98年度論文成果電子檔 98-133

中國機械工程學刊第三十卷第三期第249~257頁(民國九十八年)
Journal of the Chinese Society of Mechanical Engineers, Vol.30, No.3, pp.249~257 (2009)
-249-
5
1
P
1
2
3
4
Design of Variable Coupler Curve Four-bar
Mechanisms
Ren-Chung Soong * and Sun-Li Wu **
Keywords: Variable coupler curve, Four-bar
mechanism, Continuous path
generation.
ABSTRACT
This paper presents a method for designing a
variable coupler curve four-bar mechanism with one
link replaced by an adjustable screw-nut link and
driven by a servomotor. Different desired coupler
curves can be generated by controlling the angular
displacement of the driving link and adjusting the
length of the adjustable links for continuous path
generation. This paper also presents a derivation of
the adjustable link lengths and the specified angular
displacement of the driving link corresponding to the
desired coupler curves. The conditions for generable
desired coupler curves are also described. The
examples and experiments described in this paper
confirm the feasibility and effectiveness of the
proposed method.
INTRODUCTION
There are two types of path generation. One is
point-to-point path generation, in which the coupler
curves only specify discrete points on the desired
path. The other is continuous path generation, in
which the coupler curves specify the entire path, or at
least many points on it. Because the coupler curves of
linkage mechanisms are functions of their link
lengths, the only way to generate different continuous
coupler curves with a single linkage mechanism is to
make the length of at least one of its links adjustable.
One way of doing this is to replace the normal links
with screw-nut links driven by servomotors, as shown
in Fig. 1. The different desired coupler curves can
then be obtained by controlling the length of the
adjustable links and the angular displacement of the
driving link.
The investigation of new synthesis methods for
path generation using linkage mechanisms has been
the subject of some research attention in recent years.
Tao and Krishnamoothy (1978) developed graphical
synthesis procedures of adjustable mechanisms for
generating variable coupler curves with cusps and
Paper Received February, 2009. Revised April, 2009.
Accepted May, 2009. Author for Correspondence: Ren-Chung
Soong
* Associate Professor, Department of Mechanical and
Automation Engineering, Kao Yuan University, Kaohsiung
82141, TAIWAIN, R.O.C.
** Assistant Professor, Department of Electrical Engineering,
Kao Yuan University, Kaohsiung 82141, TAIWAIN, R.O.C.
1
2
3
4
5
P
1
(a)
(b)
Fig. 1. Adjustable mechanisms

J. CSME Vol.30, No.3 (2009)
-250-
symmetrical coupler curves with a double point.
McGovern and Sandor (1973) used complex number
methods to synthesize adjustable mechanisms for
path generation. Kay and Haws (1975) developed a
design procedure for a path generation mechanism
with a cam link, which provided accuracy over a
range of motion. Angeles et al. (1998) proposed an
unconstrained nonlinear least-square optimal
synthesis method for RRRR planar path generators.
Hoeltzel and Chieng (1990) proposed a pattern
matching synthesis method based on the
classification of coupler curves according to moment
variants. Watanabe (1992) presented a natural
equation that expressed the curvature of the path as
an equation of the arc length and was independent of
the location and orientation of the path. Ullah and
Kota (1994, 1997) presented an optimal synthesis
method in which the objective function was
expressed as Fourier descriptors. Shimojima et al.
(1983) developed a synthesis method for straight-line
and L-shaped path generation using fixed pivot
positions as adjustable parameters. Unruh and
Krishnaswami (1995) proposed a computer-aided
design technique for infinite point coupler curve
synthesis of four-bar linkages. Kim and Sodhi (1996)
introduced a method of path generation that made the
desired path pass exactly through five specified
points and close to other points. Chuenchom and
Kota (1997) presented a synthesis method for
programmable mechanisms using adjustable dyads.
Chang (2001) proposed a synthesis method for
adjustable mechanisms to trace variable arcs with
prescribed velocities. Zhou et al. (2002) proposed an
optimal synthesis method with modified genetic
optimization algorithms by adjusting the position of
the driven side link for continuous path generation.
Russell and Sodhi (2005) presented a design method
for slider-crank mechanisms to achieve multiphase
path and function generation.
In this paper, we propose a new design method for
continuous path generation by four-bar mechanisms
that incorporates a screw-nut link called an adjustable
link. Different desired coupler curves can be
generated by appropriately adjusting the length of the
adjustable link and controlling the angular
displacement of the driving link. Examples and
experiments are provided to demonstrate this design
method.
REQUIRED DRIVING LINK
ANGULAR DISPLACEMENT
CORRESPONDING TO THE DESIRED
COUPLER CURVE
The coordinate system of a four-bar linkage is
shown in Fig. 2.
The speed trajectory of the driving link, and the
lengths of links 1 or 4 can be adjusted to generate
new coupler curves. Figure 2 shows that the
relationship between the angular displacement of the
driving link 2θ and the coordinate of the coupler
point ( yx PP , ) can be written as
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=+ −
x
y
P
P1
2 tan)( μθ (1)
and the vector loop equation can be written as
R2 + R5 – RP = 0. (2)
Separating Eq. (2) into two scalar component
equations in the x- and y-directions yields
0)cos(cos 3522 =−++ xPrr βθθ and (3)
0)sin(sin 3522 =−++ yPrr βθθ (4)
where ir and iθ represent the length and angular
displacement of the ith link, respectively. Adding Eqs.
(3) and (4) after squaring both sides gives
2
2222
222
5 )sincos(2 rPPrPPr yxyx ++−+= θθ ,
(5)
which, after rearrangement, gives
Fig. 2. The coordinate system of a four-bar linkage
θ2
θ3
θ4
β
μ
P
X
Y
2r
1r
5r
rp
3r
R2
R3
R4
R1
RP
R5

R.C. Soong and S.L. Wu: Design of Variable Coupler Curve Four-bar Mechanisms.
-251-
0
2
)sincos(
2
2
2
222
5
22 =
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −−−
++
r
rPPr
PP
yx
yx θθ .
(6)
To reduce Eq. (6) to a form that can be solved more
easily, we substitute the half angle identities to
convert the 2cosθ and 2sinθ terms to
2tanθ terms:
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
+
−
=
)
2
(tan1
)
2
(tan1
cos
22
22
2
θ
θ
θ
;
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
+
=
)
2
(tan1
)
2
tan(2
sin
22
2
2
θ
θ
θ
.
This results in the following simplified form, where
the link lengths ( 2r and 5r ) and the known value
( yx PP , ) terms have been collected as constants A, B,
and C: 0)
2
tan()
2
(tan 222
=++ CBA
θθ
where
x
yx
P
r
rPPr
A −
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −−−
=
2
2
2
222
5
2
, yPB 2= , and
x
yx
P
r
rPPr
C +
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −−−
=
2
2
2
222
5
2
. The angular
displacement of the driving link can then be
calculated as
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −±−
= −
A
ACBB
2
4
tan2
2
1
2θ (7)
and the corresponding 3θ can be obtained from Eq.
(4):
β
θ
θ
θ −
−
−
= −
)
sinP
cosP
(tan
22y
22x1
3
r
r
. (8)
ADJUSTABLE LENGTH OF LINKS 1
AND 4
From Figure 2, the vector loop equation can be
written as
R2 + R3 – R1 – R4 =0 . (9)
If we assume that the length of link 4 can be
adjusted, then we separate Eq. (9) into two scalar
component equations and rearrange as follows:
113322444 coscoscoscos)( θθθθ rrrrr −+=Δ+
(10)
113322444 sinsinsinsin)( θθθθ rrrrr −+=Δ+
(11)
where 4rΔ is the length of adjustable link 4.
By dividing Eq. (11) by Eq. (10) to
eliminate )( 44 rr Δ+ , the angular displacement of link
4, 4θ , can be expressed as
)
coscoscos
sinsinsin
(tan
113322
1133221
4
θθθ
θθθ
θ
rrr
rrr
−+
−+
= −
(12)
Then 4rΔ can be calculated as
4
4
113322
4
cos
coscoscos
r
rrr
r −⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −+
=Δ
θ
θθθ
(13)
Assuming that the length of link 1 can be adjusted,
we separate Eq. 9 into two scalar component
equations and rearrange them as follows:
111332244 cos)(coscoscos θθθθ rrrrr Δ+−+=
(14)
and
111332244 sin)(sinsinsin θθθθ rrrrr Δ+−+=
(15)
We then square both equations and add them to
eliminate one unknown, say θ4. The adjustable length
of link 1, denoted as 1rΔ , can then be expressed as
1
2
1
2
4
r
CBB
r −
−±−
=Δ (16)
where
)sinsincos(cos2
)sinsincos(cos2
31313
21212
θθθθ
θθθθ
+−
+−=
r
rB
and
)sinsincos(cos2 323232
2
3
2
2
2
4
θθθθ ++
++−=
rr
rrrC
. The
corresponding 4θ is

J. CSME Vol.30, No.3 (2009)
-252-
)
cos)(coscos
sin)(sinsin
(tan
1113322
11133221
4
θθθ
θθθ
θ
rrrr
rrrr
Δ+−+
Δ+−+
= −
(17)
CONDITIONS FOR GENERABLE
COUPLER CURVES
The coupler curves that can be generated must
satisfy both the following conditions:
2525 rrrrr p +≤≤− and (18)
( ) ( ) 2
5
2
22
2
22 sincos rrPrP yx =−+− θθ (19)
where 22
Yxp PPr += . In Fig. 2, we assume that
2r and 5r are not adjustable. Therefore, as long as
the desired continuous coupler curves are in the area
between the two concentric circles with radii
25 rr − and 25 rr + , they can be generated by
controlling the angular displacement of the driving
link and adjusting the length of links l or 4.
EXAMPLES
Burrs have always been a problem for steel pipe
manufacturers. Burrs frequently form on
cross-sections when pipes, especially thick ones, are
cut, as shown in Fig. 3. Eliminating burrs in pipes
with circular cross-sections is relatively easy, but this
is much more difficult for non-circular cross-sections.
Since pipe manufacturers generally produce pipes
with various different cross-sections, clearing burrs
from pipes is very important.
(a)
(b)
(c)
(d)
In following examples, we use the four-bar linkage
shown in Fig. 2 with the dimensions shown in Table 1
to generate the coupler curves shown in Fig. 4 by
controlling the angular position of the input link and
the length of links 1 or 4. The intended application is
the removal of burrs from pipes.
Table 1 Four-bar linkage dimensions
1r 2r 3r 4r 5r
dimension 22.2 cm 10 cm 20.6 cm 23.3 cm 30.6 cm
(c)
Fig. 3. Burrs on the cross-section of steel pipes

-253-
0 5 10 15 20 25 30 35 40 45 50
0
5
10
15
20
25
30
35
40
45
50
The X coordinate of the desired coupler curves(cm)
TheYcoordinateofthedesiredcouplercurves(cm)
The quarter circle with diameter(r5
-r2
)
The quarter circle with diameter(r5
+r2
)
Desired circle curve
Desired ellipse coupler curve
Desired square coupler curve
Example 1.
Generation of a circular coupler curve with the
center at (25, 18.5) and radius = 8.5 cm, as shown in
Fig. 4.
Example 2.
Generation of an elliptical coupler curve with the
center at (25, 18.5), long axis = 10 cm, and short axis
= 6 cm, as shown in Fig. 4.
Example 3.
Generation of a square coupler curve with four
vertexes p1 (17.5, 22.5), p2 (17.5, 13), p3 (27.5, 13),
and p4 (27.5, 23), as shown in Fig 4.
Figure 5 shows the desired coupler curves
generated for all examples and Fig. 6 shows the
required angular displacements of the driving link
corresponding to the desired coupler curves. Figures
7 and 8 show the required lengths of links 1 and 4,
respectively, corresponding to the desired coupler
curves for all examples.
10 15 20 25 30 35 40
5
10
15
20
25
30
X coordinate of coupler point (cm)
Ycoordinateofcouplerpoint(cm)
Example 1
Example 2
Example 3
0 10 20 30 40 50 60 70 80
-120
-100
-80
-60
-40
-20
0
20
The number of points on the coupler curve
Angulardisplacementofthedrivinglink(degree)
Example 1
Example 2
Example 3
0 10 20 30 40 50 60 70 80
-2
0
2
4
6
8
10
12
Thelength-adjustablemagnitudeofthelink1(cm)
Example 1
Example 2
Example 3
0 10 20 30 40 50 60 70 80
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
Example 1
Example 2
Example 3
Fig. 4. The desired coupler curves for examples
Fig. 5. The desired coupler curves in all examples
Fig. 6. The required angular displacement of the
driving link for all examples
Fig. 7. The length-adjustable magnitude of the link
1 for all examples
Fig. 8. The length-adjustable magnitude of the link 4
for all examples

J. CSME Vol.30, No.3 (2009)
-254-
EXPERIMENTS
1 Experimental Setup
Figure 9 shows the schematic of a planar
PC-based controlled variable coupler curve four-bar
mechanism used in our experiments. The setup
included the four-bar mechanism, two AC
servomotors with encoders and drivers, and a belt.
One AC servomotor was used to control the angular
displacement of the driving link while the other drove
the screw to adjust the length of link 1
simultaneously.
The hardware specifications of the control system
were as follows:
(1) Intel Pentium IV 400-MHz microcomputer
with 512 MB RAM;
(2) Motion control card (PCI-8164; Adlink
Technology, Inc.);
(3) AC servomotors (400 W; Mitsubishi Co.) and
drivers (MR-J2S-A; Mitsubishi Co.); and
(4) Incremental encoder (10,000 pulses per
revolution).
2 Implementation
Three experiments corresponding to the examples
listed in Section 6 were conducted, but only link 1
was adjusted. The desired (command) and actual
coupler curves, the angular displacement of the
driving link, and the corresponding length of link 1
are shown in Figs. 10, 11, and 12, respectively. The
experimental results in this section agreed with the
design results in Section EXAMPLES. These
examples and experiments thus confirm the practical
feasibility of the proposed design method.
Time (s)
0 5 10 15 20 25 30 35
Theangulardisplacementofthedrivinglink(degree)
-100
-80
-60
-40
-20
0
20
Command
Actual
(a) Angular displacement of the driving link
Time (s)
0 5 10 15 20 25 30 35
0
2
4
6
8
10
12
Command
Actual
(b) The length-adjustable magnitude of the link 1
Xcoordinateof coupler point (cm)
16 18 20 22 24 26 28 30 32 34
10
12
14
16
18
20
22
24
26
28
Command
Actual
(c) The coupler curves
Fig. 9. The variable coupler curve mechanism
Fig. 10. The experimental results of the Example 1

-255-
Time (S)
0 5 10 15 20 25 30 35
-120
-100
-80
-60
-40
-20
0
20
40
Command
Actual
Time (s)
0 5 10 15 20 25 30 35
-4
-2
0
2
4
6
8
10
12
14
Command
Actual
10 15 20 25 30 35 40
10
12
14
16
18
20
22
24
26
Command
Actual
Time (s)
0 2 4 6 8 10 12 14
-120
-100
-80
-60
-40
-20
0
Command
Actual
Time (s)
0 2 4 6 8 10 12 14
-2
0
2
4
6
8
10
Command
Actual
16 18 20 22 24 26 28 30
12
14
16
18
20
22
24
Command
Actual
Fig. 11. The experimental results of the Example 2 Fig. 12. The experimental results of the Example 3

J. CSME Vol.30, No.3 (2009)
-256-
CONCLUSIONS
The proposed approach was based on a variable
coupler curve four-bar mechanism in which one link
was replaced by a screw-nut link driven by a
servomotor. The different desired couple curves could
be generated by controlling the angular displacement
of the driving link and changing the length of the
adjustable link. The derivations of the adjustable link
length and the specified angular displacement of the
driving link corresponding to the desired coupler
curves were presented along with the conditions
required to achieve the desired generable coupler
curves. The examples and experiments confirmed the
feasibility of this design method, which is suitable for
cases that require several different coupler curves
within a specific area for practical applications.
ACKNOWLEDGMENT
This research was supported by the National
Science Council Taiwan, R.O.C, through the grant
NSC 94-2212-E-244-003.
REFERENCES
Angeles, J., Alivizators, A., and Akhras, A., “An
Unconstrained Nonlinear Least-square
Method of RRRR Planar Path Generators,”
Mechanism and Machine Theory, Vol. 23, No.
5, pp. 343–353(1998).
Chuenchom, T., and Kota, S., “Synthesis of
Programmable Mechanisms Using Adjustable
Dyads, ”ASME Journal of Mechanical Design,
Vol. 29, No. 5, pp. 232–237.
Chang, C.F., 2001, “Synthesis of Adjustable Four-bar
Mechanisms Generating Circular Arcs with
Specified Tangential Velocities,” Mechanism
and Machine Theory, Vol. 36, No 3, pp.
387–395(1997).
Hoeltzel, D.A., and Chieng, W.H., “Pattern Matching
Synthesis as An Automated Approach to
Mechanism Design,” ASME Journal of
Mechanical Design, Vol. 112, No. 2, pp.
190–199(1990).
Kay, F. J., and Haws, R. E., “Adjustable Mechanisms
for Exact Path Generation,” ASME Journal of
Engineering for Industry, Vol. 97, No. 2, pp.
702–707(1975).
Kim, H.J,. and Sodhi, R.S., “Synthesis of Planar
Four-bar Mechanisms for Generating A
Prescribed Coupler Curve with Five Precision
Points,” Journal of Applied Mechanisms and
Robotics, Vol. 3, No 2, pp. 9–13(1996).
McGovern, J. F., and Sandor, G.N., “Kinematic
Synthesis of Adjustable Mechanism (Part 1:
Function Generation),” ASME Journal of
417–422(1973).
McGovern, J. F., and Sandor, G.N., “Kinematic
Synthesis of Adjustable Mechanism (Part 2:
Path Generation),” ASME Journal of
423–429(1973).
Russell, K., and Sodhi, R. S., “On The Design of
Slider-crank Mechanisms. Part II: Multi-phase
Path and Function Generation,” Mechanism
and Machine Theory, Vol. 40, No. 3, pp.
301–317(2005).
Shimojima, H., Ogawa, K., and Sato, O., “Kinematic
synthesis of adjustable mechanisms,” Bulletin
of JSME, Vol. 26, No. 214, pp 627–632(1983).
Tao, D. C., and Krishnamoothy, S., “Linkage
Mechanism Adjustable for Variable Coupler
Curves with Cusps,“ Mechanism and Machine
Theory, Vol. 13, No. 6, pp. 577–583(1978).
Tao, D. C., and Krishnamoothy, S., “Linkage
Mechanism Adjustable for Variable
Symmetrical Coupler Curves with a Double
Point,” Mechanism and Machine Theory, Vol.
13, No 6, pp. 585–591(1978).
Ullah, I., and Kota, S., “A more Effective
Formulation of the Path Generation
Mechanism Synthesis Problem,” ASME
Design Technical Conference, Minneapolis,
USA, 11-14 September, DE-Vol. 70, pp.
239–244(1994).
Ullah, I., and Kota, S., “Optimal Synthesis of
Mechanisms for Path Generation Using
Fourier Descriptors and Global Search
Methods,” ASME Journal of Mechanical
Design, Vol. 119, No. 4, pp. 504–510(1997).
Unruh, V., and Krishnaswami, P., “A computer-aided
Design Technique for Semi-automated Infinite
Point Coupler Curve Synthesis of Four-bar
Linkages,” ASME Journal of Mechanical
Design, Vol. 117, No. 1, pp. 143–149(1995).
Watanabe, K., “Application of Nature Equations to
The Synthesis of Curve Generating
Mechanisms,” Mechanism and Machine
Theory, Vol. 27, No. 3, pp. 261–273(1992).
Zhou, H., and Cheung, E.H.M., “Analysis and
Optimal Synthesis of Adjustable Linkages for
Path Generation,” Mechatronics, Vol. 12, No.
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-257-
可變耦桿點曲線之四連桿
機構設計
宋仁羣
高苑科技大學機械與自動化工程系
吳上立
高苑科技大學電機工程系
摘要
本文提出一種可變耦桿點曲線之四連桿機構
設計方法，將四連桿機構中之一根連桿由一根可調
整桿長以伺服馬驅動之螺桿取代。藉由控制輸入桿
之角位置及調整可調整桿之桿長，產生所欲之不同
型態的連續耦桿點曲線。本文並提出對應不同之期
望耦桿點曲線，輸入桿之角位置及可調整桿之桿長
調整量的數學推導結果。本文亦提出可被產生之耦
桿點曲線之條件。最後以設計實例及實驗驗證本設
計方法之可行性及有效性。

98年度論文成果電子檔 98-133

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98年度論文成果電子檔 98-133