A simple model predicts energetically optimised jumping in dogs

It is generally accepted that, in the absence of other imperatives, animals minimise the energy used for locomotion. Evidence from a range of investigations indicates that animals adopt speeds (Ralston, 1958; Hoyt and Taylor, 1981), step length–frequency combinations (Zarrugh et al., 1974; Zarrugh and Radcliffe, 1978; Cavagna and Franzetti, 1986; Minetti and Saibene, 1992; Minetti et al., 1995), step widths (Donelan et al., 2001) and gaits (Hoyt and Taylor, 1981; Alexander, 1984; Griffin et al., 2004) similar to those that minimise the net cost of transport when travelling over flat, level ground. The natural environment of terrestrial animals, however, is typically not uniformly flat but contains obstacles; for example, holes, steps, rocks and vegetation. Animals that encounter obstacles in the path of travel must either re-route to avoid them or execute a manoeuvre such as jumping to traverse the obstacle. There is evidence that the principle of minimising energy use might extend beyond the physiology of locomotion to locomotor behaviours (Minetti, 1995) and beyond regular gait to manoeuvres (Moraes and Patla, 2006).

Jumping over an obstacle in the path of travel during locomotion is a manoeuvre in the locomotor repertoire of many species of terrestrial animal. It allows traversal of a raised obstacle or a length of ground without physical contact, change of direction or significant reduction in forward velocity but at the cost of raising the body’s centre of mass (CoM). The manoeuvre has been studied extensively in species that use jumping as a mode of progression (e.g. Marsh and John-Alder, 1994; Peplowski and Marsh, 1997; Aerts, 1998; Azizi and Roberts, 2010) and there is a sizable body of literature relating to its use in the athletic pursuits of humans and horses (e.g. Alexander, 1990; Hay, 1993; Seyfarth et al., 2000; Dutto et al., 2004; Bobbert and Santamaría, 2005). The focus for much of this work has been on the anatomical mechanisms and physiological processes that allow animals to produce the high power required for the exceptional performance observed in some species. Whether animals control the manoeuvre to minimise energy use is not known.

Although jumping is an inherently costly manoeuvre, the mechanical work required to jump over an obstacle depends on the trajectory followed. Consider a simple obstacle formed from a raised horizontal bar at a fixed distance from a defined take-off position. Any one of a theoretically infinite number of trajectories could be used to pass over the obstacle with minimum adequate clearance; only a single unique trajectory, however, would minimise overall mechanical energy. Both horses and dogs use jump trajectories that differ depending on the size and shape of the obstacle being traversed (Pfau et al., 2011; Birch et al., 2016; Lewczuk et al., 2007), suggesting that they might have the capacity to control take-off to follow an energetically optimum trajectory.

We hypothesised that domestic dogs (Canis lupus familiaris Linnaeus 1758) would utilise trajectories that minimised the mechanical work done during the take-off stride when jumping over a raised obstacle from a constrained take-off position. The optimum jump trajectory was predicted based on the height of the obstacle and the horizontal distance from the obstacle at take-off. The predicted trajectories were then compared with the trajectories recorded for dogs when jumping, over a large range of constrained take-off distances.

A simple planar model was developed to calculate the mechanical energy required at the instant of take-off for the traversal of a raised obstacle.

In Fig. 2, the energy associated with ballistic trajectories that pass over the height constraint is plotted as a function of apex position at a spatial resolution of 10 mm. Although there are an infinite number of trajectories that would pass over the height constraint, their apices within the space defined by the axes of Fig. 2 are constrained to the region shown. Because of symmetry, the x coordinate of the apex must be >d/2. It can be observed that there exists a single, unique trajectory that minimises the energy required to pass over the height constraint. The optimisation arises as a result of the trade-off between maximum GPE gained and KE due to velocity in the direction of travel. The trade-off is apparent from Eqn 1 in which the GPE increases and KE decreases with increasing y. It can be shown analytically that a single minimum exists for any combination of h and d (h, d>0).

We applied this simple model to the energetics of jumping manoeuvres used by dogs during locomotion as follows. (1) The trajectories of jumps made by dogs were represented using a Cartesian reference frame in which horizontal distance in the direction of travel was represented on the x-axis and vertical distance or height was represented on the y-axis. We assumed that motion outside this x–y plane was negligible. Ground level was defined as y=0. (2) An obstacle in the form of a horizontal raised bar over which dogs jumped was modelled as a height constraint as described above. The location of the obstacle was used to define x=0, with x increasing in the direction of travel (Fig. 3). Hence, the intersection with ground level of a line projected vertically downwards from the obstacle defined the origin of the reference frame. (3) The location and mass of the dog were represented by a single point mass equal to body mass and coincident with the location of the body CoM. (4) Take-off was defined as the instant at which the animal lost contact with the ground at the onset of the aerial phase of the jump. (5) The effect of aerodynamic drag was assumed to be negligible and so the horizontal component of velocity (v_x) was assumed to be constant throughout ballistic flight. (6) The contribution of angular kinetic energy of the body to the total mechanical energy was relatively insignificant and not incorporated into the model. Estimation of angular kinetic energy based on the morphometric data of Fedak et al. (1982) suggested that this component comprised <7% of the energy cost of a jump. (7) Approach velocity v_0,x was defined as the mean horizontal velocity of the ultimate stride before take-off. (8) Jump length was approximated as twice the horizontal distance travelled by the CoM between take-off and apex location.

In the context of jumping during locomotion, kinetic energy associated with approach velocity (see point 7 above) contributes to the kinetic energy at the point of take-off. We hypothesised that animals would choose a trajectory that minimised the energy added during the take-off stride regardless of approach velocity. Thus, the cost function E_mech+ used to find the minimum energy trajectory was evaluated as the energy of the jump trajectory (E_mech, Eqn 1) minus the kinetic energy due to approach velocity. It was observed in the experimental data that the horizontal velocity during the jump was always greater than the approach velocity. The mechanical cost of transport based on energy added at take-off C_mech+ was defined as the mechanical energy added at take-off E_mech+ divided by jump length. Mass-specific values of these parameters were obtained by dividing by body mass.

The parameters defined in Fig. 3 were obtained using motion capture for each jump performed by a group of dogs over a set of obstacle positions. Our approach was to compare the measured trajectory defined in terms of apex position with the optimum trajectory modelled as the apex position that minimised E_mech+. The optimum trajectory was obtained numerically by evaluating the energy required for potential apex positions at a spatial resolution of 10 mm. Calculations were performed in MATLAB (R2009a, MathWorks, Natick, MA, USA) using the following formulae (Eqns 2–5).

Examples of trajectories that minimise E_mech+/m and C_mech+/m for the take-off position and height constraint used for Figs 1 and 2 are shown in Fig. 4.

The model so far described has considered the CoM passing over the obstacle as the condition for successful traversal. In reality, the entirety of the body, within which a jumping dog's CoM is located, must pass over the obstacle without making contact. The minimum possible height of the CoM at the instant when it is vertically above the obstacle (x=0; indicated as point ‘c’ in Fig. 5A) must thus be the sum of obstacle height and vertical distance from the CoM to the ventral surface of the animal's thorax. This parameter, the minimum height that must be reached by the CoM for a successful traversal, is termed effective obstacle height (EOH; Fig. 5B). An example E_mech+/m optimisation showing the additional constraint imposed by the body is shown in Fig. 5A. Increasing the height by which the CoM must be raised between take-off and crossing the obstacle (EOH−d_y) shifted the optimum apex position in the direction of travel (positive x) and in the upwards (positive y) direction, whilst increasing v_0,x or d_x shifted the optimum apex position in the direction of travel and in the downwards (negative y) direction. For initial conditions within the experimentally observed range, a different but similar trajectory minimised C_mech+/m.

Data were collected from five dogs (Table 1). Ethical approval for the study protocol was provided by the University of Bristol Animal Services Unit. The cranio-caudal position of the CoM during standing was measured directly in dogs P1, P2 and P3 using a balance board. The identified CoM location relative to anatomical landmarks in these dogs was used to estimate cranio-caudal CoM position in P4 and P5, neither of which would stand still enough for a reliable measurement to be made. The dorso-ventral position of the CoM was estimated in all dogs as the midpoint of a vertical line drawn on the surface of the trunk between the dorsal and ventral midlines at the cranio-caudal position of the CoM while the dog was standing. A retro-reflective marker was placed on the lateral trunk at the position of the CoM. Markers were also placed on the feet to allow the instant of take-off to be identified.

An obstacle was constructed on a 12 m-long track in a gait laboratory. The obstacle comprised a single raised bar (the obstacle bar) and a variable number of base bars which were laid on the track surface on the approach side of and parallel to the obstacle bar. The base bars were used to constrain the minimum distance in front of the obstacle bar (obstacle length) from which the dogs could take off when jumping over the obstacle (Fig. 5B). The height of the obstacle was set for each dog to be comfortably traversable from a range of take-off positions. Each dog jumped obstacles of a range of lengths presented in ascending order (Table 2) whilst kinematic data were recorded at 200 frames s⁻¹ using an optical motion capture system (Oqus, Qualisys AB, Gothenburg, Sweden).

CoM position at take-off (d_x, d_y), CoM apex position (a_x, a_y) and approach velocity (v_0,x) were recorded for each trial, where v_0,x was defined as the mean horizontal velocity of the CoM in the ultimate stride before take-off. Take-off was identified as the first frame of the obstacle traversal stride in which no part of the animal was in contact with the ground. EOH was calculated for each dog by summing of the height of the obstacle bar and the vertical distance from the ventral surface of the dog's trunk to the CoM marker during standing (Fig. 1A). EOH was used as the obstacle height (b_y) input to the model.

For each trial, the CoM position at take-off (d_x, d_y) and the EOH (0, b_y) were used to predict the apex position (a_x, a_y) of the unique trajectories that would minimise E_mech+/m and C_mech+/m for the jump. The corresponding apex position in the experimental data was identified as the position at which the CoM marker attained maximum height. The E_mech+/m associated with the measured trajectory was calculated from the take-off velocity, and C_mech+/m was obtained by dividing E_mech+/m by twice the horizontal distance travelled by the CoM between take-off and the apex location. Measured and predicted values of a_x and a_y were divided by EOH to calculate dimensionless parameters (relative apex position) â_x and â_y for comparison across dogs. The model was evaluated by plotting measured values of â_x and â_y against predicted values that optimised E_mech+/m and C_mech+/m, respectively. Measured values of the two energetic parameters were also plotted against their predicted optimum values. Linear regression models were fitted to the data in each plot and the residuals examined to evaluate the appropriateness of the fitted model. Hypothesis tests of the regression models against the line y=x were used to evaluate statistically the deviation from the predicted values. r² values were used to indicate the proportion of linear variation in the measured parameters that may be attributed to variation in their predicted value.

Data for a total of 62 trials were collected from 5 dogs. All dogs performed 12 trials each except for subject P2, which performed 14 trials.

Experimentally modifying the obstacle length caused changes in d_x and hence in the predicted optimum trajectories. For all trials, the measured positions of the trajectory apices were close to those predicted for least mechanical energy and least mechanical cost of transport (Fig. 6A,B,D,E).

Dogs systematically changed their apex position as predicted by the model in response to changes in obstacle length: as obstacle length increased, jump CoM apices became higher and horizontally more distant from the obstacle bar (see Figs S1, S2 and Table S1 for all individual results). In all cases, linear models were appropriate for the data. Changes in predicted values if minimising E_mech+/m accounted for 72% of the measured variation in â_x, 39% of the measured variation in â_y and 71% of the measured variation in E_mech+/m (Fig. 6A–C). The empirical models for â_x and â_y data did not deviate significantly from the theoretical model [â_x, P=0.91; â_y, P=0.06 (intercept); and â_x, P=0.25; â_y, P=0.17 (slope)]. Changes in predicted values if minimising C_mech+/m accounted for 72% of the measured variation in â_x, 41% of the measured variation in relative â_y and 66% of the measured variation in C_mech+/m (Fig. 6D–F). The empirical models for â_x and â_y data did not deviate significantly from the theoretical model [â_x, P=0.52; â_y, P=0.06 (intercept); and â_x, P=0.14; â_y, P=0.18 (slope)]. The empirical models for E_mech+/m and C_mech+/m differed significantly in intercept (P<0.01 and P<0.01, respectively) and slope (P<0.01 and P=0.03, respectively) from the theoretical model predictions.

There was no significant relationship between v_0,x and obstacle length. A trend towards a small positive relationship was observed although not significant at α=0.05 (P=0.06). Dogs did not thus appear to be strongly regulating approach speed for the jump and the systematic change in the horizontal component of jump velocity with obstacle length may be assumed to originate largely from the take-off rather than from the approach.

The aim of the experiment was to provide evidence that the dogs used a control law for jumping that minimised external work. In controlling take-off position, the experimental design equated the minimisation of work to the selection of a unique ballistic trajectory which could easily be measured from the kinematics of the body CoM. Estimation of CoM position was inevitably the largest potential source of error in the collection of experimental data. Given the inter-breed variability in dog morphology and the lack of appropriate morphometric data, it is unlikely that using a dynamic estimate based on a multi-segment model (Amit et al., 2009; Nielsen et al., 2003; Colborne et al., 2005) would have improved the accuracy. Using the method for estimating CoM position described in Channon et al. (2012), excluding the distal limbs, for comparison, we found negligible non-systematic differences in measured/optimal â_x and â_y of <0.1 and <0.04, respectively, with an effect on E_mech+/m of <0.15 J kg⁻¹. By changing the take-off position over a relatively large range, the expected systematic change in the position of the CoM at the apex of the ballistic trajectory was much greater than the uncertainty in the measurement of CoM position and the effect was clearly apparent in the data.

The empirical models for â_x and â_y were not significantly different from the theoretical predictions for minimum energy and for minimum cost. There was evidence of a small bias towards greater measured than predicted values, especially for â_y, and a trend towards a smaller slope (Fig. 6A,B,D,E), but these may be attributed at least in part to the dogs including a safety margin in their clearance of the obstacle. Our theoretical prediction was based on the absolute minimum height to pass above the obstacle not including any additional clearance. The relative size of the bias on â_x and â_y suggests that clearance was achieved mostly by an increase in apex height.

The empirical models for E_mech+/m and C_mech+/m differed significantly from the theoretical predictions, but as these parameters were constructed from the data for â_y and the square of the data for â_x, the statistical difference is perhaps unsurprising. In absolute terms, the difference between the theoretical and empirical models resulted from differences in apex position of the order of a few centimetres and comparable to the magnitude of uncertainty of CoM position. The clustering of C_mech+/m data (Fig. 6F) for some individual dogs may be explained by the correlation between potential energy gain and jump distance for those dogs. We note, however, that the slope of the regression model for C_mech+/m was dominated by data from a single dog and there was a relatively large amount of random variation in the data for another. We conclude that the data do not show that the dogs behaved consistently to minimise C_mech+/m. In contrast, the data for E_mech+/m (Fig. 6C) indicate consistent behaviour of all dogs with the variation in theoretical predictions explaining 71% of the variation in the measured values. We propose that it is unlikely that dogs would have exhibited this behaviour or anything close to it were it not to minimise the work of jumping or some closely related energetic parameter: it differed considerably from the behaviour required to minimise the height gained by the CoM during the jump, which would have also minimised added GPE and required that the apex position was invariant above the obstacle (i.e. at x=0 in Fig. 2, and in middle row panels of Figs S1 and S2). We conclude that dogs closely followed trajectories that minimised E_mech+/m.

One might reasonably assume that a selection pressure to minimise energy use would act to reduce metabolised energy and not mechanical work (Alexander, 1989). Both would be strongly correlated if muscle efficiency and the elastic strain energy exchanges remained fairly constant across the range of jumps performed but that cannot be assumed. Neither can we exclude the possibility that the observed optimisation is a secondary correlate to a closely related target parameter such as power.

We propose that a notable feature of our model is that it allows an absolute prediction of a unique expected behaviour based on very few assumptions, which minimises an energetic parameter.

Most of the evidence for energetic optimisation of locomotion is based on either the free selection of a behaviour shown empirically to minimise metabolic cost or the observation that experimentally imposed changes to preferred locomotor behaviour result in increased cost. The theoretical basis for empirically determined cost minima is as yet incomplete.

Successful obstacle traversal requires anticipatory control based on remote sensing using vision (Mohagheghi et al., 2004). It is unclear how the dogs acquired the complex perception and control necessary to exhibit the behaviour observed in this study. It is unlikely that any of the dogs would have encountered obstacles of similar geometry during their everyday activity or with the frequency for skill acquisition through practice. In some measure, the behaviour might be innate and part of an extended phenotype for the species. We note that it is a remarkable example of the effectiveness of vision-based anticipatory control and raises the possibility that other similar constrained ballistic tasks might be optimised in the same way.