Take-off and landing forces in jumping frogs

Anurans use a saltatorial (jumping) mode of locomotion. A jumping cycle can be divided into four subphases: propulsion, flight, landing and recovery. We studied the landing phase during locomotion in Rana esculenta by measuring the ground reaction forces during propulsion and landing over a range of distances. Landing performance affects locomotor ability in jumping frogs. Landing and recovery together take up one third of the locomotor cycle. Peak landing forces are on average almost three times larger than propulsive forces. The forelimbs appear to be fully extended when they make contact with the substrate and absorb the first impact peak. The height of this peak varies depending on arm positioning and jumping distance. Since the stiffness of the arms stays constant over the full jumping range, it is possible that this is a limiting factor in the ability of the forelimbs to work as dampers. A spring-dashpot model is used to model the effect of arm angle at touch down. Damping during landing is performed by placing the forelimbs at an optimal angle to cancel frictional forces effectively.

The approach to quantifying terrestrial locomotor performance in adult anurans has thus far been based on (1) variables characterising the propulsive phase, such as take-off velocity (e.g. Marsh, 1994; Marsh and John-Alder, 1994;Choi et al., 2000, 2003) and ground reaction forces (GRFs) measured using a force plate (e.g. Calow and Alexander, 1973; Wilson et al., 2000; Nauwelaerts and Aerts, 2003),or (2) direct measures of jumping distance (e.g. Wermel, 1934; Zug, 1972; Emerson, 1978; Hirano and Rome, 1984; Lutz and Rome, 1994; Marsh, 1994). Although these variables are considered appropriate as general performance parameters,conceptual concerns arise when examining locomotion in an ecological context. For instance, escaping from a predator means realizing a distance as large as possible between predator and prey, as fast as possible. This implies that factors other than jumping distance, take-off velocity or GRFs can become crucial in saltatory locomotion. One of these factors is the larger landing forces that inevitably result from increased take-off velocities and GRFs during propulsion. Valuable time could be lost during damping of the impact force and the recovery that follows. Simply jumping further by taking off faster may not represent the optimal ecological solution. The idea that recovery may be as important as take-off comes from the observation that during endurance tests frogs seem to have a preferred jumping distance that was smaller than their maximal jumping distance (S.N., unpublished data). This observation led to the hypothesis that it may not be ideal to increase the jumping distance beyond a certain limit because of its consequences on recovery.

A jumping cycle can be divided into four sub-phases: propulsion, flight,landing and recovery. To escape optimally, all phases must be of short duration with large displacements of the center of mass (COM). Anurans take off by extending their hindlimbs and using their forelimbs for landing. The forelimbs touch the ground and form a pivot, about which rotation of the body occurs. This can play a crucial role in supporting and stabilizing the frog as it lands (Peters et al.,1996). Since the forelimbs are considerably shorter than the hindlimbs and provide less deceleration distance, impact forces on the front limbs are expected to be high. The landing phase could therefore be a limiting factor in the jumping capacity of a frog for jumps where the frogs remain terrestrial. By determining the durations of the four sub-phases of a jumping cycle, we will answer the following question: (1) what fraction of the total locomotor cycle does the landing phase take up during saltatorial locomotion in frogs? By recording the GRFs we can answer: (2) are peak landing forces higher than peak propulsive forces in jumping frogs?

The forelimbs are not only used during landing(Peters et al., 1996), but also must perform a number of other functions. During feeding, they are used to manipulate and bring prey towards the mouth(Gray et al., 1997) and during wiping behavior, forelimb movements help to protect the skin by keeping it moist. To realize these functions, the forelimbs need to be flexible. During the breeding season, male frogs use forelimbs in combat to repel rival males(Peters and Aulner, 2000). Males also grip females with their forelimbs during amplexus(Duellman, 1992). This activity demands strong isometric force production from the forelimb muscles with the forelimbs in a flexed position. If the forelimbs also play a major role during landing by damping the kinetic energy from the flight phase, the extensor muscles of the forelimbs are expected to work as dampers. By flexing the elbows, the impact forces can be reduced by increasing the braking distance. Performing various tasks that have conflicting demands might restrict the capacity of the forelimbs to perform landing effectively. By recording the landing forces of jumping frogs and the timing of forelimb action for a range of distances, we can answer an additional question: (3)does stiffness of the arms increase with jumping distance? Lastly, based on high-speed video recordings of the landing phase and predictions made by a spring-dashpot model, we can answer the final question: (4) does arm angle at touchdown influence the landing phase?

Five adult male Rana esculenta L. were acquired from Centre d'élevage d'Animaux (Bray-et-Lû, France), ranging in size from 48-73 g (62±8 g, mean ± s.d.). Animals were kept in a light- and temperature-controlled room (8 h:16 h light:dark; T=20°C) in a large glass tank (150 cm×75 cm×75 cm) filled with wet sand and fed crickets once a week. In addition, 11 adult males, held in similar conditions,were used in the landing experiment to determine arm angles.

The animals were placed upon a small force plate [AMTI MC3A-6-100, AMTI,Watertown, MA, USA; Sensitivities vertical GRF (F_v)=1.35μV/(V N) and horizontal GRF (F_h)=5.4 μV/(V N),natural frequency 300 Hz] with a top plate of 11.7 cm×12 cm and were induced to jump onto a second, larger force platform [AMTI OR6-6-2000;Sensitivities F_v=0.08 μV/(V N) and F_h=0.34 μV/(V N), natural frequency 1000 Hz] with a top plate of 50.8 cm×46.4 cm. Some cross-talk always exists when using strain-gauge type force plates, but the effect was minimized by including the correction terms in the calibration matrix. Slipping was prevented by covering the force platforms with water-resistant parquet sandpaper (P15U). To reduce mechanical noise, the small force plate was placed in a plastic container filled with wet sand. The two force plates were positioned 10-30 cm apart, to ensure a full jump distance range. Both force plates measured the three-dimensional GRFs during jumping. The output of the strain gauges was sent to an amplifier and A/D converted at 1 kHz (AMTI MCA strain gauge amplifier). Digital traces of the force platform and a signal from a LED(light emitting diode), used to synchronise force measurements with video recordings, were read into a PC. Forces were subsequently smoothed using a fourth order zero phase shift Butterworth filter in a custom-made Labview program.

We examined landing behaviour by videotaping the jumps (high-speed video camera, Redlake, MASD Inc., San Diego, CA, USA) at 250 Hz and zooming in on the second force plate. Simultaneous lateral and frontal views were obtained by placing a mirror at a 45° angle next to the second force plate. The propulsive phase was also visible in the mirror view. In order to obtain the total jumping distance for each jump, a Sony 50 Hz camera was placed above the set-up, in order to provide a simultaneous view on both force plates. The set-up was lit using a Tri-lite light (Cool Light Co., Inc., Hollywood, CA,USA; 3×650 W).

Five jumps for each animal, resulting in a total of 25 jumps, were selected for further analysis. The selection of the trials was made on the basis of the jumping distance, aiming to obtain the largest possible distance range for each animal.

Three key timing variables were obtained from the high-speed video recordings: (1) timing of first contact of the arms with the force plate, (2)timing of body contact with the plate and (3) timing of full recovery (i.e. frog is back in launch position).

Horizontal forces F_h and vertical forces F_v were plotted for each jump and the resultant force F_r was computed. The angle of the resultant force was calculated for both the propulsion and landing phases.

Force profiles were used to determine the durations of the propulsion,flight and landing phases. Landing was defined as the phase from the first contact on the force plate to the time when the vertical GRF equalled the weight of the animal. Recovery was defined as the phase between the end of landing, based on the force profiles, and full recovery, based upon the high-speed video recordings.

Accelerations in two directions were determined from the force profiles. To obtain vertical acceleration a_v, body weight BW was subtracted from F_v and divided by total body mass M_b. The first and second time integrals of the accelerations yielded the velocities v_h and v_v and the displacements d of the COM. Calculations for landing were done by starting with v_h and v_v at take-off, taking gravity into account(v_v-gt, where gis the gravitational acceleration and t the duration of flight). Instantaneous power (P) profiles were calculated by multiplying force F by v. The time integral of the force profiles provided the impulses (I) and work (W) was obtained by integrating the power profiles. Total work for propulsion and landing was calculated by summing the absolute values of the maxima of the work profile in two directions. In addition, work done by only the forelimbs(W_f) was analysed separately.

Stiffness of the forelimbs K_f was calculated as GRF during landing (F_l,r) divided by resultant displacement of the COM (d_r) during landing.

Muscle-specific mechanical power was calculated from the GRFs and divided by the forelimb muscle mass M_m,f, determined as a fraction of M_b (4%, measured on two dead animals). Similar calculations were done for the hindlimbs for propulsion (hindlimb muscle mass M_m,h taken as 15% of total M_b,measured on 40 dead animals as part of a different study; S.N., unpublished data).

Total jumping distance D_j was measured from the dorsal 50 Hz video recordings, by digitising the snout at the start and end of the jump using a NAC-1000 XY coordinator (NAC Image Technology, Inc.,Tokyo, Japan) connected to a PC.

We used a simple model of a mass mounted on a linear spring and a velocity-dependent dashpot to test the effect of arm angle θ_fat the moment of touchdown on the course of the consequent landing phase. Data from a real sequence where the arms perform the whole deceleration action were used to parameterize the model, i.e. to obtain stiffness and damping coefficients. The simulation was ended when the mass was about 1 cm above the ground surface, which coincided with the body making contact with the force plate. The landing phase of the chosen sequence lasted 33 ms and ended with the body COM positioned approximately 1 cm above the force plate with a vertical velocity of zero. These landing conditions could be reproduced in the model with a stiffness value K of 70 N m^-1 and damping coefficient c of 1.4 Ns m^-1 with an arm angle of 125°at touchdown, so these stiffness and damping coefficients were used in all simulations. At touchdown, the model showed some residual v_h (about 50 cm s^-1), which in reality would be dissipated by frictional forces between body and the force plate surface, and by deformation of the body. The high-speed video recordings showed that when the forces become large, the body sometimes slipped forward during landing,illustrating this phenomenon. The forces in the spring-dashpot were similar to the GRFs measured in the real sequence: peak vertical force was approx. 3 N(F_l,v,max) and peak horizontal force(F_l,h,max) was approx. 2 N, and they both had decelerating actions throughout the landing phase. The model was used to test the effect of arm angle θ_f, increasing θ_f from 110 to 140° in 1° increments. Simulations were carried out for five jumping heights between 0.20 m and 0.40 m and for five horizontal velocities between 0.73 m s^-1 and 2.19 m s^-1 (or the initial value,determined from the real sequence; ±25% and ±50%).

To test whether frogs optimise θ_f according to the spring-dashpot model, we recorded additional video material on the landing behaviour of frogs using a NAC-1000 high speed video camera at 500 Hz. We recorded a total of 90 landing phases from 11 frogs. These recordings were used to obtain arm angles at touchdown and 25 ms prior to touchdown. The change in position of the COM (calculated as the midpoint on a line drawn between snout tip and cloaca) during this period of time (25 ms) was used to estimate v_h and v_v at touchdown. Height h was calculated from v_v.

Peak forces F_max were compared among phases(propulsion-landing) and within phases using a paired t-test.

A correlation matrix of all selected variables with total jumping distance was built to test direct relationships. A second correlation matrix, with the peaks of all measured forces, was used to test within-phase (propulsion and landing) and between-phase relationships. In addition, a third correlation matrix was built to detect correlations between the durations of the four sub-phases of a jump.

We found two types of force profiles during landing, depending on whether the peak force was mediated by only the forelimbs or by a combination of forelimbs and trunk. We used a t-test to test for differences in peak forces and jumping distances between the two types of profiles.

To test whether frogs adjust their arm angle at touchdown, linear regressions were performed on arm angle vs height and vshorizontal velocity.

Typical force profiles (Fig. 1) show the components of the GRF in the three directions during propulsion and landing. The jumping cycle was divided into four phases: (1)propulsion, (2) flight, (3) landing and (4) recovery. Propulsion usually started with a slow increase in GRF F_p,h in the horizontal direction (direction of movement) and F_p,v in the vertical direction, often with a small time lag between F_p,h and F_p,v, with F_p,h preceding F_p,v. Peak forces, however, were reached simultaneously and did not differ significantly from each other (forces averaged over all jumps are

N and

\(\overline{F_{\mathrm{p},\mathrm{h},\mathrm{max}}}=1.32{\pm}0.06\)

N, P=0.494). Propulsion lasted on average 0.187±0.008 s. After the flight phase (⁠

\({\bar{t}}=0.180{\pm}0.010\)

s), during which no GRFs are recorded, the landing phase(⁠

\({\bar{t}}=0.078{\pm}0.007\)

s) was initiated by the arms touching the force plate. Peak landing forces

\(\overline{F_{1,\mathrm{max}}}\)

differed significantly from propulsive forces (P<0.001) and were about three times greater than propulsive forces

\(\overline{F_{1,\mathrm{max}}}\)

(⁠

\(\overline{F_{1,\mathrm{h},\mathrm{max}}}=2.17{\pm}0.12\)

N and

\(\overline{F_{1,\mathrm{v},\mathrm{max}}}=5.02{\pm}0.32\)

N). For landing forces,

\(\overline{F_{1,\mathrm{h},\mathrm{max}}}\)

differed significantly from

\(\overline{F_{1,\mathrm{v},\mathrm{max}}}\)

(P<0.001). The end of the landing phase was defined as the moment when horizontal forces become small and the vertical GRF(F_v) was approximately equal to weight (BW). This moment coincided with the start of the recovery phase(⁠

\({\bar{t}}=0.150{\pm}0.019\)

s), the end of which was determined by high-speed recordings. On average, the duration of landing and recovery comprised 38±4% of the total cycle.

Based on the timing of body contact, we recognized two types of force profiles during landing (Fig. 1). In type I,

were reached while only the arms were in contact with the force plate. The timing of body contact in a type II profile preceded the timing of

\(\overline{F_{1,\mathrm{max}}}\)

⁠. Forces were on average higher in type II profiles, but this did not correspond to a large difference in jumping distance (Table 1).

For the experiment in which GRFs were recorded, jumping distances D_j ranged between 0.32 m and 0.76 m. These distances were distributed equally over the five frogs and within each frog's jumping distance range. For the arm angle experiment, the animals jumped distances between 0.04 m and 0.79 m.

The angle of F_p (θ_p,r) was fairly stereotyped (Fig. 2). At the start of propulsion phase, the resultant GRF (F_r) was directed at an angle of 88° to the horizontal (viewed from the right).θ _p,r decreased exponentially to around 50°, after which it increases again to an angle between 51° and 75°. At this stage, the forces became so small that the angle calculation became unreliable and usually dropped sharply to zero. In seven sequences, the angle profile decreased almost linearly from 88° to a value between 40° and 60°. However, we could not detect any differences between these seven sequences and the others, so we assumed that this slight difference in profile was due to undetermined variation. Typically, the angle profile during landingθ _l,r had a U shape, with a plateau around 120° and the edges increasing steeply. The timing of the angle and magnitude of the GRF in relation to the posture of the animal is exemplified in Fig. 3 (propulsion) and Fig. 4 (landing). As previously stated, the forces during propulsion were stereotyped, while there was considerably more variation in the landing force profile.

A correlation matrix was built using the peak forces of the force components during both propulsion and landing(Table 2). Both within-phase correlations and between-phase correlations were expected between the horizontal and vertical components. Apart from the expected correlations that would occur in a purely elastic, passively behaving structure like a forward bouncing ball, F_p,v,max also correlated significantly with F_l,h,max.

There were no significant difference in impulses of propulsion and landing,although mean impulse I during landing was mathematically smaller in both directions.(⁠

Ns vs

\(\overline{I_{1,\mathrm{h}}}=0.067{\pm}0.020\)

Ns;

\(\overline{I_{\mathrm{p},\mathrm{v}}}=0.154{\pm}0.034\)

Ns vs

\(\overline{I_{1,\mathrm{v}}}=0.143{\pm}0.049\)

Ns).

Power and work profiles for propulsion and landing were calculated for each sequence (Fig. 1). In absolute values, peak power was on average higher during landing in both directions:

W vs

\(\overline{P_{1,\mathrm{h}}}=-1.79{\pm}1.06\)

W and

\(\overline{P_{\mathrm{p},\mathrm{v}}}=0.92{\pm}0.42\)

W vs

\(\overline{P_{1,\mathrm{v}}}=-3.79{\pm}1.82\)

W. The values were negative for the landing phase because the energy was absorbed during this phase. Absolute total work was slightly larger during landing as opposed to the total work of propulsion, due to larger vertical work during landing. On average,

\(\overline{W_{\mathrm{p},\mathrm{v}}}=0.056{\pm}0.021\)

J and

\(\overline{W_{\mathrm{p},\mathrm{v}}}=0.056{\pm}0.021\)

J,while the corresponding values for landing were

\(\overline{W_{1,\mathrm{h}}}=-0.042{\pm}0.015\)

J and

\(\overline{W_{1,\mathrm{v}}}=-0.067{\pm}0.032\)

J. The work delivered by the front limbs was on average

\(\overline{W_{1,\mathrm{h},\mathrm{f}}}=-0.024{\pm}0.011\)

J and

\(\overline{W_{1,\mathrm{v},\mathrm{f}}}=-0.053{\pm}0.030\)

J, which meant that the front limbs delivered 55±20% of the total negative work in the direction of the movement and 78±22% of the total vertical negative work. Muscle mass specific power for propulsion was on average 195±16 W kg^-1 and maxima up to 338 W kg^-1; for landing the muscle mass specific power was on average-2292± 218 W kg^-1 with maxima up to -5000 W kg^-1.

Mean stiffness of the hindlimbs

during propulsion was on average 14±1 N m^-1, while mean front limbs' stiffness

\(\overline{K_{\mathrm{f}}}\)

was on average 100±5 N m^-1.

To determine which variables change with distance, a correlation matrix of all variables was built (Table 3). The F_p,h,max increased with total jumping distance while F_p,v,max remained constant over the range of jumping distances. In contrast, both F_l,h,max and F_l,v,max increased with jumping distance. More positive W_p,h and more negative W_l,h,f were delivered with increasing distance. Stiffness of the arms K_f was constant over the jumping distance range.

Correlations within the phase durations and with distance D_j are shown in Table 4. There was a significant positive correlation between the duration of propulsion and the duration of landing. This was due to a shared negative correlation with jumping distance. Flight duration was positively correlated with distance. Finally, the duration of recovery was only correlated with F_h of landing (not shown).

A lateral view of the path of the model mass at landing is shown in Fig. 5 for three simulations with different arm angles. Fig. 5A, with the arms at an angle θ_f of 125° from the horizontal, is based upon a real sequence in which the data were used to tune the model (see Materials and methods). The spring-dashpot becomes shorter and rotates to a vertical position during landing. To show the considerable effect of the arm angle on the course of the landing, two scenarios are shown where only the arm angle has been changed. An increase of the arm angle to 140° yields a totally different result(Fig. 5B): the arms will rotate in the opposite direction to the 125° scenario and impact velocity on the body will be much greater, especially for the vertical velocity (1.5 m s^-1). Decreasing the arm angle to 110° results in a landing where the arms will rotate over the vertical and will flip backwards, and horizontal velocity will be high at body impact(Fig. 5C).

We also investigated the effect of changing the input parameters, height and horizontal velocity, mimicking the effect of different take-off conditions. The optimal angle at which the arms are best placed clearly depends on these conditions. The optimal angle was defined as the angle for which impact will be smallest, with impact being expressed as the velocity squared (⁠

⁠) at the moment the mass would hit the ground. This was done because landing means dissipating kinetic energy. The corresponding angle at the point of intersection between horizontal

\(v_{\mathrm{h}}^{2}\)

and vertical

\(v_{\mathrm{v}}^{2}\)

was chosen as the optimal angle(Figs 6 and 7). Varying the height of the jump will principally influence the vertical velocity at impact, but horizontal velocity will also be affected to a lesser degree(Fig. 6). The optimal angle decreases linearly with increasing height. The linear equation through these data points can be used to predict the optimal angles over a wider range of heights. We plotted the angles that were observed during additional high-speed recordings of landing and superimposed the predicted equation for optimal angle versus height (Fig. 6). There is considerable scatter in the measured data, butθ _f increased significantly with height (P=0.048),but not with horizontal velocity (P=0.29). The predicted value ofθ _f is an overestimation when height is taken as a crucial factor (P<0.001, d.f.=89). On the other hand, when the optimal angle is calculated from its relationship to horizontal velocity(Fig. 7), the measured data do not differ from the predicted data (P=0.45, d.f.=89).

At the start of propulsion, the GRFs increase slowly during the positioning of the trunk, first in the fore-aft direction while F_vremains at BW, and then more rapidly in both directions when the leg segments start to extend. Peak F_r is attained when hip and knee extensions are almost complete. Thus, frogs make full use of the acceleration distance (Nauwelaerts and Aerts,2003). The GRF decreases rapidly during the roll-off of the digits(the metatarsals and the phalanges). The horizontal force pattern is similar to the force profile of a jumping galago(Günther et al., 1991). The vertical force profile, however, is different because it is more symmetrical. Both species (Rana and Galago) do not use a countermovement while jumping, making the first part of the force profile similar.

Halfway through the propulsion phase, the GRF runs behind the centre of gravity, building up an angular impulse that results in a clockwise angular momentum at take-off (looking at an animal that jumps from left to right). Due to conservation of angular momentum, the body rotates around its centre of gravity during flight. This makes the frog land on its forelimbs instead of its hindlimbs. In Galago, this mid-air rotation is avoided by dorsi-flexion of the tail, which counterbalances the torque of the trunk's inertia (Günther et al.,1991). The fact that there is considerable variation in landing forces probably means that the frogs are not always performing the landing phase optimally, causing the landing phase to be less predictable, although we could detect some general patterns.

Landing and recovery phases together comprise more than one third of the jumping cycle during which the frog is essentially not moving forward. Combined with the one third that is taken up by the flight phase, the saltatory way of locomotion may cause the frog's trajectory to be highly predictable and therefore easy to intercept by a predator. Taking smaller jumps can increase the maneuverability because the animal would be able to change direction during the propulsive phase and would, in theory, spend less time on the same spot during landing and recovery. However, the frogs do decrease landing duration when they jump further, suggesting that they are not working against their limits to damp the kinetic energy from flight and they are motivated to decrease all phase durations. The duration of recovery was not correlated with distance, but was correlated with

during landing, which does suggest that a larger jump will take up more recovery time if the arms are not placed optimally (see further).

Peak landing forces are on average almost three times larger than propulsive forces and landing phase duration is more than two times shorter. Both peak forces are probably overestimations of the maximal forces in the field because muddy surfaces decrease the peak forces through damping. Compliant surfaces are known to decrease the vertical force peaks through energy absorption (Demes et al.,1995; Giatsis et al.,2004). When we normalize the force data by dividing the forces by total BW, we obtain a 2.7 factor for the propulsive forces, similar to the 3.5×BW reported by Hirano and Rome(1984). The normalized landing forces amounted to 9.2×BW. Only studies on primates(Demes et al., 1995; Demes et al., 1991) and birds(Bonser and Rayner, 1996) are available for comparison of the landing forces. Although take-off forces for birds are in a range similar to the frogs' propulsive forces, their landing forces are much smaller (1.8×BW). Birds are capable of altering their landing mechanics with their wings (Green and Cheng, 1998). Primates, on the other hand, attain much higher forces during take-off (9.6-10.3×BW), but have lower landing forces(6.7-8.4×BW: Demes et al.,1999). It has been stated that primates change their posture during flight to improve aerodynamic performance(Demes et al., 1991) and possibly their landing conditions. Many primates and birds use the same limbs for take-off and landing, making the potential acceleration and deceleration distance the same. In frogs, the forelimbs are much shorter than the hindlimbs. This means that the potential acceleration and deceleration distance is different, causing the landing forces inevitably to be larger if propulsion and landing have the similar duration.

Since the stiffness of the arms K_f stays constant over the full jumping range, it is possible that this is a limiting factor in the ability of the forelimbs to work as dampers. Human legs and other mammalian locomotor limbs also maintain a constant stiffness, regardless of speed during normal running (Glasheen and McMahon,1995; Farley et al.,1993), but the stiffness of the leg spring doubles when humans hop in place at different frequencies (Farley et al., 1991). Because of the constant stiffness regardless of speed, the forelimbs of a frog can be regarded as locomotor limbs, unlike the human arm that operated differently by increasing its stiffness with speed(Glasheen and McMahon, 1995). Limb stiffness depends on the torsional stiffnesses of the joints and the geometry of the musculoskeletal system(Farley et al., 1998; Ferris and Farley, 1997). It is also affected by muscle activation (Morasso et al., 1999) and GRF alignment(Farley et al., 1998). In our definition of stiffness, we used the resultant GRF. This force does increase with jumping distance, so the arms compress more with distance. This is easy to achieve by bending the elbows outwards. There is, however, a limit to how much they can bend before the frog's trunk hits the ground. By controlling arm stiffness, frogs can absorb the first impact peak with their forelimbs. However, this is limited by forelimb length and, during long jumps, the highest peak will be absorbed by a combination of forelimbs and trunk. In addition to elbow flexion, the pectoral girdle is probably used as a damper. In frogs, the scapula is divided into two parts. The two scapula parts articulate through a joint that could contribute to damping during landing. In addition, the upper part (suprascapula) is mainly cartilaginous(Shearman, 2005). Ranids have a firmisternal pectoral girdle in which the epicoracoids are fused midventrally. It has been hypothesized that a firmisternal girdle is less useful for landing than an arciferal (overlapping halves) girdle because the cartilages rotates in the horizontal during landing in the latter girdle,allowing the animal to decelerate over a greater distance(Emerson, 1988). However, the coracoid of the firmisternal girdle was not found to be loaded anywhere close to the breaking strength of the bone and seems to perform as well as an arciferal girdle (Emerson,1983). Also, no correlation was ever found between pectoral morphology and jumping ability in frogs(Emerson, 1984). A thorough comparative functional morphological study during landing in frogs seems to be necessary to further elucidate the function of the pectoral girdle.

Maximal power generated by the hind limb muscles exceeds 300 W kg^-1 muscle, which is near the theoretical maximum, meaning that all the muscle fibers of all hind limb muscles should be recruited and contributing directly to the mechanical power, unless some form of power amplifier is present. It has been hypothesized that these supra-maximal powers result from the rapid release of strain energy from elastic elements(Marsh, 1999; Roberts and Marsh, 2003). Such power amplifiers have been shown in other animals(Aerts, 1998; Bennet-Clark, 1975) and usually rely on a preloading of elastic components by muscular action(Roberts and Marsh, 2003),causing muscle tension to built up followed by a rapid release and a sudden increase in velocity and therefore in power. The maximal powers generated during landing are even more extreme. Since the muscles work eccentrically during braking, the muscle specific power output is expected to be 1.5-2 times higher (Rijkelijkhuizen et al.,2003; James et al.,1996) than the power output during concentric work. Our data far exceed this maximum. Rotational stiffness was not considered. The protactor muscles in the forelimbs are very small and we therefore assume that most of the damping was done passively. The high power output, together with the fact that stiffness does not increase with jumping distance, seems to confirm the presence of a passive viscous damper that can either be the pectoral girdle(see earlier) or the frog's body through deformation.

If the take-off and landing conditions are the same, as for a bouncing elastic ball, we would expect there to be correlations, due to ballistics,between the force components of the same direction because the angle of the GRF during take-off would be complementary to that for landing. Although these correlations are observed in jumping frogs, the unexpected, additional correlation between the Z-force of propulsion and the Y-force of landing indicates that this landing angle of the GRF is actively changed. The angular momentum and the difference in stiffness between the front- and hindlimbs are the most plausible causes of this angle change. The role of the front limbs during landing proved to be considerable. Two types of jumps were observed, depending on whether the impact peak was mediated by the arms or by the body. Although the peak forces were on average larger when the body mediated the forces, no clear arm function limits could be detected in peak force, power or work. A mechanical limit is highly likely,but could be hard to detect because of its interaction with the positioning of the arms. The spring-dashpot model shows a spectacular effect of arm angle on the course of the landing phase (Fig. 5). In the simulation that was based on a real sequence, landing with an arm angle of 125° caused a rotation of the arm during landing to a vertical posture at body contact, purely due to the spring-dashpot action. Positioning the arms at an angle of 110° or 140° at touchdown results in a higher impact and a body posture with the arms flapped backwards and forwards respectively, thus hampering recovery. Frogs increase the arm angle with the height of the jump. When we define the optimal angle as that for which the kinetic energy at the moment of body contact is minimal in horizontal and vertical directions, the optimal arm angle is found to increase with horizontal velocity and to decrease with height. The measured arm angle of the landing frogs follow the predictions of the optimal arm angle calculated from the horizontal velocity. Avoiding remainders of horizontal kinetic energy at body contact, that would have to be cancelled by frictional forces, seems to be of major importance.

List of symbols and abbreviations
 
• a

  acceleration of the COM

 
• a_v

  vertical acceleration of the COM

 
• BW

  body weight

 
• c

  damping coefficient

 
• COM

  center of mass

 
• d

  displacement of the COM

 
• d_r

  resultant displacement of the COM

 
• D_j

  jumping distance

 
• F

  force

 
• F_f

  GRF on the forelimbs

 
• F_h

  horizontal GRF

 
• F_l

  GRF during landing

 
• F_max

  peak GRF

 
• F_p

  GRF during propulsion

 
• F_r

  resultant GRF

 
• F_v

  vertical GRF

 
• F̄

  GRF averaged over all sequences [plus all combinations of subscripts:phase (p, propulsion/l, landing), direction (h, horizontal/v, vertical/r,resultant), max.]

 
• GRF

  ground reaction force

 
• g

  gravitational acceleration

 
• h

  height of the jump

 
• I

  impulse

 
• \(\overline{I_{\mathrm{p},\mathrm{h}}}\)

  horizontal impulse during propulsion averaged over all sequences

 
• \(\overline{I_{1,\mathrm{h}}}\)

  horizontal impulse during landing averaged over all sequences

 
• \(\overline{I_{\mathrm{p},\mathrm{v}}}\)

  vertical impulse during propulsion averaged over all sequences

 
• \(\overline{I_{1,\mathrm{v}}}\)

  vertical impulse during landing averaged over all sequences

 
• K

  stiffness

 
• K_f

  stiffness of forelimbs

 
• K_h

  stiffness of hindlimbs

 
• K̄ _f

  stiffness of forelimbs averaged over all sequences

 
• K̄ _h

  stiffness of hindlimbs averaged over all sequences

 
• M_b

  body mass

 
• M_m,f

  forelimb muscle mass

 
• M_m,h

  hindlimb muscle mass

 
• P

  power

 
• \(\overline{K_{\mathrm{f}}}\)

  horizontal peak power during propulsion averaged over all sequences

 
• \(\overline{K_{\mathrm{h}}}\)

  horizontal peak power during landing averaged over all sequences

 
• \(\overline{P_{\mathrm{p},\mathrm{h}}}\)

  vertical peak power during propulsion averaged over all sequences

 
• \(\overline{P_{1,\mathrm{h}}}\)

  vertical peak power during landing averaged over all sequences

 
• t

  duration

 
• T

  temperature

 
• v

  velocity of the COM

 
• v_v

  vertical velocity of the COM

 
• v_h

  horizontal velocity of the COM

 
• W

  work

 
• W_f

  work by the forelimbs

 
• W_p,h

  horizontal work during propulsion

 
• W_l,h,f

  horizontal work during landing by the forelimbs

 
• W_p,v

  vertical work during propulsion

 
• W_l,v

  vertical work during landing

 
• \(\overline{P_{\mathrm{p},\mathrm{v}}}\)

  horizontal work during propulsion averaged over all sequences

 
• \(\overline{P_{1,\mathrm{v}}}\)

  horizontal work during landing by the forelimbs averaged over all sequences

 
• θf

  arm angle at touchdown

 
• θr

  angle of F_r

 
• θp,r

  angle of F_r during propulsion

 
• θl,r

  angle of F_r during landing

The authors would like to thank Joe Carragher and Rick Essner for their comments on the manuscript, Ilse Arnauts for conducting a pilot study and Jan Scholliers for taking care of the animals. Also thanks to Cheryl Wilga for giving me the time to finish up this project. This study was supported by grants of IWT-Vlaanderen to S.N and GOA-BOF to P.A.