Supplementary Materials for
A bioinspired revolving-wing drone with passive attitude stability and
efficient hovering flight
Songnan Bai et al.
Corresponding author: Pakpong Chirarattananon, pakpong.c@cityu.edu.hk
Sci. Robot. 7, eabg5913 (2022)
DOI: 10.1126/scirobotics.abg5913
The PDF file includes:
Sections S1 to S7
Figs. S1 to S15
Table S1
References (83–89)
Other Supplementary Material for this manuscript includes the following:
Movies S1 to S5
S1
S1.1
Flight dynamics
Translational dynamics
The translational dynamics of the revolving-wing robot is obtained by modeling the robot as a
rigid body with mass m under the gravitational field g. Let X̂ŶẐ denote the inertial frame W
and x̂ŷẑ be the body-fixed frame B, situated at the center of mass of the robot (Fig. 1A). The
ŷ axis aligns with the thrust direction of the propeller 1 (Fig. 1B). The ẑ axis aligns with the
direction of the total lift force of the revolving wings. With basis vectors e1 , e2 , and e3 , the
position of the robot P is governed by
mP̈ = R (e2 tp,1
e2 tp,2 + fw )
mge3 ,
(S1)
where R denotes a rotation matrix mapping the body frame to the inertial frame, tp,1 and tp,2
are the magnitudes of the propellers’ thrusts (Fig. 1B), and fw is the total aerodynamic force
attributed to the revolving wings, defined in the body frame, given by
fw = e3 CT ⌦2z
(S2)
bRT Ṗ.
The first term of fw denotes the thrust force produced by revolving wings (see Eq. 3), aligned
with the ẑ axis. The other term represents the drag induced by of the translation of the revolving wings. The drag is primarily caused by the drag difference of the advancing and retreating
blades when the robot travels (56, 71, 83), assumed to be linear with the body-centric translational velocity (RT Ṗ) with the coefficient b.
S1.2
Attitude dynamics
The attitude of the robot is fully described by Euler’s rotation equations in B
˙ + ⌦ ⇥ (I⌦) = ⌧prop + ⌧w ,
I⌦
(S3)
where, I = diag (Ix , Iy , Iz ) is the body-frame inertial tensor, vector ⌦ =
⇥
⌦ x ⌦y ⌦ z
⇤T
is the angular velocity, and the right-hand side presents the applied torques. The torques are
from (i) the actuation torque provided by the propellers’ thrusts,
propellers’ induced torque,  (tp,1
rm (tp,1 + tp,2 ) e3 and the
tp,2 ) e2 for the thrust-to-torque coefficient , collectively
termed
⌧prop =
rm (tp,1 + tp,2 ) e3 +  (tp,1
tp,2 ) e2 ,
(S4)
and (ii) the sum of torques produced by the revolving wings, ⌧w .
The translation of the revolving wings with relative large areas results in several contributing
factors, including torque from the rotating motion ⌧b , from the aerodynamic force fw from Eq.
S2, ⌧t , and from the dissymmetry of lift ⌧d , such that
⌧w = ⌧b + ⌧t + ⌧d .
(S5)
The term ⌧b combines attitude damping torque and possible contribution from the wings. It is
modeled as a torque in the opposite direction of the angular velocity ⌦. Since for the revolvingwing robot ⌦z
⌦x , ⌦y , we employ a linear law to model ⌧b in x̂ and ŷ directions, whereas
the torque along ẑ is taken from Eq. 3:
⌧b =
⇥
c⌦x c⌦y CQ ⌦z |⌦z |
⇤T
(S6)
,
where c is the linear coefficient. The term ⌧t is computed from the aerodynamic force from the
wings fw and the vertical offset between the center of mass of the robot and center of pressure
of the revolving wings lcp as
⇣
⌧t = lcp e3 ⇥ fw = lcp e3 ⇥ e3 CT ⌦2z
= lcp be3 ⇥
⇥
e 1 e2 e3
⇤T
RT Ṗ = lcp b
⇥
e2
bRT Ṗ
⌘
e1 0
⇤T
(S7)
RT Ṗ,
(S8)
where lcp > 0 when the center of pressure of the revolving wings is above the center of mass
of the robot. The term ⌧d captures the dissymmetry of lift (84) of the revolving wings from
the translating motion. The relative airflow experienced by the advancing wing is elevated
whereas the relative wind speed acting on the retreating blade is reduced. The resultant torque
is approximately linear against the translational speed and revolving speed as
⌧d =
where
⌦z
⇥
e 1 e2 0
⇤T
(S9)
RT Ṗ.
is the corresponding coefficient. Together, Eqs. S3-S9 describe the attitude dynamics
in the body-fixed frame.
S1.3
Hovering solution
The equilibrium hovering condition is defined as Ṗ, P̈ = 0. When this is applied to the complete
flight models detailed by Eqs. S1 and S3, the hovering solution (denoted by ⇤ ) can be found as
ẑ⇤ = Ẑ,
t⇤p,1 = t⇤p,2 =
1
CQ ⌦⇤2
z ,
2rm
and
⌦⇤ =
⇥
⌦⇤x ⌦⇤y ⌦⇤z
⇤T
=
⇥
p
0 0
mg/CT
That is, the robot stays upright with a constant revolving speed ⌦⇤z .
S1.4
⇤T
.
(S10)
Reduced flight dynamics
Under the assumption of small deviations from the hovering condition (ẑ⇤ = Ẑ) from Eq. S10,
the two-dimensional vector ⇠ = [⇠x , ⇠y ]T with ⇠x , ⇠y ⌧ 1 in Fig. 3A is defined to represent
the robot’s inclination as ẑ ⇡ [⇠y , ⇠x , 1]T . Let
be the yaw angle of the robot, the rotation
matrix can be approximated as
2
cos
R ⇡ 4 sin
⇠y
3
⇠y
⇠x 5 .
1
sin
cos
⇠x
(S11)
As a result, the translational dynamics of the robot from Eqs. S1 and S2 can be separated into
the altitude and lateral dynamics as
mz̈ = (tp,1
tp,2 ) ⇠x + CT ⌦2z
and
mp̈ = (tp,1
tp,2 )

sin
cos
bż
(S12)
mg
+ CT ⌦2z I⇥ ⇠
bṗ,
(S13)
in which p = [e1 , e2 ]T P = [x, y]T is the horizontal position of the robot and I⇥ is a 2 ⇥ 2
skew-symmetry matrix:
I⇥ =

0 1
1 0
.
(S14)
Meanwhile, the attitude of the robot near the hovering condition can also be re-examined under
the notion of ⇠. Abstracting the revolving robot as an axisymmetric spinning disk with an
inertia I = diag (Id , Id , Iz ) and a constant angular rate ⌦⇤z as depicted in Fig. 3A, the angular
momentum of the robot in W is
L = Id ⇠˙x x̂m + Id ⇠˙y ŷm + Iz ⌦z ẑ,
(S15)
where x̂m ⇡ [1, 0, ⇠y ]T and ŷm ⇡ [0, 1, ⇠x ]T are unit vectors along the axes of the non-inertial
frame {x̂m ŷm ẑb } shown in Fig. 3A. The time derivative of L is equated to the sum of the
torques applied to the robot (⌧prop + ⌧w from Eq. S3) projected onto the inertial frame W,
⇣
L̇ = Id ⇠¨x x̂m + Id ⇠¨y ŷm + Iz ⌦˙ z ẑ + Iz ⌦z ⇠˙y x̂m
⌘
⇠˙x ŷm = R (⌧prop + ⌧w ) ,
⇠˙y ẑb , ŷ˙ m = ⇠˙x ẑb , and ẑ˙ b = ⇠˙y x̂m
where we have used the fact that x̂˙ m =
(S16)
⇠˙x ŷm . Focusing
on ⇠x and ⇠y , we take the projection of L̇ along x̂m and ŷm :
x̂Tm L̇ = Id ⇠¨x + Iz ⌦z ⇠˙y = x̂Tm R (⌧prop + ⌧w ) ,
(S17)
T
ŷm
L̇ = Id ⇠¨y
(S18)
T
Iz ⌦z ⇠˙x = ŷm
R (⌧prop + ⌧w ) ,
which can be consolidated as
Id ⇠¨ + Iz ⌦⇤z I⇥ ⇠˙ =
⇥
x̂m ŷm
⇤T
(S19)
R (⌧prop + ⌧w ) ,
Substituting in the expression of ⌧prop and ⌧w from Eqs. S4-S9 and R from Eq. S11, Eq. S19
becomes
✓

sin
cos
sin
˙
c⇠+lcp b I⇥ ṗ + ż
tp,2 )
cos
sin
cos
✓

◆
cos
sin
⌦z ṗ ż
I⇥ ⇠ + O |⇠|2
sin
cos

sin
c⇠˙ + lcp bI⇥ ṗ
⌦z ṗ
=  (tp,1 tp,2 )
cos

cos
sin
( lcp b
⌦z I⇥ ) ⇠ + O |⇠|2
+ż lcp b
sin
cos
|
{z
}
¨ z ⌦z I⇥ ⇠˙ =  (tp,1
Id ⇠+I

⇠
◆
(S20)
⌧
where we have applied the fact that ⌦x cos
⌦y sin
= ⇠˙x and ⌦x sin
+ ⌦y cos
= ⇠˙y .
Lastly, the yaw dynamics is obtained from the projection of Eq. S16 along ẑ,
Iz ⌦˙ z = ẑT R (⌧prop + ⌧w ) =
rm (tp,1 + tp,2 )
CQ ⌦z |⌦z | .
(S21)
Together, Eqs. S12, S13, S20 and S21 capture the translational and attitude dynamics of the
revolving-wing robot near its hovering condition (ẑ⇤ = Ẑ).
S2
Passive attitude stability
Under certain design parameters, the samara-inspired robot exhibits attitude stability such that
it stays approximately upright (ẑ ! Ẑ) when only the altitude (Eq. S12) is actively controlled.
Understanding the underpinning principles of the observed stability facilitates the development
process, making it possible to parameterize the robot design for passive stability. To shed light
on this phenomenon, the dynamics of ⇠, which is tightly coupled with ṗ, is investigated.
To apply the linear system analysis, we focus on the near hovering condition, or when only
the altitude and yaw rate are controlled (Eq. S12 in the case that z̈, ż = 0). This is concurrently satisfied with the the hovering solution from Eq. S10, or tp,1 = tp,2 =
mg
2rm
1
C ⌦⇤2
2rm Q z
=
(CQ /CT ). Under such scenarios, the dynamics of ⇠ and ṗ from Eqs. S13 and S20 simplify
to
(S22)
mp̈ + bṗ = mgI⇥ ⇠,
Id ⇠¨ + Iz ⌦⇤z I⇥ ⇠˙ + c⇠˙ = (lcp bI⇥
⌦⇤z ) ṗ.
(S23)
To facilitate the analysis, we regard the two equations above as two interconnected multipleinput multiple-output (MIMO) linear subsystems (Fig. S12). In the frequency domain with
the Laplace variable s and L {·} denoting the Laplace transform operator, the feedback system
in Fig. S12 is mathematically described by
L {ṗ} = H (s) L {⇠} ,
(S24)
H (s) =
mg/b
I⇥ ,
(m/b)s + 1
(S25)
L {⇠} =
G (s) L {ṗ} ,
(S26)
G (s) = Id s2 + Iz ⌦⇤z I⇥ s + cs
1
( lcp bI⇥ + ⌦⇤z ) .
(S27)
From this perspective, G (s) is treated as a primary system that takes L {ṗ} as an input. The
input comes from the output of G (s) itself after passing through the controller H (s). Furthermore, the controller is a simple first-order low-pass filter with the cutoff frequency b/m and DC
gain mg/b.
To make the stability analysis tractable, we first assume that the cutoff frequency is sufficiently high with respect to the system’s dynamics, such that H (s) can be approximated as
H (s) ! H̄ (s) = (mg/b) I⇥ . In the time domain, this is equivalent to dropping the inertial
term in Eq. S22. This implies the translational drag dominates—a reasonable supposition for
the revolving-wing robot with relatively large aerodynamic surfaces and small mass. As a result,
we obtain
(S28)
bṗ = mgI⇥ ⇠.
Combining Eqs. S23 and S28 yields a two-dimensional second-order system of ⇠:
Id ⇠¨ + Iz ⌦⇤z I⇥ ⇠˙ + c⇠˙ =
⌦⇤z
lcp mg⇠
mg
I⇥ ⇠.
b
(S29)
As a linear time-invariant system, the Routh–Hurwitz stability criterion is employed to evaluate
the system stability. Apart from trivial conditions that are readily met, there are two important
requirements:
Iz2 ⌦2z + c2
Iz ⌦2z
< lcp ,
+
mgId
bc
Iz ⌦2z
Id gm⌦2z
+
bc
b2 c2
and
2
< lcp .
(S30)
(S31)
The requirements for meeting these constraints are discussed in the main text.
On the other hand, if the cutoff frequency b/m of the low-pass filter H (s) is only moderately
high, the major difference between H (s) and H̄ (s) is the phase delay of the output. When the
closed-loop system in Fig. S12 with the controller H̄ (s) is stable (as the conditions in Eq.
S31 are satisfied), it likely remains stable when H̄ (s) is replaced by H (s) as long as the phase
margin of the system is sufficiently large. By this notion, in practice, we may employ Eq. S31
as guidelines for designing the revolving-wing robot with passive stability.
It is worth emphasizing that in the analysis above, the coupled attitude and lateral dynamics
of the robot is considered. The provided stability proof, therefore, is radically distinct from a
simplified analysis based on Euler’s rotation equations found in (31, 36). Therein, the authors
demonstrate the convergence of the angular velocities (not the attitude) of the vehicles subject
to negligible external forces and torques. In contrast, the system analysis in here proves the
attitude stability in the presence of additional aerodynamic forces and torques.
S3
Robot assembly
In the assembly process, we adopted a custom-made jig with an adjustable angle to accurately
yield the desired pitch angle of the wings (Fig. 4B), matching the 3D printed connectors.
The jig also ensures the wings are kept entirely flat and symmetrical during the procedure.
Closed-up and alternative views of the wings and the robot are shown in Fig. 4A and
C. The photos verify that the wings remain entirely flat after the assembly. The 3D printed
connectors are visible in the closed-up photo (Fig. 4A). Note that the ribs towards the wing
tips are oriented differently due to the tapered wing design.
S4
Wing rigidity
The aerodynamic modeling and optimization of the wings in this work assume the fabricated
wings remain sufficiently rigid without major deformation when they are subject to aerodynamic load. That is, any possible deformation is minor and do not visibly affect the aerodynamic
forces, permitting the wings to be treated as rigid in the modeling. To validate the assumption,
we first visually assessed the rigidity of the structure of the fabricated carbon fiber-reinforced
polyimide wings (including chordwise, spanwise and pitch angle stiffness). Then we compared
the aerodynamic forces of the polyimide wings with that of rigid fiberglass wings. Afterwards,
we re-conducted the test using the entire robot with the motors and propellers mounted on the
polyimide wings. The added experiments were performed to investigate the impact of the spinning propellers.
For the first evaluation, we mounted the fabricated pair of polyimide wings (without motors
and propellers) on the rotating platform driven by a servo motor (Fig. 2B). The wings were
commanded to revolve at different angular speeds, up to 37.7 rad/s (cf. the speed in hover
is ⇡18.8 rad/s). The resultant motion was recorded by a high-speed camera (MotionBLITZ
EoSens mini 2) at 600 frames per second. The footage reveals unnoticeable or negligible wing
deformation in the spanwise and chordwise directions when the rotation speed is below 32
rad/s. In this range, the leading edge remained horizontal and the wing pitch angle was close
to the designated value of
= 19.5 (Fig. 4, S13 and Movie S4). On the other hand, at
⌦z = 37.7 rad/s (far higher than the flight condition), significant deformation was observed in
the form of wing flapping (Fig. S13 and Movie S4). This was a consequence of the periodic
spanwise and chordwise bendings combined.
For comparison, we fabricated fiberglass wings with the same geometry and assembled them
on the platform at the same blade pitch angle (see Fig. 4D). To support the heavier wings,
additional carbon fiber rods were used to make up the spinning platform. High-speed videos of
the spinning fiberglass wings were recorded. In comparison, the degrees of deformation of rigid
wings and reinforced polyimide wings are indistinguishable when ⌦ = 18.8 rad/s (Fig. 4D).
To quantify the aerodynamic properties, we plotted the measurements of force T and torque
Q of both polyimide and fiberglass wings with respect to the spinning rate ⌦. Under the rigid
wing assumptions, both force and torque are expected to be quadratic functions of ⌦ (i.e.,
T = CT ⌦2 and Q = CQ ⌦2 for constant CT and CQ ). Significant deformation would render this
quadratic condition violated (CT and CQ become dependent on ⌦). As shown in Fig. S15,
the experimental data fits the rigid-wing model with very high R squared values for both polyimide wings (CT : 99.75%, CQ : 99.31%) and fiberglass wings (CT : 99.95%, CQ : 99.85%)
for ⌦ < 30 rad/s. Furthermore, best-fitted values of CT and CQ (for ⌦ < 30 rad/s) of polyimide
and fiberglass wings are nearly identical. The differences in aerodynamic coefficients between
both types of wings are negligible: 1% for CT and 2% for CQ . The results attest that, in terms
of aerodynamic forces, the flexibility of polyimide wings is insignificant for the spinning rates
of interest.
In the next set of tests, we mounted the entire robot with polyimide wings on the same
setup. The servo motor was commanded to rotate at the rate 18.8 rad/s (robot’s hovering speed)
without powering the propellers. The footage from the high-speed video displays no detectable
wing deformation in both spanwise and chordwise direction (Fig. S14). The wings remained
flat and rigid during the test. The measurements of thrust and torque generated (yellow star
in Fig. S15) also show little deviation from the previous wing sets polyimide wings and
rigid wings. This verifies that the motors and propellers do not cause any observable structural
deformation.
Next, we conducted the entire robot test with propellers on. The propellers were driven with
the same voltages and commands the robot uses during the hovering flight. Again, no significant
structural deformation was observed (Fig. S14). In particular, the wing pitch angle stayed
near its nominal value of 19.5 . In this case the torque measurement records a near-zero net yaw
torque (green dot in Fig. S15). This is because the drag torque induced by the wing Q is
approximately cancelled out by the counter torque generated by the propellers 2tp rm . The thrust
measurement (green star in Fig. S15), on the other hand, indicates a detectable increase with
respect to other measurement points. We hypothesize that this could be attributed to the wingpropeller interaction. Since the propellers are located above the wings adjacent to the leading
edges, the downstream wake could lower the pressure above the wings and positively impact the
lift. Due to the complexity of the flow interaction, the wing-propeller interaction is not further
scrutinized in this work.
S5
Position controller
The proposed revolving-wing vehicle features a unique flight mode with a high degree of underactuation. To accomplish precise position control, we developed a flight controller based on the
nonlinear dynamics inversion (NDI) technique (81, 85). The controller was devised based on
the reduced flight dynamics, which decouples the altitude from the lateral position. The altitude
and lateral dynamics are separately considered, but simultaneously controlled (as opposed to
being controlled in a cascaded fashion). Thereafter, the outputs from the controller are mapped
to the motor commands in a cycle-averaged manner (70) to deal with the underactuation.
S5.1
Altitude control
Assuming the robot is approximately upright (⇠x ! 0, this is reasonable as ⇠x is simultaneously
controlled by the lateral position controller), the altitude of the robot is governed by Eq. S12.
For a given altitude setpoint zd (t), we define the altitude error z̃ = z
zd . To eliminate the
error, we attempt to impose the following condition on the close-loop dynamics of z̃:
z̃ (3) +
¨+
z,2 z̃
˙+
z,1 z̃
z,0 z̃
= 0,
(S32)
where the notation ·(i) denotes the i-th order derivative. The system is provable asymptotically stable when the tuning parameters
z,i ’s
satisfy the Routh-Hurwitz stability criterion. The
derivatives of z̃ are computed from Eq. S12 with ⇠x = 0 as
z̃˙ = ż
CT 2
⌦
z̃¨ =
m z
(S33)
żd ,
b
ż
m
g
z̈d ,
(S34)
z̃ (3) =
CT
2⌦z ⌦˙ z
m
Substituting the results back into Eq. S32 produces
✓
CT
b
b
CT 2
(3)
2⌦z ⌦˙ z
z̈ zd + z,2
⌦z
ż
m
m
m
m
b
z̈
m
zd .
g
◆
(3)
z̈d
(S35)
+
˙+
z,1 z̃
z,0 z̃
= 0.
(S36)
Thus, to realize Eq. S36 and minimize the altitude error, we regard ⌦˙ z as a virtual control input.
Provided the feedback of the revolving rate ⌦z and the altitude z is available, the control law is
✓
✓
◆
◆
b
C
m
b
T
(3)
2
˙
z̈ + zd
⌦
ż g z̈d
(S37)
⌦˙ z =
z,2
z,1 z̃
z,0 z̃ .
2CT ⌦z m
m z m
To obtain the actual control input, we leverage the yaw dynamics from Eq. 5 or S21. This lets
us treat the total propelling thrusts tp,1 + tp,2 as the actual control input according to
tp,1 + tp,2 =
Iz ˙
⌦z
rm
CQ
⌦z |⌦z | ,
rm
(S38)
Subsequently, the actual altitude control law is the expression of tp,1 + tp,2 from Eq. S37 and
S38.
S5.2
Lateral position control
The lateral position control is complicated by the fact that the translational and attitude dynamics are highly coupled. To regulate the position of the robot, we must simultaneously stabilize
the attitude. This is achieved by considering both the lateral dynamics and reduced attitude dynamics at the same time. To begin, we recall the lateral dynamics (Eq. 7 and S13) and reduced
attitude dynamics (Eq. 8) in the vector form
mp̈ =
bṗ + CT ⌦⇤2
z I⇥ ⇠ + (tp,1
Id ⇠¨ + Iz ⌦⇤z I⇥ ⇠˙ =  (tp,1
tp,2 ) ŷ
(S39)
tp,2 ) ŷ,
c⇠˙ + blcp I⇥ ṗ
⌦⇤z ṗ,
(S40)
where ŷ = [ sin , cos ]T is the projected direction of the body axis (see Fig. 1B). Notice
that (i) both position p and attitude ⇠ states are present in both equations; and (ii) the differential
thrust term (tp,1
tp,2 ) is also visible in both equations. This implies we may treat the horizontal
force f = (tp,1
tp,2 ) ŷ as the virtual control input to stabilize both p and ⇠ at the same time.
This is needed because without considering Eq. S40, the attempt to use f to control p alone
could destabilize the attitude dynamics.
pd and design the close-loop
To do so, we define the horizontal position error as p̃ = p
dynamics of p̃ as
p̃(4) +
p,3 p̃
(3)
+
where, similar to the altitude control,
¨+
p,2 p̃
p,i ’s
p,0 p̃
= 0,
(S41)
are tunable control gains. The stability of p̃ is
p,i ’s
guaranteed when Eq. S41 is realized and
˙+
p,1 p̃
satisfy the Routh-Hurwitz criterion. The higher
order derivatives of p̃ can be derived from Eq. S39 as
¨=
p̃
p̃(3) =
p̃(4) =
b
CT ⌦⇤2
1
z
ṗ +
I⇥ ⇠ + f
m
m
m
p̈d ,
CT ⌦⇤2
b
1
(3)
z
p̈ +
I⇥ ⇠˙ + ḟ pd ,
m
m
m
b (3) CT ⌦⇤2
1
(4)
z
p +
I⇥ ⇠¨ + f̈ pd .
m
m
m
(S42)
(S43)
(S44)
It can be seen that the term ⇠¨ appears in the expression of p̃(4) . This allows us to incorporate
the attitude dynamics (Eq. S40) into Eq. S41. In other words, the desired closed-loop dynamics
(Eq. S41) encompasses both lateral and attitude dynamics.
To determine the control input f that renders Eq. S41 true, we observe that in addition to f ,
¨ p̃(3) , and p̃(4) . To deal with this,
the terms ḟ and f̈ also appear in Eq. S41 after substituting in p̃,
we apply the concept taken from the incremental nonlinear dynamic inversion (INDI) method.
That is, we focus on the change of f . In the discrete-time implementation of sample time
t,
we have
ḟt ⇡
f̈t ⇡
ft
ft
2ft
ft
t
1
,
(S45)
1 + ft 2
,
t2
(S46)
where the subscripts are time indices. With these representations, we are able to compute ft that
make the closed-loop dynamics follow Eq. S41 such that p̃ converges to zero. This guarantees
⇠ ! 0 at the same time.
However, as f = (tp,1
tp,2 ) ŷ is a two-dimensional vector with only a single degree-of-
freedom independent input tp,1 tp,2 (as
in ŷ is not directly controlled), this means the desired
control input f cannot be truly realized. To work around the restriction from the under-actuation,
we exploit the difference in the timescales of the yaw rotation ( ˙ = ⌦z ) and the dynamics. The
lateral position control is achieved on the cycle-average basis. To elaborate, we propose the
following differential thrust command:
tp,1
tp,2 = 2ŷT f ,
(S47)
tp,2 ) ŷ = 2ŷT f ŷ
(S48)
such that
(tp,1
=f+

1
|
2 cos2
sin 2
{z
1
sin 2
2 sin2
f.
(S49)
}
For the revolving-wing robot with substantial yaw rate ⌦z = ˙ , its timescale (t ⇠ 2⇡⌦z 1 ) is
anticipated to be an order of magnitude smaller than that of the translational dynamics, attitude
dynamics, or the controller output f . At this time scale (⇠ 2⇡⌦z 1 ), the averaged value (denoted
by h·i⌦z 1 ) of
f is h
f i ⌦z 1 ⇡ h
i⌦z 1 f . Since h
i⌦z 1 = 0, the proposed differential
command guaranteed
(tp,1
tp,2 ) ŷ ⇡ f .
(S50)
Subsequently, the desired close-loop lateral dynamics (Eq. S41) is realized on the cycle-average
basis. The compromise on the control performance is attributed to the underactuated property
of the proposed two-rotor vehicle.
S5.3
Individual thrust commands
In summary, the proposed altitude controller produces the desired total thrust tp,1 +tp,2 (Eq. S38)
under the assumption that the robot is nearly upright. The lateral position controller ensures that
the attitude of the robot is stabilized together with its horizontal position. This generates the
desired differential thrust tp,1 tp,2 (Eq. S47). Together, each controller constrains one degree of
freedom out of two available independent inputs (tp,1 and tp,2 ). Hence, it is then straightforward
to compute tp,1 and tp,2 .
S5.4
Feedback availability
For the altitude controller, the yaw rate and vertical acceleration were obtained from the onboard
IMU. The altitude and its rate were provided by the motion capture system.
For the lateral position controller, the feedback of p, ṗ, ⇠ and ⇠˙ was taken from the motion
capture system. The heading angle
was estimated by integrating the yaw rate from the on-
board gyroscope at 500 Hz. The angle was periodically corrected using the measurement from
the motion capture system at 50 Hz to resolve the integration drift.
S6
S6.1
Figure of merit and power loading
Figure of merit
The figure of merit of the robot is the ratio of the aerodynamic power of the rotating wings Pa,w
to the mechanical power of the propellers 2⌧p ! :
⌘r =
Pa,w
.
2⌧p !
(S51)
The figure of merit of the revolving-wing subsystem ⌘w differs from ⌘r as its mechanical power
is the work done by the spinning propellers 2tp rm ⌦
⌘w =
Pa,w
.
2tp rm ⌦
(S52)
The figure of merit of the propeller subsystem ⌘p is computed in a similar fashion despite the
presence of the non-zero free stream velocity vs = rm ⌦. This is because the free stream flow
does not appreciably affect the rotor efficiency (56). In this case, the aerodynamic power is the
product of the thrust and the flow velocity vi
⌘p =
vs . Hence,
tp · (vi rm ⌦)
.
⌧p !
(S53)
Lastly, combining Eq. S51-S53 produces the figure of merit of the robot in terms of ⌘w and ⌘p
⌘r =
rm ⌦/vi
·⌘w ⌘p .
1 rm ⌦/vi
|
{z
}
(S54)
In other words, the figure of merit of the robot, which indicates its aerodynamic efficiency, is
primarily influenced by that of the wings and the propellers. It is maximized at ⌘r⇤ = ⌘w ⌘p when
rm ⌦/vi = 0.5.
For the proposed robot, the exact value of the figure of merit of the wings can be calculated
p
using the thrust and torque coefficients. This is because Pa,w = T 3/2 / 2⇢Aw with T = CT ⌦2
and Aw being the propeller disk area and 2tp rm ⌦ = CQ ⌦3 . Since CT = 5.59 ⇥ 10
and CQ = 4.61 ⇥ 10
5
4
Ns2 /rad2
Nms2 /rad2 , we obtain
3/2
⌘w = p
2
where Aw = ⇡ rtip
CT
1
= 0.36,
2⇢Aw CQ
(S55)
2
with rtip = 0.30 m and rroot = 0.06 m. Employing the same
rroot
approach, the figure of merit of the propellers is
3/2
as cT = 1.80 ⇥ 10
8
1
cT
⌘p = p
= 0.33,
2⇢Ap c⌧
Ns2 /rad2 , c⌧ = 1.31 ⇥ 10
10
(S56)
Nms2 /rad2 , and Ap = ⇡0.022 m2 .
The values of ⌘w and ⌘p are in the anticipated region for centimeter-sized actuator disks (9).
Overall, ⌘r = 0.12, lower than ⌘w and ⌘p , but it terms of power loading, the robot benefits from
stacking the two propeller disks together as shown below.
S6.2
Power loading
Since the wingspan of the robot is significantly larger than the diameter of the propellers, the
wings enjoy a higher P Lm compared with that of propellers thanks to the significantly reduced
DL. To be more precise, we can calculate both P Lm,w and P Lm,p in the hovering state of the
robot as follows.
The mechanical power loading of the revolving-wing subsystem is the ratio of thrust T =
mg generated to the mechanical input power CQ ⌦3 . We compute ⌦ from T = mg = CT ⌦2 . As
a consequence
P Lm,w =
mg
= 52.3 g/W.
CQ (mg/CT )3/2
(S57)
To evaluate P Lm,p , we first find the propelling thrust by equating it with the torque of the
revolving wings: 2tp rm = CQ ⌦2 . We get tp = 52.4 mN. The mechanical input of each propeller
is approximated in a similar fashion. Hence
P Lm,p =
tp
= 8.2 g/W.
c⌧ (tp /cT )3/2
(S58)
Lastly, we obtain the ratio P Lm,w /P Lm,p to be 6.4. Therefore
P Lm,p=r ⇡ ⌘p P Lm,w ⇡ 2P Lm,p .
S7
(S59)
Mapping Example
The multi-ranger deck (82) was integrated into the revolving-wing robot to exemplify the potential of the robot in the mapping task. The deck features five laser time-of-flight sensors for
measuring the distances in the four horizontal and dorsal directions. In the demonstration, only
two oppositely aligned horizontal sensors were engaged as the other two were occluded by other
components on the robot (Fig. S9).
For the example flight, the robot was commanded to fly inside the 3.6 ⇥ 3.3-m arena surrounded by three walls (the same environment in Fig. 3E, schematically shown in Fig. 6A).
The robot was controlled to hover near the center of the arena at an altitude of 1 m using the
devised position controller. The distance readings from the two active sensors were logged to
the ground station at 10 Hz. The data were taken over a 18-s period.
To construct the environmental map, distance measurements over 3 m were deemed unreliable and discarded. This is because the sensors give out saturated measurements of ⇡ 3
4
m when nothing is detected. The map presumes the robot was approximately upright so that
the wall location was given by the location of the robot and the measured distance, taking into
consideration the instantaneous direction of the sensor. In other words, the knowledge of the
robot’s absolute position from the motion capture system and its heading angle was used to
generate the presented point cloud (Fig. 6B).
Although the outlined method is rudimentary and still reliant on external feedback from
the motion capture system, it serves as a proof of concept and illustrates the potential of the
revolving platform. To eliminate the dependency on the motion capture system, a technique
based on simultaneous localization and mapping (SLAM) algorithms similar to an example
involving a rapidly yawing multirotor robot in (81) can be further developed and employed.
Figure S1: Application of blade-element momentum theory. (A) A wing, revolving at the
rate ⌦, is split into infinitesimally thin blades specified by the spanwise location r and elemental
thickness dr. (B) The locally perceived airspeed and elemental lift and drag forces on the
elemental blade. Note that for an ideal flat plate, ↵ and ↵e are nominally identical. In practice,
a slight wing deformation and wing ribs may lead to the dissymmetry about the chord line,
creating a minor difference between ↵ and ↵e .
Torque (Nm)
0.4
Thrust (N)
AR=5.71
=
12 °
23 °
32 °
39 °
46 °
0.2
0.06
0.04
0.02
0
0
0
2000
Torque (Nm)
0.6
Thrust (N)
AR=4.43
=
8°
16 °
28 °
37 °
47 °
1000
0.4
0.2
2000
0.6
Torque (Nm)
Thrust (N)
1000
0.4
0.2
500
Torque (Nm)
0.5
500
0.5
Torque (Nm)
Thrust (N)
0.4
0.2
0
500
1000
0
500
1000
0.15
0.1
0.05
0
500
1000
Torque (Nm)
0.6
Thrust (N)
1000
0.1
1000
0.6
0
AR=3.74
500
0.2
0
500
0
=
16.5°
20 °
28.5°
37 °
44 °
0
0.05
Torque (Nm)
Thrust (N)
1
0
AR=1.63
1000
0.1
1000
0
=
8°
15 °
23 °
31 °
45 °
500
0.15
0
0
AR=2.09
0
0.05
0
=
6°
13 °
25 °
36 °
46 °
2000
0.1
1000
1
Thrust (N)
AR=2.88
1000
0
0
8°
18 °
30 °
40 °
45 °
0
0.15
0
=
2000
0
0
AR=4.04
1000
0.05
0
=
9°
20 °
30 °
45 °
0
0.1
0.4
0.2
0
0.1
0.05
0
0
200
2
400
(rad2/s 2)
600
0
200
2
400
600
(rad2/s 2)
Figure S2: The raw data of the thrust and torque measurements from seven tested flat wings.
The left-hand side lists the photographs of the wings and the measured blade pitch angles. The
plots on the right-hand side show the raw measurements of thrust and torque (dots) and the best
fitted (linear regression) lines. Different colors indicate different blade pitch angles.
A
B
2
CL
CD
CL/C D
CL,CD
1.5
1
0.5
0
0
30
45
(°)
0
D
6
4
C3/2
/C D
L
C3/2
/C D
L
C
15
7
6
5
4
3
2
1
0
2
0
15
30
45
(°)
6
4
2
0
0
15
30
(°)
45
0
1
2
3
4
5
6
7
CL/C D
Figure S3: Empirically fitted lift and drag coefficients: CL (↵) = 0.130 + 1.672 sin (2↵) and
CD (↵) = 0.046 + 1.142 (1 cos (2↵)).
(A) lift and drag coefficients versus angle of attack. (B) Glide ratio versus angle of attack. (C)
power factor versus angle of attack. (D) power factor versus gliding ratio.
A
B
26.4
6.3
5.4
7.6
26.5
dC T /dr
1.2
0.8
0.4
0
= 47°
37°
28°
16°
8°
10
2.5
2
1.5
1
0.5
0
-3
10 -4
3.6
2
dC Q/dr
2
dC Q/dr
(N s /rad )
10 -4
2.4
45
36
27
18
9
0
-3
(N s 2/m rad 2)
10
1.6
(°)
= 46°
39°
32°
23°
12°
1.6
0.8
2
(°)
(N s 2/m rad 2)
(N s /rad )
AR=4.43
2
dC T /dr
45
36
27
18
9
0
9.3
7.6
AR=5.71
0
0
9
18
2.4
1.2
0
27
0
9
r (cm)
27
D
11.5
6.6
11.5
4.7
C
30.4
30.4
AR=2.88
AR=4.04
= 45°
27
18
9
9°
dC T /dr
0
-3
6
4
2
0
10 -3
1.5
2
dC Q/dr
2
10
8
-3
(N s /rad )
10
= 45°
40°
30°
18°
8°
-3
3
1.2
45
36
27
18
9
0
(N s 2/m rad 2)
10
6
2
(N s /rad )
(N s 2/m rad 2)
0
dC T /dr
(°)
35°
25°
0.8
0.4
0
0
8
16
r (cm)
24
32
2
(°)
36
dC Q/dr
18
r (cm)
1
0.5
0
0
8
16
24
32
r (cm)
Figure S4: The model predictions of the local angle of attack and the distribution of thrust
and torque coefficients along the wingspan. (A) to (D) present the predictions for wings No.
1 to 4.
B
9.1
11.5
19.7
30.4
30.4
AR=1.63
(°)
(N s 2/m rad 2)
dC T /dr
10 -3
dC Q/dr
(N s 2/rad 2)
1
0
0
31°
23°
15°
8°
18
0
0
2
= 45°
27
9
0.005
(N s 2/rad 2)
dC T /dr
36
= 46°
36°
25°
13°
6°
0.01
dC Q/dr
(N s 2/m rad 2)
(°)
AR=2.09
45
36
27
18
9
0
8
6.6
A
16
24
32
10 -3
8
4
0
10 -3
1.5
1
0.5
0
0
r (cm)
8
16
24
32
r (cm)
C
7.8
6.0
30.0
AR=3.74
=
44°
37°
28.5°
12
20°
16.5°
(°)
36
24
dC Q/dr
(N s 2/m rad 2)
(N s 2/rad 2)
dC T /dr
0
10 -3
8
4
0
10 -3
1.2
0.8
0.4
0
0
10
20
30
r (cm)
Figure S5: The model predictions of the local angle of attack and the distribution of thrust
and torque coefficients along the wingspan. (A) to (C) present the predictions for wings No.
5 to 6 and the optimized wing.
A
B
original wing: 155.8 cm2
the enlarged wing
enlarged wing area: 11.6 cm2
C
25
1.5
1
x (m)
(°)
20
15
10
5
0.5
0
-0.5
0
-1
0
20
40
60
80
100
120
1
0
20
40
0
20
40
60
80
100
120
60
80
100
120
2.2
z (m)
y (m)
0.5
0
-0.5
2
1.8
-1
-1.5
1.6
0
20
40
60
80
100
120
time (s)
time (s)
Figure S6: Passive stability of an asymmetrical prototype. (A) A drawing of the modified
wing with an added area towards the wing tip. (B) Photo of the asymmetric robot made with one
modified wing. (C) The trajectory of the inclination angle and the position of the asymmetric
robot in an uncontrolled flight.
Revolving-wing drone
Revolving-wing drone
with 650-mAh battery
Crazyflie
Benchmark quadcopter
battery voltage (V)
4.5
3.5
2.5
1.5
0
200
400
600
800
1000
1200
1400
1600
time (s)
Figure S7: Battery voltages during the endurance flight tests.
A
B
power (W)
power measurement
module
robot
4.8
in hovering
4.6
4.4
4.2
4
-5
0
5
10
15
20
25
30
15
20
25
30
15
20
25
30
moment
arm
servo motor
load cell
Thrust (N)
0.6
0.4
0.2
0
-5
Torque (Nmm)
robot's weight
0
5
10
0
5
10
40
20
0
-20
-40
-5
(rad/s)
Figure S8: HIL experiment for power measurements. (A) The HIL setup. (B) Measurements
of power, vertical thrust, and torque at different revolving speeds.
Multi-ranger deck
sensor 2
flight controller
sensor 1
Figure S9: Photo of the robot equipped with laser range sensors for constructing a map of
the environment. The arrows denote the measurement directions.
camera controller
flight controller
camera
Figure S10: Photo of the robot equipped with a camera. The monochrome camera has the
resolution of 1280 x 800 pixel and the field of view of 75 degrees. A Raspberry Pi Zero W was
employed as a camera controller.
A
69-mm 2-blade
propeller
20 mm
37.5-mm
2-blade
propeller
20-mm
4-blade
propeller
7x20 mm
coreless
motor
B
C
Figure S11: Coefficient identification of the motor model. (A) Three propellers and the DC
motor used for the identification of the motor parameters. The colored dots denote the corresponding datapoints shown in (B) and (C). (B) The current and voltage measurement against
the motor spin rate. the plane represents the best fitted parameters Rm and km . (C) The residual
errors of ! predicted by the linear regression.
Figure S12: The block diagram representing the open-loop flight dynamics. The diagram
depicts the connection between the attitude and translational dynamics of the revolving-wing
robot when only the altitude is actively controlled.
= 0 rad/s
19.5 o
0
o
= 18.8 rad/s
19.5 o
0
o
= 37.7 rad/s
19.5 o
0
o
Figure S13: Rigidity test for the rotating wings. Snapshots from high-speed videos of the
wings revolving at different speeds. The left view demonstrates the spanwise stiffness (the
orientation of the leading edges). The right view shows the chordwise stiffness (angle of the
wing pitch compared with the reference line at 19.5 . The wing deformation became evident
when the robot rotated at high angular velocity (37.7 rad/s, twice the regular rate during hover).
= 0 rad/s, Propellers off
19.5 o
0o
leading edge
motor
trailing edge
propeller
= 18.8 rad/s, Propellers off
19.5 o
0o
= 18.8 rad/s, Propellers on
19.5 o
0o
Figure S14: Rigidity test for the entire robot. Snapshots from high-speed videos of the robot
at rest and rotating at the hovering speed (⇠18.8 rad/s), with the propellers on and off. The
left view demonstrates the spanwise stiffness (the orientation of the leading edges). The right
view shows the chordwise stiffness (angle of the wing pitch compared with the reference line at
19.5 ).
1.2
fiberglass wings
polyimide wings
Thrust (N)
0.9
robot's weight
0.6
0.3
0
0 10 2
20 2
30 2
40 2
Torque (Nmm)
80
60
entire robot with
propellers off
40
outliers at high
revolving speed
20
0
entire robot with
propellers on
-20
0 10 2
20 2
30 2
2
2
40 2
2
(rad /s )
Figure S15: Thrust and torque measurements of the rotating wings and robot. The plots
show the values of thrust and torque obtained during the wing rigidity tests. The blue dashed
lines fit the data of the carbon fiber reinforced polyimide wings at non-extreme rotational rates
linearly, indicating the absence of wing deformation and flapping (the thrust and torque coefficients are constant). This is in contrast to the datapoints at ⌦ > 30 rad/s, which deviate from
the trend (the blue dashed lines). The red dashed lines linear fit the data of the rigid fiberglass
wings. They coincides with those of polyimide wings. The datapoints corresponding to the
rotating robot with unpowered propellers (yellow) display no notable deviation from the trend
lines. The datapoints corresponding to the rotating robot with powered propellers (green) offer
a hint of possible wing-propeller interaction.
flight time
(s)
894
1470
1200
600
120
300
240
210
528
540
380
241
480
780
480-600
power in
hover (W)
4.39±0.12
5.81±0.08
4.8
5.64
3.27
3.44
5.05
7.57
11.4
3.75
-
power loading
motor
(g/W)
type
8.00±0.23
brushed
7.37±0.11
brushed
brushless
brushless
brushless
5.73
brushless
5
brushless
5.81
brushed
brushless
4.6
brushed
brushless
2.48
brushless
4.24
brushed
3.44
brushed
5.33
brushed
brushed
brushed
Table S1: Endurance and power consumption of sub-100-g MAVs.
mass
(g)
Revolving-wing drone
35.1
with a 650-mAh battery
42.8
Robotic samara-I
75
Robotic samara-II
38
Robotic samara-III
9.5
X-winged ornithopter
27.5
Delfly Nimble
28.2
Nano Hummingbird
19
NUS-Roboticbird
31.0
KUBeetle-S
15.8
Quad-thopter
37.9
Purdue Hummingbird
12.5
Crazyflie (including markers) 32.1
Benchmark quadcopter
39.2
XQ-139µ QuadSparrow
20
DJI Tello
80
Parrot Mambo
63
robot
this work
this work
(31, 86)
(31, 86)
(31, 86)
(10)
(13)
(12)
(75)
(72)
(73)
(14, 74)
this work
this work
(87)
(88)
(89)
source