Synthesis of Coupler Curves of Spherical Four-bar ... - IFToMM

12th IFToMM World Congress, Besançon (France), June18-21, 2007
Synthesis of Coupler Curves of Spherical Four-bar Mechanism
Through Fast Fourier Transform
Jinkui Chu * Jianwei Sun †
Key Laboratory for Precision and Non-traditional Machining Technology of Ministry of Education;
Key Laboratory for Micro/Nano Technology and System of Liao-ning Province
DaLian 116023, P. R. China
DaLian 116023, P. R. China
Abstract—In this paper , the mathematic representation of
coupler curve of the spherical four-bar mechanisms is proposed.
Base on the expression, the relationship between coupler curve
and the basic dimensional types is discovered, and the formulas
which can compute the position for coupler point, true size and
installing dimensions of the spherical four-bar mechanism is
deduced. These formulas make the spherical four-bar mechanism
path generation with prescribed timing be possible. Finally, an
example is given to prove it
.
Keywords: Spherical four-bar mechanism, Path generator,
Harmonic component, Dimension synthesis
mechanism, and use these formulas to approach the
problems of path generation with prescribed timing.
II. Mathematical description of basic geometric
constructions
I. Introduction
Path generation problems in planar has been extensively
studied. It seems that investigations specifically dealing
with path generating problems in spherical kinematics are
relatively few. In recent years, there is a tendency to use
spherical mechanism; more and more authors used
different means to solve this problem. Tong [1] discussed
the syntheses of planar and spherical four-bar path
generators by using the pole method. Chiang [2] presented
a discussion of the spherical four-bar path generator to
generate a coupler curve passing through four given path
points. Moreover, Bagci [3] proposed an approach to treat
path generating by means of geometric methods. The
Symmetrical coupler curve and singular point
classification in planar and spherical swinging-block
linkages have been discussed in Shirazi [4].Chu [5] study
the coupler curve of planar four-bar linkages in the
frequency field by using the complex vector and frequency
method. And base on the theory, the synthesis of coupler
curves of planar four-bar linkages through fast Fourier
transforms was investigated
[6-8] . In this paper, by
discussing the relationship between coupler curve of the
spherical four-bar mechanism and it’s harmonic
composition, we determine the basic dimensional type of
spherical four-bar mechanism and deduce the formulas
which may compute the position for coupler point, true
size and installing dimensions of the spherical four-bar
*E-mail: chujk@dlut.edu.cn
† E-mail: avensun@tom.com
Fig. 1. Model of spherical four-bar mechanism
As shown in fig.1, point O is the centre of a unit sphere.
And Oxyz is a left-handed rectangular coordinate system
with origin at O, the sphere of radius R which may be
considered of dimensionless length, α = central angle of
the driving crank AB, β = central angle of the swing link
CD, γ = central angle of the coupler link BC, ξ =
central angle of the framework AD. P is the arbitrary point
on coupler link BC, and the corner dimension is θ
p
, use
θ ,θ 0 1
to denote the central angle of the BP and rotation
angle of driving lever AB respectively. Consider moving
coordinate ( O, e1,
e2
, e ) , e ,e
3 1 2
coincide with OB and
the tangent of AB respectively, e
3
= e1
× e2
. Coordinate
system e1 , e2
, e3
is defined relative to the fixedcoordinate
system Oxyz by
e
1
= icosα
+ sinα(
jcosθ1
+ ksinθ1)

12th IFToMM World Congress, Besançon (France), June18-21, 2007
= i cosα
+ jsinαcosθ1 + ksinαsinθ1
(1)
e
2
= jcos(
θ
1
+ π / 2) + ksin(
θ1
+ π / 2)
= − j sinθ1 + k cosθ1
(2)
e
3
= isinα
− cosα
( jcosθ1
+ ksinθ1)
= isinα
− jcosαcosθ1 − kcosαsinθ1
(3)
Where: θ
1
= Q + ωt
; Q is initial angle of the
driving crank
Positions of point P may be expressed in terms of the
coordinate system Oe e as follows:
1 2e3
x
pe
= R cosθ 0
(4)
y
pe
= −R
sinθ 0
sin( θ
2
+ θ
p
) (5)
z
pe
= R sin θ
0
cos( θ
2
+ θ
p
) (6)
We obtain the positions of point P in terms of the
fixed-coordinate system Oxyz as follows:
r = x e + y e + z e
Where ( e
p pe 1 pe 2
1
, e2
, e3
, xpe
, y
pe
, z
pe
pe
3
) was defined by
Equation (1-6). Thus
r = i cosθ cosα
+ sinθ
cos( θ + θ )sinα
] R
p
[
0
0 2 p
[ cosθ 0cosθ1sinα
+ sinθ
0sinθ1sin(
θ
2
+ θ
p
− sinθ 0cos(
θ
2
+ θ
p
)cosαcosθ1
[ cosθ 0sinθ1sinα
− sinθ
0cosθ1sin(
θ
2
+ θ
p
− sinθ 0cos(
θ
2
+ θ
p
)cosαsinθ1
+ j
+ k
The y axis and z axis are defined as the real axis and
imaginary axis respectively. Above equation reduces to
r = i cosθ cosα
+ sinθ
cos( θ + θ )sinα
] R
p
[
0
0 2 p
+ R [ cosθ
0
sinα
− sinθ
0
(cos( θ
2
+ θ
p
)
]R
)
]R
) cosα
+ k sin(
θ + θ ))]e
Where k = −1
When the mechanism lies in standard installing position,
and the angular velocity of input link AB is constant
parameter ω (the input angle θ 1
can be expressed as
θ
1
= ωt ), the rpx
which is the projection of rp
on the x
axis may be expressed as follow:
r t)
= Rcosθ cosα
px
(
0
t =
2
p
kθ
+ R sinθ
cos( θ ( t)
θ ) sinα
(7)
θ /
1
ω
0 2
+
Where:
The standard installing position means that the angle
Q between AB and CD is 0°
(defined the angle Q is
initial angle of the driving crank); the frame AD lies on x-y
plane; the centre of sphere is located at the origin of
coordinate.
p
1
Similarly for the projection of on the y-z plane we
have
r t)
= R[cosθ
sinα
− sinθ
(cos( θ ( t)
θ ) cosα
pyz
(
0
0 2
+
p
kθ1
+ k sin(
θ
2
( t)
+ θ
p
))]e (8)
Figure.2 is a spherical four-bar mechanism (mechanism
lies in general installing position), where O
x
, O
y
, Oz
is
the distance between O and O'
, θ
x
is defined as the
angle between the y' axis and the y axis measured in
accord with the right-hand convention from y axis and the
y ' axis about the x'
axis.
Fig. 2. General position of spherical four-bar mechanism
In this case, base on the Equation (7),
r p
r p
which is the
projection of on the x axis, can be expressed as follow:
rpx
( t + t′
) = Ox + Rcosθ 0cosα
+ R sinθ
0
cos( θ
2
( t + t′
) + θ
p
) sinα
(9)
Where: t ' = Q / ω
Similarly, base on the Equation (8), for the projection of
r p
on the y-z plane we have
r
r px
kμ
kθ
x
t + t′
) = R' e + e R[cosθ
sinα
- sinθ
0
(cos( θ
2
( t + t′ ) + θ
p
) cosα
kθ1
+ k sin(
θ ( t + t′
) + θ ))]e
pyz
(
0
Where
R'
2
y
2
z
= O + O and
2
p
μ = arctan
O
O
z
y
(10)
Equations (9), (10) will be used to discuss the
relationship between coupler curve of the spherical
four-bar mechanism and its harmonic composition

12th IFToMM World Congress, Besançon (France), June18-21, 2007
III. Frequency analysis of the spherical four-bar
mechanism’s coupler curve.
The θ
2
( t)
+ θ
p
is a periodic function whose
periodicity is 2 π , so Equation (9) is also a periodic
function, and it’s periodicity was determined by the
function of cos(
θ
2
( t + t′ ) + θ
p
) .Defined the one
dimensional Fourier expansion of cos(
θ
2
( t ) + θ
p
) can
be expressed as follow:
∑ ∞ kϕ
knω
2
( t)
+ θ
p
) = c
ne
e
n=
−∞
n t
cos(
θ (11)
c
Where
n
and ϕ
n
are the Fourier series’ amplitude
and phase angle respectively.
Substituting Equation (11) in Equation (9) yields
r t + t′
) = Ox + Rcosθ cosα
px
(
0
+ Rsin
θ sinα
0
∑ ∞ n
n=
−∞
c e
k(
ϕ + nQ)
Merge the 0 item of the Fourier series, we can obtain
r px
( t + t′
)
kϕ0
= ( O + Rcosθ
cosα
+ Rc
sinθ
sinαe
)
x
0
0
∑
0
n
e
knωt
k ( ϕ n + nQ) knωt
+ sinθ0sinα
cne
e
n ≠0
kϕ −1
c−
1
e , we have
1 −kϕ
∑ ∞ c
− 1 n k ( ϕn
−ϕ−
) n t
cos(
2
( ) +
p
)e = e
1 k ω
θ t θ
e
−1
n=
−∞ c
−1
Equation (11) dividing by
c
R (12)
Equation (12) dividing by Rc
−1sinθ
0sinαe
have
r ( + ′
px
t t )
k ( Q− −1
)
e
ϕ =
R c
-1sinθ
0sinα
1
(O
x
+ Rcosθ
0cosα
R c
-1sinθ
0sinα
kϕ0 −k
( ϕ −
+ Rc
sinθ
sinαe
) e
0
+
0
∑
k ( ϕ − 1 −Q)
1
−Q)
(13)
, we
c n k ( ϕ n −ϕ
− 1 + (n + 1)Q) knωt
(14)
n ≠0
c−
1
By comparing the equation (13) with the equation (14),
we can find that they have the same amplitude (except the
0 item) and the equation (13)’s phase angle subtracted
from equation (14)’s phase angle gives ( n + 1)Q .
Theθ 2
( t)
+ θ
p
is a periodic function whose periodicity
is 2 π , so Equation (10) is also a periodic function, and the
periodicity was determined by the function of
cos(
θ ( t ) + θ )cosα
+ ksin(
θ ( t)
+ ) .Defined
2 p
2
θ
p
e
e
the two-dimensional Fourier expansion of the expression
can be expressed as follow:
cos(
θ ( t ) + θ )cosα
+ ksin(
θ ( t)
+ )
=
∑ ∞ n
n = −∞
c'
2 p
2
θ
p
n ω
c' e e
k ϕ ' kn
t
(15)
Where
n
and ϕ'
n
are the Fourier series’ amplitude and
phase angle respectively.
Substituting Equation (15) in Equation (10) yields
r
′
kμ
kθ
x
pyz
( t + t ) = R' e + e
0
R[cosθ
sinα
− sinθ
(
0 ∑ ∞ c'
n
n = -∞
e
k ( ϕ ' n + nQ)
e
knωt
Merge the 1 item of the Fourier series, we can obtain
r
( t + t′
) = R' e
kμ
pyz
kϕ
' 0 k ( θx
+ Q) kωt
R( cosθ
0sinα
− c'
0
sinθ
0e
)e e
+
− R sinθ
∑
)] e
k ( ϕ'
n + (n+
1)Q+
θ x ) k (n+
1) ωt
0
c'
n
e e
n≠0
Merge the 0 item of the Fourier series and defined
n = n + 1, we have
r
pyz
( t + t′
) = (R' e
+ R
− Rc'
sinθ
e
kμ
k ( ϕ'
-1 + θ x )
−1
0
kϕ
' 0 k ( θx
+ Q) kωt
( cosθ
0sinα
− c'
0
sinθ
0e
)e e
∑
k ( ϕ ' n -1 + nQ+
θ x ) knωt
0
c'
n-1
e e
n≠0,1
kϕ − 2
e
′
−
, we have
)
kθ1
− R sinθ
(16)
Equation (15) dividing by c'
2
1 -kϕ′
e
- 2
(cos( θ
2
( t)
+ θ
p
)cosα
+ ksin(
θ
2
( t)
+
p
)) =
c′
θ
-2
∑ ∞ c' n k ( ϕ ' n −ϕ
' − ) n t
e
2 k ω
e
(17)
n=
−∞ c'
−2
k ( ϕ′
2 −2Q)
Equation (16) dividing by Rc−2sinθ0e
−
, we can
obtain
r ( + ′
pyz
t t )
k (2 Q − − ′ 2
e
ϕ ) 1
kμ
= - [(R' e
R c′
-2sinθ
0
Rc'
-2
sinθ
0
k ( ϕ '-1 + θx
) -k
( ϕ '-2
-Q+
θx
)
− R c' sinθ
e )]e
−
c'
−1
0
1 kϕ
' 0 k (2Q-ϕ
'-2
) kωt
[(cosθ
0sinα
− c'
0
sinθ
0e
)]e e
−2
sinθ
0
∑
c' 2
n −1
k ( ϕ ' n −1
−ϕ
' − + (n + 1)Q) knωt
+ e
e (18)
n ≠0,1
c'
−2
By comparing the equation (17) with the equation (18),
we can find that they have the same amplitude (except the
0 item and the 1 item) and the equation (17)’s phase angle
subtracted from equation (18)’s phase angle gives
( n + 1) Q + ϕ'
n − 1-ϕ
'
n
.

12th IFToMM World Congress, Besançon (France), June18-21, 2007
These findings indicate that the Fourier series of the
general position of spherical four-bar mechanism’s coupler
curve mostly depend on equation (13) and equation (17),
and the equations were determined by mechanism’s basic
dimensional types which includeα , γ , β , ξ and θ p
. We
can calculate coupler rotation-angle θ
2
of the mechanism
basic dimensional type, and obtain corresponding
harmonic component’s database by utilizing one
dimension and two dimensions FFT. Finally, constitute a
numerical atlas database by put the harmonic component’s
database and mechanisms’ basic dimensional types
together. By comparing the harmonic characteristic
components of the given coupler curve with the harmonic
characteristic component of every group in the numerical
atlas and utilizing fuzzy identification in Ref.[10], the
weighting hamming distance can be figured out.
According to fuzzy theory, weighting hamming distance
can measure the similarity degree of the referred model
and identified object. The two parts will more approximate
when the distance is smaller. If the distance is equal to
zero, the two are the same. The basic dimensional types of
linkage are determined by this method.
IV. Compute the position for coupler point, true size
and installing dimensions of the spherical four-bar
mechanism
Defined
r
px
kζ knωt
∑ ∞ n
ne
e
n = −∞D
= is the harmonic
component of the given coupler curve
and
r
pyz
kζ ' knωt
∑ ∞ n
'
n
e e
n=
−∞D
r p
project on x axis,
= is the harmonic
component of the given coupler curve
plane.
(1)Central angle of the driving crank α
From equations (12) (16), we know that
D'
r p
project on y-z
n
R sinθ
0
= (n ≠ 0,1) (19)
c'
n−1
D
sin sin =
n
R θ
0
α (n ≠ 0,1) (20)
c
n
Substituting Equation (19) into Equation (20) yields
D
nc'
n
= arcsin(
D' c
)
−1
α (n ≠ 0,1) (21)
n n
The other basic dimensional types can be obtained by
the proportional relation.
(2) Initial angle of the driving crank Q
From equations (11) (12), we can obtain
Q = ζ −
(22)
1
ϕ 1
(3) Turn angle θ
x
From equations (15) (16), we know that
ζ ' = −Q + θ
(23)
- 1
ϕ'
-2
From equation (23) we can obtain
x
-1
x
θ = ζ ' + Q − ϕ'
(24)
(4) The sphere of radius R
From equation (16), we know that
kζ
' 1 kϕ'
0 k(
θx
+ Q)
D' 1
e = R(cosθ 0
sinα
− sinθ0c'
0
e ) e (25)
From equation (25) we can obtain
2
2 2
R = A + ( D n
/ c ) ) / sin α (26)
Where
D
(
n
A = [ D'
1
cosζ
'
1−
'
n
c'
0
cos( ϕ '
0
+ θ
x
+ Q)
/ c'
n−1
]/ cos( θ
x
+ Q
θ
(5) Central angle
0
From equation (19), we can obtain
-2
D'
= arcsin(
Rc'
n
θ
0
(27)
n−1
(6) Translational component on x axis Ox
From equation (12), we know that
D0 cosζ
0
= O x
+ cosθ
0sinα
+ c0sinθ0sinαcosϕ0
(28)
From equation (28), we can obtain
O
x
= D0cosζ
0
− cosθ
0sinα
− c0sinθ0sinαcosϕ0
(29)
(7) Translational component on y axis and z axis O
y
, Oz
From equation (16), we know that
kζ
' 0 kμ
k ( ϕ '-1
+ θ x )
D'
0
e = R' e − c'
−1
sinθ
0e
(30)
From equation (30), we can obtain
O
y
= D'
0
cosζ '
0
+ c'
−1
sinθ
0cos(
ϕ'
−1
+ θ
x
) (31)
O
z
= D'
0
sinζ '
0
+ c'
−1
sinθ
0sin(
ϕ'
−1
+ θ
x
) (32)
Base on the equations (21)(22)(24)(26)(27)(29)(31)(32),
we can compute the position for coupler point, true size
and installing dimensions of the spherical four-bar
mechanism.
V. Illustration
Figure 3 shows the given coupler curve, (a) the given
coupler curve; (b) the coupler curve project on x-y plane;
(c) the coupler curve project on x-z plane; (d) the coupler
curve project on y-z plane. Suppose a spherical four-bar
linkages is to be synthesized to generate the given coupler
curve
Spherical four-bar mechanism path generation will be
done by using the follow steps. First, based on those
sampling points (in this paper, we choose 64 points and
the sampling points are denoted by “+”), the
)
)

12th IFToMM World Congress, Besançon (France), June18-21, 2007
corresponding harmonic component of the given couple
curve can be obtained by utilizing one dimension and two
dimension FFT. Second, carry out identification between
the harmonic characteristic component got from first step
and the corresponding component in the numerical atlas.
Decide the basic dimensional types of mechanism by using
fuzzy identification method. Finally, the true size and
installing dimensions of the spherical four-bar mechanism
can be computed by the basic dimensional types and the
equations (21) (22) (24) (26) (27) (29) (31) and (32).The
results are:
α = 23.0795° γ = 47.1624°
β = 53.1831° ξ = 57.1969°
θ = 68.0000° θ = 29.7188°
p
0
θ = 15.5601° Q = 30.0578°
x
O = 9.9804mm O = -4.9766mm
x
y
O = -5.9621mm R = 2.7231mm
z
Fig. 3. Given coupler curve of spherical four-bar mechanism
Figure 4 shows the fitting comparison between the
given coupler curve and the actual mechanisms’ coupler
curve. The given coupler is denoted by “—” and the actual
mechanisms’ coupler curve is denoted by “·”.
VI. Conclusion
The spherical four-bar mechanism is the most basic and
useful spherical hinge which can be used to many
machines. Because the spherical mechanism have more
parameters, the path generation is harder than plane
mechanism’s. The usual method used for path generation
only solve the finitely separated path-points or a segment
of coupler curve, the method we have proposed makes the
closed coupler curve generation be possible.
VII. Acknowledgements
This project was supported by the National Natural
Science Foundation of China (Grant No. 50475153) and
the Foundation of Ministry of Education (NCET-04-0266)
Reference
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Fig. 4. Fitting comparison between the given coupler curve and theactual
coupler curve of spherical four-bar mechanism