Exploring the limits to turning performance with size and shape variation in dogs

Locomotion is central to an animal's behaviour and its ability to catch prey, avoid predators, find mates and successfully reproduce (Arnold, 1983; Garland and Losos, 1994; Moore and Biewener, 2015; Wilson et al., 2018). One would expect faster prey to successfully escape their predators, and a faster predator to successfully capture prey (Wilson et al., 2013a,b). However, if speed was the only determinate of predator–prey success, the cheetah (Acinonyx jubatus) would be the most successful predator on land, yet this is not the case. Only 58% of cheetah hunts end in successful prey capture (Sharp, 1997; Hilborn et al., 2012), suggesting other attributes of locomotor performance may be equally important.

Manoeuvrability, or turning performance, is an important component of animal locomotion (Clemente and Wilson, 2016; Wilson et al., 2018; Amir Abdul Nasir et al., 2017). High manoeuvrability is characterised by high turning rates and small turn radii, meaning quick sharp turns. In contrast, low turning rates at large turn radii characterise low manoeuvrability (Jindrich and Qiao, 2009). The relative importance of turning performance was illustrated by Wilson et al. (2018), who measured this along with muscle power, top speed, acceleration and deceleration frequency in two predator–prey pairs on the African savannah. They showed that predators were more athletic and superior in horizontal speed and acceleration, and that prey rely heavily on turning performance for successful escape and survival (Wilson et al., 2018).

Compared with maximal speeds, less is known about the biomechanical challenges faced by a turning animal and the morphological characteristics associated with high manoeuvrability. Turning performance may depend on the animal's ability to overcome four key challenges associated with high-speed turns: (1) withstanding additional forces during a turn, (2) maintaining sufficient friction to avoid slipping, (3) minimising toppling moments by leaning into the turn and (4) rotating the body into the new direction of travel (Fig. 1).

When running in a straight line, fore–aft forces are generated to (tangentially) accelerate and reach maximal speeds (Weyand et al., 2000; Roberts and Scales, 2002; Biewener and Patek, 2018). However, turning requires the additional ‘centripetal force’, which is proportional to the squared horizontal velocity (v²) over the radius of the turn (r) and acts perpendicular to the gravitational force (g) (Fig. 1A). The animal must apply this centripetal force towards the centre of the turn (with respect to the heading) to deflect its centre of mass (COM) and accelerate laterally (Jindrich and Qiao, 2009; Wilson et al., 2013b; Biewener and Patek, 2018). The constant limb force hypothesis suggests that force generation limits locomotion performance, thus any increase in centripetal forces should result in a decrease in the possible generation of tangential forces (Greene, 1985; Weyand et al., 2000).

Several studies have supported this force limit. Horses are constrained by limb forces when turn radii exceed 30 meters and must decrease horizontal speeds to apply the centripetal force (Tan and Wilson, 2011). Similar results have been found in wild cheetahs, lions (Panthera leo), zebras (Equus quagga) and impalas (Aepyceros melampus), where centripetal acceleration increases with decreased tangential acceleration and vice versa (Wilson et al., 2018). Humans, mice and quolls (Dasyurus hallucatus) all decrease horizontal speeds when turning, and the increased ground contact time seen in humans and mice supports a limb force limit to speed (Walter, 2003; Usherwood and Wilson, 2006; Alt et al., 2014; Wynn et al., 2015; Churchill et al., 2016). The greyhound (Canis lupus familiaris), however, does not increase ground contact time, but is capable of accelerating laterally without slowing down, suggesting that, at least for this breed, force is not a limitation to turning (Usherwood and Wilson, 2005).

Yet, turning performance for most cursorial animals appears constrained by this force limit (Weyand et al., 2000; Tan and Wilson, 2011; Wilson et al., 2013a). As a result, larger animals might be expected to be more force limited than smaller animals as they must support relatively more mass during locomotion (Iriarte-Díaz, 2002; Dick and Clemente, 2017). However, turning performance is limited not only by limb force production, but also frictional properties and the risk of slipping (Alexander, 1982; Tan and Wilson, 2011; Wynn et al., 2015; Wheatley et al., 2018).

During small radii turns, animals usually reduce turning speeds owing to a friction limit (Alexander, 2002; Jindrich and Qiao, 2009; Tan and Wilson, 2011). The frictional coefficient (μ) can be estimated from tangential velocity (v), turning radius (r) and gravity (g); μ=v²/gr (Eqn 5; Fig. 1B), suggesting a trade-off between turning radius and speed (Alexander, 1982). When horses turn with radii less than 30 m, tangential velocity correlates with the friction coefficient, thus suggesting that turning performance in horses is limited by limb force production at larger radii and friction at smaller radii (Tan and Wilson, 2011).

Surface properties, foot size and morphology also affect grip limits. A high friction surface allows for higher speeds without slipping and crashing compared with a smoother surface (Wheatley et al., 2018). Animals have also evolved footpads and claws to increase their frictional limits. Cheetahs have non-retractable claws to increase traction on the surface, and similarly, the relatively high friction coefficient for quolls may be explained by their claws and footpad design (Russell and Bryant, 2001; Wilson et al., 2013a; Wynn et al., 2015; Wheatley et al., 2018). Generally, smaller animals have relatively larger ground surface contact area in relation to body mass (Chi and Roth, 2010). As mass increases, the friction limit would be expected to decrease owing to the decrease in relative surface area of the feet. The probability of slipping for a small animal is thus lower compared with a larger animal (Wynn et al., 2015). Further, foot morphology and body shape may become even more important to maintain friction as animals utilise other turning strategies such as lateral leaning.

Multiple species display an inward lean when turning. This lean promotes a lateral shift in foot placement relative to the animal's COM, which may increase turning performance by better aligning the limbs with the axis of centripetal force generation (Jindrich et al., 2006; Jindrich et al., 2007; Kuznetsov et al., 2017; Biewener and Patek, 2018). However, leaning comes with the risk of lateral toppling. To avoid falling over sideways, body weight (BW) must not exceed the product of medially ground-directed forces (mGRF), i.e. centripetal forces, and the leaning angle (BW<mGRF×tanθ; Biewener and Patek, 2018: Fig. 1C). Therefore, we expect individuals that are capable of increasing the angle of inward leaning to be capable of faster turns.

Centripetal forces deflect the animal's COM, but the body must also be rotated into the new direction of travel when turning. When doing so, the body experiences a resistance to rotation about the COM, called rotational inertia. To assess how well animals overcome this experienced inertia and rotate their body into a new heading, the heading angle and body axis angle can be compared (Fig. 1D). If the difference in angles is a positive value, the animal is under-rotating its body or lagging behind the turn, whereas a negative value indicates over-rotation (Jindrich et al., 2006; Jindrich and Full, 1999). Ostriches (Struthio camelus) turn with relatively small under- and over-rotations, closely aligning their body axis with deflection, whereas humans are found to over-rotate when turning at high speeds (Jindrich et al., 2006, 2007). Mice use the strategy of over-rotation between strides, possibly to enable production of greater fore–aft forces in the new direction (Walter, 2003).

Body mass and shape affect the rotational inertia of animals. From a point mass, rotational inertia is the product of mass (m) and radius squared (r²) (Shankar, 2014). As volume scales with m^3/3, and radius with m^1/3, rotational inertia should scale with body m^5/3 in geometrically similar animals.

Angular acceleration, which powers the turn, would be proportional to torque applied to the body divided by its rotational inertia. The torque equals muscle force multiplied by moment arm, which scale as m^2/3 and m^1/3 respectively, meaning torque scales with m^3/3. Thus, in geometrically similar animals, angular acceleration is expected to scale with m^3/3/m^5/3, which equals m^−2/3 (Carrier et al., 2001). Thus, distribution of mass and body shape may limit turning performance as increased inertia may result in decreased turning speeds (Carrier et al., 2001). Animals that differ in morphology are likely to use different strategies to reduce their inertia when turning (Eilam, 1994; Walter, 2003), yet there is limited knowledge of the relationship between mass, morphology and rotation in high-speed turning.

The present study seeks to quantify the four key challenges to turning performance in domestic dogs (C. lupus familiaris). Domestic dogs are well suited for the study as they allow us to investigate how large ranges in body mass and morphometrics affect turning performance trade-offs within a single species. A decrease in horizontal speed was expected with sharper turns for all dog sizes, but we wanted to identify any potential optimum body mass range or shape characteristics associated with high turning performance. We hypothesise that increases in mass should result in reduced turning performance as measured by lateral accelerations, friction limits and rotational inertia; however, the slope of this relationship may be modulated by variation in body proportions.

We filmed 20 domestic dogs sourced from the Agility Dog Club of Queensland (ADCQ). Both pure breed and crossbreed dogs participated, but breeds known to suffer from respiratory distress and diseases such as pugs, French bulldogs and English bulldogs were excluded from the study. The dogs were analysed with body mass as a continuous variable, but for general descriptions we refer to three broad body size groups: small (<10 kg, N=4), medium (10–25 kg, N=13) and large (>25 kg, N=3). A complete list of the dogs with details can be found in Table S1. All experimental protocols and methods were approved and carried out in compliance with the ARRIVE guidelines under the approval of the University of the Sunshine Coast Animal Ethics permit (ANA/19/156).

We challenged the dogs to run around an oval shaped course (20×10 m in size) defined by turning markers at each end. The racetrack was on a flat grass field and all trials were completed in dry weather conditions. Dogs were encouraged to run at high speed toward a turn by owners, then attempted to turn the corner at the highest speed possible, thus the radii of the turn were determined by the dogs turning ability. Each dog ran 5–10 turns per attempt, with multiple attempts over several days. The total number of attempts for a dog on a single day depended on running motivation and endurance. We positioned two to three GoPro cameras (GoPro HERO5 BLACK v2.06, frame width 1280 pixels, frame height 720 pixels, 240 frames s⁻¹, and GoPro HERO7 BLACK v1.80, frame width 1280 pixels, frame height 960 pixels, 240 frames s⁻¹) in a half circle at each turn. Recordings were started and stopped using a GoPro Smart Remote control.

To obtain a 2D birds-eye field of view (FOV), we used a drone (DJI Spark, SZ DJI Technology Co. Ltd, mass 300 g, 44 dB at 30 m) hovering between 20 and 35 m above the ground (Movie 1). The drone view covers a relatively large area, which allows the dogs to reach relatively high horizontal speeds and turning speeds at small and large turning radii. Drone videos were recorded at 30 frames s⁻¹, frame width 1920 and frame height 1080, using the onboard camera (f/2.6 wide-angle lens, 25 mm focal length, 1/2.3-inch CMOS sensor, two-axis mechanical gimbal). The drone can hover accurately up to 35 m height with a vision positioning system (VPS) owing to its flight autonomy system. We tested the accuracy of our drone by digitising a stationary point throughout trials. We found that the mean variation in position was 17.7±8.7 mm stride⁻¹. Mean stride length was 1239 mm (minimum=267 mm), meaning the error was ∼1.4% (maximum error ∼6.6%).

In addition to drone footage, we used one to two Basler acA1920-150um cameras (black/white, 200 frames s⁻¹, frame width 1500–1800, frame height 600–700) to film maximal horizontal speeds in 2D and 3D. Each dog had two attempts to reach their maximal speed and was encouraged to run by their trainer or owner, perpendicularly past the cameras.

For all camera views, we used the digitising tool DLTdv7 (Hedrick, 2008) in MATLAB R2019b (version 9.7.0, The MathWorks, Natick, MA, USA) to track the dogs in each frame. Videos in 3D were calibrated using the ‘wand method’ and the easyWand5 package in MATLAB (Theriault et al., 2014), and 2D videos had the FOV calibrated by measuring an object of known length. We also included the maximal horizontal speeds during turning attempts for each dog. The dog’s nose was tracked throughout the videos and the horizontal speed for each stride was calculated.

To calculate the maximum frictional coefficient possible for each individual dog, we created a sequence of coefficients from 0.01 to 1.99 in 0.02 increments. For each stride, we calculated the expected angular velocity via rearrangement of Eqns 4 and 5, including the measured horizontal speed during the stride. The expected angular velocity (ω_exp) was then compared with the calculated angular velocity (ω_actual) (Eqn 3). We calculated the frictional coefficient for each dog as the 95% upper limit of the turns where ω_exp<ω_actual.

We estimated the 3D toppling angle from the GoPro videos placed at each turn (Movie 2). Cameras were synchronised using a GoPro remote (Smart Remote, GoPro) and confirmed by the timing of footfall strikes occurring simultaneously in multiple videos. Cameras were calibrated using a wand of known length, waved throughout the FOV, and a static axis was input into the easyWand5 (Theriault et al., 2014).

Leaning angle will always be between 0 and 90 deg, with 90 deg meaning no inward leaning (legs are positioned straight down under the body) and decreasing angles indicating that leaning increases. We chose to use both the outer forelimbs and hindlimbs as leg vectors of interest as they are suggested to produce the largest lateral forces during turning (Söhnel et al., 2021).

Values close to 0 indicates that the body is closely aligned with heading direction, values <0 indicate over-rotation and values >0 indicate under-rotation.

Body mass for each dog was collected using a digital scale (Propert ‘Reflections’, no. 3177) and recent veterinary visit records. To obtain shape measurements, we photographed each dog standing naturally upright on all four legs, with a measurement tape in the FOV as calibration reference. We used ImageJ (Schneider et al., 2012) to calibrate each photo and measure the morphological characteristics. We digitised the hip, knee, ankle, metatarsophalangeal (MTP) joint and the toetip, along with the equivalent joints on the forelimb. Segment lengths were estimated from the 2D straight-line distance between these joints. Total limb length was the sum of these joints, and total body length was the distance between the shoulder and hip marker. We further estimated the effective limb length (ELL), following Day and Jayne (2007), expressed as a ratio of the shoulder or hip height to the total forelimb and hindlimb length, respectively, where shoulder and hip height were measured from each joint to each corresponding toe point.

All data analysis was completed using RStudio (ver. 4.0.1, 2020). To explore how change of heading, angular velocity and turn radii each were constrained by horizontal speed, the relationship between tangential and centripetal acceleration, and the effect of change in heading of relative body rotation, each pair of interest was plotted against each other using a linear mixed effects model (LME) with subject included as a random factor.

For each dependent variable, the three best-achieved scores (either maximum or minimum depending on the factor) were extracted for each dog; we then used the mean maximum (or minimum) for the top three results for each subject. The effect of body mass on each key limitation was explored by fitting generalized additive models (GAMs) or generalized additive mixed models (GAMMs) from the ‘mgcv’ package in R (Wood, 2011). The effect of body shape on turning performance was explored by fitting three different linear models (LMs). The first included absolute values of forelimb height, hindlimb height and body length, the second included the residual segments lengths from body mass, and the third included the ELL of the forelimb and hindlimb. In each case non-significant terms were removed using the step.R function from the ‘stats’ package in R, which uses stepwise model selection based on AIC scores. Only results from the resulting minimal model are reported, but full models are available in the Supplementary Materials and Methods. All data and R code used in statistical analysis, along with complete statistical models, can be found at https://doi.org/10.6084/m9.figshare.19612971.v5.

Before analysing the four limitations to turning, we determined how body mass and body shape change with each other. As might be expected, body mass had a positive linear relationship with all shape morphometrics (Table S2). Estimates for the effective limb length did not vary with body mass for either the hindlimbs (GAMs, F_1.39,1.63=0.92, P=0.523) or the forelimbs (GAMs, F_1,1=0.90, P=0.354).

We were able to estimate forces for 1011 strides from 19 dogs, with a median turn radius of 3.63 m. We first determined the extent to which limb forces limit turning ability among all dogs. Change of heading was negatively correlated with horizontal speed (LME, F_1,989=934, P<0.001; Fig. 2A). This result is reflected when comparing acceleration of the body. Centripetal acceleration is negatively correlated with tangential acceleration (LME, F_1,989=146, P<0.001; Fig. 2B), showing that dogs which push sideways to initiate a turn are less able to accelerate forward, suggesting a strong force limitation to turning ability.

This limitation to turning ability was not consistent with body size among dogs. GAMs showed that there was a complex pattern of maximum horizontal speed with body mass, with the fastest dogs being medium to large sizes, peaking just above 30 kg (F_1.88,1.98=4.84, P=0.026; Fig. 2C). Similarly, maximum centripetal acceleration was highest for dogs within a body mass range of 15–20 kg (F_1.88,1.98=3.94, P=0.040; Fig. 2D). Maximal tangential deceleration was similarly lowest within the 15–20 kg body mass range (F_1.90,1.99=4.84, P=0.023; Fig. 2E) and although maximal tangential acceleration was similarly high for medium sized dogs in the 15–20 kg range, this pattern was not statistically supported (F_1.80,1.96=1.96, P=0.172).

We also explored the effect of body shape on maximal speed, tangential and centripetal force generation. Maximum speed was significantly related to the absolute length of the forelimbs and the body (F_2,17=4.81, P=0.022), with longer forelimbs and a shorter body length associated with higher speeds (Fig. 3A,B). The residual segment lengths of the hindlimb metatarsals were negatively related to maximum speed (P=0.035), whereas the radial segment of the forelimbs was positively related to maximum speed (P=0.012), but the overall model was not significantly associated with speed (F_5,14=2.39, P=0.091; Fig. 3D,E). The estimates of effective limb length were not significantly related to maximum speed (F_1,18=4.08, P=0.058).

The pattern for acceleration was much more complex (Fig. 4). Maximum tangential acceleration and centripetal acceleration were both significantly related to the absolute length of the forelimbs and hindlimbs (F_2,16=5.31, P=0.017 and F_2,16=12.9, P<0.001, respectively), but the pattern differed among the limbs. In these cases, longer forelimb lengths and shorter hindlimb lengths were associated with higher accelerations (Fig. 4A,B). This relationship was also reflected in tangential deceleration, with longer forelimbs and shorter hindlimbs being associated with the ability to slow quickly (F_2,16=3.74, P=0.046; Fig. 4C). The pattern could at least partially be repeated when exploring residual segment lengths, with significant associations between centripetal acceleration and relatively longer humerus and radius segments, and shorter tibial and metatarsal segments (F_6,12=4.97, P=0.008; Fig. 4A). Similarly, longer radial segments and shorter toe and metatarsal segments were linked with tangential acceleration (F_7,11=3.16, P=0.043; Fig. 4B). There was no significant association with relative segment length with tangential deceleration (F_2,16=2.79, P=0.091). Finally, when we explored the effective limb length, there was a significant positive relationship with hindlimb ELL and centripetal acceleration, but not forelimb ELL (F_1,17=11.5, P=0.003), as well as a weaker positive relationship between tangential acceleration and hindlimb ELL, but similarly not forelimb ELL (F_1,17=4.57, P=0.047; Fig. 5). More upright limbs appeared associated with higher accelerations (Fig. 5). Tangential deceleration was not significantly related to either forelimb or hindlimb ELL (F_1,17=2.17, P=0.158).

We found that both turn radius and angular velocity were constrained by horizontal speeds (Fig. 6A,B). Among all dogs, friction coefficients appear to constrain turning ability at low radii (<5 m), with a maximum friction coefficient of 1.3 constraining most of the strides. At higher speeds and larger turning radii, strides move away from this limit, suggesting force limitation may be more important. However, there was considerable variation in the bounding frictional limit among body size ranges. Both large and small dogs appear to be bound by smaller frictional coefficients than medium sized dogs. To explore this further, we estimated the frictional coefficient for each dog individually. To do this, we calculated what frictional bound contained 95% of the strides for each subject (Fig. 6C). These frictional coefficients ranged from 0.57 to 1.53, suggesting considerable variation within species (Table S5). We compared this variation with body size using GAMs and, like the results for force limitations above, we show a significant non-linear pattern with size, where medium bodied dogs were able to produce higher frictional coefficients than both smaller and larger individuals (F_1.92,1.99=6.39, P=0.001; Fig. 6D).

The effect of body shape on friction limits was explored in an LM. As for the results for force generation above, there was a differential pattern between the forelimbs and hindlimbs, with absolute longer forelimbs associated with higher frictional coefficients, and shorter hindlimbs associated with higher coefficients (F_2,16=15.4, P<0.001; Fig. 6E,F). This result was also supported when residual limb segments were explored, with relatively longer humeral and radial segments associated with greater frictional coefficients in the forelimbs (F_6,12=5.59, P=0.005). There was further support for relatively shorter tibial, metatarsal and phalange segments with higher coefficients in the hindlimbs. Finally, higher frictional coefficients were associated with more upright hindlimbs (Fig. 6G), as measured via ELL, but no association was supported for the forelimbs (F_1,17=9.66, P=0.006).

We were able to measure the lean angle during midstance for 92 turns, from 12 dogs. The mean±s.d. lean angle for all hindlimbs and forelimbs analysed was 53.8±8.9 deg, with lower values indicating greater inward leaning. The mean angle between the limb and the ground was significantly higher for forelimbs (56.3±8.3 deg) than for hindlimbs (51.9±8.9 deg), suggesting that the forelimbs are held more upright during the turn (LME, F_1,79=6.49, P<0.013; Fig. 7A). There was no significant relationship with lean angle and body size, when all runs were included in a GAMM including foot with subject as a random factor (F_1.65,1.65=2.02, P=0.273), nor was there a significant effect when only the minimum lean angle for each dog was included in a GAM (F_1.87,2.27=0.94, P=0.533; Fig. 7B).

There was little effect of body shape on the minimum lateral leaning angle. There was no association with lean angle and the absolute lengths of the limb for either the forelimb (F_1,10=2.81, P=0.125) or the hindlimb (F_2,9=2.37, P=0.149). Nor was there an association with the effective limb length with lean angle in the forelimbs (F_1,10=1.23, P=0.293) or hindlimbs (F_1,10=0.79, P=0.393). Of the segment lengths, only the relative length of the radial segment was associated with forelimb leaning, with longer segments associated with greater leaning in the forelimbs (F_1,10=6.61, P=0.028). Hindlimb segments were not associated with leaning in the hindlimbs (F_3,8=1.30, P=0.339).

To explore body rotation, we compared the axis of the body with a vector describing the direction of travel. If the dogs reorientate their body into the new direction of travel faster than the heading vector, the result would be negative values from this comparison, which we call over-rotation. Alternatively, if the body axis is unable to keep up with the heading vector, we will obtain positive values, which we call under-rotation. Among all trials, dogs tended to over-rotate their bodies (mean=−15.9±24.9 deg). This over-rotation was linearly correlated with the change of heading, whereby larger changes in heading were associated with the body axis leading the turn more (LME, F_1,972=441, P<0.001; Fig. 8A).

Thus, to explore the effects of size, we included the change of heading in a GAMM including subject as a random factor. This showed a significant non-linear effect, and as for results above, it suggested that medium sized dogs aligned their body axis better with the heading vector at any change in heading (GAMM, F_1.88,1.88=5.62, P=0.017; Fig. 8B). Medium sized dogs aligned their body axis with the heading vector at a mean deviation of −14.6±24.5 deg. In comparison, both smaller and larger dogs showed significantly higher over-rotations with mean values of −18.2±24.3 and −20.7±29.2 deg, respectively.

To explore the effect of body shape on body rotation, we similarly included the change of heading in an LME model. There was a significant positive effect of absolute body length with body rotation (LME, F_1,17=5.07, P=0.038), but not forelimb or hindlimb length, suggesting that dogs with absolute longer bodies were better able to reorient their body into the new direction of movement (Fig. 7C). This finding was also supported when exploring relative body lengths, with dogs with relatively shorter body lengths showing greater over-rotation during turns (LME, F_1,17=5.21, P=0.035).

Among segment lengths, dogs with relatively longer tibial segments were associated with less over-rotation (LME, F_1,16=10.7, P=0.005) but no other association was significant (Fig. 7D). Finally, when the effective limb length was explored, no strong association was supported for hindlimb ELL (LME, F_1,16=1.29, P=0.272), but a significant negative relationship existed between forelimb ELL and reorientation (LME, F_1,16=7.03, P=0.017; Fig. 8E). Dogs with relatively crouched forelimbs were better able to maintain their body axis in the new direction of movement.

The ability to make quick and stable changes in heading may have significant ecological consequences, particularly during predator–prey interactions (Arnold, 1983; Garland and Losos, 1994; Moore and Biewener, 2015; Wilson et al., 2018). Yet, unlike speed, we have little idea of what underlying morphological limitations depict performance. Several studies have proposed biomechanical models that influence turning ability. Turning performance may be limited by the ability to generate centripetal forces in addition to tangential forces (Jindrich and Qiao, 2009; Wilson et al., 2013b; Biewener and Patek, 2018), maintain sufficient friction between the feet and substrate to avoid slipping and crashing (Alexander, 1982; Wheatley et al., 2018), lean inwards to counter medially directed ground reaction forces without toppling over sideways (Biewener and Patek, 2018), and re-align the body axis with the new heading (Walter, 2003; Jindrich et al., 2006, 2007), yet how these aspects are influenced by size and shape is not yet understood. To explore these limits, we measured the limitations to turning performance in domestic dogs (C. lupus familiaris) and explored the variation with body mass and shape.

Generally, predictions suggest that manoeuvrability should decrease with increased size, based on a series of morphological predictors such as flexed limb posture, COM height and foot area among geometrically similar animals (Biewener and Patek, 2018). Our results suggest a more complex pattern with size, with estimates of turning, including centripetal acceleration, the coefficient of friction and the rotational lag, all suggesting that the greatest turning ability was achieved by intermediately sized dogs (∼15–20 kg). This pattern of intermediate sized animals producing greater performance estimates is not uncommon and has been reported in a wide variety of taxa, including mammals (Garland, 1983), lizards (Clemente et al., 2009, 2012) and swimming frogs (Clemente and Richards, 2013), yet the underlying cause remains complex (for a review, see Dick and Clemente, 2017). Whether this body mass range represents the global maximum should also be considered with caution. Greyhounds may weigh ∼30 kg and can be considered turning specialists (Hudson et al., 2012), supporting the possibility of a wider peak for force generation relative to body mass. In any case, the ability to produce quick turns appears to increase with increasing body size, at least initially among dogs.

Like humans and mice, dogs appear to over-rotate their body during a turn (Jindrich et al., 2006, 2007; Walter, 2003). A large rotational inertia might be expected to result in increased under-rotation, as the body lags behind the new heading angle. Thus, given the expected scaling of rotational inertia of body mass^5/3, we might expect a reduction in over-­rotation as size increases, which we see from small to medium sized dogs, but it does not explain the larger over-rotation for larger dogs. This suggests that rotational inertia might not be the main limitation to turning in dogs, and that turning performance cannot be predicted exclusively on rotational inertia. This might suggest that body rotation does not greatly limit turning performance, and instead, these dogs may be acting near their frictional limits, and both the coefficient of friction and centripetal acceleration may be useful measures of turning ability.

Few estimates exist which allow us to draw a confident conclusion as to how the coefficient of friction might change over a larger body size range. The friction coefficient (µ) estimated for individual dogs ranged from 0.57 to 1.53, with a mean of 1.23 (mean mass=16 kg). Polo horses and racehorses (∼450 kg) had an estimated µ of 0.6 when turning on a horizontal grass surface (Tan and Wilson, 2011), with zebras (∼400 kg) falling within the 0.6 limit of horses (Wilson et al., 2018). Cheetahs (∼65 kg) have an estimated µ of 1.3, and northern quolls (∼0.6 kg) have a relatively high friction coefficient between 3 and 4 (Wynn et al., 2015; Wilson et al., 2018). The high friction limits for cheetahs and quolls might be explained by their footpad design and claws (Russell and Bryant, 2001; Wilson et al., 2013a; Wynn et al., 2015). Direct comparison of these friction limits is difficult as surface properties will also affect the estimated frictional coefficient, for example the yellow-footed antechinus (∼0.035 kg) showed friction coefficients of 0.8 and 1.3 for low- and high-friction surfaces, respectively (Wheatley et al., 2018). However, combined, these estimates do support a general decrease in turning ability over a much larger body size range, with substantial scatter around the line (Fig. 9). This variation is likely due to changes in shape among species and individuals, but also, as we have shown, the differential role in the forelimbs and hindlimbs during turns in quadrupedal species.

Our results supported a significant difference toward the contribution to turning ability in the forelimbs and hindlimbs. Centripetal acceleration and coefficient of friction estimates were more strongly related with the lateral leaning angle of the hindlimbs (R²=0.56 and 0.57, respectively), yet showed a weaker association with the lateral leaning angle for the forelimbs (R²=0.15 and 0.18, respectively; Fig. 10). Decreasing the lateral leaning angle would benefit turning because the limb is better able to produce horizontal forces, relative to vertical forces, which can contribute to establishing a new movement direction. The tight association for the hindlimbs might suggest a more important role in lateral force production for these limbs, when compared with the forelimbs.

The different role of the limbs in quadrupeds has been largely studied with regard to straight-line movement. In general, forelimbs produce net braking forces, whereas hindlimbs produce net propulsion. This has been shown not only in dogs (Lee et al., 1999, 2004; Bertram et al., 2000; Usherwood and Wilson, 2005; Walter and Carrier, 2007), but also in a variety of terrestrial taxa including cockroaches (Jindrich and Full, 1999; Full et al., 1991), chipmunks and ground squirrels (Biewener, 1983), horses (Merkens et al., 1993; Dutto et al., 2004) and geckos (Chen et al., 2006) during trotting and galloping gaits. Based on these results, and the greater fraction of body weight the forelimbs support, it was hypothesised that the forelimbs may be better suited to providing lateral impulses during a turn (Alexander, 2002; Moreno, 2010). This finding was supported during jump turns in dogs (Söhnel et al., 2021, 2020), circling horses (Clayton et al., 2014; Clayton and Hobbs, 2019) and turning goats (Moreno, 2010). Yet among both dogs and goats, it was noted that the hindlimbs were still capable of producing substantial net lateral linear impulse to redirect the COM.

The steeper lean angle in the hindlimbs, and the tighter relationship between lean angle with centripetal acceleration, might suggest an important role for the hindlimbs in producing lateral forces during the turns included in this study. Further, investigation on goats highlighted that the hindlimbs produce forces that create yaw moments away from the turn, which alternate and balance, the yaw moments generated by the forelimbs into the turn (Moreno, 2010). If the hindlimbs are producing much larger lateral forces than previously suggested, the presence of these forces might produce a yaw moment away from the turn, reducing the over-rotation, as observed for the medium sized dogs in the current study. Further investigation using force platforms would be required to confirm this.

Finally, we expected that more crouched postures would be associated with increased turning ability (Biewener and Patek, 2018). This was supported for estimates of body rotation, where more flexed forelimbs were associated with an increased ability to maintain the body axis in the direction of movement. However, the opposite trend was supported from the ELL of the hindlimb, where more upright limbs were linked with increased centripetal acceleration, tangential acceleration and coefficients of friction. This might be a consequence of the estimations for ELL, which were calculated from static neutral lateral views of the dogs, but our findings also appear to support previous estimations for dog postures. When jumping mechanics were compared between beginner and experienced dogs, experienced dogs tended to keep limbs stiffer leading into jumps (Söhnel et al., 2020). Similarly, when the gaits of labrador retrievers and greyhounds were compared, they showed that the greyhounds, which perform well during racing turns (Usherwood and Wilson, 2005), generally exhibited a more upright posture, particularly in the stifle and tarsal joints, during voluntary runway trotting (Colborne et al., 2005). Given the role the hindlimbs might play during acceleration, or even in lateral force production, a more upright hindlimb might reduce soft tissue stress, allowing it to support higher forces, and might benefit turning. As for the coefficient of friction above, a larger sample including more variation in limb posture might be required to completely understand the influence this trait might have on manoeuvrability over larger body size ranges.

This study has highlighted the differential morphological response to turning in the hind and forelimbs. Overall, it appears that turning and acceleration may benefit from longer more crouched forelimbs, but shorter stiffer hindlimbs. The extent to which this pattern extends into other quadrupeds remains to be resolved, but must be complicated by the body-size-related changes in posture and morphology already tightly linked with size (Biewener, 1989). Our results may also help to explain why turning specialists such as felids do not appear to follow the same posture trends seen in other quadrupedal mammal groups (Day and Jayne, 2007; Dick and Clemente, 2017). Further research differentiating these proximal causes of body shape with size might be a fruitful area for further research.

We thank the Agility Dog Club of Queensland (ADCQ) and the ADCQ president Ruth Raymond for accommodating us, for being nothing but friendly and welcoming, and for supporting this research by donating their time, resources and amazing dogs. We thank Jasmine Annett and Tasmin Proost for their time in helping to collect data, and Dominique Potvin for reviewing earlier versions of the manuscript.