1. 249~257() 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 2. 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 2 and the coordinate of the
coupler point ( yx PP , ) can be written as =+ x y P P1 2 tan)( (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 and (3) 0)sin(sin
3522 =++ yPrr (4) where ir and i 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
+++= , (5) which, after rearrangement, gives Fig. 2. The coordinate
system of a four-bar linkage 2 3 4 P X Y 2r 1r 5r rp 3r R2 R3 R4 R1
RP R5 3. 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 . (6) To reduce Eq. (6) to a form that can be solved more
easily, we substitute the half angle identities to convert the 2cos
and 2sin terms to 2tan terms: + = ) 2 (tan1 ) 2 (tan1 cos 22 22 2 ;
+ = ) 2 (tan1 ) 2 tan(2 sin 22 2 2 . 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 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 2 (7) and the corresponding 3 can be obtained from Eq.
(4): = ) 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)( rrrrr +=+ (10) 113322444 sinsinsinsin)( rrrrr +=+
(11) where 4r is the length of adjustable link 4. By dividing Eq.
(11) by Eq. (10) to eliminate )( 44 rr + , the angular displacement
of link 4, 4 , can be expressed as ) coscoscos sinsinsin (tan
113322 1133221 4 rrr rrr + + = (12) Then 4r can be calculated as 4
4 113322 4 cos coscoscos r rrr r + = (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 rrrrr ++= (14) and 111332244 sin)(sinsinsin rrrrr
++= (15) We then square both equations and add them to eliminate
one unknown, say 4. The adjustable length of link 1, denoted as 1r
, can then be expressed as 1 2 1 2 4 r CBB r = (16) where
)sinsincos(cos2 )sinsincos(cos2 31313 21212 + += r rB and
)sinsincos(cos2 323232 2 3 2 2 2 4 ++ ++= rr rrrC . The
corresponding 4 is 4. J. CSME Vol.30, No.3 (2009) -252- )
cos)(coscos sin)(sinsin (tan 1113322 11133221 4 rrrr rrrr ++ ++ =
(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 =+
(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 5. R.C. Soong and S.L.
Wu: Design of Variable Coupler Curve Four-bar Mechanisms. -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 The number
of points on the coupler curve
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
The number of points on the coupler curve
Thelength-adjustablemagnitudeofthelink4(cm) 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
6. 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
Thelength-adjustablemagnitudeofthelink1(cm) 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
Ycoordinateofcouplerpoint(cm) 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 7.
R.C. Soong and S.L. Wu: Design of Variable Coupler Curve Four-bar
Mechanisms. -255- Time (S) 0 5 10 15 20 25 30 35
Theangulardisplacementofthedrivinglink(degree) -120 -100 -80 -60
-40 -20 0 20 40 Command Actual (a) Angular displacement of the
driving link Time (s) 0 5 10 15 20 25 30 35
Thelength-adjustablemagnitudeofthelink1(cm) -4 -2 0 2 4 6 8 10 12
14 Command Actual (b) The length-adjustable magnitude of the link 1
X coordinate of coupler point (cm) 10 15 20 25 30 35 40
Ycoordinateofcouplerpoint(cm) 10 12 14 16 18 20 22 24 26 Command
Actual (c) The coupler curves Time (s) 0 2 4 6 8 10 12 14
Theangulardisplacementofthedrivinglink(degree) -120 -100 -80 -60
-40 -20 0 Command Actual (a) Angular displacement of the driving
link Time (s) 0 2 4 6 8 10 12 14
Thelength-adjustablemagnitudeofthelink1(cm) -2 0 2 4 6 8 10 Command
Actual (b) The length-adjustable magnitude of the link 1 X
coordinate of coupler point (cm) 16 18 20 22 24 26 28 30
Ycoordinateofcouplerpoint(cm) 12 14 16 18 20 22 24 Command Actual
(c) The coupler curves Fig. 11. The experimental results of the
Example 2 Fig. 12. The experimental results of the Example 3 8. 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
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