The mechanics of azimuth control in jumping by froghopper insects

Jumping is a ubiquitous behaviour used by many animals for escape, predation, rapid locomotion and, in insects, take-off into flight. Some jumps may simply be the fastest way to escape from hostile surroundings when the overriding demand is to move as rapidly as possible away from a threatening stimulus. On other occasions jumps may need to be directed toward a particular target in the environment as when, for example, a grasshopper jumps from one grass stem to another. Then the accuracy of the movement will depend on the precise control of three variables; the velocity of the movement and its direction in both the azimuth and elevation planes. An inaccurate jump might cause a collision with a plant leaf or stem that would prevent the establishment of flight, or, at worst, cause the insect to ricochet at high speed toward a hungry predator. Furthermore, jumping movements in most insects are so fast, taking less than 1 ms in froghopper insects (Burrows, 2003; Burrows, 2006a) to a few tens of milliseconds in grasshoppers (Bennet-Clark, 1975; Brown, 1967), that there is no time for sensory feedback to make any adjustments. The challenge of jumping is therefore to produce a movement that is both fast and when necessary accurate, without the assistance of sensory feedback.

In locusts, jumping is constrained by the orientation of the hind legs on either side of the body which move in separate planes to each other, and by the structure of the joints and muscles. A jump is powered by muscles in the enlarged hind femora that move the femoro-tibial joints which have their own specialised mechanics for increasing the lever ratios of the flexor and extensor muscles depending on joint position, and for locking the tibiae when they are fully flexed (Heitler, 1974). When jumping to a target, a locust first peers by moving its head from side to side to determine the desired trajectory (Eriksson, 1980; Sobel, 1990). The azimuth direction of the jump is then set by adjusting the orientation of the body to the target by movements of the forelegs (Santer et al., 2005). Elevation is set by rotating the coxae of the hind legs so that the force generated by the extension of the hind tibiae is exerted along a line connecting the distal end of the tibia with the proximal end of the femur (Sutton and Burrows, 2008). The velocity is determined by the force exerted by the femoral muscles that move the hind tibiae, which are controlled by a motor pattern of three phases (Burrows, 1995; Heitler and Burrows, 1977). Normally the two hind legs extend at the same time or within a few milliseconds of each other. The mechanical arrangement of the hind legs means, however, that directed jumps can still be produced even if one leg extends some time before the other, or if only one hind leg is used (Bennet-Clark, 1975; Santer et al., 2005; Sutton and Burrows, 2008). This control strategy may be used by other insects with the same arrangement of their hind legs, such as stick insects (Burrows, 2008), bush crickets [Orthoptera, Tettigoniidae (Burrows and Morris, 2003)], and flea beetles [Coleoptera, Alticinae (Brackenbury and Wang, 1995)].

By contrast, the fastest of the jumping insects, froghoppers (Hemiptera, Auchenorrhyncha, Cercopidae), have their power-producing hind legs slung underneath their body so that both move in the same plane (Burrows, 2003; Burrows, 2006a; Burrows, 2006b). In further contrast, the movements of the hind legs are powered by huge muscles in the thorax that depress the trochantera. These different mechanisms nevertheless allow jumps to be directed toward a target in both elevation and azimuth planes (Brackenbury, 1996). Elevation is now controlled by the front and middle legs that set the attitude of the body relative to the ground before take-off (Burrows, 2006a). Velocity is controlled by a three-phase motor pattern (Burrows, 2007c) that delivers power in a catapult mechanism through the hind legs.

An unresolved question for froghopper jumping is how the azimuth is controlled. We have therefore analysed how the forces in the coxo-trochanteral joints are likely to be delivered to the ground by the mechanics of the hind legs. The mechanics suggests two possible mechanisms of azimuth control: one in which it is determined by the initial placement of the hind tibiae, and a second in which it is controlled by the amount of delay between the propulsive movements of the two hind legs. Predictions from this analysis were then tested using high-speed imaging of the natural jumping performance before and after experimental interventions. We show that the azimuth of the jump is controlled by initial placement of the metathoracic tibiae, which thus determine the vector of the thrust that propels jumping.

To analyse leg positions and the subsequent jump trajectories, adult froghoppers, Philaenus spumarius (Linnaeus 1758), were caught near Cambridge, UK. They belong to the order Hemiptera, sub-order Auchenorrhyncha and to the family Cercopidae. To analyse the detailed movements of the hind legs during natural jumping, individuals were allowed to jump from a glass slide coated with a thin layer of transparent Sylgard (Dow Corning, Midland, MI, USA) to provide a good purchase for the legs. When jumping on an uncoated glass slide, both hind legs slipped so that they were unable to propel an effective jump. Images of jumping, which occurred spontaneously or was elicited by a light touch with a fine paintbrush, were captured from a ventral perspective at a rate of 5000 frames s⁻¹ and an exposure time of 0.04 ms, with a Photron Fastcam 1024 PCI camera [Photron (Europe) Ltd, Marlow, Bucks., UK]. The positions of the hind legs, angular movements, and jump trajectories were measured in 23 natural jumps by 23 froghoppers. The azimuth of each tibia was measured by using the femoro-tibial joint and the distal end of the tibia as landmarks. The orientation of the body was measured by using the anterior tip of the head and the posterior tip of the abdomen as landmarks. The jump trajectory was measured by passing a line through the centre of mass in each successive image. The angular velocity was measured by recording the yaw of the body in each image, and taking the mean angular velocity over five successive images.

To determine whether there were any differences in the timing of the left and right hind legs during natural jumping, we analysed jumps from three species of froghopper that all share the same mechanism of jumping (Burrows, 2006a); Philaenus spumarius, Aphrophora alni (Fallén 1805) and Cercopis vulnerata (Rossi 1807). The latter two species were also caught near Cambridge, UK. The jumps occurred from either a substrate of high density foam, and were viewed from the side, or from transparent Sylgard, and were viewed ventrally. The analysis was then extended to 157 restrained jumps by 15, Aphrophora alni. Each froghopper was mounted ventral side up in Plasticine so that the hind legs were able to move freely. The sequence of rapid movements of both hind legs in these restrained froghoppers closely resemble those used in natural jumping (Burrows, 2006b). The movements of the proximal joints of the hind legs were viewed ventrally by mounting the Photron camera on a Leica MZ16 stereo microscope. Images were captured at rates of 30,000 frames s⁻¹ and with an exposure time of 0.02 ms. This second and larger species of froghopper was used here because it was less prone to desiccation under the bright lights needed at these very high frame rates. Images from all experiments were fed directly to a computer and analysed with CorelDraw 11 (Ottawa, Canada) to determine the kinematics.

To estimate the effect of only one hind leg propelling a jump, we captured images of 13 jumps by 13 Philaenus. In seven of them, the left hind tibia was removed at the femoro-tibial joint, and in the other six the right hind tibia was removed. Jump trajectories in the azimuth plane that were parallel to the midline of the body were assigned a value of zero; those toward the side of the intact hind leg were assigned positive values, and those toward the side with the missing tibia were assigned negative values. Take-off time was designated as 0 ms. Data are given as mean ± standard deviation (s.d.).

To estimate the effects on the final jump trajectory of differences in the timing of the movements of the left and right hind legs, and independently, their different placements, we built a kinetic model (details in  Appendix) that approximated a froghopper (parameters based on Aphrophora alni) as linkages each powered by a spring (Fig. 1). The model was built in Mathematica, and the equations were solved using a Runge–Kutta integrator with a timestep of 5 μs.

The ground reaction force generated by one hind leg is therefore in a direction parallel to the longitudinal axis of the tibia of that leg. For example, a jump powered solely by depression of the left trochanter should produce a ground reaction force parallel to the left tibia. In natural jumping, if both tibiae were placed symmetrically and both coxo-trochanteral joints were depressed synchronously, then the lateral components of the forces of the two tibiae will be equal and opposite, producing a trajectory with no azimuth. If, however, the froghopper were to place its tibiae asymmetrically relative to the midline of its body, the body trajectory should be predicted by the average of the angles subtended by each tibia relative to the midline (θ_Rti, θ_Lti; Fig. 1B).

Because the force vector from each hind leg acts from the point of ground contact and is parallel to that leg, the ground reaction force cannot generate a vector that goes through the centre of mass of the body: one hind leg acting on its own should cause the body to spin. The only way for a hind leg not to generate a torque would be for its tibia to be pointed at the centre of mass, a physiologically difficult position for the froghopper to adopt. The left hind leg thus generates a clockwise torque on the body and the right leg an anti-clockwise torque. If both tibiae adopt symmetrical angles relative to the midline and both coxo-trochanteral joints are depressed synchronously, then they would generate equal and opposite torques on the body with the consequence that the body should not spin once airborne. If, however, the two coxo-trochanteral joints do not depress at the same time, the body should rotate after take-off.

Adjusting the angles adopted by the femoro-tibial joints and hence the placement of the hind tarsi on the ground before a jump would therefore be a mechanism that a froghopper could use to control the azimuth of a jump to the left or right of the body midline. Alternatively, it could also control azimuth by extending the hind legs at different times during a jump.

This analysis makes four testable predictions. (1) Symmetric placement of the hind tibiae and synchronous extension of both hind coxo-trochanteral joints should generate jump trajectories along the longitudinal axis of the body accompanied by only small amounts of body spin. (2) The azimuth of a jump should be predicted by the initial angles of the hind tibiae relative to the midline of the body; a jump should be toward the side to which the tibiae point. (3) The azimuth of a jump in which only one hind leg participates should be parallel to the tibia of that leg, and be accompanied by spin of the body in the yaw plane. A correlate of this prediction is that a jump powered by the left hind leg should rotate clockwise while a jump powered by the right hind leg should rotate anti-clockwise. (4) Adjusting the times at which the two hind legs depress could be an alternative or additional mechanism for determining the azimuth direction of a jump.

To test these predictions we therefore analysed the kinematics of natural jumping as revealed by high speed imaging.

Prior to a jump, does a froghopper adopt a stereotyped posture, or does it adjust its posture and position of its hind legs? To determine the initial position of the hind legs, we analysed high-speed videos of natural jumps by the froghopper, Philaenus spumarius. Before a natural jump, a protrusion on a dorsal femur engaged with a protrusion of a coxa effectively locking the femur relative to the body (Fig. 2) (Burrows, 2006b). The result was that, prior to a jump, a froghopper could not change the angle between a femur and the longitudinal axis of the body (θ_f in Fig. 1B) and the angle between a trochanter and a femur (θ_tr in Fig. 1B). In the jumps analysed here, θ_f was 28±11.2 deg., but θ_tr was not measurable from the data here, but is about 150 deg. (Burrows, 2006b). The femoro-tibial joint is not locked, allowing a froghopper to adjust the angle of the tibiae and hence the placement of the tarsi on the ground either symmetrically or asymmetrically relative to the longitudinal midline (Fig. 3).

For a jump in which a froghopper initially placed its hind tibiae symmetrically about the longitudinal axis of the body and thus the tarsi symmetrically on the ground, the following sequence of movements occurred. As the trochantera and femora began to depress at the start of a jump, the effect was to push each femoro-tibial joint more laterally (Fig. 2). The tarsi remained at the same positions on the ground so that the femoro-tibial joints were extended and the front of the body was accelerated and raised from the ground. As the depression of the trochantera continued, the positions of both femoro-tibial joints now moved medially and the femoro-tibial angles were further extended. This meant that the line of action of each tibia changed continuously as the hind legs accelerated the body. At take-off both tibiae were parallel to the long axis of the body and once airborne the body was accelerated with 22 Hz of rotation in the yaw plane (Fig. 2). The implication is that the extension of the tibiae resulted from the forces applied to depress the trochanter, rather than from a contraction by the extensor tibiae muscle.

Before a jump, the net azimuth angles adopted by the tibiae of the left and right hind legs ranged from −30 to 40° (Fig. 3). The femoro-tibial angles adopted by a left and a right hind leg could be varied independently and were only weakly correlated with each other (R²=0.22).

The azimuth trajectory of most jumps varied over a window of 60 deg., some 30 deg. to the left and right of either side of the longitudinal axis of the body (Fig. 4). The final azimuth of a jump was correlated with the mean initial azimuth of the two hind tibiae (R²=0.68, slope=1.29, y-axis intercept=−0.05 deg.). If the initial mean azimuth of the tibiae was positive (they pointed to the right) then the jump azimuth was to the right, and if the initial mean azimuth of the tibiae was negative (they pointed to the left), then the jump azimuth was to the left. This indicates that trajectory is largely explained by the average of the two starting, tibial azimuths.

To test how closely coupled the movements of the two hind legs were during natural jumping, high speed images were analysed. The jumps occurred from either a substrate of high density foam, and were viewed from the side, or from transparent Sylgard, and were viewed ventrally. We analysed jumps from three species of froghopper all of which have the same jumping mechanisms (Burrows, 2006a); 48 jumps from 30 Philaenus spumarius, 41 jumps from 8 Aphrophora alni, and 35 jumps from 7 Cercopis vulnerata. This includes 33 jumps from a previous investigation by Burrows (Burrows, 2006a) in which the hind legs on both sides could be clearly seen, and which were conducted using the same methods as used here. In 123 of these 124 jumps no difference could be detected in the start of the depression movements of the hind legs.

To test whether any differences in the timing of the two hind legs may have been masked in natural jumping by having to lift the weight of the body, jumping movements were induced in Aphrophora alni restrained on their backs in Plasticine, but with the hind legs free to move. Both hind legs would then kick rapidly against the resistance of only the air in a sequence of movements that closely resembled those used in natural jumping. Images of these restrained jumps were captured at 30,000 frames s⁻¹. In 157 jumps by 15 Aphrophora, the mean difference between the start of trochanteral depression in the two hind legs was 0.032±0.022 ms (mode and median 0.033 ms or 1 frame), and was never greater than 0.067 ms (2 frames).

In the absence of an experimental method that could manipulate the timing of the movements of the two hind legs to these levels of resolution, we used a model ( Appendix) to quantify the effects of leg asynchrony on the final jump trajectory. The values modelled extended from the small measured values (Fig. 5A, vertical grey bar) in restrained jumps to the total acceleration time (2 ms) of a jump propelled by one hind leg. A jump propelled by one hind leg has half the force of a jump propelled by both hind legs, so that it takes twice as long for the hind leg to depress fully. The modelling showed that the largest measured difference in timing (67 μs) would produce an azimuth offset from the longitudinal midline of only 0.4 deg. (Fig. 5A). With an asynchrony of 2 ms, the jump azimuth was increased to 14 deg. By contrast, using asymmetrical tibial positions within the range observed in natural jumping and making the tibiae depress symmetrically, the model could produce the full range of observed azimuths (up to 25 deg.; Fig. 4, Fig. 5A).

The angular velocity of simulated jumps increased as timing differences were increased, reaching 77 Hz at 2 ms of asynchrony (Fig. 5B). By contrast, changing tibial positions while both hind legs moved at the same time resulted in a spin of no more than 5 Hz, or 6% of that experienced with a 2 ms timing difference (Fig. 5B).

The simulations also show that controlling azimuth with asynchronous movements of the hind legs is energetically inefficient. The kinetic energy (1/2mV²) dropped from 88 μJ, when the hind leg movements were synchronous, to 29 μJ (a 67% decrease) when one leg was depressed 2 ms after the other (Fig. 5C). By contrast, if azimuth was changed by adjusting the initial position of the hind tibiae then the kinetic energy remained constant. This is because moving the hind legs asynchronously causes two losses of energy. First, asynchronous jumps have less kinetic energy because the leg moving last will not initially be in contact with the ground and thus does not transmit its force directly to the ground. Second, asynchrony generates body spin and a loss to rotation of as much as 17% of the energy. By contrast, jumps powered by synchronous movements of the hind legs lose only about 1% of energy to spin. The projected take-off velocity of a jump also fell from 2.5 m s⁻¹ when the hind legs moved synchronously to 1.4 m s⁻¹ when they moved asynchronously. The fall resulted from a loss of energy due to spin and from a reduction in the total energy available.

In only one of the 124 natural jumps analysed was one hind leg seen to depress before the other (Fig. 6), emphasising the rarity of such events. In this jump the left hind leg moved 1 ms before the right hind leg which caused the body to rotate before the right leg could make any contribution. As predicted from the modelling above, the movement of the left hind leg caused a fast clockwise rotation (78 Hz). Furthermore, the time to take-off was 2.2 ms or about twice that taken when a jump was propelled by two hind legs depressing synchronously.

To obtain more experimental data on how the azimuth is controlled in jumps propelled by one hind leg, we removed the left hind tibia in one group of seven Philaenus and the right tibia in another group of six, and then correlated the initial position of the remaining tibia with the resulting azimuth at take-off (Fig. 7, Fig. 8A). Jumps powered by one hind leg generated trajectories that were linearly related (slope=1.15, R²=0.66) to the initial azimuth of the sole remaining tibia (Fig. 8A). In 8 of the 13 jumps analysed the trajectory was toward the side of the functional hind leg. The trajectory of a jump powered by one hind leg was thus largely explained by the starting azimuth of the tibia. All 13 jumps powered by one hind leg were accompanied by rotation rates of 161±55.7 Hz (N=13) compared with rotation rates of 28±23.6 Hz (N=23) when jumps were propelled by both hind legs (Fig. 8B). For jumps powered only by the left hind leg the spin was clockwise and for jumps powered only by the right hind leg it was anti-clockwise.

Froghoppers control the azimuth of their jumps by altering the angles of the left and right hind tibiae relative to the midline of their body and then synchronously and rapidly depressing both hind trochantera. The mean angle of the two hind tibiae determines the azimuth of a jump which can deviate from the longitudinal axis of the body by about ±30 deg. Images of natural and experimentally manipulated jumps show that the azimuth is strongly correlated with the initial position of the hind tibiae, and kinetic modelling has shown how this happens and why it is an energetically favourable method to control azimuth.

To jump, a froghopper uses huge muscles in the thorax to generate large torques on both the thorax and the trochantera. The torque is not, however, transmitted to the tibiae because the muscles moving the femoro-tibial joints are small. A consequence of this arrangement is that the ground reaction force generated by one hind leg is parallel to the longitudinal axis of its tibia. This means that if both hind tibiae are placed symmetrically and both coxo-trochanteral joints depress synchronously, then the lateral components of the forces generated by the left and right hind legs will be equal and opposite, and the azimuth of a jump will be parallel to the long axis of the body. If, however, the hind tibiae are placed asymmetrically, the trajectory of a jump is determined by the average of the angles subtended by each tibia relative to the midline. During natural jumping the mean, initial tibial angles of both hind legs and the azimuth of a resulting jump are strongly correlated. It follows that in froghoppers with only one hind leg, the azimuth of a jump will be determined by the angle of the remaining tibia. An undesirable consequence of using just one hind leg is that the body now spins on average six times faster than when powered by both hind legs, dissipating enormous amounts of energy and resulting in less effective jumps. This may explain why froghoppers with just one hind leg are rarely found in nature.

Modelling indicates that jump azimuth could theoretically be controlled by extending the hind legs asynchronously, but analysis of froghoppers executing restrained jumps reveals a remarkable synchrony in the movements of both hind legs (mean timing difference of only 32 μs). Furthermore, in only one of 124 natural jumps analysed was the timing difference visible. This means that if the tibia are placed symmetrically and both legs are synchronously depressed, then the azimuth of the jump will be parallel to the longitudinal axis of the body. There are severe limitations if the froghopper attempts to control azimuth by adjusting the timing differences between the movements of its hind legs. First, the largest (67 μs) difference in timing observed in natural jumping would only contribute some 0.4 deg. to azimuth control. Second, modelling shows that increasing the time differences up to the 1 ms of the acceleration period for a natural jump, can achieve greater azimuth control but only at the expense of progressively increasing body spin and energy losses that would substantially degrade take-off velocity. As timing differences increase past that point, the jump essentially becomes powered by just one hind leg so that the trajectory is determined by the tibial angle of that leg and high rates of body spin are unavoidable.

These results may explain why froghoppers so rarely depress their two legs asynchronously when jumping. They may also explain the requirement for controlling the power producing muscles on the two sides of the body with closely synchronised sequences of motor spikes (Bräunig and Burrows, 2008; Burrows, 2007c). Synchrony of the motor spikes to the left and right muscles is likely to produce closely matching levels of force in each and increase the probability that both hind legs will move at the same time. Controlling azimuth by adjusting the timing of the movements of the two hind legs to within a fraction of a millisecond would, however, still be a challenging task for neural control. Instead, changing the initial position of the tibiae is both a simpler and more energetically efficient mechanism.

The azimuth of a jump by a froghopper is strongly correlated with the initial azimuth of the tibiae; for natural jumps powered by both hind legs the slope of the relationship was 1.29 and for experimental jumps powered by a single hind leg it was 1.15. If the force vector was perfectly parallel to the tibia, the slope should be 1 for both data sets. What is the cause of the small discrepancies? The calculations were simplified by making three assumptions. First, we assumed that the angle of a tibia relative to the midline remains constant during a jump. High speed images of the hind leg movements clearly show, however, that the depression of the trochantera and femora (resulting from the release of stored energy in the pleural arches) cause the azimuth of the tibiae to change continuously during the acceleration phase of a jump (Fig. 2) (Burrows, 2006a). As the coxo-trochanteral joints depress, the femoro-tibial joints are forced laterally, thus increasing the angle of the tibiae relative to the longitudinal axis of the body. Continuing depression then causes the femoro-tibial joints to move medially, decreasing the angle. These changing orientations of the tibiae mean that their force vectors also change progressively. Second, we assumed that the magnitude of force vectors of the tibiae were constant throughout a jump. In fact they will change as the springs of the pleural arches progressively recoil. Third, no torque was assumed to be transmitted through the femoro-tibial joints, but some may be. The calculations made here take no account of these three factors and yet initial positions of the tibiae alone predict linear intercepts of 0 deg. when plotted against jump azimuth. The experimental data show intercepts of 0.05 deg. and 2.16 deg. for two-legged and one-legged jumps, respectively, that are thus close to the expected value given the simplifying assumptions made. A more accurate predictor could be developed from the diagrams in Fig. 1 but this would require, for only a very small gain, detailed, real-time measurements of the stiffness and forces delivered by each pleural arch, of changes in moment arms of the trochantera and moment of inertia of the froghopper, and ground forces, or three-dimensional measurements of the kinematics to determine kinetic energy (both in translation and in rotation).

Brackenbury (Brackenbury, 1996) found that froghoppers and leafhoppers could jump to targets with a resolution of 7 deg. × 7 deg., and suggested that the constraints on the targeting were due to both visual and biomechanical factors. In terms of jump azimuth, however, we find no evidence to support a biomechanical limit on the resolution of a jump because trajectories of a jump are controlled by adjusting the initial positions of the hind tibiae. This suggests that the limitations are a consequence of the sensory system which are discussed by Brackenbury (Brackenbury, 1996).

The strategy proposed here for controlling azimuth should apply to insects that have: (1) jumps propelled by legs that move in the same plane; (2) jumps that are powered by muscles that move the coxo-trochanteral joint; (3) leg segments that remain approximately rigid during a jump and do not bend.

Leafhoppers (Burrows, 2007a; Burrows, 2007b; Burrows and Sutton, 2008) and possibly members of some of the other families in the Auchenorrhyncha meet these criteria and could thus be tested experimentally to see if their control of azimuth is the same. In Hackeriella (Hemiptera, Coleorrhyncha) the hind legs that propel jumping are also slung beneath the body but can propel jumps with greater asynchrony than that seen in either frog- or leafhoppers (Burrows et al., 2007). Fleas propel jumping with muscles that act at the coxo-trochanteral joints of the hind legs (Bennet-Clark and Lucey, 1967) like a froghopper, but have their legs placed at the sides of their bodies like a locust. It is not known how they control the direction of their jumping.

The strategies for jumping in froghoppers and orthopterans such as grasshoppers and locusts appear to be distinct. First, locusts generate the forces required for jumping by contractions of muscles that move the femoro-tibial joints of the hind legs. Second, the hind legs move in separate planes to each other on either side of the body. This arrangement means that the force vectors that propel a locust jump remain in a constant direction that is determined at the joint between the coxa and the fused trochanter and femur. Rotations of as much as 90 deg. at this joint set the angle of elevation for a jump (Sutton and Burrows, 2008), in contrast to a froghopper where elevation is set by movements of the front and middle legs. Consequently, a locust, which experiences neither changes in the direction of force vectors as its hind legs move, nor an intrinsic torque on the body caused by those vectors, does not have to balance the forces between the two hind legs. It therefore has no difficulty jumping with just one hind leg, or when both hind legs extend with some delay between them (Bennet-Clark, 1975; Santer et al., 2005; Sutton and Burrows, 2008). This is in contrast to the froghopper, which experiences changing force directions and intrinsic torques on the body as it hind legs depress synchronously. Azimuth in froghoppers is controlled by the positions of the hind tibiae, whereas in grasshoppers it is controlled by the front legs shifting the orientation of the whole body (Santer et al., 2005).

Despite the differences in strategies, the jumping by both froghoppers and grasshoppers illustrate common underlying principles. In both, control of three key parameters, velocity, elevation and azimuth, are compartmentalised. Velocity is controlled by the force generated at the coxo-trochanteral joints of froghoppers (Burrows, 2007c; Burrows et al., 2008) and at the femoro-tibial joints of grasshoppers (Bennet-Clark, 1975). Elevation is controlled by the front and middle legs of froghoppers (Burrows, 2006a) and at the hind coxo-femoral joints in grasshoppers (Sutton and Burrows, 2008). Finally, azimuth is controlled at the femoro-tibial joints in froghoppers and by the front legs in grasshoppers (Santer et al., 2005). The separation of these functions, albeit in different joints or legs, should allow a simpler neural control with each parameter being set independently once a target has been identified.

Easily directed trajectories appear to be common in high speed motions. The controllability of the jumping movements described have parallels with the tongue strike of a toad (Mallett et al., 2001), the strike of a squid tentacle (Van Leeuwen and Kier, 1997) and suction feeding (Van Wassenbergh and Aerts, 2009; Wainwright et al., 2007) in which there is a simple transformation from the initial position to the final trajectory. Ease of control of high speed motions allows a nervous system to prepare, in real time, controlled motions quickly and easily in response to visual or tactile stimuli (Card and Dickinson, 2008).

Further analyses of the strikes of mantid shrimps (Burrows, 1969; Patek and Caldwell, 2005), praying mantids (Gray and Mill, 1983) and snakes (Smith et al., 2002) would usefully reveal whether they are also mechanically configured to produce easily controlled trajectories. High-speed motions represent a clear example where the biomechanics and the neural control work together to generate successful behaviours both for predators and for prey (Chiel and Beer, 1997; Dickinson et al., 2000).

The equation of motion for the froghopper model is based on the same assumptions that were used for modelling the control of elevation in jumping of locusts (Sutton and Burrows, 2008). The following assumptions were made about a froghopper. The hind legs are small relative to the size of the body (Burrows, 2006a; Burrows, 2006b), so that the femora and tibiae were assumed to have no mass. The leg segments do not appear to bend during a jump (Burrows, 2006b), allowing the coxa, femur and tibia to be modelled as rigid bodies. The trochanter moves with the femur during a jump (Burrows, 2006b), allowing it to be modelled as a part of the femur. The body of the froghopper was assumed to be a uniform disk (model parameters in Table A1).

Based on these assumptions and the data from free-body diagrams of a froghopper (Fig. 1) nine equations were derived to calculate the accelerations (, Ÿ, ) in terms of joint torques (τ_L, τ_R):

The forces are in terms of lateral (l) and anterior (a) components, and the femoral and tibial angles are defined relative to the midline of the froghopper.

Length_s was determined by the angle of the femur relative to the midline of the froghopper. To generate the torques (τ_L, τ_R) generated by each spring, the force was multiplied by the perpendicular moment arm of the spring about the coxo-trochanteral (plus femur) joint.

For each timestep, these 20 equations were solved and integrated using Mathematica 5.0 with a RK4 integrator, and a timestep of 5 μs.

G.P.S. was supported by the Marshall Sherfield Commission and the Human Frontiers Science Program (HFSP). We would like to thank Jo Riley for her help during the course of this work and our Cambridge colleagues for many helpful discussions and their constructive comments on this manuscript.

 
• a

  anterior

 
• b

  body

 
• c

  coxa

 
• f

  femur

 
• G

  ground reaction force

 
• I

  inertia

 
• l

  lateral

 
• Length_f, Length_ti, Length_m,Length_c

  lengths of the femur, tibia, moment arm of the trochanteral depressor muscle, coxa

 
• L

  left

 
• m

  trochanteral moment arm

 
• M

  mass

 
• N

  normal reaction force at the coxo-trochanteral joint

 
• R

  right

 
• ti

  tibia

 
• tr

  trochanter

 
• , ÿ

  horizontal and vertical components of acceleration

 
• X,Y

  position of the centre of mass on the x- and y-axes

 
• X_{R contact}, Y_{R contact}

  the contact point of the right tibia with the ground

 
• θ_c

  angle of the coxa with respect to the horizontal

 
• θ_f and θ_ti

  angles between the femur or tibia and the midline

 
• θ_tr

  angle between the trochanter and the femur

 
• τ

  torque