Physics-based simulations of aerial attacks by peregrine falcons reveal that stooping at high speed maximizes catch success against agile prey
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 λξ is subsequently transformed into an angular velocity vector 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 by the navigation constant N to obtain the commanded change in the angle of the falcon’s velocity (see Eq 1), and the cross product is taken with the velocity of the falcon to obtain the commanded acceleration . The dynamics and control function depends on the morphological parameters μ1, … μn 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).