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An Interpretable Data-Driven Model of the Flight Dynamics of Hawks

Lydia France, Karl Lapo, J. Nathan Kutz

Abstract

Despite significant analysis of bird flight, generative physics models for flight dynamics do not currently exist. Yet the underlying mechanisms responsible for various flight manoeuvres are important for understanding how agile flight can be accomplished. Even in a simple flight, multiple objectives are at play, complicating analysis of the overall flight mechanism. Using the data-driven method of dynamic mode decomposition (DMD) on motion capture recordings of hawks, we show that multiple behavioral states such as flapping, turning, landing, and gliding, can be modeled by simple and interpretable modal structures (i.e. the underlying wing-tail shape) which can be linearly combined to reproduce the experimental flight observations. Moreover, the DMD model can be used to extrapolate naturalistic flapping. Flight is highly individual, with differences in style across the hawks, but we find they share a common set of dynamic modes. The DMD model is a direct fit to data, unlike traditional models constructed from physics principles which can rarely be tested on real data and whose assumptions are typically invalid in real flight. The DMD approach gives a highly accurate reconstruction of the flight dynamics with only three parameters needed to characterize flapping, and a fourth to integrate turning manoeuvres. The DMD analysis further shows that the underlying mechanism of flight, much like simplest walking models, displays a parametric coupling between dominant modes suggesting efficiency for locomotion.

An Interpretable Data-Driven Model of the Flight Dynamics of Hawks

Abstract

Despite significant analysis of bird flight, generative physics models for flight dynamics do not currently exist. Yet the underlying mechanisms responsible for various flight manoeuvres are important for understanding how agile flight can be accomplished. Even in a simple flight, multiple objectives are at play, complicating analysis of the overall flight mechanism. Using the data-driven method of dynamic mode decomposition (DMD) on motion capture recordings of hawks, we show that multiple behavioral states such as flapping, turning, landing, and gliding, can be modeled by simple and interpretable modal structures (i.e. the underlying wing-tail shape) which can be linearly combined to reproduce the experimental flight observations. Moreover, the DMD model can be used to extrapolate naturalistic flapping. Flight is highly individual, with differences in style across the hawks, but we find they share a common set of dynamic modes. The DMD model is a direct fit to data, unlike traditional models constructed from physics principles which can rarely be tested on real data and whose assumptions are typically invalid in real flight. The DMD approach gives a highly accurate reconstruction of the flight dynamics with only three parameters needed to characterize flapping, and a fourth to integrate turning manoeuvres. The DMD analysis further shows that the underlying mechanism of flight, much like simplest walking models, displays a parametric coupling between dominant modes suggesting efficiency for locomotion.
Paper Structure (16 sections, 3 equations, 4 figures, 1 table)

This paper contains 16 sections, 3 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methods
  3. Data and Flights
  4. Dynamic Mode Decomposition (DMD)
  5. Rank Choice
  6. Marker and Mode Reconstruction
  7. Results and Discussion
  8. DMD Modes for Flapping Flight
  9. Consistency across Individuals
  10. Double Frequency Mode
  11. Full Flight DMD Modes
  12. Turning Flights
  13. Reconstruction Accuracy
  14. Broader Connections to Locomotion (Walking) Models
  15. Central Pattern Generators
  16. ...and 1 more sections

Figures (4)

  • Figure 1: Dynamic Mode Decomposition on hawk flapping flight reproduced with a model with 3 terms. (a) A reconstruction of eight wing and tail landmarks in 3D of a Harris' hawk flying between two perches using motion capture. (b) Restricting data to just flapping behaviour lasting 0.7s after take-off, a binned mean was taken from 67 flights by the same individual (bin size = 0.005s, shaded area ±1 standard deviation). (c) Using DMD, we approximate the flapping behaviour as the sum of three dynamic modes, which contain both a spatial, $\phi_k$, and temporal $e^{\omega_k t}$ component. (d) Projecting the effect of each mode onto the hawk shape, with time going from yellow to purple: the first mode (k(1) freq = 4.51 Hz) contains the circular wingbeat oscillation with a gradual decay; the second mode shows a close to double frequency of the first (k(2) freq = 8.76 Hz) showing a small oscillation on top of the main wingbeat; the third mode shows a gradual increase in wingspan and tail dropping (k(3) freq= 0.00Hz). (e) From the flapping flight, three dynamic modes combine to reconstruct the original flapping behaviour with total RMSE error within 1.2% the hawk's maximum wingspan (total absolute error = 12mm).
  • Figure 2: DMD fit to 9m straight flights by juvenile (J) and adult (A) hawks' flapping behaviour for 0.7s after take-off from a perch. Left plots: binned mean wingtip positions relative to the center of mass in z (pink: left wing, red: right wing) overlaid with the DMD model fit (black: right wing, black dashed: left wing). Right plots: the projection traces show the influence of each dynamic mode on the wings and tail shape. Individual variation in flight is high, with bilateral asymmetry, and the same hawk showing a change in technique after three years of maturation. Despite individual variations, the three modes show similar spatial patterns. The dominant mode showed the circular wingbeat with folding, and another mode close to double this frequency showing modulation of the wings on top of the wingbeat. A slower, non-oscillating mode also showed an increase in wing and tail area, though achieved in different ways.
  • Figure 3: Turning modes. (a) 9m flights between two perches with a midpoint obstacle were recorded by one hawk (Toothless), (b) with marker data split into left (36 flights) and right (30 flights) and a binned mean taken at every 5mm. As before, the modes describe the (c) flapping wingbeat oscillation, and (d) an increase in wingspan and pitching up, but the DMD modes now include significant banking rotation, which are mirrored for left and right turns. (e) An asymmetric mode creates left-right imbalances likely driving the turn.
  • Figure 4: Reconstruction error calculated from every DMD run for each individual flight (rather than averaged flights). DMD was run on every flapping flight sequence from the dataset detailed in France2025, including multiple perch-perch distances. Root mean squared error (RSME) calculated between the original marker position and the DMD fit. Histograms pool all experimental conditions and show (a) the mean RMSE across markers per frame, pooled from all individuals; (b) the RMSE per marker per frame, pooled from all individuals; (c) the mean RMSE across markers per frame and boxplots binned by horizontal distance to the perch (0.2m), pooling all individuals and using all 9m flights with and without an obstacle; d) the mean RMSE per frame across markers and separated by individual, juvenile and adult as indicated. The wingspan is about a meter, indicating the errors for DMD fits to individual flights are reasonably small.