Effect and correction of control delay in longitudinal dynamics of insect hovering flight.

Sensor-to-actuator delay is inevitable in any complex control system, be it one for a free-flying insect or a mimicking insectlike robotic flyer. In this work, we analyze the effects of control delay (latency) on the hovering performance of a model insect flyer, as exemplified by the hummingbird hawkmoth $\mathrm{Re}\ensuremath{\sim}3000$, and determine how control coefficients or gains may be modified to ameliorate the adverse effects of latency. The analyses are based on a simplified or reduced dynamic model of the hovering flyer. The longitudinal dynamics of the hovering flyer comprises the coupled forward (backward) and vertical translations and pitch rotation of the flyer, with kinematical wing actions being governed by proportional-differential (PD) closed-loop control. Keeping to the same PD control coefficients as a stable reference zero-delay case, the flight system becomes overly responsive at a small control delay, eventually diverging when delay approaches around one wing cycle. Stable hovering may be regained for control delay of up to several wingbeats by suitably reducing or softening the PD control coefficients. The results of the analyses are validated by a series of time-based simulations using the simplified dynamic model and a high-fidelity three-dimensional computational fluid dynamics with fluid structure--body interaction model of the hovering flyer. The simulations also show that noncyclic asymptotic oscillations about the mean equilibrium hovering state are enhanced with larger control delay. The analyses and simulations have helped us to gain a better understanding of the effects of control latency in insect free flight, which may be relevant for the design of mimetic insect flyers.