It was thought that the wing hinge position can be tuned to stabilize an uncontrolled fly. However here, our Floquet stability analysis shows that the hinge position has a weak dependence on the flight stability. As long as the hinge position is within the fly’s body length, both hovering and ascending flight are unstable. Instead, there is an optimal hinge position, $h^{\ast }$, at which the ascending speed is maximized. $h^{\ast }$ is approximately half way between the centre of mass and the top of the body. We show that the optimal $h^{\ast }$ is associated with the anti-resonance of the body–wing coupling, and is independent of the stroke amplitude. At $h^{\ast }$, the torque due to wing inertia nearly cancels the torque due to aerodynamic lift, minimizing the body oscillation thus maximizing the upward force. Our analysis using a simplified model of two coupled masses further predicts, $h^{\ast }=(m_{t}/2m_{w})(g/\unicode[STIX]{x1D714}^{2})$. These results suggest that the ascending speed, in addition to energetics and stability, is a trait that insects are likely to optimize.
Using Triples to Assess Symmetry Under Weak Dependence*