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Figures
- Fig. 1.
A phase-based approach to analysis of gait incorporating considerations of quasi-static stability suggests that dogs exhibiting a gait that becomes more trot-like when walking on rough terrain is due at least in part to concern for stability. (A) Top-down view of the limb phases and limb phase differences of a quadruped. Legs are coloured circles (hind-left limb, θ0, orange; fore-left, θ1, green; fore-right, θ2, cyan; rear-right, θ3, magenta) and arrows denote computation of phase differences, φi (as in Eqn 1). Illustration of the longitudinal quasi-static stability margin for the case where limbs 0, 1 and 3 are on the ground. The dog is for illustrative purposes; our predictions are based on modelling, not observations of foot touch-downs. The position of the centre of mass is indicated by a black and white circle; dotted lines show the distances above the ground. The polygon of support is shown in black at the bottom of the figure; the purple line is the quasi-static longitudinal stability margin. (B) Scatter plot of gait usage by dog (371 strides, 6 subjects, marker shape as in Table 1) and terrain condition (blue, flat; red, rough), with crosses at the stereotyped walk (single-foot in lateral sequence), trot and pace. The lateral couplet walk lies to the left of walk, towards pace, approximately where the majority of blue data points are. The black line is the path connecting the single-foot walk and trot (a theoretical construct, not derived from the data).
- Fig. 2.
Kernel density plots and boxplots for each dog of the values of the trot projection, λ, for flat (left; blue) and rough (right; red) terrain. The gait is most trot-like when the projection value (λ) is zero; stereotypical (single-foot in lateral sequence) walking has a projection value of –π/2. On the rough terrain each subject is generally more trot-like. Kernel density bandwidth estimated by the Scott (2009) method implemented in the SciPy (http://www.scipy.org/) library. Here, frequency is the frequency density of strides with respect to the observed projected distance to trot. Markers in top left of each sub-plot denote the subject as in Table 1 (371 strides, 6 subjects).
- Fig. 3.
Contour plot of longitudinal quasi-static stability margins against projected distance to trot, λ, duty factor and fore-hind touch-down spatial separation a, in the neighbourhood of the stereotypical (single-foot in lateral sequence) walk. λ is varied on the x-axis, duty factor on the y-axis. Rows have different fore–hind touch-down spatial separation a, with (A) 0.5, (B) 1.0 and (C) 2.0. Lighter green is higher stability. The left panels display the minimum stability margin observed throughout a stride; the right panels display the average. The solid black region is where during the stride the quadruped has at least one period with no feet on the ground and the stability margin is undefined. Trot lies at zero on the x-axis, and the stereotypical (single-foot in lateral sequence) walk at –π/2. Irrespective of a and over the full range of duty factors, a shift towards trot is more desirable than away from trot. The box plots above the contour plots are of the observed λ, for all dogs (n=6; blue, even ground; red, uneven terrain). The gait on rough terrain shifts towards trot, where the longitudinal static stability margins are higher.
- Fig. 4.
Scatter plot of projection on the line between trot and walk against speed. Projection is in radians, the marker shape denotes dog (371 strides, 6 subjects, as in Table 1), the colour of the points indicates terrain (blue=flat, red=rough).
- Fig. 5.
Scatter plot of projection on the line between trot and walk against duty factor. Projection is in radians, the marker shape denotes dog (371 strides, 6 subjects, as in Table 1), the colour of the points indicates terrain (blue=flat, red=rough). This pattern of duty factors is consistent with a shift in the limb phasing towards trot rather than a change to a trotting gait.
- Fig. 6.
Scatter plot of ϕ1 against speed. This is the phase difference between fore-left and hind-left. Projection is in radians, the marker shape denotes dog (371 strides, 6 subjects, as in Table 1), the colour of the points indicates terrain (blue=flat, red=rough).
- Fig. 7.
Scatter plot of ϕ2 against speed. This is the phase difference between fore-right and hind-right. Projection is in radians, the marker shape denotes dog (371 strides, 6 subjects, as in Table 1), the colour of the points indicates terrain (blue=flat, red=rough).
- Fig. 8.
Scatter plot of ϕ3 against speed. This is the phase difference between hind-right and hind-left. Projection is in radians, the marker shape denotes dog (371 strides, 6 subjects, as in Table 1), the colour of the points indicates terrain (blue=flat, red=rough).
- Fig. 9.
Mechanism by which a shift towards trot enhances quasi-static stability. Each row corresponds to footfall patterns associated with (A) a diagonal couplet walk, (B) the single-foot lateral sequence walk and (C) the lateral couplet walk (assuming a fixed duty cycle of 0.75). The single-foot lateral sequence walk is a walk in which footfalls are spaced evenly at one-quarter cycle apart, whilst the diagonal couplet walk has footfalls on ipsilateral legs more closely together, typically at 15% of a stride (and hence closer to pace, and further from trot; Hildebrand, 1968). Time evolves moving left to right across the footfall patterns, and the animal is walking from left to right across the page. The contact patterns have been separated horizontally to ease readability. As we move from bottom to top on the figure we move towards a more trot-like gait. Both the diagonal and lateral couplet walks are inferior in terms of quasi-static longitudinal stability to the single-foot walk because of the presence of the phases with only two limbs in ground contact (highlighted in red). However, for the lateral couplet walk, the pattern is especially unstable because for phases of the stride only contra-lateral pairs of limbs are in ground contact, leading to high pitch instability. For diagonal couplet walks, diagonal pairs of limbs are in contact during these unstable phases. As a result, while both these walks are longitudinally unstable, the lateral couplet walk is much more unstable.
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