Flapping strategies for flying formations

TL;DR

The paper investigates how in-line flying flyers can maintain a fixed inter-flyer distance using wake-mediated interactions. It develops a follower-wake model with memory, reformulates it as an iterated ordinary differential equation system, and non-dimensionalises it to reveal key parameters $\gamma$ and $\chi$ that govern propulsion and wake dissipation. By deriving exact separation-enforcing flapping rules for a sinusoidally flapping leader, it analyzes two canonical phase strategies (in-phase and out-of-phase), showing that stability and energy efficiency compete and identifying goldilocks zones where both criteria are satisfied. It further extends the analysis to non-band-limited leader forcing, demonstrating persistence of spectral content downstream and richer dynamics at higher $\gamma$, with implications for UAV and robotic swarms where constant spacing and stability are critical.

Abstract

Long arrays of identical, self-propelling flapping flyers are inherently unstable and thus unlikely to exist without active control mechanisms. One approach to enable long in-line formations is to enforce a constant separation between the group members. The objective then becomes to determine the flapping strategies the flyers should adopt to achieve a certain separation. Using an aerodynamic model of vortex wake production and inter-flyer effects, we explore different flapping strategies for followers given the motion of the leader. The choice of tactic is dependent upon the aerodynamic, kinematic, and physical parameters of the system, and reflects an interplay between efficiency and stability. We find that whether a flyer flaps in or out of phase with its upstream neighbour, together with the target separation, strongly affect the flapping amplitude and, therefore, the energetic cost of group flight. In certain regimes, group flight is energetically favourable compared to isolated flight, while in others, flying in formation becomes less efficient. We also identify "goldilocks zones", ranges of separation in which one of the in- or out-of-phase motions can be simultaneously energetically efficient and dynamically stable. Outside these regions, energetically favourable flight is unstable and therefore unlikely to occur.

Figures (14)

• Figure 1: Schematic diagrams demonstrating the flyer-wake model. (a) Each flyer generates a wake signal with speed $W_1(x,t)$, directly related to its flapping velocity $V_1(t)$, with the wake flow thereafter decaying in time. (b) An interaction model used to simulate the group dynamics with one-way interactions. The next flyer (i.e. the follower) experiences a thrust force that depends on its flapping velocity $V_2(t)$ relative to the ambient wake $W_1(X_2(t),t)$ at position $x=X_2(t)$. (c) Problem setup of a linear formation of $N$ flapping flyers indexed by $n$, each undergoing heaving-and-plunging motions with velocity $V_n(t)$ and free forward flight motions with speed $U_n(t)$. Flyer $n$ and its upstream neighbour $n-1$ are separated by a distance $d_n(t)=X_{n-1}(t)-X_n(t)\geq 0$.
• Figure 2: Flapping velocities $V_n(t)$ for each flyer $n=2,3,\dots, N$ to maintain a constant distance $d$ (equal to 0.3, 0.6, and 0.9; increasing from left to right) with its upstream neighbour for $\gamma=1$ (panels (a)--(c)) and $\gamma=10$ (panels (d)--(f)), given that the leader in the group flaps sinusoidally with velocity $V_1(t)=\cos(2\pi t)$ (grey curves). Here the group size is $N=20$ and $\chi=1$. The flapping velocities of flyers $n=2$ to 10 are shown in shades of blue to green and the last member in the group $n=N=20$ is displayed as a dashed red curve.
• Figure 3: Same as in figure \ref{['fig:Vn_gamma_d_inphase']}, but with sign-modified flapping velocities $(-1)^{n+1}V_n(t)$, obtained using \ref{['eq:GeneralFormulaVn']}.
• Figure 4: Plots of the first Fourier coefficient of $V_\infty(t)$ for the in-phase flapping velocities ($\sigma_n=+1$; blue) and the out-of-phase flapping velocities ($\sigma_n=(-1)^{n+1}$; red) versus fixed flyer separation $d$. Analytical approximations to each one, expected to hold for $\gamma\lesssim 1$ are plotted as dashed lines. Results are shown for $\chi=1$ and for $\gamma$ equal to (a) 0.1, (b) 1, and (c) 10.
• Figure 5: (a) Stability diagram in $(d,\chi)$ space. We plot whether the sign of $S/\sigma_2$ as given in \ref{['eq:signofS_stability']} is positive (blue, in-phase is stable) or negative (red, out-of-phase is stable). (b) Magnitude of flapping velocities for in-phase ($J_2$) and out-of-phase ($Q_2$) flapping as a function of $d$, with "goldilocks" regions of both stability and efficiency highlighted in light blue and light red shades. Lines are drawn solid where they are stable and dotted where they are unstable. (c) Same as panel (b) but with $J_\infty$ and $Q_\infty$ instead. In panels (b,c), $\chi=1$.