A universal strategy for visually guided landing

Abstract

Landing is a challenging aspect of flight because, to land safely, speed must be decreased to a value close to zero at touchdown. The mechanisms by which animals achieve this remain unclear. When landing on horizontal surfaces, honey bees control their speed by holding constant the rate of front-to-back image motion (optic flow) generated by the surface as they reduce altitude. As inclination increases, however, this simple pattern of optic flow becomes increasingly complex. How do honey bees control speed when landing on surfaces that have different orientations? To answer this, we analyze the trajectories of honey bees landing on a vertical surface that produces various patterns of motion. We find that landing honey bees control their speed by holding the rate of expansion of the image constant. We then test and confirm this hypothesis rigorously by analyzing landings when the apparent rate of expansion generated by the surface is manipulated artificially. This strategy ensures that speed is reduced, gradually and automatically, as the surface is approached. We then develop a mathematical model of this strategy and show that it can effectively be used to guide smooth landings on surfaces of any orientation, including horizontal surfaces. This biological strategy for guiding landings does not require knowledge about either the distance to the surface or the speed at which it is approached. The simplicity and generality of this landing strategy suggests that it is likely to be exploited by other flying animals and makes it ideal for implementation in the guidance systems of flying robots.

Keywords: flight control; insect; three-dimensional surface; vision.

Fig. 1

Flights of bees approaching the checkerboard pattern (gray vertical line and Inset) as viewed from the side (A) and from above (B). The black line in each panel shows the mean trajectory. The feeder tube is at 0 mm. N denotes the number of individuals; n denotes the number of flights.

Fig. 2

The relationship between approach speed and distance from the surface of bees approaching a checkerboard (A), concentric ring (B), or sector (C) pattern. Light gray lines represent individual flight data; dark gray lines indicate the mean of all flights and starred lines represent linear regressions of the mean data, as specified by the equation in each panel. N denotes the number of individuals; n denotes the number of flights.

Fig. 3

(A) Variation of speed with distance to the surface as honey bees approach a four-arm spiral that is stationary, expanding at 0.5 or 1 rps (black, blue, and red data, respectively). Solid lines show the mean response; shaded areas indicate the SEM. Dotted lines indicate linear regressions of the mean data as specified by the equations. (B) Maximum image angular velocity that bees experience as they approach the spiral at each rotational speed. The black dotted line indicates the expected variation of the maximum angular velocity with the distance to the surface if the bees were to approach a stationary spiral at a constant speed corresponding to their initial approach speed. The green dashed box depicts the range of mean maximum angular velocities that are experienced during the landing phase in all conditions. (C) Data from experiments with details as in A but using contracting spirals. The data for the stationary spiral (black line) is repeated from A to facilitate comparison. In this case, the linear regressions for 0.5 (blue dotted line) and 1 rps (red dotted line) have been fitted to data for distances lower than 210 mm and 120 mm, respectively, as explained in the text. (D) Data from experiments with details as in B but using contracting spirals.

Fig. 4

Variation of speed with distance to the surface for spirals of different spatial frequency when the spiral is stationary, contracting or expanding at 1 rps, and when it has three, four, or six arms (green, black and red lines, respectively). Solid lines show the mean response; shaded areas indicate the SEM. The number of flights for the stationary, expanding, and contracting spirals is, respectively, 19, 22, and 21; 21, 22, and 20; and 20, 23, and 22 for the three-, four-, and six-arm spirals, respectively.