When most people think of solving a problem, they start at the beginning and work through to the end. But not Ty Hedrick and Tom Daniel. When they were curious to find the range of wing beat movements that could sustain a hawkmoth's hovering flight, the team, based in Seattle Washington, realised that they'd have to think backwards and deal with hovering flight as an`inverse problem': start with the hovering and work back to find out which combinations of wing beats could keep the moth in place above a flower(p. 3114). Tackling this problem turned out to be an enormously challenging computational task,requiring a 16-processor cluster, to discover which wing beat combinations can keep a moth hovering.
So why work on hovering and not some other manoeuvre? Hedrick explains that hovering is probably the insect's best-understood form of flight. Not only is hawkmoth hovering well characterised in the literature, but the equations that govern hovering flight are well understood and it was relatively simple for Hedrick to incorporate them into a mathematical simulation.
Having developed the mathematical moth, Hedrick chose 10 wing motion parameters, which the insects probably vary during flight, that he could adjust during the course of each simulation, and began running over 125 million simulations to see which combinations of wing beat parameters kept the moths hovering and which sent them crashing. Starting each simulation with a unique set of wing beat parameters and running it over a flight of 41 wing beats, Hedrick used a genetic algorithm to slightly modify a subset of the parameters from beat to beat, keeping the rest at their initial value, to simulate the wing beat variability inherent in the real world.
After weeks of number crunching, Hedrick and Daniel found the wing beat parameters that seemed to influence the moth's ability to hover most,including wing beat amplitude and the timing of the forward wing beat relative to the downwards wing beat. The team also found that allowing three of the wing beat parameters to vary during the course of a simulation produced the optimal hovering performance. However, increasing the numbers of variable parameters to 4 or more didn't improve the simulation's performance at all,which wasn't entirely surprising given that Hedrick and Daniel allowed their moth simulation to hover with only three degrees of freedom. The team turned up an enormous range of wing beat parameter combinations that kept `math-moth'aloft and Hedrick suspects that `there's a whole universe of possibilities available to hovering moths, and the real ones probably live in a small set of those possibilities'.
Hedrick admits that he didn't compare his simulated moths with the real thing until the end of the project when all the tests were run and the data were in; `we didn't want to prejudice our results' he says. But after comparing math-moth's performance with the moths in the literature he admits that there was a `big sigh of relief' when the simulations behaved just like the real thing.