kinematic analysis of a flapping-wing micro-aerial-vehicle with watt

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 1st International Conference on Biomimetics And Orni-

thopters (ICBAO-2015), held by Tamkang University, Tamsui, Taiwan,

during June 28�30, 2015.

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. Ki-

nematic analysis must be performed to determine posi-

tion, 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 sec-

tion 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�,�,�, and� con-

stitute a Watt four-bar mechanism that makes joint C

moving along an approximate straight line. A Stephen-

son type III six-bar linkage is formed by adding links�

and� 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 revolu-

tion. 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

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 posi-

tion of this linkage, and he solved the equations numeri-

cally. Watanabe and Funabashe [10,11] obtained solu-

tions 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 si-

milar construction. Since these solution techniques in-

volve 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 vari-

ables �3, �4, �5, and �6, for a given input angle �2. The

Stephenson III linkage is composed of a four-bar linkage

that includes links ��, and a two-bar chain that

contains links��. Note that the separation of this six-

bar linkage this way has been suggested by Watanabe and

356 Chao-Hwa Liu and Chien-Kai Chen

Figure 2. A Stephenson type III six-bar linkage.Figure 1. An illustrative diagram for a flapping wing MAV.

Funabashe [10,11]. Yet they used this idea in determin-

ing 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 na-

ture.

In the first method, the point D in Figure 2 is con-

sidered as the intersection of two curves. The first curve

is the circle of radius L3, whose center G may be deter-

mined from input angle �2. The second curve, called cou-

pler curve, is the curve generated by D as the four-bar link-

age �� is in motion. Figure 3 illustrates a portion

of the coupler curve for an arbitrary four-bar linkage. Us-

ing 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 Stephen-

son III six-bar linkage are obtained by locating intersec-

tions 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, �3, �4, �5, and �6 may be calculated by geometric

relations.

In the second method, we choose link (or link)

as the driving link to the four-bar linkage BCDEF in Fig-

ure 2. For each given value of �5, values of �4, �6, and the

corresponding positions of the point D may be calcu-

lated using analytical expressions that may be found in

a textbook such as [8]. Then, for this particular value of

�5, the distance between D and the known position of G

is calculated. The purpose now is to determine the va-

lues of �5 so that the distance DG equals to L3. The pro-

cedure used in this study to determine these values is as

follows. We first incrementally search for intervals of �5

within which the value (DG � L3) changes sign, then the

bisection method is used in these intervals to locate va-

lues of �5 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 non-

dimensioned 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 �2 =

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)


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


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


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.

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Manuscript Received: Sep. 9, 2015

Accepted: Oct. 23, 2015


Figure 15. Acceleration difference between link and link�.

kinematic analysis of a flapping-wing micro-aerial-vehicle with watt

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