FREE VERTICAL MOMENTS AND TRANSVERSE FORCES IN HUMAN WALKING AND THEIR ROLE IN RELATION TO ARM-SWING

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1 The Journal of Experimental Biology 24, 47 8 (21) Printed in Great Britain The ompany of Biologists Limited 21 JEB FREE VERTIAL MOMENTS AND TRANSVERSE FORES IN HUMAN WALKING AND THEIR ROLE IN RELATION TO ARM-SWING YU LI*, WEIJIE WANG, ROBIN H. ROMPTON AND MIHAEL M. GUNTHER Department of Human Anatomy and ell Biology, The University of Liverpool, Liverpool L69 3GE, UK * mj2@liverpool.ac.uk Accepted 19 October; published on WWW December 2 We present force plate data on vertical free moments (force couples in the horizontal plane between the foot and the ground) and on transverse force during unloaded walking in different modes and at different speeds (including running) by adults of both sexes and by children, and examine loaded walking by adult males and one boy. Free moments in slow and normal-speed walking are characterised by a lateral peak in the accelerative phase of stance, but the peak during running, and in some cases of fast walking, occurs in the deceleration phase. Free moments are strongly affected by arm fixation in males, but less so in females. The pattern, but not the scale, of free moments is affected by loading position and side, but load magnitude has little effect if the loaded weight is treated as part of the body. Transverse force is more variable than sagittal force. In males, the transverse force curve shows a Summary marked trough at mid-stance, whereas in females this trough is rarely seen. The transverse force of males also differs from that of females in response to arm fixation, showing a local medial inflection at three-quarters of the stance phase that is not present in females. Adults differ from children younger than 9 years in the presence of a very short, medially directed peak following heel-strike. Analysis of the effects of arm fixation and the timing of forces suggests strongly that arm-swing and free moments tend to reinforce each other in balancing trunk torques induced by the lower limbs. Both are of reduced importance in slow walking. Key words: biomechanics, force couples, human gait, kinetics, torque, transverse force, walking. Introduction In human walking, external forces act in three dimensions to accelerate the body. Sagittal forces are involved in propulsion and deceleration, and vertical forces both support the body weight and play an important role in the exchange of potential and kinetic energy (avagna et al., 1977). Both of these have been well researched. Transverse forces are relatively small and have received relatively little attention. Transverse and sagittal forces can both, however, change the conditions of rotation of the body about a vertical axis, as can a further force component, the vertical free moment. Rotational forces have long been thought to be implicated in arm-swing (e.g. Elftman, 1939), and the form of anatomical structures such as the heel pads (Blechschmidt, 1934) has also suggested a functional link to rotation, but technical difficulties have precluded more detailed study. Similarly, the difficulty of measuring transverse forces may explain why these have received little attention other than in normative studies (see, for example, hao et al., 1983; Schneider and hao, 1983). Each of the force components changes over the course of a gait cycle. For a level movement at a constant speed, the acceleration conditions in each direction are the same at the end as at the beginning of a cycle (although forces and accelerations fluctuate during the course of the cycle). The integrals of each of the forces over a time period are constant (zero for horizontal forces and mg for the vertical force (where m is mass and g is the gravitational constant). If we assume that the left and right limbs act symmetrically, the vertical and sagittal components for the lower limb will undergo velocity changes and return to the starting conditions half-way through a gait cycle. In other words, their cycle period is only half that of the gait circle. However, the cycle period of transverse forces and free vertical moments is the same as the gait cycle period. The free vertical moment is a force couple and, as such, has the same action on any vertical axis regardless of its position. However, the rotation effect of the horizontal forces (transverse and sagittal) will depend on the position of the axis. For a free body, the axis that passes through the centre of mass gives the smallest moment of inertia, and this axis is therefore of particular importance. Its exact position is not readily determined, but it is possible to analyse the direction and relative value of the torque about this axis. Arm-swing also produces a vertical force couple. A common feature of arm-swing and the major vertical moment during

2 48 Y. LI AND OTHERS Table 1. Details of subjects Weight Identity Sex Age (N) M A 618 Y M A 67 R M A 83 I M A 777 M M A 813 GI F A 6 IN F A 476 LI F A 6 GW F A 67 JU F A 3 QI F A 1 EA F JE F 9 4 JH M 9 28 JI M 7 21 JS M 7 21 YU M 7 23 M, male; F, female; A, adult; for children, a number indicates the age in years. walking is that they both affect the angular acceleration of the trunk (and of the lower limbs) in the vertical direction. In practice, it is difficult to measure the moment produced by a swinging arm, but a method for measuring the free moment on the ground is available. This study examines the characteristics of transverse forces and vertical free moments and the effects of torque produced by horizontal forces during the single- and double-support phases of the gait cycle. Our findings are based on experimental force plate measurements on male and female human adults and children, in a variety of modes of loaded and unloaded walking. Finally, we address the significance of armswing by examining the consequences of its enforced absence, and attempt to analyse aspects of the interaction between armswing and torque during walking. Materials and methods Seventeen subjects (Table 1), including five adult males, six adult females and six children, were observed in the first set of experiments. The age of the adults ranged from 2 to 4 years and that of the children from 7 to 1 years. No subject had obvious limb abnormalities. Subjects walked or ran barefoot along a wooden walkway, into which a Kistler 9281B force plate was inset, and they were asked to make foot contact with the force plate as close to its centre as possible. Details of the experimental arrangement are given by Li et al. (Li et al., 1996). The force plate was used to record the foot contact forces in different locomotor modes, including running, walking at subject-defined slow, comfortable and fast speeds and walking with fixed arms. In fixed-arm walking, subjects were asked to allow their upper arms to hang Direction of locomotion y x y Fig. 1. oordinates of the measurement system. M z, moment about the vertical axis. vertically, with the forearms placed horizontally across the front and back of the trunk. Because of practical difficulties, not every subject participating in the experiments performed all the locomotor modes. The effects of loading were examined in a smaller series of subjects: four adult males and one boy. In this experiment, the adult subjects were asked to walk over the force plate while carrying 1 and 2 kg weights. The boy carried only the 1 kg load. Three modes of load-carrying were investigated. The subjects were asked: (i) to support the weight on their left or right shoulder; (ii) to support the weight on the left or right side of the trunk; and (iii) to support the weight around their neck. The first two modes represent asymmetric loading, and the third represents symmetrical loading. If subjects stepped onto the force plate using the right foot, the sign of the transverse force (the x axis in the coordinate system used here, see Fig. 1) was corrected by multiplying by 1. Therefore, a positive value for the x force represents a medially directed component and a negative value a lateral component. The three-dimensional contact forces were measured directly by the force plate, and the free moment (M z ) about the vertical axis was calculated from the equation: M z = M z F y a x + F x a y, (1) where M z is the vertical torque about the centre of the force plate, F is the recorded force and a is the distance between the position of the centre of pressure and the centre of the force plate. Subscripts x and y denote components in the threedimensional coordinate system illustrated in Fig. 1. More detailed descriptions of these components are provided in the Kistler 9281B user s manual. alculated moments also have the convention that a positive value represents a medial direction, and vice versa, as a consequence of pre-treatment of the x force in cases of right foot contact. The direction of a medial moment is here defined following the anatomical usage of medial or lateral rotation towards the trunk axis. Thus, a x z z M z

3 Horizontal torque and arm-swing during walking 49 medial moment may be expressed in linear terms as a moment tending to move anterior points medially. Results Transverse forces in adults and children Fig. 2A shows the transverse forces exerted by the subjects in each locomotor mode. Apart from the initial reversal of force from medial to lateral observed previously (Giakas et al., 1996), the curves for transverse forces we recorded for male adults (Fig. 2A) resemble the typical pattern of vertical force curves, with two humps, separated by a trough at approximately mid-stance. In females, with one exception (QI), the trough at mid-stance is not so clearly expressed (Fig. 2B). The general pattern for females is that the force is larger in the first half of the curve and gradually diminishes in the second half, with some fluctuations. For both males and females, transverse reaction force curves in adults typically displayed a sharp initial medial peak of opposite sign to the main (lateral) force. All adult subjects except LI showed this pattern in normal walking. Kinematic recording show that this female had a much larger carrying angle than the other subjects (here the carrying angle is the angle between the tibia and a line extended from the thigh), so that her feet contacted the ground further away from the mid-sagittal plan than in the other subjects (Fig. 3). The pattern for young children typically lacks the medial force peak at the beginning of stance (Fig. 2). Two 7-year-old boys, YU and JI, and two 9-year-olds, boy JH and girl JE, lacked any medial force in the mean values. One 7-year-old boy, JS showed a very small medial force occasionally, but only in fast walking (This property is not shown in the mean values, because other records without the medial force have larger weight.) Only one 1-year-old girl, EA, showed the adult pattern. Table 2 lists the standard deviations of the transverse and vertical forces for each locomotor mode. Standard deviations were calculated as the mean standard deviation of each force component during the stance phase (see Table 2 legend). Variation in the medially directed transverse force is in general smaller than that for the sagittal component. However, the mean force in the transverse direction is only approximately one-quarter of the magnitude of the mean sagittal force and approximately one-twentieth of the mean vertical force (Table 2). Thus, in relative terms, the transverse force component is more variable than the sagittal force component. The r 2 values give a better indication of the variability of forces than do the standard deviations because they take into account only the trends, not the value itself. The r 2 value for sagittal (and vertical; results not shown) forces were nearly always greater than.98, indicating a very uniform shape for the force curves over a series of trials. In contrast, r 2 values for transverse forces occasionally fall below.8 (Table 2). Relative transverse forces for children also differ from those of adults. When expressed as a proportion of the vertical force (see F x / F z in Table 2), the transverse forces of most adults are around 3 6 %, whereas those of children are 7 % or higher, with exception of the oldest child, EA. Vertical moment Apart from a brief medial peak shortly after heel-strike, vertical free moments during stance are predominantly in a lateral direction and tend to be greater during the second half of the stance phase, when the centre of pressure is approximately below the metatarsal heads (Fig. 4). The vertical moment during stance clearly differs between males and females: all male subjects show considerably larger lateral rotation values in fixed-arm walking than in other modes, but in females, with the exception of GW, the moments in fixedarm walking are similar to those in other modes. The peak laterally directed vertical moment consistently occurs at approximately three-quarters of the stance phase (around the 3th of the 41 data point). At this point, the standard error of the mean (S.E.M.) is approximately one-third of the standard deviation (S.D.) for N=1 trials (e.g. for fixedarm walking in subject, the S.D. at the 31st data point is 1.67, and the S.E.M. is 1.67/ 1=1.67/3.16=.3). Relationship between vertical moments and locomotor mode and speed The vertical moment in running is highly variable; a typical curve for running is shown in Fig.. The largest vertical moments recorded during running are similar to those recorded during walking (compared with Fig. 4). However, the timing of the peak vertical moments differs substantially between running and walking at slow and comfortable speeds. In walking, there tends to be a small medially directed (positive) moment at the beginning of stance, which becomes laterally directed before mid-stance and peaks at approximately threequarters of stance, corresponding to the accelerative part of the sagittal force. In running, in contrast, the largest moment occurs in the first half of the stance phase, when the sagittal force is a braking force. In fast walking, there is also an earlier lateral peak during the first half of stance, which in some cases exceeded in magnitude that in the second half of stance; in these cases at least, fast walking closely resembles running. Transverse forces recorded in running humans were found to vary very substantially between individuals and they are not presented here. Influence of arm-swing omparison of Figs 2 and 4 suggests that transverse force is less sensitive to the presence or absence of arm-swing than is the vertical force couple. Fig. 4 confirms that arm fixation exerts more influence in males than in females, producing a larger lateral moment in the second half of stance. The absence of arm-swing in adult males also affects transverse forces at the time of approximate contralateral heel-strike, just over three-quarters of the way through the stance phase. A consistent second peak in lateral transverse force occurs at this point in slow, comfortable and fast walking. onversely, in

4 Y. LI AND OTHERS Table 2. Standard deviations, correlation coefficients and relative magnitude of horizontal forces Transverse Sagittal F x/ F z F y/ F z Subject Mode S.D. r 2 (%) S.D. r 2 (%) Fixed-arm walk Fast walk omfortable walk Slow walk I Fixed-arm walk Fast walk omfortable walk Slow walk Y Fixed-arm walk Fast walk omfortable walk Slow walk M Fixed-arm walk Fast walk omfortable walk Slow walk R Fixed-arm walk Fast walk omfortable walk Slow walk GI Fixed-arm walk Fast walk omfortable walk Slow walk GW Fixed-arm walk Fast walk omfortable walk Slow walk IN Fixed-arm walk Fast walk omfortable walk Slow walk JU Fixed-arm walk Fast walk omfortable walk Slow walk LI Fixed-arm walk Fast walk omfortable walk Slow walk QI Fixed-arm walk Fast walk omfortable walk Slow walk EA Fast walk omfortable walk Slow walk F JE Fast walk omfortable walk Slow walk JH Fixed-arm walk Fast walk omfortable walk Slow walk

5 Horizontal torque and arm-swing during walking 1 Table 2. ontinued Transverse Sagittal F x/ F z F y/ F z Subject Mode S.D. r 2 (%) S.D. r 2 (%) JS Fast walk omfortable walk Slow walk F JI Fixed-arm walk Fast walk omfortable walk Slow walk YU Fixed-arm walk Fast walk omfortable walk Slow walk Forces are grouped by locomotor mode. Each force curve is divided into 4 equal intervals, producing 41 data points. If a group has k force curves, the k+1 th curve is calculated as the mean of the other k curves. The correlation coefficient is then calculated for each of the k curves with its counterpart on the k+1 th, based on the 41 data points. There will therefore be k correlation coefficients, and their mean value is listed. The S.D. values listed for each group are the mean of the 41 standard deviations calculated over k trials for each of the 41 data points. F x/ F z is defined as ( 1 Fx dt)/( 1 Fz dt), and Fy/ Fz is defined similarly. The integral is from t= (heel-strike) to t=1 (toe-off). Fx, Fy and F z are the forces in the x, y and z directions as defined in Fig. 1 (transverse, sagittal and vertical, respectively) and t is time (proportion of stance phase). fixed-arm walking, during this second lateral peak, a small medial deflection occurs (see arrows in Fig. 2). Loading and the free vertical moment If the body weight of individuals walking with an added load is taken as the sum of the actual body weight plus the load, the maximum vertical free moment expressed per unit body weight in loaded walking does not differ from that in unloaded walking (result not shown). However, there is evidence that the shape of the curves for free moments does differ. Using the general method for Fourier analysis of Hamming (Hamming, 1973), Alexander and Jayes (Alexander and Jayes, 198) showed that five Fourier coefficients, a 1, a 3, a, b 2 and b 4 serve as economic and unique descriptors of the shape and intensity of ground reaction force curves. Following the method of Alexander and Jayes (Alexander and Jayes, 198), we derived these coefficients for vertical free moments in loaded and unloaded walking. An analysis of variance showed that loading position (i.e. shoulder versus side of the trunk) and loading side (on the stance- or swing-foot side) both have a significant effect on the Fourier coefficients. For loading position, four of the five Fourier coefficients are significantly different (at P<.) between the two positions (b 4 non-significant, P=.8); for loading side, all five coefficients are significantly different (P<.1). Discussion Free vertical moments and transverse force in relation to arm-swing It is well known that humans tend to swing their arms during walking. At comfortable and fast walking speeds, the contralateral arm swings forward with the swinging leg (lower limb), while at very slow speeds, both arms swing forwards or backwards simultaneously at twice the stride frequency (Webb et al., 1994). Arm-swing in the normal human gait was originally believed to be a passive pendulum movement (Gerdy, 1829), but Elftman (Elftman, 1939) challenged this, arguing that arm-swing is not a pure pendulum action, but is influenced by muscle forces. Fernandez Ballesteros et al. (Fernandez Ballesteros et al., 196) found from electromyographic studies that, while retraction of the arm was always accompanied by muscle activity, protraction of the arm was not, suggesting that passive, elastic forces arising from soft tissue may also contribute to arm-swing. Since they found that muscle activity continued even when the arms were restrained, they suggested that arm-swing was part of a centrally controlled pattern of locomotion, a conclusion echoed by others (Jackson et al., 1978; Jackson, 1983), who regarded arm-swing as a retention of neurally determined activity patterns exhibited in quadrupedal locomotion, a concept first proposed by Gray (Gray, 1944). The above studies thus suggest that arm-swing may be modelled as a pendulum entrained under neuromuscular control to a rate that varies from the natural frequency of the arm according to the speed of walking (see Webb et al., 1994). Elftman (Elftman, 1939) investigated the angular movement and acceleration of the arm during arm-swing, and found these values to be the same as for the trunk, but in the opposite direction. He proposed, therefore, that arm-swing might counteract trunk torques in the horizontal plane induced by lower limb swing. Principal components analyses of the Fourier coefficients of ground reaction forces performed by Li et al. (Li et al., 1996) show that the absence of arm-swing in

6 2 Y. LI AND OTHERS Force, N/N body weight (%) A I M Force, N/N body weight (%) R 1 Y Fast walking omfortable walking Slow walking Walking with fixed arms - -1 Force, N/N body weight (%) B GI 1 GW 1 IN Force, N/N body weight (%) JU 1 LI 1 QI Fig. 2. Transverse forces measured from (A) male subjects, (B) female subjects and () children during steady motion in four locomotor modes. In each figure part, the upper case letter identifies the individual (see Table 1). For each individual, each pair of curves in the same colour represents the upper and lower standard deviations about the mean (note that the means are not plotted, but lie half-way between the plotted lines). Of all trials performed by the subjects in the given modes, N=1 except for YU and JI for whom N=6. Positive force values indicate medially directed forces, negative values indicate laterally directed forces. The arrows indicate the medial deflection in the second lateral peak that occurs in fixed-arm walking by males.

7 Horizontal torque and arm-swing during walking 3 Force, N/N body weight (%) EA 1 JE 1 JH Force, N/N body weight (%) 1 JI 1 JS 1 YU Fig. 2. ontinued fixed-arm walking has little effect on vertical and sagittal contact forces. Therefore, the influence of arm-swing on the kinetics of gait must be exerted primarily through its effects on transverse force and/or on the vertical free moment about the ground contact point of the foot. These features were investigated here, and the effects are clearly greater on vertical free moments. Fixed-arm walking in adult males leads to a clear increase in the magnitude of the free moment over those that occur when arm-swing is permitted. During stance, the transverse force F x, the sagittal force F y and the free moment (couple) M z rotate the body about a vertical axis passing through the body centre of mass. If a subject walks against the sagittal axis of the force plate, with a left foot contact, the directions of the moments are as indicated in Table 3. It is clear that in most situations the free moment is in the same direction to the torque produced by the sagittal force but in the opposite direction to that produced by the transverse force (Table 3) except during the first half of the stance phase for fast walking. Although the sagittal force (F y ) has a maximum value four times larger than the transverse force (F x ), its lever arm about the centre of gravity (G) of the whole body (i.e. the distance from the G to the centre of pressure of the stance foot) is in most cases smaller (except at mid-stance, when the lever arm of the transverse force tends to be zero). As a result, the moments about the G produced by the two force components tend to be similar in value. Most importantly, the torques produced by the two horizontal force components are nearly always opposite in direction. The exception is during a very A H Mid-sagittal plan B H Fig. 3. Definition of the carrying angle of the lower limb. H is the hip joint, V is the vertical line passing through point H to the floor. The carrying angle α is the angle between the tibia and a line extended from the long axis of the thigh. The foot contact position is affected by this angle: a large angle will place the foot contact point further away from the mid-sagittal plane, reducing the medial transverse force component in the foot contact resulting from the horizontal strut effect defined by Gray (1944). (A) Normal case, foot contact is at the medial side of the hip joint; (B) the situation in female LI, foot contact is further away from the mid-sagittal plane at a position much closer to the vertical line V passing through the hip joint. V α Normal carrying angle V α A very large carrying angle

8 4 Y. LI AND OTHERS Vertical free moment (N m N -1 body weight) A I R Y Fast walking 1-3 omfortable walking Slow walking Walking with fixed arms M Vertical free moment (N m N -1 body weight) B GI GW JU LI QI Fig. 4. The free moment about the vertical axis for (A) males, (B) females and () children during steady motion in four locomotor modes. In each figure part, the upper case letter identifies the individual concerned (see Table 1). For each individual, each pair of curves in the same colour represents the upper and lower standard deviations about the mean (note that the means are not plotted, but lie half-way between the plotted lines). Of all trials performed by the subjects in the given modes, N=1 except for YU and JI for whom N=6. Positive force values indicate a medially directed moment, negative values indicate a laterally directed moment.

9 Vertical free moment (N m N -1 body weight) Horizontal torque and arm-swing during walking EA JE JH JI JS YU Fig. 4. ontinued Table 3. Pattern of forces and free moments about the vertical axis Factors Before mid-stance After mid-stance (G behind the foot) (G in front of the foot) F x <*, moment lateral <, moment medial F y <, moment medial >, moment lateral M z Medial first, then changes Lateral to lateral; tending to be all lateral in fast walking F x, transverse force; F y, sagittal force; M z, free vertical moment. The moments produced by F x and F y are referred to the centre of gravity (G) of the body about the vertical axis. *There is a very short period of medial force (see text) at foot contact, which brings about a medial moment. Here, this force is not taken into consideration. short period immediately after heel-strike, when the transverse force is exerted in a medial direction and the torques produced by the sagittal and transverse forces are therefore in the same direction. The details of the sagittal force in human walking are not presented here because they can be found elsewhere (e.g. Winter, 1991; Li et al., 1996). We have previously shown (Li et al., 1996) that during double support (e.g. after left heel-strike but before right toeoff), sagittal force can reach.2 times body weight (up to 1 N for a 6 kg subject). If we assume the lever arm of the sagittal force about the G (L y ) to be. m (Fig. 6A), it Vertical free moment (N m N -1 body weight) Fig.. A typical curve showing the free moment about the vertical axis for running. Data are from one trial by male subject I. follows that the torque produced by the sagittal force of a single foot will be approximately 7. N m, marginally larger than the vertical free moment produced during single foot support. Thus, taking both feet into consideration, the largest moments will be exerted during double support, which in normal human walking lasts for approximately one-fifth of the gait cycle. During double support, both feet have a relatively large moment arm, L x, for force F x (Fig. 6A). We have seen that, apart from a very brief medial peak at heel-strike, F x is overwhelmingly lateral (Fig. 2). Assuming, as in Fig. 6A, that

10 6 Y. LI AND OTHERS Fig. 6. Formation of vertical torques during the double support phase. The ellipse represents a cross section of the upper part of the trunk. In this figure, both forces and torques are shown as the ground acting on the feet. G, G, centre of gravity of the whole body at t 1 and t (t 1>t ), where t is time; F x, F x, transverse force at t 1 and t ; F y, F y, sagittal force at t 1 and t ; L x and L x: lever arm for F x and F x respectively; L y and L y, lever arm for F y and F y respectively. The foot node r in F xr, F yr, F xr and F yr indicates the right foot (otherwise the left). (A) The torques produced by the transverse force and the sagittal force are in different directions. As a result, the torques produced by the two horizontal force components always cancel each other out. In other words, the horizontal force vector always points to the vertical line passing through G. For each force component, the curved arrow indicates the rotation effect of that force about the centre of gravity of the body. A red arrow represents a clockwise rotation, and a blue arrow represents L x A F y L y F x F xr G an anti-clockwise rotation. (B) The forces and their lever arms are so arranged that no large torque about the body centre of gravity is produced during double support. At heel-strike (t ), the centre of gravity is at G (in blue), the centre of pressure of the foot is at the heel, with transverse force F x and sagittal force F y; their lever arms are L x and L y respectively. After a small fraction of time (time changes from t to t 1), the centre of gravity moves to a position indicated as G, both sagittal and transverse forces for the front foot (left here) increase, but their lever arms decrease (L x and L y). For the rear foot (right foot here), the lever arm for horizontal force increases (not displayed for simplicity), but the forces themselves decrease. In B, blue indicates time t and black indicates t 1. () The free moments on the front and rear feet during double support are in the same direction, indicated by red arrows representing clockwise moments. F yr B L x L F x y F y F x F x L y L y G F xr G F yr F xr F yr G the left foot is in front (as the heel-strike foot), the ground reaction forces in the transverse direction on the two feet will both tend to turn the trunk to the right (clockwise, as seen from above in the z direction). The force F y, causing braking of the front foot but acceleration of the rear foot, will tend to turn the body anticlockwise, to the left. Therefore, during double support, the torques produced by the two horizontal force components tend to cancel each other out. From Fig. 6B, and again assuming that the left foot is in front, for each of the horizontal force components (F x and F y ), torque about the centre of gravity of the body is also minimised because large forces and a large lever arm do not occur simultaneously. For example, after heel-strike, as F x gradually increases (Fig. 2), the trunk is moving forward, so the lever arm of the left foot is reduced (compare L x and L x in Fig. 6B). For the same reason, the lever arm of the right foot, situated posterior to the trunk, increases, but the force under the right foot will be declining. A similar situation occurs for F y : after left heel-strike, as the sagittal force increases, the trunk moves towards the stance foot, in this case to the left (see, for example, arlsoo, 1972), minimizing the moment produced by the left foot. Although the distance between the right foot and the body centre of gravity increases, the force under the right foot approaches zero. After mid-stance, the horizontal forces and their lever arms increase simultaneously. However, the sagittal and transverse forces are both small and they have opposite effects on the vertical moment. Fig. 6 shows the situation during double support with the left foot about to enter toe-off. For comfortable and slow walking, the left foot produces a lateral free moment that peaks just before the commencement of double support. The ground reaction torque rotates the body to the right. After right heelstrike, during double support, the right foot will be exerting a medial moment on the supporting surface: the effect is the same as for the left foot, but the value is relatively smaller. Thus, the free moments sum between left and right feet and balance the body about a vertical axis. The above discussion concerns double support. As shown above, during this period, torques produced by the horizontal forces are larger than the maximum value of the free moment. However, free moments still play a major role during double support. onsider the situation during single support by the left lower limb before and after mid-stance. Before mid-stance, as Table 3 indicates, the torques produced by the two horizontal forces will be in different directions. After mid-stance, the centre of gravity of the body advances beyond the centre of pressure (the force action point). Because of the difference in period, the sagittal force changes direction, but the transverse force does not. The directions of the torques they produce remain opposite to each other. Again, the free moment is the main element involved in balance about the vertical axis. Our results (Fig. 4A) indicate that fixed-arm walking by male adults results in larger peaks in lateral free moments during the later part of stance, immediately before double support. At this

11 Horizontal torque and arm-swing during walking 7 point in normal walking, just before right heel-strike, the acceleration of the left arm is in a posterior direction, in opposition to its displacement, while the reverse is the case for the right arm. In the horizontal plane, the arm movements have an anticlockwise angular acceleration if seen from above. If, as earlier studies (e.g. Fernadez Ballesteros et al., 196) indicate, the posterior acceleration of the arm is brought about by the action of muscles arising from the trunk, the reaction to the accelerative force will produce a clockwise moment applied to the trunk. (In the sagittal plane, in which the swing of the two arms has a phase difference of half a cycle, the armswings have no overall effect.) As Elftman (Elftman, 1939) suggested, the arm-swing exerts a moment of the same magnitude as the vertical rotation of the trunk induced by leg swing, but in the opposite direction. The free moment of the ground reaction to the stance foot and the moment arm-swing exerts on the trunk are in one direction, but the leg swing and lower trunk rotation it induces are in the other. To balance the moment produced by anterior swing of the right lower limb, the right arm swings in the opposite direction, i.e. posteriorly, producing a torque in the same direction as the ground reaction force couple, and therefore reinforcing it. In a sense, arm-swing may be regarded in this instance as an external source of a vertical couple, where the boundary between the human body and its environment is the shoulders. When arm-swing is restricted, our data (Fig. 4A) suggest that the moment produced by the foot is increased, particularly in males, to compensate for the loss of the action of the moment of arm-swing in balancing lower limb swing. The peak vertical moment occurs at three-quarters of the stance phase, before touch-down of the opposite foot. At this point, if the arms were swinging, they would produce the largest moment. (The periodic arm-swing kinematics can be modelled as a sine-type function. For function sinx, the peak values occur when x=kπ/2, k=, 1, 2,...,. Here, k is any an integer, and kπ/2 represents the end of arm-swing.) Fig. 2A indicates that, for adult males, there is a medial inflection in the second lateral peak of fixed-arm walking. The vertical rotation effect of this change in direction is the same as the vertical free moment. When this occurs, the centre of mass of the body is already in front of the stance foot, so that this relative medial shift of force on the supporting surface introduces a reaction force tending to rotate the body medially. For the left foot, the body will be turned towards the right, and both the free vertical moment and arm-swing at this time will have the same effect. Note that the above argument concerns the relative medial inflection in the lateral force during fixedarm walking compared with normal walking, not the absolute value of the force. A pilot experiment we conducted, albeit with a small sample size, indicated that, when subjects turn through 9 at a predetermined point, the free moment exerted when the turn occurs reaches the same maximum value as in normal walking, although the shape of the curve differs. This suggests that, in human bipedalism, free moments are produced mainly to achieve balance in normal walking rather than for changing direction. Unfortunately, with a single force plate, it was not possible to carry out confirmatory experiments measuring the vertical moment in a sudden turn at a randomly selected point. Free vertical moments and loading We found that if the body weight is corrected with the magnitude of loads, the maximum vertical free moment per unit body weight did not differ between loaded and unloaded walking. However, loading position and loading side both had a significant effect on the form of the free moment curves. Thus, while loading per se does not increase the vertical free moment for the male subjects we examined, asymmetric loading does change the pattern of the free moments: contralateral loading has a major influence. For example, when the left foot begins its stance phase, one of its roles is to rotate the lower trunk to the left. The force plate then records a medial moment, indicating that the reaction torque of the ground is lateral (or leftwards, for the left lower limb). If loading is ipsilateral, the lever arm (the distance between the load and a vertical line passing through the point of left foot contact) is smaller than in cases where loading is on the right side (contralateral). As a result, an increased moment is required for contralateral loading. Two explanations may exist for the difference between trunk-side loading and shoulder loading. First, trunk-side loading gives a larger lever arm for the load in a vertical rotation about the body centre of mass. Second, shoulder loads are applied at a higher point, which has a different phase from lower trunk (appozzo, 1982), and hence requires a different vertical moment. Effects of gender on free vertical moments and transverse force Our finding that vertical moments and transverse force are less sensitive to arm-fixation in adult females than in male adults is likely to be attributable to the narrower shoulders and lighter arms of a female subject which should make arm-swing less effective: but this remains speculative. hampman and Kurokawa (hampman and Kurokawa, 1969) found that in normal walking, males exhibited greater mean body rotation than did females, although no significance test was provided. Our data support their finding in that upper body (trunk and arm) rotation will be more important for males than for females. Fig. 4 shows that males exhibit greater differences in moments between locomotor modes. This suggests that vertical moments generally play a more important role in locomotion for males than for females. Horizontal strut effects (as defined by Gray, 1944) alone would produce transverse forces in a medial direction, since foot contact occurs medial to the hip joint. However, the transverse force is largely lateral, so joint torques must be heavily involved. ontraction of the gluteus medius produces an abductor torque at the hip joint, and this effect is registered by the force plate as a lateral force. The force registered depends partly on the torque exerted by the muscle in the frontal plane. According to Vaughan et al. (Vaughan et al., 1992), the gluteus medius activity of adults remains slight until after heel-strike, at which time a medial peak may be experienced; for this short

12 8 Y. LI AND OTHERS period (after foot contact and before the muscle is fully activated), it is indeed a horizontal strut lever effect. Age differences in transverse force Only one child (EA, 1 years old) showed the characteristic adult pattern of transverse force, with an initial medial peak after heel-strike. The proportions and the kinematics of the lower limbs of the children, and their muscle activities, could be responsible for the differences in patterns from those seen in adults. (For example, the gluteus medius may be used more extensively to control motion while overall body control is still in the process of development; electromyographic studies of the ontogeny of walking in children might help to evaluate this possibility.) While it has been argued that children adopt an adult gait patternt as early as years (McGraw, 194; Scrutton, 1969) or even younger (Sutherland et al., 198), Li et al. (Li et al., 1996) found differences between adults and children up to 9 years old in the path of the centre of pressure under the foot. The existence of differences in transverse forces between adults and children up to and including 9 years old in the present study is important, because transverse forces are a directly measured rather than derived variable. In fact, Table 2 indicates that children have larger relative transverse forces than adults, although in this study we do not attempt to explain this phenomenon. From a sample of only six children, our data do not permit us to determine the age of maturation of these gait variables. It is unlikely, however, that maturation will occur as young as 4 or years, or even a couple of year older because our 7- and 8-yearold subjects showed no sign of an adult -type transverse force. In conclusion, we present information on the pattern of vertical free moments during walking and show that, although transverse forces are more variable than vertical or sagittal forces, they do have a characteristic pattern that differs between adults and children. While it was not possible to perform a full analysis of the mechanics of rotation of the body, our analysis of the effects of arm fixation and the timing of forces suggests strongly that arm-swing and free moments tend to reinforce each other and that both act to balance trunk torques induced by the lower limb. Arm-swing is more influential in males, however, and both arm-swing and the free moment under the foot are less important at slow walking speeds. This research was funded by the Natural Environment Research ouncil and the Biotechnology and Biological Sciences Research ouncil. We thank two anonymous referees for helpful comments. 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