Introduction

Kangaroos and other macropods are unique in both their morphology and their locomotor style. At slow speeds they use a pentapedal gait, where the forelimbs, the hindlimbs, and the tail all contact the ground, while at faster movement speeds they use their distinctive hopping gait (Dawson and Taylor 1973, O’Connor et al. 2014). Their uniqueness extends into their energetics of locomotion. As far back as the 19^th century, researchers noticed that the metabolic cost of running in quadrupeds and bipeds, like dogs, horses and humans, increased linearly with speed (Zuntz 1897, Taylor et al. 1970, Heglund et al. 1982, Taylor et al. 1982). To explain why metabolic rate increased at faster running speeds among quadrupeds and bipeds, Kram and Taylor (1990) refined the ‘cost of generating force’ hypothesis (Taylor et al. 1980). They reasoned that the decrease in contact time with increased speed, reflects an increase in the rate of generating muscle force, and the rate of cross-bridge cycling (Kram and Taylor 1990). This was supported for a diverse range of running animals, suggesting that metabolic rate was inversely proportional to contact time (Kram and Taylor 1990). Yet hopping macropods appear to defy this trend. On treadmills, both red kangaroos (∼20 kg) and tammar wallabies (∼5 kg) showed little to no increase in the rate of oxygen consumption with increased hopping speed (Dawson and Taylor 1973, Baudinette et al. 1992, Kram and Dawson 1998). The underlying mechanisms explaining how macropods are able to uncouple hopping speed and energy cost is not completely understood (Thornton et al. 2022).

The ability to uncouple speed and energy expenditure in macropods is related to the behaviour of their ankle extensor muscle-tendon units, which store and return elastic strain energy (Morgan et al. 1978, Biewener et al. 2004b, McGowan et al. 2005). In tammar wallabies, ankle tendon stress increased with hopping speed, leading to a greater rise in elastic strain energy return than muscle work, which increases the proportion of work done by tendon recoil while muscle work remains near constant (Baudinette and Biewener 1998, Biewener et al. 1998). Size-related differences in ankle extensor tendon morphology (Bennett and Taylor 1995, McGowan et al. 2008), and the resultant low strain energy return, may explain why small (<3 kg) hopping macropods and rodents appear not to be afforded the energetic benefits observed in larger macropods (Thompson et al. 1980, Biewener et al. 1981, Biewener et al. 1998) (but see Christensen et al. (2022)). However tendon morphology alone is insufficient to explain why large macropods can increase speed without cost, while large quadrupeds with similar tendon morphology cannot (Dawson and Webster 2010). The most obvious difference between macropods and other mammals is their hopping gait, but previously proposed mechanisms to explain how hopping could reduce the metabolic cost of generating muscle force, such as near-constant stride frequency (Heglund and Taylor 1988, Dawson and Webster 2010) or respiratory-stride coupling (Baudinette et al. 1987), do not distinguish between small and large macropods, nor galloping quadrupeds (McGowan and Collins 2018).

Postural changes are another mechanism which could contribute to reduced energetic costs by altering the leverage of limb muscles. The effective mechanical advantage (EMA) is the ratio of the internal muscle-tendon moment arm (the perpendicular distance between the muscle’s force-generating line-of-action and the joint centre) and the ground reaction force (GRF) moment arm (the perpendicular distance between the vector of the GRF to the joint centre). Smaller EMA requires greater muscle force to produce a given force on the ground, thereby demanding a greater volume of active muscle, and presumably higher metabolic rates. In humans, an increase in limb flexion and decrease in limb EMA recruits a greater volume of muscle and increases the metabolic cost of running compared to walking (Biewener et al. 2004a, Kipp et al. 2018, Allen et al. 2022). Elephants— one of the largest and noticeably most upright animals, do not have discrete gait transitions, but rather continuously and substantially decrease EMA in limb joints when moving faster, which has also been linked to the increase in metabolic cost of locomotion with speed (Ren et al. 2010, Langman et al. 2012). EMA has also been observed to change with size in terrestrial mammals. Smaller animals tend to move in crouched postures with limbs becoming progressively more extended as body mass increases (Biewener 1989). Yet, rather than metabolic cost, this postural transition appeared to be driven by the need to reduce the size-related increases in tissue stress (Dick and Clemente 2017, Clemente and Dick 2023). As a consequence of transitioning to more upright limb postures, musculoskeletal stresses in terrestrial mammals are independent of body mass (Biewener 1989, 2005).

Macropods, in contrast, maintain a crouched posture during hopping. Kram and Dawson (1998) explored whether kangaroos transitioned to a more extended limb posture with increasing speed as a potential mechanism for their constant metabolic rate, but did not detect a change in EMA at the ankle across speeds from 4.3 to 9.7 m s^-1. Further, ankle posture and EMA appears to vary weakly (McGowan et al. 2008) or not at all with body mass in macropods (Bennett and Taylor 1995, Snelling et al. 2017). As a consequence, stress would be expected to increase with both hopping speed and body mass. High tendon stresses are required for high strain energy return (strain energy ϑ stress²) (Biewener and Baudinette 1995). Given that tendon recoil plays a pivotal role in hopping gaits, it is perhaps unsurprising that tendon stress approaches the safe limit in larger kangaroos. The ankle extensor tendons in a moderately sized male red kangaroo (46.1 kg) operate with safety factors near two even at slow hopping speeds (3.9 m s^-1) (Kram and Dawson 1998). Tendon stresses are also unusually high in smaller kangaroos. Juvenile and adult western grey kangaroos, ranging in mass from 5.8 to 70.5 kg, all hop with gastrocnemius and plantaris tendon safety factors less than two (Snelling et al. 2017). High tendon stress may not only be a natural consequence of their crouched posture and tendon morphology, but also adaptively selected. Considering this, kangaroos may adjust their posture to increase tendon stress and its associated elastic energy return. If so, there would likely be systematic variation in kangaroo posture with speed and mass—which is yet to be fully explored.

In this study, we investigate the hindlimb kinematics and kinetics in kangaroos hopping at their preferred speed. Specifically, we explore the relationship between changes in posture, EMA, joint work, and tendon stress across a range of hopping speeds and body masses. To do this, we built a musculoskeletal model of a kangaroo based on empirical imaging and dissection data (Fig. 1a). We used the musculoskeletal model to calculate ankle EMA throughout the stride by capturing changes in both muscle moment arm and the GRF moment arm, allowing us to explore their individual contributions. We hypothesised that (i) the distal hindlimb, and the ankle joint in particular, would be more flexed when hopping at faster speeds and larger sizes, and (ii) the change in posture and moment arms would contribute to the increase in tendon stress with speed, and may thereby contribute to energetic savings by increasing the amount of positive and negative work done by the ankle.

Figure 1:

Results

Ground reaction forces

The stance begins with a breaking phase (negative horizontal component), followed by a propulsive phase (positive horizontal component). At the transition from negative to positive, the GRF was vertical, and this occurred at 42.9±26.9% of stance. The peak in vertical GRF, however, occurred at 46.4±5.3% of stance (Fig. 2a,b).

Figure 2:

Larger kangaroos and faster speeds were associated with greater magnitudes of peak GRFs (Fig. 2a,b; Suppl. Table 2). When peak vertical GRF was normalised to body weight, peak forces ranged from 1.4 multiples of body weight (BW) to 3.7 BW per leg (mean: 2.41±0.396 BW) and increased with speed (Suppl. Fig. 2a). Smaller kangaroos may experience disproportionately greater GRFs at faster speeds (Suppl. Fig. 2b, Suppl. Table 2).

Kinematics and posture

Kangaroo hindlimb kinematics varied with both body mass and speed. Higher masses and faster speeds were associated with more crouched hindlimb postures (Fig. 3a,c). Size-related changes in posture occurred throughout stance and were distributed as small changes among all hindlimb joints rather than a large shift in any one joint (Fig. 3d,f; Suppl. Table 3). The hip, knee and ankle tended toward more flexion, and the metatarsophalangeal (MTP) toward greater plantarflexion in larger kangaroos.

Figure 3:

In addition to, and independent of, the change in posture with mass, there were substantial postural changes due to speed (Suppl. Table 3). Unlike with mass, speed-related changes occurred predominantly during the braking phase, and the change was concentrated in the ankle and MTP joints, with little to no change in the proximal joints (Fig. 3c,e,g; Suppl. Table 3). There was a significant decrease in ankle plantarflexion at initial ground contact and increase in dorsiflexion at midstance at faster speeds (Fig. 3g; Suppl. Table 3). Maximum ankle dorsiflexion occurred at 44.8±4.5% of stance, and tended to occur 3.9±0.7% earlier in stance with each 1 m s^-1 increase in speed (P<0.001). MTP range of motion increased with speed due to an increase in MTP plantarflexion prior to midstance rather than a change in dorsiflexion (Fig. 3g).

Effective Mechanical Advantage

Effective mechanical advantage (EMA) decreased as the limb became more crouched. The change in EMA with mass and speed was substantial, particularly in the braking period (Fig. 2c,d). We evaluated the change in EMA at midstance, as it was approximately the point in the stride where GRF (and therefore tendon stress) was highest. EMA at midstance decreased with body mass, although we did not detect an effect of speed (Fig. 2c,d; Suppl. Table 5). However, examining whether the decrease in EMA was caused by a decrease in the muscle moment arm, r, or an increase in the external moment arm, R, revealed a more nuanced picture.

Speed, rather than body mass, was associated with a decrease in r at midstance (Suppl. Fig. 3a,b; Suppl. Table 5). A kangaroo travelling 1 m s^-1 faster would decrease r by approximately 4.2%. The muscle moment arm reduced with speed for the full range of speeds we measured (1.99 – 4.48 m s^{- 1}), with the maximum r occurring when the ankle was at 114.4±0.8°. Maximum ankle dorsiflexion ranged from 114.5° in the slowest trials, to 75.8° in faster trials (Suppl. Fig. 3g). The timing of the local minimum r at midstance coincided with the timing of peak ankle dorsiflexion.

We observed an increase in R with body mass, but not speed, at midstance (Suppl. Fig. 3c,d; Suppl. Table 5). We expected R to vary with both mass and speed because the increase in MTP plantarflexion prior to midstance reduced the distance between the ankle joint and the ground, which would increase R since the centre of pressure (CoP) at midstance remained in the same position (Fig. 6, Suppl. Table 2). Increasing body mass by 4.6 kg or speed by 1 m s^-1 resulted in a ∼20% reduction in ankle height (Suppl. Fig. 3e,f). Although, posture alone may not account for the change in R, as the metatarsal segment also increased in length with body mass (1 kg BM corresponded to ∼1% longer segment, P<0.001, R²=0.449).

Joint moments

The change in the muscle moment arm, r, with speed has further implications for Achilles tendon stress, because force in the tendon is determined by the ratio of the ankle moment to r (Fig. 6). As such, the decrease in r with speed would tend to increase tendon force (and thereby tendon stress), as would an increase in ankle moment. We found that speed, rather than mass, was associated with an increase in the maximum ankle moment (Suppl. Fig. 4; Suppl. Table 4, see supplementary material for details on other hindlimb joint moments).

Tendon stress

Peak tendon stress increased as minimum EMA decreased (Fig. 7a). Small decreases in EMA correspond to a nontrivial increase in tendon stress, for instance, reducing EMA from 0.242 (mean minimum EMA of the slow group) to 0.206 (mean minimum EMA of the fast group) was associated with an ∼18% increase in tendon stress. Tendon stress increased with body mass and speed (Suppl. Table 5). Maximum stress in the ankle extensor tendons occurred at 46.8±4.9% of stance, aligning with the timing of peak vertical GRF and maximum ankle moment. Body mass did not have a significant effect on the timing of peak stress, but peak stress occurred 3.2±0.8% earlier in stance with each 1 m s^-1 increase in speed (P<0.001), reflecting the timing of the peak in ankle dorsiflexion.

Joint work and power

An analysis of joint-level energetics showed the majority of work and power in the hindlimb was performed by the ankle joint (Fig. 4; Suppl. Table 8). The ankle performed negative work in the braking period, as kinetic and gravitational potential energy was transferred to elastic potential energy by loading the ankle extensor tendons. An increase in negative ankle work was associated with the increase in tendon stress in the ankle extensors (Β=0.750, SE=0.074, P<0.001, R²=0.511). Simple linear regression indicated that the increase in negative work was associated with speed but not body mass (Fig. 5a; Suppl. Fig. 5e,f; Suppl. Table 6).

Figure 4:

Figure 5:

Elastic potential energy was returned to do positive work in the propulsive period as the ankle started extending prior to midstance. Positive ankle work increased with both mass and speed (Fig. 5a; Suppl. Fig. 5e,f; Suppl. Table 6), but since it increased at the same rate as negative work, critically, there was no change in net ankle work with speed (Fig. 5b; Suppl. Fig. 6e,f; Suppl. Table 6).

Discussion

How does posture contribute to kangaroo energetics?

The cost of generating force hypothesis (Taylor et al. 1980) implies that as animals increase locomotor speed and decrease ground contact time, metabolic rate should increase (Kram and Taylor 1990). Macropods defy this trend (Dawson and Taylor 1973, Baudinette et al. 1992, Kram and Dawson 1998). It is likely that the use of their ankle extensor tendons set them apart from other mammals, yet the underlying mechanisms for this were unclear (Bennett and Taylor 1995, Bennett 2000, Thornton et al. 2022). We hypothesised (i) the distal hindlimb would be more flexed when hopping at faster speeds and larger masses, mainly due to the ankle, and (ii) the change in posture and EMA would contribute to the increase in tendon stress.

Tendon stress depends on the muscle moment arm, ankle moment, external moment arm, and ground reaction force (GRF), assuming tendon properties remain unchanged (Fig. 6). We detected an increase in peak ankle moment with speed independent of increases in body mass, despite the change in the external moment arm, R, with mass, likely due to a dominant effect of the GRF (ankle moment = GRF · R). Peak GRFs varied with speed as well as mass. Both the increase in ankle moment and a decrease in r with speed tend to increase tendon stress (stress = tendon force / cross-sectional area; tendon force = ankle moment / r). Although the increase in peak GRF with speed explains much of the increase in tendon stress, the significant relationship between tendon stress and ankle EMA suggests that the increase in stress is not solely due to the greater GRFs, but that stress may also be modulated by changes in posture throughout the stride (Fig. 7a).

Figure 6:

Figure 7:

Previously, macropods were reported to have consistent ankle EMA due to the muscle moment arm, r, and the external moment arm, R, scaling similarly with body mass (Fig. 7b) (Bennett and Taylor 1995). Our results, however, suggest posture is not as consistent or static as previously reported if speed is taken into account; we found larger and faster kangaroos were more crouched, leading to lower ankle EMA (Fig. 2c,d). We were able to explore the individual contributions to ankle EMA using our musculoskeletal model, to demonstrate that changes in the ankle and metatarsophalangeal (MTP) range of motion resulted in changes in r with speed, and R with mass, respectively. These effects were additive; combined, there was a marked decrease in ankle EMA, particularly during the early stance when the ankle extensor tendons are loaded (Fig. 2c,d). As such, the change in ankle EMA appears to be a mechanism that contributes to the increase in tendon stress (Fig. 7a) and ankle work (Fig. 5a).

Kangaroos appear to use changes in posture and EMA to absorb and return energy during stance. A comparison among all the hindlimb joints suggests the ankle was primarily responsible for both the storage and release of work during the stride (Fig. 4). Positive work increased with mass and speed, consistent with other studies in other species (Cavagna and Kaneko 1977). However, critically, the amount of negative work absorbed during the stride also increased with speed, and net ankle work did not change with speed, indicating the increase in negative work absorbed matched the increase in positive work that is required to move forward (Fig. 5b). Indeed, we observed all trials had a similar positive net ankle work (mean: 0.67 ± 0.54 J kg^-1). The consistent net work observed among all speeds suggests the ankle extensors are performing similar amounts of ankle work independent of speed. Previous studies using sonomicrometry have shown that the muscles of tammar wallabies do not shorten considerably during hops, but rather act near-isometrically as a strut (Biewener et al. 1998). Further, these small changes in muscle fibre length did not vary with hopping speed, despite large increases in muscle tendon forces (Biewener et al. 1998). Isometric muscle contractions are efficient (Smith et al. 2005). The cost of generating force hypothesis suggests that faster movement speeds require greater rates of muscle force development, and in turn greater cross bridge cycling rates, driving up metabolic costs (Taylor et al. 1980, Kram and Taylor 1990). The ability for the ankle extensor muscle fibres to remain isometric and produce similar amounts of work at all speeds may help explain why hopping macropods do not follow the energetic trends observed in quadrupedal species. Their ability to do this requires the amount of negative work to increase with speed. Modifying the ankle EMA as speed increases allows an increase in tendon stress (Fig. 7a) enabling the storage of more energy during the first half of stance. Since the muscle remains near isometric, no additional work is performed by the muscle, but the overall positive work increases, likely owing to the greater return of elastic strain energy, and enabling faster hopping speeds (Fig. 6). Thus, changes in EMA provide a likely mechanism to explain the uncoupling of metabolic energy expenditure with locomotor speed observed in large hopping macropods (Dawson and Taylor 1973, Baudinette et al. 1992, Kram and Dawson 1998).

Why are macropods unique?

No other mammals are known to achieve the same energetic feat as macropods, despite similar tendons or stride parameters (Thornton et al. 2022), but macropods are unique in other ways. Kangaroos operate at much lower EMA values than other mammalian species (Fig. 7b). Mammals >18 kg tend to operate with EMA values 0.5 to 1.2. The only mammals with comparable EMA values to kangaroos are rodents <1 kg (Fig. 7b) (Biewener 1990), such as kangaroo rats which have relatively thicker tendons that may be less suited for recovering elastic strain energy (Biewener and Blickhan 1988) (but see: Christensen et al. (2022)). Fig. 7a shows the non-linear relationship between tendon stress and EMA in kangaroos, quadrupeds and humans. The range of EMA estimates for other mammals suggest they operate in the region of this curve where large changes in EMA would only produce small changes in stress. This implies EMA modulation in other species is not as effective a mechanism to increase tendon stress with increased running speed as that observed in kangaroos.

Were macropods performance or size limited?

The morphology and hopping gait that make kangaroos supremely adapted for efficient locomotion likely also have several performance limitations. One possible consequence is a predicted reduction in manoeuvrability ((Biewener 2005). The high compliance of the tendon would limit the ability to rapidly accelerate, owing to the lag between muscle force production and the transmission of this force to the environment. A second important limitation is the requirement for kangaroos to operate at high tendon stresses. Previous research has suggested that kangaroos locomote at dangerously low safety factors for tendon stress, predicted to be between 1-2 for large kangaroos (Kram and Dawson 1998, McGowan et al. 2008, Snelling et al. 2017, Thornton et al. 2022). Our new insights into the mass and speed modulated changes in EMA suggest that these safety factors may be even lower, likely limiting the maximum body mass that hopping kangaroos can achieve. This suggests that previous projections of tendon stress may have overestimated the body mass at which tendons reach their safety limit (‘safety factor’ of 1). Snelling et al. (2017) estimated the maximum body mass to remain above this limit was approximately 160 kg, but even if we consider this is a conservative prediction, it is far lower than the projected mass of extinct macropodids (up to 240 kg (Helgen et al. 2006, Janis et al. 2023)). Thus we expect there must be a body mass where postural changes shift from contributing to stress to mitigating it (Dick and Clemente 2017).

This study highlights how EMA may be more dynamic than previously assumed, and how musculoskeletal modelling and simulation approaches can provide insights into direct links between form and function which are often challenging to determine from experiments alone.

Methods

Animals and data collection

Hopping data for red and eastern grey kangaroos was collected at Brisbane’s Alma Park Zoo in Queensland, Australia, in accordance with approval by the Ethics and Welfare Committee of the Royal Veterinary College; approval number URN 2010 1051. The dataset includes 16 male and female kangaroos ranging in body mass from 13.7 to 26.6 kg (20.9 ± 3.4 kg). Two juvenile red (Macropus rufus) and 11 grey kangaroos (Macropus giganteus) were identified, while the remaining three could not be differentiated as either species. Body mass was determined in several ways, primarily by measurements of the kangaroos standing stationary on the force plate. If the individual did not stop on the force plate, and forward velocity was constant, then body mass was determined by dividing the total impulse across a constant velocity hop cycle (foot strike to foot strike) by the total hop cycle time and further dividing by gravitational acceleration. Finally, if neither of these approaches were sufficient, we interpolated from the relationship between leg marker distances (as proxy for segment lengths) and body mass. These methods produce estimates of body masses close to stationary measurements, where both were available.

Experimental protocol

Kangaroos hopped at their preferred speed down a runway (∼10x1.5 m) that was constructed in their enclosure using hessian cloths and stakes (Suppl. Video 1). The runway was open at both ends with two force plates (Kistler custom plate (60x60 cm) and AMTI Accugait plate (50x50 cm)) set sequentially in the centre and buried flush with the surface. The force plates recorded ground reaction forces (GRF) in the vertical, horizontal and lateral directions.

A 6-camera 3D motion capture system (Vicon T160 cameras), recorded by Nexus software (Vicon, Oxford, UK) at 200 Hz, was used to record kinematic data. Reflective markers were placed on the animals over the estimated hip, knee, ankle, metatarsophalangeal (MTP) joints; the distal end of phalanx IV; the anterior tip of the ilium; and the base of the tail (Fig. 1a). Force plate data was synchronously recorded at 1000 Hz via an analogue to digital board integrated with the Vicon system.

Data analysis

We calculated the stride length from the distance between the ankle or MTP marker coordinates at equivalent time points in the stride. The stance phase was defined as the period when the vertical GRF was greater than 2% of the peak GRF. We determined ground contact duration and total stride duration from the frame rate (200 Hz) and the number of frames from contact to take-off (contact duration) and contact to contact (stride duration), respectively, and calculated the stride frequency. Stride parameter results are detailed in the supplementary material.

In most trials, the stride before and after striking the force plate was visible, providing a total of 173 strides. If two strides were present in a trial, we took the average of the two strides for that trial. Trials were excluded if only one foot landed on the force plate or if the feet did not land near-simultaneously, to give a total of 100 trials. We did not include trials where the kangaroo started from or stopped on the force plate. We assumed GRF was equally shared between each leg and divided the vertical, horizontal and lateral forces in half, and calculated all results for one leg. We normalised GRF by body weight. We assumed the force was applied along phalanx IV and that there was no medial or lateral movement of the centre of pressure (CoP) (Fig. 1b). The anterior or posterior movement of the CoP was recorded by the force plate within the motion capture coordinate system.

We calculated the hopping velocity and acceleration of the kangaroo in each trial from the position of the pelvis marker, as this marker was close to the centre of mass and there should have been minimal skin movement artefact. Position data were smoothed using the ‘smooth.spline’ function in R (version 3.6.3). The average locomotor speed of the trial was taken as the mean horizontal component of the velocity during the aerial phase before and after the stance phase.

Building a kangaroo musculoskeletal model

We created a musculoskeletal model based on the morphology of a kangaroo for use in OpenSim (v3.3; Seth et al. (2018)) (Fig. 1a). The skeletal geometry was determined from a computed tomography (CT) scan of a mature western grey kangaroo (MorphoSource.org, Duke University, NC, USA). Western grey kangaroos are morphologically similar to eastern grey and red kangaroos (Thornton et al. 2022).

We extracted the skeletal components from the CT scan using Dragonfly (Version 2020.2, Object Research Systems (ORS) Inc., Montreal, Canada) and partitioned the hindlimb into five segments (pelvis, femur, tibia, metatarsals and calcaneus, and phalanges). The segments were imported into Blender (version 3.0.0, https://www.blender.org/; Amsterdam, Netherlands) to clean and smooth the bones, align the vertebrae, and export the segments as meshes. We imported the meshes into Rhinoceros (version 6.0, Robert McNeel & Associates, Seattle, WA, USA) to construct the framework of the movement system.

All joints between segments were modelled as hinge joints which constrain motion to a single plane and one degree of freedom (DOF), except the hips which were modelled as ball-and-socket joints with three DOF. The joints were restricted to rotational (no translational) movement. The joints were marked with an origin (joint centre) and coordinate system determined by the movement of the segment (x is abduction/adduction, y is pronation/supination, z is extension/flexion). The limb bones and joints from one leg were mirrored about the sagittal plane to ensure bilateral symmetry.

The segment masses for a base model were determined from measurements of eastern grey and red kangaroos provided in Hopwood (1976). Cylinders set to the length and mass of each segment were used to approximate the segment centre of mass and moments of inertia.

We scaled the model to the size and mass of each kangaroo using the OpenSim scale tool. A static posture was defined based on the 3D positions of the markers at midstance. Each segment was scaled separately, allowing for different scaling factors across segments. The markers with less movement of the skin over the joint (e.g. the MTP marker) were more highly weighted than the markers with substantial skin movement (e.g. the hip and knee markers) in the scaling tool.

Joint kinematics and mechanics

We used inverse kinematics to determine time-varying joint angles during hopping. Inverse kinematics is an optimisation routine which simulates movement by aligning the model markers with the markers in the kinematic data for each time step (Suppl. Video 2). We adjusted the weighting on the markers based on the confidence and consistency in the marker position relative to the skeleton. The model movement from inverse kinematics was combined with GRFs in an inverse dynamics analysis to calculate net joint moments for the hip, knee, ankle and MTP joints throughout stance phase. Joint moments were normalised to body weight and leg length.

We calculated instantaneous joint powers for each of the hindlimb joints over the stance phase of hopping as the product of joint moment (not normalised) and joint angular velocity. To determine work, we integrated joint powers with respect to time over discrete periods of positive and negative work, consistent with Dick et al. (2019). For the stance duration of each hop, all periods of positive work were summed and all periods of negative work were summed to determine the positive work, negative work, and net work done at each of the hindlimb joints for a hop cycle. Joint work was normalised to body mass.

Posture and EMA

We evaluated overall hindlimb posture to determine how crouched or upright the hindlimbs were during the stance phase of hopping. The total hindlimb length was determined as the sum of all segment lengths between the joint centres, from the toe to ilium (Fig. 1a). The crouch factor (CF) was calculated as the distance between the toe and the ilium marker divided by the total hindlimb length. Larger CF values indicated extended limbs whereas smaller CF values indicated more crouched postures.

We calculated the effective mechanical advantage (EMA) at the ankle as the muscle moment arm of the combined gastrocnemius and plantaris tendon, r, divided by the external moment arm, R (perpendicular distance between the GRF vector and the ankle joint) (Fig. 1b). We dissected (with approval from University of the Sunshine Coast Ethics Committee, ANE2284) a road-killed 27.6 kg male eastern grey kangaroo and used this, combined with published anatomy on the origin and insertion sites, to determine the ankle extensor muscle-tendon unit paths on the skeleton in

OpenSim (Bauschulte 1972, Hopwood and Butterfield 1976, Hopwood and Butterfield 1990). The value r to the gastrocnemius and plantaris tendons for all possible ankle angles were determined from OpenSim and scaled to the size of each kangaroo, while R was calculated as the perpendicular distance between the ankle marker and the GRF vector at the CoP.

Tendon stress

Ankle extensor tendon forces were estimated as the time-varying ankle moment divided by the time varying Achilles tendon moment arm. The sum of the gastrocnemius and plantaris tendon cross-sectional areas were scaled to kangaroo body mass for each kangaroo by interpolating literature values (Snelling et al. 2017). Forces were divided by tendon cross-sectional area to calculate tendon stress. We excluded the third ankle extensor tendon (flexor digitorum longus), which stores ∼10% of strain energy of the ankle extensors in tammar wallabies (Biewener and Baudinette 1995).

Statistics

We used multiple linear regression (lm function in R, v. 3.6.3, Vienna, Austria) to determine the effects of body mass and speed on the stride parameters, ground reaction forces, joint angles and CF, joint moments, joint work and power, and tendon stress. We considered the interaction of body mass and speed first, and removed the interaction term from the linear model if the interaction was not significant. Effects were considered significant at the p<0.05 level. Species was not used as a factor in the analysis as there was no systematic difference in outcome measures between kangaroo species.

Acknowledgements

We thank the Brisbane Alma Park Zoo for hosting and facilitating this work; particularly Dena Loveday and Heather Hesterman. Matthew Brown provided access to the CT scan, the collection of which was funded by the Texas Vertebrate Paleontology Collections. Megan Johnston and Rachel Lyons from Wildcare Australia provided road-killed kangaroos for muscle dissection. We also thank Alex Muir from Logemas for loan of the 3D motion capture system.

Funding

This work was supported by: Australian Government Research Training Program Scholarship and Comparative; ISB Neuromuscular Biomechanics Technical Group Student Grant-in-Aid of Research to LHT; Biotechnology and Biological Sciences Research Council Grant (BB/F000863) to JRH; Journal of Experimental Biology Travelling Fellowship to CPM; Australian Research Council Discovery Project Grant to CJC and TD (DP230101886).

Supplementary Results

Stride parameters

Preferred hopping speed ranged from 1.99 to 4.48 m s^-1. Larger kangaroos tended to hop at slightly faster speeds (Β=0.048, SE=0.018, P=0.009, R²=0.057), and due to this weak relationship between body mass and speed, both variables were considered in multiple linear regression models to determine their relative effects on the outcome measures (see Table 1).

Faster speeds were associated with a greater magnitude of acceleration in the braking period of the stance phase, i.e. minimum horizontal acceleration and maximum vertical acceleration (Suppl. Fig. 1a). There was no significant relationship between body mass and acceleration; however, there was a significant interaction between body mass and speed on maximum vertical acceleration, whereby smaller kangaroos had a greater change in vertical acceleration between slower and faster hopping speeds than larger kangaroos (Table 1).

Body mass and speed had different effects on ground contact duration (Table 1). There was a slight increase in contact duration in larger kangaroos. A stronger, opposing relationship was found with speed, and as speed increased, contact duration decreased. The relatively tight correlation between contact duration and hopping speed (R²=0.73) could prove useful for predicting speed when contact duration can be accurately measured. In Suppl. Fig. 1b we combined our data with red kangaroo data from Kram and Dawson (1998) to extend the predictive range of both studies.

Larger kangaroos hopped with longer strides and lower frequencies than smaller kangaroos (Table 1), and stride length also increased with speed (Suppl. Fig. 1c). We found a significant decrease in stride frequency with mass and an increase with speed, if we did not consider the interaction term (Suppl. Fig. 1d). However, a significant interaction between body mass and speed suggests that larger kangaroos relied more on increases in stride frequency to increase hopping speed compared to smaller kangaroos (Table 1).

Ground reaction forces

It is commonly assumed that the GRF is vertical and at a maximum at midstance or 50% of stance (Bennett and Taylor 1995, Kram and Dawson 1998, McGowan et al. 2008, Snelling et al. 2017) but our results suggest it occurred earlier, at 42.9±26.9% of stance, and the wide range suggests such assumptions should be used with caution.

Kinematics and posture

Kangaroos were maximally crouched at midstance, with crouch factor (CF) reaching a minimum at 50.1±4.2% of stance. Crouch factor (CF) at initial ground contact decreased at faster speeds, although the limb was similarly flexed during midstance (P=0.295). Consequently, CF changed less at faster speeds than slower speeds.

Larger kangaroos had lower hip and knee ranges of motion (ROM) compared to smaller kangaroos (Fig. 3d; Suppl. Table 3). In the distal hindlimb, ankle ROM increased with body mass, largely owing to an increase in dorsiflexion at midstance (Fig. 3f; Suppl. Table 3). The ROM of the MTP joint did not change with body mass; however, there was both an increase in plantarflexion and a decrease in dorsiflexion, resulting in a shift to larger MTP angles with mass (Fig. 3f; Suppl. Table 3).

Joint moments

Despite the small changes in joint rotation in the proximal hindlimb, we detected a decrease in magnitude of the dimensionless hip extensor and knee flexor moments with mass (Suppl. Fig. 4a, Suppl. Table 4), and an increase in magnitude of both joint moments with speed during the braking period of stance (Suppl. Fig. 4b). Maximum hip extensor moment and knee flexor moment were significantly influenced by the interaction between body mass and speed, suggesting that larger kangaroos increased the magnitude of the moments at a faster rate with speed compared to smaller kangaroos.

Joint work and power

The MTP was the only joint which did predominantly negative work. The MTP did more negative work with speed, while net MTP work decreased with both mass and speed (Suppl. Fig. 5g,h; Suppl. Fig. 6g,h; Suppl. Table 6). The MTP transition from negative work to positive work at ∼80% stance, as the back of the foot started to leave the ground. Conversely, the knee did almost no negative work, and the hip did very little (Fig. 4; Suppl. Fig. 5; Suppl. Fig. 6). Net work slightly increased with mass and speed in the hip, while at the knee, net work increased only with speed (Suppl. Table 6).

Supplementary Figures

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Tables

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