T. Subbian
Graduate Student.
D. R. Flugrad, Jr.
Four-Bar Path Generation
Synthesis by a Continuation
Method
Assistant Professor.
Mechanical Engineering Department,
Iowa State University,
Ames, IA 50011
A different approach for the synthesis of four-bar planar path generating mechanisms
is presented. A continuation method is used to solve the system of nonlinear equations
derived for the path generating problem. A brief description of the method is
providedfollowed by the development of equations representing thefour-bar linkage.
The implementation of the method for five position path generation is discussed in
detail and the solutions for two examples are presented.
Introduction
Synthesis of four-bar path generating mechanisms has been procedure that theoretically assures convergence without any
accomplished in the past by both graphical and analytical meth- initial solution estimates and that also produces the complete
ods. Lindholm (1969) applied the point position reduction solution set for the given system of nonlinear equations, protechnique to synthesize five and six position path generation vided they are cast in polynomial form. It is used here as a
linkages. Suh and Radcliffe (1983) used Newton's method in tool to solve the five position synthesis problem. The procedure
arriving at a solution. An analytical approach developed by can effectively be extended to six or more precision points
Sandor and Erdman (1984) successfully reduces the system of without difficulty. However, the computational expense inequations associated with the five position path generation creases significantly.
problem with prescribed timing to a quartic polynomial expresSimilar methods were implemented by Freudenstein and Roth
sion. Solutions for the quartic equation can then be determined (1963) for synthesis of geared five bar mechanisms and by Tsai
numerically.
and Morgan (1985) for the analysis of five and six degree of
Closed form solutions are effective up to four precision freedom manipulators. Roth and Freudenstein (1963) made
points and can be used as a tool to solve five specified points, use of a variation of the method discussed here for the synthesis
but for six points and beyond the nonlinear equations are of nine position, path generating, geared five-bar and fourdifficult to handle. Graphical methods have been applied for bar mechanisms. Nine points on the coupler path of an arsix precision points but with only limited success. Furthermore, bitrary mechanism were chosen as the starting points. A numthe solution set obtained by graphical methods is not complete. ber of subproblems were solved to move from these points
This leads to the application of numerical methods that can toward the nine prescribed positions, thereby developing the
theoretically be used to solve for a maximum of nine prescribed desired mechanism. The method to be used here starts with a
precision points. However, the practical limit has been five or simple set of equations involving the link lengths and moves
six specified positions. On the other hand, if exact precision toward the set of equations for the five prescribed precision
is not required, least square methods (Sarkisyan et al., 1973) points. In doing so, it moves from the starting solution of a
and selective precision synthesis (Kramer and Sandor, 1975; simple mathematical system to the desired result. The method
Kramer, 1979) have been used to minimize the deviation from is set up to determine all possible solutions for the system of
equations at hand while Roth and Freudenstein (1963) looked
a path described by more than nine points.
Numerical methods presently used for precision point syn- at one solution only.
thesis of path generating mechanisms exhibit two major shortThe continuation method and its application to the synthesis
comings:
of path generating mechanisms is described in detail in the
• Convergence depends on reasonably good approximations following sections. Nonlinear loop closure equations are developed and modified for efficient implementation. The nufor the solutions.
9
Current numerical methods converge to a single solution merical method is then applied to determine the solutions for
dependent on the initial estimates, and thus do not yield •two five position synthesis problems.
all designs that satisfy the constraints.
The continuation method (Morgan, 1987) is a mathematical
Continuation Method
Continuation methods constitute a family of mathematical
procedures used to solve systems of nonlinear equations. These
methods are particularly useful for dealing with sets of
Contributed by the Design Automation Committee for publication in the
JOURNAL OF MECHANICAL DESIGN. Manuscript received December 1989.
polynomial equations. To implement the approach, one starts
Journal of Mechanical Design
MARCH 1991, Vol. 113/63
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Similarly, for vectors Z 3 and Z 4 ,
Ziie'^J - 1) + Z 4 (e^ - 1) = 5j
(6)
Where Zk for k = 1, 2, 3, 4 are complex link lengths that can
be represented using real and imaginary terms in the form Zk
= Zkx + iZky. Upon substitution for Zk in terms of real and
imaginary quantities and expansion of the exponentials using
Euler's equation, the expressions may be written as
(Z u .+ iZly)(cos4>j - 1 + /sin0,)
• + (Z to + iZ2y)(cos7j - 1 + /sin?,) = 8jx + i8jy
(7)
and
(Zix + iZ3y)(cos\l/j - 1 + /sim/'y)
with a system of equations for which the solutions are known
and then marches along a path toward the solutions of the
original system. For simplicity, the procedure for a system of
two equations in two unknowns is described here.
Consider two polynomial equations given by,
F\(zuz2)
=0
(!)
F2{zu z2) = 0
To implement the method a simple system of two equations
in two unknowns is first considered (Morgan, 1987),
Gi = C 1 1 z 1 1 -C 1 2 = 0
(2)
2
G2 = C2lz2 -C22 = 0
The terms Cn, Cl2, C2U and C22 are randomly chosen complex
constants, and dl and d2 are the degree of functions Fx and
F2, respectively. The necessary homotopy functions are then
obtained by combining the two systems of equations (1) and
(2).
Hl(zl,z2,t)
= tFl + (\-t)Gl
=0
(3)
H2(zi,z2, t) = tF2 + {\-t)G2 = Q
Here t is called the homotopy parameter. When t = 0, the
homotopy functions reduce to the simple set of equations and
when / = 1, they represent the original system. Therefore, by
increasing t from zero to one, and by solving a number of
intermediate subproblems along the way, the solutions for the
original system are found. There are a variety of ways to move
from a t value of zero to one. In the present approach, basic
differential equations (BDE's) (Garcia and Zangwill, 1981) are
formed, and ordinary differential equations involving the variables with respect to t are determined. These differential equations are integrated numerically to determine the solution for
the given system of equations. The solution is then refined
using Newton's method. The procedure is repeated using all
the various combinations of solutions for the simple assumed
equations as starting points.
+ (Z4x + /Z4>,)(cosYy- - 1 + /sinYy) = 5jx + i8jy
(8)
To transform the equations into polynomial form, cos^y and
sin^ are treated as two independent variables, C4>j and Sfy,
respectively. In order for this to work, of course, a constraint
equation representing the relationship between sine and cosine
must be introduced. This is done for the angles ipj and T,- as
well. Upon substitution of these values into equation (7) and
separation into real and imaginary components, the following
expressions result,
Z\xC<l>j -Zlx — ZlyS(j)j + Z^CJj — Z-te — Z2yS7j = 8jx
ZlxS4>j + ZiyC<t>j -Zly
Z^J
+ Z^J
= 8j+Zi + Z2
(4)
Z 1 ( e ' ^ - l ) + Z 2 ( e ' ^ - l ) = 5y
(5)
or
+ Z2yCYj - Z2y = 8jy
zlxS\i,j+z3yapj - ziy+z4xsyj+z^cij
- z4>,= sjy
(1 o)
The constraint equations mentioned above are represented by,
Gj>] + S4>j=l
(11)
cy] + syj=i
(12)
aij+stf=i
(13)
Multiplication of the first equation (9) by Zly and the second
by Zix followed by subtraction of the two eliminates the C<t>j
terms,
(Z\x + Z\y)S4>j + Z^Z^Slj
- Z^Z^CJj
+ ZlxZ2yCjj + ZiyZ2yS7j
— Z[xZ2y + ZlyZ2x = Zlx8jy — Zly8jX
S<t>j = (Zlx8jy — Zly8jx — Z\xZ2xS7j + Z^Z^Clj
- zlyz2ysyj+zlxz27
(14)
— ZlxZ2yCYj
- Z 1 ^,Z 2A .)/(Z 1X +z ly )
(i 5)
The equation for C<$>j is obtained from equation (9) by following a similar procedure to eliminate S<j>j,
C4>j = (Zly8jy + Zlx8Jx + (Z\x + Z\y) - Z^Z^Slj
-
Z^Z^CJJ
— Z\yz2ycyj+zlxz2ysjj+zl7z2>,+Z1XZ2A.)/(Z1X+zly)
(16)
By substituting for S<j>j and C</>; in equation (11) and by rearranging the results we get,
2(.ZixZ2y - Z ^ Z ^ + Z2y8jx -
Z^j^Syj
~ 2(%2x + ^2y + Z ^ Z ^ + Z l7 Z 2y + Z^jx + Z2y8jy)Cyj
~ ^Ziy. — Z£j2y — LZ^Z-^
— LZ\yZ2y — 2Z2xoyx
-2Z2y8jy-2Zly8Jy-2Zlx8jx-8Jx-8jy
(17)
To simplify the relationship further, the following quantities
are defined,
A\j = Z lxZ2y - Z xyZ^ + Z2y8jx -
Z^jy
By = Z2x + Z2y + Z u Z 2jr + Z ^ Z ^ + Z ^ j . + Z2y8jy
D,j = 2Zlx8jx + 2Zly8jy + 8% + 8%
6 4 / V o l . 113, MARCH 1991
(9)
^ixGwj — Zix — Z}yS\//j + ZAxClj — Z 4x — Z4ySyj = 8jx
=
Development of Equations
A four-bar mechanism in two finitely separated positions
can be represented by the vectors shown in Fig. 1. Two loop
closure equations are written, one for each dyad (vector pair).
For vectors Zx and Z 2 we can establish a loop as follows,
+ Z^Jj
Similarly for equation (8) we obtain
(18)
Then equation (17) can be written as,
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2Ay SVj-Wtj Clj= - 2 5 „ - A y
(19)
A parallel development is pursued using equation (10) to eliminate i/j. This yields
2AySyJ-2ByCYJ=
-2By-D2j
(20)
The degree of these equations should be at least equal to the
degree of the original system. The equations considered for
the five position synthesis problem are,
G{ = C u Z lx — C12 = 0
G2 = C2iZ[y — C22 = 0
where,
Ay = Z3xZ4y — Z}yZ4x + Z4y5jX — Z4x6jy
G3 - C3(L3x — C32 = 0
By = Z4x + Z4y + Z3^Z4x + ZfyZfy + Z4xojx + Z4y5jy
Dv = 2Z3x5JX + 2Z3y5jy + 5% + 5]y
G4 = C 4 1 Z ^ - C 4 2 = 0
where the Cyi's'and Cy2's are randomly chosen complex constants. Each equation results in four complex solutions determined as follows,
Equations (19) and (20) are then solved for 57, and CV),
SYj = (By Dy-By
Dy)/(2(Ay
CYj=l+ (Ay Dy-Ay
By-Ay
Dy)/(2(Ay
By) )
By-Ay
By) )
(22)
Substitution for ST) and C7,- in equation (12) produces the final
function, a polynomial in the complex variables Z,, Z 2 , Z 3 ,
and Z 4 ,
Fj= {By Dy-By
Dy)2 + A(Ay By-Ay
X (Ay Dy-Ay
By)
Dy) + (Ay Dy-Ay
Dy)2 = 0
(23)
In the expression above j represents the individual displacements from an initial position. Therefore, a five position path
synthesis problem yields four equations. Since four complex
Z variables are involved, the maximum number of positions
is limited to nine. In the sections to follow equation (23) is
used in conjunction with the continuation method to solve five
position synthesis problems.
The Five Position Synthesis Problem
Having developed the equations for the path generation
problem, we need to implement the continuation method to
solve them. For a five position synthesis problem a set of four
complex polynomial equations is required. Since the number
of variables is greater than the number of equations, we have
the freedom of selecting values for four of the eight complex
components. Here, the coupler link vectors, Z 2 and Z 4 , are
specified to be, e + if and g + in, respectively, with particular
numerical values assigned to e,f, g, and h. The four equations
to be solved then reduce to the following form,
Fj= (By Dy-By
Dy)2+4(Ay
X (Ay Dy-Ay
By - Ay By)
Dy) + (Ay Dy-Ay
Dy)2 = 0
(24)
where j = 1, 2, 3, 4 and
Ay=fZix-
eZly +fbjx- ebjy
By = eZlx + / Z „ + e2+f2 + e6jx +/S y >
D[j = 2bjXZU + 2bjyZXy + 8jX + 8jy
Ay = flZ3x - gZ3y + hbjX - gdjy
By =gZix + hZiy + g2 + h2 + g8jx + hbjy
Dy = 28jXZ3x + 2djyZ3y + djX + 8jy
Upon substitution and simplification, it is apparent that the
degree of each of the equations is 4.
If we rule out the possibility of an infinite number of finite
solutions as well as an infinite number of solutions at infinity,
Bezout's theorem guarantees that (Morgan, 1987) the total
number of finite solutions and solutions at infinity should add
up to 256 counting multiplicities. This number is equal to the
product of the degrees of each of the individual equations; in
this case 44 = 256. Therefore, this system of equations will
produce a maximum of 256 complex solutions, real solutions
and/or solutions at infinity.
To implement the continuation method, it is necessary to
start with a set of polynomial equations (one for each of the
four unknowns Zlx, Zly, Z3x, and Z3y) which are easy to solve.
Journal of Mechanical Design
(25)
(21)
Zix = r\M[cos(2irk/4 + a,/4) + /sin(27rA:/4 + a,/4)]
(26)
where (rlt o^) are polar coordinates of the ratio Cl2/Cn and
k = 0, 1, 2, 3. Similarly Zly, Z3x, and Z3y can be determined.
The solutions thus obtained are grouped to form 256 combinations satisfying the simplified system of equations. These
provide 256 starting points for the continuation procedure.
Homotopy functions are defined next using the original
equations, the simplified equations and the homotopy parameter t as follows,
Hj = tFj+(l-t)Gj
= 0, j= 1,2,3,4
(27)
When ? = 0 the homotopy functions reduce to the simplified
set of equations, and when t= 1 they represent the equations
to be solved to obtain a four-bar linkage passing through the
five prescribed positions. Hence, on varying the parameter t
from zero to one we move away from the simple system to the
original system of equations. To do this we need to follow the
paths of the unknowns as we increment the t variable.
For path following, we determine the first order derivatives
of the variables with respect to / using the BDE's discussed
earlier (for example dZlx/df). Knowing the derivatives we can
integrate these first order differential equations with solutions
of the simplified system as initial values to obtain the desired
four-bar mechanism.
In order to determine the derivatives we need to evaluate
the extended Jacobian matrix of the homotopy function set as
indicated below,
3Zix
dHi
dZly
dHt
3Z3x
dHi
dZ3y
dHx
dt
dH2
dZu
dH2 dH2
BZly dZix
dH2
dZ3y
dH2
dt
dH3
3Zlx
dH3
8Zly
dH3
dZ3x
dH3
dZ3y
dH3
dt
dHA
dZu
dH4
dZly
dH4
dZ3x
dH4
dZ3y
dH4
dt
Use of this Jacobian matrix allows the ordinary differential
equations to be written as,
dZiX/dt, dZiy/dt,
dZ3x/dt
and dZ3y/dt
= ((-iy'+1
det(DHu]))/Den for./ = 1,2, 3,4, respectively. The det(DHm)
represents the determinant of the Jacobian matrix, DH, with
the y'th column deleted, and Den is the determinant with the
fifth column deleted.
Each of the solutions for the simplified system is a starting
point for a path, or an initial condition for the integration of
the differential equations. As the assumed system has 256
combinations of solutions, there are 256 paths and they proceed
to all of the solutions of the original system with multiple paths
converging toward repeated solutions. If the original system
has solutions at infinity, paths will diverge toward those solutions.
Thus by integrating the first order differential equations
MARCH 1991, Vol. 1 1 3 / 6 5
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Table 1
Prescribed precision points for example 1
Precision
Point
1
2
3
4
5
X
Y
0
0
-0.4535 -0.1730
-0.8385 -0.5228
-1.0840 -0.9358
-1.1794 -1.2957
Table 2 Continuation method solutions satisfying the prescribed conditions for example 1
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Z\j.
0.0009
-5.1960
-1.1852
1.9835
0.2998
-1.9668
-1.8773
-2.0930
-0.9050
-0.8583
0.6044
0.4295
1.9769
-2.2828
-2.5162
0.1328
0.1401
1.0298
0.0467
-0.0567
-0.0121
0.3723
-0.5502
0.7951
1.3553
Z\y
Zix
0.9997
-0.6386
1.8622
0.4997
-0.1413
1.6996
5.9712 -21.9874
1.6070
-1.8717
-0.8894
2.0674
-0.9879
2.0039
-0.7404
2.1363
-0.2183
2.2164
0.2599
2.5453
-0.8544
-1.4799
1.4901
1.1392
-4.9389
2.1435
-0.5300
2.1898
-0.3240
2.2195
-2.5004
2.1082
0.8555
0.8157
-1.0562
3.5144
0.9137
0.9355
1.0265
1.0408
0.9483
0.9808
0.8183
0.6657
-1.5196
2.0264
-0.5396
-8.6096
-0.7603
4.3469
Z3y
1.8974
0.4437
0.8254
23.4323
3.0586
1.4717
1.6386
1.3139
0.7913
2.5985
4.8112
-0.1488
0.0899
1.1878
1.1085
0.0314
-0.2319
-3.7175
-0.2420
-0.2781
-0.2618
-0.2566
0.0965
12.1160
-4.6379
COMMENTS
Grashof,crank rocker
Grashof.crank rocker
Grashof,double crank
Grashof, double rocker
Non Grashof,double rocker
Non Grashof,double rocker
Non Grashof,double rocker
Non Grashof,double rocker
Grashof
Grashof
Grashof
Grashof
Grashof
Grashof
Grashof
Non Grashof
Non Grashof
Non Grashof
Non Grashof
Non Grashof
Non Grashof
Non Grashof
Non Grashof
Non Grashof
Non Grashof
Fig. 4 Mechanism 3 in its first position with the coupler curve
(example 1)
Fig. 2 Mechanism 1 in its first position with the coupler curve
(example 1)
obtained, we can determine all the solutions (complex, real,
and those at infinity) for the original system of equations.
Among these, the real solutions turn out to be the only useful
ones.
Fig. 5 Mechanism 4 in its first position with the coupler curve
(example 1)
uation method. To synthesize the mechanism the links Z2 and
Z4 were chosen to be 1.1344 + zT.3975 and -1.7287 + /'0.5016,
respectively, and the complex link vectors Zx and Z3 were
Example 1
computed. A total of 256 real, complex and solutions at infinity
A four-bar mechanism was designed to pass through the were obtained from the 256 starting points. Of these only
five precision points listed in Table 1 by applying the contin- twenty-five solutions which are listed in Table 2 were real.
Examples
66 / V o l . 113, MARCH 1991
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Fig. 6 Mechanism 5 in its first position with the coupler curve
(example 1)
Fig. 8 Mechanism 7 in its first position with the coupler curve
(example 1)
Fig. 7 Mechanism 6 in its first position with the coupler curve
(example 1)
Thirty-three pairs of complex conjugate solutions were obFig. 9 Mechanism 8 in its first position with the coupler curve
tained, and the rest either diverged to infinity or resulted in
(example 1)
very large link lengths that caused the procedure to be aborted.
Of the twenty-five real solutions found, the first five did not
exhibit any branch or order defects and so these are the only from the point where it is close to position 1, it will pass through
useful solutions. Three more solutions came close to satisfying 2, 3, 4, 5 in the specified order. It should be noted, however,
the prescribed conditions, and they are discussed here as well. that when the coupler is at this new starting point the configOf the eight useful solutions only the first four satisfy Grash- uration is different from that shown in the figures. That is,
of's criteria. All eight are shown in Figs. 2 through 9. Among the configuration is not consistent with the arbitrarily specified
them, the first two solutions resulted in crank rocker mech- values for Z and Z . Even so .these three linkages might still
4
anisms and the third was a double crank mechanism. Mech- be usable to 2solve the
path generation problem.
anism 4, a double rocker mechanism, has very large link lengths.
The above designs then provide us with eight four-bar mechHowever there are situations in which such dimensions might anisms for the task. Figures 2 through 9 give the mechanisms
be feasible, and so it is listed here as a valid solution. Solution in their initial position with the coupler curve superimposed.
5 turned out to be a double rocker mechanism as well.
More mechanisms can be obtained, giving the designer more
Mechanisms 6, 7, and 8 at the outset looked like multiple options, by changing the link vectors Z and Z . For each of
2
4
solutions (repeated solutions) but on further analysis were iden- the link vectors chosen the procedure is carried
out for the 256
tified as being distinct and geometrically isolated. These so- starting points.
lutions are discussed here despite the fact that they do not
quite satisfy the specified order for the precision points. From
Example 2
Figs. 7, 8, and 9 it is clear that the coupler passes through the
specified points in the order 1, 5, 4, 3, 2 and then comes close
The procedure was also applied to design a mechanism passto passing through 1 again. If the device is driven backwards ing through five points lying on a straight line (Table 3). The
Journal of Mechanical Design
MARCH 1991, Vol. 113/67
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Table 3
Prescribed precision points for example 2
Precision
Point
1
2
3
X
0
-3
-6
Y
0
0
0
4
-9
0
5
-12
0
Table 4 Continuation method solutions satisfying the prescribed conditions for example 2
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2i,
21.725
37.915
21.806
39.436
23.716
-44.669
35.249
42.292
17.125
11.872
-0.016
-22.630
-3.394
-0.184
-27.774
-9.345
1.433
12.722
4.627
0.148
Z\y
-29.334
-53.216
-58.840
-55.532
-64.374
50.802
-49.166
-59.879
-21.851
-12.500
4.074
23.825
3.400
4.108
30.043
-10.556
3.784
-15.122
-9.862
6.971
z„
-17.410
-51.574
2.966
-49.675
2.479
55.339
-53.763
-41.478
-12.011
-8.037
2.128
10.239
-1.106
3.298
16.779
-2.861
1.503
-7.535
2.871
1.863
Zzy
-26.083
-34.365
-17.521
-32.166
-17.364
-4.733
-38.150
-25.550
-24.651
-25.530
-9.682
-16.170
-9.701
-13.797
-14.634
-32.327
-10.734
-23.276
-18.844
-14.603
COMMENTS
Grashof,double rocker
Grashof,doubIe rocker
Grashof,double rocker
Grashof, double rocker
Grashof,double rocker
Grashof,double rocker
Grashof,double rocker
Grashof,double rocker
Non Grashof,double rocker
Non Grashof,double rocker
Non Grashof,double rocker
Non Grashof,double rocker
Non Grashof,double rocker
Non Grashof,double rocker
Non Grashof,double rocker
Non Grashof,double rocker
Non Grashof,double rocker
Non Grashof,double rocker
Non Grashof,double rocker
Non Grashof,double rocker
Fig. 11 Mechanism 9 in its first position with the coupler curve (example 2)
Fig. 10 Mechanism 1 in its first position with the coupler curve (example 2)
links Z2 and Z4 were chosen to be 9.7226 + il. 1722 and 2.572,3
+ (17.5314, respectively. Twenty usable solutions were obtained without any order or branch defects. These are listed
in Table 4. Of these twenty, eight were Grashof double rocker
mechanisms, and the rest were non-Grashof double rocker
linkages. No usable crank rocker solutions were obtained. One
of the Grashof double rocker mechanisms is shown in Fig. 10
with its coupler path and precision points. Figure 11 depicts
a non-Grashof double rocker mechanism. A couple of the
6 8 / V o l . 113, MARCH 1991
Fig. 12 Mechanism 19 in its first position with the coupler curve (example 2)
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solutions produced quite unexpected results. One of them is
displayed in Fig. 12 which shows the coupler point passing
through all five precision points in the proper order, but a full
loop is executed between the third and fourth points. For that
reason this particular design might not be acceptable.
Computations for the above examples were carried out in a
VAX/VMS (11/785) environment. The average CPU time for
each starting point was 26.65 seconds for the first example and
46.56 seconds for the second. When the solution diverged to
infinity the CPU time was 60 seconds on average for example
1 and 100 seconds for example 2. For converging solution paths
the CPU times were considerably less. The values were approximately 20 and 30 seconds for the first and second problems respectively.
Conclusions
The continuation method, as indicated by the above examples, can be used to systematically solve a five position path
generation problem. Newton's method could be applied, with
a random number generator to pick the initial guesses, but we
would not be assured of a complete solution set. Thus the
continuation method is more reliable than the mathematical
procedures currently available.
When a set of polynomial equations is solved using the
continuation method, we should obtain real solutions, complex
solutions, and solutions at infinity. As mentioned earlier, the
real solutions are the only useful ones for mechanical design
applications. The solutions at infinity and complex solutions
are of no use. If the procedure is working properly, the complex
solutions should appear with their conjugates. This was observed in solving the two example problems. Further work is
planned to perfect the technique for path generation linkage
Journal of Mechanical Design
synthesis. Extension of the continuation method to a wide
variety of other synthesis problems is also a very promising
possibility.
References
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MARCH 1991, Vol. 113/69
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