Optimized kinematics enable both aerial and aquatic propulsion from a single three-dimensional flapping wing

Flapping wings in nature demonstrate a large force envelope, with capabilities far beyond the traditional limits of static airfoil section coefficients. Puffins, murres, and other auks particularly showcase this effect, as they are able to generate both enough thrust to swim and enough lift to fly, using the same wing, purely by changing the wing motion trajectory. The wing trajectory is therefore an additional design criterion to be optimized along with traditional aircraft parameters and could open the door to dual aerial-aquatic robotic vehicles. In this paper we showcase one realization of a three-dimensional flapping-wing actuation system that reproduces the force coefficients necessary for dual aerial-aquatic flight. The wing apparatus oscillates by the root and employs an active upstream and downstream sweep degree of freedom. We analyze two types of motions in detail: aerial motions where the wing tip moves upstream during the power stroke of each flapping cycle and aquatic motions where the wing tip moves downstream during the power stroke. We design these aerial and aquatic flapping-wing trajectories using an experiment-coupled optimization routine, allowing control of the unsteady forces throughout each flapping cycle. Additionally, we elucidate the wakes of these complex wing trajectories using dye visualization, correlating the wake vortex structures with simultaneous experiment force measurements. After optimization, the wing trajectories generate the large force envelope necessary for propulsion in both fluid media and furthermore demonstrate improved control over the unsteady wake.

Figure 1

Comparison of actuator areas in aerial and aquatic avians. Dual aerial-aquatic wings are of intermediate planform size compared to larger flying-only wings and smaller swimming-only wings. Selected birds have approximately equal body mass, ranging from 500 to 1500 g. Photographs show (a) European Herring Gull Larus argentatus [22] with flying-only wings, (b) and (c) Horned Puffin Fratercula corniculata both flying [23] and swimming [24] using dual aerial-aquatic wings, and (d) Little Blue Penguin Eudyptula minor [25] with swimming-only wings. Photos were reproduced under Creative Commons (see [22, 23, 24, 25]).

Figure 2

Dihedral, sweep, and stroke angles in auks. Auks, similar to many other birds, move their wings at an inclined stroke angle, which combines both dihedral and sweep motion. (a) Dihedral $\mathrm{{\rm Y}}$ as seen from the front. (b) Sweep $\mathrm{\Lambda }$ and stroke angle $\beta$ as seen from the side. The stroke angle can be greater or less than ${90}^{\circ }$. Animal art has been modified from public domain content [29].

Figure 3

Stroke angle control. (a)–(d) Variable stroke angles $\beta$ of the wing tip observed in auks with a variety of flying and swimming gaits, along with the vectorial force throughout the flapping cycle. (e) and (f) Mean forces on the auk in air and water, produced by both the wings and the environment. Animal art has been modified from public domain content [29] and airfoil trajectories with permission from Izraelevitz and Triantafyllou [18].

Figure 4

Towing apparatus. (a) Photograph of flapping-wing apparatus in the MIT Towing Tank, with red and blue dye. (b) Definition of three Euler rotations: dihedral $\mathrm{{\rm Y}}$, sweep $\mathrm{\Lambda }$, and pitch $\theta$. (c) Mechanism for pitching the wing tip and enforcing pitch along the wingspan.

Figure 5

Optimization routine example. Aquatic trajectory ($\text{St}=0.4$, $\beta ={135}^{\circ }$, and ${\alpha }_{\text{max}}={15}^{\circ }$) after successive experimental trials, driving down the error between the desired motion-relative lift (black) and experimental motion-relative lift.

Figure 6

Experiment trajectory optimization. (a) Mean horizontal force for each trial of each trajectory. (b) Mean vertical force for each trial of each trajectory. (c) Force direction Lissajous curves over the course of the flapping cycle.

Figure 7

Aquatic motion trajectory. Experiments of a trajectory employing a backward-moving downstroke (stroke angle $\beta ={135}^{\circ }$), maximum angle of attack ${\alpha }_{\text{max}}={15}^{\circ }$, and flapping frequency given by $\text{St}=0.4$ are shown for the (a) trajectory of the wing-tip pitch $\theta (R,t)$, (b) thrust coefficient (envelopes on each curve indicate two standard deviations over six runs), (c) vertical force coefficient, and (d) global motion of the wing tip, with superimposed wing-tip orientation and instantaneous total force.

Figure 8

Aquatic motion dye visualization. (a) Cartoon of camera placements around the wing (not represented at scale). (b) Diagram of visualized wake, consisting of a twisted vortex ring with central jet. (c) Snapshots from the top, rear, and left cameras of the trajectory at the transition between the downstroke and upstroke ($\varphi =3\pi /4$ to $\varphi =5\pi /4$). The leading-edge vortex (LEV), trailing-edge vortex (TEV), and wing-tip vortex (WTV) are labeled wherever distinct. Flow is in the positive $x$ direction. All images have been enhanced in both lightness and contrast by 10% for clarity.

Figure 9

Dye visualization of control action. Comparison of the open-loop trajectory to the optimized trajectory after three iterations. Snapshots are from the left camera (wing pressure side), near the end of the upstroke ($\varphi =7\pi /4$ and $\varphi =2\pi$). All images have been enhanced in both lightness and contrast by 10% for clarity.

Figure 10

Aerial motion trajectory. Experiments of a trajectory employing a forward-moving downstroke (stroke angle of $\beta ={45}^{\circ }$), maximum angle of attack ${\alpha }_{\text{max}}={45}^{\circ }$, and flapping frequency given by $\text{St}=0.4$ are shown for the (a) trajectory of the wing-tip pitch $\theta (R,t)$, (b) thrust coefficient (envelopes on each curve indicate two standard deviations over six runs), (c) vertical force coefficient, and (d) global motion of the wing tip, with superimposed wing-tip orientation and instantaneous total force.

Figure 11

Aerial motion dye visualization. (a) Camera placements for aerial motion. (b) Wake diagram, now with a $z$-oriented jet. (c) Snapshots of dye visualization. Similar to Fig. 8, flow is again in the positive $x$ direction and all images have been enhanced in both lightness and contrast by 10% for clarity.

Figure 12

Dihedral-only trajectory. Experiments of an unswept wing employing only a dihedral motion (stroke angle $\beta ={90}^{\circ }$), maximum angle of attack ${\alpha }_{\text{max}}={45}^{\circ }$, and flapping frequency given by $\text{St}=0.4$ are shown for the (a) trajectory of the wing-tip pitch $\theta (R,t)$, (b) thrust coefficient (envelopes on each curve indicate two standard deviations over six runs), (c) vertical force coefficient, and (d) global motion of the wing tip, with superimposed wing-tip orientation and instantaneous wing force.

Figure 13

Stroke angle mechanism. Photograph of setup for constraining the dihedral and sweep motions to follow a tilted stroke angle.