Journal of Applied Science and Engineering, Vol. 18, No. 4, pp. 355-362 (2015)
DOI: 10.6180/jase.2015.18.4.06
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 suggested and tested. For example, Yang [1] utilized a fourbar crank-rocker linkage in his flapping wing device.
Galinski and Zbikowski [2] developed a mechanism in
*Corresponding author. E-mail: chaohwa@mail.tku.edu.tw
This paper is the extension from the authors’ technical abstract presented in the 1st International Conference on Biomimetics And Ornithopters (ICBAO-2015), held by Tamkang University, Tamsui, Taiwan,
during June 28-30, 2015.
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 desired motion. The mechanism developed by McIntosh et
al. [4] includes planar four-bar linkages, spatial cam mechanisms, and slotted arms. Zhang et al. [5] used a mechanism with a spatial single crank double rocker mechanism. Yang, et al. [6] demonstrated the design, fabrication, 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 mechanism 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-
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Chao-Hwa Liu and Chien-Kai Chen
age. However, a Watt four bar linkage can only generate
an approximate straight-line, the purpose of this study,
therefore, is to determine the difference, or the phase lag,
between the two wings. Phase lags include differences of
the two wings in position, velocity, and acceleration. Kinematic analysis must be performed to determine position, velocity, and acceleration of the mechanism.
The paper is organized as follows. The construction
of the mechanism is introduced in the next section; then
the procedure for position analysis is discussed in section 3. Velocity and acceleration analysis procedures are
explained in section 4. Analysis results are shown in
section 5, followed by the conclusion in section 6.
2. Construction of the Mechanism
Figure 1 is an illustrative diagram for the flapping
wing MAV proposed by Cheng [7]. The mechanism has
10 links and 13 pin joints. Links , ˆ, ‰, and Š constitute a Watt four-bar mechanism that makes joint C
moving along an approximate straight line. A Stephenson type III six-bar linkage is formed by adding links ‚
and ƒ to the Watt straight line mechanism. The reason
for adding these two links is obvious, since not a single
link in the Watt mechanism is able to make a full revolution. With two links added, input to this mechanism is
the continuous rotation of link ‚ generated by a motor.
Finally, links „ … and links † ‡ are the right wing and
the left wing mechanisms, and flapping of both wings
are produced by motion of joint C. Symmetric motion of
the two wings is achieved if joint C moves along a perfect
Figure 1. An illustrative diagram for a flapping wing MAV.
straight line along the vertical direction, which can only
be generated by more complex mechanisms [8]. In the
current design, the Watt four-bar linkage is used to reduce
number of links and total weight, the cost of using this
simple linkage is that this mechanism can only produce
approximate straight lines. In the subsequent analysis,
we determine the phase lag of the two links when link ‚
rotates with a constant angular velocity.
3. Position Analysis
3.1 Stephenson Type III Six-bar Linkage
Position analysis of the Stephenson type III six-bar
linkage is performed first, from which the position of joint
C in Figure 1, as a function of position of the input link,
may be determined. Previous study includes Jawad [9]
who obtained algebraic equations that govern the position of this linkage, and he solved the equations numerically. Watanabe and Funabashe [10,11] obtained solutions for positions of various Stephenson six-bar linkages
by solving six order algebraic equations. Watanabe et al.
[12] extended this study to 23 planar linkages with a similar construction. Since these solution techniques involve equations with page-long coefficients, in this study
we developed different methods to obtain solutions, as
introduced below.
Figure 2 shows a general Stephenson type III six-bar
linkage driven by link ‚. When dimension of all links
are known, the purpose is to determine position variables q3, q4, q5, and q6, for a given input angle q2. The
Stephenson III linkage is composed of a four-bar linkage
that includes links …„†, and a two-bar chain that
contains links ‚ƒ. Note that the separation of this sixbar linkage this way has been suggested by Watanabe and
Figure 2. A Stephenson type III six-bar linkage.
Kinematic Analysis of a Flapping-wing Micro-aerial-vehicle with Watt Straight-line Linkage
Funabashe [10,11]. Yet they used this idea in determining limiting positions of the linkage, not in determining
positions in general. In the following, we will discuss
two techniques based upon this separation, one leads to
closed-form solutions and the other is numerical in nature.
In the first method, the point D in Figure 2 is considered as the intersection of two curves. The first curve
is the circle of radius L3, whose center G may be determined from input angle q2. The second curve, called coupler curve, is the curve generated by D as the four-bar linkage …„† is in motion. Figure 3 illustrates a portion
of the coupler curve for an arbitrary four-bar linkage. Using the notations used in Figure 3, the coupler curve is a
sixth degree polynomial whose equation is [13].
(1)
Closed-form solutions for position analysis of a Stephenson III six-bar linkage are obtained by locating intersections of the two curves, and they are found by MATLAB
in this study. Each real solution so obtained is a position
of the joint D in Figure 2. As long as the location of D is
known, q3, q4, q5, and q6 may be calculated by geometric
relations.
In the second method, we choose link … (or link †)
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as the driving link to the four-bar linkage BCDEF in Figure 2. For each given value of q5, values of q4, q6, and the
corresponding positions of the point D may be calculated using analytical expressions that may be found in
a textbook such as [8]. Then, for this particular value of
q5, the distance between D and the known position of G
is calculated. The purpose now is to determine the values of q5 so that the distance DG equals to L3. The procedure used in this study to determine these values is as
follows. We first incrementally search for intervals of q5
within which the value (DG - L3) changes sign, then the
bisection method is used in these intervals to locate values of q5 which make (DG - L3) nearly zero.
As a check of these two procedures, we analyzed the
Stephenson III six-bar linkage in Figure 2 with the nondimensioned lengths: a = 5, b = 13, c = 8, d = 4, e = 6.5, f
= 3, L2 = 1, L3 = 6, L5 = 1, L6 = 3, and input angle q2 =
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 faster, the following results for position analysis are calculated 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 q2 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 intersection of the following two circles, the circle centers at
Table 1. Comparison of results obtained using two
procedures; differences are underlined
Closed-form
Figure 3. A part of the coupler curve generated by the point D
on the coupler of a four-bar linkage.
Numerical
1 sol -21.0981401119277° -21.0981401120935°
2nd sol -9.34660961319297° -9.34660961312838°
1st sol
28.1602559557637° 28.1602559560016°
q4 nd
2 sol 22.7638442767922° 22.7638442768305°
1st sol
189.74678710913°
189.74678711034°
q5 nd
2 sol
113.48012093066°
113.48012092989°
1st sol
145.052820929300° 145.052820929443°
q6 nd
2 sol 124.750245471184° 124.750245470924°
q3
st
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C. H. Liu and Chien-Kai Chen
C with a radius L4, and the circle centers at E whose radius 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)
(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 disconnected and reassembled.
The left wing also has two solutions, given by [13]
where R is the known distance between C and E, and
(6)
(3)
The second solution as shown in Figure 6 contains the
following two angles
(7)
and
(8)
(4)
(9)
4. Velocity and Acceleration Analysis
Velocity analysis is performed after the position analysis, 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.
Figure 4. The Stephenson III six-bar linkage in the MAV.
4.1 Stephenson Type III Six-bar Linkage
The Stephenson III linkage illustrated by Figure 4
has the following two closed loops:
Figure 5. The first solution for the position analysis of the
right wing mechanism.
Figure 6. The second solution for the position analysis of the
right wing mechanism.
Kinematic Analysis of a Flapping-wing Micro-aerial-vehicle with Watt Straight-line Linkage
AB + BC + CG = AI + IG
(10)
HF + FG = HI + IG
(11)
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(20)
Upon differentiation, we may obtain
The x and y components of equation (10) are
(21)
(12)
where
(13)
(22a)
Similarly vector equation (11) has the following two
components
(22b)
(14)
(15)
Differentiating these four equations once with respect
to time, one may obtain
Angular velocities q& 4 and q& 5 may be found from equation (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 obtain [13]
(16)
(24)
in which
where the matrix M1 is given by
(25)
(17)
With positions of all links known, for a given input velocity q& 2 one may solve equation (16) for unknown velocities q& 3 , q& 8 , q& 9 and q& 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)
Figure 7. Notations of the right wing mechanism for velocity
and acceleration analysis.
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Chao-Hwa Liu and Chien-Kai Chen
From these equations the velocities q& 6 and q& 7 can be obtained.
Unknown accelerations may be calculated when positions, velocities, and driving accelerations &&
q 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
and
(29)
5. Results and Discussions
(26)
where
(27)
After accelerations &&
q 3 , &&
q 8 , &&
q 9 , and &&
q 10 are obtained from
equation (26), accelerations for the right wing and the
left wing may be found from
(28)
Figure 8. Notations of the left wing mechanism for velocity
and acceleration analysis.
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 particular configuration obtained from position analysis
when q2 = 0. Angular position q5 of the right wing is
shown in Figure 10. Note that the angle q5 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 always differ by a constant. The angular stroke of the wing,
also called flapping range, is the maximum difference in
q5. For this mechanism its value is 70.6°. To check the
symmetry between the two wings, we compare the angle
q ¢5 in Figure 7, to the angle q7 in Figure 8; also we compare and the angle q4 in Figure 7 to the angle q 6¢ in Figure 8. The differences between these angles are shown
in Figure 11. The maximum difference in this figure is
between q ¢5 and q7, its value is 0.0825°, and it occurs
when q2 = 90°. The coupler curve generated by the point
Figure 9. Configuration of the flapping wing MAV when q2
= 0.
Kinematic Analysis of a Flapping-wing Micro-aerial-vehicle with Watt Straight-line Linkage
C on the Watt linkage approximates a vertical straight
line. The position q2 = 90° corresponds to one extreme
position for joint C where it begins to deviate further
from a straight line, causing a larger degree of asymmetry of the two wings at this position.
Results for velocity and acceleration analysis are
obtained with the constant input velocity w2 = 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 q2 =
113°. In Figure 13 we show the velocity difference between link … and link ‡. Its maximum value is 4.8085
´ 10-5 rad/s, which occurs when q2 = 114°. The difference 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 q2 = 90°. Finally,
the difference in acceleration between link … and link ‡
is shown in Figure 15. The maximum difference occurs
Figure 10. Angular position of link ….
Figure 11. Phase lag of the two wings.
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when q2 = 100°, whose value is 1.820966 ´ 10-4 rad/s2.
6. Conclusions
In this study we show that a Stephenson III six-bar
Figure 12. Velocity difference between link „ and †.
Figure 13. Velocity difference between link … and link ‡.
Figure 14. Acceleration difference between link „ and link †.
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Chao-Hwa Liu and Chien-Kai Chen
Figure 15. Acceleration difference between link … and 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 suggested. While one technique may lead to closed-form solution without any iterative procedure, the numerical technique is much faster and converges to nearly the same
value obtained by using closed-form solution. The numerical technique is used for position analysis of a flapping 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 insignificant.
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.
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Manuscript Received: Sep. 9, 2015
Accepted: Oct. 23, 2015