 for both colliion (CO) and puh-off (PO): W CO = C CO ( el ) 2 + D CO, (9) W PO = C PO ( el ) 2 + D PO. (1) Similar model-baed fit were applied to the impler prediction of Eqn5: W CO = C CO ( ) 2 + D CO, (11) W PO = C PO ( ) 2 + D PO, (12) where the prime ymbol (C and D ) indicate the ue of peed and tep length a gait-baed predictor. Thee fit were applied to all data except the condition with highet tep length (1.6*; ee Fig. 3),

2672 P. G. Adamczyk and A. D. Kuo 1.2 1..8.6.4.2 Natural walking (m 1 ) 1 1.5 2 CV CS CF Actual Target 1.2 1..8.6.4.2.3.4.5.6.7 Fig.3. Walking peed and tep length meaured in four et of condition (24 total condition). Each condition wa pecified a a target peed and tep length (open ymbol) that ubject attempted to match (actual combination achieed are hown with filled ymbol) while walking oer ground. Set included natural walking (circle) acro a range of peed uing ubject preferred tep length; walking at contant tep length (CS; diamond); walking at contant tep frequency (CF; quare); and walking at a contant peed oer a range of tep length (CV; triangle). The highet tep length pecified in the CF condition et wa difficult for ubject to utain without running (Donelan et al., 22a); thee data were excluded from model fit to COM work (trial marked with plu ign). Walking peed and tep length were non-dimenionalized uing leg length, L, and graitational acceleration, g, a bae unit; equialent dimenional mean alue are hown on the top and right axe. N=1 ubject. (m) pre.7.6.5.4 Pre-tranition elocity walking peed (m 1 ) 1. 1.5 2. Human data Linear model fit 2. 1.5 1..3 Slope 1.3 R 2 =.99.2.2.3.4.5.6.7 Fig. 4. Pre-tranition center of ma (COM) elocity walking peed ( ). Data point hown are for all ubject (N=1) walking in natural walking, contant tep frequency and contant tep length condition (contant peed condition are excluded). Model-baed fit how that pretranition COM elocity increaed approximately linearly with walking peed (a predicted by Eqn 6), R 2 =.99. Walking peed can therefore predict the magnitude of COM elocity at initiation of the tep-to-tep tranition. Velocitie are hown in dimenionle unit (left-hand and bottom axe) and m 1 (right-hand and top axe). pre (m 1 ) which wa deemed unuitable becaue ubject were unable to utain it for appreciable time (Donelan et al., 22b). We performed an additional et of model-baed fit for the duration of the tep-to-tep tranition. We performed linear regreion fit of duration baed on the three definition for the tranition (τ el, τ work, τ DS ) to walking peed for the natural walking condition. Thee are purely data-drien fit, becaue the imple walking model do not predict the duration of COM redirection. We performed all regreion uing dimenionle ariable to account for difference in ubject body ize (Adamczyk et al., 26). We ued bae unit of ubject ma, M, graitational acceleration, g, and tanding leg length, L. Velocity wa therefore made dimenionle by the diior (gl).5, and work and energy by MgL. Step length wa non-dimenionalized by leg length, L. Model fit are preented in dimenionle unit, but SI unit are alo hown, uing the mean non-dimenionalizing factor. For example, the mean non-dimenionalizing factor for work wa MgL=63.6 J. We alo accounted for inter-ubject ariation in kinematic and energetic by computing the offet D in each equation eparately for each ubject and then aeraging thee acro ubject. Statitical tet were performed on all fit to determine ignificant dependencie. We computed the 95% confidence interal from each fit, uch that a confidence interal not including zero indicated ignificant change with α=.5. RESULTS We found the kinematic and mechanic of the COM to behae approximately a predicted by imple walking model. Pre-tranition COM elocity increaed approximately linearly with walking peed. Both the angular redirection of COM elocity and the angular difference between leg impule increaed approximately linearly with tep length. Work performed on the COM during the tep-to- tep tranition alo increaed in rough proportion to the elocitybaed and gait-parameter-baed predictor. The duration of the tepto-tep tranition, howeer, appear to be longer than double upport. Detail of reult are preented below. Baeline alue for the nominal walking condition were a follow. The nominal walking peed wa an aerage of 1.27±.1m 1 (mean ±.d.), or.42±.13 in dimenionle unit. The preferred tep length, *, aeraged.714±.3 m, or dimenionle.766±.39. The directional change, el, in COM elocity wa.324±.53rad, or 18.5±3.deg.; the angle, imp, between leg impule wa.328±.28 rad, or 18.8±1.6 deg. Mean pre-tranition COM elocity,, wa 1.23±.2 m 1, or dimenionle.48±.14. Negatie colliion work, W CO, for each tep of normal walking wa.25±.32jkg 1, or dimenionle.23±.4. Poitie puh-off work, W PO, performed on the COM during the tep-to-tep tranition wa.242±.43 J kg 1, or dimenionle.26±.5. Pre-tranition COM elocity,, increaed ignificantly with increaing walking peed (Fig.4). Data were fit well (R 2 =.99) by the linear prediction of Eqn 6. Both el and imp increaed ignificantly with tep length, a predicted by Eqn 7 and 8, acro all condition where tep length wa aried (Fig.5). Dimenionle coefficient were C =1.3±.9 (CI, 95% confidence interal) and D =.11±.6 for pre-tranition elocity, C el =.37±.37 (CI) and D el =.88±.38 for elocity change angle (R 2 =.68; Fig.5A), and C imp =.477±.21 and D imp =.37±.22 for impule angle (R 2 =.92; Fig.5B). The amount of work performed on the COM during the tep-totep tranition alo increaed ignificantly with predicted quantitie (Fig. 6). In term of elocity-baed predictor (Fig. 6A), W CO increaed approximately linearly a with Eqn9 (R 2 =.74). W PO increaed approximately a with Eqn1 (R 2 =.59). In term of peed

Redirection of center of ma elocity 2673 A COM redirection angle tep length A Colliion work elocity-baed predictor.6 (m).4.8 1.2 Human data Model fit 3 (. pre el ) 2 (m 2. 2 ).5 1.1 R 2 =.74 6 el (rad).4.2 R 2 =.68.4.8 1.2 el 2 1 el (deg.) WCO.5 Human data Model fit.5.1 Velocity-baed predictor (. pre el ) 2 4 2 WCO (J) imp (rad) B Leg impule angle tep length (m).4.8 1.2.6.4.2 imp R 2 =.92 PO CO.4.8 1.2 Fig. 5. Angular difference in the direction of (A) center of ma (COM) elocity at beginning and end of the tep-to-tep tranition, and (B) trailing leg and leading leg ground reaction impule during the tep-to-tep tranition, both a a function of tep length. Data hown are for all ubject (N=1) walking in the natural walking, contant tep frequency and contant walking peed condition, where tep length wa aried oer a range of about.2 to 1.2 m (contant tep length condition are excluded). Model-baed fit (Eqn 7 and 8) indicate that both angular difference increaed approximately linearly with tep length. Inet how definition for angular change ( el ) in COM elocity and for angular difference ( imp ) between ground reaction impule performed by each leg, puh-off (PO) and colliion (CO). Step length are hown normalized by leg length, L (bottom axe), a well a in unit of meter (top axe). and tep length a gait-parameter-baed predictor (Fig.6B), W CO increaed approximately linearly a in Eqn11 (R 2 =.82). W PO alo increaed approximately a in Eqn12 (R 2 =.65). The dimenionle coefficient uing elocity-baed predictor were C CO =.84±.68 (CI), D CO =.9±.4, C PO =.57±.64 and D PO =.17±.4. Coefficient uing peed and tep length were C CO =.128±.8 (CI) and D CO =.8±.3, C PO =.81±.9 and D PO =.17±.3. The duration of the tep-to-tep tranition wa dependent on whether it wa defined baed on GRF, elocity or work. The COM hodograph howed that the elocity and work-baed definition conitently exceeded double upport (meaured by the GRF) a a function of walking peed (Fig.7A). The fraction of a tride, τ DS, pent in double upport decreaed with peed, but the duration baed on angular change in COM elocity and on COM work were both greater and relatiely contant (Fig. 7B). At the nominal peed of 1.25m 1, double upport wa about 14% of a tride, wherea τ el wa 27%, and τ work wa 2%. Double upport decreaed with 3 2 1 imp (deg.) B Colliion work gait-baed predictor WCO.1.5 (. ) 2 (m 4. 2 ) 2 4 6 R 2 =.82.2.4.6.8 Gait-baed predictor (. ) 2 Fig. 6. Negatie center of ma (COM) work, W CO, performed by the leading leg during the tep-to-tep tranition model prediction. Data hown are for all ubject (N=1) walking in all 24 experimental condition. (A) The imple walking model predict (Eqn 9) that work per tep to redirect the COM will increae with ( el ) 2, where i the magnitude of COM elocity at initiation of the tep-to-tep tranition, and el i the angular change the COM elocity undergoe during the tranition (ee Fig. 5A). Data matched the model (olid line) reaonably well (R 2 =.74). (B) A impler predictor i the quared product of walking peed and tep length ( ), deried from predicted linear relationhip (Eqn 11). Data matched the model (olid line) reaonably well (R 2 =.82). The COM negatie work required for gait i well predicted by the trend deried from our imple dynamic walking model. Trial with the highet peed and longet tep (marked by plu ymbol) were excluded from model-baed fit becaue ubject could not conitently maintain thoe gait without running (ee Fig. 3). For both A and B, two et of axe are hown, with left-hand and bottom axe in dimenionle unit, and right-hand and top axe in dimenional unit. increaing walking peed, from about 17% to 9% of a tride cycle, with dimenionle lope of.36±.4 (CI) and offet of.42±.2 (R 2 =.82). The other duration, for elocity change and work performed on the COM, both increaed by ery light amount. τ el ranged from 27% to 28% of a tride, with lope.9±.9 (CI) and offet.36±.4 (R 2 =.8). τ work ranged from about 19% to 21% of a tride, with lope.9±.8 (CI) and offet.51±.4 (R 2 =.7). (The latter two R 2 alue were low due to the near-zero lope.) DISCUSSION Simple model of human walking predict how the COM will be redirected, auming each leg i fairly rigid during it tance phae. Thee model hae predicted trend in oerall mechanical work and 6 4 2 WCO (J)

2674 P. G. Adamczyk and A. D. Kuo Vertical COM elocity Duration (fraction of tride) A COM hodograph.2.2 Mean elocity Intantaneou elocity el.5 (m 1 )=.75 Velocity Work Force Velocity (m 1 ) 1 1.5 2 Velocity change COM work 1. 1.25 1.5 1.75 2..2.4.6 Forward COM elocity B Tranition duration.5.4.3.2.1 (m 1 ).75 1. 1.25 1.5 1.75 2. Double upport τ el τ work τ DS.2.3.4.5.6.7.5 Fig. 7. (A) Plot of center of ma elocity component (referred to a COM hodograph) for a ariety of walking peed, howing agittal plane elocity fluctuation. Aeraged elocity data i hown for all ubject (N=1) in natural walking condition, with tandard deiation denoted by haded area, and mean walking peed denoted by mall filled circle. Hodograph how maximum angular redirection el of COM elocity (broken line for 2m 1 condition), a well a the pre- and pot-tranition elocitie at the point (mall circle) where COM elocity i directed mot downward and mot upward. The beginning of puh-off work and end of colliion (mall triangle) occur oer lightly horter duration, and the double upport period between heel-trike and toe-off (HS and TO; mall quare) occur oer a till horter duration. Both angular redirection and pre-tranition elocity increae conitently with walking peed, requiring greater work to redirect COM elocity. (B) Duration of tep-to-tep tranition in natural walking (in term of fraction of a tride) a a function of walking peed. Velocity change refer to the duration (τ el ) of the angular redirection of COM; COM work refer to the duration (τ work ) oer which the trailing leg perform poitie work and the leading leg perform negatie work in the time interal urrounding double upport. Double upport refer to the period (τ DS ) when both leg contact the ground. metabolic energy rate with change in peed, tep length and tep width (Donelan et al., 21; Donelan et al., 22a; Donelan et al., 22b), but they hae not preiouly been teted in term of the actual motion of the COM. The preent tudy inetigated whether the rigid leg aumption applie ufficiently to predict the angular redirection of the COM and the aociated work performed on the COM by the indiidual leg. We found that, acro a wide range of walking condition, human ubject appear to redirect the COM TO HS.5 Velocity (m 1 ) largely a the model predict. For example, mechanical quantitie uch a the work performed on the COM during the tep-to-tep tranition change largely a a function of only walking peed and tep length. Thee ame model do not, howeer, predict the duration of the tranition, which wa obered to exceed double upport and therefore indicate non-rigid tance leg behaior. The amount of COM redirection appear to be determined mainly by walking peed and tep length. Thee two gait parameter influence, repectiely, the magnitude and direction that decribe the COM elocity. In both model and human, the tep-to-tep tranition occur at greater COM elocity than the inerted pendulum phae (ee Fig. 1B,D). Between thee phae, at initiation of the tepto-tep tranition, the elocity magnitude wa found empirically to nearly equal walking peed (Fig.4), een for tep length and frequencie differing greatly from the nominal walking condition. The cloe model fit (R 2 =.99) demontrate that pre-tranition elocity magnitude i determined almot entirely by walking peed and much le by tep length or frequency. By contrat, el i determined primarily by tep length (Fig. 5). Thi direction change can be attributed to the impule applied along the leg, with angle imp alo increaing with tep length. Howeer, el depend not only on leg impule but alo on the effect of graity and the diplacement of the COM during the tep-to-tep tranition. Thee factor caue el to increae le than imp with increaing tep length, although the relatie amount cannot be predicted by our imple model, which incorporate neither duration nor diplacement in the tep-to-tep tranition. Thee effect partially account for the obered poorer model fit for el than for imp (R 2 =.68.92, repectiely). The model cannot predict what additional apect of gait contribute to thee quantitie, but it i clear that tep length i a major determinant of COM elocity direction change, and walking peed i a major determinant of the COM elocity at the beginning of the tep-to-tep tranition. COM redirection require that work be performed on the COM. We found tep-to-tep tranition work to increae imilarly with both elocity- and gait-parameter-baed predictor (ee Fig. 6). The prediction of Eqn 4 and 5 again aume a perfectly impulie tepto-tep tranition, which differ from reality. Colliion work per tep increaed omewhat more than puh-off work (C CO >C PO for both elocity- and gait-parameter-baed predictor), perhap becaue poitie work by mucle i more trictly rate-limited than negatie work, which can be performed in part by paie tiue. The puhoff limitation mean that, at higher peed and tep length, ome of the work to offet the colliion mut be performed at other part of the tride (e.g. Kuo et al., 25). It i alo intereting to note that the gait-baed predictor peed and tep length yielded lightly better fit than the elocity-baed predictor (e.g. R 2 =.82.74 in Fig. 6). The error of multiple prediction (Fig. 2B) could conceiably hae accumulated to yield a ery poor fit oerall, but thi wa eidently not the cae. Cloer inpection reeal that the fit are poorer for ery long tep (oer 1m) and fat peed (about 2m 1 ), combination not normally employed by human in walking. Other hae uggeted that pendular mechanic do not apply at uch tep length (Bertram et al., 22). Data throughout the range of condition appear to exhibit light nonlinearity, perhap with an exponent lower than the quare of the product of peed and tep length. An arbitrary nonlinear regreion would almot urely fit better than the model prediction, but would lack explanation. The imple model, while imperfect, proide both a reaonable fit and a mechanitic explanation. Our reult are conitent with preiou report of COM motion. Thee include report of el for a nominal gait (Adamczyk et al.,

Redirection of center of ma elocity 2675 26), of COM work for walking at arying tep length but fixed tep frequencie (Donelan et al., 22a) and of COM work at arying tep width but fixed tep length and frequencie (Donelan et al., 21). The preent tudy how that een when gait parameter are aried coniderably, tep length and walking peed largely account for the angular redirection of the COM, the pre-tranition magnitude of COM elocity and the work performed on the COM during the tep-to-tep tranition. Similar pendulum mechanic are thu preered acro a wide range of gait parameter, a i aumed in a number of theoretical tudie of walking (Kuo, 21; Sriniaan and Ruina, 26; Uherwood et al., 28). One uch model (Ruina et al., 25) uggeted that COM redirection work could be ery different if human omehow alter the relatie timing of GRF aociated with puh-off and colliion. The empirical reult here how quite conitent timing (Fig.7) and work (Fig.6) of the tepto-tep tranition. We alo computed an additional metric to etimate the oerlap of puh-off and colliion, a propoed by Ruina et al. (Ruina et al., 25), and found puh-off to alway precede colliion with imilar functional oerlap (ee Appendix for detail). Een though the duration of double upport change with peed, the work of COM redirection i performed in a ery conitent manner acro a wide range of walking pattern. Conitent COM motion alo ha implication for energy expenditure. Metabolic energy expenditure ha been found to increae linearly with COM work, a a function of tep width or length (Donelan et al., 21; Donelan et al., 22a), when tep frequency i kept fixed. Thi control for poible energetic cot, uch a for moing the leg back and forth, that may increae with tep frequency (Doke et al., 25). We obered conitent COM mechanic not only for a wide range of tep frequencie but alo during natural walking, when ubject walked at preferred (uncontrained) tep frequency. The preent reult ugget that COM work may effectiely quantify the contribution of tep-to-tep tranition to energetic cot, for both natural and contrained walking. Two key outcome ariable in thi tudy magnitude and direction of COM elocity may be iualized from the COM hodograph plot (Fig.1; Fig.7A). A plot of ertical forward COM elocity yield a cloed, counter-clockwie cure for each tep, facilitating comparion of gait feature releant to the tep-to-tep tranition. The left-mot portion correpond to the inerted pendulum phae, a the COM rie and low prior to mid-tance, and then peed up while falling forward. The right-mot portion correpond to the tep-totep tranition, a the COM i accelerated and redirected upward by the trailing leg puh-off, and i decelerated and redirected upward by the leading leg colliion. The COM hodograph for the natural walking condition (Fig. 7A) how that the pre-tranition elocity and angular redirection both change conitently with peed. It i notable that, a walking peed increae, a greater proportion of the redirection occur before and after double upport. Thi i confirmed by the decreaing duration of double upport a a function of peed, relatie to the duration of actual elocity change (Fig.7B). It i alo notable that the elocity profile during the tep-to-tep tranition i quite conitent acro ubject, yet alo exhibit a different tereotypical pattern for each peed. Thi conitency among healthy human may alo make the COM hodograph helpful for examining abnormal or impaired gait. It might, for example, help determine whether a gait impairment ha mot effect on the inerted pendulum phae or the tep-to-tep tranition. Our analyi emphaize the tep-to-tep tranition a a key iue in pendulum-like walking. Thi contrat with preiou tudie (e.g. Caagna and Kaneko, 1977; Willem et al., 1995) that quantified fluctuation of kinetic and potential energy but could not predict their dependence on gait parameter. Energy fluctuation are urely important indicator of pendulum mechanic but cannot quantify work performed to redirect the COM elocity between pendulum-like tep. Preiou tudie hae alo plotted the COM trajectory in pace a a cloed path (Margaria, 1976), but again with little predictie ability. Our reult how that trend in both COM work and elocity change can be predicted a a function of gait parameter while alo remaining compatible with pendulum mechanic. Some important gait feature cannot be examined uing the imple model preented here. The model rigid leg perform a perfectly impulie tep-to-tep tranition and therefore can predict neither the duration nor the actual trajectory of COM diplacement a it i redirected. The duration of the tep-to-tep tranition, a defined by angular redirection or COM work (Fig. 7), appear coniderably longer than that of double upport. We originally ued COM work to indicate the imultaneou poitie and negatie work performed during double upport (Donelan et al., 22b) but later found puh-off work to begin earlier and colliion work to end later than that period (Donelan et al., 22a). The model perform minimum work by puhing off impuliely jut prior to colliion wherea human cannot produce ideal impule and mut perform more work oer finite time. The model cannot predict how the trailing leg begin extending prior to double upport or how the leading leg top flexing after double upport. Our reult here how that een though double upport capture much of the tep-to-tep tranition and it dependence on peed and tep length, it alo underetimate the total angular change in COM elocity by about 19%, and the aociated work by about 25% at the nominal walking peed. Thi behaior alo partially explain why the model predict trend in work but not the actual amount, equialent to the coefficient of the model fit (Eqn 6 12). The additional feature neceary to model thee effect may include axial leg compliance a ha already been incorporated in other tudie (e.g. Alexander, 1991; Buczek et al., 26; Geyer et al., 26). The ariable tudied here appear to be phyiologically releant. We preiouly hypotheized that work performed to redirect the COM elocity may explain a portion of the metabolic energy cot of walking. We hae found COM work to predict metabolic cot under controlled condition for ariable uch a tep length (Donelan et al., 22a), tep width (Donelan et al., 21) and the hape of rigid foot bottom (Adamczyk et al., 26). The preent tudy how that a imple dynamic walking model, depite it limitation, remain ueful for predicting trend in COM elocity and work. The trend may be predicted largely by imple gait parameter, uch a tep length and tep frequency, that can be controlled experimentally. Auming thee control do not induce abnormal gait, they make it poible to control for tep-to-tep tranition cot and thereby examine other contributor to the oerall energetic demand of normal walking. Finally, we alo peculate that the obered duration of the tep-to-tep tranition may be phyiologically adantageou. It allow the work for COM redirection to occur oer an extended time, poibly reducing the peak force and power requirement of mucle. Concluion Human leg are not perfectly rigid during either the inerted pendulum phae or the tep-to-tep tranition of walking. Depite thee deiation from model aumption, change in magnitude and direction of COM elocity are predicted well by dynamic walking model. Greater walking peed lead to greater COM elocity magnitude, and greater tep length lead to greater redirection angle. Thee ariable in turn predict work performed on the COM a major contributor to metabolic energy expenditure a a function of walking peed and

2676 P. G. Adamczyk and A. D. Kuo tep length. There are alo ubtantial limitation to rigid-legged model, namely their inability to predict the duration and diplacement of the tep-to-tep tranition. Empirical oberation how that the duration exceed double upport, with the non-rigid ingle upport leg contributing ubtantially to the tep-to-tep tranition. APPENDIX Step-to-tep tranition duration We examined two other definition for the duration of the tep-totep tranition in addition to that baed on center of ma (COM) elocity, τ el (ee Fig.A1). The firt, τ DS, wa baed on the double upport period a determined from the ertical ground reaction force (Donelan, 22b). The econd, τ work, wa baed on the interal oer which the trailing and leading leg perform work on the COM (ee Fig. A1B) (Donelan, 22a). Here we compare the leat-quare fit between model and data for all three of thee definition (ee TableA1). We found the oerall trend predicted by the model to be largely independent of the particular definition ued. All definition produced fit that indicated approximately linearly increaing trend. Thoe baed on COM elocity and work rate, howeer, agree better with each other and with data, yielding R 2 alue ranging.6 to.99. The definition baed on ertical GRF yielded imilar trend but omewhat poorer fit, with R 2 alue a low a.31. Thi i becaue the fraction of a tride pent in double upport decreae at fater tep frequencie or peed, o that double upport capture le of the total COM elocity redirection el (ee Fig.7) and COM work (Fig. A1). Preiou tudie hae alo reported that COM redirection occur oer more than double upport alone (Donelan et al., 22a). Puhoff poitie work begin prior to oppoite leg heel trike, and colliion negatie work end after oppoite leg toe-off. Thi caue etimate baed on double upport to underetimate the work, depite uccefully capturing the broad trend of work increaing with tep length. Both COM elocity and COM work are better delineator of the tep-to-tep tranition. Relatie timing of puh-off and colliion Ruina et al. preented a theoretical model of the tep-to-tep tranition in which the relatie oerlap of the puh-off and colliion impule can be aried within their infiniteimal duration [ee Appendix in Ruina et al. (Ruina et al., 25)]. They defined an oerlap parameter ( o ) to repreent how imultaneouly the puhoff and colliion impule occur. We computed an adapted form of o for human, modified to account for tranition of finite duration. We defined each leg net impule (P* PO, P* CO ) a the time integral of it three-dimenional GRF during the whole tep-to-tep tranition. A parameter q PO (t) quantifie the fraction of the net puhoff impule that ha been completed prior to time t. We defined thi puh-off fraction a the projection of the completed puh-off impule, P PO (t), along the net impule direction ector, P* PO, normalized by the magnitude of the net impule P* PO : q PO (t) = P PO(t) ˆP PO *. P PO * We defined q CO (t) imilarly with repect to the colliion impule P* CO. The oerlap parameter o i defined by the integral of the colliion fraction with repect to the puh-off fraction: o = 1 q CO dq PO. (A1) (A2) Vertical elocity (m 1 ) COM work rate (W) Vertical GRF (N).2.1.1.2 3 2 1 6 4 2 A Walking peed.8 1. 1.2 1.4 1.6 Forward elocity (m 1 ) B C Velocity Force Work Poitie puh-off work (PO) Work-baed tranition Negatie 1 colliion work 2 (CO).2.4.6.8 1. Time () 8 Double upport Puh-off impule Colliion impule HS TO.2.4.6.8 1. Time () Fig. A1. Three poible definition for duration of the tep-to-tep tranition, baed on (A) center of ma (COM) elocity, (B) COM work and (C) ertical ground reaction force (GRF), illutrated with a typical ubject nominal gait at 1.25 m 1. (A) Sagittal plane COM hodograph how trajectory of ertical and forward component of COM elocity. The tep-totep tranition may be alternatiely defined baed on the maximum angular excurion of COM elocity (τ el ; marked by circle), by interal of poitie and negatie work performed on the COM (τ work ; marked by triangle) and by the period of double upport from GRF (τ DS ; marked by quare). (B) The trailing leg perform poitie puh-off work on the COM, tarting lightly before double upport and ending with toe-off. The leading leg perform negatie colliion work, tarting with heel-trike and ending lightly after double upport end. Thee interal of poitie and negatie work were ued to define a work-baed tep-to-tep tranition (haded region), with the alternatie definition yielding different interal for computing work (circle and quare). (C) The double upport period ered a the third definition for the tep-to-tep tranition. Ground reaction force were integrated during the tranition to determine the impule in each leg, for puh-off and colliion (haded region, howing ertical component only; HS denote heel-trike and TO denote toe-off). Each definition i baed on a different ignal, but COM elocity and COM work both indicate that the tep-to-tep tranition occur oer a duration exceeding double upport. The oerlap parameter can ary from (puh-off entirely before colliion, a aumed for the model in the preent tudy) to 1 (colliion entirely before puh-off), with.5 indicating imultaneou impule. Subject in the current tudy demontrated ery conitent alue of o acro all condition. Uing tep-to-tep tranition baed on COM elocity, o wa.47±.26 (mean ±.d.), indicating that the puh-off impule occurred coniderably earlier than the colliion impule. There were light, tatitically ignificant decreaing trend in o with increae in peed, tep length and tep frequency. The

Redirection of center of ma elocity 2677 Table A1. Comparion of three different ignal for defining the tep-to-tep tranition, applied to een model prediction equation Equation (equation number in parenthee) =C +D (6) el=c el+d el (7) imp=c imp+d imp (8) W CO=C CO( el) 2 +D CO (9) W PO=C PO( el) 2 +D PO (1) W CO=C CO( ) 2 +D CO (11) W PO=C PO( ) 2 +D PO (12) Signal Slope C Offet D R 2 COM elocity 1.3±.9.11±.6.99 COM work rate.998±.7.5±.5.99 Vertical GRF 1.94±.18.17±.12.98 COM elocity.37±.37.88±.38.69 COM work rate.259±.25.69±.26.75 Vertical GRF.1±.38.217±.39.31 COM elocity.477±.21.37±.22.91 COM work rate.47±.19.8±.2.93 Vertical GRF.69±.24.67±.25.93 COM elocity.84±.68.9±.4.75 COM work rate 1.153±.88.8±.4.77 Vertical GRF 1.388±.181.9±.4.53 COM elocity.57±.64.17±.4.62 COM work rate.74±.89.17±.4.6 Vertical GRF.688±.147.19±.3.4 COM elocity.128±.8.8±.3.82 COM work rate.129±.8.8±.3.81 Vertical GRF.86±.9.1±.3.65 COM elocity.81±.9.17±.3.66 COM work rate.82±.9.17±.4.63 Vertical GRF.46±.8.19±.3.49 Signal were baed on center of ma (COM) elocity, the interal oer which poitie and negatie work i performed on the COM, and ertical ground reaction force (GRF) indicating double upport. The bet-fit coefficient (C and D) are reported, along with R 2 alue. tronget trend wa for peed and had a bet linear fit of o =.124+.11, R 2 =.49. Other tep-to-tep tranition definition captured horter duration, which led to higher alue of o. Thi change wa expected, becaue retricted tep-to-tep tranition timing cut off the beginning of puh-off and the end of colliion. For timing baed on COM work rate, o wa.8±.39 (mean ±.d.); for timing baed on ertical GRF, o wa.151±.27. Trend line were imilar with timing baed on COM work rate; howeer, trend differed for timing baed on ertical GRF, howing that o decreaed for increaing tep frequency, increaed with increaing tep length and had no trend with repect to peed. Regardle of the timing bai, o wa alway le than.24, indicating that the puh-off impule alway occurred coniderably earlier than the colliion impule. LIST OF SYMBOLS AND ABBREVIATIONS C, C caling coefficient of linear model fit to data CF contant tep frequency COM center of ma CS contant tep length CV contant peed D, D offet coefficient of linear model fit to data f* preferred tep frequency at deignated walking peed * GRF ground reaction force L leg length M body ma P* CO net colliion impule P* CO direction ector of the net colliion impule P CO (t) cumulatie colliion impule that ha occurred before time t within the tep-to-tep tranition P* PO net puh-off impule P* PO direction ector of the net puh-off impule P PO (t) cumulatie puh-off impule that ha occurred before time t within the tep-to-tep tranition q CO (t) cumulatie fraction of the net colliion impule completed prior to time t within the tep-to-tep tranition q PO (t) cumulatie fraction of the net puh-off impule completed prior to time t within the tep-to-tep tranition tep length o oerlap parameter for puh-off and colliion impule * preferred tep length at deignated walking peed * mean walking peed * deignated nominal walking peed 1.25m 1 mid COM elocity at the middle of the model tep-to-tep tranition, between the puh-off and heel-trike impule COM elocity at the end of the tep-to-tep tranition COM elocity at the beginning of the tep-to-tep tranition W work W CO colliion negatie work W PO puh-off poitie work directional change in elocity in imple model imp angular change in ground reaction impule during tep-to-tep tranition el angular change in center-of-ma elocity during tep-to-tep tranition g graitational acceleration τ DS duration of tep-to-tep tranition baed on the double upport period a determined from ground reaction force τ el duration of tep-to-tep tranition baed on COM elocity change τ work duration of tep-to-tep tranition on the interal of COM work for puh-off and colliion Thi work wa upported in part by NIH grant R44HD5576. Depoited in PMC for releae after 12 month. REFERENCES Adamczyk, P. G., Collin, S. H. and Kuo, A. D. (26). The adantage of a rolling foot in human walking. J. Exp. Biol. 29, 3953-3963. Alexander, R. M. (1991). Energy-aing mechanim in walking and running. J. Exp. Biol. 16, 55-69. Alexander, R. M. (1992). A model of bipedal locomotion on compliant leg. Philo. Tran. R. Soc. Lond. B Biol. Sci. 338, 189-198. Bertram, J. E. A., Pardo, J., D Antonio, P. and Lee, D. V. (22). Pace length effect in human walking: groucho gait reiited. J. Mot. Beha. 34, 39-318.