Kinematic Analysis of a Flapping-wing Micro-aerial-vehicle with Watt Straight-line Linkage Chao-Hwa Liu* and Chien-Kai Chen Department of Mechanical and Electro-mechanical Engineering, Tamkang University, Tamsui, Taiwan 251, R.O.C. Abstract In this study kinematic analysis of a particular flapping wing MAV is performed to check the symmetry of the two lapping wings. In this MAV symmetry is generated by a Watt straight line mechanism. After appended by two more links to provide a continuous input, and it becomes a Stephenson type III six-bar linkage. Together with the two wings the vehicle has 10 links and 13 joints. Since a Watt four-bar linkage can only generate approximate straight lines, the deviation from an exact straight line causes phase lags of the two wings. The goal of this study is to determine the phase lags. To achieve this goal a forward kinematic analysis of the Stephenson III linkage is performed, which refers to the procedures that may determine the position, velocity, and acceleration of the MAV. Among these procedures, position analysis involves equations that are highly nonlinear and deserves special attention. The authors developed two solution techniques for the forward position analysis of Stephenson III mechanisms: an analytic procedure which leads to closed-from solutions; and a numerical technique to obtain approximate solutions. We use the numerical technique to perform kinematic analysis because solutions obtained by the two methods agree almost exactly but the numerical method is much faster. We analyzed the MAV with the same dimension as the real model, and found it to have very good symmetry with negligible phase lags between the two wings. Key Words: Micro-aerial-vehicle, Forward Position Analysis, Watt Straight-line Linkage, Stephenson Type III six-bar Linkage 1. Introduction Recently many research efforts have been made on design and construction of Micro Aerial Vehicles (MAV). Insect-like MAVs generally have two flapping wings, and various mechanisms to drive the wings have been sug- gested and tested. For example, Yang [1] utilized a four- bar crank-rocker linkage in his flapping wing device. Galinski and Zbikowski [2] developed a mechanism in which a double rocker linkage is driven by a crank rocker mechanism. Galinski and Zbikowski [3] made use of a double spherical Scotch yoke mechanism to generate de- sired motion. The mechanism developed by McIntosh et al. [4] includes planar four-bar linkages, spatial cam me- chanisms, and slotted arms. Zhang et al. [5] used a mech- anism with a spatial single crank double rocker mecha- nism. Yang, et al. [6] demonstrated the design, fabrica- tion, and performance test of a 20 cm-span MAV, which has a flapping angle up to 100°. Recently, a model of a flapping wing MAV is proposed [7], the basic structure of which consists of a Stephenson type III six-bar mecha- nism that includes a Watt four-bar linkage, and two wing mechanisms. Symmetry of the two wings is due to the straight line motion generated by the Watt four-bar link- Journal of Applied Science and Engineering, Vol. 18, No. 4, pp. 355-362 (2015) DOI: 10.6180/jase.2015.18.4.06 *Corresponding author. E-mail: [email protected]This paper is the extension from the authors’ technical abstract pre- sented in the 1 st International Conference on Biomimetics And Orni- thopters (ICBAO-2015), held by Tamkang University, Tamsui, Taiwan, during June 28-30, 2015.
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Kinematic Analysis of a Flapping-wing
Micro-aerial-vehicle with Watt Straight-line Linkage
Chao-Hwa Liu* and Chien-Kai Chen
Department of Mechanical and Electro-mechanical Engineering, Tamkang University,
Tamsui, Taiwan 251, R.O.C.
Abstract
In this study kinematic analysis of a particular flapping wing MAV is performed to check the
symmetry of the two lapping wings. In this MAV symmetry is generated by a Watt straight line
mechanism. After appended by two more links to provide a continuous input, and it becomes a
Stephenson type III six-bar linkage. Together with the two wings the vehicle has 10 links and 13 joints.
Since a Watt four-bar linkage can only generate approximate straight lines, the deviation from an exact
straight line causes phase lags of the two wings. The goal of this study is to determine the phase lags.
To achieve this goal a forward kinematic analysis of the Stephenson III linkage is performed, which
refers to the procedures that may determine the position, velocity, and acceleration of the MAV.
Among these procedures, position analysis involves equations that are highly nonlinear and deserves
special attention.
The authors developed two solution techniques for the forward position analysis of Stephenson
III mechanisms: an analytic procedure which leads to closed-from solutions; and a numerical
technique to obtain approximate solutions. We use the numerical technique to perform kinematic
analysis because solutions obtained by the two methods agree almost exactly but the numerical method
is much faster. We analyzed the MAV with the same dimension as the real model, and found it to have
very good symmetry with negligible phase lags between the two wings.
Key Words: Micro-aerial-vehicle, Forward Position Analysis, Watt Straight-line Linkage, Stephenson
Type III six-bar Linkage
1. Introduction
Recently many research efforts have been made on
design and construction of Micro Aerial Vehicles (MAV).
Insect-like MAVs generally have two flapping wings, and
various mechanisms to drive the wings have been sug-
gested and tested. For example, Yang [1] utilized a four-
bar crank-rocker linkage in his flapping wing device.
Galinski and Zbikowski [2] developed a mechanism in
which a double rocker linkage is driven by a crank rocker
mechanism. Galinski and Zbikowski [3] made use of a
double spherical Scotch yoke mechanism to generate de-
sired motion. The mechanism developed by McIntosh et
al. [4] includes planar four-bar linkages, spatial cam me-
chanisms, and slotted arms. Zhang et al. [5] used a mech-
anism with a spatial single crank double rocker mecha-
nism. Yang, et al. [6] demonstrated the design, fabrica-
tion, and performance test of a 20 cm-span MAV, which
has a flapping angle up to 100�. Recently, a model of a
flapping wing MAV is proposed [7], the basic structure
of which consists of a Stephenson type III six-bar mecha-
nism that includes a Watt four-bar linkage, and two wing
mechanisms. Symmetry of the two wings is due to the
straight line motion generated by the Watt four-bar link-
Journal of Applied Science and Engineering, Vol. 18, No. 4, pp. 355�362 (2015) DOI: 10.6180/jase.2015.18.4.06
300�, Convergence criterion for numerical procedure is
that convergence is achieved when �DG � L3� < 10-10.
Two solutions are found for this problem and they are
given in Table 1. One may observe that results obtained
from the two procedures agree up to at least 7 digits after
the decimal point. Since the numerical procedure is fas-
ter, the following results for position analysis are calcu-
lated by this procedure.
3.2 Wing Mechanisms
Figure 4 shows the Stephenson III six-bar linkage in
the flapping wing MAV. As a value of input angle �2 is
given, positions of all other links may be found by the
method just been discussed, and from these positions the
coordinates of C, namely Cx and Cy, can be calculated.
When the position of C is found, joint D of the right wing
mechanism (Figure 1) can be located since it is the inter-
section of the following two circles, the circle centers at
Kinematic Analysis of a Flapping-wing Micro-aerial-vehicle with Watt Straight-line Linkage 357
Table 1. Comparison of results obtained using two
procedures; differences are underlined
Closed-form Numerical
1st sol �21.0981401119277� �21.0981401120935��3 2nd sol �9.34660961319297� �9.34660961312838�
1st sol 28.1602559557637� 28.1602559560016��4 2nd sol 22.7638442767922� 22.7638442768305�
1st sol 189.74678710913� 189.74678711034��5 2nd sol 113.48012093066� 113.48012092989�
1st sol 145.052820929300� 145.052820929443��6 2nd sol 124.750245471184� 124.750245470924�Figure 3. A part of the coupler curve generated by the point D
on the coupler of a four-bar linkage.
C with a radius L4, and the circle centers at E whose ra-
dius is L5. In general, there are two intersections, as
shown in Figures 5 and 6. The first solution shown in
Figure 5 is given by
(2)
where R is the known distance between C and E, and
(3)
The second solution as shown in Figure 6 contains the
following two angles
(4)
(5)
Note that, generally the two solutions do not intersect.
Once the right wing is assembled in either of the two
ways, it remains the same configuration unless it is dis-
connected and reassembled.
The left wing also has two solutions, given by [13]
(6)
(7)
and
(8)
(9)
4. Velocity and Acceleration Analysis
Velocity analysis is performed after the position analy-
sis, that is, when all link positions have been found. The
order for velocity analysis is similar to that for position
analysis; the Stephenson III six-bar linkage must be treated
first, since both wings are driven by it.
4.1 Stephenson Type III Six-bar Linkage
The Stephenson III linkage illustrated by Figure 4
has the following two closed loops:
358 C. H. Liu and Chien-Kai Chen
Figure 4. The Stephenson III six-bar linkage in the MAV.
Figure 5. The first solution for the position analysis of theright wing mechanism.
Figure 6. The second solution for the position analysis of theright wing mechanism.
AB + BC + CG = AI + IG (10)
HF + FG = HI + IG (11)
The x and y components of equation (10) are
(12)
(13)
Similarly vector equation (11) has the following two
components
(14)
(15)
Differentiating these four equations once with respect
to time, one may obtain
(16)
where the matrix M1 is given by
(17)
With positions of all links known, for a given input ve-
locity �� 2 one may solve equation (16) for unknown ve-
locities �� 3, �� 8 , �� 9 and �� 10.
4.2 Wing Mechanisms
Referring to Figure 7, one may notice the vector loop
closure equation for the right wing
AB + BC + CD = AE + ED (18)
The x and y components of this equation are
(19)
(20)
Upon differentiation, we may obtain
(21)
where
(22a)
(22b)
Angular velocities �� 4 and �� 5 may be found from equa-
tion (21). Finally, the loop equations for the left wing is
(see Figure 8)
AB + BC + CK = AJ + JK (23)
Following the same procedure as before, one may ob-
tain [13]
(24)
in which
(25)
Kinematic Analysis of a Flapping-wing Micro-aerial-vehicle with Watt Straight-line Linkage 359
Figure 7. Notations of the right wing mechanism for velocityand acceleration analysis.
From these equations the velocities ��6 and �� 7 can be ob-
tained.
Unknown accelerations may be calculated when posi-
tions, velocities, and driving accelerations ��� 2 are known.
Equations for calculating accelerations are obtained by
differentiating velocity equations (16), (21), and (24).
For the Stephenson III six-bar linkage, the equation is
(26)
where
(27)
After accelerations ��� 3, ��� 8 , ��� 9 , and ��� 10 are obtained from
equation (26), accelerations for the right wing and the
left wing may be found from
(28)
and
(29)
5. Results and Discussions
The dimension for the MAV, as illustrated by Figure
1, is as follows [7]: AB = L2 = 2 mm, BC = L3 = 16 mm,
CD = L4 = 6.5 mm, CK = L6 = 6.5 mm, DE = L5 = 3 mm,
JK = L7 = 3 mm, GI = L8 = 10 mm, HF = L9 = 10 mm, GF
= L10 = 4 mm, and GC = FC = 2 mm. These lengths are
used in the subsequent analysis. Figure 9 shows the par-
ticular configuration obtained from position analysis
when �2 = 0. Angular position �5 of the right wing is
shown in Figure 10. Note that the angle �5 defined in
Figure 7 is not the flapping angle. The flapping angle
generally refers to the angle by which the wing extends
outward from the body (see Figure 1). The two angles al-
ways differ by a constant. The angular stroke of the wing,
also called flapping range, is the maximum difference in
�5. For this mechanism its value is 70.6�. To check the
symmetry between the two wings, we compare the angle
�� 5 in Figure 7, to the angle �7 in Figure 8; also we com-
pare and the angle �4 in Figure 7 to the angle ��6 in Fig-
ure 8. The differences between these angles are shown
in Figure 11. The maximum difference in this figure is
between �� 5 and �7, its value is 0.0825�, and it occurs
when �2 = 90�. The coupler curve generated by the point
360 Chao-Hwa Liu and Chien-Kai Chen
Figure 9. Configuration of the flapping wing MAV when �2
= 0.Figure 8. Notations of the left wing mechanism for velocity
and acceleration analysis.
C on the Watt linkage approximates a vertical straight
line. The position �2 = 90� corresponds to one extreme
position for joint C where it begins to deviate further
from a straight line, causing a larger degree of asym-
metry of the two wings at this position.
Results for velocity and acceleration analysis are
obtained with the constant input velocity �2 = 1 rad/s.
The difference in angular velocity between link � and
link is shown in Figure 12. The maximum value in
this figure is 8.8048 � 10-5 rad/s and it occurs when �2 =
113�. In Figure 13 we show the velocity difference be-
tween link and link�. Its maximum value is 4.8085
� 10-5 rad/s, which occurs when �2 = 114�. The differ-
ence in acceleration between link� and link is shown
in Figure 14. The maximum difference is 3.583525 �
10-4 rad/s2, which occurs at the position �2 = 90�. Finally,
the difference in acceleration between link and link�
is shown in Figure 15. The maximum difference occurs
when �2 = 100�, whose value is 1.820966 � 10-4 rad/s2.
6. Conclusions
In this study we show that a Stephenson III six-bar
Kinematic Analysis of a Flapping-wing Micro-aerial-vehicle with Watt Straight-line Linkage 361
Figure 11. Phase lag of the two wings.
Figure 12. Velocity difference between link� and.
Figure 13. Velocity difference between link and link�.
Figure 14. Acceleration difference between link� and link.
Figure 10. Angular position of link.
linkage may be viewed as two separate mechanisms in
forward position analysis. Based on this separation, two
techniques for position analysis of this linkage are sug-
gested. While one technique may lead to closed-form so-
lution without any iterative procedure, the numerical te-
chnique is much faster and converges to nearly the same
value obtained by using closed-form solution. The nu-
merical technique is used for position analysis of a flap-
ping wing MAV whose basic structure is Stephenson III
six-bar linkage. We find the two wings of this MAV are
highly symmetric and phase lags of the two wings are in-
significant.
Acknowledgements
The authors gratefully acknowledge that this study
was supported by the National Science Council of ROC
under Grant NSC 101-2632-E-032-001-MY3.
References
[1] Yang, L. J., U.S. Patent 8,033,499B2 (2011).
[2] Galinski, C. and Zbikowski, R., “Materials Challenges
in the Design of an Insect-like Flapping Wing Mecha-
nism Based on a Four-bar Linkage,” Materials and
Design, Vol. 28, pp. 783�796 (2007). doi: 10.1016/j.
matdes.2005.11.019
[3] Galinski, C. and Zbikowski, R., “Insect-like Flapping
Wing Mechanism Based on a Double Spherical Scotch
Yoke,” Journal of the Royal Society Interface, Vol. 2,
No. 3, pp. 223�235 (2005). doi: 10.1098/rsif.2005.
0031
[4] McIntosh, S., Agrawal, S. and Khan, Z., “Design of a
Mechanism for Biaxial Rotation of a Wing for a Hover-