Rapid frequency modulation in a resonant system: aerial perturbation recovery in hawkmoths

Abstract

Centimetre-scale fliers must contend with the high power requirements of flapping flight. Insects have elastic elements in their thoraxes which may reduce the inertial costs of their flapping wings. Matching wingbeat frequency to a mechanical resonance can be energetically favourable, but also poses control challenges. Many insects use frequency modulation on long timescales, but wingstroke-to-wingstroke modulation of wingbeat frequencies in a resonant spring-wing system is potentially costly because muscles must work against the elastic flight system. Nonetheless, rapid frequency and amplitude modulation may be a useful control modality. The hawkmoth Manduca sexta has an elastic thorax capable of storing and returning significant energy. However, its nervous system also has the potential to modulate the driving frequency of flapping because its flight muscles are synchronous. We tested whether hovering hawkmoths rapidly alter frequency during perturbations with vortex rings. We observed both frequency modulation (32% around mean) and amplitude modulation (37%) occurring over several wingstrokes. Instantaneous phase analysis of wing kinematics revealed that more than 85% of perturbation responses required active changes in neurogenic driving frequency. Unlike their robotic counterparts that abdicate frequency modulation for energy efficiency, synchronous insects use wingstroke-to-wingstroke frequency modulation despite the power demands required for deviating from resonance.

Distinct forms of resonant optimality within insect indirect flight motors

“…Physically, this is significant: an indirect flight motor only needs to operate in the broad vicinity of the global-resonant state to attain almost all of its energetic benefit. This provides some explanation for insects' ability to afford variation in wingbeat frequency [9,14–16], as we discuss in §5.…”

“…Estimates of insect Q-factors are thus of key importance to understanding this resonant distinctness in the context of real insect flight motors: for instance, in identifying which resonant frequency these flight motors operate at, why they do so, and how wingbeat frequency variation affects flight motor resonance. It is fascinating to consider that effects which have often been attributed to flight motor elastic properties, such as the discrepancy between measured thoracic resonant frequencies [11], and wingbeat frequency variation behaviour [9,22–26], may be just as much the effect of flight motor damping—as we discuss further in §§5.3–5.4. There is significant scope for further extension of the current analysis.…”

“…The operation of sets of pleurosternal and tergopleural muscles has been proposed as a mechanism for time-varying control of the thoracic resonant frequency [9,[22][23][24][25][26], potentially allowing wingbeat frequencies to vary while maintaining a state of continued resonance. Alternatively, behaviour involving intentional frequency control has been motivated in terms of the trade-off between deviating from resonance and achieving other goals: for instance, performing a manoeuvre [9], or engaging in courtship [18]. One of the impediments to understanding these effects, and candidate explanations, is the lack of any clear information on what resonance actually represents in an insect indirect flight motor-and what costs, exactly, would be associated with deviating from resonance.…”

Stability and manoeuvrability in animal movement: lessons from biology, modelling and robotics

“…It had previously been hypothesized [9,10] that insects that operate their wings at the resonance of the entire thorax are unlikely to modulate wingbeat frequency on short time scales due to the inherent control constraints and power demands imposed by deviations from the energetically favourable mechanical resonance. In their study, 'Rapid frequency modulation in a resonant system: aerial perturbation recovery in hawkmoths', Gau et al [11] tested whether hawkmoths modulate wingbeat frequency and amplitude in response to sudden vortex ring perturbations. The authors discovered that hawkmoths were able to modulate both frequency and amplitude on short time scales, with a 32% change in frequency on a single-wingstroke time scale.…”

“…In their study, ‘Rapid frequency modulation in a resonant system: aerial perturbation recovery in hawkmoths', Gau et al . [ 11 ] tested whether hawkmoths modulate wingbeat frequency and amplitude in response to sudden vortex ring perturbations. The authors discovered that hawkmoths were able to modulate both frequency and amplitude on short time scales, with a 32% change in frequency on a single-wingstroke time scale.…”