Optimal leap angle of legged and legless insects in a landscape of uniformly distributed random obstacles

Abstract

We investigate theoretically the ballistic motion of small legged insects and legless larvae after a jump. Notwithstanding their completely different morphologies and jumping strategies, some legged and legless animals have convergently evolved to jump with a take-off angle of 60°, which differs significantly from the leap angle of 45° that allows reaching maximum range. We show that in the presence of uniformly distributed random obstacles the probability of a successful jump is directly proportional to the area under the trajectory. In the presence of negligible air drag, the probability is maximized by a take-off angle of 60°. The numerical calculation of the trajectories shows that they are significantly affected by air drag, but the maximum probability of a successful jump still occurs for a take-off angle of 59–60° in a wide range of the dimensionless Reynolds and Froude numbers that control the process. We discuss the implications of our results for the exploration of unknown environments such as planets and disaster scenarios by using jumping robots.

Pronking and bounding allow a fast escape across a grassland populated by scattered obstacles

“…Incidentally, a leap angle of 60∘ represents a good trade-off between the need of reaching simultaneously a jump height and range as large as possible during an escape. Larvae of species like fruit flies and gall midge and insects like froghoppers have evolved to leap with a take-off angle of 60∘ and are able to escape by trespassing natural obstacles of sizes much larger than that of their bodies [16].…”

“…We assume that the modulus of the velocity v 0 at the end of the propulsive phase of each step or leap is fixed, while the take-off angle θ formed with the horizontal plane can change. We consider the real case of mammalians of medium–large size with a large Reynolds number and a small Froude number, a condition that allows neglecting the drag exerted by air during the jump [ 15 , 16 ]. A fast horizontal speed of locomotion and the ability of overpassing a series of obstacles are the two key factors that determine the survival of a prey during an escape run.…”