Determinants of the center of mass trajectory in human walking and running

Humans walk at low speeds and run at high speeds. The transition from walking to running is obvious because the two gaits are distinctly different from each other. As a human switches from a walking gait to a running gait, the kinematics and kinetics of locomotion change abruptly. For example, when a person switches from a walk to a run, the time of foot–ground contact decreases by 35 % and the peak ground reaction force increases by approximately 50 % (Nilsson et al. 1985; Hreljac, 1993; Minetti et al. 1994). In addition, the trajectory of the center of mass is completely different for walking and running gaits in humans and other animals (Cavagna et al. 1976, 1977; Heglund et al. 1982; Blickhan and Full, 1987; Farley and Ko, 1997). During walking, the center of mass reaches its highest point at the middle of the stance phase. In contrast, during running, it reaches its lowest point at the middle of the stance phase. This difference in the pattern of vertical movement of the center of mass has been proposed as the defining difference between a walking gait and a running gait (McMahon et al. 1987).

This distinction between running and walking is reflected in the simplest mechanical models used to describe the gaits (Farley and Ferris, 1998). Running animals are often modeled as simple spring-mass systems (Blickhan, 1989; McMahon and Cheng, 1990; Blickhan and Full, 1993b; Farley et al. 1993; Alexander, 1995; Farley and Gonzalez, 1996). The spring-mass system consists of a point mass equal to the runner’s body mass and a compliant spring that connects the mass to the point of ground contact (‘leg spring’). As the foot hits the ground during running, the leg spring compresses as a result of joint flexion, and the mass moves downwards. At the middle of the stance phase, the leg is maximally compressed, and the mass reaches its lowest point. In contrast, walking is often modeled as an inverted pendulum system (Alexander, 1995). This model consists of a point mass equal to the walker’s body mass and a rigid strut that connects the mass to the point of ground contact. During the stance phase of walking, the mass vaults over the rigid strut, reaching its highest point at the middle of the stance phase (Cavagna et al. 1976, 1977; Margaria, 1976).

The inverted pendulum model accurately predicts the general pattern of mechanical energy fluctuations of the body during walking (Cavagna et al. 1963, 1976, 1977; Margaria, 1976). In moderate-speed walking, the kinetic energy and gravitational potential energy of the center of mass are nearly 180 ° out of phase. Between touchdown and mid-stance, the forward velocity of the center of mass decreases as the trunk arcs upwards over the stance foot. In this phase, kinetic energy is converted to gravitational potential energy. During the second half of the stance phase, the center of mass moves downwards as the forward velocity of the center of mass increases. In this phase, gravitational potential energy is converted back into kinetic energy. This exchange of kinetic energy and gravitational potential energy during walking is similar to the energy exchange of an oscillating pendulum (Cavagna et al. 1976, 1977; Margaria, 1976). An ideal inverted pendulum system has perfect exchange between gravitational potential energy and kinetic energy. In a human walker, energy exchange by the inverted pendulum mechanism reduces the mechanical work required from the muscular system by a maximum of 70 % (Cavagna et al. 1976). One reason why a human walker does not achieve 100 % energy exchange is that the fluctuations in gravitational potential energy and kinetic energy are not matched in magnitude. For example, in high-speed walking, the fluctuations in gravitational potential energy are smaller than the fluctuations in kinetic energy (Cavagna et al. 1976).

The inverted pendulum model of walking emphasizes the advantages of a stiff-legged gait in which the center of mass arcs over the stance limb (a ‘compass gait’). An alternative view of walking emphasizes the advantages of minimizing the vertical movements of the center of mass (Inman, 1966). Inman and colleagues carefully examined the kinematics of walking and identified several mechanisms that are involved in flattening the trajectory of the trunk (Saunders et al. 1953; Inman, 1966; Inman et al. 1994). For example, the knee and ankle flex and extend in the sagittal plane during the stance phase, and the pelvis ‘lists’ or rotates in the frontal plane. They pointed out that these movements appear to reduce the vertical movements of the trunk. These observations strongly suggest that the distance between the point of foot–ground contact and the center of mass (‘virtual stance-limb length’) does not remain constant during walking as it does in the inverted pendulum model. Rather, the virtual stance limb shortens (i.e. ‘compresses’) and lengthens during the course of the stance phase. Although it is well established that compression of the stance limb plays an important role in determining the dynamics of running (McMahon and Cheng, 1990; He et al. 1991; Farley et al. 1993; Farley and Gonzalez, 1996), little is known about the magnitude and role of virtual stance-limb compression during walking. Intuitively, it seems likely that the virtual stance limb compresses more at higher speeds during walking and that there is a gradual transition between walking and running in terms of stance-limb behavior.

Inman and his colleagues compared human walking to a compass gait in which the rigid stance limb contacts the ground at a single point that remains fixed throughout the stance phase. In an actual human walker, the heel of the foot hits the ground first, and the toe of the foot leaves the ground last. In the inverted pendulum model, this can be modeled as forward translation of the point of ground contact during the stance phase (Fig. 1A). If there is forward translation of the point of ground contact, then the virtual stance limb will sweep through a smaller angle because the movement of the point of ground contact accounts for part of the forward movement of the center of mass during the stance phase. Because of the smaller angle swept by the stance limb, the vertical trajectory of the mass is flattened if the point of ground contact is allowed to translate forward during the stance phase (Fig. 1A). Thus, it is possible that the trajectory of the center of mass is flattened in a walking human because the point of ground contact rolls forward from the heel to the toe.

The goal of our study was to examine the effects of translation of the point of foot–ground contact and virtual stance-limb compression on the vertical movements of the center of mass in human walking. We began by comparing the predicted trajectories of the center of mass for a model with a rigid leg and a fixed point of foot–ground contact (‘compass gait’) and for a model with a rigid leg and forward translation of the point of ground contact (Fig. 1A). From this comparison, we quantified the reduction in the vertical displacement of the center of mass that occurred as a result of the movement of the point of force application (ΔPFA) under the foot in our subjects. Subsequently, we quantified the magnitude of virtual stance limb compression and its effect on the vertical displacement of the center of mass (Fig. 1B). We defined ‘virtual stance-limb compression’ as a reduction in the distance between the point of ground contact of the stance limb and the center of mass. Throughout the study, we compared these factors for walking and running to gain insight into the biomechanical determinants of the extremely different trajectories of the center of mass for the two gaits.

Five healthy human subjects (three females and two males) agreed to participate in this study. The mean body mass of the subjects was 56.3±9.9 kg (mean ± S.D.), mean leg length (the distance from the ground to the greater trochanter during quiet standing, L₀) was 0.88±0.03 m, and the mean age was 23±2.1 years. Approval was obtained from the university committee for the protection of human subjects and informed consent was given by all subjects.

Subjects walked at five speeds (0.5, 1.0, 1.5, 2.0 and 2.5 m s⁻¹) and ran at six speeds (1.5, 2.0, 2.5, 3.0, 4.0 and 5.0m s⁻¹) along a runway that had force platforms built into it. The speed range corresponded to Froude numbers [u(gL₀)^−0.5, where u is forward velocity and g is the gravitational acceleration] of 0.17–0.85 for walking and 0.51–1.70 for running. The only instruction given to each subject was to walk or run normally. We measured the mean speed over the 3 m section containing the force platforms using infrared photocells placed at the beginning and end of the section. Subjects were allowed to start as far back as necessary to accelerate to a constant speed before the 3 m section and were given plenty of room after the 3 m section to decelerate safely. We accepted trials in which the measured speed was within 5 % of the prescribed speed. We collected and analyzed three acceptable trials from each subject under each condition. Thus, whenever a mean value for all the subjects is stated for a given condition (e.g. 2.5 m s⁻¹ walk), it is the mean of 15 values.

Two AMTI force platforms (AMTI model LG6-4-1, Newton, MA, USA), placed in series, were used to measure the vertical and horizontal (i.e. fore–aft) components of the ground reaction force. In addition, we used the force platform to measure the moment about the medio-lateral axis of the force platform (M_x) so that we could calculate the position of force application in the fore–aft direction at each instant during the stride. Collection of force data began when the subject passed the first photocell and ended when the subject passed the second photocell. Signals from the force platforms were sampled at 1000 Hz using Labview Software and a computer A/D board (National Instruments, Austin, TX, USA). The signals from the two force platforms were summed using software before further analysis. The horizontal ground reaction force signal was integrated with respect to time in order to calculate the horizontal impulse and the change in forward velocity while the subjects were on the force platforms. We only accepted trials in which the net change in forward velocity was less than 0.15 m s⁻¹.

Video data were recorded in the sagittal plane at 200 fields s⁻¹ (JC Labs, Mountain View, CA, USA). Video and force platform data were synchronized using a circuit that illuminated a light-emitting diode in the video field and simultaneously sent a voltage signal to the A/D board when the subject passed each photocell. Strips of reflective tape were placed on the heel and toe of the subjects’ shoes to facilitate identification of heel-strike and toe-off from the video recordings. The time interval during which a foot was in contact with the ground was referred to as its ‘stance phase’. The period between successive heel-strikes (i.e. the time from touchdown of one foot to touchdown of the contralateral foot) was referred to as a ‘step’.

We calculated the vertical displacement of the center of mass during the stance phase (Δy) for each trial of walking and running. This calculation involved determining the vertical acceleration of the center of mass from the vertical ground reaction force data. Subsequently, we calculated the vertical displacement of the center of mass by double integration of the vertical acceleration over an integral number of steps as described in detail elsewhere (Cavagna, 1975; Blickhan and Full, 1993a). Sample data for the vertical displacement of the center of mass as a function of time during the stance phases of walking and running are shown in Fig. 2.

We calculated the reduction in the vertical displacement of the center of mass due to the movement of the point of force application under the foot at each speed in each gait by taking the difference between Δy_compass and Δy_PFAmodel at the instant at mid-stance when the center of mass reached its highest position (Fig. 1A; see Fig. 4A).

The actual trajectory of the center of mass for the subjects differed from the prediction of the inverted pendulum model with a translating point of ground contact because the length of the virtual stance limb did not remain constant during stance in each subject (Fig. 1B). Thus, the second major determinant of the center of mass trajectory that we examined was virtual stance-limb compression. We calculated the extent of virtual stance-limb compression (ΔL) for each instant of the stance phase of walking and running. We began by using leg length (L₀) and the angle of the leg at touchdown (θ_0,PFAmodel) to calculate the horizontal (L₀sinθ_0,PFAmodel) and vertical (L₀cosθ_0,PFAmodel) positions of the center of mass at the instant that the foot hit the ground. After twice integrating the center of mass acceleration, as described above, these values were used as the integration constants to determine the instantaneous horizontal and vertical positions of the center of mass throughout the stance phase. The length of the virtual stance limb (L) at each instant of the stance phase was then calculated from the instantaneous position of the center of mass using the distance formula. We calculated virtual stance-limb compression from the difference between the leg length at heel-strike (L₀) and the leg length at each instant of the stance phase (L).

To compare the magnitude of virtual stance-limb compression among the walking and running trials, we used the virtual stance-limb compression value at the instant when the center of mass reached its highest point for walking and its lowest point for running (ΔL_midstance; Fig. 1B). The magnitude of virtual stance-limb compression varied throughout the stance phase for both walking and running (Fig. 2). For running, maximum virtual stance-limb compression occurred at nearly exactly the same time as the center of mass reached its lowest point (Fig. 2C,D). For walking, maximum virtual stance-limb compression often occurred after the center of mass reached its highest point. Thus, by calculating virtual stance-limb compression at the instant that the center of mass had reached its highest point, we underestimated the full extent of virtual stance-limb compression for walking (Fig. 2A,B). However, one of our primary goals was to understand the role of virtual stance-limb compression in determining the maximum vertical displacement of the center of mass, and this technique was most appropriate for achieving this goal.

The actual path of the center of mass during the stance phase of walking differed substantially from that predicted by the inverted pendulum model with a rigid leg and a fixed point of ground contact (compass gait; Fig. 2A,B). For both the subjects and the pendulum model, the center of mass reached its highest vertical position at approximately the middle of the stance phase. However, the trajectory of the center of mass was much flatter for the subjects than for the rigid leg model with a fixed point of ground contact. In the typical examples illustrated in Fig. 2A,B, the net upward movement of the center of mass of the subject was less than 0.035 m between the beginning and middle of stance during walking at 1.5 ms⁻¹ and 2.5 ms⁻¹. In contrast, the rigid leg model with a fixed point of ground contact predicted an upward movement greater than 0.12 m for both walking speeds.

The actual vertical displacement of the center of mass between touchdown and mid-stance was smaller than that predicted by a rigid leg model with a fixed point of ground contact at all speeds of walking and running (Fig. 3; P<0.0001). The center of mass moved upwards (positive vertical displacement) during the first half of stance during walking at all speeds. The maximum vertical displacement increased from 0.013±0.001 m (mean ± S.E.M.) at 0.5m s⁻¹ to 0.031±0.007 m at 2.5 ms⁻¹ (P=0.037). If the virtual stance limb had behaved as a rigid strut with a fixed point of ground contact, the vertical excursion of the center of mass would have increased more steeply with speed, reaching a value of 0.191 m during walking at 2.5 ms⁻¹ (Fig. 3A).

The first factor that substantially reduced the vertical displacement of the center of mass was the heel-to-toe movement of the point of force application under the stance foot. The point of force application moved forward by an average of 0.201±0.008 m (mean ± S.E.M.) between the beginning and the end of the stance phase of walking for all subjects. During the stance phase of running, the point of force application moved forward by 0.157±0.006 m (mean ± S.E.M.). The forward displacement of the point of force application during the stance phase was independent of speed in both gaits (P>0.92). The movement of the point of force application substantially flattened the trajectory of the center of mass during the stance phase. The rigid leg model incorporating forward translation of the point of ground contact (PFA translation) had a smaller stance-limb touchdown angle (Fig. 1A, equations 1, 3). As a result, it also had a smaller vertical displacement of the center of mass (Figs 2, 3). During walking, the forward translation of the point of force application reduced the vertical displacement of the center of mass by 0.035±0.002 m at the lowest speed and by 0.069±0.002 m (means ± S.E.M.) at the highest speed (Fig. 4A). During running, the forward translation of the point of force application reduced the vertical displacement by 0.030±0.002 m at the lowest speed and by 0.050±0.006 m at the highest speed (Fig. 4A).

The second factor that reduced the vertical displacement of the center of mass was that the virtual stance limb compressed (Fig. 1B). During walking, the virtual stance limb compressed substantially at the beginning of the stance phase (Fig. 2A,B). Later in the stance phase, the virtual stance limb lengthened, reaching its maximum length at the end of the stance phase. We quantified virtual stance-limb compression at the instant when the center of mass reached its highest point for comparison among different speeds and gaits. Because the virtual stance limb was vertically oriented at that instant in the stance phase, the magnitude of virtual stance-limb compression was equivalent to the reduction in vertical displacement of the center of mass due to virtual stance-limb compression. The magnitude of virtual stance compression at the instant when the center of mass reached its highest point (ΔL_midstance) during walking increased with speed from 0.021±0.005 m at the lowest speed to 0.091±0.005 m at 2.5 ms⁻¹ (means ± S.E.M.; P<0.0001) (Fig. 4B).

During running at all speeds, the virtual stance limb reached maximum compression at approximately the same time as the center of mass reached its lowest position near the middle of the stance phase (Fig. 2C,D). During running, the virtual stance-limb compression at the instant that the center of mass reached its lowest point (ΔL_midstance) increased from 0.106±0.012 at 1.5 m s⁻¹ to 0.141±0.016 m at 5 m s⁻¹ (means ± S.E.M.; Fig. 4B).

The virtual stance limb compressed less during moderate-speed walking than during running at any speed (Figs 2, 4B). At a moderate walking speed (1.5 ms⁻¹), virtual stance-limb compression reduced the vertical displacement of the center of mass by 0.050±0.005 m. At a moderate running speed (3.0 ms⁻¹), virtual stance-limb compression reduced the vertical displacement by 0.134±0.009 m. Thus, when moderate speeds for each gait were compared, the difference in virtual stance-limb compression was a major reason for the nearly opposite patterns of vertical movement of the center of mass during the stance phase. However, it is important to realize that this comparison was not for walking and running at the same absolute speed.

At the highest walking speed, the virtual stance limb compressed by 26 % less during walking (0.091 m) than during running (0.123 m; P<0.0001; Figs 2B,C, 4B). In spite of this relatively small difference in virtual stance-limb compression, the pattern of movement of the center of mass during the stance phase was very different for the two gaits. The center of mass reached its highest point at mid-stance in walking but reached its lowest point at mid-stance in running (Figs 2B,C, 3). The major reason for this difference in the movement of the center of mass was the difference in the stance-limb touchdown angle. At a given absolute speed, stance-limb touchdown angle was greater for walking than for running (Fig. 5). For example, at 2.5 m s⁻¹, the leg touchdown angle was 30.4±1.4 ° for walking and 19.2±0.9 ° (means ± S.E.M.) for running.

In the absence of compression of the virtual stance limb, a greater stance-limb touchdown angle will result in a larger upward vertical displacement of the center of mass during the stance phase. Because the stance-limb touchdown angle was larger during walking than during running, stance-limb rotation alone would have caused a greater upward movement of the center of mass during the first half of the stance phase during walking than during running. Without virtual stance-limb compression, the rotation of the leg would have caused a 0.122 m upward movement of the center of mass during the first half of the stance phase of walking at 2.5 m s⁻¹ (Figs 2B, 3A). However, the 0.091 m virtual stance-limb compression (Fig. 4B) reduced the upward movement of the center of mass to 0.031 m. In contrast, during running at the same speed, stance-limb rotation would have caused only a 0.050 m upward movement of the center of mass during the first half of stance if there had been no virtual stance-limb compression (Figs 2C, 3B). However, because the virtual stance limb compressed by 0.123 m, the center of mass moved downwards by 0.073 m during the first half of the stance phase. It is interesting to note that, even if the virtual stance limb had only compressed by the same amount during running as during walking at 2.5 m s⁻¹ (0.091 m), the center of mass still would have moved downwards by 0.041 m during the first half of the stance phase during running. This observation makes it clear that the stance touchdown angle is the primary determinant of the trajectory of the center of mass at speeds near the gait transition speed.

The simplest models for walking and running depict the stance limb as behaving very differently in the two gaits. The inverted pendulum model for walking includes a rigid strut connecting the center of mass to the point of foot–ground contact (‘virtual stance limb’). The spring-mass model for running includes a compliant spring representing the virtual stance limb. A comparison of these models suggests that the dramatic differences in the trajectories of the center of mass between the two gaits are due primarily to differences in the amount of compression of the virtual stance limb. However, our findings show that the virtual stance limb compresses substantially during walking. Indeed, during fast walking, the virtual stance limb compresses by an amount that is only 26 % less than during running at the same speed.

Our findings demonstrate that the difference in the trajectory of the center of mass between walking and running depends as much on virtual stance-limb touchdown angle as on virtual stance-limb compression. This is most obvious when we compare walking and running at the same speed. Our findings show that the virtual stance limb compresses by only 26 % less when a person walks than when a person runs at 2.5 m s⁻¹. In spite of this relatively small difference in virtual stance-limb compression, the center of mass reaches its highest point at mid-stance in walking and reaches its lowest point at mid-stance in running. The reason is that the stance limb touches down at a substantially larger angle relative to vertical during walking than during running. Thus, at speeds near the gait transition speed, the difference in the trajectory of the center of mass between walking and running is caused primarily by a difference in virtual stance-limb touchdown angle.

Previous studies have demonstrated the importance of stance-limb touchdown angle in determining the dynamics of running (McMahon and Cheng, 1990; He et al. 1991; Farley et al. 1993; Alexander, 1995). Running, hopping and trotting animals adjust their spring-mass systems for different speeds by increasing the angle swept by the stance limb while keeping leg stiffness nearly constant (He et al. 1991; Farley et al. 1993). This simple adjustment reduces the vertical displacement of the center of mass and the ground contact time at higher speeds (He et al. 1991; Farley et al. 1993). Similarly, running, hopping and trotting robots that have spring-based legs change their forward speed by adjusting the angle swept by the stance limb while keeping leg stiffness the same (Raibert et al. 1993). These examples demonstrate the important role of stance-limb touchdown angle in determining the kinematics and kinetics of bouncing gaits in animals and robots. Our findings show that stance-limb touchdown angle is also an important determinant of whether the center of mass reaches its highest or lowest point at mid-stance and thus of whether the gait is a walk or a run.

In a walking or running human, the heel usually strikes the ground first and the toe leaves the ground last. This translation of the point of ground contact reduces the stance-limb touchdown angle (Fig. 1A). The center of mass moves forward by a given distance during a single stance phase (‘ut_c’ in Fig. 1). When forward translation of the point of ground contact contributes part of the forward movement of the center of mass during stance, the stance limb touches down with a smaller angle relative to the vertical and rotates through a smaller angle during stance. A smaller stance-limb touchdown angle results in a flatter trajectory of the center of mass. The translation of the point of ground contact under the human foot can be modeled as translation of the point of ground contact of the stance limb in the inverted pendulum model (Fig. 1A). In a human walking at a moderate speed, this movement of the point of ground contact reduces the vertical movements of the center of mass by approximately 40 %. Interestingly, in the passive dynamic bipedal walkers developed by McGeer (1990), a long semi-circular foot is included. In the passive dynamic walker, having a semi-circular foot is more stable than having a point foot and more economical than having a flat foot. Humans are plantigrade but most other mammals are not. A combination of modeling and experiments might provide insight into the effects of foot size and shape on bipedal and quadrupedal walking in animals.

In the present study, virtual stance-limb compression refers to the change in distance between the center of mass and the point of ground contact as shown in Fig. 1. We did not quantify the specific contributors to virtual stance-limb compression. It could be affected by several factors, including (1) flexion of limb joints in the sagittal plane, (2) three-dimensional movements of the pelvis and trunk, and (3) movement of the center of mass within the body. The contributions of individual joints to virtual stance-limb compression have been quantified for humans hopping in place (on the spot) (Farley et al. 1998; Farley and Morgenroth, 1998). As in forward running, the stance limb behaves like a spring during hopping in place. Thus, hopping in place serves as a relatively simple experimental system for gaining insight into the link between the behavior of the individual joints and the virtual stance limb during running. During hopping in place, the stiffness and compression of the virtual stance limb can be nearly fully explained by the angular displacements of the stance-limb joints in the sagittal plane. However, this type of analysis has not been performed for walking.

Kinematic studies of human walking offer insight into the contributors to virtual stance-limb compression. Both the knee and the ankle undergo flexion during the first half of the stance phase and are likely to contribute to leg compression (Saunders et al. 1953; Inman et al. 1994; Sutherland et al. 1994). The contribution of knee flexion to leg compression is reflected in small fluctuations in the distance between the greater trochanter and the lateral malleolus of the ankle during the stance phase of walking (Borghese et al. 1996). These fluctuations are substantially smaller than the magnitude of virtual stance-limb compression measured in our study, probably because they depend only on knee flexion. In contrast, the distance between the greater trochanter and the point of force application on the ground, a distance that depends on ankle angle as well as knee angle, yields much larger length changes (Siegler et al. 1982). These observations suggest that ankle flexion (i.e. dorsiflexion) during the first half of the stance phase contributes to virtual stance-limb compression. In addition, movement of the center of mass within the body (approximately 0.006 m; Whittle, 1997) and the actions of the contralateral limb during the brief period of double support (Saunders et al. 1953; Inman, 1966; Inman et al. 1994) are likely to contribute to virtual stance compression as defined in our study. It has long been believed that pelvic list plays an important role in flattening the trajectory of the center of mass (Saunders et al. 1953; Inman, 1966; Inman et al. 1994). However, recent research has shown that pelvic list has very little effect on the vertical movements of the trunk during walking (Gard and Childress, 1997).

Virtual stance-limb compression during walking has kinematic and energetic consequences. If the stance limb behaved like a rigid strut, the vertical movements of the center of mass during walking would be much larger and would increase steeply at higher speeds (Fig. 3A). However, because virtual stance-limb compression increases at higher speeds, the vertical movements of the center of mass and the fluctuations in gravitational potential energy are relatively small even at the highest walking speeds (Fig. 3A). At high walking speeds, the fluctuations in kinetic energy exceed the fluctuations in gravitational potential energy (Cavagna et al. 1976). As a result, only part of the kinetic energy that is lost by the center of mass during the first half of the stance phase can be converted to gravitational potential energy. Clearly, at high walking speeds, virtual stance-limb compression reduces the upward movement of the center of mass during the first half of the stance phase and, thus, reduces the pendulum-like exchange of kinetic energy and gravitational potential energy.

Although virtual stance-limb compression has the disadvantage of reducing pendulum-like energy exchange, it may have the advantage of enhancing elastic energy storage. Because the stance limb compresses during the first half of the stance phase, the loss in kinetic energy exceeds the gain in gravitational potential energy of the center of mass, thus inhibiting inverted pendulum energy exchange. However, some of the kinetic energy that is lost during the first half of the stance phase may be converted to elastic energy. Elastic energy could be stored in muscles, tendons and ligaments as the ankle and knee flex, contributing to leg compression, during the first half of the stance phase. This elastic energy could be utilized to help increase the velocity of the center of mass in the second half of the stance phase. Although elastic energy storage is certainly more important during running than during walking, empirical evidence supports the idea that elastic energy storage in the ankle extensor muscle–tendon units plays an important role in human walking (Hof et al. 1983). It seems likely that elastic energy storage during walking becomes increasingly important at higher speeds where virtual stance-limb compression limits the pendulum-like exchange of kinetic energy and gravitational potential energy of the center of mass.

An advantage of allowing the stance-limb joints to flex and extend is a reduction in the peak musculoskeletal forces associated with foot–ground impact. When knee flexion is restricted during walking, there is a substantial increase in the peak vertical ground reaction force and in the rate of rise of the ground reaction force (Yaguramaki et al. 1995; Cook et al. 1997). Conversely, exaggerated limb joint flexion leads to a decrease in the peak vertical ground reaction force (Li et al. 1996). These observations suggest that there are trade-offs between maximizing energy exchange by the inverted pendulum mechanism and reducing impact forces. However, the loss in inverted pendulum exchange incurred by allowing the virtual stance limb to compress may be offset by increases in elastic energy storage as discussed above.

Springs and damping elements have been incorporated into the legs of some models of walking in order to match the ground reaction force patterns observed in human walking and to predict correctly the relationship between speed and stride length (Siegler et al. 1982; Pandy and Berme, 1988, 1989; Alexander, 1992). The stiffness (k) of these springs is generally somewhat higher (k=12–34.5 kN m⁻¹) than the leg stiffness values reported for normal human running (k≈11 kN m⁻¹) (Pandy and Berme, 1988; He et al. 1991; Alexander, 1992; Farley and Gonzalez, 1996). It is important to realize that, although a simple spring-mass model (McMahon, 1990; McMahon and Cheng, 1990) simulates running reasonably accurately, it does not include the pendulum-like energy exchange that is a major factor in walking.

Most bipedal animals are birds, and an earlier study demonstrated that the kinematics of locomotion change less dramatically at the walk–run transition in birds than in humans (Gatesy and Biewener, 1991). However, birds do have the same pattern of vertical movement of the center of mass during walking and running as humans and other mammals (Cavagna et al. 1977; Heglund et al. 1982). In birds, the center of mass reaches its highest point at mid-stance during walking and its lowest point at mid-stance during running. Our findings on human locomotion show that the trajectory of the center of mass changes dramatically at the walk–run transition because of an abrupt decrease in the touchdown angle of the stance limb. In birds, the stance-limb touchdown angle does not decrease abruptly at the transition from walking to running (Gatesy and Biewener, 1991). We are not aware of published information about virtual stance-limb compression in walking and running birds, making a full comparison with humans impossible. However, Gatesy and Biewener (1991) point out that the posture of the stance limb of birds is very different from that of humans during walking and running. Some limb segments (e.g. the femur) that are nearly horizontal in birds are nearly vertical in humans. This postural difference could greatly affect limb compliance and the kinematics of the walk–run transition (Gatesy and Biewener, 1991).

In conclusion, the inverted pendulum model with a rigid leg qualitatively describes the mechanism of mechanical energy exchange during walking in humans and other animals. However, our findings show that virtual stance-limb compression is surprisingly large during walking, reaching similar values during high-speed walking to those during running. At these speeds, the difference in the trajectory of the center of mass between walking and running is due primarily to the difference in stance-limb touchdown angle. The unexpectedly large virtual stance-limb compression during walking could have implications for understanding the energetic cost of walking and the determinants of the gait transition speed. In addition, this finding could give insight into the design of prosthetic limbs and robotic legs that work effectively for both walking and running.

This research was supported by a National Institutes of Health grant (R29 AR44008) to C.T.F. and by a UC Berkeley Presidential Undergraduate Fellowship to C.R.L.