Amphibious Wing: Dual-Modal Propulsion

• Amphibious wings are dual-medium propulsion systems that adapt force production via flapping kinematics or adjustable wing area.
• They employ precise actuation and geometric control methods, with designs emphasizing either continuous force-envelope expansion or rapid morphological reconfiguration.
• Experimental results reveal trade-offs in thrust, lift, glide distance, and transition control, highlighting challenges in managing unsteady forces across media.

An amphibious wing is a lifting or propulsive appendage designed to function in both air and water, either by modulating force production through flapping kinematics or by reconfiguring planform area during water–air transition. In the literature considered here, two implementations define the concept with complementary emphases: a rectangular flapping wing with an active in-line motion degree of freedom intended to generate the large force envelope required for propulsion in both fluid media (Izraelevitz et al., 2014), and collapsible pectoral-fin wings actuated by soft hydraulics in an aquatic-aerial robot inspired by the flying fish, where the wings support gliding, transition, and multi-modal operation between water and air (Xiong et al., 2023). Taken together, these studies frame the amphibious wing as a coupled problem in geometry, unsteady kinematics, force synthesis, and transition control rather than as a single fixed morphology.

1. Taxonomy of amphibious-wing architectures

The available arXiv literature shows that amphibious wings are not a single canonical mechanism. One line of work uses a flapping foil actuator whose motion is explicitly designed to span aerial lift-support and underwater thrust production. Another uses collapsible membrane wings whose area changes according to mission phase, while propulsion is provided by a single 300 W brushless motor / propeller (Izraelevitz et al., 2014, Xiong et al., 2023).

System	Wing architecture	Primary demonstrated role
Flapping amphibious wing	Rectangular half-model, three actuated joints, active in-line motion	Dual aerial/aquatic force generation
Collapsible-wing aquatic-aerial robot	Pectoral-fin wings with soft hydraulic actuators	Gliding, transition, and wing-area modulation

The flapping configuration is centered on force-envelope expansion. Its defining feature is the active in-line motion degree of freedom at the shoulder, which allows the same wing to trade off axial and vertical force by varying stroke angle $\beta$. The collapsible-wing configuration is centered on morphological adaptation. Its defining feature is a soft hydraulic actuator that changes wing opening angle $\theta_w$, and thus span and area, during take-off, glide, and dive.

A plausible implication is that “amphibious wing” should be understood functionally rather than morphologically: the essential property is operation across two fluid regimes, while the mechanical realization may be either kinematic or reconfigurable.

2. Geometry, materials, and actuation mechanisms

The flapping-wing prototype is a one-half model of a full flapping vehicle wing, rectangular in planform with chord $c = 0.152 \,\text{m}$ and semispan $s = 3c$ (Izraelevitz et al., 2014). Three actuated joints define the motion. In-line motion $\theta_1$ is about the $y$-axis at the wing root with spanwise $s_1=0$ and chordwise $c_1=0.375c$, with range $\pm \sim 45^\circ$ set by stroke angle $\beta$. Flapping $\theta_w$0 is about the $\theta_w$1-axis at $\theta_w$2, with range $\theta_w$3 set by heave amplitude $\theta_w$4. Pitching $\theta_w$5 is about the $\theta_w$6-axis at the $\theta_w$7-chord location at span $\theta_w$8, with range $\theta_w$9.

The corresponding kinematic amplitudes are defined through the flapping period $c = 0.152 \,\text{m}$0 and stroke angle $c = 0.152 \,\text{m}$1:

$c = 0.152 \,\text{m}$2

with representative span

$c = 0.152 \,\text{m}$3

For the experiments, $c = 0.152 \,\text{m}$4.

The collapsible-wing robot adopts a different design language (Xiong et al., 2023). Its body length is $c = 0.152 \,\text{m}$5, folded-wing width is $c = 0.152 \,\text{m}$6, full wingspan is $c = 0.152 \,\text{m}$7, and weight is $c = 0.152 \,\text{m}$8. The hull uses a basswood skeleton with mortise-tenon joints, heat-shrink film for waterproofing, and polyimide reinforcement. Internal electronics include an STM32 flight controller, IMU, pressure sensor, and $c = 0.152 \,\text{m}$9 LiPo battery.

Its collapsible pectoral-fin wings use a 170 mm carbon-fiber tube as wing arm and a $s = 3c$0 hydrophobic silicone film membrane with excellent tensile strength and low water absorption. The maximum deployed pectoral-fin area satisfies

$s = 3c$1

Pelvic fins are fixed and provide pitch/yaw stabilization underwater. The soft hydraulic actuator is molded from silicone rubber (Mold Star 30) and has a $s = 3c$2 pre-bent channel that straightens toward $s = 3c$3 under internal fluid pressure. Its least-squares fit for bending angle is

$s = 3c$4

with $s = 3c$5 in kPa. The volume required for full extension from $s = 3c$6 is approximately $s = 3c$7 for liquid and $s = 3c$8 for gas.

3. Kinematics and governing models

For the flapping amphibious wing, the joint time histories are prescribed as

$s = 3c$9

while $\theta_1$0 is computed to impose a prescribed effective angle of attack $\theta_1$1 (Izraelevitz et al., 2014). All forces are nondimensionalized on the semi-span planform $\theta_1$2, with towing or flight speed $\theta_1$3 and fluid density $\theta_1$4:

$\theta_1$5

The quasi-steady force laws are

$\theta_1$6

For the rectangular finite-aspect-ratio wing, the spanwise lift slope follows Hoerner’s approximation,

$\theta_1$7

with $\theta_1$8 because the half-model sits on a symmetry plane.

The reported nondimensional operating point is

$\theta_1$9

At the $y$0 span section, the local velocities are

$y$1

$y$2

where $y$3. The instantaneous flow angle is

$y$4

the local foil angle of attack at the quarter-chord is

$y$5

and the effective angle at the three-quarter-chord is

$y$6

with $y$7. The pitch state is obtained from

$y$8

The collapsible-wing robot is modeled with separate inertial, body, and velocity frames (Xiong et al., 2023). The body frame origin is at the center of gravity with $y$9 forward, $s_1=0$0 right, and $s_1=0$1 down. The velocity frame is aligned with the flight path and parameterized by angle of attack $s_1=0$2 and sideslip $s_1=0$3. The rigid-body dynamics use a mass-plus-added-mass matrix,

$s_1=0$4

with Newton–Euler equations

$s_1=0$5

$s_1=0$6

In air, aerodynamic force and moment in the body frame are decomposed as

$s_1=0$7

for pectoral and pelvic fins. For each fin,

$s_1=0$8

with transformation into the body frame, and

$s_1=0$9

In water, buoyancy, weight, and hydrodynamic surface forces are included through

$c_1=0.375c$0

$c_1=0.375c$1

$c_1=0.375c$2

During transition,

$c_1=0.375c$3

$c_1=0.375c$4

Ground effect is also modeled for normalized clearance $c_1=0.375c$5:

$c_1=0.375c$6

$c_1=0.375c$7

Using the typical model attributed to Boschetti et al. for $c_1=0.375c$8,

$c_1=0.375c$9

$\pm \sim 45^\circ$0

4. Force production, performance envelopes, and measured operating points

The central result of the flapping-wing study is that in-line motion modifies the balance between vertical and axial force through the stroke angle $\pm \sim 45^\circ$1 (Izraelevitz et al., 2014). For $\pm \sim 45^\circ$2, corresponding to symmetric flapping, the wing yields mean thrust coefficient $\pm \sim 45^\circ$3, nearly zero mean $\pm \sim 45^\circ$4, and large vertical oscillations up to $\pm \sim 45^\circ$5. For $\pm \sim 45^\circ$6, described as “turtle-like” backwards down-stroke, the mean thrust rises to $\pm \sim 45^\circ$7 with minimal net $\pm \sim 45^\circ$8, greatly reducing heave oscillation. For $\pm \sim 45^\circ$9, described as “bird-like” forwards down-stroke, the mean vertical force becomes $\beta$0 with $\beta$1. The summary interprets this as thrust amplification by approximately $\beta$2 over symmetric flapping at identical $\beta$3 and $\beta$4, or net lift more than double that of a symmetric motion tuned for vertical support.

The experimental conditions for these measurements were fresh water at $\beta$5, $\beta$6, $\beta$7, and $\beta$8, with key results reported as means over five runs with $\beta$9 bands. For the turtle-like case with $\theta_w$00 and $\theta_w$01, negligible net $\theta_w$02 was kept below $\theta_w$03 except transient peaks near rotation. For the bird-like case with body $\theta_w$04 and $\theta_w$05, mean thrust was approximately zero and slightly negative in parts of the cycle.

The same data are used to estimate scaling implications. From the measured thrust in water with $\theta_w$06, steady underwater speed follows from

$\theta_w$07

With $\theta_w$08 for a $\theta_w$09 axisymmetric hull, the wing could propel $\theta_w$10 toward $\theta_w$11. In air, with $\theta_w$12, maintaining $\theta_w$13 requires $\theta_w$14 at $\theta_w$15, and the lifting balance

$\theta_w$16

gives $\theta_w$17. Scaling to $\theta_w$18 with constant $\theta_w$19 and thus $\theta_w$20 raises the supported mass to $\theta_w$21.

The collapsible-wing study reports a different performance envelope, focused on aerodynamic deployment and post-exit gliding (Xiong et al., 2023). At $\theta_w$22 and total wing area $\theta_w$23, the measured aerodynamic coefficients show a linear lift slope $\theta_w$24 for $\theta_w$25, a stall-limited $\theta_w$26, a drag minimum $\theta_w$27 at $\theta_w$28, and $\theta_w$29 at $\theta_w$30. The pitching-moment coefficient crosses zero at $\theta_w$31, which the summary identifies as providing restoring torque for $\theta_w$32.

For discharge-angle studies with fully open wings at $\theta_w$33 and initial $\theta_w$34 over calm water, the maximum glide distance is approximately $\theta_w$35 at $\theta_w$36, and peak altitude is approximately $\theta_w$37 at the same angle. At $\theta_w$38, peak altitude increases to approximately $\theta_w$39 but with reduced glide distance due to steeper climb and stall margin. Wing-area studies at $\theta_w$40 show $\theta_w$41 for $\theta_w$42, $\theta_w$43 for $\theta_w$44, $\theta_w$45 for $\theta_w$46, and $\theta_w$47 for $\theta_w$48, corresponding to $\theta_w$49 relative to the folded case. A rapid wing-fold maneuver at apex reduces descent distance from approximately $\theta_w$50 to approximately $\theta_w$51, a reduction of $\theta_w$52.

Ground effect provides an additional low-clearance benefit. With $\theta_w$53, the study reports $\theta_w$54 versus $\theta_w$55 at $\theta_w$56, a $\theta_w$57 gain; $\theta_w$58 versus $\theta_w$59 at $\theta_w$60, a $\theta_w$61 gain; and no benefit at $\theta_w$62 because fly height exceeds the $\theta_w$63 regime.

5. Control, transition, and mission-phase scheduling

In the flapping-wing system, control enters primarily through the pitching trajectory and stroke selection (Izraelevitz et al., 2014). Pitching $\theta_w$64 is not a free sinusoid but is computed to impose a prescribed effective angle of attack, which makes the system explicitly dependent on local flow-angle variation created by in-line and flapping motions. The summary identifies model-based optimization of $\theta_w$65 as an open challenge, specifically to suppress transient force spikes and improve cycle-averaged performance. This emphasizes that amphibious flapping is not only a matter of mean force production but also of managing unsteady loading during rotation and reversal.

In the collapsible-wing robot, the control architecture is stated more explicitly (Xiong et al., 2023). The desired wing angle $\theta_w$66 is set according to flight stage—take-off, glide, or dive—and actuator pressure follows a PID law:

$\theta_w$67

Using the actuator relation, $\theta_w$68. The summary also defines

$\theta_w$69

with desired angle of attack approximately $\theta_w$70 for maximum $\theta_w$71, and roll/pitch moment commands

$\theta_w$72

During water exit, $\theta_w$73 and $\theta_w$74 are scheduled to maintain positive net lift until clear of water.

The transition dynamics combine gravity, buoyancy, propulsive thrust, aerodynamic lift and drag, and hydrodynamic forces. In this sense, the amphibious wing is embedded in a full multi-domain force balance rather than acting as an isolated airfoil. A plausible implication is that transition performance depends as much on timing and coordination of wing state as on static aerodynamic or hydrodynamic coefficients.

6. Scalability, design principles, and unresolved questions

The flapping-wing study reports that force coefficients are weakly sensitive to Reynolds number in the range $\theta_w$75–$\theta_w$76, but strongly dependent on $\theta_w$77 and $\theta_w$78 (Izraelevitz et al., 2014). It recommends maintaining $\theta_w$79 for peak propulsive efficiency and scaling $\theta_w$80 when changing speed or fluid. It also notes that spanwise pitching extent must be kept narrow for large $\theta_w$81 to match rapid flow-angle variation, whereas for symmetric flapping a linear pitch distribution across a larger span is acceptable. The chord/span design $\theta_w$82 and $\theta_w$83 gave $\theta_w$84 and allowable joint ranges; future vehicles may adjust $\theta_w$85 and $\theta_w$86 to meet different Reynolds-number scaling or structural mass constraints.

The collapsible-wing study formulates a parallel set of design principles (Xiong et al., 2023). Soft hydraulic actuators permit compact folding, shock tolerance, and precise control through $\theta_w$87, but require fluid reservoirs and pumps. Liquid fill of $\theta_w$88 reduces reservoir size by $\theta_w$89 versus air, but there is a trade-off between pump flow rate $\theta_w$90 and wing-fold response time $\theta_w$91. The aerodynamic operating point remains centered on $\theta_w$92 for $\theta_w$93, which the summary presents as central for pitch stability. For ground-effect exploitation, clearance should be maintained at $\theta_w$94 to obtain up to $\theta_w$95 glide distance.

Both studies also specify unresolved engineering problems. For the flapping wing, these include robust wing structural design using a streamlined flooded shell, flexible skin, or compliant joints, and integration of body dynamics with free-flight testing to confirm vehicle stability under combined unsteady loads (Izraelevitz et al., 2014). For the collapsible-wing robot, future directions include integration of a miniature electro-hydraulic pump, optimization of wall thickness for faster actuation, variable-camber membrane sections to extend the linear $\theta_w$96 range, reinforcement fibers to raise $\theta_w$97 without early stall, and integrated strain-sensor arrays for closed-loop wing-shape feedback (Xiong et al., 2023).

A common simplification is to treat amphibious operation as requiring either a conventional aerial wing or a conventional underwater propulsor, with limited coupling between the two regimes. The studies reviewed here do not support that simplification. Instead, they show two concrete alternatives: a single flapping actuator whose in-line degree of freedom widens the force envelope across media, and a collapsible membrane wing whose geometric state is scheduled across take-off, glide, dive, and re-entry. This suggests that the central research problem is not merely dual-medium survivability, but coordinated generation and regulation of force under large changes in density, added mass, and transition topology.