Physics-based simulations of aerial attacks by peregrine falcons reveal that stooping at high speed maximizes catch success against agile prey

Abstract

The peregrine falcon Falco peregrinus is renowned for attacking its prey from high altitude in a fast controlled dive called a stoop. Many other raptors employ a similar mode of attack, but the functional benefits of stooping remain obscure. Here we investigate whether, when, and why stooping promotes catch success, using a three-dimensional, agent-based modeling approach to simulate attacks of falcons on aerial prey. We simulate avian flapping and gliding flight using an analytical quasi-steady model of the aerodynamic forces and moments, parametrized by empirical measurements of flight morphology. The model-birds' flight control inputs are commanded by their guidance system, comprising a phenomenological model of its vision, guidance, and control. To intercept its prey, model-falcons use the same guidance law as missiles (pure proportional navigation); this assumption is corroborated by empirical data on peregrine falcons hunting lures. We parametrically vary the falcon's starting position relative to its prey, together with the feedback gain of its guidance loop, under differing assumptions regarding its errors and delay in vision and control, and for three different patterns of prey motion. We find that, when the prey maneuvers erratically, high-altitude stoops increase catch success compared to low-altitude attacks, but only if the falcon's guidance law is appropriately tuned, and only given a high degree of precision in vision and control. Remarkably, the optimal tuning of the guidance law in our simulations coincides closely with what has been observed empirically in peregrines. High-altitude stoops are shown to be beneficial because their high airspeed enables production of higher aerodynamic forces for maneuvering, and facilitates higher roll agility as the wings are tucked, each of which is essential to catching maneuvering prey at realistic response delays.

Fig 1. Examples of prey motion in the no maneuver, smooth maneuver and non-smooth maneuver condition.

Fig 2. Block diagram of the feedback-loop in model-falcons.

This diagram is intended to communicate the general structure of the model. A detailed explanation of the model equations is provided in Materials and methods. The boxes denote transfer functions, and additional parameters of the functions are noted in between brackets. Most of the feedback loop is generic, except for the detailed implementation of flapping flight contained in the black box labelled “dynamics and control”. A brief summary of the feedback loop now follows, in which we walk through each of the different segments of the feedback loop summarised as “Vision”, “Guidance”, “Dynamics and Control”, “Kinematics”, and “Vector Geometry”. Vision: to determine how it should turn, the falcon first extracts the line-of-sight angle λ, which is measured subject to visual error ξ. The measured line-of-sight angle λ_ξ is subsequently transformed into an angular velocity vector ${\dot{\mathrm{\lambda }}}_{\xi }$ that denotes the estimated rate of change in the line-of-sight. The resulting signal from the visual system is fed to the guidance function every time interval τ, as denoted by the block labelled “sample … and hold”. Note that we also test an alternative implementation of visual processing delay in the model (continuous and delayed, instead of in discrete update intervals), as little is known about the nature of delay in birds. Results using either form of delay are highly similar (see S1 Fig). Guidance: the falcon’s guidance system multiplies the estimated line-of-sight rate ${\dot{\mathrm{\lambda }}}_{\xi }$ by the navigation constant N to obtain the commanded change in the angle of the falcon’s velocity $\dot{\gamma }$ (see Eq 1), and the cross product is taken with the velocity of the falcon to obtain the commanded acceleration $\widehat{\alpha }$. The dynamics and control function depends on the morphological parameters μ₁, … μ_n and manipulates the wing shape and motion to produce an acceleration α which maximizes the forward acceleration whilst meeting the commanded acceleration as closely as possible (see Materials and methods section D.2 and E for detailed model equations). Kinematics and Vector Geometry: the acceleration of the falcon α is integrated in the kinematics section and fed back to the visual system through the medium of the vector geometry needed to relate the line-of-sight angle to the updated positions of the model-falcon and model-starling. Note that the segment of the block diagram labeled “Vector Geometry” operates outside of the model-falcon, so we do not imply that the falcon cognitively represents either its own position or that of its target. In particular, the falcon has no knowledge of—and no need to know—the distance to its target; all that the falcon needs to know is the direction of its target as measured visually by the line-of-sight angle, and its own velocity, which is needed to determine the commanded acceleration from the commanded turn rate. Model-starlings have a similar control-loop, in which the segments of the feedback loop labelled “Vision” and “Guidance” are replaced by a forcing function ζ(t) that determines their (desired) trajectory (see Materials and methods section C).

Fig 3. Flight performance graphs in the flight simulator for the peregrine falcon (dark blue) and the common starling (light blue).

The double arrows denote the direction of acceleration displayed in the graph. The starling is able to outmaneuver the falcon at a given airspeed, if there exists a region under the curve of the starling that is not overlapping with that of the falcon. (a) Level acceleration versus air speed: level flight with the requirement that lift equals weight. Dashed lines denote the speed wherein torque forces constrain the maximum acceleration (mechanical constraints). Top level flight speed is reached at the point where level acceleration is zero. (b) Vertical dive acceleration (including gravity) versus air speed. At the end of the dashed lines, flapping is substituted by gliding with retracted wings in order to maximize vertical acceleration. (c) Load factor versus air speed. The load factor is defined as lift divided by weight. The maximum load factor does not scale quadratically with forward speed due to constraints in torque forces [11]. Instead, wings are retracted optimally to increase maximum load. (d) Roll acceleration versus air speed. Roll acceleration determines the speed with which the bird can redirect its lift and is calculated by estimating the whole-body inertia around the roll-axis and the maximum net torque production [11]. (e) Turning radius is calculated as the square of air speed divided by the maximum normal acceleration.

Fig 4. Catch success mapped onto initial altitude and horizontal distance from the prey for non-maneuvering, smooth maneuvering and non-smooth maneuvering prey, and for 4 values of the navigation constant N: A low extreme (N = 1), the optimal value for catching non-smooth maneuvering prey (N = 2.8), the optimal value for catching smooth maneuvering prey (N = 5.6), and a high extreme (N = 15).

The yellow asterisks depict the global optima with respect to attack position and N, showing the attack strategy which uniquely maximizes catch success for a given prey motion. The yellow crosses denote local optima for a given N and prey motion. The approximate intercept speed corresponding to the initial altitude is shown on the right of the graph. This is only an approximate relationship because the exact intercept speed depends on many factors within each hunt.

Fig 5. Catch success mapped onto intercept speed and navigation constant N for (a) non maneuvering, (b) smooth maneuvering and (c) non-smooth maneuvering prey.

The solid line denotes the optimal interception speed for a given N and the dashed line denotes the optimal N for a given interception speed. The asterisks denote the global optimum with respect to intercept speed and N for a given prey motion.

Fig 6. Relationship between catch success and various model parameters.

Graphs depict results for non-smooth maneuvering prey, because in this condition the high-speed stoop with a low N shows the most marked increase in catch success for the falcon. The upper bounds in values for reaction times and errors in vision and control are chosen such that they are different enough to show substantial variation in simulation results, but remain low enough to allow for capture. (a) Maximum catch success as a function of N in the baseline model, for smaller (τ = 0.1ms & 25ms) or larger (τ = 100ms & 150ms) response delays, assuming the optimal attack position. The margins depict the 95% confidence intervals of the GAM. The asterisks denote the global optimum with respect to the x-axis. (b) Maximum catch success as a function of N for different visual (ξ) and control error (χ). (c) Maximum catch success as a function of altitude, for the baseline and for increased error in vision. (d) Maximum catch success as a function of altitude for various values of control error. (e) Maximum catch success as a function of altitude, for various values of response delay τ.

Fig 7

(a) Look-up table for the accelerations due to the aerodynamic forces acting on the falcon. At each model time-step, the falcon maximizes forward acceleration (minimizes deceleration), given its forward speed and with the constraint that load factor is set to achieve the net commanded acceleration by the guidance law. If this constraint cannot be met (i.e. if it is unfeasible due to aerodynamics or high resulting torque forces), the closest approximation of the load factor is chosen. In the blade-element model, the falcon optimizes the wing twist, the wing’s angle-of-attack, the wingbeat frequency and the wing retraction. Inside the trapezoidal contour, the falcon flaps at maximal wingbeat frequency, and outside it the falcon glides. Above the contour, flapping results in too high torque forces on the wing. Gravity is excluded from the accelerations in the figure. (b) The partial derivative of forward aerodynamic acceleration with respect to load factor, termed the “aerodynamic deceleration penalty”.