Kinematic control of extreme jump angles in the red-legged running frog, Kassina maculata

The physics of jumping seem simple. Small vertebrates such as frogs and lizards tune their performance by shifting their take-off velocity and angle (Marsh, 1994; Toro et al., 2004). While this 2D view has expanded our understanding of the musculoskeletal system (Galantis and Woledge, 2003; Roberts and Marsh, 2003; Aerts, 1998; Aerts and Nauwelaerts, 2009), it does not fully explain the control of performance. As a jumping frog traces a simple path along the vertical plane, how do the elaborate 3D leg rotations (Nauwelaerts and Aerts, 2003; Astley and Roberts, 2014) act to carry the body upward and forward? Despite recent advances in frog biomechanics (Azizi and Roberts, 2010; Astley and Roberts, 2011, 2014), no study has yet explained how 3D multi-joint kinematics resolve into a smooth planar body motion. Furthermore, how frogs modulate joint rotations to vary take-off angle remains unclear. The present study addresses how limb joint rotations modulate trajectory in jumping frogs.

Determining how jumpers modulate limb kinematics is a step towards a broader understanding of vertebrate limb mutli-functionality. Using frog pelvic anatomy as an example, a single frog's ability to perform multiple behaviours (e.g. jumping versus walking) is a classic example linking skeletal morphology (Emerson, 1979; Reilly and Jorgensen, 2011) and musculature (Emerson and De Jongh, 1980) to locomotor variability; less understood is how muscles shift their control of limb motions across different modes of locomotion. Recent evidence suggests that variations between swimming versus jumping kinematics are not strongly dependent on muscle activation patterns. Rather, kinematic differences largely emerge from differences in the fluid versus solid substrates (Nauwelaerts and Aerts, 2003; Aerts and Nauwelaerts, 2009), despite similar muscle firing patterns (Emerson and De Jongh, 1980; Kamel et al., 1996; Gillis and Biewener, 2000; d'Avella and Bizzi, 2005). In contrast, limb motions within terrestrial modes of locomotion (walking, climbing, jumping, etc.) may be dictated by varied coordination among multi-articular leg muscles to modulate the force direction of the foot (Lombard and Abbott, 1907; Kargo and Rome, 2002). Yet, even with knowledge of how limb muscles are recruited during jumping (Emerson and De Jongh, 1980; Kamel et al., 1996; Gillis and Biewener, 2000; Gillis, 2007), we currently cannot resolve which muscles are the drivers for behavioural variation. For example, it is difficult to speculate which muscles are differentially recruited when pushing the leg rearward during walking (Barclay, 1946; Ahn et al., 2004) versus laterally for burrowing (Emerson, 1976) without knowing how individual leg joints contribute to the 3D components of overall limb motion.

In light of the complexity of 3D kinematics, how can we resolve individual joint contributions to limb motion? As an alternative to standard 3D kinematics (e.g. Euler angles), a simplified axis-angle approach may provide clarity while facilitating calculations. Briefly, a scalar ‘3D extension’ angle (θ) is defined for each joint as the shortest angle between adjacent segments. More precisely, θ is simply the 2D angle within the plane defined by two adjacent segments at a moment in time. To address the influence of 3D extension on body movement, additional information is required: the orientation of the plane itself must be calculated via its normal, n, i.e. the instantaneous axis of joint rotation (Fig. 1). For simplicity, we describe the instantaneous orientation of each joint by the angle (φ) of n with respect to the vertical (see Materials and methods). Thus, the putative role of any given joint can be assessed by two angles: θ is the magnitude of motion and φ dictates how a limb segment moves relative to the body. For any given small change in 3D extension (Δθ), φ≈90 deg suggests that 3D extension thrusts the body forward (Fig. 1A) whereas φ≈0 deg suggests that Δθ elevates (Fig. 1B).

Armed with the above framework for characterizing joint functions, the present study addresses vertical versus horizontal jumping as a subset of frog behavioural diversity. We propose a hypothetical mechanism by which frog joints influence the instantaneous body pitch angle (ψ) to control take-off angle (ψ_take-off; Fig. 1). Given that frogs push their body upwards to reach a ‘preparation angle’ during the first phase of jumps (Wang et al., 2014), we similarly expect the limb joint axes to reach a ‘joint preparation angle’ enabling frogs to aim for a steep versus shallow jump. Specifically, we expect the rotation axes of the hip, knee and ankle to be approximately parallel such that leg segments move within the same plane (i.e. the ‘extension plane’). Thus, we hypothesize two mechanisms to modulate ψ: (1) adjusting the initial leg joint configuration to set the degree of inclination of the extension plane during the preparation phase (Fig. 1C) and (2) adjusting the angle of the sacroiliac joint to tilt the body upwards about the pelvis (Emerson and De Jongh, 1980; Jenkins and Shubin, 2010).

An ideal model for the present study is Kassina maculata Duméril 1853 (African running frogs), which are grass- and reed-dwelling frogs (Wendy and Channing, 1983) that jump between perches (C.T.R., L.B.P., personal observations) as well as upwards to escape into overhanging tree leaves when frightened (Loveridge, 1976). We used high-speed video and inverse kinematics analysis to determine whether limb segment motions and joint axes of rotation predict jump angle.

Adult K. maculata frogs (28.4±3.7 g body mass; 60.0±1.2 mm snout–vent length; N=4; Table 1) were obtained from Amey Zoo (Hempstead, UK). Animals were housed in terraria at the Royal Veterinary College at 25°C under a 12 h:12 h light:dark cycle and were fed two to three times weekly.

Small plastic circular markers (∼5 mm diameter) were cut from thin plastic sheets (∼0.25 mm thick) using a screw punch (Nonaka Mfg. Co. Ltd., Japan) with a 5 mm hollow point drill bit, and then painted with white correction fluid. Markers were placed on anatomical landmarks on the body (head, vent) and left leg [hip, knee, ankle, tarsometatarsal (TMT) joint] using cyanoacrylate adhesive. Frogs were then stimulated to jump while filmed from two orthogonal views at 250 Hz, 1/1500 shutter speed using a pair of Photron SA3 high-speed video cameras (Photron Ltd, Tokyo, Japan). An inclined mirror was used to obtain a third view to track landmarks on the body and limb simultaneously. Prior to each jump, frogs were placed at the same starting position on a small force platform (see Porro et al., 2017). A range of jump angles were elicited by consistently varying the height of the frog box using pre-cut wood blocks of pre-determined thickness. Trials in which the frog turned during the jump (i.e. asymmetric leg extension) were discarded. All experimental procedures were approved by the UK Home Office (License 70/8242).

Video data from three views were calibrated using direct linear transformation and landmarks were ‘digitized’ to XYZ coordinates using open source script (Hedrick, 2008) run in MATLAB (MathWorks, Natick, MA, USA). All subsequent data processing and analysis was performed in Mathematica (Wolfram, Hanborough, UK). XYZ coordinate data were filtered by a second-order lowpass Butterworth filter using a 25 Hz cut-off frequency to smooth video tracking error.

The camera views defined a right-handed global reference frame with the y-axis pointing forward along the trackway, the x-axis pointing right and the z-axis pointing up (Fig. 1). For each video frame, digitization of seven landmarks yielded a 7×3 matrix, M (i.e. seven xyz points). Although only straight jumps were included in analysis, frogs rarely initiated their jumps exactly parallel with the trackway axis. Occasionally, frogs began with an initial angle within 10 deg of the trackway axis, but carried on moving along the same direction (i.e. frogs did not turn during the included jumps). Thus, for every video frame, M was rotated (M′) to align the points along the axis of the jump, effectively moving the camera to align with the forward axis of the jump (see  Appendix 1).

Landmarks placed on the leg were used as a proxy for rigid body motion of the limb segments. Leg segment angles were calculated using local reference frames anchored at joint centres of rotation (assumed to coincide with landmark positions) as follows. For each joint, the global Cartesian x-y-z axes were anchored at the given joint centre. XYZ coordinates of the distal tips of the leg segments (thigh, shank, and proximal foot) were converted to spherical polar coordinates (i.e. [x, y, z]→[r, θ_xy, θ_z]). For example, the thigh segment angle was calculated using the hip as the origin and the distal tip of the segment (i.e. knee) as the point of interest, P, in the hip local reference frame. Converting P into spherical polar coordinates yields a length (r=segment length) and two angles: (1) the angle θ_xy in the horizontal XY plane (retraction) and (2) the angle θ_z with respect to the global vertical axis (adduction). Note that for convenience, we offset θ_z by −90 deg so that angles could be based from the horizontal (i.e. a segment oriented horizontally has an adduction angle of 0 deg; see Fig. 1).

(and similarly for the knee and ankle joints), where Â_h,z is the z-coordinate of the normalized hip rotation axis vector. Note that the XY plane (horizontal) was defined here as 0 deg. Note also that because the knee rotation axis direction is opposite that of the hip and ankle, the knee orientation angle was negated.

To statistically compare temporal patterns among trials, time series data were normalized by percent of jump contact time (t_c=t_end−t_start), where t_end was defined as the final video frame where the foot is seen in contact with the surface. Because jumps are not periodic behaviours, we used the body centre of mass (COM) motion to mark the onset of jumps. COM resultant velocity was estimated as the velocity of a body landmark near the hip. For each trial, the time-varying magnitude of the COM resultant velocity vector was fit to a six-order polynomial. The time value for the first zero crossing (i.e. start of body velocity) of the polynomial equation was used as an estimate of t_start. Translational velocity values are reported in body lengths s⁻¹ (absolute velocity/snout–vent length).

To explain more thoroughly, IK analysis relates changes in joint angles (q) to the linear velocity of the limb endpoint. In a robotic arm, for example, IK computes the joint angles necessary to achieve a desired linear velocity of the hand towards a target. For a frog limb at a given posture, one can calculate a small change (Δq) required to move the head in a given direction. Importantly, IK does not account for masses and moments-of-inertia of the limb segments, hence it ignores the required forces/torques in the system. Rather, IK assumes that the muscles (or electric motors in a robot) can produce sufficient force (or torque) to achieve the desired kinematics. Regardless, IK can be used as a modelling tool to hypothetically address how a frog might extend its joints to adjust its instantaneous body velocity (hence body angle, ψ; Fig. 1) and its rate of change (hence dψ/dt). Because of the many possible ways in which a frog limb can move, there likewise exist many solutions for Δq to achieve a desired instantaneous ψ or dψ/dt. Implicitly, IK calculates the minimal required Δq, and thus reveals a plausible hypothesis for how joint rotations might influence COM motion. The steps to compute joint angle changes in response to a hypothetical change in ψ are as follows.

where v_target is a column vector of x, y, z coordinates, Δθ are changes in hip, knee and ankle 3D extension angles and J′_T is a 3×3 matrix computed as the pseudoinverse of the translational portion of a modified Jacobian matrix (i.e. a matrix that converts between joint angular velocities and end-of-limb linear velocity; see  Appendix 2). With simple geometry (e.g. a single bone rotating about a hinge-like joint), one can easily see the relationship between angular rotation and linear excursion at the tip of the limb. However, with multi-joint systems (such as robotic arms, frog legs, etc.), the cumulative motions caused by in-series joints can only be known through messy calculations. Consequently, how individual joint rotations contribute to motion at the end of the limb (e.g. the hand of a robotic arm or the body of a frog) is not obvious. To address this, Jacobian matrices are used in kinematics and dynamics analysis because they allow one to translate joint movements into end-of-limb movements and vice versa.

where J′ is the 6×3 pseudoinverse of the modified Jacobian matrix (see Appendix 2). For the present analysis, v_target was first selected using ψ=1 rad (≈60 deg) to represent the steep jump trials (Eqn 7). Then, dψ/dt_target was determined from the Δq due to the selected v_target.

Descriptive statistics were performed on data from n=50 jumps pooled over N=4 animals. Data were arbitrarily divided into three categories based on jump take-off angle (see above) for data plotting and descriptive statistics. After dividing jump angle values into quantiles, low jumps were classified as below the first quantile, middle jumps between the first and third quantiles and high jumps above third quantile. ANCOVAs were used to compare several separate performance variables for their influence on increasing jump angle. A significance threshold of P=0.05 was used for the regression component (jump angle versus performance variable), individual effects among frogs (frog) as well as interactions (frog×jump angle). Statistical analyses were performed in Mathematica.

Jump performance varied widely among trials, with take-off angles ranging from 0.27 deg to nearly 70 deg pooled across all frogs. The average minimum jump angle (minimum angle averaged across all frogs) was 7.61±6.68 deg and the maximum was 58.43±8.24 deg, while peak resultant velocities ranged from 16.49±3.64 to 31.47±3.27 body lengths s⁻¹ (mean±s.d., N=4 frogs; Fig. 2). The distribution among pooled jump angle data suggested data partitioning into low (<20 deg) versus high jumps (>50 deg) with the remainder designated as middle jumps (see Materials and methods). Each individual frog represented nearly the entire performance range (see Table S1) and there was no observed effect of individual frog on the performance variables measured (Table 2).

Despite variation in jump angle and take-off velocities, limb and body kinematics were qualitatively similar. The sacroiliac angle change was statistically significant, but modest in magnitude and rate compared with ‘3D extension’ angles of hip, knee and ankle joints (Fig. 3B), which propelled frogs toward peak translational velocity by the time of take-off (Fig. 4A–C). At take-off, the resultant COM velocity increased significantly with jump angle (ANCOVA, P<0.0001) as a function, not surprisingly, of the rising vertical component magnitude (Table 2, Table S1). Surprisingly, despite the lack of strong sacroiliac rotation, steep jumps were marked by rapid pitching rotational velocity of the body segment (dψ/dt) that was less apparent for low and intermediate jumps (Fig. 4D–F).

Segment orientations for the thigh, shank and PF were calculated in local joint reference frames anchored at the hip, knee and ankle, respectively (see Materials and methods). Retraction (about the local Z-axis) and adduction (about the local Y-axis) carried the thigh, shank and PF segments backward and downward. Seen from dorsal (‘top’) view, segments mainly retracted as the distal ends of limb segments moved posteriorly (see Figs 1 and 3). Segment retraction patterns were qualitatively similar across jump angle. Although limb segments began at smaller retraction angles (i.e. more ‘flexed’ posture) for shallower jumps, jump angle was independent of total minimum–maximum angular excursion (retraction) for all segments except for the thigh (ANCOVA; Table 2, Table S1). In all trials, the thigh and proximal foot retracted in concert causing the two to remain parallel in dorsal view as they retracted. In contrast, the shank orientation remained nearly constant during the earliest ∼80% of jumps before retracting (straightening) slightly to align with the body midline in the final stage of take-off (Fig. 5A–C).

Unlike the observed retraction patterns, segment adduction (downward rotation towards the ventral body midline) varied qualitatively among segments and jump angles (Fig. 5D–F). Initial adduction angles varied differentially among segments (ANCOVA, P<0.05; Table 2). More importantly, however, increasing jump angle was strongly predicted by greater thigh and shank adduction excursion (Table 2, Table S1). Most notably, average shank orientation remained nearly horizontal except for medium and steep jumps, which were marked by increasingly strong adduction during the final ∼20% of take-off time (Fig. 5F).

Limb segment motions were described using the axis-angle convention (see Materials and methods; Fig. 1). Briefly, the relative orientation of two adjacent segments (e.g. thigh and shank) was defined by two parameters: the 3D extension angle (i.e. the shortest angle between two segments) and the 3D axis of rotation. The former determines the ‘straight-line’ displacement between joints whereas the latter determines the direction.

Frogs showed a similar pattern of joint orientation among jumps. Generally, joint axes were time-varying, indicating that the effects of 3D extension shifted during jumping (Fig. 6). Hip and knee rotation axes began nearly vertical, but reoriented to ∼45 deg throughout jump duration, suggesting a functional transition from pure retraction to retraction+adduction. In contrast, the ankle rotation axis began at a lower angle (closer to horizontal) and increased through time, signifying a shift from adduction to retraction+adduction. A striking general pattern emerged across all jumps: during the final ∼20% of jumps, hip, knee and ankle joint axes converged towards a common angle; consequently, all hind limb joints rapidly shifted into alignment (Figs. 3A inset, 6A–C).

Changes in axis orientations co-varied significantly with jump angle. Most notably, at higher jump angles the final orientation angles of hip, knee and ankle joints decreased significantly (ANCOVA, P<0.0001; Table 2, Table S1). Specifically, shallow jump angles appear to result from nearly vertical rotation axis orientations (φ closer to 90 deg) whereas steep jumps correlated with more horizontal orientation angles (φ closer to 0 deg; see Fig. 1A versus B). Functionally, this shift from high orientation angles (shallow jumps) to low orientation angles (steep jumps) suggests a putative mechanism for modulating take-off angle via control of rotation axes (see Discussion). In addition to take-off kinematics, the pre-jump limb posture in the early phase of the jump was also a predictor for jump angle. For increasing jump angles, the hip and knee orientations became less vertical, suggesting an increasing bias towards vertical COM motion.

IK analysis predicted differential roles for the hip, knee and ankle joints. This analysis predicts how hypothetical extension of an individual joint might have the potential to act to move the frog at a given point in time. Near time=0%, ankle 3D extension showed potential for increasing the body velocity angle of kinematic models (Fig. 7A) whereas combined ankle and knee 3D extension showed the strongest effects late (time=80%; Fig. 7B). IK was similarly used to model the roles of joint extension on body rotational velocity (dψ/dt), which is partially caused by simultaneous knee 3D extension combined with the decreasing angle of the knee rotation axis (Fig. 7C; see Discussion).

The present study addressed how frog joint kinematics shift to enable flexibility of jump angles. Because prior work found that forelimb preparation angle predicts take-off angle (Wang et al., 2014), we hypothesized an analogous ‘joint preparation angle’ whereby leg joints form a ‘3D extension plane’ whose inclination is preset to determine final take-off angle (ψ_take-off; Fig. 1). Additionally, we expected sacroiliac extension to modulate take-off angle. Current findings support aspects of each hypothesis, but neither entirely. Regardless, general patterns emerged across all animals and jumps. Frog jumps were powered by limb segment retraction (rearward rotation pushing the body forward) and adduction (downward rotation elevating the body; Nauwelaerts and Aerts, 2003; Astley and Roberts, 2014) as well as rotational pitching velocity (dψ/dt). Within this observed behavioural variation, we sought joint kinematic evidence explaining two mechanisms: (1) how do leg joints influence body velocity angle (ψ) via relative amounts of forward thrust versus elevation; and (2) how do joints influence upward pitching rotational velocity of the body (dψ/dt)? Our three key findings are as follows. Firstly, although we found that K. maculata adjust aspects of initial joint posture, jump angle does not appear to be solely predetermined by initial limb configuration. In addition to adjusting initial pose, frogs appear to dynamically modulate their leg kinematics late during jumps by adjusting shank adduction as well as the inclination of hip, knee and ankle rotation axes. Secondly, we found evidence for a ‘3D extension plane’ governing take-off angle, but only in the final 20% of jump duration as the joint rotation axes rapidly orient into alignment. The influence of this alignment is not immediately clear, but could be investigated with forward dynamics modelling in future work. Thirdly, body rotational velocity (dψ/dt) was negligible in low and intermediate jumps, but contributed strongly to the steepest jumps.

Regardless of jump angle, K. maculata showed stereotyped kinematics, perhaps giving insight into the underlying musculoskeletal mechanics of jumping. After the onset of body motion, jumps proceeded in two phases: a slow ‘preparatory’ phase followed by a fast ‘launch’ phase. Similar to ranid frogs, the preparatory phase (roughly first ∼60–80% of duration) is marked both by forelimb contact (Wang et al., 2014) and slow acceleration of the COM (Fig. 4). In contrast, the launch phase is characterized by rapid retraction and adduction via hip, knee and ankle 3D extension (Fig. 5). Additionally, the joint axes of rotation rapidly reorient to align just before take-off (Fig. 6).

The striking consistency of joint rotation patterns, even for the most extreme jump angles, suggests an underlying kinematic control mechanism that is adjusted to vary take-off angle. To escape, arboreal frogs must achieve the correct mixture of upwards and forwards COM motion to reach a particular perch. Ideally, extension at each joint would transmit the desired combination of elevation and thrust to the body at every instant in time. More likely, however, because of the 3D posture of frogs, each joint does not necessarily move the body in a useful direction. While one joint may contribute to elevation, another joint may simultaneously act to depress the COM. Adding to this complexity, a particular joint's kinematic influence depends on the instantaneous orientations of other limb segments, each varying through time. Accounting for this interdependence of joint function, IK analysis predicted time-varying function among the main leg joints. During the preparatory phase (time=0%), ankle 3D extension is predicted to be the main contributor to increased body velocity angle (Fig. 7A), which is counterintuitive given the proximal-to-distal sequence of joint extension characteristic in frog jumping (e.g. Peters et al., 1996). To explain, IK does not necessarily indicate how joints are acting, but rather how they have the potential to act (i.e. a prediction of a joint's effect if it were to increase torque or extend). Although IK predicts the ankle has high potential to influence jump angle early in the jump, we are not suggesting that the ankle must extend prior to proximal joints. IK is simply predicting that the ankle's influence on jump angle is high early in the jump, so that when the ankle begins extending (in the first ∼10% of the jump), it will exert a relatively strong effect on instantaneous body angle. In contrast, combined ankle and knee 3D extension are predicted to exert the strongest effect during the launch phase (time=80%), with the knee contribution dominating the steepest angles (Fig. 7B). From a dynamic control perspective, our model predicts that if the nervous system were aiming to elevate the body, it might increase activation to ankle extensors early in the jump followed by the addition of knee extensors later during launch. Although IK results are hypothetical, they suggest which joint motions are most ‘kinematically parsimonious’ for transmitting motion to the body. Thus, they may be interpreted as control hypotheses to be tested by further experiments (e.g. electromyography).

In addition to the greater body elevation during the steepest jumps, K. maculata also rotated their body rapidly to increase body velocity angle (Fig. 4F). Given the weak rotational velocity observed at the sacroiliac joint (Fig. 3), the origin of the rapid body rotational velocity (dψ/dt) is not clear, although the forelimbs could play a role (Wang et al., 2014). As argued above, 3D extension at any joint influences the ratio of body thrust versus elevation. Similarly, IK can predict changes in joint angles required for a change in body rotational velocity. Counterintuitively, IK predicts that dψ/dt is partially caused by simultaneous knee 3D extension combined with the decreasing angle of the knee rotation axis (Fig. 7C). As the knee rotation axis tilts away from vertical (Fig. 6), the leg joints become able to pitch the body upward while elevating the COM by the mechanism above. This can be most clearly seen during the period of greatest knee axis change coinciding with the steepest increase in rotational velocity (time ∼50 to 80%; Fig. 7C). Hypothetically, if the knee axis fails to tilt, the joint angle changes required for a steep jump (ψ≈60 deg) would cause abduction (upward rotation) of the femur and shank, consequently pitching the body down. However, as the knee axis orientation tilts downward (either via femur or shank adduction; Fig. 5), 3D knee extension effectively rotates the lower limb downward to pitch the body up. More succinctly, IK suggests that COM elevation and rotation would be mutually exclusive without appropriate reorientation of the knee axis. Further forward dynamics analysis would be required to confirm whether knee axis reorientation contributes to body rotational velocity. Moreover, forward dynamics would additionally be required to assess whether the observed kinematics are most effective compared with simulations that have been ‘optimized’ (e.g. for power generation or efficiency).

The above IK analysis offers hypothetical mechanisms by which K. maculata may fine-tune take-off angle over a continuous range. However, the pronounced kinematic difference of steep jumps (e.g. high rotational body velocity; Fig. 4F versus D,E) may suggest that the underlying musculoskeletal mechanics do not vary continuously with inclination. Rather, extreme high versus low take-off angles may represent two biomechanically distinct movements: vertical versus horizontal locomotion. As the legs adapt different postures to overcome varying amounts of gravitational loading, they perhaps demand different recruitment patterns within and among muscle groups. Analogously, from an ecological perspective, vertical versus horizontal versus diagonal jumps may also be interpreted as distinct locomotor behaviours. For example, vertical jumps are required for escape (Loveridge, 1976) versus diagonal jumps for distance (Marsh, 1994) versus horizontal ‘jumps’ to lunge after prey (Deban and Nishikawa, 1992). Hence, we propose that although traditional 2D models of vertical jumps have been used as a template for all jumps (e.g. Alexander, 1995; Galantis and Woledge, 2003; Aerts and Nauwelaerts, 2009), they may not be appropriate for the entire range of jumping. Future investigation would be required to determine whether vertical jumping and horizontal jumping represent two distinct behaviours.

There are two main limitations to the present study. Firstly, external landmarks were used to reconstruct body and limb motions; thus, joint centres of rotation could not be precisely located. More importantly, because each segment was only defined by a pair of skin markers, true rigid body motions (i.e. rotations about each segment's local reference frame) could not be calculated. Generally, any observed ‘3D extension’ may stem from a combination of local extension, adduction or long-axis rotation. However, owing to the limitations of skin marker kinematics, we cannot resolve whether an apparent downward rotation of the shank resulted from an adduction of the knee joint or internal long-axis rotation of the femur about the hip. Consequently, we can only speculate on putative contributions of extensor versus adductor versus rotator muscles to limb motion. Future use of X-ray reconstruction of moving morphology (XROMM; Brainerd et al., 2010) will be used to clarify whether adduction or internal rotation drives body elevation during jumps. Additionally, XROMM data could be used to compare a more thorough standard analysis of Euler angles with our simplified axis-angle analysis to better assess the full utility of current approaches.

Secondly, a pure kinematics analysis can only speculate on the functional roles of joints in the absence of further analysis of ground reaction forces and joint torque estimates. Although IK predicts potential contributions of joint motion, pure kinematics analysis does not measure the ‘effort’ required to jump, and thus cannot disentangle how force, work and power requirements might shift as frogs aim for steeper angles (see Porro et al., 2017). Joint kinematics emerge from the balance of muscle torques and external torques from ground reaction forces. Hence, while a purely kinematic study is an important first step to investigate putative joint function, it can be misleading without further analysis. Although we report how 3D extensions may ‘control’ either forward or upward body motion (Fig. 7), these are hypothetical effects which must be confirmed by inverse and forward dynamics analysis. For example, without knowledge of net joint torques one cannot determine whether an apparent extension movement was controlled by active torque at that joint or produced under the influence of other joints via the ground reaction force. In another example, we interpret from IK analysis that rotation of the knee joint axis (e.g. via knee adductors) drives the pitching motion to the body segment. However, another plausible interpretation is that active pitching of the body segment (e.g. due to thigh or pelvic musculature) transfers rotational kinetic energy to the knee joint, causing it to reorient passively. Kinematic analysis cannot discern between these possibilities without forward dynamics simulation. In the spirit of 2D models of human jumping (e.g. Bobbert and van Ingen Schenau, 1988), future use of inverse dynamics (Porro et al., 2017) combined with electromyography will lead to an integrated understanding of muscle function, 3D joint mechanics and jump performance for sprawled limb jumpers. Regardless of the limitations, kinematics analysis is an important complement to dynamics analysis: dynamics analysis resolves which joints drive limb motion, whereas kinematics analysis resolves how the motions themselves translate to coordinated behaviour.

The present study used K. maculata as a model to explore how frogs modulate jumping performance. It is the first known study to address how 3D leg segment kinematics leads to coordinated control of the body angle and body rotational velocity. Future studies would be required to determine whether other species employ similar mechanisms.

As frogs varied take-off angle, we found that leg segments shifted the amount of abduction/adduction relative to protraction/retraction. These ‘3D effects’ corroborate computational evidence that 2D planar joints are insufficient to modulate jump angle (Kargo et al., 2002). Specifically, we found two main kinematic mechanisms influencing take-off angle: (1) shank adduction (downward rotation) presumably pushed the body upward late during the launch phase, and (2) rapid rotational velocity (backward pitching) caused the body to tilt vertically, contributing to a steeper take-off. IK modelling suggests that knee 3D extension contributes to the second mechanism whereas the former may result from action at either the knee or ankle. To resolve whether shank adduction is caused by either ankle or knee torque would require inverse dynamics analysis (Porro et al., 2017).

We interpret the above findings as evidence for how frogs use kinematics to control performance. Importantly, as this is a single-species study, we cannot conclude whether these kinematic mechanisms are shared amongst other jumpers or whether they are unique to arboreal frogs such as K. maculata. Prior work on human biomechanics suggests that navigating an arboreal environment involves precise targeting between small areas, thus requiring greater control at all temporal phases of the jump (Günther et al., 1991). Regardless, the natural range of motion and kinematic degrees of freedom of frog limbs (Kargo and Rome, 2002) may enable most terrestrial or semi-terrestrial species to tweak jumping kinematics to adjust jump trajectory. If this is the case, we ask: was the evolution of alternative modes of frog locomotion (e.g. climbing, lunging, burrowing) a byproduct of such mechanisms that enabled the flexible control of jumping? More generally, continued comparative work applying the current analyses as well as forward dynamics and muscle experimentation may reveal general principles by which ‘specialized’ limbs can gain secondary mechanical function by ‘morphing’ the geometry and timing of joint motion.

where · denotes the dot product or matrix multiplication, ^T is the matrix transpose, B_xy is the horizontal component of the body axis vector and Y is the y-axis vector ([0,1,0]). The above transformation effectively cancelled any yaw angle, causing landmarks to move as if all jumps aligned perfectly with the y-axis.

IK is typically used to control jointed robotic manipulators (e.g. robotic arms). Specifically, the user specifies a target position and orientation for the ‘end effector’ (e.g. robotic hand or grabber at the distal end of the limb). IK is then used to update the angles of the robotic joints to translate and rotate the end effector to the desired position and orientation. Importantly, IK calculates the smallest necessary joint movement to reach the target. In the present study, we used IK as a theoretical tool to ‘back-calculate’ the hypothetical changes in 3D joint angles that would be required to move our end effector (i.e. frog body) to a target position or orientation.

A_i is the instantaneous ‘3D extension’ rotation axis (see Materials and methods) for the ith joint, expressed as a unit vector. P_i is the ith point (i.e. joint rotation centre) and P_n is the most distal point (P_s, in the present study). For example, for the hip joint, P_i=P_h (see Materials and methods) × denotes the 3D cross-product.

using the same definitions as in Eqn A5.

where J_T is the translational block (upper three rows) of J and dq/dt is the n×1 column vector of instantaneous rotation velocities.

where J_R is the rotational block (lower three rows) of J.

Finally, the pseudoinverse of J_T and J_R (J′_T′;J′_R) can be used to solve for hypothetical changes in joint angles (Eqns 8, 9; see Materials and methods). In practice, a small incremental change in joint velocities can be used (e.g. maximum rotational velocity value×sampling rate⁻¹) as a substitution for dq/dt. At an instant in time, the translational part of the Jacobian matrix (J_T) is used to solve for angular changes that would produce an instantaneous change in body position (i.e. velocity) and orientation (i.e. body pitch angle). As components of instantaneous velocity yield forward (V_thrust) and upward velocity (V_elevation), one can calculate the instantaneous angle (ψ) as the angle between these components (Fig. 1C). Hypothetically, if the frog aims to achieve a particular body angle at an instant in time, we can solve how joint angle changes would dictate an instantaneous ψ (Eqns 7–9; see Materials and methods). Furthermore, we can calculate a target orientation axis about which the body rotates. In the present study, the x-axis (body pitch axis) was chosen. Similar to the above, if the frog aims to pitch about the x-axis, we can solve how joint angles would cause an instantaneous dψ/dt.

We greatly thank Alastair Wallis for animal care. Enrico Eberhard helped greatly with the experimental setup. We also thank Steve Amos, Emily Sparkes and Timothy West for providing infrastructural, administrative and technical support. Hannah Safi assisted during experiments. We also thank Duncan Irschick for insightful conversations that helped stimulate aspects of this work.