Optimal leap angle of legged and legless insects in a landscape of uniformly-distributed random obstacles
Fabio Giavazzi, Samuele Spini, Marina Carpineti, Alberto Vailati
OOptimal leap angle of legged and legless insects in a landscape ofuniformly-distributed random obstacles
Fabio Giavazzi
Dipartimento di Biotecnologie Mediche e Medicina Traslazionale,Universit`a degli Studi di Milano, I-20133 Milano, Italy
Samuele Spini, Marina Carpineti, and Alberto Vailati ∗ Dipartimento di Fisica A. Pontremoli, Universit`a degli Studi di Milano, I-20133 Milano, Italy
We investigate theoretically the ballistic motion of small legged insects and leglesslarvae after a jump. Notwithstanding their completely different morphologies andjumping strategies, these legged and legless animals have convergently evolved to jumpwith a take-off angle of 60 ◦ , which differs significantly from the leap angle of 45 ◦ thatallows reaching maximum range. We show that in the presence of uniformly-distributedrandom obstacles the probability of a successful jump is directly proportional to thearea under the trajectory. In the presence of negligible air drag, the probability ismaximized by a take-off angle of 60 ◦ . The numerical calculation of the trajectoriesshows that they are significantly affected by air drag, but the maximum probabilityof a successful jump still occurs for a take-off angle of 59-60 ◦ in a wide range of thedimensionless Reynolds and Froude numbers that control the process. We discuss theimplications of our results for the exploration of unknown environments such as planetsand disaster scenarios by using jumping robots. Keywords: animal movement; jumping; insects; ballistics; leap angle; robotic exploration
I. INTRODUCTION
Some animal species have evolved the ability of jump-ing, both as a fast locomotion method and as an es-cape maneuver [1]. Compared to deambulation or lo-comotion by crawling, jumping allows achieving a fast-displacement by simultaneously overcoming natural ob-stacles distributed across the landscape [2]. The evolu-tive process leading to the optimization of the jumpingperformances is strongly influenced by the features of thehabitat, which contribute to determining the selection ofa particular take-off angle in an animal species [3].Jumping performances become remarkable in insectsthat use a catapult mechanism to amplify their muscu-lar power and achieve long-range jumps by storing elasticenergy inside their exoskeleton [4]. When the elastic en-ergy is suddenly released by unlocking a latch, the bodyis projected into the air at high speed at distances sig-nificantly larger than the size of their body. Insects havedevised drastically different methods to jump by usinga catapult mechanism. Froghoppers (
Philaenus spumar-ius ) take advantage of elastic deformation of their chiti-nous exoskeleton to achieve a rapid extension of theirhind legs [5]. As a result, the body undergoes an accel-eration as large as 400 m/s and enters a ballistic phasewith top speeds up to 4 m/s and a typical take-off angleof 58 ◦ ± . ◦ [6] (Table I).Jumping driven by a catapult mechanism can beachieved also in the absence of highly specialized limbslike those present in froghoppers. A remarkable example ∗ corresponding author: [email protected] is represented by the legless jumping of the larvae of theMediterranean fruit fly ( Ceratitis capitata ), which takeadvantage of a peculiar hydrostatic catapult mechanismthat allows them to jump to distances of the order of 12cm, namely more than ten times their body length, in afraction of a second [7]. In this case, a hydrostatic skele-ton made by flexible outer muscular bands is anchored toa flexible skin layer, which confines an inner fluid region[8]. To jump, the larva first contracts its longitudinalventral muscles to form a loop and anchors its head tothe tail by using a pair of mouth hooks. The inner pres-sure of the body is then increased by contraction of heli-cal muscular bands, and the sudden release of the mouthhooks gives rise to an abrupt straightening of the body,leading to a rapid acceleration phase ending with take-offat a velocity of about 1.2 m / s and a take-off angle of 60 ◦ [7] (Table I). A similar hydrostatic catapult mechanismis adopted by the larvae of gall midge ( Asphondylia sp. )[9]. In this case, latching is achieved by micrometer-scalefinger-like microstructures distributed across the widthof body segments. The sudden release of this latch leadsto a take-off velocity of 0.85 m/s at a take-off-angle of63 ◦ ± ◦ (Table I).Notwithstanding their drastically different morpholo-gies, froghoppers and the larvae of the fruit fly and gallmidge have convergently evolved to jump with a take-off angle close to 60 ◦ . The understanding of the reasonsbehind this peculiar choice could shed new light on theselective pressure exerted by the geometrical and statis-tical features of the environment, and drive the develop-ment of bio-inspired robotic devices suitable for efficientexploration of territories with unknown features. More-over, froghoppers represent the key vectors for the Xylella a r X i v : . [ phy s i c s . b i o - ph ] J a n mass body effective take-off take-off substrate Ref.Species ( × − kg) length diameter speed angle ( × − m) ( × − m) (m/s)Gall midge larva 1.27 3.28 1.2 0.85 63 . ± ◦ plastic [9] (Asphondylia sp.) Fruit fly larva 17 8.5 2.8 1.17 60 ◦ - [7] (Ceratitis capitata) Froghopper 12.3 6 4 4 58 ± . ◦ - [6]( Philaenus spumarius ) 53 . ± ◦ epoxy [10]53 . ± ◦ ivy leave [10]TABLE I. Kinematic parameters of legged and legless jumpers. fastidiosa Wells bacterium, which led to a dramatic epi-demic disease on olive trees in the Mediterranean area[11]. The dispersal capabilities of froghoppers are stillnot very well known and a recent field study has shownthat they could be more effective than expected [12]. Adeeper understanding of the key factors that lead to theeffectiveness of dispersion of froghoppers could help theidentification of strategies to mitigate the fast spread ofthe dieback of olive trees.In this work, we show that a take-off angle of 60 ◦ max-imizes the probability of overcoming obstacles of randomsize and position scattered across the landscape. Wesolve numerically the dimensionless equations that de-scribe the kinematic motion of the animal in the pres-ence of air drag and show that the features of the motionare completely determined by the dimensionless Reynoldsand Froude numbers at take-off. We demonstrate thata take-off angle of 60 ◦ maximizes the probability of asuccessful jump in a very wide region of the parameterspace, largely encompassing the conditions of interest forthe jump of froghoppers and the larvae of fruit fly andgall midge. II. OPTIMAL STRATEGY FOR JUMPINGOVER A RANDOM OBSTACLE
Living beings like froghoppers and the larvae of fruitfly and gall midge have a typical size of the order of afraction of 1 cm (Table I) and are surrounded by obsta-cles whose size can largely exceed theirs. Under theseconditions, both the maximum height of the jump andits range become important in overcoming an obstacle.A leap angle of 60 ◦ represents a good compromise be-tween jump height and range because it maximizes theirproduct or, which is the same, the area below the trajec-tory. To show this, let us assume that the take-off of theanimal occurs at a fixed velocity v , and is only affectedby the gravitational acceleration g , in the absence of airfriction. Under these conditions, the motion completelyoccurs in a plane perpendicular to the ground and theequations of motion ¨ x = 0 and ¨ y = − g can be easily integrated to yield the time evolution of the componentsof the displacement from the initial position: x ( t ) = v cos( θ ) t, (1) y ( t ) = v sin( θ ) t − g t , (2)where x and y are the horizontal and vertical displace-ments, and θ is the take-off angle. The range of the jump, i.e. the distance traveled along the horizontal directionbefore landing at y = 0, is: x R ( θ ) = 2 x M sin( θ ) cos( θ ) , (3)where x M is the maximum range achieved for a take-offangle of 45 ◦ : x M = v /g. (4)By integrating the trajectory across x from the initialposition x = 0 to the range of the jump x R the areabelow the curve as a function of θ (Fig. 1) is: A ( θ ) = (cid:90) x R y ( x ) dx = 23 x M sin ( θ ) cos( θ ) (5) FIG. 1.
Area below the trajectory as a function oftake-off angle.
The area exhibits an absolute maximum at θ = 60 ◦ . FIG. 2.
Success of a jump.
Obstacles can be eitherrepresented by vertical fences (left column) or steps (rightcolumn). If the top-left edge of the obstacle is located abovethe trajectory the jump fails (top row), while when it is belowthe jump is successful (bottom row). and derivation of Eqn. 5 with respect to θ immediatelyshows that the area is has an absolute maximum for θ =60 ◦ . From Eqn. 5 one can appreciate that the area belowthe trajectory is proportional to the product between therange (Eqn. 3) and top height h ( θ ) = x M sin ( θ ) / i) the landscape is populated by obstacles,which can be either vertical fences (Fig. 2, left column),or steps (Fig. 2, right column); ii) a jump is consid-ered not successful when the trajectory passes below thetop left edge of an obstacle (Fig. 2, top row), and suc-cessful when it passes above it (Fig. 2, bottom row) iii) the height of the obstacles is uniformly distributed in therange [0 , H ], where H > x M /
2, and their position is uni-formly distributed in the range [0 , L ], where
L > x M ; iv) the initial velocity v of the jump is fixed.Due to the uniform distribution of height and posi-tion of the obstacles, the probability P ( θ ) of a successfuljump can be calculated directly as the area of the regionof possible positions of the top-left edge of the obstaclebelow the trajectory (Fig. 3, solid light-blue area belowthe trajectory), normalized by the area of the region ofall the possible positions (Fig. 3, rectangle of base L andheight H ): P ( θ ) = A ( θ ) H L . (6)Equation 6 shows that the area A ( θ ) below the trajec-tory represents a direct measure of the probability of asuccessful jump and, in combination with Eqn. 5, that the maximum probability is achieved for a take-off angleof 60 ◦ (Fig. 1). III. EFFECT OF AIR FRICTION
The results derived in the previous section are obtainedunder the implicit hypothesis of negligible air friction.However, as discussed by Vogel [13], the bio-ballisticsof small projectiles like the insects listed in Table 1 isstrongly affected by air drag, which becomes increas-ingly important as the size of the projectile is diminished.Under such circumstances, the presence of air drag candetermine a significant decrease of both the range andheight of the jump and in turn of the area below the tra-jectory. Indeed, Eqn. 5 for the area below the trajectoryhas been obtained in the absence of air friction. We willshow that, although air drag affects significantly the tra-jectories of the insects listed in Table 1, the optimal take-off angle is affected only marginally by its presence. Onefirm result of the model reported above is represented byEqn. 6, which states that the probability of a success-ful jump is proportional to the area under the trajectory.This result is valid under generic conditions, both in thepresence and in the absence of air drag. Therefore, to cal-culate the probability of a successful jump under realisticconditions one just needs to determine the area below thetrajectories in the presence of air drag. The effect of in-ertia and viscous drag on the body are combined into thedimensionless Reynolds number:Re = ρ l vη , (7)where ρ and η are respectively the density of air (1.2kg/m at 20 ◦ C) and its shear viscosity (1.8 × − Pa s),and l is a typical effective diameter of the body (TableI). When Re < v ,while for Re > v . Following FIG. 3.
Probability of a successful jump.
The solidlight-blue region below the trajectory marks the regions ofpossible positions of the top edge of the obstacle in a successfuljump, while the green rectangle all the possible positions.
Vogel [13], we introduce a drag coefficient that takes intoaccount the transition between these regimes: C d (Re) = 24Re + 61 + Re / + 0 . . (8)The modulus of the drag force acting of the body can becalculated as D = 12 C d ρ S v , (9)where S = π ( l/ is the cross sectional area that thebody offers to the air flow. Following Landau and Lifs-chitz [14], gravitational effects can be accounted for byintroducing the dimensionless Froude number:Fr = v l g (cid:48) , (10)where g (cid:48) = g ( ρ p − ρ ) /ρ is the reduced acceleration ofgravity [15] and ρ p is density of the insect.The equations of motion can be written in terms of thedimensionless variables (˜ x, ˜ y ) = ( x, y ) /x M and ˜ t = t/t ,where t = v /g is a characteristic time of flight:¨˜ x = − µ ˙˜ x (11)¨˜ y = − µ ˙˜ y − , (12)where µ = 34 C d (Re · ˜ v ) · Fr · ˜ v (13)is a dimensionless drag coefficient that only depends onthe dimensionless velocity ˜ v and on the Reynolds andFroude numbers at take-off, defined by Re = ρlv /η and Fr = v /lg (cid:48) , respectively. According to Eqns. 8and 13, for small Re , the drag coefficient attains a con-stant value µ ∼
18 Fr / Re and Eqns. 11 and 12 canbe solved analytically. For larger Re , the nonlinear de-pendence of µ from ˜ v makes the equation hard to solveanalytically, but solutions can be still found numerically.We integrated the equations numerically by using theEuler’s method detailed in [16] to obtain the area underthe trajectory as a function of take-off angle for a setof parameters mirroring the three representative cases ofleaping insects and larvae detailed in Table 1. The nu-merically calculated dimensionless areas ˜ A ( θ ) are shownin Fig. 4, where dashed and dotted lines represent re-sults obtained in the presence of air drag, while the solidline those obtained without air drag. One can appreciatethat, although air drag determines a decrease of ˜ A ( θ ),the maxima of the curves are located in a narrow rangeof angles 59 ◦ < θ < . ◦ , very close to the value θ = 60 ◦ predicted analytically in the absence of air drag.To assess the robustness of these results, we systemat-ically investigated the optimal leap angle, correspondingto the maximum of ˜ A ( θ ), over a wide range of the twocontrol parameters Re and Fr . As shown in Fig. 5, the FIG. 4.
Numerically calculated area below the trajec-tories in the presence of air drag. dashed line: gall midge;dotted line: fruit fly larva; dashed-dotted line: froghopper.The parameters used to process the trajectories mirror thosereported in Table 1. The solid line represents the area in theabsence of air drag, calculated from Eqn. 5. parameters of all the jumpers considered in this work fallwell within a large domain of the parameter space wherethe optimal leap angle is almost indistinguishable fromthe drag-free ideal case θ = 60 ◦ . For small Reynoldsnumbers (Re (cid:46) ) this domain is identified by the sim-ple condition µ (cid:28)
1, which corresponds to Fr (cid:28) Re / FIG. 5.
Optimal leap angle as a function of theReynolds and Froude numbers at take-off.
Black sym-bols: gall midge (downward triangle); fruit fly larva (cir-cle); froghopper (square). The dotted line corresponds toFr = Re /
18, which is equivalent to the condition µ = 1 inthe low Reynolds number limit (see Eqns. 8 and 13) IV. DISCUSSION
We have demonstrated that the fact that a take-off an-gle close to θ = 60 ◦ maximizes the probability of successof a jump in the presence of uniformly distributed ran-dom obstacles represents a robust result that does notdepend on the details of the model adopted. This re-markable result can be profitably used to optimize thefeatures of autonomous robots used for the explorationof environments populated by unknown obstacles, suchas other planets [17], or Earth regions than can be dan-gerous for human beings, like nuclear disaster sites orearthquakes’ scenarios [18, 19]. A well-known approach isto use rover robots that have usually large size and mass,move by using wheels or tracks, and navigate thanks toa high degree of technology and a closed-loop control oftheir movements. Although these characteristics allowa fine control of the navigation, they accomplish a lim-ited capability of mapping extended territories. A dif-ferent approach is the use of colonies of small and agilerobots, typically tens [20], with simple design and func-tions and open-loop control, which can map extendedterritories. As the size of robots decreases, they likelyhave to overcome obstacles whose size is comparable oreven larger than their own one [21, 22]. A bio-mimeticapproach suggests that jumping has a great potentialityof success [23–25] in rough terrains. In fact, in recentyears a large number of researches has focused on the re-finement of miniaturized robots inspired to jumping or-ganisms, like for example froghoppers [26], locusts [27],insects [24, 28, 29], or even soft worms [30, 31]. The finalchoices made in the design of a jumping robot result froma complex balance among constraints connected to take-off, air flight, and landing. Flight issues have often todo with posture adjustment, landing with stability andtake-off with the mechanisms of energy storing and fastconversion in kinetic energy for jumping [26, 27, 30]. Toour knowledge, the probability of success in overcomingan obstacle of random size and position has not been considered yet during the design process, and the take-off angle is often chosen a priori [30, 32].The results reported in this work could inspire a differ-ent approach in the design of miniaturized robots, whichcould be particularly effective for the challenging case ofgroups of small robots that collectively explore a roughterrain. Under these conditions the implementation ofa robot hardwired to jump at an angle of 60 ◦ would al-low attaining optimal performances in the exploration ofunknown rough regions, sided by an extremely simpleconceptual design. DATA ACCESSIBILITY
The Matlab code used for the numerical integrationof the equations of motion in the presence of air drag isprovided as Electronic Supporting Material.
AUTHORS’ CONTRIBUTIONS
FG designed the study, developed the model,performed numerical analysis and helped draft themanuscript; SS conceived the study, contributed to thedevelopment of the model and critically revised themanuscript; MC participated in the design of the studyand helped draft the manuscript; AV conceived the study,designed the study, coordinated the study, contributed tothe development of the model, performed numerical anal-ysis and drafted the manuscript. All authors gave finalapproval for publication and agree to be held accountablefor the work performed therein.
ACKNOWLEDGEMENTS
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