Running Springs: Speed and Animal Size

Running, hopping, trotting and galloping animals bounce along the ground using springs to store and return elastic energy (Cavagna et al. 1964, 1977; Heglund et al. 1982b). These springs include muscles, tendons and ligaments which alternately stretch and recoil, storing and releasing elastic energy, while the feet are on the ground (Alexander, 1988). The long and compliant Achilles tendon is an example of a spring which stores and returns elastic energy during locomotion. In a hopping kangaroo or a running human, the Achilles tendon alone conserves as much as 35% of the mechanical energy required for a stride (Alexander, 1988).

The concept of musculoskeletal springs has been the basis for several simple spring-mass models for running (Alexander, 1992; Alexander and Vernon, 1975; Blickhan, 1989a; Full and Blickhan, 1992; Cavagna et al. 1988; He et al. 1991; Ito et al. 1983; McGeer, 1990; McMahon and Cheng, 1990; McMahon and Greene, 1979; Thompson and Raibert, 1989). These simple models, which employ a single linear leg spring and a mass, accurately predict the mechanics of running gaits. The body’s complex system of active muscles, tendons and ligaments therefore behaves much like the single linear spring employed in the models (Farley et al. 1991).

In the simplest application of a spring-mass model, the spring-mass system only moves vertically, simulating hopping or running in place. This model makes accurate predictions of the mechanics of a human hopping in place (Farley et al. 1991). In this purely vertical model, the stiffness of the leg spring is the most important determinant of the amount of time that the feet are on the ground (ground contact time). Experimental evidence shows that the stiffness of the leg spring changes by more than twofold when humans hop in place at different hopping frequencies (Farley et al. 1991). This leg spring stiffness can be adjusted by changing the amount that the knee flexes while the feet are on the ground (Greene and McMahon, 1979). Because of the adjustable leg spring stiffness, humans can hop in a spring-like manner over the large range of ground contact times associated with a twofold range of frequencies (Farley et al. 1991).

In forward locomotion, unlike hopping in place, the leg spring is not oriented vertically during the entire time that the feet are on the ground. Because the body moves forward while the feet are on the ground, the leg spring sweeps an arc (Fig. 1). As a result, the vertical motions of the body and the ground contact time depend on a combination of the leg spring’s stiffness and the angle through which the leg spring sweeps (McMahon and Cheng, 1990). For example, as a human runs faster, although the time that each foot is on the ground decreases dramatically, the stiffness of the leg spring does not change substantially (He et al. 1991). The shorter ground contact times at higher speeds are achieved by having the leg spring sweep a greater angle while the foot is on the ground. The greater angle at higher speeds leads to an increased compression of the leg spring and an increased force in the leg spring. Consequently, the vertical excursion of the center of mass is reduced and the ground contact time decreases at higher speeds (McMahon and Cheng, 1990).

The first goal of this study is to examine how the properties of the spring-mass system change with speed in trotting and hopping animals. We hypothesize that the stiffness of the leg spring will be independent of speed but that the angle swept by the leg spring will be larger at higher speeds.

The second goal of this study is to examine how the properties of the body’s spring system vary with animal size. The mechanics of bouncing gaits change even more dramatically with body size than with speed. For example, when a horse and a white rat trot at moderate speeds, the horse’s feet are on the ground four times longer than the rat’s. We will examine whether these dramatic differences in the mechanics of locomotion in small and large animals are the result of differences in the stiffness of the leg spring or in the angle that it sweeps.

In the first part of the study, we examined how the properties of the spring system changed with trotting or hopping speed. We focused on trotting and hopping because these are symmetrical gaits which can be modeled as simple spring-mass systems. Our experimental animals were three dogs (Canis familiaris), two horses (Equus caballus), one African pygmy goat (Capra hircus) and one red kangaroo (Megaleia rufa) (Table 1). Each animal either ran across a ground-mounted force platform or ran on a treadmill with a force platform mounted in it. We obtained the largest possible range of trotting and hopping speeds for each animal.

In the second part of the study, we examined how the properties of the spring system changed with animal size. The additional animals studied in this part were three white rats (Rattus norvegicus), three kangaroo rats (Dipodomys spectabilis) and one tammar wallaby (Macropus eugenii) (Table 1). Each of the additional animals ran across a ground-mounted force platform at a speed near the middle of its trotting or hopping speed range. These speeds were considered to be physiologically equivalent because the duty factors (the ratio of the ground contact time of one foot to the stride time) and Froude numbers were similar in all of the animals (Table 1; Alexander and Jayes, 1983; Heglund and Taylor, 1988; Perry et al. 1988). On average, the duty factor was 0.411±0.017 (S.E.M.) and the Froude number was 1.5±0.095 for all the animals. Errors, where stated, are standard error of the mean.

For the dogs, goat and horses, the vertical ground reaction force was measured as the animal trotted on a treadmill. The treadmill had a force platform equipped with strain gauges (model OR6-5-1, Advanced Mechanical Technology, Newton, MA) mounted under the tread. The natural frequency of vertical vibration of the force platform mounted in the treadmill was 160Hz, and the cross-talk from a horizontal force to the vertical was less than 1%. The signal-to-noise ratio for the measurement of the ground reaction force for a 700N runner was approximately 100:1. The treadmill-mounted force platform is described in greater detail elsewhere (Kram and Powell, 1989). The force signal was sampled at 1000Hz for the dogs and goat, and at 500Hz for the horses. During the treadmill runs, the animals were either filmed (horses, Photosonics 1Pl, 100frames s⁻¹) or videotaped (dogs and goats, NAC video recorder, 200 fields s⁻¹) in lateral view.

The vertical and horizontal (fore–aft) ground reaction forces were measured as the white rats trotted and the kangaroo rats hopped across a series of five small ground-mounted force platforms equipped with strain gauges. The platforms were 0.25 m×0.25m and had natural frequencies of 180Hz in the vertical direction and 200Hz in the horizontal direction. The cross-talk from a horizontal force to the vertical was less than 3%. The force signal was sampled at 1000Hz and was digitally filtered with a cut-off frequency of 100Hz. The design and specifications of these force platforms have been described in detail elsewhere (Heglund, 1981). The average forward speed of the animals was calculated from the time between breaking two light beams located over the series of force platforms (0.5m apart). The horizontal force was integrated to find the net change in the forward velocity during the stance phase. If this change was greater than 5%, the trial was not accepted. During the trials, the animals were filmed in lateral view at 200frames s⁻¹ (Photosonics 1Pl).

The tammar wallaby and the red kangaroo hopped across a ground-mounted Kistler force platform (model 9261). The natural frequency of vertical vibration was greater than 200Hz, and the force signal was sampled at 1000Hz. The forward velocity of the animals was calculated from the time between breaking four light beams placed 1m apart, equally spaced before and after the force platform. To check that a particular stride was at a steady speed, we integrated the horizontal force to find the net change in the forward velocity of the center of mass during the stance phase. If it changed by more than 5%, the trial was not accepted. During the trials, the animals were filmed in lateral view at 200frames s⁻¹ (Photosonics 1Pl).

The stiffness of the leg spring (k_leg) changed only slightly with speed in all of the animals (Fig. 2A–D). Over a threefold range of trotting speeds in the dog, k_leg increased by 16% from 4.24 to 4.92kNm⁻¹. In the goat and the horse, k_leg increased by 25% and 29% respectively, and in the kangaroo, k_leg decreased by 21% over the range of speeds measured. The stiffness of the leg spring (k_leg) was calculated from the ratio of the peak force (F) to the peak displacement of the leg spring (ΔL, equation 1). On average for all the animals, F increased by 35%, ΔL increased by 20% and k_leg increased by 12 % between the lowest and highest speeds.

The effective vertical stiffness (k_vert) increased dramatically at higher speeds in spite of the relatively slight changes in k_leg (Fig. 2A–D). The effective vertical stiffness increased at higher speeds because half the angle swept by the leg spring while it was in contact with the ground (θ) increased (Fig. 2E–H). When θ was greater, the vertical excursion of the center of mass during the ground contact phase (Δy) decreased, and k_vert increased (equation 4). On average for all of the animals, θ increased by 31%, Δy decreased by 35% and k_vert more than doubled between the lowest and highest speeds. In the trotting dog, θ increased from 28.6° to 40.4° and k_vert increased by 228% (Fig. 2). In the goat, θ increased from 25.1° to 28.5° and k_vert increased by 76% over the range of trotting speeds. In the horse, θ increased from 24.7° to 29.2° and k_vert increased by 170%. Finally, in the kangaroo, θ increased from 38.7° to 55.3° and k_vert increased by 48%. It is important to notice that for a given k_leg, k_vert depends on cosθ, a function that is sensitive to small changes in θ in the range used for trotting and hopping (equation 5).

The stiffness of the leg spring increased dramatically with body mass (k_leg∝M^0.67, Fig. 3C). For a 1000-fold increase in body mass (e.g. horse versus white rat), the stiffness of the leg spring increased by approximately a 100-fold. This finding is similar to that of Full and Blickhan (1992), who used the mechanical work rate of the center of mass to calculate the stiffness of the leg spring in a variety of animals. In this study, we calculated the stiffness of the leg spring from the ratio of the peak force (F) to the peak displacement (ΔL) of the leg spring. The peak force increased in direct proportion to body mass (F∝M^0.97, Fig. 3A), and ΔL also increased with body mass (ΔL∝M^0.30, Fig. 3B).

Half the angle swept by the leg spring during the ground contact phase (θFig. 1) was nearly the same in large and small animals (θ ∝M^{− 0.034}; Fig. 4A). For example, θ was 29 ° for the kangaroo rat and 26° for the horse. Because of the similar geometry of the leg spring during the ground contact phase in animals of different sizes, the effective vertical stiffness (k_vert) increased nearly in parallel with k_leg across the range of body masses (Fig. 4B, k_vert∝M^0.61). The effective vertical stiffness was about 70-fold greater for a horse than for a white rat.

The ground contact time (t_c) increased with body mass in nearly the same way as half of a resonant period of vertical vibration (T/2) (Fig. 5; t_c=0.0994M^0.19, r²=0.927; T/2=0.0611M^0.19, r²=0.911). Half of a resonant period of vertical vibration is the amount of time that the spring-mass system would remain on the ground if the landing velocity (the vertical velocity at the instant before the system touches the ground) were large enough for the vertical inertial acceleration to be much greater than the gravitational acceleration. The scaling exponents for t_c and T/2 were virtually identical but the coefficient was 38% higher for t_c than for T/2. On average, t_c was 38% longer than T/2 for the animals in this study.

The work rate of the leg spring was similar to the mechanical work rate of the center of mass for all of the animals regardless of body size (Fig. 6). The mass-specific work rate of the leg spring was nearly independent of body size (P_spring∝M^0.11, Fig. 6), and the mechanical work rate of the center of mass (P_com) was also nearly independent of body size (P_com∝M^0.12) (calculated from Heglund et al. 1982a). It is important to note that these comparisons were made at physiologically equivalent speeds and not at the same absolute speed (Table 1). Heglund et al. (1982a) showed that the mechanical work rate of the center of mass increases with speed in a similar way in mammals of all sizes and, as a result, is independent of body mass at the same absolute speed. In our study, the larger animals used higher absolute speeds (u) than the smaller animals (u∝M^0.12), so the mechanical work rate of the center of mass increased slightly with body mass. Our finding that the mechanical work rates of the center of mass and of the leg spring are similar at physiologically equivalent speeds in all the animals supports our assumption that the leg spring behaves as a simple linear spring. This finding also provides evidence that the spring-mass model accurately describes the mechanics of locomotion over a large size range of animals.

The spring-mass model we use in this study is the simplest possible model for describing the mechanics of bouncing gaits (Fig. 1). This model lumps the mass of the whole body at the hip and represents the entire musculoskeletal spring system as a single linear spring (the leg spring). In reality, the musculoskeletal spring system consists of a complicated system of muscle, tendon and ligament springs which act across nearly every joint in the body (Alexander, 1988). Because of the difference between the simplicity of the spring-mass model and the complexity of the actual musculoskeletal spring system, it is remarkable how well the model describes and predicts the mechanics of locomotion (Blickhan, 1989a; Full and Blickhan, 1992; McMahon and Cheng, 1990; He et al. 1991). In the context of this simple spring-mass model, we have been able to find general principles about how the musculoskeletal spring system operates in trotting, hopping and running mammals. In each of these bouncing gaits, the stiffness of the leg spring is nearly the same at all speeds. The spring system is adjusted to operate at higher speeds by increasing the angle swept by the leg spring while it is in contact with the ground rather than by increasing the stiffness of the leg spring. Consequently, at higher speeds, the leg spring experiences greater vertical forces, which result in larger compressions of the leg spring. Furthermore, it sweeps through a greater angle and the center of mass follows a flatter trajectory. The combination of these effects results in an increased vertical stiffness and a decreased period of ground contact.

Our findings also show that the resonant period of vertical vibration is greater in larger animals in spite of their stiffer vertical springs. This is possible because, as body size increases, the spring stiffness increases less steeply than body mass. The predicted half-period of vertical vibration (T/2) is proportional to (M/k_vert)^0.5. Because k_vert is proportional to M^0.61 (Fig. 4), T/2 is proportional to (M^1.0/M^0.61)^0.5 or M^0.19 (Fig. 5). The increase in the predicted half-period of vertical vibration in larger animals parallels the increase in the measured ground contact time (t_c∝M^0.19).

The results of this study also show that mammals of different sizes move in a dynamically similar manner. Dynamic similarity is a paradigm that proposes how it is possible for animals of different sizes to move in ways that are easily recognizable as similar (Alexander, 1989; Alexander and Jayes, 1983). For example, if a horse and a rat trot in a dynamically similar manner, their feet will spend a similar fraction of a stride period on the ground and they will swing their limbs through similar angles. In the context of the spring-mass model, dynamic similarity between a large and small animal predicts that if they are moving at similar Froude numbers (forward velocity normalized for differences in leg length; Table 1), they will have (1) an equal compression of the leg spring relative to the rest length of the spring (ΔL/L₀), (2) an equal angle swept by the leg spring while it is on the ground (2θ), (3) an equal fraction of the predicted half resonant period of vibration spent in contact with the ground [t_c/(T/2)] and (4) an equal ratio of peak force in the leg spring to the weight of the body (F/Mg).

Our findings support the hypothesis that mammals of different sizes move in a dynamically similar manner at physiologically equivalent speeds. This conclusion is based both on our results about the mechanics of locomotion and on the observation that the leg lengths of the animals in this study are proportional to M^0.34 (Table 1). First, we find that the compression of the spring (ΔL) is about 25% of leg length in animals of all sizes (ΔL/L₀∝M^0.30/M^0.34; Fig. 3B, Table 1). Second, as predicted by dynamic similarity, half the angle swept by the leg spring (θ) is similar in animals of all sizes (0∝M⁻ 0.034; Fig. 4A). This result is not significantly different from an earlier study which found that the angle swept by the hindlimb was proportional to M⁻ 0.10 (McMahon, 1975). Third, the ratio of the ground contact time (t_c) to the predicted resonant period of vertical vibration (T/2) does not change in a regular way with body mass [t_c/(T/2) ∝M^0.19/M^0.19; Fig. 5].

The final prediction of dynamic similarity, that the peak force in the leg spring relative to the weight of the body should be independent of animal size, is on average supported by our findings. We find that for all of the animals running at physiologically equivalent speeds, the peak force in the leg spring is nearly proportional to body weight (F/mg∝M^0.97/M^1.0; Fig. 3A). However, it is important to note that the peak force in the leg spring is about twice as great in the bipedal hoppers (four times body weight) as in the quadrupedal trotters (twice body weight). This peak force refers to the peak vertical force during a ground contact phase and, therefore, is the sum of the forces exerted by both legs in contact with the ground. Our observation of higher peak ground reaction forces in hoppers than in trotters or runners agrees with the findings of earlier studies (Blickhan, 1989b; Full and Blickhan, 1992; Perry et al. 1988). Perry et al. (1988) found that the peak forces in the ankle extensor muscles of a hopping kangaroo rat are about twice as great as in a galloping white rat at their preferred speeds. However, because the kangaroo rat has twice the cross-sectional area of ankle extensors as the white rat, the muscle stresses are similar in both animals.

In spite of the simplicity of the spring-mass model, it describes and predicts many aspects of the mechanics of hopping, trotting and running in animals that vary in body mass over three orders of magnitude. Our observation that the stiffness of the leg spring is nearly constant at all speeds applies equally well to trotting, hopping and running animals. We conclude that a constant-stiffness leg spring may be a basic and invariant principle of running.

The authors wish to thank C. Richard Taylor and Tom Roberts for their help throughout the study and Robert Full for his comments on the manuscript. This work was supported by a National Institutes of Health Grant (no. 2RO1 AR18140) to C. Richard Taylor, a National Institutes of Health Postdoctoral Fellowship to C.T.F. and a National Defense Science and Engineering Graduate Fellowship to J.G.