E - IFToMM

New approach for optimum synthesis of six-bar dwell
mechanisms by adaptive curve fitting*
Huimin Dong
Delun Wang
Dalian University of Technology
Dalian University of Technology
Dalian China
Dalian China
∗ Abstract—A new approach for optimum synthesis of six-bar
dwell mechanisms is presented in this paper. At first, a new
approach of adaptive segment curve fitting is presented in terms
of invariants of circle or line. Meanwhile, an approximate
segment circle point and an approximate segment slide point are
defined. Based on these definitions and the adaptive curve fitting
approache, the unified mathematical model is established for the
synthesis of circular-arc and straight-line dwell mechanisms.
Particularly, the non-dwell portion is also considered in the
approach. The co-operation of global searching and local
searching is adopted to satisfy the prospective accuracy.
Examples under various given conditions illustrated show that
the technique is simple, efficient and accurate.
Key words:Six-bar dwell mechanism, Adaptive curve
fitting, Function generation
I. Introduction
Dwell mechanisms have applications in many industries,
especially for automation. Linkage-type dwell mechanisms
are less expensive to manufacture and maintain, and can be
superior to cams at high speed. Commonly used linkagetype
dwell mechanisms, as Fig.1 shows, are composed of a
basic four-bar mechanism that generates either the
circular-arc or straight-line portion of the coupler curve
and adding an output dyad with rocker output option or
slider output option to convert uniform motion of the input
link into periodic and intermittent motion of the output
link. However linkage-type dwell mechanisms only give
"approximate dwells". Since exact dwells are often not
necessary and are rarely obtained in practice due to
vibration, manufacturing errors and backlash, this makes
the use of approximate dwells acceptable. Several
researchers have made contribution to studying on linkagetype
dwell mechanisms in the past [1-12]. Sandgren[1] led
to an optimum method with constraint equations into a
class of six-bar linkages to produce single- and multipledwell
mechanisms, in which the objective function was the
length l 5 of the link shown in fig.1b with invariable and the
optimal parameters were all dimensions in the mechanism.
For Stephenson’s six-bar inversion three with pin jointed,
Shimojima and Jizai [2] presented the conditions for
dwelling at edge points of rocking, the relationships
between the dwell periods and mechanisms parameters
∗ E-mail : donghm@dlut.edu.cn
E-mail : dlunwang@dlut.edu.cn
*Supported by National Natural Science Foundation of China under grant
No. 50475154
were approximated by polynomials in the order of less
than five, and the existing domains of parameters. Both
methods mentioned [1, 2] have more optimal parameters.
With computer technique rapidly developing, the atlas
method based on path curvature theory was more
employed for designing linkage-type dwell mechanisms.
Kota et al. [3, 4] presented several "rules of thumb" based
on path curvature theory, which includs systematic
analysis and classification of straight-line, circular-arc and
symmetrical coupler curve generating mechanisms. Based
on those research, Kota et al.[5] developed MINN-
DWELL, transforming design experience into design
catalogues--preprogrammed building blocks for the
intelligent and efficient design of dwell linkages. And
more in 1991 Kota [6], taking Stirling Engines as an
example, presented the generic design models, as well as a
new concept in which single-input control the multipledwell-output.
Venkatesh[7] developed RECDWELL
helping to design both circular-arc and straight-line dwell
mechanisms which is simple, efficient and accurate.
Beier[8], according to the axiom, for any curve if
curvature center invariable the curvature radii is constant,
presented a method looking for circular-arc or straight-line
portion of the coupler curve, which reducing the design
time. Cao Qinglin et al[9] studied the atlas with
symmetrical path carefully, and applied their research to
design twice dwell mechanisms. Yu Hongying[10], using
the methods of “linear scanning “and “tangent envelop”,
separately synthesized the basic four-bar linkage and the
output dyad, set up four-bar linkages database with
circular-arc segment coupler curve, carried out synthesis of
six-bar dwell linkages with pin jointed. For improving the
design efficiency and reducing the randomicity of initial
selecting, several researchers have turned to study on the
method and system of the conceptual design in computer
aided design dwell mechanisms[11], [12]. All above
researches’ work pushed the development of six-bar dwell
mechanisms in different aspect.
Ref.[13] presented an adaptive curve fitting method for
four-bar planar mechanism synthesis based on the
theoretics of curvature, by means of searching a invariants
of curve, and established the unified mathematic model,
and solved such problems as selecting initial point,
capturing global optimal solution.
We modify the adaptive curve fitting method [13] in this
paper according to the structural characteristic of
Stephenson’s six-bar inversion three, and give the
definitions of an approximate segment circle point and an
approximate segment slide point. Based on these

definitions and the approach, establish the unified
mathematical model for the synthesis of circular-arc and
straight-line dwell mechanisms. Particularly, consider the
non-dwell portion in the approach. Adopt the co-operation
of global searching and local searching to satisfy the
prospective accuracy. Examples under various given
conditions illustrate that the technique is simple, efficient
and accurate.
E
E
E
A
E
A
B
A
B
B
E
F
C
F
E
C
C
G
D
G
D
D
Fig 1 Constructions of Stephenson Ш six-bar dwell linkages
E
(c)
II. Representation of coupler curve
As shown in Fig.1, the dwell motion condition of sixbar
linkages is either the straight-line or circular-arc
portion of the path traced by the coupler point E in a basic
four-bar linkage. According to the geometry characteristic
of four-bar linkages, in coordinate system (o f , x f , y f ) shown
in Fig.2, the point E( l e , β ) can be easily expressed as
⎧x
E
= l1
cosϕ
+ le
⋅ cos( γ + β )
⎨
(1)
⎩ y
E
= l1
sinϕ
+ le
⋅ sin( γ + β )
Obviously, the trace of point E ( x E
, y E
) is nonlinear
functions aboutϕ . That is, any point on the coupler plane
(a)
l e
(b)
A
E
l 5
F
E
l 5
B
l1
l e
G
A
B
l 1
l e
A
l6
F
B
l 1
l 2
l4
l 2
G
l 4
C
l 3
C
D
l 3
C
l 2
l 3
l 4
D
D
of basic four-bar linkages can be expressed as nonlinear
functions about input angleϕ , whose properties depend on
the position on the coupler plane.
y f
o f (A)
B
l e
φ
x f
E
β
C
γ
Fig.2 Coupler curve of planar four-bar linkage
III. The approach of adaptive segment curve fitting
The first step of synthesis of dwell mechanisms with the
type of Stephenson’s III six-bar is for searching coupler
point E in a basic four-bar linkage, which generates
approximate straight-line or circler-arc portion. Based on
the approach of adaptive curve fitting to search
approximate characteristic point presented in Ref.[13], this
paper presents an adaptive segment curve fitting method to
find approximate local characteristic point.
A. Adaptive segment circle fitting
As shown in Fig.3, there is the set of discrete points E i
of any coupler point in the basic four-bar linkage, a circle
is chosen to fit the segment curve E k1 Ek
2 . Based on the
adaptive fitting circle[13], the optimal mathematical model
of a local adaptive fitting circle can be written as
min max f z
f
where f k
( z)
k
z
k1
≤k≤k2
i
( )
D
2
2
( z) = ( x − x ) + ( y − y ) )
Ek
T
[ x , y , r] , z ∈ R
0 0
Z
0
Ek
− r
z =
is the fitting error between a discrete point E i
and the adaptive fitting circle, and
0
1
2
(2)
T
z = [ x , y , r ] are the
optimal parameters, in witch ( x0 , y
0
) is the coordinates of
the circle center and r is the radius of the fitting circle.
It is necessary to indicate that, the length of segment
curve fitted lies on the intermission angle ∆ ϕ m , the
position depends on the k 1 and k 2 , select a different segment
get a different fitting circle and different fitting error. Here,
we give the definition of the local adaptive fitting circle.
Definition A.1 Based on the properties of the set of
discrete points E i in the given path and the given
intermission angle ∆ ϕ , a circle is adaptively located by
m
the circle fitting in the principle of minimum maximum
fitting error in all segments, witch is called a local adaptive
fitting circle.
0
0

A local adaptive fitting circle have the same properties
with an adaptive fitting circle in Ref [13]. For a basic fourbar
linkage, every tracing point of a moving body is
corresponding to a local adaptive fitting circle, whose
difference of those is only the maximum error. Therefore,
we can define the approximate segment circle point.
Definition A.2 For a basic four-bar linkage, a tracing
point of a moving body is corresponding to a local
adaptive fitting circle, whose maximum fitting error in
normal direction reaches the minimum with respect to
other tracing points in its neighborhood, within the range
of the intermission angle ∆ ϕ m , called approximate
adaptive segment circle point, or an approximate segment
circle point for short.
An approximate segment circle point has three
meanings about minimum maximum: one is the adaptive
segment fitting circle, which leads to the maximum fitting
error with a circle get minimum, the second one is to
search for a best segment in the given path, and the third
one is to search for a best approximate segment circle
point on the moving body, which traces a segment of a
curve closest to a circle-arc, or the maximum fitting error
gets minimum in its neighborhood.
The minimum maximum fitting error of adaptive
segment fitting circle is a nonlinear function of the
coordinates of the tracing point on the moving body.
Therefore, there must exist extreme values of the
minimum maximum fitting error for all adaptive fitting
circles with respect to their tracing points of moving body.
Theorem A.1 There must exist several approximate
segment circle points on the moving body in
nondegenerated planar motion.
E k1
E k
r
(x
y 0 ,y 0 )
F
x F
o f
E k2
f k (z)
Fig.3 Local adaptive fitting circle of coupler curve
As shown in Fig.1a and 1b six-bar dwell mechanisms,
the point E is an approximate segment circle point on the
coupler plane of a basic four-bar linkage. The center
coordinate x , ) and radius r of that are the coordinate
( 0 y0
of the point F ( y )
of the link EF.
x in dwell period and the length l 5
F
,
0 F 0
B. Adaptive segment straight-line fitting
As shown in Fig.4, there is the set of discrete points E i
of any coupler point in the given intermission angle ∆ϕ
m of
a basic four-bar linkage. Fitting the segment
curve Ek1 Ek
2 needs to employ an adaptive fitting line,
which is similar to an adaptive segment fitting circle.
Hence, we can also give the definition of a local adaptive
fitting line.
y f
o f
E k1
x f
E k
f k (z)
E k2
Fig.4 Local adaptive fitting line of coupler curve
Definition B.1 Based on the properties of the set of
discrete points E in the given path and the given
i
intermission angle ∆ ϕ m , a straight-line is adaptively
located by the line fitting in the principle of minimum
maximum fitting error in all segments, which is called a
local adaptive fitting line.
The optimal mathematical model of an adaptive line can
be written as
min max f z
where ( z)
f
z
j
j1
≤ j≤
j2
j
( )
2
( z) = y
( )
E
− kxE
− b / 1+
k
j
j
T
[ k,
b] ,z∈
RZ
1
2
(3)
z =
f j
is the fitting error between a discrete point of
the given path, the adaptive fitting line: k and b are the
optimal parameters.
The properties of adaptive segment fitting line are
similar to those of adaptive segment fitting circle. In the
same way, we can give the definition of an approximate
adaptive segment slide point.
Definition B.2 For a basic four-bar linkage, a tracing
point of a moving body is corresponding to the adaptive
segment fitting line, whose maximum fitting error in
normal direction reaches the minimum with respect to
other tracing points in its neighborhood, within the range
of the intermission angle ∆ ϕ m , called approximate
adaptive segment slide point, or an approximate segment
slide point for short.
The minimum maximum fitting error of adaptive
segment fitting line is also a nonlinear function, which
solving process is same with that of the approximate
segment circle point and solutions must be existent.

Theorem B.1 There must exist several approximate
segment slide points on the moving body in
nondegenerated planar motion.
The point E in Fig.1c is an approximate segment slide
point on the coupler plane of a basic four-bar linkage,
which is corresponding to dwell position of the output link.
C. the synthesis of output dyads
After finding the segment characteristic point E on the
coupler plane of a basic four-bar linkage in the dwell
period angle ∆ ϕ , the next step is the synthesis of output
m
dyads to satisfy non-dwell function requirement, such as
having RRR, RPR as well as RRP dyadic configuration.
C. 1 Rocker output option
Fig.1a and 1c show the most commonly used six-bar
type dwell mechanism with rocker output. Based on
getting the approximate segment characteristic point E,
circle point or slide point, on the coupler plane of a basic
four-bar linkage, the synthesis of the output dyad RRR and
RPR is to search the fixed pivot.
According to the given transmission function in the nondwell
section, under the precondition of all parameters in
the basic four-bar linkage known, the object function of
the optimal mathematical model can be expressed as
Ⅱ
Ⅱ
f z = max ψ −ψ
(4)
Ⅱ
ψ
j
Ⅱ
j
j
( )
Ⅱ
j
where is the output angle (output function of the
rocker), ψ is the desired output function of the rocker,
the optimal parameters are ( ) T
coordinate ( x
G
, y G
) of fixed pivot.
j
z = x , y G G
which are the
For the dyad RRR shown in Fig.1a, the output angleψ ,
according to the principle of the distance between the two
pivots of the link keeping constant, can be found as follow
equations
2
( x − x ) + ( y − y ) 2
l6 =
G F0 G F 0
(5)
and
2
2 2
( ⎪⎧
x − x ) + ( y − y ) = l
E
E 5
⎨
2
2 2
( ⎪⎩ x − xG) + ( y − yG)
= l6
(6)
substituting equation (5) into equations (6) and rearranging
them yields
⎧xF
= xE
+ pa1
− qa2
⎨
⎩y
F
= xF
+ pa2
− qa1
(7)
Where
a = x − x , a = y − y a
2
= a
2
a (8)
1 G E 2 G E
,
3 1
+
2
Ⅱ
j
2 2
⎧ ( a + −
⎪ =
3
l5
l6
)
p
2a3
⎨
(9)
2
⎪ l
2
q = M ⋅
5
− p
⎪⎩
a3
Equation (9) has two solutions, obtained from
the ± condition of the coefficient M, which depends on the
initial position of ( x
F
y ). If y
F
>= yG
, then
0
,
F 0
Ⅱ
x
F
− xG
ψ
j
= arccos(
) (10a)
2
2
( x
F
− xG
) + ( y
F
− yG
)
otherwise
x
F
− xG
ψ Ⅱ j
= 2π
− arccos(
) (10b)
2
2
( x
F
− xG
) + ( y
F
− yG
)
For the dyad RPR shown in Fig.1c, the point E is the
segment slide point on the coupler plane of the basic fourbar
linkage, whose line segment is corresponding to the
rocker position with dwell. According to the geometry
Ⅱ
properties of the output dyad RPR, the output angle ψ
j
,
can be found as follow equation
y = kx b
(11)
where ( )
F
y F
F F
+
x , are the coordinates of fixed pivot F; k and
b are the parameters of point E at the position of the
adaptive segment fitting line. If y
E
>= kxF
+ b , then
Ⅱ
x
E
− x
F
ψ
j
= arccos(
) (12a)
2
2
( x
E
− x
F
) + ( x
E
− kxF
− b)
otherwise
x
E
− x
F
ψ Ⅱ
j
= 2π
− arccos(
) (12b)
2
2
( x − x ) + ( x − kx − b)
E
C. 2 Slider output option
The output link is a slider in the six-bar type dwell
mechanism shown in Fig.1b which transforms the rotating
input motion to the translating output and dwell. After
selecting an approximate segment circle point on the
coupler plane as pivot E, the synthesis of the output dyad
with slider output is to search the tilted angle θ of the
guideway according to the given transmission function in
the non-dwell section. As with the section C. 1 the object
function of the optimal mathematical model in the nondwell
portion can be constructed as
Ⅱ
Ⅱ
f θ = max s − s
(13)
j
( )
Ⅱ
Ⅱ
where s
j
is the output displacement of the slider F, s Ⅱ j
is
the desired output displacement, θ is the optimal
parameter which is the tilted angle of the guideway with
respect to the x f . axis. The output displacement of the slider
F can be found as follow equations
F
j
j
E
F

2
2 2
⎪⎧
( xF
− x
E
) + ( y
F
− y
E
) = l5
⎨
(14)
⎪⎩
y
F
= ( xF
− x
F
) tgθ
+ y
0 F0
the solutions are
⎧
2
⎪x
=
( −b
+ M b − 4ac)
F
⎨
2a
(15)
⎪
⎩
y
F
= ( x
F
− xF
) tgθ
+
0 2
y
F0
where
2
⎧a
= 1 + tg θ
2
⎪
⎪b
= 2dtgθ
2
− 2xE
⎨
(16)
2 2 2
⎪c
= xE
+ d − l5
⎪
⎩
d = y
F
− xF
tgθ
2
− y
0
0
E
As with the equation (9), this also has two solutions, obtain
from the M ± condition on the radical, which also depends
on the initial position of ( x
F
, y )
0 F 0
. If set the point F 0 as
the fixed point on the guideway line, then
Ⅱ
s
j
= ( xF
− xF
) ⋅cosθ
+ ( y − ) ⋅sinθ
0 F
y
j
j F
(17)
0
IV. Unified mathematical model of the synthesis
A similar process to that used for finding the output
function of the output dyad can be applied to the problem
of dwell portion, since it is easy to convert the fitting error
in the formula (2) and (3) to the function error. For the
desired output function of in the dwell and non-dwell
portion, the object functions can be established
respectively. let
F
Ⅰ Ⅰ
( X ) = maxψ
i
−ψ
i
Ⅱ Ⅱ
( X ) = maxψ −ψ
1
(18a)
F2
i i
(18b)
where F 1
( X ), F 2
( X ) are the output function errors in the
dwell and non-dwell portion, which express angular
displacement error or linear displacemen error depending
on the type of the mechanisms in Fig.1.
Generally, it is need to take both output function errors
in dwell and non-dwell potion into account in the practical
application of dwell mechanisms. Therefore, we employ
linear weighted method to construct the evaluating
function as follow
F( ) = λF1 ( X ) + ( 1−
λ) F2
( X )
where λ is weight number with λ ∈ [ 0,1.0]
U (19)
. The formula
(19) describes the optimal problem with single objective.
The solving process of that in detail can refer to Ref.[13].
V. Examples
According to the approach above introduced, the
software, which carries out the approximate function
synthesis for planar six-bar dwell mechanisms, is
programmed. For illustrating the effectiveness of the
approach, the results of different synthesis examples are
given.
Example 1, design a six-bar dwell linkage with rocker
output option for 40 0 rocker motion over 240 crank
degrees with dwell for the remaining 120 0 , which function
between the input crank angle and output rocker angle is
listed in table 1. The result of the synthesis with those of
o
the minimum transmission angle γ
min
=15 , maximum
common link ratio T = max
10 , weight number λ = 0. 8 , and
the dimension of the basic for-bar linkages selected in
table 2, shows in table 3. Fig 5 illustrates the result of the
synthesis with high precision has been carried out.
o
o
o
o
ϕ ( ) ψ ( ) ϕ () ψ ( )
0 0.0 180 40.0
20 2.68 200 40.0
40 10.0 220 40.0
60 20.0 240 40.0
80 30.0 260 37.3
100 37.3 280 30.0
120 40.0 300 20.0
140 40.0 320 10.0
160 40.0 340 2.68
TABLE 1 Desired function (Example 1)
l
1
(mm) l
2
(mm) l
3
(mm) l
4
(mm)
100 150 220 220
TABLE 2 Dimensions of the basic four-bar linkage
l e (mm) β(°) e 1 (°)
RR 236.0391 203.0333 0.2274
RP 272.0157 350.137 0.440
x G (mm) y G (mm) e 2 (°)
RR 269.8405 223.1493 7.2667
RP 347.6064 -0.8521 3.148
TABLE 3 Optimum result
e , e express the maximum error in dwell and non-dwell portion.
1
2
Fig. 5 Desired and resulting function
Example 2, design a six-bar dwell linkage with slider
output option used in the industrial sawing machine to
enlace button neck with thread, which require that
maximum stroke is 80mm, slider motion over 320 crank
degrees with dwell for the remaining 40 0 , which the

function between the input crank angle and output slider
displacement is listed in table 4. The result of the synthesis
o
with those of the minimum transmission angle γ = 40 ,
min
maximum link ratio T = max
6 , weight number λ = 0. 8 , and
the dimension of the basic for-bar linkage selected in table
5, shows in table 6. Fig 6 illustrates that the result of the
synthesis with high precision has been carried out.
ϕ (° S ϕ (°)
S
ϕ (°) S )
0 0 127 46.61 250 80
7 0.82 134 50.57 255 80
14 1.54 140 53.99 260 80
20 2.26 147 57.58 268 79.23
27 2.98 154 61.18 283 76.35
33 4.06 160 64.06 292 72.39
40 5.14 168 67.67 300 66.64
47 6.75 188 74.85 315 49.85
54 8.73 199 77.00 320 41.22
66 13.23 206 78.33 326 30.4
73 15.93 213 79.05 332 18.56
80 19.52 220 79.76 340 7.77
87 23.12 225 80 348 3.27
100 30.85 230 80 356 0.93
108 35.35 235 80
114 39.0 240 80
TABLE 4 Desired function
l e
(mm
)
68.2
l
1
(mm) l
2
(mm) l
3
(mm) l
4
(mm)
50 100 100 130
TABLE 5 Dimensions of the basic four-bar linkage
β
(°)
281.
7
l 5
(mm
)
238.
8
θ
(°)
84.
3
x F
(mm
)
-
73.9
y F
(mm
)
126.
7
e 1
(mm
)
0.32
1
TABLE 6 Optimum result
e
1 , e2
express the maximum error in dwell and non-dwell portion.
S
O
ϕ ( )
Fig. 6 Desired and resulting function
e 2
(mm
)
6.29
7
VI. Conclusion
This paper presents a new optimization approach of
six-bar dwell mechanism synthesis by adaptive segment
curve fitting, in which the definitions of approximate
segment circle points and approximate segment slide
points are given. Here, we introduce the approximate
properties and optimization process of the six-bar dwell
mechanisms, and prove the existence of the best solution
and the convergence of the proposed optimization
algorithm. In addition, we establish the unified
mathematical model, which can describe the approximate
segment circle point and the approximate segment slide
point, and develop the “saddle-point planning”
programming.
By employing the approach, the approximate function
synthesis of six-bar dwell mechanisms can be optimized in
the unified mathematical model and algorithm. Numerical
examples show that the approach can ensure the desired
precision of the output function in dwell and non-dwell
period.
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