The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.OS2.008

Genliang CHEN1 Hao WANG2 Weidong YU3 Yong ZHAO4

State Key Laboratory of Mechanical System and Vibration,

Shanghai Key Laboratory of Digital Manufacture for Thin-walled Structures,

Shanghai Jiao Tong University, Shanghai 200240, P.R. China.

Optimal Synthesis of the Slider-crank Mechanism for Path Generation Based on the

Generalized Kinematic Mapping of Constrained Plane Motions

Abstract: This paper presents a new strategy for the

optimal synthesis of the planar slider-crank mechanism for

continuous path generations. Based on the generalized

kinematic mapping of constrained plane motions, the

desired path for the coupler link is mapped into the

three-dimensional projective space as a spatial surface,

which is regarded as the target kinematic manifold. Also,

the potential motions of the coupler link constrained by the

crank and the slider are projected on the target manifold in

the image space, whose difference can be regarded as an

alternative description for the kinematic conflict of the

mechanism with the design requirement. Thus, a metric for

the structure error between the candidate mechanism and

the ideal one for the desired path is set up in terms of the

area of a ruled surface, the distance surface intercepted

from the target manifold, which is also selected as the

objective function for the optimal synthesis. To obtain the

final optimum based on the initial design variables with

high convergence rate, the gradient descent algorithm is

introduced. A numerical example for the optimal synthesis

of the planar slider-crank mechanism for an elliptic

trajectory is studied to validate the effectiveness of the

proposed approach. Keywords: Optimal synthesis, Planar slider-crank mechanism,

Continuous path generations, Generalized kinematic mapping,

Gradient descent algorithm

I. Introduction

The planar slider-crank mechanism, as shown in Fig. 1,

has been widely applied in industrial practice during the

past century, such as the internal combustion engines, the

shaping machines and the pumps. Generally, it is used as

three different types: the function generators from the input

to the out-put links, the rigid body guidance through the

study of rigid body motion and the path generators by

referring to the coupler curve [1, 2]. In order to implement

these functions, the dimensional synthesis is carried out to

determine the dimensions of each mechanical element in

the mechanisms so that the required tasks can be achieved.

In the case of path generation, it is subjected to the task that

one point on the coupler link traces out the desired

trajectory curve as close as possible.

The tasks for path generation can be divided into two

types: the point-to-point path generation and the

continuous path generation. The former is to specify only a

small number of points on the path, and the trajectory

between any two adjacent points is not required strictly. It

1 leungchan@sjtu.edu.cn 2 wanghao@sjtu.edu.cn 3 yuweidong@sjtu.edu.cn 4 zhaoyong@sjtu.edu.cn

is a very classical problem in the field of mechanism and

machine theory, for which many effective analytical

synthesis methods have been developed [2-9]. The latter

focuses on the whole path or a large number of points on

the trajectory generated by the mechanism. However, due

to the limitation of independent design variables for the

mechanism, the maximum number of precision points is

also limited and small. For this reason, optimization

techniques are usually employed to specify the optimal

dimensions of the mechanism for continuous path

generation, which regards the difference between the

desired and the generated paths as the objective function to

be minimized.

Therefore, the key issue to the dimensional synthesis of

planar linkages for continuous path generation is to

designate the objective function, also called as structural

error, to evaluate the overall difference between the

generated and the desired paths. Conventionally, it is

generally formulated as the sum of the square distance

between the generated and the desired paths over a number

of comparison points [10-14]. However, it is difficult to

select the corresponding comparison points on the two

continuous paths since the overall difference between them

is required to reflect from these corresponding comparison

points. To facilitate the selection of corresponding points

on the two paths for comparison, a timing requirement is

imposed on the desired path. As the coupler point moves

from one position to the next on the desired path, the input

crank is required to rotate through a specified angle.

However, the timing requirement is an artificial constraint

introduced more for ease of formulation of the structural

error than for any practical relevance, as indicated by Kota

et al. [15].

Fig.1. Planar Slider-crank Mechanism.

For the sake of avoiding making point-by-point

comparison between the desired and the generated paths,

some alternative indices have been proposed as metrics of

the structural error to evaluate the overall difference of the

paths. Cheung et al. [16] introduced the concept of

orientation structure error of the fixed link for crank-rocker

mechanisms to reflect the overall difference between the

desired and generated path. In their model, the desired

curve of the coupler was considered as the kinematic input

of the mechanism. The orientation of the fixed link is

released from the design variables, whose variation is

defined as the alternative metric of the structure error

between the generated and the desired paths. Therefore,

according to the kinematics of the mechanism, the

difference between the two curves has been transformed to

a scalar function regarded as the objective function to be

minimized. In the same manner, they introduced the

position structural error of the fixed guider for the

dimensional synthesis of slider-crank mechanisms [17].

Taking account the design specifications in the industrial

practice, Figliolini et al. [18] addressed attentions to the

shape and the overall size of the coupler curves for the

dimensional synthesis of slider-crank mechanisms for

automatic machinery.

On the other hand, the metric of the distance norm

between the desired and generated paths may be

overestimated when they are translated with respect to

each other. For this reason, Fourier descriptors are

introduced by Kota et al. [15, 19] to represent the coupler

curves, which are invariant with respect to the translation,

rotation and scaling of the curves. Another advantage of

this method is the reduction of the size of the design space

of the dimensional synthesis problem because the

unknown variables are divided into two sets, one set of

design variables determines the shape of the generated

curve and the other defines the size, position, and

orientation. On this basis, many researchers [20-23] have

contributed their efforts to push forward the techniques for

the dimensional synthesis of planar mechanisms for

continuous path generation.

In addition, structural error is a highly non-linear and

multi-modal function in a multi-dimensional design space.

Finding the global optimum of this function is a formidable

problem. So, success is not always guaranteed. As a result,

evolutionary algorithms are usually employed to carry out

the optimal synthesis, such as the Neural Networks [20],

the Genetic Algorithms [14, 24] and the Particle Swarm

Optimization [25]. Additionally, some other methods [14,

26-28] have also been proposed to implement the

dimensional synthesis of planar mechanisms for path

generation.

This paper presents a new strategy for the optimal

synthesis of the planar slider-crank mechanism for the task

of continues path generation. The main idea is to measure

the overall “difference” between the candidate mechanism

and the ideal one for the desired path in an alternative

manner with intuitive physical insights. As a result, the

method of the generalized kinematic mapping of

constrained plane motions [34] is utilized to transform the

dimensional synthesis of planar mechanisms into the

three-dimensional projective space as a spatial geometry

problem. Then, the concept of target kinematic manifold is

introduced to represent the desired path for the coupler link

more comprehensively in the image space. On the basis of

this expression, not only the position of the coupler point

but also the orientation of the link is imposed into the

displacement to represent the four-bar motion completely.

In consequence, the distance between the constraint

manifold and the target one for the coupler link can be

defined in the three-dimensional image space, which not

only reflects the distance of the coupler point from the

desired one, but also indicates the way how to move the

coupler link onto the desired path at the same time.

In order to simplify the geometric derivation, an

alternative representation for the structure error between

the candidate mechanism and the ideal one is introduced as

the conflict of the kinematic constraints of the crank and

the slider ends on the target manifold. Then, the ruled

surface intercepted from the target manifold by the circular

and the linear constraints is defined as the distance surface

for the problem of dimensional synthesis. The area of this

surface is modeled up as a metric for the overall difference

between the generated and the desired trajectory of the

coupler motion, which is selected as the objective function

of the optimization problem for the dimensional synthesis.

In order to obtain the final optimum based on the initial

design variables, the gradient descent algorithm is

introduced in our model.

The rest of the paper is organized as follows. Firstly,

section II gives a brief introduction to the generalized

kinematic map-ping of constrained plane motions, which is

the basis of the new strategy for the optimal synthesis

proposed in this paper. Then, the position analysis of the

studied slider-crank mechanism is presented in section III.

Thereafter, in section IV the kinematic manifolds for the

coupler link is generated and analyzed in the image space

as two typical kinematic constraints, the circular constraint

and the linear one. Then, a metric for the structural error of

the candidate mechanism is set up on the target manifold of

the coupler, which is selected as the objective function for

the problem of dimensional synthesis. In section VI, the

gradient descent algorithm is discussed briefly as the

searching algorithm. Then, a numerical example is studied

in section VII to validate the effectiveness of the proposed

method. Lastly, some conclusions are drawn in section

VIII.

Fig.2. Plane displacements of a

constrained moving body.

II. The Generalized Kinematic Mapping of

Constrained Plane Motions

As shown in Fig. 2, the plane displacement ( , , )a b D

of the moving body A with respect to the fixed frame Σ

can be generally given by the homogeneous linear

transformation as

cos sin

sin cos

0 0 1

X a x

Y b y

Z z

(1)

where ( , )a b and are the translational and rotational

components of the displacement. Vectors ( , , )Tx y z and

( , , )TX Y Z represent the homogeneous coordinates of

the point Q in the local frame E attached to A and the

fixed one , respectively.

According to the definition introduced by Grünwald[30]

and Blaschk[31], the plane displacements can be mapped

onto the image points in a three-dimensional projective

space in terms of the kinematic mapping [32, 33] as

1 11 2 3 4 2 2

1 1

2 2

1

2

1

2

: : : ( sin cos )

: ( cos sin )

: 2sin

: 2cos

X X X X a b

a b

(2)

Then, there exist a general closed curve ( ) on the

plane, under which the motion of body A is constrained at Q , as shown in Fig. 2. In other words, the fixed point Q

on A will always stay on the curve when it moves.

By substituting Eq.(2) and the constraint curve into

Eq.(1) and after some algebraic reduction, the image

surface of the constrained (under ) plane motion of body

A can be obtained in the three-dimensional projective

space as

3 4 3 41

4 3 4 32

1 1( )

2 2

X X X XX x

X X X XX y

F (3)

where ( , )Tx y is the Cartesian coordinate of the point

Q in the local frame E .

Fig.3. Constraint manifold of the plane

motion constrained by the planar closed

curve

Equation (3) is regarded as a general form of the image

surface in the projective space corresponding to the plane

motion constrained by the curve . Since the constraint

curve is general in a parametric form. It is obvious that the

image surface expressed in the general form is independent

from the concrete forms of the constraint curves. In other

words, it can be stated that the generalized kinematic

mapping of constrained plane motions is invariant from the

exact types of the constraints. Furthermore, the image

surface can be normalized conveniently by letting 4X 1

under the assumption of , which is illustrated in Fig.

3.

Furthermore, some intuitive geometric interpretations

between the constrained motion on the Euclidean plane

and its image surface in the projective space have been

revealed in our previous research[34].

As shown in the figure, there are two kinds of special

curves can be extracted from the spatial surface. One is the

intersection curves with horizontal planes, denoted as

0 in Fig. 3 (here 3X cot ), which can also be

regarded as contours of the surface at different levels.

These curves perform a pure translation along the

constraint curve on the plane with the specified rotation

angle of 02 . Moreover, the shapes of these curves

are all similar to the original constraint one, which can be

transformed from the original constraint curve according

to an affine mapping consisting of a translation, a rotation

and a scaling consequently. The other kind of curves is the

straight lines denoted by 0 , as shown in the figure.

These lines correspond to the motions of pure rotation

about a fixed point on the constraint curve specified by 0 .

The directions and the intersection points (with the zero

horizontal plane in the projective space) of these lines can

be determined by both the coordinates of the concerned

point on the moving plane and the rotating center on the

constraint curve according to Eq.(3). Therefore, these

image surfaces can be regarded as ruled surfaces in the

projective space which are generated by sweeping an

arbitrary line along any of the curves. Thus, the curves are named as the directrix, also called the base

curve, of the ruled surface and the lines are called the

rulings or generators.

Fig.4. Geometric presentation for the

kinematics of the slider-crank mechanism.

Based on the above properties, a surface coordinate

system, the - frame, can be established for each image

surface of constrained plane motions, as shown in Fig. 3.

Obviously, the two coordinates are independent of each

other and can be regarded as alternative representation of

the rotation pivot and angle of the displacements along the

plane motion. Thus, for each point on the image surface,

the corresponding plane displacement can be obtained

directly from its - coordinates according to the

generalized kinematic mapping (Eq.(3)), vice versa. The

image surface in the projective space is also called the

constraint manifold [33, 35] corresponding to the plane

motion of the moving body under the constraint of the

closed curve on the plane.

Due to the space limitation, this section only presents a

brief introduction about the generalized kinematic

mapping of constrained plane motions. The detailed

discussion on this approach can be found in [34].

III. Position Analysis of the Mechanism

As illustrated in Fig. 1, there are totally eight design

variables for the slider-crank mechanism, namely the

position of the pivot for the fixed revolute joint A:

0 0( , )T

A x y r , the length of the crank AB: 1l , the

geometric dimensions for the coupler link BCM:

1 2( , , )T

c c c cx x y D and the position and orientation of the

rail for the prismatic joint: 0 0 0( , )Td p . Here, it should

be noted that the positions (orientations) of the links are

related to the initial reference frame O fixed on the

ground, and the geometric dimensions of the coupler link

are expressed in the local frame M attached to it.

The kinematics of this mechanism can be transformed to

a geometry problem as shown in Fig. 4. The rotation angle

of the crank 1 is considered as the kinematic input of the

mechanism. Then, the position of the pivot of the revolute

joint B can be obtained as

1

1

1

cos

sinB A l

r r (4)

Thus, according to the planar projective geometry, the

position of the point D can be derived conveniently as

0 00 0 ( )T

D L L B r r u u r r (5)

where 0 0 0 0(cos , sin )T

L d r is the position vector of

the point 0L and 0 0 0( sin , cos )T u represents the unit

direction vector for the rail of the prismatic joint.

Therefore, the position of the pivot of the revolute joint C , namely the location of the slider, can be obtained as

0C D sl r r u (6)

where 2

2 1( ) ( ) ( )T

s c c D B D Bl x x r r r r denotes a

half of the length of the chord CC . And 1 is the

coefficient distinguishing different assembly

configurations of the mechanism.

Finally, the position of the coupler point M can be

obtained as

1M B c c Rr r ρ (7)

where 1 1( , )T

c c cx y ρ is the position vector of the revolute

joint B in the local frame M . cR represents the rotation

matrix of the coupler link with respect to the initial frame

O , which can derived as

2 2

2 2

cos sin

sin cosc

R (8)

where 2 denotes the rotation angle of the coupler, as

shown in the figure, which can be determined according to

the direction of the line BC .

Consequently, when the crank rotates through a full turn,

the path of the coupler point M can be traced out as a

closed curve according to the above steps in an analytical

way. As an example, the curve K in Fig. 4 is a potential

generated path of the studied mechanism.

IV. Constraint Manifold of the Mechanism

The coupler link in the slider-crank mechanism, which

in a sense can be regarded as the end-effector, moves under

the constraints from the fixed ground through two ends.

One is the crank end constraining the position and

orientation of the coupler link through a revolute joint, and

the other is the slider end through a prismatic joint.

In particular, the crank end imposes a circular constraint

on the coupler link through the revolute joint B. At the

same time, the motion of the coupler link is also

constrained on the line specified by the slider end at the

joint C. Then according to the generalized kinematic

mapping of constrained plane motions presented in the

above section, the constraint manifolds caused by the two

ends can be obtained conveniently via substituting the

constraint curve ( )F in Eq.(3) with the specified circle

and line, respectively.

Since the images of the constraint manifolds are ruled

surfaces, their algebraic equations can be expressed in a

unified form as

1

2

3

( , ) ( ) ( )

X

X

X

f b u (9)

where b is the directrix of the ruled surface, also called

the base curve, and u is the director curve as discussed in

the above section.

Theoretically, each of the curves on the manifold

can be selected as the directrix. Here, in order to simplify

the algebraic derivation, the curve with / 2 (namely

0 , the coupler translates along the constraint curve

with no rotation) is designated as the base curve. Then, for

the kinematic manifold under the circular constraint

through the crank end, the directrix can be derived as a

circle on the horizontal plane at 3 0X .

1 1

0 1 cos1( )

1 0 sin2A cl

b r ρ (10)

where 1 is the rotation angle of the crank.

Otherwise, the director curve can be derived as the

direction vector of the generators of the ruled surface.

1 1

cos1

sin2

2

A cl

r ρu (11)

and 3X in Eq.(9).

Fig.5. Constraint manifolds of the coupler

link caused by the crank and the slider

ends and their intersection

In the same manner, the directrix and the director curve

for the kinematic manifold under the linear constraint can

be obtained as

00 2

0 11( )

1 02L c

b u r ρ (12)

00 21

2 2

L c

u r ρu (13)

As a consequence, both of the two constraint manifolds

can be obtained based on the above equations and the

dimensions of the mechanism, which are illustrated in Fig.

5. Furthermore, it can be approved that the constraint

manifold caused by the crank end is a hyperboloid of one

sheet and the other caused by the slider end is a hyperbolic

paraboloid[36].

In the end, the final constraint manifold for the coupler

link can be obtained as the intersection of the two

generated kinematic manifolds, which is a spatial closed

curve in the projective image space, as shown in the figure.

It is known as the image curve of four-bar motion in

kinematic literatures [32, 33]. And the coupler will travel

along this curve when the crank rotates through a full turn.

It should be noted that the final constraint manifold for the

coupler is an overall description, including both the

position and orientation, for the link’s motion in the image

space. And it is invariant of the selection of the coupler

point.

V. Metic for the Structure Error on the Kinematic

Manifold

In the above section, the constraint manifold for the

coupler link has been obtained as a spatial closed curve in

the three-dimensional projective space.

On the other hand, by regarding the desired path as a

kinematic constraint of the coupler, a ruled surface (similar

to the one illustrated in Fig. 3), which is defined as the

target manifold for the coupler, can also be obtained in the

image space according to the generalized kinematic

mapping discussed in section II. Any point on this surface

represents a displacement of the coupler satisfying the

design requirement, namely, the coupler point of the

mechanism lies on the desired path. If the whole constraint

manifold is totally on the target one, the candidate

mechanism can generate the desired path for the coupler

and we can say that it is an ideal one for the path generation

synthesis. Otherwise, the difference between the generated

and the desired paths can be mapped into the image space

as the closeness of the constraint manifold to the target one

for the coupler link. Comparing with the structure error set

up on the Euclidean plane, the model in the image space

not only concerns with the point-to-point distance between

the generated and the desired paths, but also specifies the

way how to move the coupler link from the current

configuration to some others meet the design requirement.

However, it is a metric of distance between a spatial

curve and a surface in the three-dimensional projective

space, which is not very convenient to be set up properly.

Therefore, the structure error between the candidate

mechanism and the ideal one is modeled up in the image

space in an alternative way.

As discussed in the above section, due to the kinematic

constraints caused by the crank and the slider ends, the

displacements of the coupler link will be restricted on the

circular and the linear constraint manifolds simultaneously.

To completely meet the design requirement, all these

displacements should be bound to the target manifold as

well. Then, by intersecting the circular constraint manifold

and the target one, a spatial curve can be obtained as the

feasible motion of the coupler under the kinematic

constraint through the crank end. So does the one through

the slider end. Thus, two spatial curves are derived on the

target manifold as shown in Fig. 6, which reflect the

perfect motions of the coupler link satisfying the kinematic

constraints due to the crank and the slider ends,

respectively. If these two curves are completely matched

with each other, the candidate mechanism also can

generate the desired path for the coupler which means it is

an ideal one for the design requirement. So, the difference

between these two curves is regarded as an alternative

metric of the structure error between the candidate

mechanism and the ideal one in the image space.

Both of the feasible motions for the coupler are on the

target manifold which is a ruled surface in the projective

space. Then, for each point on one curve, there always

exists a corresponding point on the other curve to make the

line segment between these two points collinear with the

generator of the ruled surface passing through either of

them, as the line ( )t δ and the point s cA A- illustrated in

Fig. 6. On the other hand, for each generator on the target

kinematic manifold, a unique pair of intersection points

can also be specified on the two curves for feasible

motions, respectively.

Therefore, for each pair of the corresponding points, a

distance can be defined on the kinematic manifold in terms

of the length of the associated generator, which can be

represented as

( ) ct s (14)

where c , s are the parametric coordinates of the

intersection points, as the one indicated in (9).

For the whole desired path, it generates a new ruled

surface in the image space which actually is the specific

part of the target manifold intercepted by the two curves, as

illustrated in Fig. 6. It is called the distance surface

between the motions of the coupler constrained by the

crank and the slider ends on the target manifold, which can

be represented as

( , ) ( ) (1 ) ( )c st t t r b b (15)

where [0,1]t . Vectors ( )c b and ( )s b represent the

feasible motions of the coupler link to the crank and the

slider ends, respectively.

Finally, the area of the distance surface can be defined as

the structure error between the candidate mechanism and

the ideal one for the desired path. The area function can be

represented as 1 2

0 0( ) (1 ) ( ) ( ) ( )s c s s ce A dt t t d

b b b b (16)

Fig.6. Projection structure error on the

target manifold

VI. Optimization Algorithm

Since the structure error of the mechanism has been

projected on the target manifold as the area of the distance

surface, the problem of dimensional synthesis for

continuous path generation can be set up as a constraint

optimization problem as

min ( )

s.t.

se f

D

X

X (17)

where X represents the vector of design variables

consisting of the dimensions and location of the

mechanism. D is the design space which specified

according to a set of constraints, such as the limitation of

the link dimensions, the Grashof’s criteria, and so on.

To obtain the final optimum based on the given

parameters with high convergence rate, the gradient

descent algorithm is introduced to find the local minimum

of the structure error established on the target kinematic

manifold [37].

The gradient descent algorithm is also called steepest

descent. If the multivariable function ( )f X is defined and

differentiable in a neighborhood of a point 0A , then

( )f X decreases fastest in the direction of the negative

gradient of f at 0A , 0( )f A . Then, point 1A is

calculated through the following equation,

1 0 0( )f A A A (18)

where is step size, then

0 1( ) ( )f fA A (19)

Considering the sequence 0 1 2, , ,...A A A which can be

calculated by the following equation,

1 ( ),n 0n n n nf A A A (20)

we have

0 1 2( ) ( ) ( ) ...f f f A A A (21)

Applying the algorithm to our model of optimal synthesis

of the slider-crank mechanism for continuous path

generation, the problem can be set up conveniently. If

( )nf A converges to a value which is less than the

structure error limit we set, then nA will be chosen as the

desired design variables.

VII. A Numerical Example

In the above sections, a metric for the structure error

between the candidate mechanism and the ideal one for the

desired path has been established based on the generalized

kinematic mapping of constrained plane motions. Thus,

the closeness of the generated and the desired paths can be

evaluated by the area of the distance surface on the target

manifold. In this section, a numerical example for the

dimensional synthesis will be provided to demonstrate the

effectiveness and efficiency of the proposed method.

Unlike some literatures synthesizing the planar

mechanisms for some paths similar to their inherent ones,

in the numerical example we assigned a much more

general closed curve, an ellipse, as the target for the

dimensional synthesis of the slider-crank mechanism. The

major and the minor axis of the ellipse are 2 1.6a and

2 1b . And the center of the ellipse is located at the origin

of the initial reference frame where the major axis is

parallel to the X-axis of the initial frame. Thus, it can be

expressed as

cos

: , 0, 2sin

x at y b

(22)

Then, based on the gradient descent algorithm presented

in the above section, the dimensional synthesis can be

carried out for the desired path.

The feasible motions to the crank end and the slider end

can be expressed as

3 0 111

1

3 012

3 3

1 cos1( )

1 sin2

c

c

X x xXl

X y yX

X X

R (23)

3 21 0 0

0

32 0 0

33

' '

' '

cos sin11( d )

12 sin cos

c

c

X xX

X yX

X X

R(24)

where R is rotation matric defined as

cos sin

sin cos

R , and

0 0

'

2

denotes the

direction angle of the constrained line.

Meanwhile, the image surface of the ellipse curve is

expressed as

3 21

0

3 22

3 3

1 mcos1( )

1 n sin2

XX

XX

X X

C (25)

where 0

C is the center of the ellipse curve.

By substituting Eq.(25) into Eq.(23) and Eq.(24),

respectively, the expressions of the 2( )s b and 2( )c b can

be obtained. Because the analytical solution of the area

function se is difficult to be obtained, so discrete numerical

method is utilized to calculate sE , where 2 2

sis i ciE . sE and se are equivalent

because the variation tendencies of them are the same. ci

and si can be obtained according to 2( )s b and

2( )c b .Then gradient matrix is obtained as

0 0 2 00

( )'

T

s s s s s s

c c

E E E E E Ef

x y y x d

X (26)

where only 6 design variables’ derivatives are calculated

because only 6 of the 8 variables are independent.

Normally, if the 8 design variables are specified randomly,

the intersection of the target manifold and the circular

constrained manifold is not a piece of surface, as illustrated

in Fig.7. Then the crank can’t rotate a whole circle and

degenerate to a rocking bar.

Fig.7. The intersection of the target

manifold and the circular constrained

manifold

So once the target path is specified, 1l and

1 become

determined values to meet the constraint condition of the

slider-crank mechanism. The value of 1l +

1 equals to the

longest distance between the point A in Fig.4 and the point

on the target path. The absolute value of 1l -

1 equals to

the shortest distance between the point A and the point on

the target path. In this way, 1l and

1 can be obtained by

solving a quartic equation which represents the problem of

the longest and shortest distance between a point out of the

ellipse curve and the point on the curve.

Back on track, based on the initial design variables, by

substituting Eq.(26) into Eq.(20) repeatedly, the ideal

design variables nA are obtained.

The initial design variables and the optimization results

are illustrated in TABLE I and TABLE II. The projection

structure error on the target manifold before optimizing is

illustrated in Fig.8 and the one after optimizing is

illustrated in Fig.9. The generalized coupler curve of the

optimal is drawn in Fig.10 comparing with the desired and

the initial ones. Fig.11 shows the fast convergence of the

goal function to a near optimal solution in only 14

iterations. The elapsed time is 24.366292 seconds on a

workgroup of DELL OPTIPLEX980 (Intel(R) Core(TM)

i7 CPU 870 @2.93GHz, Ram 8.00GB)

0 0( , )Tx y 1l 1 2( , , )T

c c cx x y00

'( , )Td

(-0.62, -3.05) 0.52 (1.78, 6.89, -2.56) (-1.30, 0.01)

TABLE I. Initial design variables for

the dimensional synthesis

0 0( , )Tx y 1l 1 2( , , )T

c c cx x y00

'( , )Td

(-0.66, -3.00) 0.52 (1.64, 6.88, -2.60) (-1.15, 0.03)

TABLE II. Optimization results for

the dimensional synthesis

Fig.8. Projection structure error on the

target manifold before optimizing

Fig.9. Projection structure error on the

target manifold after optimizing

Fig.10. Generated coupler curve of the

optimization slider-crank mechanism to

the dimensional synthesis.

Fig.11. The changing of the optimal value

in the iterative process

VII. Conclusions

In this paper, we proposed a new strategy for the

dimensional synthesis of the slider-crank mechanism as a

generator for continuous paths. An effective and

meaningful metric for the structure error between the

candidate mechanism and the ideal one is set up in the

image space in terms of the conflict between the constraint

manifolds on the target one. Unlike the point-to-point

position comparison, the distance representation in the

three-dimensional projective space reflects the inherent

characteristics of the four-bar motions which perform

coupling movement combined both with translation and

rotation. Therefore, it is a more general description for the

structural error and states the problem of dimensional

synthesis for planar mechanisms in a simple geometric

interpretation with intuitive physical insight. The gradient

descent algorithm is employed to find the final optimum of

the dimensions of the mechanical elements for the design

requirement. The numerical result has shown the

effectiveness of the proposed approach.

Acknowledgement

This research is jointly supported by the National

Science Foundation of China (NSFC) under Grant

51305256 and Grant 11472172, and the National Basic

Research Program of China (973 Program) under research

grant 2014CB046600.

References

[1] C. H. Suh, C. W. Radcliffe, 1978. Kinematics and Mechanism Design, Wiley, New York.

[2] R. Hartenberg, J. Denavit, 1964. Kinematic Synthesis of Linkage, McGraw-Hill, New York.

[3] A. Erdman, 1981. “Three and Four Precision Point Kinematic Synthesis of Planar Linkages”, Mechanism and Machine Theory, 16 (5), pp. 227-245.

[4] C. Wampler, A. Morgan, A. Sommese, 1992. “Complete Solution of the Nine-Point Path Synthesis Problem for Four-Bar Linkages”, Journal of Mechanical Design, 114 (1), pp. 153-159.

[5] Z. Luo, J. Dai, 2007. “Patterned Bootstrap: A New Method That Gives Efficiency for Some Precision Position Synthesis Problems”, Journal of Mechanical Design, 129 (2), pp.173-183.

[6] A. Perez, J. McCarthy, 2004, “Dual Quaternion Synthesis of Constrained Robotic Systems”, Journal of Mechanical Design, 126 (3), pp. 425-435.

[7] X. Huang, et al., 2009. “Solving a Planar Four-Bar Linkages Design Problem”, In Proceedings of the 2009 IEEE International Conference on Information and Automation, June, 2009, Zhuhai/Macau, China.

[8] Brunnthaler K, Pfurner M, Husty M. Synthesis of planar four-bar mechanisms[J]. Transactions of the Canadian Society for Mechanical Engineering, 2006, 30(2): 297.

[9] Husty M L, Pfurner M, Schröcker H P, et al. Algebraic methods in mechanism analysis and synthesis[J]. Robotica, 2007, 25(06): 661-675.

[10] R. Fox, K. Willmert, 1967. “Optimum Design of Curve-generating Linkages with Inequality Constraints”, Journal of Engineering for Industry, 89 (1), pp. 144-152.

[11] J. Angeles, A. Alivizatoss, R. Akhras, 1988. “An Unconstrained Nonlinear Least-square Method of Optimization of RRRR Planar Path Generators”, Mechanism and Machine Theory, 23 (5), pp. 343-353.

[12] J. Cabrera, A. Simon, M. Prado, 2002. “Optimal Synthesis of Mechanisms with Genetic Algorithms”, Mechanism and Machine Theory, 37 (10), pp. 1165-1177.

[13] R. Sancibrian, F. Viadero, P. Garcia, A. Fernandez, 2004. “Gradient-based Optimization of Path Synthesis Problems in Planar Mechanisms. Mechanism and Machine Theory, 39 (8), pp. 839-856.

[14] A. Smaili, N. Diab, N. Atallah, 2005. “Optimum Synthesis of Mechanisms Using Tabu-gradient Search Algorithm”, Journal of Mechanical Design, 127 (5), pp. 917-923.

[15] I. Ullah, S. Kota, 1997. “Optimal Synthesis of Mechanisms for Path Generation Using Fourier Descriptors and Global Search Methods”, Journal of Mechanical Design, 119 (4), pp. 504-510.

[16] H. Zhou, E. Cheung, 2001. “Optimal Synthesis of Crank-rocker Linkages for Path Generation Using the Orientation Structural Error of the Fixed Link”, Mechanism and Machine Theory, 36 (8), pp. 973-982.

[17] H. Zhou, K. Ting, 2002. “Adjustable Slider-crank Linkages for Multiple Path Generation”, Mechanism and Machine Theory, 37(5), pp. 499-509.

[18] G. Figliolini, M. Conte, P. Rea, 2008. “Analysis and Synthesis of Slider-crank Mechanisms for Automatic Machinery”, In Proceedings of the 2008 ASME Design Engineering Technical Conferences, Aug. 2008, Brooklyn, NY, USA.

[19] I. Ullah, S. Kota, 1994. “A More Effective Formulation of the Path Generation Mechanism Synthesis Problem”, In Proceedings of the 1994 ASME Design Engineering Technical Conferences, September, 1994, Minneapolis, USA.

[20] A. Vasiliu, B. Yannou, 2001. “Dimensional Synthesis of Planar Mechanisms Using Neural Networks: Application to Path Generator Linkages”, Mechanism and Machine Theory, 36 (2), pp. 299-310.

[21] Y. Liu, R. Xiao, 2005. “Optimal Synthesis of Mechanisms for Path Generation Using Refined Numerical Representation Based Model and AIS Based Searching Method”, Journal of Mechanical Design, 127 (4), pp. 688-691.

[22] A. Smaili, N. Diab, 2007. “A New Approach to Shape Optimization for Closed Path Synthesis of Planar Mechanisms”, Journal of Mechanical Design, 129 (9), pp. 941-948.

[23] J. Buskiewicz, R. Starosta, T. Walczak, 2009. “On the Application of the Curve Curvature in Path Synthesis”, Mechanism and Machine Theory, 44 (6), pp. 1223-1239.

[24] G. Roston, R. Sturges, 1996. “Genetic Algorithm Synthesis of Four-bar Mechanisms”, Artificial Intelligence for Engineering, Design, Analysis and Manufacturing, 10 (5), pp. 371-390.

[25] R. McDougall, S. Nokleby, 2008. “Synthesis of Grashof Four-bar

Mechanisms Using Particle Swarm Optimization”, In Proceedings of the 2008 ASME Design Engineering Technical Conferences, Aug. 2008, Brooklyn, NY, USA.

[26] K. Watanabe, 1992. “Application of Natural Equations to the Synthesis of Curve Generating Mechanisms”, Mechanism and Machine Theory, 27 (3), pp. 261-273.

[27] J. McGarva, 1994. “Rapid Search and Selection of Path Generating Mechanisms from a Library”, Mechanism and Machine Theory, 29 (2), pp. 223-235.

[28] X. Zhang, J. Zhou, Y. Ye, 2000. “Optimal Mechanism Design Using Interior-point Methods”, Mechanism and Machine Theory, 35 (1), pp. 83-98.

[29] N. Nariman-Zadeh, M. Felezi, A. Jamali, M. Ganji, 2009. “Pareto Optimal Synthesis of Four-bar Mechanisms for Path Generation”, Mechanism and Machine Theory, 44 (1), pp. 180-191.

[30] J. Grünwald, 1911. “Ein abbildungsprinzip welches die ebene Geometrie und kinematik mit der räumlichen Geometrie verknüpft”, Sitzber. Ak. Wiss. Wien., 120, pp. 677-741.

[31] W. Blaschke, 1911. “Euklidische kinematik und nichteuklidische geometrie”, Zeitschr. Math. Phys., 60, pp. 203-204.

[32] O. Bottema and B. Roth, 1990. Theoretical Kinematics. Dover Press, New York.

[33] J. M. McCarthy, 1990. An Introduction to Theoretical Kinematics. MIT Press, Cambridge, MA.

[34] G. Chen, Z. Lin, H. Wang, 2010, “Generalized Kinematic Mapping of Constrained Plane Motions and Its Application to the Accuracy Analysis of General Planar Parallel Robots”, IEEE Transactions on Robotics (under review).

[35] W. Jun, A. Purwar, Q. Ge, 2010. “Interactive Dimensional Synthesis and Motion Design of Planar 6R Single-loop Closed Chains via Constraint Manifold Modification”, Journal of Mechanisms and Robotics, 2 (3), pp. 031012 (8 pages).

[36] M. Hayes, P. Murray, C. Chen, 2004. “Unified Kinematic Analysis of General Planar Parallel Manipulators”, Journal of Mechanical Design, 126 (5), pp. 866-874.

[37] Sancibrian R, Viadero F, Garcıa P, et al. Gradient-based optimization of path synthesis problems in planar mechanisms[J]. Mechanism and machine theory, 2004, 39(8): 839-856.