Optimal Walking in Terms of Variability in Step Length

Optimal Walking in Terms of Variability in Step Length Noboru Sekiya, PT, MA ' Hiroshi Nagasaki, PhD Hajime Ito, PT, MA Ta keto Furuna, PT R esearchers studying human movement have attempted to define skilled motor performance by employing a range of kinematic variables, such as movement speed, movement time, timing accuracy, etc. In recent research, temporal and spatial intrasubject variability in kinematic variables is regarded as an important measure of motor skill and practice effects. Intrasubject variability of kinematic variables is an index of movement consistency or stability and thereby provides us with a measure to evaluate motor skill for a given task (8). The variability was examined in human gait, in particular, for its temporal and spatial parameters. It was expected that stability of gait could be assessed using the variability measure, especially for older persons (3-5) and physically disabled subjects (1). In fact, there was a negative correlation between variability in step width and balance performance for women 60 years of age and over (5) and also an increased variability in step length for hospitalized fallers compared with nonfallers (4). The results suggest that the variability measure can be an index of stability of gait performance. These studies, however, measured variability in the gait parameters at the preferred speed of each subject. Maruyama and Nagasaki (8) reported The optimal condition in speed, step rate, and step length of human walking has been reported in terms of temporal consistency, energy cost, and attentional demand. No study, however, has been conducted on the optimal condition in terms of spatial variability of walking. This study examined whether there is an optimal walking speed with minimum intrasubject variability in step length and step width during free walk (experiment 1) and whether there is an optimal step rate with minimum step length variability during walking with imposed step rates (experiment 2). Wearing shoes with ink-applied felt squares attached to the heels, healthy students walked on a flat walkway (0.6 x 16 m) at five different speeds with a freely chosen step rate in experiment 1 and walked at three different speeds with five different step rates in experiment 2. Free walk was found to have the fewest variable errors (VEs) in step length approximately at preferred walking speed. Variable error in step width increased linearly with an increase in walking speed. Under imposed step rates, VEs in step length were the fewest when walking with step rates close to those in free walk. Our everyday walking is performed most frequently at preferred speed and/or with freely chosen step rate, thereby optimizing the consistency of gait performance. Intrasubject variability in step length may be a useful measure for evaluation of walking. Key Words: gait, step length, variability, step width ' Associate Professor, Department of Physical Therapy, College of Medical Sciences, Showa University, 1865, Tokaichiba-cho, Midori-ku, Yokohama, Kanagawa, 226, lapan Chair, Department of Kinesiology, Tokyo Metropolitan Institute of Gerontology, Tokyo, japan ' Professor, Department of Physical Therapy, lbaraki Prefectural University of Health Sciences, Ibaraki, japan ' Researcher, Department of Kinesiology, Tokyo Metropolitan lnstitute of Gerontology, Tokyo, lapan for treadmill walking of normal sub jects that intrasubject variability in the duration of every walking phase decreased with increased speed, thereby supporting the hypothesis that faster movement is more consistent than slower movement. Furthermore, they showed that the coeficients of variation (CV) were the lowest at the specific step rates (cadence) that were linearly dependent on walking speed. The relationship between speed and step rate for walking with the lowest temporal CV was very close to the speed-rate relationship found in free walk. They suggested that stability of gait in terms of CV for the duration of the walking phase was optimal for walking with a freely chosen step rate at any given speed. There is also an optimal method of walking, both in terms of energy cost and attentional demand. At preferred walking speed, energy consumption per unit of distance was at a minimum (13). Furthermore, freely chosen step rates required the least oxygen consumption at a given speed (6,17). Kurosawa (7) measured reaction time to a probe during treadmill walking and demonstrated that walking at a subject's preferred speed called for a minimum attentional de- Volume 26 Number 5 November 1997 JOSPT

RESEARCH STUDY start / Isnm steps \ heel I mark finish step width ; SWi = WRi - WLi step length; SLn FIGURE 1. Walkway. WR,, WL, = Distance from the edge of the walkway to heel marks. SL,, = Step length. n = Number of measured steps. mand, while walking at higher or lower speeds required more attention. Thus, an optimal relationship between speed and step rate exists in terms of temporal consistency, energy efficiency, and attentional demand. No study, however, has been conducted on the optimality in terms of spatial variability of walking. The purpose of this study was to examine whether there is a walking speed with the least variability in step length and step width during free walk (experiment 1) and whether there also exists an optimal relationship between step rate and step length in terms of the step length variability (experiment 2). As a result, this research may provide an index of optimality for gait training in physical therapy and rehabilitation. EXPERIMENT 1 METHODS Subjects Twenty-two students with no neurological or orthopaedic disorders [lo males and 12 females, mean age = 25.9 (SD = 4.1) and 20.6 (SD = 1.4) years, respectively] participated in the experiment. Mean height was 172.8 cm (SD = 5.3) for males and 157.0 cm (SD = 5.0) for females. Walk Test The subjects were instructed verbally to walk on a flat walkway (0.6 m X 16 m) (Figure 1) at five differ- ent speeds: preferred, slow, slowest, fast, and fastest speeds. The subjects regulated these different speeds themselves relative to their preferred walking speeds. They walked in their comfortable way under each speed condition with no instructions on step length and step rate, ie., free walk. The subjects performed one trial after several practicq trials in each condition. In order to cancel out possible order effects, half of the subjects walked in the order mentioned above, and the remaining half walked at preferred speed first, followed by fast, fastest, slow, and slowest. The subjects wore shoes with inkapplied felt squares (15 X 15 X 3 mm) attached to the heels (2). Ink was supplied for each trial so that heel positions were clearly recorded on the walkway. JOSPT Volume 26 Number 5 * November 1997

M of SW (cm) (an) (mlc Experimental V (m/min) SL (m) ve SR (stepshin) sw (an) Conditions X SD X SD X SD X SD X SD X SD X Male Slowest Slow Preferred Fast Fastest Female Slowest Slow Preferred Fast - Fastest V = Velocity; SL = Step length; SR = Step rate; SW = Step width; SUSR = Step length divided by step rate; VE = Variable error. TABLE 1. Means and standard deviations of walking parameters in each experimental condition. - Walking speed was measured in the middle 10 m of the walkway. Step length, step width, and number of steps were measured using the heel marks recorded toward the middle of the walkway from the first step beyond the 3m mark to that beyond the 13-m mark. Figure 1 illustrates the measurement of step length and step width, to the nearest millimeter, from heel marks on the walkway with a grid graduated in centimeters. The measurements were taken from the midpoint of each heel mark. Using the data from total steps of 9 to 23, mean speed (V), step rate (cadence, SR), step length (SL), and step width (SW) as well as intrasubject variable error [VE, ie., standard deviation (SD) in each trial] and coefficient of variation (CV = mean/sd) in step length and step width were computed for each trial under each speed condition. Since there was no difference in these means between right and left steps, the results were described below in terms of the mean values for both legs. Analyses Analyses of variance with repeated measure on speed and variability were used to compare group means. In case of missing data, all of the data of the subject were excluded in the analyses. Regression analyses were used to relate VEs or CVs in step length or step width to walking speed and step rate. Table 1 shows means and standard deviations of walking parameters under the five speed conditions. There was one missing value in the fastest speed condition. A 2 X 5 analysis of variance (gender X speed condition) with repeated measures showed a main effect of speed condition on speed (F(4,76) = 259.8, p < 0.01), but no effects of gender (F(1, 19) = 1.3, p > 0.05) nor speed X gender interaction (F(4.76) = 0.4, p > 0.05). The post hoc tests showed that walking speed was different between every two conditions (Fisher's PLSD, p < 0.05), indicating that the subjects walked at five different speeds as instructed by the experimenters. Step length and step rate also increased consistently with increase in speed for the pooled data (r = 0.94, r = 0.96, respectively). Step width, however, did not show any speed dependency. The ratio of step length to step rate [SL/SR or walk ratio; Nagasaki et a1 (9)] was nearly constant at 0.006 m/steps/min regardless of walking speed for males and also for females at preferred, fast, and fastest speeds in accordance with our previous study (16). In Figure 2, VEs in step length and step width averaged across subjects are illustrated as a function of the speed condition. The VEs in step length had a U-shaped relation to speed with the lowest VE at the preferred speed. A 2 X 5 analysis of variance (gender X speed condition) with repeated measures (Table 2) showed a main effect of speed condition (F(4,76) = 6.59, p < 0.01), but no effects of gender (F(1,19) = 0.39, p > 0.05) nor speed X gender interaction (F(4.76) = 1.71, p > 0.05). Figure 3 indicates a U-shaped relationship of VEs in step length against step rate or step length for data from all subjects. Regression FIGURE 2. lntrasubject variable error WE) in step length and step width as a function of the speed condition. Volume 26 Number 5 November 1997 JOSF'T

RESEARCH STUDY Source Sum of Mean Square df Sauare Gender 1.633.633 0.39 0.54 Subject (Gender) 19 30.969 1.63 speed 4 24.891 6.223 6.59 0.0001 Speed x Gender 4 6.463 1.616 1.71 0.1 6 Speed x Subject (Gender) 76 71.851 0.945 Total 104 134.807 Dependent variable: Variable error in step length. TABLE 2. Analyses of variance on variable error in step IenRth. analyses showed that the quadratic regressions fit the data better than linear ones for step rate and step length. The regression equations were as follows: and The VEs had a minimum at a step rate of 117 steps/min and at a step length of 0.679 m, which was in the vicinity of preferred step rate (107 steps/min) and step length (0.73 m). These minimums were similarly observed for CVs related to speed, step length, and step rate. The VEs in step width, on the other hand, increased linearly as speed (Figure 2) (F(4,76) = 10.5, p < 0.01) and step rate increased (r = 0.42). The analyses for CV showed similar results. In short, free walk at a variety of speeds was found to have the least variability in step length approximately at speed, step length, and step rate of preferred walking. Step width variability increased linearly with increased speed. EXPERIMENT 2 METHODS Subjects Ten female students without neurological or orthopaedic disorders [mean age = 23.3 years (SD = 3.3), mean height = 156.9 cm (SD = 4.6)] participated in the experiment. They were different from those who participated in experiment 1. Walk Test The walkway M in Step length F subjects walked on the same (0.6 m X 16 m) as in experi- 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 m Step length (SL) B FIGURE 3. lntrasubject variable error (VE) in step length as a function of A) step rate and B) step length. ment 1, first at a preferred speed with a freely chosen step rate (preferred walk). The preferred walk was followed by forced walking, in which the subjects walked with imposed step rates at each speed verbally instructed: preferred, slow, and fast. The subject had a small electric metronome (84 X 54 X 8 mm, 25 g) on her chest or waist and was instructed to walk with steps in time to metronome sounds under each speed condition. The metronome sounds were set at 80, 100, 120, 140, and 160 beats/min and given to the subjects in this order. Half of the subjects walked in the order of speeds mentioned above, and the remaining half walked in the following order: preferred, fast, and slow. For each subject, data were recorded for one walking trial after several practice trials were performed for each condition. The same measurement procedure as in experiment 1 was used in experiment 2, except that step width was not measured. RESULTS Table 3 shows means and standard deviations of gait parameters in every walking condition. There was one missing value in the preferred speed, 160 step rate condition. The mean step rates indicated that the subjects walked under each speed condition in time to metronome sounds as requested by the experimenters. At a given step rate, the subjects walked at different speeds as instructed (F(2.16) = 143.2, P < 0.01), but walking speed drifted slightly higher with increase in step rate under a given speed condition. Thus, step length did not decrease in a monotonic fashion for each speed condition. Figure 4 shows mean intrasubject variable error in step length in each experimental condition. The VEs showed a tendency to reach a minimum at a step rate of 120 steps/min and the minimum VE was smaller at JOSPT Volume 26 Number 5 November 1997 269

RESEARCH STUDY Metronome (beablmin) 80 100 120 140 160 Preferred X SD X SD X SD X SD X SD X SD Fast SL 0.867 0.070 0.904 0.079 0.930 0.062 0.883 0.067 0.865 0.097 VE of SL 3.6 0.86 2.6 0.74 2.5 0.74 3.3 1.57 3.2 0.57 Preferred SL 0.611 0.085 0.650 0.067 0.687 0.069 0.679 0.063 0.633 0.096 0.714 0.0071 VE of SL 2.4 0.84 2.5 0.64 2.0 0.58 2.5 0.89 2.9 0.68 1.8 1.OO SR 80.1 0.7 100.4 0.8 120.8 0.8 140.4 0.7 158.2 3.1 116.2 6.2 V 49 6.9 65.2 6.8 83.0 8.4 94.7 9.2 100.2 15.8 83.2 10.9 SVsR 0.0076 0.0010 0.0065 0.0007 0.0057 0.0006 0.0048 0.0004 0.0040 0.0006 0.0062 0.0006 Slow SL 0.534 0.077 0.487 0.080 0.443 0.065 0.447 0.077 0.427 0.088 VE of SL 2.9 0.72 3.0 1.09 2.7 0.67 3.0 0.87 2.6 0.87 SR 79.9 0.7 100.1 0.5 120.4 0.5 139.7 1.1 159.1 1.2 V 42.6 6.2 48.7 8.0 53.3 7.8 62.4 10.9 67.8 14.3 SVSR 0.0067 0.0010 0.0049 0.0008 0.0037 0.0005 0.0032 0.0005 0.0027 0.0006 SL = Step length (m); VE = Variable error km); SR = Step rate (stepshin); V = Velocity (dminl; SUSR = Step length divided by step rate (dstepdmin). TABLE 3. Means and standard deviations of walking parameters in each experimental condition. the preferred speed than at slow and fast speeds. The CVs also had minimums at 120 steps/min except at slow speed. In order to approximate VE as a function of both step length (SL) and step rate (SR), a second-order polynomial regression analysis was performed using VE as a dependent variable and SL and SR as independent variables. The regression equa- in Step length hst preferred slow 80 100 120 140 160 stsps/mh Step rate preferred walk FIGURE 4. lntrasubject variable error length as a function of step rate. in step tion for data averaged across subjects was as follows (r = 0.65, N = 15): Using this equation, Figure 5 illustrates VE as a function of step length and step rate. Each ellipsoidal curve indicates a contour of an equal VE. The ellipsoidal contours were drawn by substituting each VE value (2.5 to 5.0 cm at 0.5 cm intervals) into the above equation. Figure 5 clearly demonstrates that VEs in step length were dependent not only on speed but also on step length and/or step rate. During free walk, in which speed may vary to keep the walk ratio (SL/SR) invariant at 0.006 m/steps/ min, VE reaches a minimum at a speed about 80 m/min. Alternatively, when walking with forced step rates under a given speed, Figure 5 indicates that VE reaches a minimum at a walk ratio of about 0.006 m/steps/ min. DISCUSSION When one performs a discrete me tor task repeatedly or performs a cyclic motion, such as walking and finger tapping, intrasubject variability (VE or CV) in the kinematic variables is thought of as an index of consistency or stability of the movement (8,14). The variability in movement duration and amplitude may be called temporal and spatial consistencies, respectively. The present study examined the spatial consistency in step length and step width during free walk at a variety of speeds (experiment 1) and also during forced walking with imposed step rates at a given speed (experiment 2). The results of experiment 1 demonstrated that the variability in step length exhibited a minimum at a walking speed, step rate, and step length that were close to those of preferred speed walking. Free walk as examined in experiment 1 was characterized by an invariant ratio of step length divided by step rate (walk ratio), except for walking at extremely slow speeds. The results of experiment 2 suggest that free walk with the invariant walk ratio has the least Volume 26 Number 5 November 1997 JOSPT

m 1.5 1.o Step length 0.5 50 100 I so 200 steps/min Step rate FIGURE 5. Contours of variable e m WE) in step length as a function of step length and step rate. variabilities in step length (Figure 5). Maruyama and Nagasaki (8), on the other hand, reported that the temporal variability in phase durations within a walking cycle was a decreasing function of speed. The speed range examined by Maruyama and Nagasaki, however, was limited to below 100 m/min because walking was done on a treadmill, and their data suggest that the variability would reach a minimum at about 120 m/ min when examined at speeds beyond 100 m/min. Increased walking speed produced a linear increment in step width variability in contrast to the step length variability. Gabell and Nayak (3) suggested that step length is determined predominantly by gaitpatterning mechanisms, whereas step width is largely determined by balance-control mechanisms. Heitrnann et a1 (5). in fact, showed a significant negative correlation between balance performance and variability in step width but not between balance performance and step length. It is suggested, therefore, that the linear increase in the step width variability with walking speed found in the JOSPT Volume 26 Number 5 November 1997 present study was related to a speeddependent balance function during gait but not to rhythmicity control of walking. Experiment 2 of this study examined walking at three different speeds with imposed step rates. The result indicated ratedependent minimum variabilities of step length; the variability is the least when one walks by keeping the ratio of step length to step rate to about 0.006 m/steps/ min. This invariant relationship between step length and step rate was the characteristic found for free walk in experiment 1. Maruyama and Nagasaki (8) reported that the consistency in terms of CV in the duration of walking phases was optimal for walking with freely chosen step rate. The vertical head displacement is reported to have a minimum standard deviation at around the preferred step rate (6). Taken together, it is evident that free walk is optimal not only in terms of temporal variability but also in spatial variability. Cyclic motions have been reported to have optimal rates in terms of temporal variability. Nagasaki and Nakamura's research (1 1) on finger It is evident thaf free walk is optimal not only in terms of temporal variability but also in spatial variability. -*-."..-.-..- -- *.--- tapping revealed that relative variability (CV) in the intertap intervals reached a minimum at a tapping frequency of about 3 Hz. A push-pull movement of an arm paced by a metronome showed a similar tendency (10). It is also reported that stepping movement is the most consistent in terms of variability of the lateral component of ground reaction force and also of step duration when stepping at rates similar to those of preferred stepping and walking (15). These ratedependencies of the variability in cyclic motions may be contrasted to discrete motor tasks in which it is generally argued that faster movements are temporally more consistent (12). Since walking speed is determined by the product of step length and step rate, an infinite set of step lengths and step rates may exist in walking at a given speed. When walking freely, however, we are used to coordinating step length and step rate so as to keep the walk ratio invariant. It is suggested that the invariant walk ratio is a result of the constraints that require one to walk with the maximum energy efficiency (9, 18). Free walk that keeps the walk ratio invariant is actually optimal in terms of energy consumption per distance (17.18). As a result, our everyday walking may be performed most frequently at preferred speed and/or with invariant walk ratio regardless of speed. Thus, our everyday walking is optimizing temporal and spatial consistency and attentional demand. These optimality criteria in.-*.-

RESEARCH STUDY.. -...-. -.- - terms of energy efficiency, temporal and spatial variability, and attention would provide us with useful measures for evaluating walking skill. Previous studies examined intrasubject variability for walking of physically disabled subjects (1) and older persons (3-5). Although the variability measured was expected to relate to disability or aging, the results were inconsistent among authors. The variability of stride length in Parkinson patients was more marked than controls and increased as a function of Yahr's clinical stages (1). There was a negative correlation between variability in step width and balance performance for women of 60 years and over (5) and also an increased variability in step length for hospitalized fallers compared with nonfallers (4). Gabell and Nayak (3), however, did not find the effects of age on variability in step length and step width in walking. The inconsistency in the previous studies may be due to the fact, at least in part, that none of the studies considered that the variability of walking is speed and rate dependent. Our present study showed in healthy subjects that the variability of walking is clearly dependent both on walking speed and rate and thereby provided speed- and ratedependent optimality criterion for spatial variability of walking. It is thus suggested that the criterion of spatial consistency of walking is useful for evaluating gait in clinical and practical settings. CONCLUSION The present study found that the variability in step length was minimum at a preferred walking speed and also at free walking with the invariant ratio of step length divided by step rate (walk ratio). Together with findings in previous studies, it is suggested that free walk with the invariant walk ratio is optimized in terms of energy efficiency, temporal and spatial consistency, and attentional demand. Thus, step length variability, determined at a variety of walking speeds and step rates, may provide a criterion for evaluating walking skills in clinical and practical settings. JOSPT REFERENCES 1. Blin 0, Ferrandez AM, Serratrice G: Quantitative analysis of gait in Parkinson patients: Increased variability of stride length. J Neurol Sci 98:9 1-97, 1990 2. Boenig DD: Evaluation of a clinical method of gait analysis. Phys Ther 57: 795-798, 1977 3. Gabell A, Nayak USL : The effect of age on variability in gait. J Gerontol 39: 662-666, 1 984 4. Guimaraes RM, lsaacs B: Characteristics of the gait in old people who fall. Int Rehabil Med 2: 1 77-180, 1 980 5. Heitmann DK, Gossman MR, Shaddeau SA, Jackson JR: Balance performance and step width in noninstitutionalized, elderly, female fallers and nonfallers. Phys Ther 69:923-93 1, 1989 6. Holt K, Jeng SF, Ratcliffe R, Hamill J: Energetic cost and stability during human walking at the preferred stride frequency. J Mot Behav 27:164-178, 1995 7. Kurosawa K: Effects of various walking speeds on probe reaction time during treadmill walking. Percept Mot Skills 78:768-770, 1994 Maruyama H, Nagasaki H: Temporal variability in the phase durations during treadmill walking. Human Move Sci 11:l-14, 1992 Nagasaki H, Ito H, Hashizume K, Furuna T, Maruyama H, Kinugasa T: Walking patterns and finger rhythm of older adults. Percept Mot Skills 82:435-447, 1996 Nagasaki H, Nakamura R: Rhythmic control of a push-pull movement. Percept Mot skills 5 i:747-75 1, 1980 11. Nagasaki H, Nakamura R: Rhythm formation and its disturbances; A study based upon periodic response of a motor output system. J Human Ergo1 1 1: 127-142, 1982 12. Newel1 KM: The speed-accuracy paradox in movement control: Errors of time and space. In: Stelmach GE, Requin J (eds), Tutorials in Motor Behavior, pp 501-509. Amsterdam: North- Holland Publishing Company, 1980 13. Ralston HI: Energy-speed relation and optimal speed during level walking. Int Z Ph ysiol Arbeitsph ysiol 7:277-283, 1958 14. Schmidt RA: Motor Control and Learning (2nd Ed), Champaign, IL: Human Kinetics Publishers, 1988 15. Sekiya N, Nagasaki H, /to H, Furuna T: The ratedependency of force variability in stepping movement. In: Proceedings of the Fifth General Assembly of the Asian Confederation for Physical Therapy, Taipei, Taiwan, R.O.C., September, 1993, pp 226-230. Taipei, Taiwan: Physical Therapy Association of The Republic of China, 1993 16. Sekiya N, Nagasaki H, Ito H, Furuna T: The invariant relationship between step length and step rate during free walking. J Hum Mov Stud 30:241-257, 1996 17. Zarrugh MY, Radcliffe CW: Predicting metabolic cost of level walking. Eur / Appl Physi0138:2 15-223, 1978 18. Zarrugh MY, Todd FN, Ralston HJ: Optimization of energy expenditure during level walking. Eur J Appl Physiol 33:293-306, 1974 Volume 26 Number 5 November 1997 JOSPT