robotics.sciencemag.org/cgi/content/full/5/38/eaay1246/DC1

Supplementary Materials for

Soft biohybrid morphing wings with feathers underactuated by wrist and

finger motion

Eric Chang, Laura Y. Matloff, Amanda K. Stowers, David Lentink*

*Corresponding author. Email: dlentink@stanford.edu

Published 16 January 2020, Sci. Robot. 5, eaay1246 (2020)

DOI: 10.1126/scirobotics.aay1246

This PDF file includes:

Supplementary Text Fig. S1. Rubber band force versus length properties. Fig. S2. Templates used for rubber band selection and tuning. Table S1. PigeonBot feathers were 90% from the same individual; the wind tunnel model feathers were 100% from the same individual. Table S2. Rubber bands used to connect PigeonBot feathers. Table S3. Rubber bands used to connect wind tunnel model feathers. Table S4. Bill of materials for constructing PigeonBot. References (45, 46)

Supplementary Materials

PigeonBot preliminary flight tests

During preliminary tests, we flew our morphing wing prototypes on the PigeonBot test platform

(Fig. 2A) under manual teleoperated control to simplify qualitative performance evaluation. In

preparation for evaluating our feathered biohybrid morphing wings, we first test flew three

‘static’ pigeon planforms; fully extended, intermediate, and tucked wings. The fully extended

planform is shown in Fig. 2E,1. These three planforms were based on the three planforms

reported by Pennycuick for pigeons gliding in a wind tunnel (21). We tested these planforms

with the aerial robot platform to determine the necessary tail volume (the product of tail area and

arm; see result in Fig. 2A,C) given the pigeon wing planform area and chord. This helps ensure

that PigeonBot flies sufficiently stably across all morphed states with minimal autopilot input for

the elevator.

Biohybrid wing prototypes

To construct a variable sweep biohybrid wing, we first tested a foamboard ‘swing wing’ to

demonstrate that PigeonBot could be manually controlled during dynamic wing morphing (Fig.

2E,3). We then glued primary feathers to a foamboard hand wing skeleton (Fig. 2E,4) and varied

its sweep in flight, after which we also glued secondary feathers to a foamboard arm wing

skeleton and tested the feathered ‘swing wing’ in flight (Fig. 2E,5). We then created our first

underactuated morphing wing prototype (Fig. 2E,6). Each flight feather was connected with a

single rotational degree of freedom to a more sophisticated multi-layered wooden wing skeleton

with an articulated wrist. The feathers were heat shrunk to LEGO minifigure hands (LEGO Part

PigeonBot’s soft biohybrid morphing wing was developed through systematic bioinspired

prototyping and flight testing with successively increasing complexity. Following the

preliminary flight tests in which we test flew three 'static' pigeon wing planforms, we created our

first biohybrid wing prototype by simply gluing pigeon flight feathers in their fully extended

position to a foamboard skeleton (Fig. 2E,2).

983) that snapped on and released easily from cylindrical carbon fiber tubes that formed the pin

joints, while enabling each feather to rotate freely. The feathers were elastically connected via a

custom-made rubber band that was woven around each calamus (base of the feather shaft) of the

20 remiges, with the ends of the rubber band connected to the wing root and hand bone. This

design enabled each wrist to underactuate 20 remiges in each wing half. However, without an

active finger joint, this design did not extend or tuck as much as pigeons when gliding (21).

Additionally, when perturbed by a gust or maneuver, the feathers started to slide along the rubber

band and cluster in flight, complicating flight control. We integrated learnings from these

prototypes in our final soft underactuated biohybrid morphing wing design (Fig. 2E,7).

where 𝜖 is the strain of the postpatagium. When a feather is deflected over an angle about its

feather pin joint, 𝜃, the strain, 𝜖, as a function of width, 𝑤, is

𝜖(𝑤) = 𝑤 sin 𝜃

𝑑0. (2)

The associated moment, d𝑀, generated by a thin slice of the postpatagium, 𝑑𝑤, is

𝑑𝑀 = 𝑑𝐹 ∗ 𝑤 = 𝐸𝑤2𝑡∗sin 𝜃

𝑑0𝑑𝑤. (3)

We define the torsional stiffness, T, as the moment per angle deflection, or

𝑇 = 𝑑𝑀

𝑑𝜃 . (4)

Building off the small angle approximation, sin 𝜃 ≈ 𝜃, and integrating along the width of the

postpatagium we find,

𝑇 = ∫𝐸𝑤2𝑡

𝑑0𝑑𝑤

𝑊

0. (5)

Evaluating the approximated torsional stiffness for estimated values; 𝑑0 = 5 mm, 𝑡 = 1 mm, 𝑊 =

20 mm, and 𝐸 = 1-100 kPa (45), we find that the torsional stiffness ranges from 0.0005 – 0.05

Nm/rad due to the wide range of reported elastic moduli.

We model the torsional stiffness of our underactuated biohybrid wing by linearly

approximating the stiffness of the rubber band, 𝑘. In this analysis we assume the adjacent

feathers are approximately parallel and that the rubber band is attached at radial distance, 𝐿,

measured with respect to the center locations of the 2D pin joints. The moment, 𝑀, about a

feather pin joint for some feather deflection angle, 𝜃, is

𝑀 = 𝑘𝐿2 sin 𝜃. (6)

Again, applying small angle approximation, sin 𝜃 ≈ 𝜃, the torsional stiffness, T, is

𝑇 = 𝑘𝐿2. (7)

Postpatagium versus elastic band stiffness

We model the postpatagium as a linearly elastic smooth muscle with modulus 𝐸 connecting two

parallel feathers with 2D pin joints with an initial feather spacing, 𝑑0, thickness, 𝑡, and width

along the shaft of the feather, 𝑤. The force, dF, generated by a thin slice of the

postpatagium, 𝑑𝑤, is

𝑑𝐹 = 𝐸𝜖𝑡 ∗ 𝑑𝑤, (1)

For our wing-half, the total number of links including the wing base is 𝑛 = wing base + 1 hand

bone + 1 finger bone + 20 feathers = 23 links. The associated number of kinematic pairs 𝑗 = 1

hand bone + 1 finger bone + 20 feathers = 22. Summing the number of degrees of freedom, 𝑓𝑖,

per kinematic pair, i, across all kinematic pairs, j, we find ∑ 𝑓𝑖𝑗𝑖=1 = (2 skeletal joints + 19 feather

joints) × 1 + (1 rigid connection between the finger bone and P10) × 0 = 21. Thus the degrees of

freedom per wing half is 3(23 − 22 − 1) + 21 = 21, or 42 total.

Estimated fundamental natural frequency of the elastic ligament

We estimate the order of magnitude of the fundamental natural frequency, 𝑓0, of the elastic

ligament using Rayleigh’s method based on the ratio of potential and kinetic energy (28)

𝑓0 =1

2𝜋√𝑋𝑇𝐾𝑋

𝑋𝑇𝑀𝑋, (9)

where 𝑋 is the assumed displacement vector prescribing harmonic motion and 𝐾, 𝑀, are the

generalized stiffness and mass matrices respectively. To estimate the order of magnitude of 𝑓0

for our biomimetic elastic ligament, which underactuates the flight feathers, we formulate order

of magnitude estimates for each element of the 𝑋, 𝐾, 𝑀 matrices through nondimensionalization

𝑂[𝑓0] = 𝑂 [1

2𝜋√𝑋𝑇𝐾𝑋

𝑋𝑇𝑀𝑋 ], (10)

in which we nondimensionalize the elements, 𝐾𝑖𝑗, of the stiffness matrix 𝐾, as

𝐾𝑖𝑗 = 𝑘 ∗ 𝐾𝑖𝑗∗ , (11)

where 𝑘 is the characteristic stiffness scale and 𝐾𝑖𝑗∗ is the variable nondimensionalized stiffness

with 𝑂[1] (for non-zero matrix elements). Similarly, we nondimensionalize each element, 𝑀𝑖𝑗, of

the generalized mass matrix, 𝑀, as

𝑀𝑖𝑗 = 𝑚 ∗ 𝑀𝑖𝑗∗ , (12)

Evaluating this equation with approximate values of 𝑘 = 100 Nm-1 (Fig. S1) and 𝐿 = 20 mm, we

obtain a torsional stiffness of 0.04 Nm/rad. The calculated stiffness due to the rubber bands is

therefore within the stiffness range calculated for the postpatagium modeled as elastic smooth

muscle.

Degrees of freedom of the underactuated wing

The degrees of freedom, DoF, embodied in the wing-half also follow from the Kutzbach-Grübler

equation for planar mechanisms (46)

𝐷𝑜𝐹 = 3(𝑛 − 𝑗 − 1) + ∑ 𝑓𝑖𝑗𝑖=1 . (8)

where L and R are the characteristic length scales that contribute to potential and kinetic energy

respectively, and 𝑋𝑃𝐸,𝑖𝑗∗ and 𝑋𝐾𝐸,𝑖𝑗

∗ are nondimensionalized variable displacements with 𝑂[1] (for

non-zero matrix elements). Substituting Eqns. 11-14 back into Eqn. 10, we obtain

𝑂[𝑓0] = 𝑂 [1

2𝜋√

(𝐿∗𝑋𝑃𝐸∗ )

𝑇(𝑘∗𝐾∗)(𝐿∗𝑋𝑃𝐸

∗ )

(𝑅∗𝑋𝐾𝐸∗ )

𝑇(𝑚∗𝑀∗)(𝑅∗𝑋𝐾𝐸

∗ ) ]. (15)

Since each non-zero element of the nondimensionalized matrices, 𝑋𝑃𝐸,∗ 𝑋𝐾𝐸

∗ , 𝐾∗, and 𝑀∗ are

𝑂[1], we estimate the order of magnitude of the fundamental natural frequency as

𝑂[𝑓0] = 𝑂 [𝑐𝑜𝑛𝑠𝑡 ∙1

2𝜋√

𝐿2𝑘

𝑅2𝑚 ] =

1

2𝜋

𝐿

𝑅√

𝑘

𝑚, (16)

because

𝑂 [√𝑋𝑃𝐸

∗ 𝑇𝐾∗𝑋𝑃𝐸

∗

𝑋𝐾𝐸∗ 𝑇

𝑀∗𝑋𝐾𝐸∗

] = 𝑂[𝑐𝑜𝑛𝑠𝑡] = 1. (17)

Based on established natural frequency analyses for coupled spring-damper-mass systems, it is

reasonable to assume the constant, const, is close to 1 (28). Based on the characteristics of our

elastic ligament and the pigeon feathers used, we calculate the magnitude of the fundamental

natural frequency, 𝑓0, as follows. We approximate 𝑘 ≈ 100 N/m based on the approximate slope

of Fig. S1, and 𝐿 ≈ 0.02 meter as the distance from the spring attachment to the feather pin joint.

We sampled three left P1 pigeon primary feathers (in the middle of the wing) and found their

average mass to be 0.118 ± 0.001 gram and length to be 0.122 ± 0.002 meter. Based on these

values we estimate the feather mass as 𝑚 ≈ 0.0001 kg and the distance between the pin joint

and the center of mass of the feather as 𝑅 ≈ 0.06 meter, half the feather length. Substituting

these values into Eq. 16, we find

where m is the characteristic mass scale and 𝑀𝑖𝑗∗ is the variable nondimensionalized stiffness

with 𝑂[1] (for non-zero matrix elements). Finally, we distinguish the displacement vector in the

numerator of Eq. 9 associated with the potential energy in the elastic ligament, 𝑋 𝑃𝐸, and the

displacement vector in the denominator of Eq. 9 associated with the kinetic energy due to the

velocity of the feather masses, 𝑋𝐾𝐸. We nondimensionalize each element, 𝑋𝑃𝐸,𝑖𝑗, of the

displacement vector 𝑋𝑃𝐸 and each element, 𝑋𝐾𝐸,𝑖𝑗, of the displacement vector 𝑋𝐾𝐸 as

𝑋𝑃𝐸,𝑖𝑗 = 𝐿 ∗ 𝑋𝑃𝐸,𝑖𝑗∗ , (13)

and

𝑋𝐾𝐸,𝑖𝑗 = 𝑅 ∗ 𝑋𝐾𝐸,𝑖𝑗∗ , (14)

2.17. Superfluous points in the motion tracking that did not correlate to a placed marker on the

feather or bone were not labeled or included. The labeled marker position data were

subsequently filtered with a fourth order low-pass Butterworth filter with a cutoff frequency of 8

Hz to smooth jitter. Tracking error arises because we operate at the limit of the motion tracking

capability using a high number (55) of tiny (2.4 mm) markers on feathers. The flight feathers are

also IR reflective, and secondary feathers overlap during wing tucking, which caused marker

occlusion when the wing is tucked. Due to occlusion of the secondary feathers during flexion, we

do not have the same number of data points for each wrist angle, since fewer feathers are visible

at lower wrist angles. We binned the motion capture data into 0.5° bins of wrist and finger

angles. This resulted in a non-fixed number of data points in each bin when calculating the mean

and standard deviation whenever secondary feather markers were occluded (during wing

tucking).

Computational bone markers for principal component analysis

To perform the principal component analysis on the skeletal motion that underpins avian wing

morphing, we created computation markers along each of the four major wing bones (humerus,

ulna, radius, manus) to define a bone axis for each bone as in Stowers et al. (14). We define the

bone axes to be right-handed coordinate systems comprising the principle axes of each bone,

with the x-axis corresponding to the minimal principle axis pointing distally. We placed the

origin at the center of mass and place the computational marker points 30 mm out along the

for a total of 4 markers per bone. The coordinates of all the data were centered on the

mean, and not normalized.

al

𝑂[𝑓0] =1

2𝜋

𝐿

𝑅√

𝑘

𝑚= 50 𝐻𝑧. (10)

Pigeon wing measurements, additional details

We fixed the body of the cadavers to a custom stand and animated the wings following the path

of least resistance through wing flexion and extension cycles defined by wing planforms for a

gliding pigeon (21). The wings were animated by hand at 0.5 Hz per flexion/extension cycle,

using a metronome to ensure equivalent data between all trials. Each of the four main bones and

20 feathers were tracked with motion tracking marker clusters (three for a reference frame, three

per bone, and two per feather) for a total of 55 markers. The markers were manually labeled by

visually inspecting motion capture data overlaid on video data using Qualisys Tracking Manager

x, y,

and z-axes

mode to get the robot in position, an intermediate mode with a selection of robot inputs held

constant or under closed-loop control by the autopilot before starting the trial, and finally, a wing

asymmetry mode with all robot inputs held constant during the trial. We began data collection in

the manual mode by hand-launching and manually teleoperating the robot with wings spread by

controlling its throttle, elevator, and rudder. We trimmed the control surfaces by adjusting the

neutral position of the elevator and rudder so that the robot flew straight and level at each

symmetric wing planform before starting each wing asymmetry trial. To reduce outdoor wind

effects on the results, we flew each trial alternating opposite directions (approximately north and

outh). Once positioned by eye to begin a wing asymmetry trial, we engaged an intermediate

flight mode meant to standardize the initial conditions of each trial. In this intermediate flight

mode, the pilot adjusted the rudder to maintain a level orientation while the motor throttle of the

propeller was set and locked to 68% and the elevator was under autopilot control to maintain a

level pitch orientation Finally, we engaged the wing

mode, which commanded a predetermined asymmetric wing planform while

locking the rudder at its neutral position, continuing to lock the throttle at 68%,

with autopilot elevator control. We ended each trial by returning the robot to

teleoperation with symmetric wings when we determined that the robot’s position or

was no longer safe for reliable operation near the ground.

s

.

PigeonBot static stability analysis

We calculated the static longitudinal flight stability of PigeonBot in different wing

configurations by simulating the aerodynamic characteristics of the wing in XFLR5 6.47 using

the Wortmann FX 60-126 airfoil in the biohybrid wing (based on in vivo airfoils of gliding

pigeons) and the measured center of gravity of PigeonBot. We ran a vortex lattice method

(VLM2) analysis at PigeonBot’s approximate cruise speed of 10 m/s. The predicted static margin

was 10.3% with extended wings and 16.2% with tucked wings, which shows that PigeonBot is

longitudinally stable throughout the wing’s morphing range, as observed in flight.

PigeonBot control and data logging system

The PigeonBot control and data logging system consists of an electric motor (T-motor F20),

speed controller (Castle Creations Talon 15), and propeller (APC 6x4) for thrust, an elevator and

rudder actuated by servos (HK5330) and pushrods, an autopilot system (mRobotics PixRacer

R14) to log instrumentation data, a radio receiver (FrSky R-XSR) and transceiver (Micro

HKPilot Telemetry radio) for communication, and a battery (Zippy Compact 850mAh 2S 25C).

The autopilot system had redundant sensors consisting of a 3-axis accelerometer/gyroscope

(Invensense ICM_20608-G), a 3-axis accelerometer/gyroscope/magnetometer (Invensense MPU-

9250), a barometer (MEAS MS5611), a magnetometer (Honeywell HMC5983), a GPS module

(u-Blox Neo-M8N and LIS3MDL and IST8310 dual compasses), and an airspeed sensor (3DR

Airspeed Sensor).

PigeonBot flight data collection procedure

To test the robot’s response to wing morphing asymmetry, we utilized three sequential flight

modes to keep the initial robot pose for each trial as consistent as possible: first a fully manual

,

[ArduPilot v3.8.3 FBWA (FLY BY WIRE_A)]

asymmetry flight

simultaneously

and continuing

orientation

manual

Fig. S1. Rubber band force versus length properties. We characterized the force versus length

properties for each rubber band type used between each feather for wind tunnel and free-flight

models ( = 3 bands per type, shaded regions show standard deviations). The different force vs.

length profiles for each band enabled the feather motion to be tuned by selecting different bands

between different feathers.

Fig. S2. Templates used for rubber band selection and tuning. Templates were printed at 1:1

scale with feathers oriented at the same angles recorded from the wing animation motion capture

study. We tuned the rubber band selection in between each feather so that the feather angles

matched the templates in extended (A) and tucked (B) poses.

n

Table S1. PigeonBot feathers were 90% from the same individual the wind tunnel model

feathers were 100% from the same individual. We substituted S1 and P10 in the left and right

wings from another individual because the original feathers were not fully developed pin

feathers. Feathers for the wind tunnel model all came from the same individual.

;

Left wing Right wing

Feather Individual Feather Individual

S10 A S10 A

S9 A S9 A

S8 A S8 A

S7 A S7 A

S6 A S6 A

S5 A S5 A

S4 A S4 A

S3 A S3 A

S2 A S2 A

S1 B S1 B

P1 A P1 A

P2 A P2 A

P3 A P3 A

P4 A P4 A

P5 A P5 A

P6 A P6 A

P7 A P7 A

P8 A P8 A

P9 A P9 A

P10 B P10 B

Table S2. Rubber bands used to connect PigeonBot feathers .

Feather Pair Left wing Right wing

P10-9 1/4” Light 1/4” Light

P9-8 1/4” Light 1/4” Light

P8-7 5/16” Light 3/8” Medium

P7-6 1/4” Medium 1/4” Light

P6-5 1/4” Medium 1/4” Medium

P5-4 1/4” Medium 1/4” Medium

P4-3 1/4” Medium 1/4” Medium

P3-2 3/16” Medium 1/4” Medium

P2-1 1/4” Medium 3/16” Medium

P1-S1 5/16” Light 3/8” Medium

S1-2 1/4” Medium 1/4” Medium

S2-3 1/4” Medium 1/4” Medium

S3-4 1/4” Medium 1/4” Medium

S4-5 1/4” Medium 1/4” Medium

S5-6 1/4” Medium 1/4” Medium

S6-7 1/4” Medium 1/4” Medium

S7-8 1/4” Medium 1/4” Medium

S8-9 1/4” Medium 1/4” Medium

S9-10 1/4” Medium 1/4” Medium

S10- 3/16” Medium 3/16” Medium

Table S3. Rubber bands used to connect wind tunnel model feathers .

Feather Pair Band

P10-9 1/4” Light

P9-8 5/16” Light

P8-7 5/16” Light

P7-6 1/4” Medium

P6-5 1/4” Medium

P5-4 1/4” Medium

P4-3 1/4” Light

P3-2 1/4” Light

P2-1 1/4” Medium

P1-S1 5/16” Light

S1-2 1/4” Medium

S2-3 1/4” Medium

S3-4 1/4” Medium

S4-5 1/4” Medium

S5-6 1/4” Medium

S6-7 1/4” Medium

S7-8 1/4” Medium

S8-9 1/4” Medium

S9-10 1/4” Medium

S10- 3/16” Medium

Table S4. Bill of materials for constructing PigeonBot .

Item Part Quantity

Paper covering 0.2mm water resistant paper 1

Rubber bands Prairie Horse Supply Orthodontic Elastic Rubber

Bands (3/18” medium, 1/4” light, ¼” medium, 5/16”

light, 3/8” medium

40

3D printed bones Custom, Shapeways 5

Wrist shaft 12mm x 2mm Dowel Pin 2

Finger shaft 10mm x 2mm Dowel Pin 2

Brass bushings 3.5mm x 2mm x 1.4mm 16

Wrist servo Turnigy T541BBD digital 2

Finger servo HK282A 2

Heat shrink HS3A-0125 adhesive heat shrink 40

Teflon sheet 0.003” Teflon sheet 8

3D printed pin joints Custom, Ultimaker 2+ 40

Feather pin joint shaft 1mm Dowel Pin 40

Feather elastic band pin Sewing pins 40

Servo pushrod 1mm music wire 4

Ribs 1/8” balsa 4

Fuselage Flite Test Water-Resistant Foam Board 1

Tail 3mm Depron 2

Tail leading edge 3mm x 0.5mm carbon fiber strip 3

Tail servos HK-5330 2

Tail servo pushrod 0.5mm music wire 2

Motor T-motor F20 1

Speed controller Castle Creations Talon 15 1

Propeller APC 6x4 1

Autopilot mRobotics PixRacer R14 1

Radio receiver FrSky R-XSR 1

Radio transceiver Micro HKPilot Telemetry Radio 1

Battery Zippy Compact 850mAh 2s 25C 1

GPS u-Blox Neo-M8N and LIS3MDL and IST8310 dual

compasses

1

Airspeed sensor 3DR airspeed sensor 1