THE KINEMATIC ANALYSIS AND SYNTHESIS OF CONIC CONSTRAINT PAIRS IN COPLANAR MOTION
By JACK WESLEY SPARKS
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1970 To Willie Wesley Sparks and Perry Lynn Sparks ACKNOWLEDGEMENTS
The author expresses his appreciation to Dr. Delbert
Tesar for the suggestion and supervision of this dissertation.
The author is indebted to C.A . Morrison and J.A.
Samuels for the financial support provided by their research contracts with the Southern Service Company and the Office of Civil Defense.
Appreciation is given to the following committee members for their guidance and supervision,
Dr. D. Tesar
Dr. R.B. Gaither
Dr. J. Mahig
Dr. J. M. Vance
Dr. R. G. Selfridge.
Finally, the author extends his deepest appreciation to Ills father and mother, to his family and to his wife,
Cheri , for their financial support and encouragement.
1 1 r TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS i i i
LIST OF TABLES vii
LIST OF FIGURES viii
ABSTRACT . . x
CHAPTER
I GENERAL BACKGROUND . 1
Conic Sections. 2
.Analysis of the Coupler Curve Function for Conic Constraint Pairs ... 4
Analysis of the Coupler Curve Function for Burmsster Constraints. ... 4
Linkage Synthesis for Burmester nd Conic Constraint Pairs 5
Kinematic Synthesis for Burmester Constraints 6
Kinematic Synthesis for Conic Constraints 6
The Application of Coplanar Synthesis . , 7
II BURMESTER THEORY THE STATE OF THE ART. .... 15
Burmester Constraints Variables Vs. Parameters. 17
Burmester Theory The Study of Five Multiply Separated Positions in Coplanar Motion 13
Nine Precision Point Synthesis for Burmester Constraints .... 32
IV Page
III CONIC THEORY THE STATE OF THE ART 35
Conic Theory Variables Vs. Parameters ... 37
Six Position Conic Theory . 39
The Six Position Conic Section Point Curve 44
Verification of Circularity 47
The Describing Parameters of the General Conic 48
The Describing Parameters of the
Parabola...... 51
Seven Position Conic Theory 56
IV ANALYSIS OF THE GENERALIZED CONIC CONSTRAINT PAIR 63
Higher Order Contacts . . 67
Analysis of Burmester Constraints 69
Locating the Pole of the Moving Plane ... 72
Analysis of Arbitrary Points in the Moving Plane 7 3
Conversion to Real Time 74
V LINKAGE SYNTHESIS FOR THE GENERALIZED CONIC
CONSTRAINT SET . 76
Linkage Synthesis for the Burmester Constraint Pair 79
Coupler and Fixed Link Synthesis for Conic Constraints 83
Coupler and Fixed Link Synthesis for Burmester Cons tx dints 86
VI SIX MULTIPLY SEPARATED POINT-POSITIONS IN COPLANAR MOTION 90 Page
Algorithm or Six Multiply Separated
Point-Positions in Coplanar Motion. . . .
VII NINE MULTIPLY SEPARATED PRECISION POINTS IN COPLANAR MOTION
Synthesis cf Nine Symmetrical Precision Points 109
APPENDIX
A. EXPANSION OF THE CIRCLEPOINT EQUATION BY DETERMINANTS .114
LIST OF REFERENCES. CITED 117
OTHER REFERENCES 119
BIOGRAPHICAL SKETCH 121
v 1
2- LIST OF TABLES
3- 3- TABlE Page 4- 1 MOTION COEFFICIENTS FOR THE
BURMESTER CONSTRAINTS . . 25
1 DESCRIBING COEFFICIENTS OF THE CONIC. . . 40
2 MOTION COEFFICIENTS FOR THE CONIC CONSTRAINTS 43
1 COEFFICIENTS OF ANALYSIS FOR THE CONIC - CONSTRAINTS ...... 70
VI LIST OF FIGURES
FIGURES Page
1-1 Plane Intersecting a Cone in Space. . . 3
1-2 Posthole Digger Employing a Straight-
Line Mechanism . 9
1-3 Function Generator (Decimal to Log Converter) 10
1-4 Dwell Mechanism 11
1-5 Conic Constraint Mechanism (Adjustable
Step- Indexing Mechanism) . 12
1-6 Automatic Feed Mechanism Employing a Doub 1 e Dwe 1 1 C am 13
1-7 Geared Mechanism (Cycloidal Crank With Adjustable Amplifier) .... 14
2-1 Class I (Burmester Constraints) .... 16
2-2 Class II (Conic Constraints) ...... 16
2-3 Variables Vs . Parameters for Coplanar Synthesis 18
2-4 Translation to the Pole Reference . . . 20
2-5 Rotation and Scaling of the Fixed Reference System. .... 21
2-6 The Generalized Reference System. . . . 23
2-7 Five Multiply Separated Positions for
Case PPP-PP . 30
2-8 oint Synthesis toir Cnso
PPP-P-P . . . . 31
3-1 Typical Conic Constraints 3b
3-2 Properties of a Conic in the Fixed
Plane . . 38
viii 3- FIGURES Fage 4- 3 A Conic Constraint Pair Satisfying
of . 4- Seven Positions the Moving Plane. . 62
5- 1 Coupler Curve Function for Conic Constraints 66 5- 2 Coupler Curve Function for Burmester 6- Constraints 71
6- 1 Conic Set Synthesis While Satisfying Seven Precision Points 78
7- 2 Burmester Link Synthesis While Satisfying Five Precision Points .... 81
1 Six Multiply Separated Point-Positions in Coplanar Motion 92
2 Mechanism Satisfying Six Multiply Separated Point-Positions in Coplanar Motion 98
1 Deviation of a Burmester Set for Different Input 100
7-2 Mechanism Satisfying Nine Multiply Separated Precision Points in Coplanar Motion 108
7-3 Symmetrical Displacements of die Moving Plane 112
ix Abstract of Dissertation Presented to the Graduate Council in Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy
THE KINEMATIC ANALYSIS AND SYNTHESIS OF CONIC CONSTRAINT PAIRS IN COPLANAR MOTION
By
Jack Wesley Sparks
December, 1970
Chairman: Dr. Delbert Tesar Major Department: Mechanical Engineering
A comprehensive study of the analysis and synthesis of
conic constraint pairs in coplanar motion is presented in a
generalized form. A state of the art is given for
Burmester Theory and for the Generalized Conic Theory. A
seven position algorithm for the Generalized Conic Theory
provides two quartics relating the sixteen generalized
co n i c po i n t s ..
An analysis algorithm is shown to satisfy both, the
Burmester mechanisms and the conic mechanisms and does not
require iterative techniques for solution. The conversion
to real time is shown to be an elementary transformation.
Algorithms for prescribing fixed and moving pivots while satisfying five precision points provide extensions
to the well-known graphical procedures of the past. These . .
algorithms are presented for both Burmester and Conic
Theory
An extension of Burmester Theory to the synthesis of six point-positions of the moving plane is presented with a theorem of uniqueness for six positions of the moving plane. The nine precision point problem is reduced to a solution of five equations with five unknown variables
Tabulated coefficients for the analysis and synthesis algorithms provide a systematic and simplified means of computation for all case studies.
xx CHAPTER I
GENERAL BACKGROUND
In 1967 the author completed research which coalesced
the study of finite (positional) and infinitesimal
(instantaneous) kinematics by presenting an algorithm for
five multiply separated positions (any multiple combination
of finite and/or infinitesimal positions) in coplanar
motion. The work is similar to that outlined by Bottema
[1] for the finite case and requires the resolution of
four linear and two non-linear algebraic equations based
on circular constraints in the fixed plane. A generalized
computer program capable of solving all seven cases was
the culmination of this work. Since its completion there
have been few research publications associated with
generalized positional synthesis for Burmester or Conic
constraints. Consequently, as in many disciplines of
study, kinematics appeared to have reached a plateau in
its capabilities in coplanar synthesis. Often when this
happens the only format to follow is to survey the field
of discipline with new conceptual ideas to see if this might precipitate different concepts which will reflect
upon and advance the particular area of study. Success is
primarily dependent upon taking an interrogative viewpoint
in the light of what is known and what has created such a .
standstill. t This attitude generally allows one to gain
new insight concerning his problem and most often
establishes a solution.
The technique employed in synthesizing six point-
positions and nine precision points, presented herein,
is a result of taking an interrogative look at coplanar
synthesis and applies a new concept to obtain the resulting
algorithms. The algorithms presented for six point-
positions and for nine precision points are capable of
solving all case studies for multiply separated point- positions in coplanar motion.
Conic Sections
It was Rene Descartes who in 1637 first applied algebra
to geometry making the study of conic sections a part of elementary mathematics, As shown by Descartes, there are
three general classes of curves obtained by the intersection of a plane and a cone as illustrated in Figure (1-1) and defined as follows:
1) An ellipse is formed wTien the plane section cuts only one nappe of a cone and is not parallel to an element of the cone. A circle is obtained as a special ellipse when the plane is perpendicular to the centerline and resolves to a point at the vertex
2) A parabola is obtained as a plane section becomes parallel to an element of the cone. A line is formed as a limiting case.
3) A hyperbola is formed in both nappes of the cone. Two intersecting lines are obtained as a limiting case. 3
TYPES OF CONICS
1) ELLIPSE
2 ) PARABOLA
3) HYPERBOLA
4} CIRCLE (SPECIAL ELLIPSE)
Figure (.1-1) Plane Intersecting a Cone in Space 4
Analysis of the Coupler Curve Function for Conic Constrain t Pa irs
In recent years many kinematic researchers have expended considerable effort to define and analyze the output function (coupler curve) of the simple four -bar mechanism. Since this study is concerned with conic constraint pairs in motion, an algorithm is presented to coalesce and generalize the analysis of the conic mechanisms with the mode of synthesis.
A.s will be shown in the text, the analysis of the circular constraint system employs the same algorithm as the generalized conic constraint system. However, it becomes advantageous to discuss the two systems separately because of the difference in the complexity of the solutions
The basic concern for the development of an analysis algorithm was to acquire a method of verifying the results provided by the synthesis program. The author found that his efforts were markedly reduced as a result of this particular investigation and formulation.
Analysis o f the Coupler Curve Function for Burmester Con straints
From a kinematic standpoint, analysis of coplanar linkages is certainly a well-known area of study. Works by Freudenstein [2] have provided explicit results for all types of circular (Bunn.es ter) constrained systems. There- fore, it is not the intention of this paper to initiate new areas of research for analysis but simply to develop a i
5 discipline or format which is compatible with the described mode of synthesis. Methods will be discussed for analyzing positions as well as orders of contact (tangent, curvature, etc.) for Burmester constraints.
L nkage S yn thesis for Burmester and Conic Cons traint P air s
Recent works by Sparks and Tesar [3] have provided a mode of synthesizing a Burmester pair while works by
Hain [4] have provided a graphical mode of synthesizing the ground link for Burmester constraint pairs. It is the author's intention to provide a method which includes the above results and also provides the capability of synthe- sizing a coupler link cr ground link while satisfying five prescribed multiply separated precision points for
Burmester constraints.
A similar algorithm for synthesizing seven multiply separated precision points is presented for the generalized conic pairs. It might be noted that for conic constraint pairs, one can only synthesize the coupler link since conics
(excepting circular constraints) have no physical center physically represented by a fixed pivot.
One might say that the dependence of dynamic synthesis lies solely within this area of research since one must know at least one constraint pair to synthesize the absolute dynamic characteristics. This particular area of research represents a distinct discipline of kinematic synthesis and consequently diversification in kinematic utilization. i b s s
6
Kin ematic Synthes i for Burmester Cons r a i n t
At present several different methods are employed for
synthesizing 3,4 and 5 coplanar positions of the moving plane [1,2,3,4,5,61. Here, emphasis will be placed only on synthesis of 5 coplanar positions since this problem represents the present capability of coplanar synthesis.
Early works by Hain [4] have placed the present capability
for certain case studies at s.ix precision points with somewhat limited application and questionable versatility as is often the consequence with graphical procedures.*
An analytical approach by Veldkamp [7] has provided some implicit results with iterative techniques which often can be time consuming and fruitless.
It is therefore the intention of this paper to provide algorithms for synthesizing six multiply separated point- positions and nine multiply separated precision points in coplanar motion for Burmester constraints.
K nemat io Synthesis for Con ic Constraints
i
Considering conical constraint pairs when synthesizing coplanar positions adds two positions to the capability represented by Burmester constraints. Present wf ork by
Freudenstein , Bottema tend Koetsier [3] has provided a useful foundation for conic constraints and has made the
*It might be noted_ that a position of the moving plane r prescribed by a,|3 Y (rectangular coordinates of the origin of the moving plane) and y (angle of the moving plane) where a precision point is prescribed only by the coordinates a and . . :
7 kinematician more versatile in his ability to solve problems
This paper will provide an algorithm for synthesizing
seven coplanar positions employing conic constraint pairs.
The primary emphasis of this work will not be to develop
an algorithm for synthesizing point-positions using conic
constraints, as is the case of Burmester contraints, due
to the complexity of such an endeavor. The author will, however, provide avenues to such research and therefore provide a foundation for future research.
The Application of Coplanar Synthesis
The areas of application of coplanar linkages have recently become very broad and divergent. This is primarily a result of requirements imposed upon design by competition and society to create safe, reliable, efficient and inex- pensive mechanical systems. This has brought about considerable demand for kinematicians and their research work
Present research in the application of coplanar mechanisms generally falls into one of the following categories Multiply (3) Coupled Multi-link ' Mechanisms Separated Mechanisms Geared Mechanisms Positions in (4) Dynamic Velocity Coplanar Synthesis Acceleration Motion V Jerk
Figures (1-2. . . 1-7) represent a small portion of the many applications of such mechanisms. These applications are self-evident and require no further elucidation at this 9
39NVU
PATH
Mechanism
.Straight-Line
a
Employing
Digger
Posthole
(1-2)
igure
fM T
OUTPUT / INPUT
\rrnrvTTvQvV
INPUT/ OUTPUT
/ 0 ^— O.l 0
i Y = LO G X 10
Function Generator (Decimal to Log Converter) 11
Figure (1-4) Dwell Mechanism )
12
MATERIALS HANDLING BELT
/ /
Figure (1-5) Conic Constraint Mechanism (Adjustable S tep - Indexing Me chanism 13
CAM
Figure (1~6) Automatic Feed Mechanism Employing a Double Dwell Cam 14
Figure (1-7) Geared Mechanism (Cycloidal Crank With Adjustable Amplifier) * .
CHAPTER II
BUR.MEG TER THEORY THE STATE OF THE ART
Fundamentally there is only one constraint function used by the kinematician to constrain the moving plane.
7\s one would gather from Chapter I, this is the conic constraint function. The reason is that for convenience kinematicians almost always employ functions which are easily constructed physically and provide utility. These requirements dictate the use of conic constraint functions
It might be mentioned, however , that one may use any analytic function as a means of contraining the moving plane ,
For convenience kinematicians generally separate circular constraints from conic constraints since the analytics for the latter are significantly more complex.
Figures (2-1, 2) represent what will be referred to as the two classes of mechanisms. In theory the two classes are the same; however, the Burmester Theory represents a degenerate study of the conic constraint algorithm.
Notice the basic difference in the fixed pivots and curvature
*An analytic function is a function that can be expressed by a Taylor Series. 16
8URMESTER BURMES TER CIRCL
Figure (2-1) Class I (Burmester Constraints)
COUPLER . s .
in Figure (2-2). As will be shown in what follows, the conic coupler curves are of a higher order, making their versatility somewhat greater.
Burma star C onstra int Variables Vs. Parameters
In solving a problem it is often advantageous to know the number of variables associated with the problem, and the number of parameters specified. Therefore, it becomes imperative to describe the variables and parameters for solving or developing an algorithm of synthesis for a four-bar mechanism (Burmester constraint system)
It is seen from Figure (2-3) that there are eight variables associated with two generalized Burmester constraint pairs
( (u i , v t ), (Hi, K x ) } ; { (u 2 , v 2 ), (H 2 , K 2 )>.
For a determinant or closed system, the number of variables must equal the number of parameters.* From Figure (2-3) each position of the moving plane requires three parameters hence, it appears that one can only synthesize 8/3 position
This is contrary to the Five Position Theory; consequently, one must lock further into the problem.
In most analytical systems it becomes desirable to work with normalized parameters. Thus normalization of
Common vernacular relates a closed system to be a system of an explicit and closed set IS
• N UMB E R OF VA R I AB L E S 3
EACH POSITION REQUIRES 3 PARAMETERS [a u8,y)
Figure (2-3) Variables v s « ameters for Coplanar Synthesis y .
19
the system by translating the pole P 0 i to the origin
Figure (2-4) removes parameters ai and 3 i . Rotating the plane and scaling so that a 2 = 1 and 8 2 = 0 (Figure
2-5), leaves the following parameters
Yl r Y 2 I « 3 ' 3 3 r Y 3 r <* 4 ' 3 4 , Y4 for the problem specifications. Hence it is seen that
number of variables - number of parameters
Therefore it is shown that the system becomes a functionally closed system; i.e. a system which is in agreement with the variables vs. parameters requirements.
A simple equation governing all coplanar constraint systems can be expressed as follows
+' Number of positions - number of variables 7 . 3
Hence, one can proceed to synthesize Five Multiply
Separated Positions in Coplanar Motion.
Burmester Theory l the_ Study of Five Multip Separated Positio ns in Cop l anar Motion
Using the cartesian frame of reference (rectangular
coordinate system) , the transformation of the coordinates of A(u,v) as a point in the moving plane E, to the coordinates of the point A(U,V) of the f ixed system 1’ is
'j.Lvenf~r -> -v% V rVr
U ~ UCCSy - VS illy + a - ( 2 1 ) V -- usinY + vcosy t 3. 20
VA
Figure (2-4) Translation to the Pole Reference 21
Figure (2-5) Rotation and Scaling of ’die Fixed Reference System Q
22
From equations (2-1) it is evident that the origin e of
E(u,v) has the coordinates, e(a,(3) in X as seen in Figure
- ( 2 6 ) .
The intention of this section is to introduce the seven coefficients which are sufficient to define the motion for five prescribed positions (£) constrained by
Burraester circles. The results can be in terms of either crank and rocker or slider crank systems. Writing the general form of the circle constraint equation G in E gives
2 2 G(U,V) = (U + V ) + 2Q/J + 2 V + 0 Q c 2 3 - ( 2 2 ) where
G(U,V) = 0.
For a multiply separated position differentiation with respect to the position parameters Y evaluated for a specified Y^ gives
k 2 2 G (U,V) = d (Q (U + V ) + 2Q U + V + ] = 0 £ , 0 X 2Q 2 Q 3 dy"K Y =Y jr (2-3)
From equations (2-1) , it is seen that U and V are functions of y, so that
U « dk (u}i V = dk {V} % k df y=Ya dY K V (2-4)
Writing the difference, G, G , gives the generalized form £ q for the circular constraint function as vA
- - 7 77777/7
Figure (2~6) The Generalized 'Reference System ' .
24
(U - O 2 2 2 2 G f V) {U U + V V } + £ o £ 0 £ o - - Q {2U 2U } + Q { 2V 2V } = 0. 1 £ 0 2 & 0 (2-5)
The coefficients for each position £ become
2 2 A d a. K, { 6 } 0 £ dy
A = dj,k cosy+ gsiny} {' a 1 k a dY V , d ^{-asxny+ 3 cosy) 2 0 iy v - A = d { cosy 1} 3 k £ dy
A siny} i { U dy K
A = d
; V«) £ dY
-Jc £ d Y Y =Y*. (2-6)
The necessary form of the A for all cases of m£ multiply separated positions for up to five positions have been presented in Table (2-1), It should be noted that all
.A are of ,a functions the ^position parameters,™a and Y „ m£ £ £ '£'
For five mu 1 tip
can 1:>e expres sed for
! A . Q AO u + A „ ' c£ a i£ o - .+ A (u Q vQ 2 4.£ 5 A/ * 6^4 (2-7) a> , 'n —\/
«>C _ to r o — o r o < ca 'll ca Cl ca
- << _ to - o — o I o < & d d d d
&< •r-\ V >- fa o rH O * to 1 j II 10 T~H C* v C » J *n fO •I"k fa H O T*fa. 3 § «>- ' fa* P5 < '0 •H £-' o CO CO u l « fa c o o CJ •H K •H >- K to tH o o •n •n - r\ CO >- — o r* “ O K CO £? r-H 4-> 3 » o o — o •r~j o to ca. 1 § CO •r-t •r-% i vO - o N — 4-» Q., D i'J oa CO •r-> CM o i d CQ + O Cl ! ca < •n o + I o to I o + r: p / ca i ca W >- V -'r~» fan n — • H 3 o Eh ca the ith CO i 1 r-> — > Pi d ca to V > O 'J O P fa d > to O *H e t-> E-* d ra fa CO ^ fa to *H H d H *r~> Q U ‘o—i - r H - O 7> CJ r o ca. fa fa •H V- 1 ca c3 d to CO CO. V J G - a c> o + + o o e H o — r ’» Eh r% d +-> U O d •H dn to o o fa, ,-C - H !l o ^5 O — Cl to ij- £ Q 4 4 4 26 after substituting equations (2-1) for their respective U and V terms. By rearranging equation (2-7) in the two distinct forms [Ao + Ai u + A 2 v]Q 0 + [A ] (uQ + vQ ) f 3 2 Aj 36 X/ 1 + [~A4 u + As ]Qi + [A u + A ] 2 “ 0 £ 4£ 6 l = 1,2,3, - ( 2 8 ) and [A + A u + A,^vlQ + [A* 1 (uQ 2 - vQ ) 0 £ 1£ 0 x + [ A u + A + + A 0 3 5 ] Q j I%v S£ ]Q 2 = £ ^ a = 1,2, 3, provides two determinants (Ao + Ai u + A v) A 3 ( ~A + (A4 ! u + 2 V AsJ Ae^) 20 ^ £ £ 4 £ £ i = 1,2, 3, and (2-9) (A + + A, (A 2 0 o A u A 2 v) u + A ) (A £V + A 0 | i £ £ 3 £ 5£ 3 g£ ) h I S- - 1,2, 3, both of which must be identically zero. The two determinants appear to yield characteristic equations that are of third order. This is, however, not the case since the determinant provides a nullity of the third order term. Thus these determinants provide two conics of u and v in the moving plane which can be expressed as T 7 V (V + V + - 1 + 12 u 13) V V l4 u' 5 U Vi6 0 (2-10) V?.lV 2 + (V U + + V 2 + 22 7 2 3 ) v 2 4 u V 25 u+ V 2S 0 1 ! ) 2 7 where '' a A,+ 1 “K H £ -%l V 1 2 “ Al> + A3 As a K % £ N K £ £ hI V 1 3 A As + Ao A3 " A A £ 4 «.| K U £ | £ £ Vl 4 - |*. A, a t i h %l Vl 5 a As As A + Ac A3 As A l «t i £ »ii 1 £ £ £ <*l 7 1 6 As As As K t £ t| and (2- v 21 “ A A A A l U ‘* 5 £ »J v 22 " A, Ai A5 A, + A 2 A 4£ A3 A l i t £ tl ! £ £ *tl ” V 23 ^4 As a A Alf A A ! ° *£ 3ill £ £ h| I £ £ 7*1 V 2 4 = A A 4 £ 2 3 A 1 n £ “.tl 7 2 5 Ai A4 « A + A A A3 = « 5 0 A ° At /V X/ £ 4 p ;.\ «J | -V £ 7 2 6 A As As - ° i * % £ J. The intersections of these conics are the :Burmester poin Eliminating v from equation (2 -10) gives the quartic 3 2 Em" + E 2 u + E 3 u + E 4 u + E S = 0 (2- .in u for the: Burmester points. The coeffi cients are = 2 - 2 Ei (c-h) f(c -h) (a-f) + h ( a-f) E 2 = (h-c) [2 (p-d) + f (b-g) ] + (a- f) If (P-d) 1 g (h- c) + p (a -f) + 2h (b-g j -- 2 E 3 (d-p) + 2 -r h(b-g) + (a-f)[f(q~e) + g(p~d) + q(a-f) i- 2p(b -g)] 1 [ 1 , 4 ' 28 E 4 = (b-g) [f (q-e) + g(p-d) + 2 q(a~f) + p(b-g)] + (q-e) 2 (p-d) + g(a-f)] 2 2 E s - (e~q) + g (q-e) (b-g) + q (b-g ) ( - 3 V 47- 1J- * t where a = V21 V12 f = V 1 1 V22 = = b V 2 1 V 1 3 g Vi 1 V23 — 1 - c V 2 V 1 4 h = V 1 V24 ( 2 14 ) = = d V 2 1 Vis p V i 1 V25 e = V 2 1 Vi 6 c? - V11 V 2 s - The four solutions to equation ( 2 12 ) are u_. By 2 removing the term v from equations (2-10) , the other pin coordinate, v , is given as v (h-c) U 2 n + (p-d)un t (q-e n ) (a-f)u (b-g) n + - ( 2 15 ) The location of the centerpoints in the fixed plane can be related from Ei F - E 9 ( 2 16 ) wnere Q II + Al + a V A 2 £ n = - E As t A 3£U A 4 v £ £ n £ n r a + U + 6 A4 a 3 v £ l £ n £ n - ( 2 17 ) £ = 1,2 n - 1 , 2 3 , . g ) . 29 Since the centerpoint coordinates are given by - (Qi/Qo n n - ( 2 K: . - (Q /Q ) n 2 0 n n -L / *- ,3,4 they may be expressed as H. = (D 1 F 2 - D F 1 n 2 ) (EiF 2 - E 2.F 1 n (2-19) K. (E D _E P n 1 2 y 1 ) (EiF 2 "E 2 F 1 ) n n = 1 , 2 , 3 , 4 . It might be mentioned here that this mode of solution is adaptable for arbitrary point synthesis as well as linear point synthesis as illustrated by Figures (2-7,3). Although Figure (2-8) illustrates five positions on the U - coordinates the quartic (2-12) degenerates for this condition and also for five positions on the V - coordinate. This can be verified bv observina that for A - 0, 1,2, 3,4, si = V 1 4 , Vl5, Vis, V 2 1 , v 2 3 , V 2 6 , 0 which causes a, b, c, e, g, q = 0 and the quartic to vanish. For five positions on the ~ 0 U-coordinate A s — V 1 1 , V 1 3 , Vis, V 2 4 , V 2 5 , V26 0 which causes b, e, f, g, h, p, q = 0 and the quartic to vanish. A simple rotation of the positions about the origin so that Aj /A = constant removes this f 6? condition and causes only E s to vanish in the quarcic. 30 PPP-PP Case for Positions Separated Multiply Five (2-7) ure 31 Figure (2-8) Linear-Point Synthesis for Case PPP-P-P a . . 32 Ni ne Precision Point Synthesis ?or Burme s ter Constraints Previous wor3cs by Roth [9] represent a vector approach in locating the fixed pivots (Burinester centers) and the circlepoints (Burmester points), while synthesizing nine finite precision points in coplanar motion. So that the modes of presentation of Roth's work will be analogous to the cartesian frame of reference, this work is trans- formed here into rectangular coordinates. If from Figure (2-3) , one assumes two Burmester pairs exist, a position j must satisfy the two circular constraint functions following from equation (2-7) as Constraint 1 {g. siny. + g . cosy. = g. 1 i 1 J 2 J J 3 Constraint 2 {f slny + f . cosy . - f j. i J 2 j j 3 / \ t \ r-~ g 4 a , siny g j i 'J2 j j 3 I \ = l f r cosy . f 2 - 21 _ j i 3 2_ 3 3 ( ) V / where =* - - g . (P. K ) u - ( . II ) V j j i i 3 1 1 i = - . - I< d-i (a H ) u + ( 3 . ) V J 2 J l i 1 1 l = 9 ( H ) u + ( K ) v J. l i l l -{a . II + 3 K - (a? + 1 J 5 j - 2 ( 2 22 ) l 2 1 23 and • - f . *= (8 _ - u K )u (-CL H ) V 1 j 1 2 2 j 2 - f = toj _ H ) u + - K ) v j 2 2 2 2 < H ) u + ( K ) V h, 2 2 2 -{ a H } 8 . K - (a- 1 2 1 2 Since the identity sin g 9 a , 1 1?- ' j 1 n 1 1 2 f . f . 1 1 1 1. 1 2 (2-24) or 2 _ f-s - + fj (g^ (gj ^ g-;^ ) 13 1 2 1 3 1 2 11 1 3 1 1 1 3 - (gy . f . f . gJ . ) li 12 1 1 (2-25) From equation (2-25) one can write the positional matrix for positions j = 0,1, 2, 3, 4, 5, 6, 7,8 as [C] o 8x6 k V U V 1 V : u V (2-26) V7here u v 2 2 2 u 2 V 2 1 (2-27) V i and = f(H , K , H , K , a,, 0.) i 1 2 2 J j j - 0,1, 2,3,4, 5, 6, 7,8. From equations (2-26,27) it is evident that there are eight unknowns { (u , v ) ; (H , K ) } ; { (u , v ) ; (H , K ) } . 1 i 1 1 2 2 2 2 Thus one can write the eight precision point equations and theoretically satisfy his solution for nine finitely separated precision points. CHAPTER III CONIC THEORY THE STATE OF THE ART Early works by Beris [10] , Beyer [11] and Hackmueller [12] have provided the kinematician with the capability of using conic constraints for prescribing displacements of the moving plane. Later works Toy Sandor and Freudenstein [13] and Woo and Freudenstein [17] provided an extension of the earlier authors' works that had employed infinitesimal displacements. A recent publication by F reudens tein , Bottema and Koetsier [B] provides the kinematician with a concise method of synthesizing six finite positions of the moving plane. This paper also treats all degenerate cases of the conics for finite synthesis through six position theory. The intention here is to formulate a generalized method for synthesizing six multiply separated positions of the moving plane by defining the conic section point curve. It is also the author's objective to present a new algorithm for synthesizing seven multiply separated positions of the moving plane. This algorithm produces two quartics in u and v which relate the 16 conic points for conic constraints. 35 Figure (3-1) Typical Conic Constraints . s Conic Theory Var i ab 1 e s Vs . Parameter Before synthesizing a conic constraint pair, it is advantageous to analyze the number of variables vs the number of parameters for the system. It. is seen from Figure (3-2) that there are fourteen variables associated with two generalized conic constraint pairs: { (u ,v (H , K ) , (0 )} ] 1 3 . 1 { (u ,v ) , (H , K ) , (0 )}. 2 2 2 2 After normalizing and removing a ,3 , a and (3 by 1 2 2 translation , rotation and scaling, as outlined for the Burma s ter constraints, the number of synthesizable positions becomes number of positions = number of _variables + 7 number of positions = .14 + 7 3 number of positions = 7. The maximum number of positions (coordinates and angles) which can be synthesized using conic constraints becomes seven. Hence one can proceed to synthesize Seven Multiply Separated Positions in Coplanar Motion for conic constraint pairs 38 y EQUATIONS OF TRANSFORMATION (TRANSLATION AND ROTATION) - U { V K ) S ! N 0 *H {U-H5COS3 V = (V-K)COS© - (U -M) sir Figure (3-2) Properties of a Conic in the Fixed Plane 39 Six Posi tion Conic Theory The equation for a generalized conic with its center at the origin can be expressed as 2 2 Z U -r Z V + Z V + Z = 0 12 3 4 (3-1) where Z , , , are defined for the conics as illustrated 12 3 4 in Table (3-1) . If the center of the conic is not at the origin, the inverse transformation U = (V - K) sinO + (U - H) cosG (3-2) V = (V - K) cosO - (U - II) sin0 gives the new coordinates as shown in Figure (3-2) for an ellipse. Substituting equations (3-2) into (3-1) gives 2 2 2 Z {(V - 2VK + K ) sin 0 + 2 (UV - UK - VH + KH) sinScosO l 2 - 2 2 + (U 2UII f H ) cos 0 } + 2 - 2 2 - - - Z { (V 2VK + K ) cos 0 2 (UV VK VII + KH) sinQcosQ 2 + (U 2 - 2UH + H 2 )sin 2 0} + Z { (V - X) cos9 - (U -- H) sinG} + Z =0. 3 4 (3-3) Rearranging and collecting terms gives 2 2 2 - TJ {Z cos q + Z sin ©} + UV{2 (Z Z )sin 0 cos 0 } 1 i 2 : 2 { 2 2 - - +U -2H ( Z cos q + Z sin o) 2K(Z Z ) sin©cos 0 - Z sin©} 12 12 3 2 2 2 +V {Z s in e + Z cos 0 } (3-4) 1 2 2 2 - - TV{-2K(Z sin © f Z cos 0 ) 2H(Z Z )sin 0 cos© + Z cos©} 12 12 3 {H 2 (Z cos 2 + Z sin 2 + K 2 (Z sin 2 + Z cos 2 + + 12o 0 ) 12© ©) - -- = 2KH(Z Z )sin©cos© + Z (Ksin© kcos©) + Z } 0 . 12 3 4 40 a o ii_ u CM u CM ii: I < to fe O CO o o o iri Id q % % ft. CO L« I i™ TO to ro 10 iO < ° < tO ~J .J .J O < P - M* M* IS- o LI CO CD CO O ID o ^ a. : •w# -g O —o CC ~ to Is, n. CM CM tr CL OJ| CM M- cvi 'CM Ini I'J o N n O < CD O -1 o O ! | H -J il ii II II ii li Si it o ii li to to —1 r to cJ LO ~r CO CO X L. < <>< < < o -J -J CO s s IxJ tij o O LJ SjJ t CO — co co * N CL CQ m O w CC < u. 11 < a: LU -J <5 CL IjJ >- < X CC 41 ± z til a first position's origin is taken to be coincident with the origin of the fixed plane, then a - 3 Y = 0 o o o and the transformation equations U = ucosy - vsiny + a (3-5) V = usiny + vcosy + g become U = u V -• v This allows removal of 2 2 2 2 2 2 ( i (H cos 0 + Z sin 0 ) + K (Z sin 0 + Z cos 0 ) + Z _ i 2 i 2 — 2 KH fZ 55 ) sinQco^O -h Z (HsinG - Kcos0) + z > i 2 " 3 4 from the conic constraint equation (3-4) to give 2 2 2 2 (U - u )(z cos 0 + Z sin 0} + 1 2 (UV - uv){2(Z - Z )sinGcosG}+ 1 2 - 2 2 - - (U u) {-2H(2 cos 0 + Z sin 0) 2I< (Z Z ) sin3ccs0 - Z sin0}+ 12 12 3 2 2 (V - v ){ S sin 2 0 + Z cos 2 ©} 1 2 2 2 (V - v) {-2K(Z sin 0+ Z cos -- 2H(Z •- Z sinGccsO 12 0) 12 ) + Z cos0 } = 0 3 (3-6) Subscripting the transformation equations (3-5) provides U- = ucosy- - vsiny. + a. 3 3 3 , j (3-7) V. -- usiny + vcosy. ^ + g.. 3 3 3 3 Ana making the following substitutions 2 2 X = Z cos 0 t Z sin 0 1 1 2 - X = 2(Z S ) sin0cos0 2 12 X = -2 FIX - KX - Z sin0 1 2 3 . 42 X =Zsin 2 G+Z cos 2 0 4 1 2 X = -H X - 2KX + Z COS0 5 2 4 3 (3-8) allows one to write the six position conic constraint matrix as [C] {X} - 0 (3-9) or r 2 - 2 - - 2 - 2 - (u u ) (U V uv) (U u) (V v ) (V v) 1 1 1 (A 1 1 1 1 2 (U 0 e !» 0 X 2 2 (U 2 - u 2 3 0 3 y ) \ X 3 3 2 2 - 0 ® 3 <5 (u u ) X 4 4 2 ~ 2 a <3 9 V (U U ) X ' , 5 \ of (3-10) Substituting the transformation equations (3-7) into the first typical equation of (3-10) will give 2 2 • - - - (B (u v ) 23 , (uv) + 2B • (u) 2B . (v) + B •} X + l3 2 3 J J 1 5 63 9 . 2 - 2 (B (u + 23 • • - •) v ) (uv) + (B i + B A ) (u) +(B B 23 1 J 63 7 J 53 83 (v) + B . >X + 1 0 3 1 2 (B . - . (u) B . (v) + B }X + 3 3 4 J 1 1 3 3 2 - 2 {-B • (u + • + . + . + v ) 2B (uv) 2B (u) + 2B . (v) B }X 1 J 23 83 73 1 2 3 4 • (B (u) + B . (v) + B . }x =0 3j 13j5 j --= 1,2,3 ,4, 5. (3-11) For any multiply separated position a typical equation can be written as above by taking the coefficients B from Table (3-2) For a finite position the coefficients 43 CONSTRAINT COMIC THE FOR COEFFICIENTS MOTION (3-2) TABLE . . j• . n . 44 are given by 2 B . = -sin y rj = B . siny . cosy 2D D D B • cosYj - 1 3 3 siny 4 j B a . cosy r t j j 4 (3-12) B aj siny 6 j B 0 cosy 7 j j sinyj 1 3 J J B — a 2 93 3 3 _;= a . 0 i o 3 3 j = • B ^ a l i 3 3 B = 8? 123 3 Q • — 13 P . 1 3 J 3 which may be expressed similarly to the coefficients of equations (2-6) From the typical equation one can develop the Six Position Conic Constraint Matrix for six multiply separated positions as defined by positions ~ l 0 , 1 , 2 , 3 , 4 , 5 of the moving plane The Six Po s 1 1 1 o C oh i c S ection Point Curve As shown by Freudenstein , Bottema and Koetsier the Conic Section Point Curve is of seventh degree. This is verified from equations (3-11) by adding column 1 to column 4, which gives for the typical equation after subscripting for &th position . 2 - 2 - - (u v ) 2B (uv) + 2B (u) 2B (v) + B }x + u 2A 5 A 6 A 9 A I 2 - 2 ; (u v ) + 2B (uv) + (B + B ) u + (B B ) v +f 3 }X + 0 2 A i A 6 A 7' A/ sA 8 A i o A 2 (u) - B (V) + 3 >X T ' 3 A 4 A 1 1 A 3 : (B + B ) (u) + 2 (B 3 ) (v) + (B + B ) }x + 5 A 8 A 7 A 6 A 9 A 1 2A 4 (u) + B (V) + B }x = 0 „ 0 \A 3 A 1 3 A 5 A = 1,2 ,3,4,5. (3-13) The conic constraint matrix then becomes [C] {x} = 0 (3-14) where 2 2 - -- = B (u V ) 2B , (uv) + o 1 A l A 2 A sA 6 A 9 A 2 - 2 = B (u v ) + 2B (uv) + 2 2A i A 6 A 7 A 5 A 8 A + B „ l o A = B (u) - B (v) + B 3 A 3 A 4A l i A ; = 2 (3 + :B .) (n) + 2 (E + B ) 4 A sA 3 A 7 A 9 A 1 2 A = B (u) + B (V) + B „ 5 A 4 A 3 A 1 3 A A = 1,2,3, 4,5. (3-15) The expansion of the determinant of the conic constraint matrix [C] can best be accomplished by products of minors. Observing the last three columns and comparing these coefficients with those described in [3] for four position Burmester Theory and dividing column 4 by 2 it is seen that the coefficients are the same.* This allows for *The generalized four position Burmester Theory is formulated in Appendix [A] V ) simplification by expressing these minors (3 x 3) in terms of the generalized cubic from Burxnester Theory. Letting the minor for positions 3,4 and 5 be Qp 2, 4, and 5, be etc., there will be ten minors for the (3 x 's 0.2 < Qm 3) of the last three columns. The minors for the first two columns are given by a quartic of u and v and may be expressed as 2 2 2 2 2 2 ( (u - v - 2(u - + + g iM ) v )uv 4(uv) } + d v) {u2_ y2} + 3 m + ^ u g v + g ) ^ uv > + (3-16 4 m 5 m 6m 2 2 g r,.,{u } + q { } + 7"‘ 3 m + v + ^j.m) Q m m - 1,2,3 ,i0 where - ^m (E B B B } 2? 1 P 2 A {B (3 _ + B _.) + 2(B B - B B ) , r l P P P P V £ 6 7 si 2 5 ? A -B (B + B Pn ) } 1 6 A 7 A ~ = (B (B .. B - B 2(3 ‘ P m „ ) C\' 3 1 A sP eP e cu 6 P 2 A -B _(B - B P * ) } i' s A sA (3-17) -- = { 4 (B + B B ) 2 [B m 0 4 sA* sP 1 A iP + B (B 4- 3 )]} 2 P sA 7 A = 4 (B B ~ B B - 2 [B g m { ) 0 K P 1 A A 6 sA iP 2 V + B (B - B ~0 )]} 2 P 5 3 A - = { 2 [B B BE + B (3 6 m 9 A iP 2 A i 0 P 5 A *P' -B (B 3 „)] 6 A S P 7 P 2 [B B - B B + B (B 3 ) 9 P i A 2 P 10 A sP 5 A 8A ~B (B - B _p ) ] } 6 A 7 A 0 j £ . 47 - ) g { (B B + B B -(B B + B B ) 7 m i£ xaP aZ 2 P iP 10 Z sp zZ - + 2 LB (B + B ) B (3 + B )] } s£ 6 P 7 P 5P 6 Z 7 Z - 9 m - ( (3 .B + B . B ) (B B + B B J 8™ l£ loP 9 Z 2 P lP id iP 2 Z - - - +2 [B (B B ) B (B B ) ] } 6 £ sP 8 P sP 5 Z 8 Z = 9. m f2(B.,B.. n - B B } 5 £ loP sP i o £ + B (B + B _) - B (B + B )> sZ eP /P 3 P e£ v£ a = { 2 (B B - 3 B ..) nl 10 6 p 1 0 £ 6 £ 1 0 p +B (B B (B - B n D p ) -Bp ) } 9 £ sP 8- a P s£ 8 q = {B B - B 3 } ni 1 1 9 £ x 0 P 9 P 1 £ * where p 7 Z m = 3. ,2,3. . .10 The expansion of the determinant [C] expressed in terms of the products of minors becomes 1 n s 2 2 2 2 2 Z ("I) { ( u: v 2 (u - + Qm [g im ) v )uv 4(uv) } m = 1 + u 2 - 2 (g ?m + g 3iuv){u v } + (g mu + g mv + g Huv} 2 2 + g ?m fu )+ g^tv } f' (g u + v + = 0- ,in 3 ra ShIoH 1(, (3--1G) where + 2 +. . s=(l + j + j . m ) 1 2 and the cubics are taken from Appendix [A] Verification of Circularity The authors Freudenstein , Bottema and Koetsier expressed a knowledge of tricircularity of the conic section point curve; however, they give no verification of this . , } 48 property. From Appendix [A] the cubics Qm can be expressed as (H u + u2 + + im V> < H mv + H muv r H mu + H mv + H m ” Qm 4 s 6 ? 8 (3-19) By rearranging the first term of equation (3-18) the highest order of the conic section point curve becomes (X + X 2 2 2 2 u v) (u + v ) { (u + v " ) i 2 2 2 2uv[(u -V ) + uv] (3-20) where X and X are scalers dependent on the coefficients 1 2 B . It can ri1 ^ be shown by substituting u - Mv into equation (3-18) and letting v ; » that the resulting slope M is related by 2 3 (M + 1) (X M +X ) - 0 1 2 (3-21) which will give six imaginary asymtotes indicating tricircularity The D escribing Parameters of the General Conic For a particular u and vthe conic constraint matrix [C] becomes defined. The application of Cramer 1 s Rule provides for a particular u and v ) ) 49 X = L X 2 1 1 X = L X 3 2 1 (3-22) X -LX 4 ? 1 X = L X . 5 .41 Substituting equations (3-22a) and (3—22 c) into (3-8) gives (L cos 2 0 2sin0cos0) (L sin 2 0 + 2sin0cos0) i l 2 2 2 2 (L cos 0 sin 0) (L cos 0 -- sin 0) 3 3 2 By adding columns 1 to 2 and dividing i by cos 0 the system reduces to - (L 2Tan0) (L ) i i i = 0. (L - Tan 2 0 (L -1) 3 Substracting column 2 from column 1 yields the following form - ( 2TanO) (L ) i 0 . 2 ( .1 - Tan 0 (L -1) (3-24) 3 The determinant, or the angle of rotation function. becomes 2 L Tan 0 - 2 (L - l)Tan0 - L - 0. 1 3 1 (3-25) The centerpoint coordinates are given by equations (3-8c) and (3--8e) for Z /Z - 0 as taken from Table 3 1 so (3-1) for the circle, ellipse and hyperbola — r ( ) -2X -X H x 1 2 5 <; -X -2X k| X 2 b 5 l J l ) Substituting of equations (3-22) gives ( -2 •L (3-26) which allows for expressing the centerpoint coordinates 33 L — 2 Li Xj H = 4 L i 2 3 and (3-27) K = b L - 2L, 1 2 - f ?: 4L 3 - Li . For 5,-0 and Z /Z =0, the constraint equation (3-3) 3 1 becones 2 - 2 2 2 { (v 2vK + K ) sin 0 + 2 (uv - uK - vH + KH)sinOcos0 2 - 2 2 + (u 2uH + H ) cos 0} + 2 - 2 2 Z { (v 2vK + K ) cos 0 - 2 (uv - uK - vH + KH)sin0cos0 2 2 - 2 2 + (u 2uH + H ) sin 0} + Z =0 b letting 2 2 2 y - (v - vK + K )sin 0 + 2 (uv - uK - vH + KH)sin0cos0 1 + (u 2 - 2uH + H 2 )cos 2 0 2 2 2 Y - (v - 2vK + K )cos 0 - 2 (uv - uK - vH + KH)sin0cos0 2 2 2 2 + (u - 2uH + K ) sin 0 51 and combining with equation (3-23b) gives the system Y Y 1 U) 2 Mi < \ - - s 2 - 2 2 - 2 1 (L cos 0 sin 0) (L sin 0 cos ©) z 0 ’ 3 3 _ 1 l 1 l ) The inversion provides the semi-axis = 2 2 _Z_i -( L 3 sin 0 - cos 0) Z, sin20 - - 2 - + Yi(L 3 cos 20) Y 2 (L 3 cos 0 sin20) Z (L 2 - 2 ? = 3 cos 0 sin ©) Yj (L sin 2 0 - cos 2 - Y (L cos 2 0 - sin 2 3 0) 2 3 ©) (3-29) All of the properties for conic constraint synthesis of six multiply separated positions can now be expressed for the circle, ellipse and hyperbola. As will be shown in the following section, the parabola requires a different algorithm in locating the center and angle of rotation for the semi-axis. The geometric center of the parabola is at infinity which requires that L 2 - 4L = 0. 1 3 A method of relating the center as the vertex is derived and illustrated for this special conic. The De s cr ib ing Parame t er s of the Parabola For a particular u and v the conic constraint matrix [C] becomes defined and the application of Cramer's Rule provides • X — LX 2 l - XL i: X 3 2 X 1! d X 4 o. X = L X 5 4 for the particular set of coordinates (u,v) . Fi'om equation (3-1) and Table (3-1) if the condition L 2 - 4L =0 (3-31) 1 3 is satisfied then the conic is a parabola and Table (3-1) suggests that Z = 0. 4 For a parabola equations (3-8) become X = Z cos 0 i l X = 2Z sinGcosO i l X = -2HX - KX - Z sin9 3 12 3 2 X = Z sin 0 4 i X = -HX - 2KX + Z cosG 5 2 43 (3-32) This gives for equations (3-30a) and (3-30c) the forms 2sin0 - L cos0 = 0 i and sin^Q- L cos-0 3 both satisfying equation (3-31) . The angle of the transformations can be expressed as 3 3 TanO = _j_.L (3-33) *•> It will be shown later in this text that Z /Z determines 3 i the direction which the parabola opens and establishes the phasing of 0. Equations (3-32c) and (3-32e) can be grouped to form the characteristic set — / \ t A f + 2X X * I -X Z sin0\ i 2 '3 3 * y X 2X -X + Z cosOj 2 4 1" (3-34) The discriminate of equation (3-34) can be expressed by X 2 4X X 1 4 or L 4L = 0 3 as defined by equation (3-31) . If aquations (3-34) are to relate the coordinates of the center, they must be coincident. The coincidence also requires the discriminate X (—X - Z sinG) 2m 3 3 = 0 2X ( —X + Z cos0) * L. 5 3 (3-35) to be zero. The discriminate gives for Z /Z 3 1 Z 3 = COS0(LlL 4 2L 3 L 2 ) - Zi Li (1 L 3 ) (3-36) 3 57 B { ~2vX + uX } + 6 £ 1 2 B (uX + 2vX } + 7 £ 2 B {-vX + 2uX } + 3 £ 2 4 (3-42) B { X } + 3 £ 1 B (X } + 1 0 £ 2 B {X } + 1 1 £ 3 B (X } + 1 2 £ 4 B (X } =0 1 £ 5 =0,1, 2,3, 4,5,6 which conies from equation (3-11) . For seven positions of the moving plane the system can be expressed, by applying Cramer ' s Rule , as. \ 2 - 2 - (u v ) (X X ) + 2uv (X ) 2uX + vX 1 4 2 l 2 (u 2 - V 2 ' -2vX + UX l 2 uX + vX 6x7 uX + 2vX 3 5 2 4 •VX + UX —VX + 2uX 3 5 2 4 X X i X X J (3-43) where e. . are functions of the B^ coefficients 13 The first tv/o expressions of equation (3-43) are functions of onlv X , X , X and may be expressed as 1 2 4 2 - 2 -• -- {(U v ) 2e u + e v e }X i i 1 2 1 5 1 - { 2uv - (e + e )u - (e - e ) 12 13 1 2 1 4 1 6 2 2 - - !- s rS { -(u v ) e u h e v o X I 1: 1 3 1 7 and (3-44) 53 {-2uv - 2e u + e v - e }x + 2 1 2 2 2 5 1 2 2 - - i.i - {(u v ) (e + e ) - (e e )v e }X + 2 2 23 22 24 2 6 2 (2uv - e u T e V - e }X = 0. 2 4 2 3 2 2 4 ( 3-45) Though these two equations were linearly derived from the typical or generalized set, they must satisfy the characteristic set and the characteristic equation. By multiplying equation (3—44) by B and subtracting from H the £th equation (3-11) and by multiplying equation (3-45) by B and subtracting from the £th equation 2 £ (3-11) gives for the £t.h or typical equation. (f U t f V H- f }x \ £ 2 3 ; £ £ (f u + f V + f }x + 4 £ 5 £ 6 £ 2 r. (f u + f V + }x -r (3-46) 7 14 £ 8 £ 9 £ (f u + f V “h f } X I 0 X1 1 2 0 £ £ /V {f u + f V p }x 1 3 1 4 1 5 £ £ £ £ =0,1,2, 3 ,4,5,6 where f 2{B + 3 e + Be) 1 0 5 1 l 2 21 A/ £ H £ f -{ 2B + B e + Be) 2 G x 1 2 2 22 £ £ £ £ f (B B e + :B e } 3 9 1 5 2 25 £ £ H £ f (B + B + B (e + e* ) + B (e r e )) 4 6 7 1 12 13 2 2 2 2 3 £ £ £ £ £ - f (B B + B (e - e ) + B (e e ) ) 5 5 8 1 12 14 2 2 2 2 4 £ £ £ £ £ -C —_ r d JL i ^ + 3 a + B e } 6 1 0 1 6 2 26 & £ H £ r (23 + B e + Be) 7 s x 1 4 2 24 £ £ £ £ rr — ... j_ (23 - B e B e } CO 7 3 1 3 2 n 2 3 £ £ A/ (3 -47) 21 1 0 3 6 59 f = B i o 3 £ f - 3 1 £ 4 £ f - B 1 1 £ £ £ = 0 , 1 , 2 , 3 , 4 , 3 , 6 . f = B 1 3 £ 1 3 £ Note that columns 4 and 5, the coefficients of X and X , 3 5 are not altered by this manipulation. Employing the technique outlined for the Five Position Burmester Theory for compound determinants with linear orthogonal or linear parallel elements gives .c [f u + i. V + f ]X + [f u + f V + f ]X + 1 £ 2 £ 3 £ 1 4 £ 5 £ 6 £ 2 [f U + f V + f ]X + [f ]{uX + VX } + 7 H 8 £ 9 £ 4 1 0 £ 3 5 [~f V + f ] x + r f U t f ] X = 0 1 1 i 2 3 11 1 3 5 £ £ £ £ £ =1,2, 3,4, 5, and (3-48) [f u + f v + f ]X + [ f u + f v + f ] X + H 2 £ 3 £ 1 4 £ 5 £ 6 £ 2 [f u + f + f x [f + } + V ] 4 + ]{uX vX 7 £ 8 £ 9 £ 1 1 £ 3 5 [-f V + f ]X + [ f u + f ] X = 0 1 1 2 3 1 1 °£ £ £ £ 5 £ =1,2, 3, 4, 5, 6. (3-49) The two determinants (characteristic equations) of the square matrices provide two quartics in u and v relating the 16 Burmester points for the general case. The quartics can be expressed as A u 4 + (A t A v)u 3 + (A + A v + A v 2 )u 2 + 11 12 13 14 15 16 2 3 ( A + A v + A v -i- A v ) u + 17 13 13 110 . 0 3 60 2 3 4 (A + A v + A v + A v + A v 0 111 112 113 114 ) 115 (3-50) and 4 3 2 2 A u + (A + A v) u + ( A + A v + A v )u T 2 1 2 2 2 3 2 4 2 5 2 6 2 2 (A + A V + A v + A v ) u + 2 7 2 8 2 9 2 1 0 2 3 4 = (A + A V + A v + A v + /i v ) 0 2 1 l 2 1 2 2 1 3 2 1 4 2 1 5 (3-51) where the Apq's are the expansions of the respective determinant s . The Apq's can be expresse A = f f f f f f i l l x 4 7 1 0 1 2 lx £ £ £ £ £ *! A - f f f f f f |+ 1 2 1 £ 4 A 7 £ 1 c £ 1 2 £ 1 H * f-L -L. f f^ f~ f |+ 3 4 £~ 7 1 o „ 1 2 11 £ £ Aj £ i f f f fll + 1 6 7 1 1 2 o £ £ £ £ JO J f f f f f * x 4 9 1 c 1 2 1X £ £ £ £ £ J 9 O o = A -If f f f f f I 116 1 2 s 8 1 °£ 1 1 1 £ £ £ A SC (3-52) and similarly for equation (3-51) . Thus the quartics defining the 16 Burmester points can be related by a closed form technique The intersections of these quartics can be found by letting ci = A m i mi d A + A^ V ™2 ™2 ^3 = dm Am + ATn v + A. m 3 4 5 m c 1 . . 61 - A + < v + A v 2 + A rn 7 ms m 9 2 = - [ A + A V 4 A v ffii m s i mi 2 mi 3 A v 3 + A v 4 mi'i m i 5 ra (3-53) and employing Sylvester s Di alytic Me thod of elimination to give the resultant R for vn as A a d d d d o 0 \J i i 1 2 1 3 1 4 1 5 0 d d d a a 0 0 R ~ 1 1 2 i 3 1 4 1 5 (vn ) ' 0 0 d d d d d 0 i i 1 2 1 3 1 4 1 5 0 0 0 d d d d d = n i i 1 2 1 3 1 4 1 5 — V d d d d d 0 0 0 2 i 2 2 2 3 2 4 2 S 0 d d d d d 0 0 2 1 2 2 2 3 2 4 2 5 0 0 d d d d d 0 2 1 2 2 2 3 2 4 2 5 0 0 0 d d d d d o 2 1 2. 2 3 2 4 2 5 For che real points Burmester the resultant R(vn ) must be singular. Therefore various numerical methods of minimizing R may be employed to obtain the real Burmester points (maximum of 16) The describing parameters of the conic may be found using the same technique outlined for six position synthesis equations (3-22 3-41) for each of the intersections of the quartics (Burmester points) s 0 ^ rD l o Positions Seven Satisfying Pair Plane Constraint o <* 0 0 9 ft 0 O IfJ q o q Q O Moving C5 id to CM CM ro to 1 1 Conic the CM O o o G> to o to. 0> to to CM CQ. ' q 'of I A o' 1 s 1 (3-3) o o f- CM ro fO o N to CM N- N- to a d > • Ijj Figure — ANALYSIS OF THE GENERALIZED CONIC CONSTRAINT PAIR As shown by Sandor and Freudenstein [13] , synthesis is not and should not be restricted to circular (Burmester) constraints. In general, any conic constraint set provides a means of constraining the moving plane which is the requirement of synthesis. This is to say, and will be verified in what follows, that the Burmester constraints are merely c Regenerate conic constraints which utilize simplified modes of synthesis and analysis. Since the researchers Sandor and Freudenstein [13] , Freudenstein, Bottema, and Koetsier [3] and this author have algorithms for synthesizing positions in coplanar motion for conic constraints, it seems only logical that an algorithm of analysis is necessary. Therefore it is the author's intent to introduce such an analysis algorithm for any combination of the following conics Circle Ellipse Hyperbola Parabola. From equation (3-6) and equation (3--8) the conic constraint equation was given as z - 2 - - (Q u ) X + (UV uv) X t (U u)X + 1 2 3 • 2 - (V v ) X + (V - v)X = 0 (4-1) 4 5 . , 64 where U - ucosy - vsiny + a (4-2) V ~ usiny + vcosy + 3 and 2 2 X = Z cos 0 + Z sin 0 1 1 2 X = 2 (Z - Z ) sinQcosG 2 1 o X = -2H3 KX Z sinQ 3 X =Zsin 2 0+Z cos 2 G 4 1 2 X = -HX 2KX + Z cosQ 4 3 (4-3) For a known constraint set the variables V| [iO u, v, Z , Z I , H, K, Q become known.* Rearranging gives the quadratic Aa 2 +(A3+A)a + A3 2 + A6 + A = 0 1 2 3 4 5 6 (4-4) where = X A = X A = sinv{X u - 2X v} + cosv{2X u + X v} + X 3 21 1 2 3 A = X 4 4 A - siny{2X u - X v} + cosy{X u + 2X v} + X 5 4 2 2 4 5 = 2 2 - 2 (X - + + i\ -sin y{ (u v ) X ) 2uvX } 6 14 4 2 - 2 — — smycosy{ (u v ) X 2uv (X X ) } + 2 1 4 *For the circle, ellipse and hyperbola Z_ = 0 and for the parabola Z - 0 . 65 (cosy - 1} {uX + vX } + 3 5 siny{-vX + uX } 3 5 (4-5) Thus if two constraint sets are known ‘ / V ’z 2 u , V , N| r z K , 0 1 1 -r i 1 . 4- l 1 Constraint , 1 ( ) Pair ilS) \x N* u , V , Z r Z 0 2 2 .4^ 2 - 4J 2 equations (4-5) may be subscripted as follows x t m'n^ and A £ -ra'n] m - 1,2 n 1,2, 3, 4, 5,6. The two constraint sets provide two quadratics for equation (4-4) and may be expressed as / \ ' \ 2 2 A (A 3 + A ) a -(A 3 + A 3 + A ) Position i i 12 13 >={ 14 15 i 6 1 < / 2 Matrices A (A 3 + A ) a -(A 3 + A 3 + A ) _ 2 1 2 2 2 3 _ . 24 25 26 \ i (4-6) 1 in terms of y for the moving plane . Applying Cramer s PvUle to equation (4-6) squaring the second row and equating to the first row gives the quartic for 3 T-* 4 i t* 3 n o TJ* P _L Xi» jlj p r ±j p Jj p i jj o 1 2 4 5 (4-7) 66 HYPERBOLA Figure (4-1) Coupler Curve Function for Conic Constraints s. 67 in terms of y of the moving plane. The roots of the quaz'tic provide four 3 coordinates of the moving for each angle y provided. The corresponding a is taken from the inversion of equation (4--6) . The coefficients Ep are defined by equations (2-13) and (2-14) High er Order Conta c t It is often advantageous to analyze the geometric velocity, acceleration, jerk, jounce, etc., of a particular constraint pair. This is accomplished by differentiating the Position Matrices equations (4-6) with respect to y to give " f \ Geometric f f " •F i i 1 2 1 3 Velocity 1 I 1 -., Matrices f f f 2 1 _ 2 2__ l 23j (4-8) where - 2A a + < 6 + A III mi 2 m 3 m = A a + 2 3 + A (4-9) 2 m 2 V = A “ + T m 3 m 3 Differentiating of the Velocity Matrices will give the acceleration Georaetri c f f y i i fi ] Acceleration h \ Matrices f f 8 y - 1 2 1 2 2_ "J ^ 2 1/ 1 : S3 where ' - f (f 'a+ f ' £ + f ) 1 1 1 x 2 X 3 i = 1,2 Differentiating equation (4-10) provides the jerk \ \ ( . ", , ( Geometric cx* y 1 1 1 2 1 2 Jerk = < < > Matrices S'" y (4-11) 2 1 2 2 V 22, where r = 0 y (ff'cr + 2 f f a" + f ' + 2 f f b' + f(' ) u 1 1 ll X 2 X 2 13 i = 1,2. Differentiating once more provides for the jounce jl IV / Geometric f f [ a ] i i 1 2 Jounce IV I Matrices a y (4-12) - / 2 1 2 2 V \ 2 3/ where (£'' 'a' t + 3f, 'a'" + f'"3' + 1 3 x XI 11 i 2 '3" , / " 3f f + 3f . 3 + f . " . 1 1 X i 1 3 This illustrates that differentiating and applying Cramer's Rule provide a mode of analyzing all orders of contacts. Note, however that the higher order contacts are dependent on the lower order contacts. . 6D For simplicity it is desirable to formulate Table (4-1) for the derivatives f for m . This formulation allows ease in computer programing and mathematical calculations. Analysis of Burmester Constraint s As stated earlier in this text, Burmester Theory consists of a degenerate form of Conic Theory, thus' it becomes appropriate to investigate the consequence of the analysis algorithm. From equation (4-4) it is seen that for Burmester constraints A 11121424= A = A = A and A = A - 0.* 2 1 2 2 Subtracting equation (4-6b) from equation (4-6a) provides 2 2 A ct + A a + A 3 + A 8 + A *- 0 11 13 14 15 16 - - = (A A )a + (A -A ) 3 + (A A ) 0 23 13 25 15 26 16 and the linear dependence of a and 3 provides a quadratic rather than a quartic in terms of the moving plane coordinates The above relation points out that the order of the coupler curves are different. Thus one would expect *For circular constraints Zi = Z 2 and 0=0° . These conditions require that X = X and X =0 for both constraint pairs. 1 2 70 OTHTWiTTOMm ''>TKTr\''l TT U T UA J OTPirTWMW .7 /"i CTMIT^T I mn-1 — rrrTCTJT 71 Figure (4-2) Coupler Curve Function for Burmester Constraints greater versatility in synthesis using conic constraints as has been proven to be the case. Locating the Pole of the Moving Plane The pole or instant center is the position in the moving plane which has an instantaneous velocity of zero. And since the pole coordinates are given by the trans- formations U„ = u..,cosy - v^siny + a P p P ' Vp - Upsiny t Vpcosy + 3 differentiating with respect to y gives for Up = Vp = 0 a ’ = u siny + v0 vpcosy h (4-13) 3* = -UpCosy + VpSiny. Substituting equations (4-13) into equations (4-8) provides — \ - ( ( \ (f siny f cosy) (f cosy + f siny) up f i i 1 2 i l 1 2 1 3 < >-< / (f siny - f cosy) (f cosy + f siny) V f — 1P — 2 1 2 2 2 1 2 2 ( ) l 2 V (4-14) Applying Cramer's Rule gives the pole's position in the fixed plane for any y chosen. From the pole position the location of the inflection circle. Ball point and other higher order properties can be determined. .^ 73 Analysis of Arbitrary Points in the Moving Plane For arbitrary points in the moving plane, one simply applies the transformation / Positional U = ucosy - vsiny + a Transformation (4-15) V = usiny + vcosy + 3 where u and v are the coordinates of the arbitrary point. Equation (4-15) shows that the system' s algorithm is not altered since U and V are uniquely determined by the transformation As for the geometric velocity, one must differentiate, with respect to y, to give / — '=/ Geometric U -usiny - vcosy -f a> Velocity / (4-15) Transformat- v' ~ ucosy - v siny + 3i ion Differentiation of the velocity transformation gives for the geometric acceleration / / r _ / / Geometric U -ucosy + vsiny H* a Acceleration ^ (4-17) Transformation V' = -usiny - vcosy and further differentiation gives the jerk as ( ' 1 Geometric U' ' usiny + vcos y + a! Jerk (4-13) ' ’ ttt I \7 Transformation | V ucosy vsiny + 3 { The above equations show that if the geometric position, velocity, acceleration, jerk, etc., of the origin $re known, then these properties determine the position. u . , 54 Note that the sign of Z / Z is determined by 0 and 3 1 establishes the phasing. For the initial position 1--Q the constraint equation (3-4) becomes 2 2 H (Z cos G ) + i 2 K 2 (Z sin 0) + l KH(2Z sin0cos0) + i H(Z sin0) + (3-37) 3 K (-Z _ cosO) + 2 2 (u X uvX + uX + v X + vX } = 0. 1 2 3 4 5 Multiplying equation (3-32c) by H and multiplying equation (3~32e) by K and summing the two relations gives 2 2 H (Z cos ©) + i 2 2 X (Z s.in 0) + i KH(2Z sinOcosO) + i H + Z sing t 3 | K {Xs - Z3C03Q} - 0 . ” 2 “ (3-38) Subtracting the above equation from equation (3-38) provides the linear expression (X - Z sin0}H + (X + 7, cos0}K = -2X 3 3 5 S 6 where (3-39) 2 2 X = { X + uvX + uX + v X + vX > S 1 2 3 4 5 Combining this equation with (3-32e) gives the system . 55 The inversion of the equations provides H = - 4 X Z X 4 0 - (-X 5 + 3 cos0) (X 5 + Z 3 cosO) 2X4 (X3 - Z 3 sin©) -X2 (Xs + ZssinQ) and K (X - Z sinG) (~X Z + X, 3 3 5 + 3 cos0) 2X 2 2X (X Z : - sin0 ] (x + Z sin0 ) 4 3 3 2 s 3 ’ >i 3 - 2 —6 K = z + 2[L cos0 + L sin01 z - 2 [L L cos 0-L Z l ,J ? K . . 4 2 4 1 ’ 2 3 2 - 2 [L sin0 + 2cos0] ’z - [4sin 0I. 2cos 0L ] 3 z 2 4 L 1 - (3-41) where y 2 2 {u + UVL r uL + v L + vL } Z 12 3 4 1 The above coordinates relate the position of the vertex for the specific parabola corresponding to the set (u,v) and complete the description of the parabola in the fixed plane Seven Position Conic Theory In the previous section it was shown that six positions of the moving plane provide a seventh degree polynomial in terms of u and v (coordinates of the conic points) in the moving plane. The addition of a seventh position will provide a second seventh degree polynomial in u and v indicating that there are 49 intersections of the two carves. As shown by Freudens tein , Bottema and Koetsier, of these 49 theoretical solutions one must subtract 18 for the circular points at infinity, 10 for the common poles and 5 for the common Ball points, leaving 16 Bur- mester points for the general case.* To obtain the determinant of the 5x5 matrix in terms of u and v is a formidable task. Therefore it becomes desirable to express the matrix in some other format. If the system is arranged in terms of common coefficients, the generalized equation of motion (typicial equation) can be expressed as 2 - 2 - B {(u v ) (X X ) + 2uv (X ) } + l £ l 4 2 2 - 2 - - B {(u v ) (X ) 2uv (X X ) } + 2 £ 2 1 4 B (uX + VX } + 3 £ 3 5 B {-vX + uX } + 4 £ 3 s 3 { 2uX + vX } + s £ 1 2 *The circlepoints at infinity are a result of the function being tricircular as previously shown in this text . 74 velocity, acceleration, jerk, etc., of every point in the moving plane. Conversion to Real Time Emphasis here should be made that conversion to real time requires that differentiation be taken with respect to time and not the angle y. The absolute velocity can be expressed as Absolute dU = A dy + da Time dt 1 dt dt Velocity (4-19) dV = A dy d3 dt 2 dt + dt where A = -usiny - vcosy A = ucosy - vsiny. For the absolute acceleration one simply differentiates equation (4-3.3) to obtain ( 2 2 2 2 Absolute d U d y dy d a Tj me dt 2 = Ai dt2 - A 2 dt + dt 2 Acceleration ' 2 2 2 2 d V d y dy d a 2 2 2 ; dt = A dt + A dt + dt and similarly for higher order motions Observing equations (4-19,20) it is seen that the relations ’A3 d^ ot , a , d^ y dtF dtR' dt-< 75 are required in the conversion to real time. By judiciously assuming the origin at the circlepoint of constant or defined w (rate of angular rotation) then d rt „ R coso dt c dS _ Rc sinw dt dl dt where R = centerpoint to circlepoint length Rp = circlepoint to pole length. For higher order derivations differentiation must be taken with respect to time for a, 3, Y, to, Rc and Rp. . 6 CHAPTER V LINKAGE .SYNTHESIS FOR THE GENERALIZED CONIC CONSTRAINT SET Often the kinematician must have the ability to synthesize a conic constraint set while satisfying coordinate positions (precision points) in coplanar motion. This requires forfeiting the angle y of the moving plane or the absolute values of the derivatives. From Chapter III the generalized conic constraint equation was given by 2 2 - -- - (U u ) X + 2 (UV uv ) X + 2 (U u)X 1 2 3 2 2 + (V - v )X f 2(V - v)X = 0 4 5 (5-1) where U and V are defined by equations (3-7) and X^ by equations (3-8) specifying the following parameters ’ ’ Constraint Set I zi "z 2 yJ and ( fu, v, H , K, z.* f z, f ot. 3 ) L Precision Point 1 and letting each position carry the subscript .i will give for the finite case a. sin y . + a. siny.cosy. + a- siny. + a. cosy. + a. 0 1 1 r -- i ' 1 3-3 ' 1 1 3. 1 5 ] r 2 i i = 1 , 2 , 3 , 4 , 5 , (5-2) 76 1 • 3 } } 77 a transcendental equation of fourth degree where - 2 - 2 - a. = {-(X X ) (u v ) 2uvX } 1114 2 2 - 2 - - a. = (X (u v ) 2(X X ) uv 12 2 14 - a, = {a. (uX 2vX - (vX - 2uX (uX - vX ) 1 1 ) 3. ) + 3 2 1 12 4 5 3 a. = {a. (2uX + vX ) + 3 . (uX + 2vX ) + (uX + vX )} 1 4 1 1 2 1 2 4 3 3 a- = (a?X. + a 3 X + a-X + B?X + 3,-X - (uX + vX )}. ^ 1 1 5 1 1 11 2 1 3 4 1 5 3 5 (5-3) Rearranging and squaring provides 3 Y + c,. „.in y. + C. sm' + c.siny. + = 0 J-2 Yi (5-4) where c = a. 2 + a il 11 C = 2 (a. a. i 2 1 1 3 i 2 i i) C = a? + a 2 ar„ 2 + 2a. a* i 3 i 1 1 1 1 3 2 1 5 (5-5) = 2 ( a • a . °i 4 3 15 = a - a °i 5 is i Equation (5-4) has a closed form solution resulting in four angles for each position or (4) s theoretical sets of solutions satisfying the seven precision points. In prescribing infinitesimal displacements it will be shown that there is a unique a and 3 for the relative change prescribed. Differentiating equation (5-2) will give a 3' + 0 1 1 1 - ( 5 6 ) 1 I L V a P /j j •V T] 0 0.0 0.0 0* 0° 0° 0° -.5 39.° !S8.° 64.5° 24.5° 1 .3 ! - - 2 ?..o .5 515.5° 1 67.5° V - - A 3 2.0 2.3 202.5° 190.5* — - «* “51.0°'•115.5° 4 « Q^ 1 .4? - - 5 -1.2 “2.22 265° 67.5° - - “ 116.5° 67.5° 3 .68 ! .00 ELLIPSE Figure (5-1) Conic Set Synthesis While Satisfying Seven Precision Points . . 79 where the f ' s are taken from Table (4-1), In synthesizing infinitesimal positions the kinematician can specify only the relative change of the invariants by k k b- a + bi 3 = 0. 1 2 (5-7) Expressing equations (5-6) and (5-7) as a linear set for the first derivative gives (f )-a' + (f ),£' = -(f ). 1 1 1 12 ’ i 3 1 . ' ' b a + b . 6 = 0 . 1 12 (5-8) Applying Cramer's Rule provides the proper a', 8' for the order contact synthesized.* For higher order synthesis one need only differentiate equation (5-8) further as given by equations (4-10, 11, 12) take the coefficients f from Table (4-1) and proceed as ir M. outlined Linkage Syn thesis for the Burmester Constraint Pair * As one might gather from the previous chapters, Burmester pair synthesis is an elementary study of conic section constraint synthesis. From equation (2-5) the Burmester constraint equation is given by 2 2 2 2 (U - u - V - v )Q + 2 (U - u)Q + 2 (V - v)0 = 0 0 12 *These are the unique absolute values to be used in the synthesis a 1 gor i thm 80 where U and V are taken from equation (2-1) and Q /Q from IT o equation (2-16) , By specifying the following parameters Constraint Pair and (u, v, H, K, a, Precision Point and letting each position carry the subscript (i) gives for the finite case a. siny. + a. cosy. + a. ==0 1- 1 i 1 1 1 2 3 (5-9) which is a transcendental equation of second degree in y. The coefficients are expressed by a = - T U — - H) il (Pi -0 (a i V == - - a . (a, H)u (f3 K) v 1 2 JL. i — cL , ( + H) u + ( + K) v 13 2 ~{a E + K ~( ai- L 3i (5-10) Rearranging and squaring provides the quadratic 2 c- sin + c- siny + c - =0 1 y i? 17 1 3 (5-11) where (5-12) ° 81 j i ! ! 1 >o a "CD r, ^2 0 0.0 0.0 0 0® -04.0° 1 .3 .5 -o.o° 2 2,0 1.0 34.0° “31.5° 3 ~!,0 2.0 66.3® “ 167.5° 4 -1.3 1.0 71.5® "154.3° Figure (5-2) Burmester Link Synthesis While Satisfying Five Precision Points '' . 32 3c The uniqueness of a and 3 also exists for the Burmester constraints for the relative change prescribed. Differentiating equation (5-9) gives for the first order infinitesimal case (f a (f 3 (f ) ) 1i ) 1L = i i i ?. 1 3 (5-13) where the f coefficients are taken from Table (4-1) pq In synthesizing infinitesimal positions we specify only the relative change of the invariants k y b. a + b. 6 =0. 5-1 !2 (5-14) Combining equations (5-13) and (5-14) gives the linear system 3 (f ).ct' + (f ) • i i r i 2 1 ' . ' + b = 0 • b a . r 3 i i2 (5-15) Applying Cramer’s Rule provides the proper a' and 3' for the order contact synthesized.* For higher order derivatives one need only differentiate equation (5-.9) further as given by equations (4-10, 11, as 12) , take the coefficients from Table (4-1) and proceed outlined. *The corresponding f 1 s are found in Table (4-1) where Z ! and for circle. -fi.' G-0° the “z z . 4- L 4 J . , 83 Coupler arid Fixed L ink Synthesis for Conic Cons traints It often becomes desirable to synthesize the pin coordinates of the coupler link or fixed pivots of a conic constraint pair while satisfying seven multiply separated precision points in coplanar motion. The designer is often faced with such requirements when the solution to a problem dictates size limitations or optimal constraint locations. It is important here that one investigate just what can be prescribed while satisfying seven precision points in coplanar motion. For seven precision points there are are six prescrib- six angles y^ which indicate that there able variables for the seven precision point study. For higher order studies, the angles would be replaced by the relative change of the invariants k k b,- a + b- 3 = 0 (5-16) as explained earlier in this chapter and given by equation (5-7) The six prescribable variables may be any combination or variation of the following { (u ,v ) , (u , v ) , (0 ,0 ) , (a, 3)} i l 2 2 12 { (u ,v ) , (u , V ) , (K , H ) a, 3 1 1 2 2 rb 1 1 -1 1 'z.AUl { (u ,v ) , (u , V ) , [ z ! z; 1 , (a, 3 )> _ '•i. I o 1 !+ 1 s 1 2 2 - L { (K , H ) , (K , H ) , (0,0), (a, 3) > 11 2 2 12 . . 84 Thus one has gained by the versatility of specifying constraint requirements while satisfying seven multiply separated precision points. The generalized conic constraint pair equation (5-4) is rewritten here as 4 3 2 + = 0 c. sin y11• + c. s.in y. + c_. sin y. + c- siny. c- ' - 'l J-3 1 3-4 1 3-1 (5-17) where 'Zi ‘ Z2 c . , H, K, z r z , ©/ a, 8) 13 . H 4. as defined by equations (5-5) Unlike the conic constraint pair synthesis, the study herein is concerned with synthesizing variables for both constraint pairs. Therefore, for each precision point i there exist two functions of the form 2 Constraint #1 (f. sin 4 y. + f, sin 3 y- + f- sin y. + f. siny. + f - 0 i 5 4 sin 3 sin^y^ + siny.^ + Constraint #2 (g^ si.n y i + g^ '^ + gj g^ gis - 0 (5-18) where f = H K , Z'M , Z , 0 , a., } f{u , V , , J i pq i i i i L d i l k i i a. = f(U , v , H , K , , , 0 , , 8.) H3 1 x 2 2 2 MC 4-» 2 C 4- 2 2 Vq a as taken from equations (5-4) and (5-5) ...... 85 The resolution of these two trigonometric relations seems to be a formidable task. And since the algorithm requires the sets of f and g which have a common pq pq root or roots, a mode of solution becomes necessary. Employing Sylvester's Pialytic Method gives the resultant R. f f . f f f f 0 0 0 ii i 2 1 3 i„ .1 5 r> f f . f f . f 0 0 ii .12 13 1 4 i s 0 0 f . f . f f f 0 ll 1 2 i s 1 4 1 5 0 0 0 f . f f . f . f 1 1 1 3 1 4 s R- E i2 i E 0 1 g g g g g 0 0 0 Ml i2 i3 i, i 5 o g g g g g 0 0 il — i 2 i 3 i 4 is 0 0 g g g 0 il %2 i3 ^4 is 0 o 0 g g i g i4 g ^1 i 2 3 is (5-1 This is to say that the determinant R. must be singular i for each of the precision points prescribed. Therefore one can prescribe any six of the following variable s in addition to the coordinates of the seven precision points Z l {u , v , H , K , z„ , e > 1 i iiii z ? (5-20) {u , v , H , K , z , © > 2 2 2 2 2 2 by employing a numerical search routine satisfying equations (5-19) . For the infinitesimal positions equations (5-16) and the derivatives of the constraint equations (5-2) give 86 V k k (f .6 + (f = 0 If/ + ) 1 1 1 - 1 2 i 1 3 . k, k k k k .a + (f + (f = 0 ) J iS > 1t 2 1 i 2 2 2 3 k ak + b. = 0 1 3 Orders H 2 (5-21) f,- are taken from Table 4-1. where the coefficients nn Therefore the requirements for any order of K become k k (f (f . ), ) 1 1 i i Oi 1 3 k k = k (f )i = 0. R. (f )j: (£ ) . l 2 1 1 2 2 1 2 3 O. 0 1 1 12 - ( 5 22 ) Thus for any finite or .infinitesimal precision point i, the requirements imposed by the prescribed precision point's coordinates and the conic's prescribed variables equations (5-20) are given by equation (5-19) or (5-22) depending on the case study. Coupler and Fixed Link S ynthesis for Burmester Constraints As one may surmise from previous discussion, the synthesis of coupler and fixed links for the Burmester constraints represents a simplified form of the conic constraint algorithm. Rewriting the generalized Burmester constraint equation, equation (5-9) , for the finite case a. s iny • + a„ cosy . + 0 i ; 13 i i (5-23) ) . 4 . 87 where a— are defined by equations (5-10). Prescribing variables for both constraints, results in the two equations - Constraint #1 (g. ^siny.^ + cosy + g 0 g,^ i i ^ Constraint #2 (f- siny. + f. cosy. + f. =0 J-i 1 12 i 13 (5-24) where = I< a g f (u , v , H , , , 3 ij 1 1 1 1 - f (u , v , H , K , a, 8) 2 2 2 2 i 1 / 2 , 3, j 1/2,3 as taken from equations (5-10). Equations (5-24) require 2 2 q g h, ^2 r 3 ii 12. "h f . f f f ' ii i i 3 12 2 (5-25) or 2 2 . f.* - f,- ~ fi ) (g. f. - g. f. ) + (g g. f. ) (gj X ’ ' 1 1 gi 1 h 1 2 3 ’ll h 1 3 1 1 1 2 2 1 (5-26) By synthesizing the coupler link variables { (u , v ) , (u , v ) } XI 2 2 will give for the unknown variables of equation (5-26) . 38 [A] H 0 5x6 H K 2 2 K d H K (5-27) 1 V , where [B] 1x6 (5-28) and u ) f (u v u v a. , 8 . ij , , , , 1 1 2 2X1 By specifying the fixed link variables ( (H , K ) , (H , X ) } 1 i 2 2 provides from equation (5-26) [C] 5x6 u 2 v (5-29) 2 1 l ) . 89 where [D] 1x6 U V 1 1 u 1 V (5-30) 1 1 \ and f(H , K , H , 1 1 2 Therefore one need only satisfy the above requirements for synthesis of either the coupler link or fixed link. As for the infinitesimal displacements the procedure is identical to that outlined by equations (5-21) and C5-22) CHAPTER VI SIX MULTIPLY SEPARATED POINT-POSITIONS IN COPLANAR MOTION In Chapter II the generalized Burmester equation (2-7) was given as A Q + A Q u + A Q v + A (uQ + vQ ) 0 0 l 0 3 2 9 i 0 H A 1 - + A (uQ vQ ) + A Q +A Q =0. 2 l s £ i 2 (6-1) Making the following substitutions 0 = z (uQ + vQ ) Z 0 0 1 2 3 - = Q u = Z (uQ vQ ) z 0 1 2 1 4 Q v - Z 0 = z 0 2 1 5 Q = z 2 6 - ( 6 2 ) gives the linearized expressions S A 2,-0 m Ta m-0 £ (6-3) as the required circular constraint formulation. The ~ six point-positions of the moving plane are Z 0,1, 2., 3,4 ,5 where the first position, Z — 0, is superimposed with the fixed origin, The last five of these positions raav be expressed as SO S 91 [A] {z} - 0 5x7 (6-4) In prescribing six point-positions of the moving plane, as illustrated in Figure (6-1) , the angular rotation of two of the positions must be compatible so that all combinations of five positions have two common Burraester pairs. Similarly, infinitesimal cases require invariant magnitude compatibility. Theorem I There exists a unique pair of Burmester constraints (pair) for six multiply separated position in coplanar motion Proo f For six multiply separated positions, equations (6--4) can be reconstructed in terms of 3 unknowns . This can be accomplished by Gauss-Siedel reduction or by Cramer's Rule. This allows one to establish the two G O ll cl C JL 0 II g Z g Z + g Z - 0 IQ 2 1 3 2 and C6-5) h Z + t h Z 0 o 3 2 wn.-re i 92 \