Flying with damaged wings the effect on flight capacity and bio inspired coping strategies of a flapping wing robot

Flying with Damaged Wings: The Effect on Flight Capacity andBio-inspired Coping Strategies of A Flapping Wing Robot Zhan Tu, Fan Fei, Limeng Liu, Yiming Zhou, and Xinyan Deng Abstract—In

Flying with Damaged Wings: The Effect on Flight Capacity and

Bio-inspired Coping Strategies of A Flapping Wing Robot

Zhan Tu, Fan Fei, Limeng Liu, Yiming Zhou, and Xinyan Deng

Abstract—Insects wings are subject to wear and tear from

collisions and environmental disturbances during flight They

can tolerate both symmetrical and asymmetrical wing damages

while maintaining flight capability to some extent Drawing

inspiration from nature’s adaptation capabilities, we investigated

the consequences of wing damage on a flapping wing micro

air vehicle by quantifying the changes in wing kinematics, lift

generation, control torque offset, and aerodynamic damping

variations in flight tests with intact and damaged wings For the

proposed vehicle, the wing damage affected the lift generation

significantly Compared to the intact wings, the damaged ones

result in increased stroke angle amplitude in order to compensate

for lift loss and torque imbalance, which causes an increase

in power consumption accordingly Furthermore, asymmetric

damages usually require a larger amount of additional control

effort for flight stabilization compared to symmetric cases In

addition, aerodynamic damping varies as the wing areas change.

All these aspects pose challenges in flight control An adaptive

controller is proposed to cope with the wing damage induced

detrimental effects on flight capacity Flight tests were conducted

to validate the control performance As a result, the robot

can effectively overcome such challenges even in the case of

a maximum unilateral lift loss of up to ≈22% Such a result

matches the performance of hovering hawkmoths, which can

handle torque asymmetry up to 22.3±7.8% To the best of our

knowledge, this is the first demonstration of FWMAVs to handle

significant wing asymmetry in hover flight.

Index Terms—Biologically Inspired, Flapping Wing, Micro

Aerial Vehicle, Wing Damage, Adaptive Control.

I Introduction For aerial vehicles, wing damages can cause serious

aero-dynamic and flight stability consequences Studies on

tra-ditional fixed-wing and rotary-wing aircraft show that the

lifting surface loss and control surface loss typically results

in lift generation and flight stability challenges [1]–[3], and

therefore causes the drop in flight capacity As a result,

active and passive solutions have been proposed to solve the

corresponding flight control and safety issues [4]–[7]

Manuscript received October, 15, 2020; Revised January, 3, 2021; Accepted

January, 24, 2021 This paper was recommended for publication by Editor

Xinyu Liu upon evaluation of the associate editor and reviewers’ comments.

This research received no specific grant from any funding agency in the public,

commercial, or not-for-profit sectors.(Corresponding Author: Zhan Tu; Xinyan

Deng.)

Zhan Tu is with Institute of Unmanned System, Beihang University, Beijing

100083, China, and also with the School of Mechanical Engineering, Purdue

University, W Lafayette, IN 47907, USA Email: zhantu@buaa.edu.cn

Fan Fei is with Amazon.com, Inc, Seattle, WA 98109, USA, and also with

the School of Mechanical Engineering, Purdue University, W Lafayette, IN

47907, USA Email: ffnc1020@gmail.com

Limeng Liu, Yiming Zhou, and Xinyan Deng are with the School

of Mechanical Engineering, Purdue University, W Lafayette, IN

47907, USA Email: liu1936@purdue.edu; zhou663@purdue.edu;

xdeng@purdue.edu

Digital Object Identifier (DOI):

Fig 1 (a) Schematic representation of the treatments applied to each individual moth and their corresponding wing kinematics (top view), taken from [17], [20] (b) Bio-inspired wing damage test on a hovering FWMAV.

Meanwhile, Flapping Wing Micro Air Vehicles (FWMAVs) could be more sensitive to wing damage due to their unsteady aerodynamics and underactuation characteristics To date, a number of FWMAVs have achieved stable flight [8]–[15], and a systematic investigation on wing damage is needed

In nature, flying insects also undergo unavoidable wing wear and tear due to collisions and environmental disturbances

They are able to compensate effectively for such damage

to some extent For example, biological studies have been conducted to investigate the wing damage consequences to flight performance and the corresponding solutions in flying animals [16]–[20] It was found that insects are able to alter their wing kinematics or flight muscles asymmetrically to deal with wing damage consequences For example, hawkmoths can tolerate up to 20% wingtip loss [17], [20] Their particular adaption strategy of wing damage is shown in Fig 1(a) Flight resiliency under wing area loss enables the animals to counter the environmental factors causing wing wear and tear, e.g., dense branches and bushes in forest, or object collisions caused

by windy conditions

To find out whether similar adaptation principles could be adopted for FWMAVs and their effect on flight efficiency and performance, we systematically conducted wing damage experiments on a bio-inspired flying robot based on our previous design [15] The results of this study will inform us

on both the system resilience and the coping strategies for wing area loss, especially when the vehicle navigates in confined and cluttered environment In addition, it will also provide important insight into the improvement of flight control

A preferred prerequisite on such robotic platforms to imitate

flying creatures’ resilience lies in its bio-inspired decoupled

wings It is critical because the robot needs to be able to

adapt to asymmetric wing damage by varying wing kinematics

independently, which is a key component as observed in the

coping strategies in flying animals In this work, the test

platform is an FWMAV with independently actuated wings,

as shown in Fig 2 For each wing, the motion kinematics is

independently controlled by a motor To quantify the impact

of wing damage, we fabricated several intact and damaged

wings as shown in Fig 2(a) The first set-#1 is a pair of intact

wings The other two pairs have different degrees of artificial

damages Detailed morphological differences between three

pairs of wings are summarized in Table I

Similar to the hawkmoth study illustrated in Fig 1(a),

various pairing schemes of these wings were equipped on the

test platform to investigate the impact of both symmetric and

asymmetric wing area reductions In particular, we

systemati-cally studied the following aspects of the wing damage effect:

lift drop, control torque offset, and aerodynamic damping

changes The overall negative effects results in the unique

flight control challenges which have been rarely studied on

FWMAVs With the knowledge of the wing tearing effects,

a targeted nonlinear controller is proposed to cope with such

challenges The effectiveness of the proposed controller has

been validated with systematic flight experiments As a result,

the controller can ensure a stable flight even if there is a

severe lift gap (>22%) between the two wings To the best

of our knowledge, this is the first demonstration of FWMAVs

to handle significant wing asymmetry in hover flight

This work provides an important contribution to the

un-derstanding of the flight capacity change of FWMAVs with

damaged wings The proposed study procedure, qualitative

results, and flight control solution can be a valuable reference

and potentially be generalized to other flapping wing systems

with similar aerodynamic principles, aiding the design and

control of such bio-inspired flying vehicles

The rest of the article is organized as follows Section II

introduces the test platform and its dynamics model Section

III analysis the effects of wing damage on the test platform

Section IV presents the controller design that addresses the

corresponding control challenges caused by wing damage

Section V shows the experimental flight results of the proposed

vehicle under different wing damage scenarios Section VI

summarizes this work

II Test Platform Description and Modeling

A Test Platform

The test platform in this study is based on a dual-motor

actuated FWMAV proposed in [15] It is capable of equipping

asymmetric wings as shown in Fig 2(b) The wingspan of

such a vehicle with intact wings is about 170mm and the

weight is about 12.5g Inspired by the flying animals, it is

designed with decoupled wings, i.e., each wing is driven by a

dc motor independently Motor power efficiency is optimized

by reduction gears and torsional springs The wing is designed

to passively rotate due to the aerodynamic and inertial loading

[21] The Angle-of-Attack (AoA) of the wings is optimized

Fig 2 (a) Demonstration of the test wings: #1-intact wings, #2-damaged wings, bilateral 5% wingtip area loss, #3-damaged wings, bilateral 10%

wingtip area loss (b) Illustration of the test platform, which can equip asymmetric wings The clipped wing area is outlined with red line segments around the wingtip.

to about 45◦ The onboard electronic system consists of two power regulators, a microcontroller, an inertial measurement unit (IMU), and two motor drivers The detailed design of each module can be found in [15]

The proposed platform employs just two actuators for 6-DoF flight control Wing kinematic modulation technique is used

to generated control torque The particular wing kinematics is adjusted by a sinusoidal motor voltage input, namely: voltage amplitude 𝑉 ; the differential voltage amplitude of two motors

𝛿𝑉; the voltage bias 𝑉0; the split-cycle parameter 𝛿𝜎 [15] A typical motor input 𝑉𝑖 𝑛 is

𝑉𝑖 𝑛=

(

𝑉sin 𝜔 𝑛 𝑡 2𝜎
+ 𝑉0 if 0 ≤ 𝑡 ≤ 𝜎

𝑓

𝑉sin𝜔𝑛𝑡 −2 𝜋 2(1−𝜎)

 + 𝑉0 if 𝜎

𝑓 ≤ 𝑡 ≤ 1

𝑓

(1)

where 𝜔𝑛=2𝜋 𝑓 and 𝑓 is wingbeat frequency, 𝑡 is time

According to the defined four control inputs above, the vehicle can generate lift and 3-axis control torque In partic-ular, lift 𝑓𝑧 = 𝐾𝑢 1𝑢1 where 𝑢1 = 𝑉𝑖 𝑛 − 𝑉𝑠 0 and 𝑉𝑠 0 denotes motor driver starting voltage; roll torque 𝜏𝑥 = 𝐾𝑢 2𝑢2 + 𝜏𝑥 0,

𝑢2 = 𝛿𝑉; pitch torque 𝜏𝑦 = 𝐾𝑢 3𝑢3+ 𝜏𝑦 0, 𝑢3 = 𝑉0; and yaw torque 𝜏𝑧= 𝐾𝑢 4𝑢4+ 𝜏𝑧 0, 𝑢4 = 𝛿𝜎 𝐾𝑢 1,2,3,4 are linearized input coefficients, and 𝜏𝑥 0, 𝜏𝑦0, 𝜏𝑧0 are the mechanical imperfection induced trim condition

high-speed camera

wing kinematics

PC

position

& attitude

Vicon camera (6 in total)

power estimation

z

ψ, r

θ, q

φ, p

asymptotic x y

load Fig 3 Illustration of the flight test setup.

B Vehicle Dynamics

With the coordinate defined in Fig 3, the test vehicle can

be modeled by

𝑚𝒑 = 𝑹 𝒇¥ 𝑛− 𝑚𝒈 + 𝒇𝑑+𝒅𝒑

𝑱 ¤𝝎 = 𝝉𝑛−𝝎 × 𝑱𝝎 + 𝝉𝑑+𝒅𝝎, (2) where 𝑚 is the total mass; 𝒑 = [𝑥, 𝑦, 𝑧]𝑇

is the vehicle position in the inertial frame; 𝜼 = [𝜙, 𝜃, 𝜓]𝑇

is the attitude angle of the vehicle; 𝑹(𝜼) is the rotation matrix; ¤𝒑 = 𝑹𝒗𝒃

wherein 𝒗𝒃 = [𝑢, 𝑣, 𝑤]𝑇

is the translational velocity in the body frame; 𝒇𝑛 = [0, 0, 𝑓𝑧]𝑇

; 𝒈 = [0, 0, 9.8𝑚/𝑠2]𝑇

is the gravity acceleration vector; 𝑱 is the inertia matrix of the

vehicle; 𝝎 = [ 𝑝, 𝑞, 𝑟]𝑇

is the vehicle angular velocity; 𝝉𝑛 = [𝜏𝑥, 𝜏𝑦, 𝜏𝑧]𝑇

; 𝒅𝒑and 𝒅𝝎 denote external disturbances; 𝒇𝑑 and

𝝉𝑑 indicate a unique aerodynamic phenomenon in

FWMAVs-flapping counter forces/torques (FCFs/FCTs) induced

addi-tional damping wrenches [22], [23]

The stroke-averaged 𝒇𝑑 and 𝝉𝑑 are given by

𝒇𝑑 =𝑹𝑫1(𝒄𝒑)𝒗𝒃+𝑹𝑫2(𝒄𝒑, 𝑑𝑠)𝝎,

𝝉𝑑 =𝑫3(𝒄𝒑, 𝑑𝑠)𝒗𝒃+𝑫4(𝒄𝒑, 𝒄𝜼, 𝑑𝑠)𝝎, (3) where 𝑫1,2,3,4 are damping coefficient matrix in terms of

𝒄𝒑 = [𝑐𝑥, 𝑐𝑦, 𝑐𝑧], 𝒄𝜼 = [𝑐𝜙, 𝑐𝜃, 𝑐𝜓], and 𝑑𝑠, where 𝑑𝑠 is the

offset between the stroke plane and the center of gravity The

expanded form of 𝑫1,2,3,4is given by the equation (17) in [15]

At near-hovering condition, 𝒄𝒑 and 𝒄𝜼 are derived by





































𝑐𝑥=2𝜌𝑎𝑅2

𝑤𝑐¯Ψ𝑤0𝜔𝑛𝑟ˆ1

1𝐶𝐷cos2(𝜓𝑤) |𝑑𝜓ˆ𝑤

𝑑ˆ𝑡 |,

𝑐𝑦=2𝜌𝑎𝑅2

𝑤𝑐¯Ψ𝑤0𝜔𝑛𝑟ˆ1

1𝐶𝐷sin2(𝜓𝑤) |𝑑𝜓ˆ𝑤

𝑑ˆ𝑡 |,

𝑐𝑧= 𝜌𝑎𝑅2

𝑤𝑐¯Ψ𝑤0𝜔𝑛𝑟ˆ1

1

𝑑𝐶𝑁(𝛼) 𝑑𝛼

|𝛼0cos(𝛼0) |𝑑𝜓ˆ𝑤

𝑑ˆ𝑡 |,

𝑐𝜙= 𝜌𝑎𝑅4

𝑤𝑐¯Ψ𝑤0𝜔𝑛𝑟ˆ3

3

𝑑𝐶𝑁(𝛼) 𝑑𝛼

|𝛼0cos(𝛼0) cos2(𝜓𝑤) |𝑑𝜓ˆ𝑤

𝑑ˆ𝑡 |,

𝑐𝜃 = 𝜌𝑎𝑅4

𝑤𝑐¯Ψ𝑤0𝜔𝑛𝑟ˆ3

3

𝑑𝐶𝑁(𝛼) 𝑑𝛼

|𝛼0cos(𝛼0) sin2(𝜓𝑤) |𝑑𝜓ˆ𝑤

𝑑ˆ𝑡 |,

𝑐𝜓 =2𝜌𝑎𝑅4

𝑤𝑐¯Ψ𝑤0𝜔𝑛𝑟ˆ3

3𝐶𝐷|𝑑𝜓ˆ𝑤 𝑑ˆ𝑡 |,

(4)

where 𝜌𝑎 is the air density; 𝑅𝑤 is the wing length; ¯𝑐 is

wing mean chord length; 𝑟𝑛

means the n-order dimensionless moment of wing area; ˆ𝑡 = 𝜔𝑛𝑡 is the non-dimensional time;

Ψ𝑤 0is the nominal stroke amplitude; 𝛼 and 𝛼0are the effective

and geometric AoA, respectively; 𝐶𝐷 and 𝐶𝑁 are the drag

and normal force coefficient, respectively; 𝑑 𝜓 ˆ𝑤

𝑑 ˆ𝑡 is the non-dimensional flapping velocity

FCFs/FCTs has been particularly considered in this work because it directly related to the wings’ morphological change and affects the control bandwidth

III Effect of Wing Damage

In order to determine the effect of wing damage on the test FWMAV, modeling, analysis, and flight test validation were conducted Quasi-steady model [24] was used to estimated the aerodynamic forces and torques For cross-validation, the corresponding experimental force/torque measurement and free flight experiments were performed The flight test setup is illustrated in Fig 3, which has a high-speed top-view camera

to capture the instantaneous wing kinematics and a motion capture system-Vicon to provide the vehicle’s posture and position feedback The motor current is sensed by onboard sensing resistors for power consumption estimation These records can be utilized to systematically evaluate the particular flight capacity changes under different wing tear scenarios

In this section, we first determine the damage threshold that could cause insufficient lift generation on the proposed platform The result will guide the fabrication of the test wings

Then, modeling and experimental verification were performed

to evaluate lift loss, control torque offset, and aerodynamic damping changes, respectively

A Design of Artificial Wing Damage

Based on the Quasi-steady model, as the bilateral wing area loss goes up to about 15.6%, the proposed vehicle can barely generate sufficient lift force for stable flight Nevertheless, in actual flight tests, the vehicle cannot perform stable liftoff when the clipped area exceeds 15% as shown in the attached video A time sequential result is shown in Fig 4 On the other hand, 15% asymmetry also causes unmanageable flight due to the severe control torque offset It is notable that some flying animals such as hawkmoth can tolerate wing damage

up to 20% [17], [20] Compared to the test vehicle, hawkmoth may benefit from its wing materials, flexibility, and multiple groups of flight muscles for more elaborate wing kinematics

Based on the 15% threshold, three pairs of wings were fabri-cated for testing, covering 0%, 5%, 10% wing area loss These wings are shown in Fig 2(a) and their specific parameters are summarized in Table I Note, each wing was clipped vertically

at wingtip from the leading-edge to the trailing-edge, similar to the methods adopted in biological studies on hawkmoth wings [17], [20] Clipping in this way is because the distal wingtip area is the most vulnerable area which suffers wear and tear

in unavoidable collisions In addition, due to higher wingtip velocity, wingtip area generally produces significant larger aerodynamic lift force and control torques than the other wing area, which subsequently causes obvious flight capacity drop and stability issues as discussed in the following subsections

B Effect of Wing Damage on Lift Generation

Based on the wings listed in Table I, we first check their respective lift generation capability without flight control Such

Fig 4 Insufficient lift to takeoff with 15% loss of wing area.

TABLE I Parameters of Test Wings Test

wings

Wing area loss

𝑅𝑤 𝑐 ¯ 𝑟 ˆ 2 Aspect

ratio

#2 bilateral 5% 66.2mm 21.3mm 0.5295 3.1

#3 bilateral 10% 62.6mm 21.5mm 0.5293 2.9

an open-loop result can be used to determine the linearized

input coefficients 𝐾𝑢 1,2,3,4 for the controller design followed

later An ATI Nano-17 force/torque transducer was used for

measurement The result is shown in Fig 5 As the wing

damage intensifies, the maximum lift of the vehicle decreases

severely Besides the falling actuation capability, it also brings

a control problem: The linearized control input coefficient 𝐾𝑢 1

mentioned in Section II varies, represented by 𝛼𝑧𝐾𝑢

1, where

𝛼𝑧∈ [0, 1] This varying coefficient needs to be reconsidered

in the controller design

Compared to the open-loop test, the actual maximum lift

of the vehicle in free flight could be further limited since

additional control effort is required to ensure stability Such

effective maximum lift can be roughly obtained from a lifting

load test The vehicle was programmed to lift as much payload

as it can while maintaining flight stability Inspired by the

biological test [20], an asymptotic varying load, e.g., a string

of small aluminum beads was used in this test Each bead

weighs about 0.3g During the test, the vehicle performed

vertical takeoff and gradually lifted the aluminum beads

mid-air until it lost stability, which is shown in the attached video

The instantaneous wing kinematics and power consumption

were recorded The averaged results of five flight trials are

summarized in Table II, including the free flight and load

lift cases The vehicle with a pair of intact wings, i.e., #1,

demonstrates the baseline performance

From Table II, the actual maximum payload capacity drops

severely with respect to the increment of wing damage,

which matches the lift loss result obtained by static force

measurement Compared to the intact wing’s performance, just

10% of wing tip area loss will cause more than three times

effective payload capacity reduction of the proposed FWMAV,

which indicates that such damage affects the lift generation

significantly In order to compensate for such lift drop, the

stroke amplitude and total energy consumption of the damaged

wings increase accordingly

Note, in controlled flight, the proposed asymptotic load does

V (V)

0.1 0.15 0.2

0.25

Intact wing (#1) 5% clipped (#2) 10% clipped (#3)

V (V)

-3 -2 -1 0 1 2 3

V 0 (V) -0.5

0 0.5

-0.2 -0.1 0 0.1 0.2 -0.6

-0.4 -0.2 0 0.2 0.4

Fig 5 Static force/torque measurement using wing #1, #2, #3, respectively.

not generate significant instantaneous impact on the vehicle

In addition, at near-hovering condition, the payload brings a downward shift of the center of gravity, which enhances the passive stability of the system as introduced in [15] section IV.C As demonstrated in the attached video, during the load lifting test, the vehicle demonstrates stable flight straight upwards with minor attitude and position drift

TABLE II Cost of Symmetric Wing Damage

Wing area loss 0% bilateral 5% bilateral 10%

Hovering without payload

Stroke amplitude (left / right)

142.2◦ /141.0◦

142.2◦ /142.2◦

152.9◦ /154.5◦ Load lifting

Control input 𝑢 1 13.957V 12.564V 12.139V Control input 𝑢 2 0.157V 0.227V 0.331V Control input 𝑢 3 0.397V 0.358V 0.587V

C Effect of Wing Damage on Control Torque

As shown in Fig 5, the control torque generation of the vehicle is limited by the wing damage In general, as the wing area loss becomes severe, the control torques in all three axes decreased to a certain degree Similar to the lift generation, the respective variation of their input coefficients 𝐾𝑢 2,3,4 should be reconsidered in the controller design as well In terms of such performance drop, it is expected that the asymmetry of the left and right wings could lead to control challenges inevitably

A proper control method is preferred to counter such wing asymmetry induced flight stability issue

Besides the torque generation, as shown in Fig 5, the

effective control effort is also affected by the trim

condi-tion, i.e., 𝜏𝑥 0, 𝜏𝑦0, 𝜏𝑧0 For instance, compared to the other

wing pairs, the #2 wings’ obvious yaw trim variation would

significantly impact its overall control performances since

the limited negative yaw torque shrinks the overall flight

envelope Such control performance limitation is demonstrated

in Section V.A In fact, for the proposed FWMAV, yaw torque

generation is always weak due to the split-cycle control method

(corresponding to downstroke and upstroke asymmetries in

left and right wings) Even in the case of intact wing test,

yaw torque is the most limited one among those of the three

rotational axes Such a limitation is further exacerbated by

the loss of wing area and the wing fabrication induced trim

condition, resulting in the unmanageable yaw control drifting

eventually

D Effect of Wing Damage on Vehicle Dynamics

For FWMAVs, wing damage induces wings’ morphological

changes and the consequent kinematics changes Based on the

vehicle dynamics presented in Section II.B, such changes could

greatly affect aerodynamic damping during the flight

An interesting trade-off of the flapping wing system is that

when the wing length reduced, the stroke velocity amplitude

Ψ𝑤 0𝜔𝑛 increases correspondingly in order to maintain stable

flight On the proposed vehicle, only Ψ𝑤 0 increases since

𝜔𝑛 is invariant As the wing damage becomes severe, the

overall trend of 𝒄𝒑 and 𝒄𝜼 keeps decreasing For example,

the estimated 𝒇𝑑 and 𝝉𝒅 in open-loop condition are

illus-trated in Fig 6, where the intact wings produce the strongest

additional damping effect, as expected The initial condition

of this estimation is an ideal hover flight with initial 𝝎 =

[0.01, 0.01, 0.01] rad/s2 The control input and trim condition

are from Table II Since there is no flight control in this case,

𝒇𝑑 and 𝝉𝒅 diverge quickly as the attitude becomes unstable

Based on Fig 6, such intact wings response can guide the

controller design to compensate for this unique additional

damping wrench effect

0 0.005

0.01

0.015

0.02

-1 0 1

2 10 -3

Time (s)

-0.02

-0.01

0 0.01

0.02

Time (s)

-15 -10 -5 0

5 10 -4

Intact wing 5% clipped 10% clipped

Fig 6 Estimated additional damping wrench induced by flapping wings.

Although 𝒄𝒑 and 𝒄𝜼 demonstrated limited impact on total

lift and control torques according to Fig 6, they dominant the

system zeros and poles location at hovering condition, which determines the maximum control bandwidth of the vehicle [15] In this study, #3 wing provides overall smaller (≈86%)

𝒄𝒑 than #1 wing According to that, the increased system sensitivity challenges the transient performance of the flight control with #3 wing equipped

IV Flight Control Based on Section III, the main control challenge in this study is from the wing damage induced control effort change and the potential asymmetrical force and torque brought by

it Such actuation performance uncertainty induced stability issue can be addressed by targeted control strategy as presented below

As mentioned in Section III, damaged wing causes control input coefficient varying To avoid the change of controller structure and the repeated gain tuning, the varying actuation performance can be lumped into vehicle dynamics change For example, the lift reduction can be equivalent to the increase

of the modeled mass of the vehicle Consequently, the altitude dynamics of the vehicle with damaged wings is

𝓂 ¥𝑧= 𝐾𝑢 1· 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜃 · 𝑢1− 𝓂𝑔 + Δ𝑧, (5) where 𝓂 = 𝑚/𝛼𝑧 is the equivalent vehicle mass, 𝑢1 = 𝑉, Δ𝑧

is the lumped system uncertainty

For altitude control, control error is defined by

𝑒𝑧 = 𝑧 − 𝑧𝑑, 𝑒¤𝑧= ¤𝑒𝑧+ 𝑘𝑧 1𝑒𝑧= ¤𝑧 − ¤𝑧𝑒𝑞 (6) where 𝑒𝑧 is the altitude tracking error, 𝑧𝑑 is the z-axis control reference, 𝑒¤𝑧 is an auxiliary speed tracking error, 𝑘𝑧 1 is a positive control gain, and 𝑧𝑒𝑞 is the equivalent tracking target

𝑒𝑧→ 0 when 𝑒¤𝑧 converges to 0

An integral sliding surface is given by

𝑠𝑧= 𝑒¤𝑧+ 𝑘𝑧 2

∫

where 𝑘𝑧 2 is a positive control gain Based on 𝑠𝑧, the altitude error dynamics is given by

𝓂 ¤𝑠𝑧= 𝐾𝑢 1· 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜃 · 𝑢1− 𝓂𝑔 + Δ𝑧− 𝓂¥𝑧𝑒𝑞+ 𝓂𝑘𝑧 2𝑒¤𝑧 (8)

In order to counter the lift loss caused by wing damage, parameter adaptation is essential in control law design A feasible adaptation law is given by

¤ˆ

where 𝓂ˆ is the estimated model parameter for control, 𝛾𝑧

is the adaptation rate, 𝜑𝑧 is a regressor ˆ𝓂 ∈ [𝑚𝑚𝑖 𝑛, 𝑚𝑚𝑎 𝑥], where the 𝑚𝑚𝑖 𝑛 is designed to slightly less than the vehicle’s original weight, and 𝑚𝑚𝑎 𝑥 is obtained by the result of the lifting load test mentioned in Section III.B The estimation error is defined by ˜𝓂 = 𝓂 −𝓂.ˆ

Propose an extended Lyapunov function as :

𝑉𝑧(𝑒¤𝑧,𝓂) =˜ 1

2𝓂𝑠

2

2𝛾𝑧𝓂𝓂˜2 (10) Design

𝜑𝑧 = −(𝑔 + ¥𝑧𝑒𝑞− 𝑘𝑧 2𝑒¤𝑧),

𝑢1 = − 1

𝐾𝑢

1𝑐𝑜 𝑠 𝜙𝑐𝑜 𝑠𝜃

[𝑘𝑧 3𝑠𝑧+ 𝜑𝑧𝓂 + 𝜆ˆ 𝑧𝑠𝑔𝑛(𝑠𝑧)], (11)

Fig 7 Stable hovering with symmetric damaged wings The clipped wing area is outlined with red line segments around the wingtip (a), (b), and (c) top

figures are the time sequence result which respects the flying test using the wings #1, #2, #3 The bottom ones are the corresponding flight trajectory The

gray dashed lines are the control references.

where 𝑘𝑧 3 is a positive control gain Thus, ¤𝑉𝑧 = −𝑘𝑧 3𝑠2

𝑧 − (𝜆𝑧|𝑠𝑧| − Δ𝑧𝑠𝑧) In this case, the lumped system uncertainty

Δ𝑧 mainly depends on the aerodynamic damping and external

disturbances To let 𝑒𝑧 converge to 0 asymptotically, 𝜆𝑧 ≥ Δ𝑧

should be guaranteed The knowledge of 𝒇𝑑 and 𝝉𝑑 can aid

the design of 𝜆𝑧 For instance, in this work, the upper bound

of 𝒇𝑑 and 𝝉𝑑 can be estimated as demonstrated in Fig 6 In

indoor test, 𝜆𝑧 can be tuned with such particular model-based

result in the absence of external disturbances

The attitude controller follows a similar control scheme Due

to the underactuation characteristics of the test platform, the

𝜙 and 𝜃 are coupled to 𝑦 and 𝑥 axes, respectively Different

from our previous maneuver controller presented in [25], the

attitude reference 𝜙𝑑 and 𝜃𝑑 are given from a position PID

control law directly Such a method can effectively attenuate

the undesired oscillation induced by the the sensing error and

avoid the multiple derivative of the state feedback In practice,

this method works reasonably well on FWMAVs with damaged

wings because they generally suffer from actuation limitation

and high sensitivity issues in flight control

Attitude control errors is defined by

𝒆𝜼=𝜼 − 𝜼𝒅, 𝒆𝜼 ¤= ¤𝒆𝜼+𝒌𝜼1𝒆𝜼= ¤𝜼 − ¤𝜼𝒆𝒒 (12) Given the 𝒔𝜼 = 𝒆𝜼 ¤ + 𝒌𝜼2

∫

𝒆𝜼 ¤𝒅𝝉, where 𝒌𝜼2 is a diagonal gain matrix The model parameter is 𝒥ˆ =

[ ˆ𝐽𝑥 𝑥,𝐽ˆ𝑦 𝑦,𝐽ˆ𝑧 𝑧,𝜏ˆ𝑥0,𝜏ˆ𝑦0,𝜏ˆ𝑧0] At hovering condition, let 𝒥 =¤ˆ

𝜸𝜼𝝋𝜼𝒔𝜼 and ¥𝜼𝒆𝒒 − 𝒌𝜼2𝒆𝜼 ¤ = [𝛽1, 𝛽2, 𝛽3]𝑇

, the adaptation regressor and control input are

𝝋𝜼=











 ,

[𝑢2, 𝑢3, 𝑢4]𝑇

= −𝑲𝒖2 ,3,4

−1

[𝒌𝜼3𝒔𝜼+𝝋𝜼𝒥 +ˆ 𝝀𝜼𝒔 𝒈𝒏(𝒔𝜼)],

(13)

where 𝑲𝒖2 ,3,4 is a diagonal input coefficient matrix, 𝒌𝜼3 is

a positive diagonal gain matrix, 𝝀𝜼 > 𝚫𝝎 wherein 𝚫𝝎 is dominated by flapping induced aerodynamic damping in the absence of external disturbances

V Flight Test Results

In order to validate and the proposed controller, flight tests were conducted to cover all of the combinations with the fabricated wings shown in Fig 1 In particular, two categories

of flight test were performed, i.e., flying with symmetrical wing damage and with asymmetrical wing damage During the test, the vehicle is commanded to perform a stable hover flight The control reference is at 𝒑 = [0, 0, 350𝑚𝑚] and

𝜼 = [0◦,

0◦,0◦]

A Symmetric Wing Damage

Sample flight tests are illustrated in Fig 7 By taking advantages of the proposed flight controller, even under the severe wing area loss, i.e., flying with #3 wing, the vehicle can still maintain overall stable flight Such results demonstrate that

Time (s)

-0.1 0 0.1 0.2

u 4

Intact wing (#1) 5% clipped (#2) 10% clipped (#3)

Fig 8 Illustration of the yaw control input, corresponding to Fig 7.

Fig 9 Stable hovering flight with asymmetrically damaged wings The clipped wing area is outlined with red lines segments around the wingtip (a) and

(b) respect to the flying test result using wing combination of 1L+3R, 3L+1R From left to right, there are time sequence result, flight trajectory, and key

parameter adaptation The gray lines are the control references.

the proposed controller can respond to the parametric change

caused by wing damage effectively Note, in these cases, the

estimated model rarely converges to its true changes since

the reference does not match the Persistent Excitation (PE)

condition

In general, the results of symmetric wing damage can be

fairly expected As presented in Section III, the incompleteness

of wings is directly related to the vehicle’s actuation capability,

which determines the overall control performance accordingly

For example, in Fig 7(b) and (c), due to the wing damage,

the altitude control shows obvious steady state error though

the proposed control law has already compensated lift loss to

some extent To demonstrate the effectiveness of the proposed

controller, several comparative tests were conducted on the

same setup with a cascade PID control The results are

shown in the attached video With intact wings, two test

controllers show equal stable flight performance However,

with the incremental of the wing damage, the performance

of the PID controller has been severely affected When the

bilateral wing damage reaches 10%, PID control cannot even

guarantee stable liftoff

Compared to the other degree of freedoms, yaw-axis control

is special for such direct-drive FWMAV under wing damage

Yaw control performance majorly depends on the perfection

of wing up/down stroke rather than the wing area loss For

instance, in Fig 7(b), we intend to show a yaw drifting result

as a representative case In fact, based on Fig 5, the overall

yaw control effort generated by #3 wing is less than that of #2

wing However, due to the obvious yaw torque offset of the #2

wings as investigated in Section III.C, its effective yaw control

effort is limited Consequently, as shown in Fig 8, the control

effort of #2 wing drawn from split cycle is nearly saturated

in this case By comparison, there is no obvious saturation issue on the other two cases, thus, they demonstrate better yaw control performance As a result, as the hover regime shown

in Fig 7, the root mean square errors of yaw tracking of the flight trial (b) is 95.057◦, significantly larger than that of flight trial (a) and (c), which are 4.804◦ and 9.497◦, respectively

B Asymmetric Wing Damage

Unlike symmetric wing damage, asymmetric wing damage poses significant stabilization challenges Relying on the pro-posed controller with parameter adaptation, the test platform is able to maintain stable flight even with a severe lift imbalance (>22%) between the two wings, i.e., flying with 10% wing asymmetry

During the asymmetric wing damage test, different pairing methods of the test wings were performed In this section, we mainly focus on the most representative cases, namely, as the wing asymmetry reaches 10% The averaged results of 5 flight trails are summarized in Table III Here, #1 Left-side wing paired with #3 Right-side wing is abbreviated as "1L+3R", similarly hereinafter The results of two typical flight scenarios are illustrated in Fig 9

Referring to Table III, the vehicle generates asymmetric wing kinematics to maintain stability with the proposed control law Such a method is fairly intuitive and consistent with that observed in biological experiments [17] However, the accompanying large differential stroke amplitude will shrink the flight envelope, especially on the roll-axis As shown in Table III, when the wing asymmetry reaches 10%, the roll control input surges up significantly compared to that of the symmetry wings Such a phenomenon brings the inspiration back to the controller design, namely, higher saturation of roll

TABLE III Cost of Asymmetric Wing Damage

Wing area loss 0% + 10% 10% + 0%

Hovering without payload

Power consumption 6.131W 6.055W Stroke amplitude

(left / right)

128.3◦ /155.1◦

149.3◦ /117.6◦ Load lifting

Control input 𝑢1 12.391V 12.342V Control input 𝑢2 1.730V -1.746V Control input 𝑢3 0.741V 0.505V Control input 𝑢4 0.120 0.112

and pitch control input is able to withstand the weakened flight

capacity caused by wing damage

Due to such imbalanced actuation capability, stable flight

becomes a challenging task As shown in Fig 9, the proposed

controller enables the sustained stable flight of the vehicle

even facing severe asymmetric wings Actually, referring to

the test illustrated in Fig 9, the robot undergoes significant

position drift during liftoff due to the imbalanced roll torque

In addition, the wing asymmetric caused yaw torque offset

also affects heading angle stabilization in hover Despite, the

flight control can cope with such defective aspects properly

and maintain the overall stability during the flight As shown

in Fig 9, the proposed adaptation scheme can respond to

these unexpected imperfections immediately, which plays an

important role in the tracking error convergence

VI Conclusion

In this paper, we studied the consequences of wing damage

to the flight capacity of an FWMAV Three sets of test wings

were fabricated, including a pair of intact wings, and two pairs

of damaged wings with different degrees of artificial damages

Utilizing these wings and their different combinations, we

systematically studied the cost of the vehicle facing both

symmetrical and asymmetrical wing damages as well as their

impact on flight capacity, covering three main aspects: lift loss,

control torque offset, and aerodynamic damping change Such

impacts result in several specific control challenges To cope

with these challenges, an adaptive controller is proposed The

experimental results show that, with the proposed flight control

law, the test vehicle can maintain stability by modulating

the different kinematics of the wings to respond to such

undesired wing damages, a strategy similar to that observed

in hovering hawkmoths The proposed study procedures and

control methods can help guide the wing design and flight

control of bio-inspired flying robots operating in real world

environment facing wing wear and tear possibilities

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