Mechanism and Machine Theory Vol. 16, p p . 227-245, 1981
Printed in Great Britain.
0094-114XI81/030227-19502.00/0
Pergamon Press Ltd.
=,
•
Three And Four Precision Point Kinematic Synthesis of Planar
Linkagest
Arthur G. Erdman~:
Received for publication 31 October 1980
Abstract
Dyads modelled by complex numbers are presented in several different equation forms
for three prescribed positions of either motion, path generation with prescribed timing
or function generation. Different strategies are suggested depending on how the free
choices are best utilized. Interactive computer techniques are suggested for both the
three and four prescribed position cases. Several examples are presented to help
illustrate each of the design situations.
1. The Standard Form
TIlE GREATmajority of planar linkages may be conceived as combinations of vector pairs called
dyads. For example, the 4-bar linkage in Fig. 1 can be perceived as two dyads: the left side of
the linkage represented as a solid vector pair (W and Z), and the right side represented by the
dashed dyad. (Other vectors between the coupler pins AB and the fixed pivot pins AoBo are
easily determined when the dyad pairs are known.) The tracer point of the coupler link moves
along a path from position P~ to Pi defined in a fixed (but arbitrarily located) complex
coordinate system by rj and r/respectively.
All vector rotations are expressed relative to the corresponding vector position when the
dyad is drawn at the first prescribed precision point (Figs. 2 and 3). Rotation fl2 is the angular
change of vector W from the first position to the position associated with the second precision
point, f13 is the rotation from the first to the third position, etc. Angles ai are similarly the
prescribed rotations of the Z vector relative to its first position.
/y
rH ~
X
Figure 1. 4-bar: Two dyads.
tPresentted at Kinematic Geometry of MechanismsWorkshop, Sydney, Australia, 11-15 June, 1979.
SProfessor of Mechanical Engineering,University of Minnesota, Minneapolis,MN 55455, U.S.A.
227
228
0
/
i ~.__ Movingplane /g
~'Y
\
\
8
°,vo,
Figure 2. Three finitely separated positions.
Figure 3. Vector representation of a dyad.
A dyad loop closure equation form is derived by summing the vectors clockwise around the
loop containing We ~aj,Ze~% r s, rt, Z, and W (Fig. 1):
We~t~j+ Ze~j - r s + r] - Z - W = 0
(1)
W ( e i ~ j - 1) + Z ( e i a j - I) = 8 i
(2)
or
where the displacement vector along the prescribed path from P] to Pi is 8i = rj - r]. This makes
the coordinate system of Fig. 2 arbitrary--for simplicity it will be located at the first precision
point (Fig. 2).
Equation (2) is the so called Standard Form (Refs. (9) and (12)). This equation is simply the
vector sum around the loop describing the first and jth position of the left side of the 4-bar
linkage. Equation (2) is in the "standard form" if 8s and either a i or fls are prescribed (the rest
of the variables must be unknown). This requirement is fortunately consistent with the
definitions of the standard kinematic synthesis tasks--Motion Generation, (or Rigid Body
Guidance) Path Generation with Prescribed Timing, and Function Generation. The 4-bar as well
as the large majority of planar linkages may be modeled with groups of standard form equations
for these tasks. For dyad synthesis, the second column of Table 1 summarizes the prescribed
quantities in the standard form equation.
Table 1.
For 3 prescribed positions
Task
Motion Generation
Path generation/
prescribed timing
Function generation*
Alternately prescribed
dyad angles
prescribed
quantities
Arbitrarily
Picked angle
Angle varied
0° -~ 360°
82, 83, a2, ~3
3z
/33
82, 83,/32,//3
82, 83, 02, 33
82, 83 and
either az, f13
or ct3, f12
a2
a2
a3
a3
either as
or ot2
either 32
or/~3
* The delta's are prescribed along a circular arc timed with the prescribed output
link rotations.
Number of Prescribed Positions vs Number of Free Choices
Since there are only a finite number of adjustable parameters which completely describe the
linkage of Fig. 1 (the two components of each independent vector of the moving links) it has
been shown that there are only a finite number of prescribable parameters which can be
imposed in a kinematic synthesis effort. Table 2 reviews how to determine the maximim number
of positions of synthesis for the 4-bar motion generation dyad. Tracer point displacement
229
Table 2.
W(e ~ej- I)+ Z(e ~=j- 1)= 8j
Number of
Number of
Number of
positions (/) real equations real unknowns
2
3
4
5
2
4
6
8
5(W, Z,/]2)
6(above +/]3)
7(above +/]4)
8(above +/]5)
Number of
free choices
Number of
3
2
1
0
0(~)
0(#)
O(mt)
Finite
solutions
vectors 8j and coupler rotations a t are prescribed for this task. Although Table 2 is based on
the left side of the linkage of Fig. 1 (eqn 2), the right side of the linkage will yield the same
results. The table shows that for two positions there are two independent scalar equations
contained in eqn (2) (the real and imaginary parts of the equation). This system of two real
equations contains five unknowns: the vectors W and Z (which represent the left side of the
4-bar) and the input rotation f12. If three of the five unknowns are chosen freely, then the
system is easily solved for the remaining two unknowns. Since in general there are an infinite
number of choices of each of the three free variables, the number of possible solutions for the
two position problem is on the order of infinitely cubed, symbolized by 0(~) 3.
In the case of three prescribed motion positions there are two more scalar equations but
only one more scalar unknown (/33). Thus two free choices must be made and ideally 0(~) 2
solutions available. Each additional prescribed position in Table 2 adds two more scalar
equations and one more scalar unknown so that for four positions there is one free choice and a
single infinity of solutions and for five prescribed positions there are no free choices allowed
and at best a finite number of solutions. Five prescribed positions is the maximum number of
precision points possible for a dyad solution.
The system of dyad equations for three positions is
W(e e2 - 1)+ Z(e i~2- 1)= 82
W(e ia3- 1) + Z(e i l 3 - 1) -~83.
(3)
If the two remaining unknown angles are picked arbitrarily, then/32,/33, a2, a3, 82 and 83 are
known and system (3) is a set of linear complex equations that can be solved for W and Z (the
vectors representing the dyad in its first position) by Cramer's Rule. By varying the free choices,
up to two oo's of solutions for W and Z may be derived.
The results of Table 2 coincide with a geometric description of the motion synthesis for a
dyad. Figure 4(a) shows a moving plane in two positions. An arbitrary point "A" is picked in
the moving plane as a moving pivot or circle point. The perpendicular bisector of Al, A2 locates
a line of possible ground pivots or center points. With point Ao chosen, vectors Z and W
p,
W
Figure 4.
m
230
represent one of three infinites of dyads for two prescribed positions (the x and y coordinates
of A and the location of Ao along the perpendicular bisector).
For three prescribed positions (Fig. 4b), point "A" is again arbitrarily picked. Two
perpendicular bisectors are generated, one between A1, A2 and the other between A2, A3. The
intersection of these midnormals locates the center point Ao that will allow A to rotate through
its three positions. Thus there are two infinities of solutions for Z and W for three prescribed
positions of the moving plane. Therefore there is a correlation between the two methods of
modeling the two and three position motion generator dyads.
If an arbitrarily chosen fourth position were added to Fig. 4b, one would discover it hard to
find point A such that it's four positions would lie on a circle. Burmester and others who
followed[2, 4, 5, 7-9, 13-15] showed us that there in fact are points A whose four positions lie
on a circle, an infinite number. These points lie on the circle point curve and their corresponding centers lie on the center point curve.
Three Precision Point Standard Form Example
Completion of an assembly line requires the synthesis of a motion generator linkage to
transfer boxes from one conveyor belt to another as depicted in Fig. 5. A pickup and release
position plus an intermediate location are specified. A 4-bar is picked for this task (two
solutions of eqn 3). The prescribed quantities which are fed into the solution of eqn (3) are:
~ 2 = - 6 + Ili
a2=220
/~3= - 17 + 13i
a3 = 68°.
The free choices for each side of the 4-bar are
left side
right side
f12 = 90°
f12 = 40°
f13 = 198°
f13 = 73°.
The left side W rotations are chosen as large to try to obtain a crank rocker solution. Figure 6
shows the resulting 4-bar synthesized this task.
4
Conveyor#
I
Figure 5.
Ground Pivot Specification for Three Prescribed Precision Points[11]
In some cases it is advantageous to make different use of the two free choices of Table 2 for
the three finitely separated precision point dyad equations. Although the standard form allows
us to prescribe continuous crank rotations, with two free choices available, one of the vectors
231
- .
'%,
/_
z
Figure 6.
can be picked freely instead. In fact, by forming the dyad equations differently, a ground or
moving pivot location may be specified. Referring to Fig. 3, the loop closure equations can be
rewritten in the form:
W(l)+ Z(1) = - R
We'O2 + Ze~2 = - R + 82
(4)
We~O3+ Ze~"3 = - R + 83.
If
R I ~ -R
R2- - R + 8 2
R3 -= - R + 83
then
W(1) + Z(1) = R~
We~O2+ Zei'~2 = R2
(5)
We~a3 + Zei~3 = R3.
Table 2 still applies to eqn (5) so that two free choices are still available. Besides specifying 82,
83, a2, and a3, Ri will be arbitrarily picked. Thus R2 and R3 are now known.
Viewing eqn (5) temporarily as three complex equations with two complex unknowns (W
and Z), then this dependent set only has a solution for W and Z if the determinate of the
augmented matrix is identically zero:
1
eia2
eia3
1
R1
ei,~2 Rz = 0.
ei,~3 R3
(6)
Equation (6) represents a complex equation with two unknowns,/~2 and/~3. Since the unknowns
are in the first column, the determinant is expanded about this column:
(R3e~'~2- R2e ''3) + ei'2( - R3 + Rle ;'~3)+ ei~3(R2- Rle i~2)--- 0
232
or
1)1 + 1)2 ei02 + O3 eio3 = 0
(7)
where
D1 = R3e i~2 R:e i~3
-
D2 --- R l e i'3 - R3
D3 = R2 - R1e ic~2.
A simple graphical construction aids in solving eqn (7). Figure 7 shows an arbitrary solution
where the knowns 1)b !)2, and D3 are represented as vectors. Notice that 1)3, 1)~, and D: are
connected together but vector D~ is fixed. Note in eqn (7) that vectors 1)2 and !)3 are multiplied
by e i~2 and e i~ respectively. Recognize these quantities as rotation operators. If the vectors
formed a closed loop then eqn (7) will be satisfied. Thus 1)2 and Da are rotated (about their
connection with 1)]) until they meet. The rotations required to close the loop. are then/32 and/33.
Notice that there are two solutions for the triangle;/32,/33 and/32,/33. On set of solutions will be
trival however (/32= a2 and/33 = aa). Once the non-trivial set of/~'s is found (either graphically
or analytically based on Fig. 7), W and Z can be found from any two of eqns (5).
D2
Da
Figure 7.
Ground Pivot Specification (GPS) Example
An engineering student who had recently purchased a tape recorder unit for his sports car
was concerned with possible theft of his investment. Therefore, the student envisioned
synthesis of a 4-bar linkage to hide the tapeplayer behind the glove compartment when not in
use. Figure 8 shows a cross section of the area of interest including the glove compartment and
heating duct as well as the three prescribed positions for the tape unit. Since there is a small
acceptable area for possible ground pivots, the development above is used as a synthesis tool.
Two ground pivot locations are chosen (Ao and Bo) as shown in Fig. 8. The three position
vectors for the first dyad of the 4-bar are:
Rj = 2.14- 3.68i
R2 = 4.46 - 0.63i
R3 = 4.10 - 3.22i
while the rotations of the coupler (the tapeplayer) are
a2 = 50,7°
a3 = 91.9°.
233
\
\
Glove
Compartment
\
Line
X
of visibilit,y~ ~
Tope Player
Figure 8.
From eqn (7)
D1 = - 0.377 + 0.734i
D2 = - 0.493 - 0.959i
D3 = + 0.257 + 0.045i.
From the graphical solution of eqn (7)
/32 = 58.05o
/]3 = 122.650.
The first dyad can now be determined by inserting /32 and /33 into eqn (4) and solving
simultaneously. Thus:
W
= - 1.42-
1.45i
Z = 3.56 - 2.23i.
The second side of the linkage may be obtained by picking another ground pivot Bo, and
following the same procedure. Figure 9 shows one possible solution including the dyad
synthesized above.
Interactive Graphics and GPS
Even though the GPS method allows specification of the ground pivot location, most
applications require regions of acceptability. Thus the GPS technique was programmed [19] on a
Tektronix 4013 terminal so that the user could specify a range of possible ground pivots and ask
the computer to display the corresponding moving pivots. Then the user could interactively
synthesize dyads to form his/her linkage.
Mkff Vol. 16, No. 3--E
234
%
Figure 9.
interactive GPS Example
Dyads are to be synthesized for the following motion generation positions
R~ = 2.5 + 3.75i
R2 = 1.1 + 4.4i
R3 =
0.8 + 4.4i
-
a 2 = 20 °
a3 = 58 °.
The ground pivots were picked to be along the line x = - 1.17 between -5-< y < 5. Figure 10
shows the ground pivot curve, the corresponding moving pivot curve and the prescribed
precision points. Notice that the curves have letters A, B, etc. which correlate with the
Plot of M - / ( curves (set I)
M
K. . . .
6.00
OA
5B
IOC
15D
20 E
25F
30 G
35H
40 I
45J
0
4.00
0
~
/.~-5
14
21
19
24
27
31
:35
42
48
F //
2.00
0.00
-2.00
B
A
C
O
E
F
G
H
I
J
-4,00
"6.00
i
-6.00
Figure 10.
I
-4D0
,
I
-2C0
,
I
OO0
,
I
2.00
i
I
4.00
i
6.00
235
corresponding fixed and moving pivots. Notice also the lines drawn between the curves. These
were located using the cross-hairs on the Tektronix screen. They also identify corresponding
fixed and moving pivots. Any one of these lines extended from its moving pivot to the first
precision point (the square) will display a possible dyad solution. Of course the computer can
be programmed to display ground pivots in any fashion desired.
The GPS procedure allows the specification of ground pivots--an improvement over the
standard form approach for problems with tightly constrained ground pivot locations. Further
improvement is possible, however. Readjustment again of the dyad equations will yield a
solution procedure which allows unique display of numerous circle and center points at a time.
Circle and Center Point Circles for Three Prescribed Positions
Equations (4) are vector loop equations for three finitely separated positions shown in Fig. 3.
It is of interest to examine the array of ground and moving pivots described by vectors R and
- Z with respect to the first precision point.
In order to eliminate W from eqn (4), W is expressed in terms of Z and R in the first
position.
W=-R-Z.
(8)
Equation (8) is substituted into the 2nd and 3rd position eqations of (4) to yield
Z(e i°'2-
R(e ig2- 1) =
e ith) -
82
R(e ia3- 1) = 83.
Z ( e i~3 - e ~#3) -
(9)
Using Cramer's Rule the vector between the coupler tracer point
is
R=
Pl
- 63(e/"2 - ere2)- 8 2 ( e i ° 3 - e~3)
+ (e;'~2_ e~#2)+ (et#~ _ e~3) + e~S2e~3_ e~#~e~2•
and the fixed plane pivot m
(1o)
The vector between Pl and the circle point kl may be determined.
-- Z =
-
~ 2 ( e i'83 -
1) +
~ 3 ( e i'62 -
1)
e i~2 - e i~s3 -b e ia3 - e ia2 - eiO2e ia3 -b e i a 2 e i~3"
(ll)
There are two free choices to be made in eqns (10) and (11) (Table 2). Table 1 (last two
columns) suggests a strategy for choosing the remaining angles. One angle will be picked while
the other is varied between 0* and 360°. With this method in mind, eqns (10) and (11) may be
rewritten in the form
R-
aO+b
cO+d
(12)
eO+t
Z = gO +-~
(13)
where a-~ h are known and
0 = e i°
where 0 = the "angle varied" from Table 1.
It has been found[l, 16] when 0 varies from 0 to 2~r, # describes the unit circle. Equations
(12) and (13) are tantamount to the following sequence of transformations:
p(O) = aO,
a stretch rotation,
q(O) = aO + b,
a change of origin,
236
r(O) = cO,
another stretch rotation,
s(O) = cO + d,
another change of origin, and
_
q(O)
t(0)- s(0)
(15)
a "bilinear mapping".
Since both q(0) and s(0) are circles, it can be shown that t(0) is also a circle[3]. Thus, it is
seen that the loci of R(0) and -Z(0) are circles which, in the limit can become straight lines.
The complex constants a through h are found by appropriately rearranging eqns (10) and (11) in
the form of eqns (12) and (13). The centers of the circular loci Cu and Cr can be found directly
from the constants a through h[1], or by evaluating R and - Z at three 0 values, yielding three
points each which define the circles.
The circles of circle and center points were discovered during the displaying of ground and
moving pivots on the Tektronix 4013 graphics terminal, Loerch[1].
Many new observations have been made concerning the three precision point dyad synthesis
case[l, 16]. Some of these are:
1. For motion generation, all the ground pivot circles M intersect the poles P13 and P23
while all the moving pivot circles kl intersect pole P13 and image pole P~3 (Figs. 13-15,
discussed below).
2. For path generation with prescribed timing, all the M circles intersect the so called
pseudo-polest 813 and 823. The kl circles do not intersect at a single point but are all tangent to
two non-existence circles (one of which surrounds the first precision point). For example in Fig.
11, with 8j = 2 + 2i, 83 = 4 + 4i,/32 -- 0.5 rad and/33 = 1 rad, the M and K circles are displayed
for several values of a 2. The K circles (dashed) are tangent to two "forbidden regions" labeled
as "F". Notice also several possible W links are sketched in the diagram.
3. The dyad synthesis with opposite angles prescribed exhibit somewhat similar characteristics as path generation with timing. Figure 12 shows the K circles for the example
8 2 : 2 + 4i, 83 = 1 + 2i, a2 = - 1 rad and/33 = - I rad. The forbidden regions are signified by the
letter "F".
4. Superposition of two three position, motion generator subproblems with two precision
350" 17.35
3" 51,68
Figure 11.
tPoles with rotations/3 rather than a.
237
Figure 12.
20"
-20"
.70 °
+1
M Circles
{ ÷ - A Precision Point)
Figure 13.
-40"
238
K Circles
Figure 14.
K'(24)
~
'
K(23)
Rgure 15.
points common will generate exactly the classical construction for the four precision point
Burmester Curves based on the opposite pole quadrilateral[2].
5. Superposition of either two or three of the three position, path generation with prescribed
timing subproblems will yield non-existence areas for the four or five precision points problems
respectively.
The example below will illustrate a dyad motion generation example.
Motion Generation Example
In Figs 13-15 the intersections of the " + " of + :, + 2 and + 3 mark the prescribed positions
of the tracer point P, with 82 = 2 + 2i, 83 -- 5 + 2i, a2 = 60° and a3 = 120°. The M and Km circles
are generated for several values of the varied angle/~2 with/33 ranging from 0 to 2~r. These
circle-point circles and center-point circles are sufficient to display the properties of the
diagram. Figure 15 shows two dyad solutions which will form a 4-bar solution (f12= 24° and
/~2-- -23°). The simple steps required to construct a set of circular M and K loci for motion
generation are:
1. Find the circle-intersection poles: Pt3 and P23 for the M circles; P~3 and P~3 for the K,
circles.
239
2. Bisect the lines between the intersection pole pairs to find the lines of centers for the M
and K~ circles.
3. For each value of the varied angle//2, lay off the circle centers so that ,t;.P~3CMP23 = ~2
and ' ~ P I 3 C r P ~ 3 = a s - ~2.
4. Draw the circle pairs through the intersection poles with centers CM and CK. It can be
shown that the complex number expressions for the pole are:
82
P12 = ~1 - , a 2
83
e l 3 = -1 -- a 3
and
(15)
P23 = 8 3 a 2 -- 82at3,
~lf2 -- 0'3
where Pii is the vector from the origin of R,. to the pole Pij and where aj = e ia~.
Slider and Rotating Slider Block Solution
A search of the kinematics literature only yielded a couple special cases of these circles
considered by Beyer[4]. He noticed that the image pole triangle P~2P~3P~3has a circumscribing
circle that represents the locus of moving pivots with links of infinite length, or sliders. The
moving pivot can be replaced by a slide, moving along a line from its location on this circle
through the altitude intersection H of the pole triangle (See Fig. 16). The fixed pivot can be
considered at ooperpendicular to this line. A second case involves the circumscribing circle C of
the pole triangle itself, which is the locus of fixed pivots with moving pivots at oo. The infinite
links may be replaced by rotating slides at the fixed pivots; the initial sliding axis of the moving
pivots being a line from the circle, through the image H' of the altitude intersection H. The first
circle is the K circle for 4)2 = 0; the second is the M circle when a2 =//2. Figures 17(a) and (b)
show examples for these special cases.
~
2
Nt--e~
23
/
/
/
/
/
/
/
I
I
I
Figure 16.
Four Proscribed Posltion~The Standard Form
Table 2 shows that for four prescribed dyad motion positions that there is an infinite number
of solutions based on changing one free choice. Returning to the standard form, an additional
complex equation must be added to eqns (3) for the fourth position:
W(e i~'- 1) + Z(e i~'- I)= 84.
(16)
Consider eqns (3) and (16) as a set of three complex equations in two complete unknowns, Z
and W. In order for this set of equations to be valid, one of the complex equations must be
linearly dependent on the other two, that is, the coefficients of the equations must satisfy
certain "compatibility" relations. These relations will lead to the solution of the above
240
Fixed Sliding Axis
~'~,,..Gz~---- SIidirlg Pivot
Ist C~se
P23
2nd Cose
Io)
Figure 17.
equations. This system can only have a solution if the rank is 2. Thus the determinant of the
augmented matrix M of this system must be equal to zero.
(ei"2-1)(eia2-1)82I
M= (ei~3-1)(eia3-1)83 =0.
(ei~,-1)(ela4-1)84
(17)
If /32 is arbitrarily picked, then /33 and /3,, remain as unknowns. But eqn (17) is a complex
equation (containing two independent scalar equations) and may be solved for/33 and/34. Notice
the similarity between eqn (17) and eqn (6). Equation (17) may be rewritten as:
eilhA2 + e~3A3+
ei~4A4 + A I = 0
A2 = cofactor of (e i~2- 1)
A3 = cofactor of (e ia3- 1)
A4 = cofactor of (e ~a' - 1)
(18)
A 1 =-A
(19)
2 - A 3-~1t4"
Equation (18) is almost the same as eqn (7), especially once a /32 is chosen. Therefore the
graphical solution of eqn (18) with eqn (19) in mind is similar to Fig. 7 except that a
"compatility 4-bar" geometric solution exists [9, 17, 18]. The solutions for/33,/34, Z and W come
in pairs for all values of/32 that the compatibility linkage closes.
LINCAGES (Linkage INteractive Computer Analysis and Graphically Enhanced Synthesis Package)[17]
briefly
The following
describes an interactive computer package which uses graphics output
to help synthesize and analyze planar linkages. This program is available by contacting the
author at the Department of Mechanical Engineering, University of Minnesota.
The LINCAGES Package is the first such program to be generally available over telephone
lines in a time sharing mode and to be designed to make optimal use of graphical output from
either a teletype, graphics CRT or a micro computer. The computer, the designer and graphics
output are used strategically to maximize the utility of each. Synthesis programs (based on the
theory described above may be applied for either 3, 4, or 5 finitely spaced precision points for
combinations of motion, function, and path generation with prescribed timing.
In the four precision point option, LINCAGES uses 132 as an independent parameter to
generate solutions to the synthesis equations. The two solutions for each value of /32 are
assigned so that each solution dyad may be designated by a 1/2 value (0° - 360°) and a set number
(1 or 2).
241
LINCAGES Example: Motion Generation for Four Precision Points
The assembly of a filter product begins by forming the filtration material into what is known
as a filter blank. Next the filter blank is placed by hand onto a mandrel. This mandrel is part of a
machine that completes the assembly of the filter. The objective of this problem is to design a 4-bar
linkage for removing the filter blanks from the hopper and transferring them to the mandrel.
Figure 18 diagrams the kinematic synthesis objective. A gravity feed hopper holds the
semi-cylindrical filter blanks with the diametral plane surface initially at a 270 angle from
vertical. The blank rotated around until this plane surface is horizontal on the mandrel. The
position of the hopper can be moved about within the area indicated, although the angle must
remain at 27°.
At the beginning of the "pick and place" cycle, it is desirable to pull the blank in a direction
approximately perpendicular to the face of the hopper. To prevent folding the filter blank on the
mandrel, it is necessary to have the rotation of the blank completed at a position of 5 cm
approx, above the mandrel and then translate without rotation onto the mandrel. The motion of
the linkage should then reverse to remove the completed filter from the mandrel and eject it
onto a conveyor belt. After this, the linkage should return to the hopper and pick another blank.
Due to the requirement of reversibility, a double-crank or crank-rocker would have no real
advantage over a double rocker.
An acceptable linkage solution must have ground pivots and linkage motions within areas
that do not interfere with the hopper or assembler. Also, since the resulting linkage may be
driven by a dyad (to provide a crank-rocker input) the total angular travel of the input link of
the 4-bar synthesized here should be minimized so as to obtain acceptable transmission angles
for the entire mechanism, including its driving crank and connecting link which actuates the
input of the motion-generator 4-bar linkage.
This example is a typical challenging problem that faces linkage designers many times. Some
of the constraints are firm while others can vary within some specified range. This means
mathematically a number of infinities of solution possibilities. The computer graphics screen is
an ideal tool to help survey a large number of solutions.
The problem clearly requires motion generation synthesis (or rigid body guidance), in which
the position and angle of the filter blank is specified at different points. Four precision points
along a specified path and four corresponding coupler angles were chosen.
The first set of precision points chosen are shown in Table 3. The mandrel was designated
position one and the second position was picked above the mandrel with no rotation (to prevent
folding the filter blank). The third position was chosen to be halfway between the second and
fourth positions with about half the required rotation. The fourth position corresponded to the
angle and position of the hopper.
The M - K curves (center and circle point curves) for this set of precision points is shown in
Fig. 19. The solid and short dashed lines represent the portions of the center point curve from
Hopper
Mandrel
(not shown)
Figure 18.
Conveyor Belt
242
Table 3.
Position
X Coordinate
Y Coordinate
Rotations (degrees)
I
2
3
4
0
1
17
38
0
7
18
21
0
0
60
117
Plot of M - K
M I ~ K I ~
400
Curves (Both sets)
M2 ....
K2-----
3,00
20.0
\
,
I00
\
\
x
"%.~"-..
O0
-I0.0
-200
-I00
0.0
I00
.0
'300
Figure 19.
sets 1 and 2 respectively, while the long dash and dash-dot lines are the circle point curves from
sets 1 and 2. Figure 20 shows an M-K curve option where the/~2 values are correlated to their
m-kl positions on the curves by letters corresponding to the table along the left hand side of
the figure. The results of interactively locating ground pivots and moving pivots (by using the
cross-hairs on the graphics screen) corresponding to f12 = 340° and f12 = 20°, both from set 2, are
also shown in Fig. 20. As the designer locates a ground pivot with the cross-hairs, the computer
finds the moving pivot of the dyad and draws lines to represent the dyad. These two dyads
formed what looked to be an acceptable 4-bar solution to be further analyzed. The coupler
Plot of M - K
M
/(~
30.0
Curves (set 2)
20
340
OA
0
30B
60C
90D
120E
150F
180G
210H
240 I
27OJ
300K
330 L
36OM
20.0
I0.0
O0
\ "~.....~..
-I0.0
-I0.0
Figure 20.
0.0
////
I0.0
L:~).O
30.0
40.0
243
Coupler Curve
30.00
25.00
/
\
/
200O
I
~x3
/
1500
/
/
I0.00
/
5.00
I
/
!
~1111
\
000
/
/
-5.00
, I L L
. . . .
I
,
-I000
5.00
-500
0.00
,
,
i
I
i
,
i
i
I
,
,
,
•
I
. . . .
15DO
I0.00
I , I
,I
I l l
' '
25.0O
20.00
i
. . . .
35D0
30.00
40.00
Figure 21.
40¸00
s't.,
3500
3000
,,
25.00
...~,.-...',,
~
e'.. - "
~ -. . . . . .
s
N.
" "~
,~---'.~--'® I " "
20100
i
15,00
,, ///
i /"
I0.00
5.00
0.00
- 5.00
5.00
0 lO0
15,00
I0,00
25.00
20.00
This is The Linkage in the Precision Point
~0
"3000
Positions
Figure 22.
curve of this linkage is generated by another subroutine. Figure 21 shows that the coupler curve
shifts to the left between points one and two, and also approaches point 4 vertically. Both of
these are unacceptable characteristics. Although this linkage is not acceptable, there are others
which also satisfy the prescribed precision points in Table 3 as well as new choices for the
middle two precision points.
Making a long story short[18], the precision points in Table 3were changed and a number of
the options on the LINCAGES Package were used to optimize for the final solution. The final
linkage is shown in several positions in Fig. 22 while it's coupler curve is shown in Fig. 23.
Conclusions
This paper attempted to present several strategies for solving the three precision point dyad
synthesis problem. Each technique seems to have advantages and disadvantages which should
be weighed when choosing a specific technique for a problem. Also presented was an
introduction to the LINCAGES Computer Graphics Synthesis Package. Many illustrative
examples were also given.
244
Coupler
Curve
2500
A
0
B
C
D
E
F
G
H
i
J
K
L
30.0
600
900
120.0
1500
180.0
2100
240.0
270,0
300.0
3300
2000
/
/c"
~5.00
/
/
~000
/
/
/
5O0
oo(1
-5
oo
\
\
-Iooo
~
-1500 . . . .
000
~L....._
, ....
5 O0
, ....
1000
~...+
; ....
1500
! .........
2000
2500
i ....
30.00
I ....
4000
3500
Figure 23.
Acknowledgements--The author would like to acknowledgethe contribution of Richard Loerch, George Sandor and others
whose work is reported here. The support of the University of Minnesota's Educational Development Fund Program,
Department of Mechanical Engineering and the Computer Center is most appreciated. The author also thanks Lana
Schermerhorn for typing this manuscript.
References
1. R. J. Loerch, Extensions of revolute dyad synthesis for three finitely displaced positions. M.S. Thseis, University of
Minnesota, July (1977).
2. T. S. Hartenburg and J. Denavit, Kinematic Synthesis of Linkages. McGraw-Hill,New York (1964).
3. R. L. Goodstein, Complex Functions. Mcgraw-Hill,Maidenhead, England (1965).
4. R. Beyer, Kinematic Synthesis of Mechanisms, translated by Kunzel, Chapman and Hall Ltd., London.
5. K. Hain, Applied Kinematics, 2rid Edn. McGraw-Hill,New York (1967).
6. D. C. Tao, Applied Linkage Synthesis. Addison-Wesley,Reading, Mass. (1964).
7. A. S. Hall, Jr., Kinematics and Linkage Design. Prentice-Hall, Englewood Cliffs, New Jersey (1961).
8. F. Freudenstein and G. N. Sandor, Synthesis of path generating mechanisms by means of a programmed digital
computer. ASME J. Engng Ind. 81(2), May (1959).
9. A. G. Erdman and G. N. Sandor, Mechanism Design: Analysis and Synthesis. To be published by Prentice-Hall, 1982.
10. Tektronix Plot 10 Terminal Control System User Manual. Tektronic, Inc., Beaverton, Oregon (1977).
11. R. J. Loerch, A. G. Erdman, G. N. Sandor and A. Midha, Synthesis of four-bar linkages with specified ground pivots.
Proc. 4th OSUAppL Mech. Conf. Chicago, pp. 10(1)to 10-8,2-5 November (1975).
12. A. Rao, A. G. Erdman, G. N. Sandor et aL, Synthesis of multi-loop dual-purpose planar mechanisms utilizing
burmeister Theory. Proc. 2nd OSUAppL Mech. Conf. pp. 7-1 to 7-25, Stillwater, Oklahoma, 7-9 October (1971).
13. R. E. Keller, Sketching rules for the curves of burmester mechanisms synthesis. Trans. ASME, Paper No. 64-MECH5, Lafayette, IN, 19-21 Oct. (1964).
14. G. N. Sandor, A general complex-number method for plane kinematic synthesis with applications. Doctoral dissertation, Columbia University, University Microfilms,Ann Arbor, MI, 305 pp., Library of Congress Card No. 59-2596
(1959).
15. F. Freudenstein and G. N. Sandor, On the burmester points of a plane. ASME J. AppL Mech., Trans. ASME 25(3),
473-475 Sept. (1961).
16. R. J. Loerch, A. G. Erdman and G. N. Sandor, On the existence of circle point and center point circles for three point
dyad synthesis. ASME Paper No. 78-DET-44.To be published in the ]. Mech. Design. 101(4),October 1979,pp. 554-556.
17. A. G. Erdman and J. E. Gustafson, LINCAGES: Linkage INteraction Computer Analysis and Graphically Enhanced
Synthesis Package. ASME Paper No. 77-DET-5.
18. K. R. Struble, J. E. Gustafson and A. G. Erdman, Case study: synthesis of a four bar linkage to pick and place filters
using the LINCAGES computer package. Proc. 5th OgU Appl. Mech. Conf., Nov. (1979).
19. G. Rowley, Unpublished design report, University of Minnesota.
SYNTHESE CINEMATIQUE A TROIS ET QUATRE POINTS DE PRECISION DES MECANISMES ARTIOULES PLANS
Arthur 0, Erdman
R4sum4
--
Les dyades m4oaniques model4es par des nombres complexes sonZ pr4sent4es sous
plusieurs formes d'4quations diff4rentes pour trois positions prescrit4~, soit du mouvement,
245
soit de la g@n@ration de traJectoire avec synchronisation prescrite, soit de la g@n~ration
de fonction. Diff@rentes strategies sont sugg@r@es selon la fa~on dont les choix libres
sont le mieux utilis@s. Des techniques interaetives d'ordinateur sont sugg~r~es pour les
cas ~ trois et ~ quatre positions prescrites. Plusieurs exemples sont pr~sent~s pour
illustrer chacune de ces situations de conception.