Global phase-space geometry of three-dimensional gliding: terminal velocity manifolds, separatrices, and stability structure

TL;DR

This work develops a three-dimensional dynamical-systems framework for passive gliding and reveals a global phase-space skeleton comprised of a two-dimensional terminal velocity manifold ($TVM$) and a codimension-one separatrix that divides initial conditions into shallow, lift-dominated glides and steep, drag-dominated descent. By deriving a fully 3D model and analyzing equilibria and bifurcations across orientation (pitch, roll, yaw) for three airfoils—the snake-inspired bluff body, the Zimmerman/Draco airfoil, and the NACA 0012—the authors show that equilibrium stability and separatrix geometry depend on orientation and airfoil shape. The study finds that the $TVM$ persists across profiles, but the separatrix geometry and equilibrium structure vary, with bio-inspired airfoils exhibiting compact separatrices that enhance glide robustness. These results unify biological and engineered gliders within a unified geometric framework and suggest principled design criteria for robust passive gliding in future bio-inspired systems.

Abstract

We develop a three-dimensional dynamical-systems framework for passive gliding and identify the global phase-space structures that organize its motion. Extending previous two-dimensional models of non-equilibrium gliding, we show that the 3D velocity dynamics possess an attracting, normally hyperbolic invariant surface, the terminal velocity manifold (TVM), onto which all trajectories rapidly collapse before evolving slowly toward a glide equilibrium. There is also a separatrix surface associated with an invariant manifold of an unstable equilibrium within the TVM, which partitions initial conditions into qualitatively distinct descent behaviors: efficient shallow glides versus steep, drag-dominated descent. Using lift-drag data from three representative airfoils--a snake-inspired bluff body, the Zimmerman planform characteristic of Draco lizards, and the classical NACA 0012--we compute the full equilibrium surfaces, analyze their pitch-roll bifurcations, and reconstruct the TVM and separatrix geometry in three dimensions. The results reveal that (i) equilibrium stability changes with both pitch and roll, rather than pitch alone; (ii) separatrix geometry determines the dynamic accessibility of shallow glides; and (iii) bio-inspired airfoils possess compact separatrix regions that make efficient gliding robust across a wide range of initial jump conditions. This work unifies biological and engineered gliders within a single global geometric framework and establishes separatrix geometry on the TVM as a principled diagnostic for glide robustness.

Figures (10)

• Figure 1: Schematic of the global geometry of 3D gliding. Trajectories collapse rapidly onto the terminal velocity manifold (blue), where a separatrix surface (red) partitions initial conditions leading to shallow, lift-dominated glides (blue dot equilibrium) from those that fall into steep, drag-dominated descent. The intersection of the separatrix with the horizontal-velocity plane defines the minimum jump conditions required to reach an efficient glide.
• Figure 2: (a) Body-fixed frame illustrating pitch, roll, and yaw angles. (b) Unit velocity vector and associated glide and azimuthal angle definitions used in deriving the three-dimensional equations of motion.
• Figure 3: Left column: $C_L$ and $C_D$ curves for each airfoil. Right column: corresponding airfoil planform shapes. (a,b) Analytical Snake Airfoil. (c,d) NACA 0012. (e,f) Zimmerman / Draco where both the top view and side view of the Zimmerman planform are shown.
• Figure 4: Variation of the Zimmerman airfoil equilibrium with body orientation parameters $(\phi,\theta,\psi)$. (a) The full equilibrium locus shown in $(\gamma^*,\sigma^*,v^*)$, illustrating how the steady glide state bends smoothly through velocity space as the Euler angles change. (b) Dependence of the equilibrium glide angle $\gamma^*$ on roll $\phi$, showing that increasing roll leads to steeper descent. (c) Sensitivity of the azimuthal angle $\sigma^*$ to roll and yaw. Roll has the dominant effect, indicating that $\phi$ is the primary steering parameter in this model.
• Figure 5: Bifurcation diagrams for the three airfoils. Equilibrium branches are colored by stability: blue for stable nodes, purple for stable foci, and red for unstable (saddle-type) equilibria. Yaw is fixed at $\psi=0$, and each curve shows a slice of the bifurcation surface in $(\gamma^*,\theta)$ at a fixed roll angle. Roll $\phi$ varies from $0^{\circ}$ to $30^{\circ}$ in increments of $10^{\circ}$, revealing how the number, position, and stability of equilibrium glide states depend on body orientation.