Table 1.
Reported are sample sizes and mean (± s.d.) masses, hip heights and total leg lengths (sum of interarticular lengths of femur, tibiotarsus and tarsometatarsus).
Fig 1.
Morphometric scaling observed in the birds investigated in the current study.
(A) Hip height versus leg length. (B) Hip height versus body mass. (C) Length versus body mass. (D) Degree of crouch versus leg length; stick figure representations show the range of crouched (e.g., quail) to erect (e.g., ostrich) postures exhibited by birds. (E) Degree of crouch versus body mass. All comparisons are plotted on logarithmic scales. Note that in (A)–(C), the line of best fit was determined by applying a power I fit on the untransformed variables, rather than applying a linear fit to the log-transformed values. This helped produce better results for the upper end of the body mass spectrum. Regression equations are reported in Table 2.
Table 2.
Relationships between morphometric variables in the birds studied.
Fig 2.
Speed scaling of kinematic variables in birds and humans.
(A, B) Duty factor; horizontal dashed lines at β = 0.5 mark the transition from grounded to aerial locomotion. (C, D) Relative stance duration. (E, F) relative stride length. (A), (C) and (E) are for birds, (B), (D) and (F) are for humans. In (C) and (E), the relationship with speed varies with body mass, and so this has been demonstrated with several different masses. In (F), major axis fits for both walking and running have been applied; they have significantly different elevations. Regression equations are reported in Table 3.
Table 3.
Speed scaling of kinematic and kinetic variables in birds.
The relationships identified between each variable and speed is of one of three forms: y = Ax + B (linear), y = AxB (power I) or y = AxB + C (power II). The relationship may also be modulated by mass, as indicated by the numbers in brackets. For each relationship, the r2 values are reported, as well as the K statistic of Blomberg et al. [91] and associated P-value for both coefficients A and B. Values in italics are statistically significant.
Fig 3.
Speed scaling of peak force variables in birds and humans.
Forces are normalized to body weight. (A, B) Positive peak Fx. (C, D) Negative peak Fx. (E, F) Peak Fz. (G, H) Mean Fz. (I, J) Peak net force. (A), (C), (E), (G) and (I) are for birds, (B), (D), (F), (H) and (J) are for humans. Regression equations for birds are reported in Table 3.
Fig 4.
Speed scaling of temporal variables concerning the GRF in birds and humans.
Times are normalized to the duration of stance. (A, B) Time of positive peak Fx. (C, D) Time of negative peak Fx. (E, F) Time at which Fx is zero. (G, H) The magnitude of Fz when Fx is zero. (A), (C), (E) and (G) are for birds, (B), (D), (F) and (H) are for humans. In (E), the relationship with speed varies with body mass, and so this has been demonstrated with several different masses. In (F), major axis fits for both walking and running have been applied; they have the same elevation at the walk-run transition. Regression equations for birds are reported in Table 3.
Fig 5.
Speed scaling of the GRF at mid-stance.
(A, B) The anteroposterior component at mid-stance. (C, D) The vertical component at mid-stance. (E, F) The net GRF at mid-stance. (A), (C) and (E) are for birds, (B), (D) and (F) are for humans. Regression equations are reported in Table 3.
Fig 6.
Speed scaling of Fourier coefficients describing the anteroposterior component of the GRF.
(A, B) Xa2. (C, D) Xa3. (E, F) Xa4. (G, H) Xa5. (A), (C), (E) and (G) are for birds, (B), (D), (F) and (H) are for humans. Regression equations are reported in Table 3.
Fig 7.
Speed scaling of Fourier coefficients describing the vertical component of the GRF.
(A, B) Za1. (C, D) Za2. (E, F) Za3. (G, H) Za4. (I, J) Za5. (K, L) Za6. (A), (C), (E), (G), (I) and (K) are for birds, (B), (D), (F), (H), (J) and (L) are for humans. Regression equations are reported in Table 3.
Fig 8.
Change in the shape of the GRF force-time profiles with increasing speed.
(A) Pattern of change observed in birds. (B) Pattern of change observed in humans. Red profile is vertical component, blue profile is anteroposterior component. Among other things, note how the profiles for birds show marked temporal asymmetry compared to those of humans, with more force being applied in the first half of stance. For humans, force-time profiles were calculated from least squares linear fits applied to the respective data, with the exception of Za2 for walking, which was better fit by a power II model. The predicted curves for both components are based on the first ten sine coefficients (i.e., Xa1–Xa10, Za1–Za10), all of which either showed significant variation with speed or significantly non-zero values (or both).
Fig 9.
Speed scaling of mechanical energy fluctuations.
(A, B) percent congruity, distinguished by duty factor. (C, D) normalized net vertical displacement of the COM. (A) and (C) are for birds, (B) and (D) are for humans. Regression equation for (C) is reported in Table 3.
Fig 10.
The dominance of the vertical component of the GRF.
(A) Peak net force scales very strongly with peak vertical force; major axis line has a slope almost exactly equal to unity. (B) The instance of peak net force is very strongly coincident with the instance of peak vertical force; major axis line has a slope almost exactly equal to unity. In B, the outliers (hollow circles) are slow speed trials in which the two peaks were of almost the same magnitude, such that the peak net force can occur at a very different time to when the vertical component is at a maximum (inset illustrates one such trial).
Fig 11.
The association, or lack thereof, between t(Fz,peak) or t(Fnet,peak) and t(Fx = 0).
(A, B) t(Fz,peak) versus t(Fx = 0). (C, D) t(Fnet,peak) versus t(Fx = 0). (E, F) Fz(Fx = 0) compared to Fz,peak for each trial; the blue lines are lines of parity. (A), (C) and (E) are for birds, (B), (D) and (F) are for humans. In (A), the regression line has a slope of 0.7718 and an r2 of 0.1289; in (C), the regression line has a slope of 0.7627 and an r2 of 0.1194. In human walking, peak vertical or net force either occurs early or late in the stance; in running, they occur largely before mid-stance (t < 0.5), yet t(Fx = 0) largely occurs after mid-stance (t > 0.5).
Fig 12.
A comparison of peak positive and negative magnitudes of the anteroposterior component of the GRF.
(A) Fx,peak−versus Fx,peak+ in birds; the vast majority of data points fall below the line of negative parity. (B) Fx,peak−versus Fx,peak+ in humans; the data points fall neatly on the line of negative parity. In (A), the inset shows how differing durations of deceleration and acceleration phases require different peak magnitudes, in order for impulses (shaded area) to remain balanced.
Fig 13.
Comparison of predictions of the BIRDS model against the observed data for mechanical energy fluctuations.
(A) Percent congruity; r2 of predictions is 0.509, root mean square error (RMSE) is 14.895. (B) Relative net vertical displacement of the COM; r2 of predictions is -3.569, RMSE is 0.0339. For both variables, the model predicted an influence of body mass, although this is not particularly pronounced for the prediction of percent congruity.
Fig 14.
A trial of seven footfalls in which the subject (emu) underwent a gradual deceleration.
Stride length, stance duration, duty factor and the nature of the GRF (blue profile = Fx, red profile = Fz) change continuously with continuous decrease in speed. Note that speed is shown as the instantaneous speed of the back marker.
Fig 15.
Changes in the GRF, duty factor and mechanical energy fluctuations with increasing speed.
The force-time profile shape and duty factor for several different speeds have been diagrammatically mapped onto the comparison between percent congruity and speed. The predictions of the BIRDS Model for various body masses are also shown. Note how the shapes of the force-time profiles change far less beyond v* ≈ 1, yet duty factor continues to decrease.
Fig 16.
Asymmetry in the force-time profiles of the different components of the GRF.
(A, B) Differences in Fz force-time profile asymmetry between birds (A) and humans (B), as quantified by the ratio of Fourier coefficients Za2/Za1. (C, D) The force-time profile of Fx exhibits different gross shapes in birds (C) and humans (D). In birds, t(Fx = 0), t(Fx,peak+) and t(Fx,peak–) all occur earlier in the stance compared to humans. (E, F) The differences in asymmetry of the Fx force-time profile are probably due to differences in the location of the COM (black and white disk) relative to the hips (hollow circle). As the GRF vector tracks the COM, at temporally equivalent points in the stance the GRF vector will be more anteriorly inclined in birds (E) than humans (F). Note that the asymmetry results for a portion of the bird data investigated in this study have previously been reported [64]. However, these results were presented in a preliminary fashion, in raw format, not as the derived predictive relationships presented here.