Nonlinear flight dynamics and stability of hovering model insects

Abstract

Current analyses on insect dynamic flight stability are based on linear theory and limited to small disturbance motions. However, insects' aerial environment is filled with swirling eddies and wind gusts, and large disturbances are common. Here, we numerically solve the equations of motion coupled with the Navier-Stokes equations to simulate the large disturbance motions and analyse the nonlinear flight dynamics of hovering model insects. We consider two representative model insects, a model hawkmoth (large size, low wingbeat frequency) and a model dronefly (small size, high wingbeat frequency). For small and large initial disturbances, the disturbance motion grows with time, and the insects tumble and never return to the equilibrium state; the hovering flight is inherently ( passively) unstable. The instability is caused by a pitch moment produced by forward/backward motion and/or a roll moment produced by side motion of the insect.

Controlling roll perturbations in fruit flies

“…When a fly entered the filming volume, the reduction in the laser beam intensity was detected by a custom built circuit that triggered the cameras as well as a 5 ms (one wing-beat) vertical magnetic pulse generated by the two Helmholtz coils located on the floor and ceiling of the chamber. Such a short pulse is required to characterize the control mechanism on roll perturbations, since this degree of freedom is unstable on a time scale of a few wing-beats [14][15][16][17][18]. Controlling the voltage across the coils enables us to vary the magnetic field strength up to approximately 10 -2 T, which is about 1000 times stronger than the Earth's magnetic field.…”

“…Moreover, they fly at Reynolds numbers Re ¼ 10 2 -10 4 , in which flows are unsteady [5,6]. Most importantly, recent analytical and numerical analyses, as well as mechanical models, indicate that flapping flight is aerodynamically unstable, on a time scale of a few wing-beats [7][8][9][10][11][12][13][14][15][16][17][18][19]. It is, therefore, intriguing how insects overcome such control challenges and manage to fly with impressive stability, manoeuvrability and robustness, outmanoeuvring any man-made flying device.…”

“…Recent fluid dynamics simulations suggest roll is unstable due to an unsteady aerodynamic mechanism, where roll is positively coupled to sideways motion via asymmetry of the leading-edge vortex attached to each wing [14][15][16][17][18]. The coupling is driven by the wings' flapping and causes the roll angle to grow.…”

System-theoretic Interpretation of the Mode Sensing Hypothesis

“…the dynamics of the uncontrolled system) exist only as an abstract concept, which makes the hypothesis challenging to test empirically. Furthermore, where these natural dynamics have been simulated for model insects, they have been found to be unstable and hence fundamentally different to the stable dynamics of the closed-loop system [9,10,4,6,7,11,12,13,14,15,16,17,18,19]. Finally, there is an intrinsic difficulty associated with relating the complex eigenvectors characterising oscillatory modes of motion to meaningful directions in state space.…”

“…bja , at the reference condition, in the hypothetical case that the insect makes no change to its motor commands. Although there is no reason why these aerodynamic derivatives must be known in advance of any empirical investigation, it may be noted that they are precisely the same aerodynamic derivatives as have been computed numerically in a series of recent papers on insect flight stability [9,10,4,6,7,11,12,13,14,15,16,17,18,19].…”