This document summarizes research on modeling and simulating a flapping wing mechanism based on the flight of pigeons. The researchers created a 3 degree of freedom parallel mechanism to simulate the flapping, bending, twisting, and other wing motions. They analyzed the mechanism's workspace to identify singularities and implemented inverse kinematics control. Testing showed that increasing flapping amplitude and frequency increased lift, but excessive values caused issues. Future work proposed improving the flexibility of the wing model, implementing feedback control, and further optimizing flapping motion parameters like angle of attack variation.

1. 2016 – INSTITUT SUPERIEUR DE L’AERONAUTIQUE ET DE L’ESPACE 1
Flapping Wing
Sonali BATRA, Karthik CHERAL HOUSE, 2016
AESS, Aerospace Control and Guidance, ISAE-Supaero
Yves BRIERE, ISAE-Supaero
Toulouse, France
1 INTRODUCTION
flapping wing aircraft or ornithopter is an aircraft that
flies by flapping its wings. The flapping wing flight can
be considered as one of the most complex types of mo-
tion. A biomimetic flapping wing should be capable of fol-
lowing all the principles of the bird flight and must be able to
operate in all type of flight conditions like take off, landing,
gliding, soaring, faster maneuvers and flying sideways and
backwards [1].
During flapping flight, wings have several types of mo-
tion: flapping up & down, bending, moving forward relative
to air, twist and sweep. During flapping, wings develop lift
and create an additional forward and upward force, thrust,
to counteract its weight and drag [2]. It involves two stages:
the down stroke and up stroke. The angle of attack increases
during down stroke and decreases during up stroke. During
down stroke, the air is pushed downwards and rearwards.
Hence the thrust is obtained in the opposite direction in
accordance to Newton’s third law. The wing twists along its
span to maintain the correct angle of attack [3].
The flapping wing aircraft has its applications in the field
of security and defense due to high maneuverability neces-
sary for operation. Recently, the interest is more in flapping
wings that resemble insects and small birds such as pigeon.
The objective of this research is to mimic the wing move-
ment of pigeon during forward flight through parallel mech-
anism. This mechanism consists of Q servo angles and X
upper end platform angles. The primary objectives are: a)
Calculation of workspace to identify the regions where the
flapping can be performed without any singularities through
Jacobian calculation. b) Practical implementation of this
concept to control the half pigeon model constructed by Prof.
Yves Briére to determine lift produced by the flapping wing.
2 MODEL DESCRIPTION
Fig 1 & 2: Model description
The model is a planar, symmetrical-parallel manipulator,
with two platforms- fixed base and upper end platform. The
three limbs achieve three degrees of motion in x, y and z
directions. Each limb is rotated using a servomotor fixed at
the fixed platform.The wing and the manipulator is fixed at
one end of a carbon fiber rod. Its other end is connected to a
counter weight. It is hinged to a frame allowing the assembly
to move in y-direction. The flapping sequence produces lift
which raises the wing side of the rod. The assembly is
mounted on a stand which can rotate around the y-axis.
3 KINEMATICS
Workspace is the region reachable by the end effector. The
workspace depends on mechanical limits and lengths of the
links, servo and upper platform angles and range of motion
of joints. Calculation of workspace in this project is obtained
by an algorithm that computes the servo angles Q for a given
set of platform angles X. The program evaluates the Jacobian
of Q and X and decides whether the point is singular or not.
The Jacobian maps the velocities in joint space to velocities in
the end-effector. If the determinant of the Jacobian is equal to
0, it is said to be singular. Singularities directly affect the
mobility of manipulator due to loss in degree(s) of freedom
and should be avoided for proper trajectory planning.
𝐽 𝑄 𝛿𝑄 + 𝐽 𝑋 𝛿𝑋 = 0
JQ and JX are Jacobian matrices with respect to Q and X.
4 CONTROLS
Inverse kinematics is important to provide the input to
servo angles Q based on the desired orientation of upper
platform angles X.
Solution of a func-
tion f, X=f(Q) is
straight forward
however Q= f-1(X)
is difficult analyti-
cally. Based on
CLIK the loop
above is implemented. The computation time is significantly
reduced by this implementation however singularities are
not considered. A control model was prepared to generate Q
servo angles that are commanded to the flapping wing
mechanism through an interface. Nine input parameters are
used to form the three X angles of the upper platform.
Φ = 𝐴Φ sin(2𝜋𝑓𝑡) + 𝛼𝑓𝑙𝑎𝑝
Ψ = 𝐴Ψ sin(2𝜋𝑓𝑡 + φΨ) + 𝛼 𝑟𝑜𝑙𝑙
θ = 𝐴θ sin(2𝜋𝑓𝑡 + φθ) + 𝛼 𝑦𝑎𝑤
where (𝛷, 𝛹, 𝜃) represent flap, roll and yaw respectively with
amplitudes A, frequency f, phase 𝜑 and AoA 𝛼.
To implement CLIK, the three computed angles (Φ, Ψ, θ)
along with a constant initial value of Q are taken as the input
to compute the distance d between the fixed base and upper
platform. The aim of the model is to deliver the best possible
value of Q for given X without any distance error 𝛿𝑑.
A

2. 2 MASTER OF SCIENCE AESS
5 RESULTS
At each configuration of input X angles, a function is run
that finds the ideal set of Q servo configuration that is
achievable. Starting with workspace analysis, an algorithm
that computes the direct and inverse Jacobian for ideal (Q, X)
is coded which reports singularities even if one of the follow-
ing conditions is satisfied. a) det(𝐽 𝑄) = 0. b) det(𝐽 𝑋) = 0. c)
Both det(𝐽 𝑄) = 0 and det(𝐽 𝑋) = 0.
Fig 4&5: Workspace for 2D and 3D singularities with X= [60 60 60]
The aim of the control loop modeling was to achieve the
desired set of Q servo angles for given X angles and to map
them to the lift force hence generated. Five out of nine input
parameters were tuned at this stage of the research without
consideration of yaw parameters.
Fig 6&7: Lift generated with variable flap amplitude (left) and
variable AoA (right)
Through performance mapping, we were able to obtain the
ideal configuration for the input parameters.
𝐴Φ = 25° , 𝐴Ψ = 5°𝑓 = 4𝐻𝑧, 𝛼 𝑟𝑜𝑙𝑙 = −5°
Also we were able to realize the mechanical limitations of the
mechanism in terms of frequency, amplitude and AoA.
6 CONCLUSIONS
The implementation of new features (minimization func-
tion and explicit numerical Jacobian computation) in the
code has been effective in reducing the time required for
workspace generation. The workspace that was generated;
helped in understanding the zone where the flapping can be
performed effectively, without reaching the mechanical lim-
its and/or the singularity regions.
Increase in the flap amplitude and the frequency of flap-
ping increases the lift generation linearly. Increase in the flap
amplitude above certain limit induces flap-yaw coupling
which tends to decrease the lift produced. Exceeding the
upper limit of the flapping frequency, results in breakage of
the links. The variation of roll amplitude and its phase does
not significantly contribute in lift generation. Moreover it
induces vibrations in the mechanism.
Also the wing-counter weight model is a flexible body
instead of being considered as a rigid body earlier. There are
significant amount of vibrations produced which reduce the
lift.
7 FUTURE WORK
The future work is divided into solving mechanical, con-
trol implementation and optimization issues.
It is advised to implement ball that will facilitate more
movement between the links to overcome the mechanical
restrictions faced during this study. It is necessary to rebuild
the wing considered in the study to match the wing dimen-
sions of a fully-grown pigeon by attempting to make use of
NACA 0014 and NACA 0015 airfoils.
The aim is to study the effect of the angle of attack varia-
tion in pitch and yaw direction on the amount of lift pro-
duced. During the bibliography survey, it was found that, in
a flapping sequence, the upstroke is faster compared to the
downward stroke [3]. Hence effort has to be made towards
trying to imitate this behavior of flap by using the saw-tooth
wave as input instead of using the sinusoidal waves.
In order to further optimize the behavior of the flapping
wing it is extremely important to execute the feedback ap-
proach that makes use of sensors to track the achieved upper
platform angles. The feedback approach will have several
advantages including better flapping movements at higher
frequencies and more precision even at lower frequencies
and amplitudes. Eventually, it will be helpful in meeting the
bigger objective of mimicking the pigeon flight. The authors
look forward to implementing the singularity-recognition
feature in the control loop as well. Currently, singularity-
recognition has been performed in the workspace algorithm.
The future step would be to take it ahead in the Simulink
model to realize its behavior in real time.
References
[1] Joon Hyuk Park and Kwang-Joon Yoon. Designing a Biomimetic
Ornithopter Capable of Sustained and Controlled Flight. Journal of
Bionic Engineering 5 (2008) 39−47.
[2] Rayner JMV. A vortex theory of animal flight. Part 2. The forward
flight of birds. Journal of Fluid Mechanics 1979;4:731–63
[3] Joel GUERRERO. Numerical simulation of the unsteady aerodynam-
ics of flapping flight. Chapter 2 of thesis submitted during MPhil
studies University of Genoa.