Available online at www.sciencedirect.com
ScienceDirect
Procedia Engineering 00 (2014) 000–000
www.elsevier.com/locate/procedia
“APISAT2014”, 2014 Asia-Pacific International Symposium on Aerospace Technology,
APISAT2014
Flapping Wing Multi-body Dynamic Simulation
Zhong Jingyang a, Song Bifenga, Wang Jin a*
a School of Aeronautics, Northwestern Polytechnical University, Xi’an, 710072, China
Abstract
The previous study on modelling of the flapping wing micro air vehicle used to concentrate on the complicated aerodynamic
computation and the research on the dynamic simulation is often constrained to frequently use the oversimplified 6-degree of
freedom (DOF) rigid body equations. However, the motion of the wing is seriously interfered with the body’s motion. The
aforementioned 6-DOF single rigid body equations do not take the inertial coupling effects into account during the whole flight.
For this sake, the article established an available simulation model in the form of multi-body dynamic equation. First, by
applying Newton’s laws and angular momentum theorem to the vehicle, the multi-body dynamic equation can be deduced. Then,
a dynamic model is built under the circumstance of Matlab/Simulink. Finally, the modelling is verified against the data of the
wind tunnel tests from the laboratory and the feedback of the real flight tests.
© 2014 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA).
Keywords: flapping wing;multi-body dynamic;flight dynamics;simulation;
1. Introduction
The multi-body dynamics of the flapping wing micro air vehicle (FWMAV) are strongly coupled between the
wing and the body. The common approximation method to the dynamics of FWMAV is to ignore the mass of the
wings, since the mass of the wings generally takes up less than 5% of the total body mass [1], while neglecting the
mass of the wings may be problematic in dynamics and control studies [2]. Bolender [3] thinks that for the proper
development of control laws for FWMAVs, the mass of the wings should not be neglected.
* Corresponding author:. Tel.: +86-18729583720; .
E-mail address: zjynwpu@mail.nwpu.edu.cn
1877-7058 © 2014 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA).
2
Zhong Jingyang et al. / Procedia Engineering 00 (2014) 000–000
Considering the motion mechanism for a flapping flight is different from a common fixed wing micro air vehicle,
multi-body equations of motion are developed and presented by many researchers, such as Gebert et al [4] and Mao
Sun [5]. Although there are some articles studying about the flapping wing dynamic simulation, some simplifications
still have been made more or less in the simulating process.
Based on the others’ work about the multi-body dynamic equation derivation.This article mainly established an
available simulation model in the form of multi-body dynamic equation by regarding the aircraft body and the wing
as independent sectors. The verification of this model is made against the data of the wind tunnel tests from the
laboratory and the feedback of real flight tests.
2. Definition of the reference frame
The reference frames that are used for describing the position and the velocity of the vehicle are showed in figure
1[6], including the inertial frame, body frame and wing frame.
Fig 1 location of the reference frame
1) Inertial frame (Xg, Yg, Zg) is a frame whose center and the axes follows the definition in the Ref. [7]. The earth
is treated as a stationary flat in the inertial space.
2) Body frame (Xb, Yb, Zb) is a frame fixed on the fuselage with its origin located at the center of gravity of the
vehicle body excluding the wing part. The direction of arrows follow the definition in Ref. [7].
3) Wing frame (Xw, Yw, Zw) is a frame fixed on the wing with its origin located at the root of the wing and the
direction of the Yw is changing along with the flapping wing.
The definition of the transformation matrix of direction cosines between different frames follows the rules in Ref
[8].
3. Multi-body Motion Equations
3.1. Force equation
Force equations can be deduced from Newton’s 2nd Law [5]
d
 g F  dt m g v


(1)
The body can be divided into many small elements with a mass of m and each of them should obey Newton’s
2nd Law as well. Then, the expression for these elements are
 gF  m
d gv
dt
The inertial velocity of m can be written as follows
(2)
Zhong Jingyang et al. / Procedia Engineering 00 (2014) 000–000
v
g
d R
d
( g R cg  g R )  g vcg  g
dt
dt
3
(3)
Where gRcg is the vector describing the distance from the center of the inertial frame to the center of the gravity
of the vehicle body, gR is the vector from the center of the gravity of the body to any point in the body. Then, we
have
g


d v
d2 R
d v 
d2 R

F    m g cg   m g2     m g cg     m g2 
dt
dt  total 
dt  total 
dt 
total 

(4)
According to the mass relation between the wing and the body, equation (4) can be written as
g
F  mtotal
d g v cg
dt

d2
dt 2
 
g
body

d2 R
R m     m g2 
dt 
wing 

(5)
The vector gR can be expanded as follows
g
R  g R h  g R wg  g R p
(6)
Where gRh is the vector from the center of gravity of the vehicle body to the root of the wing; gRwg is the vector
from the root of the wing to the center of gravity of the wing; gRp is the vector from the center of gravity of the wing
to any point in the wing.
Substitute equation (6) to equation (5), we have


d2
   m 2  g R h  g R wg  g R p 
dt
dt
wing 

2
2
d g v cg
d
  mwg ,i 2  g R h  g R wg 
dt
dt
i 1
i
d g v cg
g F  mtotal
 mtotal
(7)
According to the relation below, the vector can be transformed from inertial frame to the body frame
d gV
dt

d bV
 b ωbd  bV
dt
(8)
Then d 2 g R h and d 2 g R wg can be written as
dt 2
d 2 g Rh
dt 2
d 2 g R wg
dt
2
dt 2

d b ω bd
d  d g Rh  d
 b Rh  b ω bd   b ω bd  b Rh 

   b ω bd  b Rh  
dt  dt  dt
dt

d w ω wg
d  d g R wg  d
 w R wg  w ω wg   w ω wg  w R wg 

   w ωwg  w R wg  
dt  dt  dt
dt
(9)
(10)
Finally, the force part of the multi-body dynamic equation can be derived as follows
 d b ω bd

 dt  b R h  b ω bd   b ω bd  b R h  

 d b v cg



 b ωbd  b vcg    mwg 
i
b F  mtotal 
d
ω


dt
w
wg

 i 1

 w R wg  w ω wg   w ω wg  w R wg   

E
 w b  dt
 
2
(11)
3.2. Moment equation
Applying the angular momentum theorem to an element m of the aircraft in the inertial frame, it can be acquired
4
Zhong Jingyang et al. / Procedia Engineering 00 (2014) 000–000

g
M
d gH
(12)
dt
By using some formula derivation techniques similar to the process of the force equation derivation, the moment
equation can be deduced as
M    mwg b v cg   b ωbd  b R h  b ω wg  b R wg    b ωbd  b I bd b ωbd 
N
b
i 1
i
N m ( R  R
 
wg b
h
b
wg )  ( b ν cg  b ωbd  b R h ) 
d 

 b I bd b ωbd    m
R ( ω  R )  E ( I
ω ) 
dt 
i 1 
 wg b h b wg b wg w b w wg w wg  i 

(13)
N  b ω bd  E ( w I wg w ω wg )  mwg b ωbd   b R h  ( b ω wg  b R wg )  



w b
 

i 1   m
ω  R  R  ν  ω  R 

 wg b bd  b h b wg   b cg b bd b h  
i
4. Model structure
The simulation model is built on the basis of the multi-body equation of motion as mentioned in the former
section and it can be mainly divided into 3 parts including initialization part, force part and moment part.
4.1. Initialization part
In the process of building the multi-body dynamic model, some parameters that reflect the relative position
relations between different frames should be given in the Initialization parts, such as the position vector from the
center of the body frame to the center of the wing frame. The transformational relation for a vector from one frame
to another is not just always multiplied by a transformational matrix because of the coupling relationship between
motion of the body and the wing. Taking the angular velocity of the wing for example, the transformation between
two frames should be made according to the following rules.
ω wg  w ω wg0  E b ωbd
(14)
ω wg  E
(15)
w
b
b w
w b
w
ω wg  b ω wg0 + b ωbd
Where w ω wg0 , reflecting the flapping motion of the wing, represents the angular velocity of the wing relative to
the body. Here gives the inner structure of the Initialization part.
Fig 2 inner structure of the Initialization part
The majority of the parameters that will be used in later calculation can be derived from this part.
Zhong Jingyang et al. / Procedia Engineering 00 (2014) 000–000
5
4.2. Force and moment part
On account of the inertia coupling effects for the multi-body system, the motion of the wing and the rotation of
the body interact with each other. Because of the coexistence of linear velocity and angular velocity in the force
equation and moment equation, these two parts need to be built under the same platform and the linear velocity and
the angular velocity of the aircraft body can be acquired from these two parts after some simulation process.
5. Simulation results
In order to verify this multi-body dynamic model, some simulations need to be conducted. Considering the lack
of aerodynamic model, wind tunnel test results from the laboratory would be the first choice to take the place of the
aerodynamic model. In the simulation process, it is assumed that the wing is flapping as sine wave with time and its
initial position is where the flapping angle is zero. To make the wind tunnel data match the angular velocity of the
wing, some translation operations should be done to the wind tunnel data first. Besides, the phase differences
between the force and moment should also be paid attention to while translating. The actually used data are showed
in the figure 3.
Fig 3 wind tunnel data of Fx, Fz,Ty and flapping angle of the wing
To simulate the flapping wing micro air vehicle in the general flight phase, the initial conditions are set as
follows: flying at 8m/s speed and 10m altitude with 10°angle of pitch.
Fig 4(a) angular velocity of the vehicle ;(b) displacement of the vehicle; (c) Vx of the vehicle
The simulation results obviously show that the angular rates of the body is different from the single body
dynamic model because there are three direction angular velocities under the assumption of three freedom input (Fx,
Fz, Ty) and it has features of the typical motion of sinusoidal mode as expected. Although the sign of the altitude is
minus which can be seen in the figure 4(b), the aircraft is actually climbing because the definition of the positive
direction for axis z is downward which has been shown in the figure 1. From the figure 4(c), it can be seen that the
aircraft is flying forward at a speed of about 8m/s with a periodic fluctuation. Like the real flight test, they are in
6
Zhong Jingyang et al. / Procedia Engineering 00 (2014) 000–000
good agreement with the simulation output and the flight test results.
6. Conclusion
The article puts a premium on establishing an available simulation model in the form of multi-body dynamic
equation by considering the inertia coupling effects of the wing. The inertial force and inertial moment between the
wing and the body are reflected in the simulation model and the multi-body dynamic equation of motion is presented
in the form of implicit nonlinear differential equation rather than the explicit equation of the single body dynamic.
The results evidence that the multi-body dynamic model can reflect the basic characteristics of the flapping wing
micro air vehicle.
Acknowledgements
The authors would like to thank the Northwestern Polytechnical University NMRL laboratory for the supply of
the wind tunnel data.
References
[1] Taylor, G. and Thomas, A., Dynamic flight stability in the desert locus Schistocerca gregaria, The Journal of Experimental Biology, Vol. 206,
2003, pp. 2803-2829.
[2] Orlowski, C. and Girard, A., Modelling and Simulation of the Nonlinear Dynamics of Flapping Wing Micro-Air Vehicles, AIAA Journal, Vol.
49, 2011. pp 969-981
[3] Bolender, M. A., Rigid Multi-Body Equations-of-Motion for Flapping Wing MAVs using Kane’s Equations, AIAA Guidance, Navigation,
and Control Conference, Chicago, Illinois, USA, August 10-13, 2009, AIAA, 2009.
[4] Gebert, G., Gallmeier, P., and Evers, J., Equations of Motion for Flapping Flight, AIAA Atmospheric Flight Mechanics Conference, AIAA
Paper 2002-4872, Monterrey, CA, 2002.
[5] Sun Mao, Wang Jikang, Xiong Yan. Dynamic flight stability of hovering insects. Acta Mech. Sinica 23, 231–246, 2007
[6]Gao Guanglin. Research on dynamic model and flight simulation for micro air vehicle [D].Xi’an: Northwestern Polytechnical University,
2011, pp 67-69
[7] Li Deming , Chen Changmin. Classical mechanics. Beijing: Press of Higher Education, 2006. [in Chinese]
[8] Bernard Etkin, Lloyd Duff Reid. Dynamics of flight stability and control. United States of America: Wiley, 1996, pp 98-100.