Trajectory optimization of flapping wings modeled as a three degree-of-freedoms oscillation system

Insects are able to create complex wing trajectories using power and steering muscles attached to the wing/thorax oscillation system. In this paper, we propose a dynamic model for such an oscillation system, and study its dynamic behavior. In particular, we model the wing as a rigid body with three degrees of freedom. The power muscle is modeled by a torque actuator and a torsional spring creating basic wing flapping (stroke) motion. Torsional springs at the wing longitudinal rotation and deviation axes are used to mimic the steering muscles. Aerodynamic forces and moments are calculated using blade-element analysis and quasi-steady aerodynamic model. Dimensional analysis shows that the dynamic behavior of the system is determined by the three spring coefficients and the input torque coefficient, and is characterized by four basic patterns of wing trajectories. By exploring the parameter space of these coefficients, we found that the wing trajectory that most similar to those of a real insect generates the best lift and power loading. Furthermore, a hybrid optimization algorithm is implemented to find the optimal stiffness coefficients that maximize the power loading. Notably, the results also indicate that the flapping trajectories with out-of-plane deviation achieve a better aerodynamic performance than those without it. The oscillatory property of this system does not only explain how insects use flight muscles to tune wing kinematics, but also allows for design simplifications of the wing driving mechanism of flapping micro air vehicles.