Simulation of flapping bird flight, part 2: Gait parametrization, limit cycle, and dynamic stability*

We introduce a method to identify gait trajectories in trimmed flapping flight. Such trajectories correspond to limit cycles in the state space, characterized by the same period as the wingbeat. Our method relies on a multiple-shooting algorithm that can simultaneously identify unknown limit cycles, and analyze their stability. Based on Floquet theory, this analysis computes the Jacobian of the identified limit cycle and assesses its stability from its eigenvalues.\\ In a first contribution, we adapt this framework to the flapping flight equations of motion, known to be not only non-linear and time-dependent, but also driven by state-dependent forcing aerodynamic loads. An aerodynamic model was developed following the quasi-steady lifting line approach reported in part 1, taking the wing morphology and prescribed kinematics as input, and returning the state-dependent aerodynamic loads as output.\\ Our results identify one instable mode, suggesting that birds continuously rely on sensory feedback to achieve steady-state flapping flight. This framework is then leveraged in the analysis of several gait configurations. In particular, we use it to perform a sensitivity analysis of the flapping gaits required to achieve several flight regimes (level, climbing and descending flight).