Upper Extremity Function in Running. 11: Angular Momentum Considerations

Transcription

1 INTERNATIONAL JOURNAL OF SPORT BIOMECHANICS, 1987, 3, Upper Extremity Function in Running. 11: Angular Momentum Considerations Richard N. Hinrichs Ten male recreational runners were filmed using three-dimensional cinematography whiie running on a treadmill at 3.8 m/s, 4.5 m/s, and 5.4 m/s. A 14-segment mathematical model was used to examine the contributions of the arms to the total-body angular momentum about three orthogonal axes passing through the body center of mass. The results showed that while the body possessed varying amounts of angular momentum about all three coordinate axes, the arms made a meaningful contribution to only the vertical component (H,). The arms were found to generate an alternating positive and negative Hz pattern during the running cycle. This tended to cancel out an opposite Hz pattern of the legs. The trunk was found to be an active participant in this balance of angular momentum, the upper trunk rotating back and forth with the arms and, to a lesser extent, the lower trunk with the legs. The result was a relatively small total-body Hz throughout the running cycle. The inverse relationship between upper- and lower-body angular momentum suggests that the arms and upper trunk provide the majority of the angular impulse about the z axis needed to put the legs through their alternating strides in running. In searching the literature on the biomechanics of running, one can find only an occasional mention of the arms and the role they play in the running process. It appears that little importance has been placed on the arm swing. In the opinion of Mann (1981) and Mann and Herman (1985), for example, the arms are simply used for balance and do not play a significant role in influencing the quality of running perfonnance. However, the present study will show that the arms serve a substantially greater purpose than simply maintaining balance in running. This is the second in a two-article sequence. In the first paper (Hinrichs, Cavanagh, & Williams, 1987), the arms were shown to provide lift and promote a more constant horizontal velocity for the runner. This paper addresses the angular momentum aspects of the arm action in running. Because the arms are not an isolated system, the angular momentum of the arms will be described in relation to that of the legs and of the head and trunk. Richard N. Hinrichs is with the Exercise and Sport Research Institute, Department of Health and Physical Education, Arizona State University, Tempe, AZ

2 RUNNING: ANGULAR MOMENTUM 243 Procedures Ten male recreational runners ranging in age from 20 to 32 years (M = 22.5 years) were used as subjects. They ranged in height from 1.61 to 1.83 m (M = 1.76 m), and ranged in mass from 51.3 to 75.9 kg (M = 64.8 kg). Each subject performed three trials on a treadmill, one at a "slow" speed of 3.8 d s, one at a "medium" speed of 4.5 rnls, and one at a "fast" speed of 5.4 d s. These represented running paces of 7,6, and 5 minutes per mile, respectively. A fourcamera three-dimensional (3D) cinematographic technique was employed using the DLT algorithm (Abdel-Azis & Karara, 1971). Each camera operated at a nominal rate of 100 framesls. Body Segment Parameters A 14-segment mathematical model of the body was used and consisted of the head, trunk, both upper arms, forearms, hands, thighs, calves, and feet. The 3D estimates of joint centers digitized from the films represented the endpoints of each segment (Hinrichs et al., 1987). An exhaustive set of anthropometric measures was taken on each subject to aid in the estimation of body segment parameters (BSPs). These measurements were used to predict segment masses, centers of mass (CM), and moments of inertia for each subject using four different combinations of BSP data from the literature. These four combinations are as follows: 1. Masses and CM from the regression equations of Clauser, McConville, and Young (1969), moments of inertia from the regression equations of Hinrichs (1985) based on the data of Chandler, Clauser, McConville, Reynolds, and Young (1975). 2. Masses and CM from the regression equations of Clauser et al. (1969), moments of inertia from the mean data of Whitsett (1963) corrected for height and body mass differences according to the method of Dapena (1978). 3. Masses and CM from the mean data of Clauser et al. (1969), moments of inertia from the mean data of Whitsett (1963) corrected for height and body mass differences according to the method of Dapena (1978). 4. Masses, CM, and moments of inertia from the regression equations of Zatsiorski and Seluyanov (1985). The set of BSP data that best produced constancy of airborne angular momentum and horizontal linear momentum were chosen for each subject. Details about the selection of BSP are included in Hinrichs (1982). Smoothing and Differentiating The raw 3D segment endpoint data were smoothed using a second-order Butterworth digital filter (Winter, Sidwall, & Hobson, 1974) with cutoff frequencies ranging from 2 to 8 Hz. Different cutoff frequencies were chosen objectively for each coordinate of each endpoint using an algorithm similar to the one described by Jackson (1979) adapted for use with a digital filter. The vertical (Z) coordinates of segment endpoints were generally found to need higher cutoff frequencies for smoothing than did the X or Y coordinates.

3 Due to the laborious nature of the 3D cine process, only one complete running cycle per trial was digitized for each subject. An additional 10 frames were digitized at the beginning and end of each trial to reduce endpoint effects of the digital filter. Finite differences (Miller & Nelson, 1973) were used whenever the analysis called for taking the first or second time derivative of the smoothed displacement data. No further smoothing was done following differentiation. Algorithm for Computing Angular Momentum The smoothed 3D coordinates of the segment endpoints were entered into a computer program that calculated the angular momentum of individual body segments and systems of segments about the body center of mass. The method used to compute angular momentum is similar to the method of Dapena (1978). Each segment was assumed to be symmetrical about its longitudinal (long) axis. Thus, any transverse axis through the segment CM became a principal axis of inertia for the segment. Rotations about segment long axes, with the exception of the trunk segment, were ignored. This was accomplished mathematically by setting all nontrunk segment moments of inertia about their long axes equal to zero. Dapena (1978) has shown this procedure to involve little error in the calculation of human body angular momentum. Nontrunk Segments. Consider, for example, the body segment shown in Figure 1. In the small time interval 2At (2 frames), the segment moves from position i - l through position i to position i f l relative to a noninertial B:xyz refer-.* er._--- _#--- Path of segment CM relative to B F i e 1 - Computing angular momentum of a body segment about the body center of mass (point B).

4 RUNNING: ANGULAR MOMENTUM 245 ence frame translating with the body CM (point B). The B:xyz reference frame does not rotate but remains parallel to the fixed 0:XYZ reference frame. The segment possesses two forms of angular momentum about point B: "local," which is inherent in the rotation of the segment about a transverse axis through its own CM, and "remote," which is inherent in the translation of the segment CM (point C) relative to the body CM (Hopper, 1973). Local angular momentum requires the calculation of the segment's angular velocity vector, o. First, an instantaneous plane of rotation is established. The unit vector ni normal to this plane in the direction of the angular velocity is where Li-l and Li+l are the vectors defining the segment long axis (proximal to distal) at frames i - 1 and i + 1, respectively. The angular displacement, Oi, of the segment during the time interval 2At is The angular velocity vector at frame i is therefore Because the axis of rotation of the segment is a principal axis of inertia, the local angular momentum vector is parallel to the angular velocity vector, where (HL)i is the segment local angular momentum vector at frame i, I is the segment's moment of inertia about a transverse axis through its CM, and oi is the angular velocity vector at frame i. The remote angular momentum of the segment is simply the moment of the linear momentum of the segment expressed in a reference frame translating with the body CM. Thus where (HR)i is the segment remote angular momentum vector at frame i, ri is the vector locating the segment CM relative to the body CM at frame i, m is the segment mass, and ii is the time derivative of the vector ri. The total angular momentum of the segment about the body CM at frame i is the sum of the local and remote terms, Trunk Segment. Equation 6 was used for the trunk segment rotations in the sagittal and frontal planes. Here the entire trunk was considered a cylinder rotating about a transverse axis. (Note that the term transverse axis shall refer to any axis that is perpendicular to the segment's longitudinal axis.) The rotation of the trunk about its long axis was also considered, however. The presence of two shoulder and two hip landmarks permitted the estimation of this component of angular momentum. The shoulder and hip points also allowed the trunk to be

5 subdivided into an upper trunk and a lower trunk, each having its own local angular momentum about their common longitudinal axis. The nature of the calculations did not require a plane of separation of the upper and lower trunk. The upper trunk was assumed to rotate with the shoulders, the lower trunk with the hips. The method for calculating the angular momentum of the upper trunk about the trunk long axis is outlined below. The procedure for the lower trunk is identical to that of the upper trunk with the right and left hip points being substituted for the right and left shoulders, respectively. The procedure starts by defining the trunk long axis vector Q at each of the three frames, i- 1, i, and i+ 1 (see Figure 2). An auxiliary "nql" reference frame is defined by the unit vectors n, q, and 1, and rotates with the trunk long axis frame i - 1 to i+ 1. The vector 1 is the unit vector along the trunk long axis at each frame. The unit vector n is normal to the plane of rotation of the trunk long axis between frames i- 1 and i+ 1. It is defined according to equation 1 in an identical manner as for all other segments. The auxiliary reference frame rotates about n between frames i- 1 and i+ 1. Finally, the unit vector q is defined by the cross-product of 1 and n. Rotation of D" plane ##,,' a DL di+, 5 Figure 2 - Computing angular momentum of the upper trunk or lower trunk about the trunk long axis.

6 RUNNING: ANGULAR MOMENTUM 247 Next a vector S locating the left shoulder relative to the right shoulder is defined at each frame. It is the rotation of this vector about the trunk long axis that is of interest here. Because S may not necessarily lie in the nq plane at a given frame, another vector D is defined by the cross-product of L and S. This vector D lies in the nq plane and its rotation in this plane between frames i- 1 and i+ 1 is representative of the rotation of the shoulders about the trunk long axis during this interval. This rotation can easily be measured if D is first expressed in the auxiliary reference frame. This is done in the following equation: where [XI is the transformation matrix relating the nql reference frame to the XYZ reference frame, and D' is the vector D transformed into the nql reference frame. The angular displacement of the vector D ' between frames i - 1 and i + 1 is then determined by the following equation: where 4 is the angular displacement, and Dli- and D'i+l are the vector D' at frames i - 1 and i + 1, respectively. The direction of the angular displacement depends on whether or not the rotation was counterclockwise (+) or clockwise (-) in the nq plane. This can be detected by looking at the cross-product of the vectors D Ii- and D For this purpose, the unit vector u' was defined and indicated the direction of the rotation such that If the rotation is counterclockwise, u' will have the coordinates (0, 0, 1) expressed in the auxiliary reference frame. If the rotation is clockwise, u' will have the coordinates (0, 0, - 1). If the constant C is set equal to the third coordinate of u', it is then possible to express the angular velocity of the upper trunk about the trunk long axis at frame i by the following equation: The angular velocity vector will be directed along li or opposite li depending on the value of C (C=+l). Finally, the angular momentum associated with the angular velocity is where is the angular momentum vector of the upper trunk about the trunk long axis at frame i, and IUT is the moment of inertia of the upper trunk about the trunk long axis. A similar analysis for the lower trunk yields

7 where (HLT)i and COLT)^ are the angular momentum and angular velocity vectors of the lower trunk about the trunk long axis at frame i, respectively, and ILT is the moment of inertia of the lower trunk about the trunk long axis. These two additional local terms (HUT and HLT) were added to the previous angular momentum of the trunk determined by equation 6 to arrive at the total angular momentum vector of the trunk relative to the body CM. Once the angular momentum of each segment was calculated, it was possible to sum over segments to amve at the angular momentum of various systems of segments such as the right arm, or both legs, or the whole body. To facilitate comparisons between subjects of different heights and masses, the absolute angular momentum (expressed in kg * m2/s) was normalized by dividing through the body mass (in kg) and the square of the subject's standing height (in m). The resulting unit is s-'. Because the resulting numbers are rather small, they have been expressed in units of 10-3s-1 (or s). Results and Discussion This study represents the first of its kind on this topic. Given the exploratory nature of this investigation, statistical treatment of the results has not been performed. Trends in the data will be described without additional support from tests of statistical significance. The results will be presented two ways, first as mean curves, and then as mean values. The mean curves represent the mean angular momentum versus time curves computed over all subjects. The time base was normalized to 100% of the running cycle time for each subject before averaging the curves together. To obtain mean values, peak values were computed from individual subject curves and then a mean was computed from these individual peak values. These mean values tend to be slightly larger in magnitude than their associated peak values from the mean curves because individual subject peaks were not exactly timesynchronous. Thus, the curve-averaging process tended to attenuate the peaks slightly. Throughout this paper, mean values of normalized angular momentum will be reported in units of 10-3s-1 with standard deviations in parentheses. AP Component The anteroposterior (AP) component of angular momentum (Hy) is shown in Figure 3. Included are the mean curves for the arms, legs, head and trunk, and total body at the medium speed. In this figure a positive Hy indicates a counterclockwise rotation of the runner as viewed from the front. Conversely, a negative H indicates a clockwise rotation of the runner as viewed from the front. he results were somewhat surprising. Rotation about the y axis occurs in activities such as a cartwheel or an Arabian (sideways) somersault in gyrnnastics, but one would not expect much of this to occur in running. However, the results showed that there is a relatively large AP component to the total angular momentum in running. The body possessed alternating phases of positive and negative Hy. This pattern was present in all body segments and represented primarily that the whole body was "swaying" from side to side with each contact. The largest values of the body H were attained during the airborne phase, over which they remained (theoretically) constant in the absence of external moments. The variations ob-

8 RUNNING: ANGULAR MOMENTUM Figure 3 - Mean curves of the AP component of angular of the arms, head and trunk, legs, and whole body at medium speed. (Note: In this and subsequent figures, circles have been drawn on stick figures to indicate left wrist, shoulder, and ankle. Shaded areas represent airborne phases.) served in the computed airborne angular momentum values reflect errors in the computational process. This is discussed later. The peak positive and negative values of the body Hy were approximately equal in magnitude and did not change appreciably as running speed increased. The mean values (representing the absolute values of the positive and negative peaks) were 18.6 f 5.5, , and 20.4 f 5.6 for the slow, medium, and fast speeds, respectively. At each footstrike the values of Hy started to change dramatically and continued changing throughout each foot contact phase. These changes indicate that the body experienced a net moment about its y axis during each contact phase. The source of this moment would have been the ground reaction force (principally the vertical component) passing to the left side of the body CM during the left foot contact, and the right side during right foot contact. The arms contributed a minimal amount to the total H,, of the body and, like all the other segments, tended to increase the total rather than reducing it

9 as was done about the z (vertical) axis (to be shown later). The arms were clearly not involved in any balance of angular momentum about the y axis. Mediolateral Component The mediolateral (ML) component of angular momentum (H,) is shown in Figure 4. Included are the mean curves for the arms, legs, head and trunk, and total body at the medium speed. In this figure a positive H, indicates a counterclockwise rotation of the runner as viewed from the right side. Conversely, a negative H, indicates a clockwise rotation of the runner as viewed from the right side. Rotation about the x axis occurs in activities such as forward and backward somersaults in diving and gymnastics. Once again, it came as a surprise to find out that the body possessed rather large amounts of H, throughout the running cycle. The results showed that most of the total H, was contained in the legs. Figure 5 shows the mean curves for H, of the left leg, right leg, and both legs at the medium speed. Note that the vertical scale is different than in Figure 4. Each leg possessed alternating phases of negative and positive H, as it moved Figure 4 - Mean curves of the ML component of angular of the arms, head and trunk, legs, and whole body at medium speed.

10 RUNNING: ANGULAR MOMENTUM Figure 5 - Mean curves of the ML component of angular of the legs at medium speed. through its functions as first stance leg and then swing leg. The positive H, values were associated with swing, the negative ones with stance, and they tended to cancel each other out. The stance leg always had a greater magnitude of angular momentum than the swing leg, however, as a result of its CM being located farther from the x axis. This gave it the upper edge in generating remote angular momentum about the x axis. (The remote terms made up approximately 90% of the total H, of the legs.) The result of adding the individual leg contributions together was a fairly constant negative H, for the combined legs. The legs were by far the dominant contributor to the total body H, (see Figure 4). The arms had a very small amount of H, throughout the cycle. One reason for this is that in addition to having less inertia, the arms are much closer to the body CM than the legs are and thus do not have the same potential to generate large H, remote terms. The second reason is that the H, in the forward swing of one arm tended to cancel out the H, in the backward swing of the other. The head and trunk did make a moderate contribution to the total H, by alternately flexing and extending. The negative "dip" during the contact phase from trunk flexion produced a nearly equal-sized dip in the whole-body H,. The

11 initial drop in the whole-body H, after each footstroke would have been linked to the ground reaction force vector passing behind the body CM during the initial braking phase of foot contact, and hence exerting a clockwise (negative) moment on the body. The subsequent rise in H, during the second half of the support phase would result from the ground reaction force vector passing in front of the body CM during the propulsive phase, and hence exerting a counterclockwise (positive) moment on the body. During the airborne phases, with no source of external moments on the body, the changes in the body H, seen in Figure 4 represent errors in the computational process and not real changes in H,. While the z and y components of the airborne angular momentum were predicted with a reasonable degree of constancy, there appears to have been some kind of systematic error in the computation of H,. It seems likely that the error lies somewhere in the estimation of body segment parameters because, in some subjects, the predicted airborne values for H, were very constant across all speeds. It also seems likely that the error occurs in the computation of the H, of the legs because it is there that the nonconstant pattern originates. The airborne variation (in units of 10-3s-') in the total-leg H, shown in Figure 5 (approximately 6 units) represents only 8 % of the magnitude of the peak H, contained in the stance leg (approximately 75 units). Yet when the total-leg H, is included in the plot of the whole-body H, (Figure 4), a similar 6-unit variation in the whole-body H, looks much larger because of the expanded vertical scale. Compared to the large H, generated by an individual leg, the values in Figure 4 are all relatively small. It should also be pointed out that these curves represent the means of 10 individual subjects. The airborne phases did not occur in the exact same locations within the cycle for each subject, thus some of the H, variation seen in the airborne phases is a result of the averaging process. The magnitudes of H, of the body increased over the entire cycle as running speed increased. The mean values for the peak negative H, of the body were f 4.5, f 5.9, and f 6.1 for the slow, medium, and fast speeds, respectively. The H, results may at first glance seem puzzling to some readers. How can the body possess a negative amount of H, throughout the running cycle without achieving some overall rotation of the whole body? The answer becomes clear when one looks carefully at the action of the legs. Figure 6 shows two multiple stick-figure plots of a composite runner on the treadmill at the medium speed. The first plot shows the first half of the cycle from left footstrike (LFS) to right footstrike (RFS); the second shows the second half of the cycle from RFS to LFS. The motion of the feet shows how the whole-body H, can be tied up prirnarily in the legs. The legs achieve an overall clockwise pattern of rotation. The rest of the body does not somersault forward even though the body as a whole has angular momentum in that direction. This type of phenomenon has been documented by Hopper (1973), Hay (1978), and others concerning the "hitch-kick" technique used by long jumpers while airborne. Both Hopper and Hay have pointed out that the long jumper possessed a certain amount of what may best be called forward-somersaulting angular momentum (same as negative H, here) while airborne. The jumper deals with this by "running" in the air to keep the angular momentum tied up in the extremities and keep the trunk from rotating forward. While this angular momen-

12 RUNNING: ANGULAR MOMENTUM Figure 6 - Multiple stick-figure plots of a composite runner on the treadmill at medium speed. tum is usually described as originating in the takeoff phase of the jump (Hay, 1978), it is clear that the jumper would generate a portion of it during the run-up. The results of this study would also suggest that the faster the run-up, the more angular momentum the jumper would have to contend with. Vertical Component The vertical component of angular momentum (Hz) turned out to be the most important of the three in terms of defining the function of the upper extremities in running. Whereas the arms were shown to make very little contribution to H, and H,,,the motion of the arms about the vertical axis produced relatively large amounts of Hz. This Hz was contained primarily in its remote form, that is, in the motion of the segment CM relative to the body CM. The farther a segment was from the z axis, the greater potential it had to generate this remote form of Hz. The arms, in this respect, were able to compete favorably with the more massive legs in generating Hz because the shoulders are generally broader than the hips, placing the arms further from the z axis than the legs. The slight abduction of the shoulders demonstrated by each subject placed the arms even further from the z axis. Moreover, the forward swing of one arm and the backward swing of the other, though tending to cancel out their combined linear momentum (Hinrichs et al., 1987), both created angular momentum in the same direction. Their effects were additive. This was also true for the legs, however, moving in the opposite direction. The Hz results for the arms are presented first, followed by the legs, head and trunk, and finally the whole body.

13 Figure 7 - Mean curves of the vertical component of angular momentum &) of the arms at medium speed. Arms. Figure 7 shows the mean Hz curves of the arms at the medium speed. In this figure (as in all the Hz plots that follow) a positive value indicates a counterclockwise rotation when the runner is viewed from the top. A clockwise rotation would create a negative Hz. Thus when the left arm swings forward and the right arm swings backward during the left foot contact phase, a clockwise or negative Hz is created in the arms. This changes to a counterclockwise or positive Hz when the right arm swings forward and the left swings backward during the right foot contact phase. This pattern was shown without exception in all subjects at all speeds. The crossover points occurred during the middle portion of each airborne phase as each arm completed its respective swing. Individually, it can be seen from Figure 7 that approximately 30-40% more Hz was generated by the forward-swinging arm than the backward-swinging arm. This was primarily a result of the forward-swinging arm being carried further from the z axis than the backward-swinging am. The magnitudes of their linear momenta were not markedly different, as can be seen in Figure 9 of Hinrichs et al. (1987, p. 237).

14 RUNNING: ANGULAR MOMENTUM 255 Although individual subjects demonstrated various asymmetries between right and left arms, on the average the magnitudes of the negative and positive peaks were virtually identical. The magnitude of these peaks increased with increasing running speed. For the slow, medium, and fast speeds, respectively, the mean values were , 9.5 * 1.5, and Legs. The Hz pattern of the legs was similar to that of the arms but in the opposite direction. Figure 8 shows the mean Hz curves of the legs at the medium speed. The left and right legs showed an interesting pattern in their relative contributions to the total Hz of the legs. Between LFS and left toe-off (LTO), the left (stance) leg possessed much less Hz than the right (swing) leg. This was because the CM of the stance leg was substantially closer to the z axis. In addition, the duration of the positive contribution to Hz of the left leg ended at LTO whereas the right leg continued nearly until RFS. However, together they created a total-leg Hz which peaked and then crossed zero at nearly identical times as the Hz of the arms but in the opposite direction. The second half of the cycle was a mirror image of the first half with the roles of the right and left legs reversed. Figure 8 - Mean curves of the vertical component of angular momentum (Hz) of the legs at medium speed.

15 Although individual subjects demonstrated various asymmetries between right and left legs, on the average the magnitudes of the positive and negative peaks were very similar. The magnitudes of these peaks increased with running speed. The mean values were 12.3 f 1.2, , and 14.7 f 2.2 for the slow, medium, and fast speeds, respectively. At the slow and medium speeds, the legs possessed approximately 40% more Hz than the arms. At the fast speed, however, this difference was reduced to 28%. Head and Trunk. A common belief is that the arms are swung in running in order to keep the trunk from rotating back and forth about the z axis. If this were true, one would not expect the trunk to possess any Hz. As it turned out, the trunk was found to have a moderate amount of Hz, possessed mainly in the rotation of the upper part of the trunk about the trunk long axis. Figure 9 shows the mean curves for the various portions of the total head and trunk Hz at the medium speed. The four parts of the total head and trunk Hz were 1. Transverse rotation of the head (z component), 2. Transverse rotation of the trunk (z component), 3. Long axis rotation of the upper trunk (z component), and 4. Long axis rotation of the lower trunk (z component). Recall that transverse refers to any axis perpendicular to a segment's long axis and does not necessarily mean side to side. In Figure 9 the first two terms above were combined since the contribution of the head was very small. The results showed the rotational patterns of the upper and lower trunk to be quite different. The upper trunk had an Hz pattern quite similar to that of the arms. The lower trunk possessed nearly no Hz at all. At each footstrike, the lower trunk tended to be rotating with the legs but did not follow the legs' pattern as closely as the upper trunk followed the arms' pattern. By far the dominant contributor to Hz of the two parts of the trunk was the upper trunk. The trunk also had a contribution from the z component of the transverse rotation of the trunk, that is, from the entire trunk rotating about an axis perpendicular to the trunk long axis. A certain amount of this was expected because the trunk long axis was generally not parallel to the z axis at all times, and hence any vector perpendicular to the trunk long axis would have a small component along the z axis. The Hz of the total head and trunk showed one negative and one positive peak per cycle, with the crossover points occurring approximately at each footstrike. The magnitudes of the positive and negative peaks were approximately equal and increased with running speed. The mean values were 2.5 f 1.0,2.8 f 0.9, and 3.4 f 1.3 at the slow, medium, and fast speeds, respectively. Total Body. When the arms, legs, and head and trunk contributions were added together, the resulting total-body Hz was rather small throughout the running cycle. This can be seen in Figure 10 where the mean curves have been plotted together for the medium speed. The body Hz oscillated from a small positive value during the left foot contact phase to a small negative value during the right foot contact phase. The body Hz remained (theoretically) constant during the airborne phases and was calculated to do so with a fairly good approximation.

16 RUNNING: ANGULAR MOMENTUM Figure 9 - Mean curves of contributions to the vertical component of angular of the head and trunk at medium speed. As with the individual curves comprising the total Hz, the second half of the cycle mirrored the first half with the magnitudes of the peak values being approximately equal. Contrary to the individual parts, the magnitudes of the peaks in the total-body Hz did not show a tendency to increase with running speed. The mean values were 2.4 _+ 1.O, 2.7 $. 1.3, and 2.5 _+ 1.2 for the slow, medium, and fast speeds, respectively. Upper Body Versus Lower Body. As first suggested by Elftman (1939) for walking and then by Hinrichs, Cavanagh, and Williams (1983) for running, the arms seem to balance the angular momentum of the rest of the body about the z axis. To show this, both Elftrnan and Hinrichs et al. combined the head and trunk and legs together to create a "body-minus-arms" (BMA) and compared the Hz of the BMA to the Hz of the arms. Upon close examination of Figure 9, however, it seems more appropriate to combine the upper trunk and arms together because their Hz patterns were so similar. When this was done, the balance became upper body versus lower body. For this analysis the upper body Hz was calculated by adding together the

17 ... r 2' 'i Body.. 3 :. ;.. \. Figure 10 - Mean curves of the vertical component of angular momentum (I-&) of the arms, head and trunk, legs, and whole body at medium speed. contributions of the arms, head, and upper trunk. The lower body Hz consisted of the Hz present in the lower trunk added to the Hz of the legs. The contribution to Hz from the transverse rotation of the entire trunk was divided equally between upper body and lower body. When the shuffling around of Hz components was complete, the nature of the balance became more clear. Figure 11 shows the mean Hz curves for the upper body and lower body at the medium speed. Although the addition of the lower trunk did not create a lower-body Hz substantially different from the already large Hz of the legs, the result of adding the upper trunk to the arms produced an upper-body Hz of a comparable magnitude to that of the lower body. Moreover, as speed of running increased, the differences in the magnitudes of the upper and lower body peaks were decreased. The mean values for the peak Hz of the upper body were 10.4 f 1.5, 11.3 f 1.6, and 13.5 * 2.3 at the slow, medium, and fast running speeds, respectively. For the lower body these values were 12.0 f 1.4, 13.1 f 2.0, and 14.4 f 2.2, respectively. While the lower body Hz was approximately 15 % larger than the upper body Hz at the slow and medium speeds, this difference was reduced to 7% at the fastest speed.

18 RUNNING: ANGULAR MOMENTUM Figure 11 - Mean curves of the vertical component of angular momentum (HJ of the upper body, lower body, and whole body at medium speed. These results suggest that one of the major roles of the arm swing with the accompanying motion of the upper trunk is to counteract the relatively large amount of Hz generated by the legs in running. This produces a relatively small amount of total-body Hz throughout the running cycle. As the legs increase their Hz at faster running speeds, so do the arms and trunk such that the total-body Hz does not increase. Mechanism for Hz Balance. While it is nice to demonstrate a tendency toward a balance between upper- and lower-body Hz in running, this does not explain it. In some subjects the "left over" total-body Hz was fairly large during portions of the contact phases. In others it was very small. This balancing obviously was more complete in some subjects than others. This section discusses the mechanism for this coordinated action of the upper and lower body. First we must answer one basic question. Is it desirable for a runner to have no overall angular momentum about the z axis, and if so, why? To answer this, one should first consider a hypothetical case of a runner with no upper body. The legs obviously need a fair amount of Hz to put themselves through one stride, and then an equal and opposite amount to finish the cycle. The legs have Hz by virtue of being distant from the z axis of the body.

19 To produce the changes in Hz over the running cycle, the body would have to get angular impulses about the z axis from an external source. These angular impulses can originate from a free moment (or couple) between the foot and the ground or from an eccentrically applied ground reaction force (GRF). If one ignores air resistance (an especially good approximation in treadmill running), the source of external torques disappears during each airborne phase. Looking at the curves presented in Figure 8, however, one notices that the largest changes in the Hz of the legs occur during the airborne phases. In this hypothetical example, therefore, the "runner" could not exchange the directions of motion of the legs during the airborne phase and hence could not run in any recognizable manner. The runner would have to go through a series of "leaps" in the air, waiting until footstrike to begin swinging the recovery leg through while using the ground as a source of angular impulse. (The runner might be able to walk, however, since he would always be in touch with the ground.) With a complete body intact, however, the lower body has an additional source of angular impulse-the upper body. In this case, the lower body can complete normal airborne phases by receiving torques from the upper body. At least during the airborne phases, the opposing action of the upper and lower body now becomes clear. To receive an angular impulse from the upper body, the lower body must also give an equal and opposite one to the upper body (Newton's Third Law). Thus the change in angular momentum experienced by the lower body is accompanied by an equal and opposite change in angular momentum of the upper body. The total-body Hz, of course, does not change. This situation is sometimes referred to as an action-reaction twist and is shown in Figure 12. The fact that the changes in upper- and lower-body Hz remain opposite each other during the contact phase indicates that the lower body receives most of its torques from the upper body even during the contact phases. This frees the feet to concentrate on pushing downward and backward and not to apply large torques to the ground. One would expect some torques to come from the ground, however, because the feet do not generally pass directly beneath the body CM. The AP component of the GRF thus would exert a moment about the z axis. Likewise, the ML component of the GRF would also generally exert a moment about the z axis. It is therefore unlikely that a runner would ever have zero angular momentum about the z axis throughout the running cycle. The small changes that occur in Hz over the contact phases reflect the angular impulses received by the body from the ground. It is likely that the amount of Hz present during the running cycle (and subsequent changes thereof) largely depends on a person's running style, particularly on hislher pattern of foot placement which would determine the moment arms of the AP and ML components of the GRF about the z axis. It may be that the optimum amount of total-body Hz for a particular person is that which minimizes the free moment exerted by the runner on the ground. This would allow the runner to mainly apply forces to the ground instead of both forces and torques. This might also reduce the tortional stresses on the legs. Whether the runners in this study were indeed doing this remains, of course, unknown. A future study could use, in addition to film, a force platform to measure the forces and torques the runner exerts on the ground. This would enable

20 RUNNING: ANGULAR MOMENTUM Torques Fiie 12 - Action-reaction twist mechanism for H, balance. a better understanding of the interaction of foot placement, ground reaction force, free moment, and changes in total-body angular momentum. Summary and Conclusions This investigation has provided an insight into the nature of the arm action in running. Based on the findings of this study and those reported in the previous article (Hinrichs et al., 1987), the following conclusions are drawn: 1. Vem'cal axis: The main functions of the upper extremities in running involve modifications of the body's translation along and rotation about the vertical (z) axis. The former involves the arms' ability to propel the body upward through their upward acceleration relative to the trunk. This has been termed lift and helps the body achieve an airborne phase. At the running speeds investigated in this study, the arms provided a small but meaningful contribution to lift (generally less than 10% of the total). As running speed gets faster and the total amount of lift becomes less (the vertical range of motion of the body generally decreases

21 with running speed), the arms should become increasingly more important to lift (Hinrichs et al., 1987). The arms' ability to modify the body's rotation about the z axis is judged here to be even more important than lift. The actions of the arms and upper trunk provided most of the angular impulse about the z axis needed to put the legs through their alternating strides in running. In the airborne phases, where the torque on the lower body is the greatest, the upper body is the sole source of this torque. If the entire upper body were not present, the legs could still achieve an airborne phase by providing lift themselves, but could not provide the torque necessary to reverse their own angular momentum once airborne (something that is necessary to prepare for the next footstrike). Thus a normal airborne phase could not be executed. If the arms were absent but the trunk still present, a person could run but probably only at a greatly reduced speed because the moment of inertia of the trunk about the vertical axis would be too small to generate the kind of angular momentum needed to balance the legs at these speeds of running, even if it were to vigorously twist back and forth. 2. Horizontal axes: The arms do not appear to make any direct contribution to the forward propulsion of the body (drive); rather, their function about the horizontal axes appears to be to reduce the excursions of the body CM from side to side and, at least in the case of "constant-speed" running studied here, in the direction of progression (Hinrichs et al., 1987). Rotationally, the arms do not appear to make any meaningful contribution to angular momentum about either horizontal axis. It is clear that the arms do more than just help a runner keep his or her balance. Maintaining balance in the usual sense may be the main function of the arms when one walks on a railroad track or skates on ice for the first time, but in running the arms do much more than this, as this study has shown. Aristotle was right when he said that "runners run faster if they swing their arms" (Aristotle, English translation 1961, p. 489). He may not have understood exactly why, but to have written about it when he did shows that he was a scientist truly ahead of his time. References Abdel-Aziz, Y.I., & Karara, H.M. (1971). Direct linear transformation from computer coordinates into object space coordinates in close-range photogrammetry. Proceedings of the Symposium on Close-Range 1-18). Falls Church, VA: American Society of Photogrammetry. Aristotle. (English translation by E.S. Forster, 1961). Progression of animals. Cambridge: Hamard University Press. Chandler, R.F., Clauser, C.E., McConville, J.T., Reynolds, H.M., & Young, J.M. (1975). Investigation of inertial properties of the human body (AMRL Technical Report ). Wright-Patterson Air Force Base, OH. (NTIS #AD-A ) Clauser, C.E., McConville, J.T., & Young, J.W. (1969). Weight, volume, and center of mass of segments of the human body (AMRL Technical Report 67-70). Wright- Patterson Air Force Base, OH. (NTIS #AD )

22 RUNNING: ANGULAR MOMENTUM 263 Dapena, J. (1978). A method to determine the angular momentum of a human body about three orthogonal axes passing through its center of gravity. Joumal of Biomechanics, 11: Elftman, H. (1939). The function of the arms in walking. Human Biology, Hay, J.G. (1978). The biomechanics of sports techniques. Englewood Cliffs, NJ: Prentice Hall. Hinrichs, R.N. (1982). Upper extremity function in running. Unpublished doctoral dissertation, Pennsylvania State University. Hichs, R.N. (1985). Regression equations to predict segmental moments of inertia from anthropometric measurements: An extension of the data of Chandler et al. (1975). Joumal of Biomechanics, 18: Hinrichs, R.N., Cavanagh, P.R., & Williams, K.R. (1983). Upper extremity contributions to angular momentum in running. In H. Matsui & K. Kobayashi (Eds.), Biomechanics ). Champaign, IL: Human Kinetics. Hinrichs, R.N., Cavanagh, P.R., & Williams, K.R. (1987). Upper extremity function in running. I: Center of mass and propulsion considerations. International Journal of Sport Biomechanics, 3: Hopper, B.J. (1964, September). The mechanics of arm action in running. Track Technique, 17: Hopper, B.J. (1973). The mechanics of human movement. New York: Elsevier. Jackson, K.M. (1979). Fitting of mathematical functions to biomechanical data. IEEE Transactions of Biomedical Engineering, BME-26: Mann, R.V. (1981). A kinetic analysis of sprinting. Medicine and Science in Sports and Exercise, 13: Mann, R., & Herman, J. (1985). Kinematic analysis of Olympic sprint performance: Men's 200 meters. Intemtional Joumal of Sport Biomechanics, 1 : Miller, D.I., & Nelson, R.C. (1973). Biomechanics of sport. Philadelphia: Lea & Febiger. Whitsett, C.E. (1963). Some dynamic response characteristics of weightless man (AMRL Technical Documentary Report 63-18). Wright-Patterson Air Force Base, OH. (NTIS #AD ) Winter, D.A., Sidwall, H.G., & Hobson, D.A. (1974). Measurement and reduction of noise in kinematics of locomotion. Journal of Biomechanics, 7: Zatsiorski, V., & Seluyanov, V. (1985). Estimation of the mass and inertia characteristics of the human body by means of the best predictive regression equations. In D.A. Winter, R.W. Norman, R.P. Wells, K.C. Hayes, & A.E. Patla (Eds.), Biomechanics IX-B (pp ). Champaign, IL: Human Kinetics.