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On a general method for the synthesis of the path approximation mechanisms
Mechanism and Machine Theory,Vol. 14, pp. 289-298 Pergamon Press Ltd., 1979. Printed in Great Britain
On a General Method for the Synthesis of the Path Approximation Mechanisms Simionescu Iont and Duca Cezar, Received 3 March 1978; in revised form 3 September 1978; for publication 5 March 1979 Abstract The purpose of this paper is to give a general method for the synthesis of the path approximation mechanisms. The method is based on matrix algebra and is adjusted for use with a digital computer. The procedure described for applying the proposed method of synthesis is made on the one-loop spatial mechanism with seven links having constrained motion, but it may be employed for the synthesis of any type of simple closed or poly-loop mechanism,
1. T,E USE of matrix algebra in the geometrical and kinematic analysis of mechanisms is advantageous because it presents the calculations in a concise form, which can easily be converted into a programming language. In this respect, several works (Hartenberg and Denavit[1], D. Mangeron and Drfigan[2], Pelecudi[3]) have laid the theoretical bases for using the transformation matrices in the kinematic study of 3-dimensional mechanisms. The development of the matrix method and its effectiveness in solving the problem of acceleration, velocity and displacement analysis of spatial mechanisms in a unique manner is owed to Uicker Jr. [4--6]. In his works, Uicker Jr. [4--6] presents a simple and concise numerical method, easily accessible to the design engineers, for the investigation of any mechanism type, with the aid of the digital computer. This paper gives an extension of the matrix method for the synthesis of mechanisms used to approximate a path described by given points. Any type of planar or spatial, simple closed or poly-loop mechanism can be designed using the method described in this paper. 2. The mechanism synthesis which theoretically fits a given path exactly, is a problem which can only be solved for a limited number of cases. In general, a defined path can be approximated by the curved trajectory of a point of a link in a mechanism if both the given path and the considered curved trajectory have a finite number of common points. The maximum number of common points which the given path may have with the curved trajectory of the considered point of the mechanism, follows from the condition of uniqueness and exactness of the synthesis solution and depends on the mechanism complexity. This condition is achieved only when the number of equations in the system whose solutions represent the unknown parameters of the mechanism is equal to the number of the unknowns. When the number of imposed common points is less than the maximum one calculated in the above manner, some of the dimensions of the mechanism links may be imposed initially in a convenient way. If, on the tResearch,The NationalInstitutefor ThermalEngines,SplaiBahluiStingnr.61 bis, lasi, Romania. :~Lecturer,PolytechnicInstitute,Calea23 Augustnr.l1, lasi, Romania. MMT VoL 14, No. 5--A
289
290
contrary, the number of imposed positions exceeds the calculated maximum, it is possible to find a solution for the problem which would approximately correspond to the requirements. 3. An affine transformation in the three-dimensional space is defined analytically by expressing x2, Y2, z2 as functions of x~, y~, z~. Expressing this by a homogeneous matrix equation, we get: 1 X2
=
Y2 , Z2
,~
x2
=A •
a22
a23
a32
a33
a42
a43
Xl 34
Yl
a
or
Z2 i
Xl
(la)
zli
Consider two consecutive links belonging to a mechanism connected by a revolute or prismatic pair. Two right-handed cartesian coordinate systems are considered to be attached to each link. Between the coordinates of a point belonging to the former link and the coordinates of the same point considered on the latter link a correspondence may be established, defined by eqn (1). The transformation matrix A contains six independent parameters, among which there is an established number of relations imposed by the kinematic pair and which are equal to the pair class. If the transformation equation is written around the entire chain formed by m links, the m + 1 coordinate system being identical with the 1 coordinate system, then the product of the transformation matrices is equal to the 4 × 4 unit matrix (2)
ApA2.A3....Ai....Am = L
This equation is known as the matrix loop equation. A convenient choice of location of the coordinate systems attached to each mechanism iink[1-5] reduces the number of independent parameters of the transformation matrix from six to four. Thus, if: The zt axis is the characteristic axis of the ith pair. The x; axis is formed by the common perpendicular from the zi-j to zt axes, then the four parameters are: st is the distance along the zt axis from the xt to xi+r axes, at is the length of the common perpendicular from the zt to Zt÷r, ai is the angle between the zt and zt+j axes, taken as positive along the xt+~ axis, Ot is the angle between the positive xt and positive xi+~ axes, as seen from positive z~. In this case, the transformation matrix At has the following form
A~=
at clos Ot cos0 Ot - sin Ot 0 cos c~t sin o~Osinat a;sinOf sinO; cosOicosat - cos Ot sin at si 0 sin at cos a;
(3)
The cylindric kinematic pair, the spherical pair or other kinematic pairs of class lower than five can be reduced to combinations of revolute and prismatic pairs, attached to conveniently placed links. The position of the P point belonging to the i - 1 link, which is defined in the system attached to this link by the xtP, ytP, ziP coordinates, can be determined in the system attached to the frame by calculating the x~P, y~P, z~P coordinates by means of the equation
= AbAp...Ai_l " YlP I z~P
1 x~P y~P ziP
(4)
291
For calculating the coordinates of the P point in a certain system of coordinate axes, x0, yo, Zo, eqn (4) becomes
YoP zoP
= Ao.A1.A2....Ai-I • xiP y~P ziP
(4a)
where: Ao represents the transformation matrix of the coordinates of a point in the system of coordinate axes attached to the frame from the system of axes chosen arbitrarily. If the position of these two systems of coordinate axes is so chosen that the xj axis is the common perpendicular from the zo to zl axes, then the A0 transformation matrix has the particular form expressed by formula (3). This system of coordinate axes, xoy0zo, will hereafter be called the reference coordinate axes system. 4. Consider a simple closed spatial mechanism (Fig. 1) formed by seven binary links with constrained motion. Each link is completely defined by three of the four parameters mentioned above, namely: Theai, si and al for the links whose ith pair is a revolute one, or The aj, a t and 0j for the links whose jth pair is a prismatic one. Such a mechanism is therefore completely defined in terms of its link dimensions by 3 x 7 = 21 sizes. The position of the P point belonging to the i - 1 link is given in the system of coordinate axes attached to the link by the three coordinates, xiP, yiP, ziP. The position of the mechanism frame with respect to the reference coordinate axes system is defined by the four parameters present in the Ao transformation matrix, i.e. a0, ao, so, 0o. In all, the number of sizes necessary to build a mechanism for approximating the path by the curved trajectory of the P point is 28. The possibilities of computing these parameters are as follows: for each position of the mechanism, corresponding to each given point on the (T) path to be approximated, we can write a matrix loop eqn (2) equivalent to six algebraic equations, and a matrix eqn (4a) equivalent to three algebraic equations which define the position of the P point. These nine algebraic equations introduce an additional seven unknowns, namely the variables of the kinematic pairs, 0i or st. Thus, for each defined position of the P point whose curved trajectory approximates the (T) path, we have nine algebraic equations containing 28 + 7 = 35 unknowns. For n positions of the mechanism, if the number of equations must be equal to the number of the unknowns: 9n = 28 + 7n, then the maximum number of points through which P passes is 14. These points are given by their coordinates in the reference coordinate axes system. This assertion is true only if the subsystem made up of the algebraic equations which follow from the matrix eqns (4a) is also compatible. It is necessary to take into consideration this ~'Z 3
Zo ~
l
j Xo
Figure 1.
~'x2
,%
292
matter in choosing the link the P point belongs to. The mechanism under consideration has six movable links. It is useless to choose for this purpose links 1 and 6, which are attached to the frame. The links 2 with 5, and 3 with 4 have a similar position in the mechanism from this point of view. If the P point is on link 3, then the subsystem of the (4a) eqns contains 3n eqns having 16+3n unknowns, therefore compatibility always exists. If the P point is on link 2, the subsystem of the (4a) eqns contains 3n algebraic eqns having only 13 + 2n unknowns and it is compatible only when n ~< 13, which, of course, constitutes a limitation of the possibilities to approximate a given path. From now on, we will consider the P point to be on link 3 of the mechanism. In solving this problem, an iterative method for possible use with a digital computer is appropriate. The Newton-Raphson method is suggested, since the transformation matrices are continuous functions and have differentials with respect to the unknowns a~, o~, S i, 0 i in a vicinity of solutions. Expanding in a Taylor's series one of the transformation matrices, for instance, the one corresponding to the ith revolute kinematic pair when the mechanism is in the nth position, we get
Ai=fjg+{OAil
"~dai+(c~Ail
\Oai, .....,~,1
(OAil
/OA, I
\d
\-~ail.~=a,) dai+ \ Os~,~,=~)dsi+t~i.[o~ =G) Oi.
(5)
where rig, ai, sg, ~. represent the initial estimates of the unknown parameters a~, a~, s~, 0~.. Noting that this problem is to be solved by employing a digital computer, the introduction of several differentiation matrix operators is very useful in performing the differentiation indicated. We make use of the following definition:
aA,
= ,~,Q,.
OOi ai=a i
Under this definition, the Qo matrix is found to be the following[5, 6]
0 0 0 0 1
Q"=
0
0 0
0 0 0 0 0 0 0 0
Likewise OOti c~i=~~
where
Q~=
0 0 0 0
oo! 0 0
0
0 00 I
aA'] = 0o ,, 00i. [ol.=~i. where
i 0 0 ~' Oo = 0 - 1
OAi [
1
0
0
0
= Qs,4i,
293
where 0 0 0 0 0 0 0 0
~ = o o o o 1 0 0 0 With these expressions, eqn (5) becomes [7]
A~ = .X.i + AiQ, dai + fi.~Q,, dal + Qofi.i dOi, + Q,,E,~ dsi. If we note
(5a)
//'
X4P = x4P Y4P z4P
then we get in the same manner
X4P = -~4P + ('gX'P \ Ox4P
"~dx4P
x4P=y4p]
+
( 8X4P I
dy4P
\~-~4P4PIy4p=~4p] aX4P
dz4P.
+ ( Oz4P Iz4P=24P)
In this case we can also introduce the differentiation matrix operator. Thus
OX, P = Q, " f(4P Ox4P x4P=x4P where
Q=
o o 0 0 0 0
'
OX4P [' = Qy " f(4P Oy4P I y4P=~,4P where
Q=
ioo i oo O0 O0
ax'P I'
=
az4P I z4p=~.4p
'
Q~ " ff4P
where
Qz =
i°°i o o 0 0
0 0
"
(6)
294 With these formulae, eqn (6) becomes: X 4 P = X4P +
Q, "
.'~4P dx4P + Q~ " X 4 P dy4P + Q~ . X 4 P dz, P.
(6al
Substituting the relation (5a) in the matrix loop eqn (2) and performing the multiplications, we get a very lengthy and complicated equation. However, since we are making use of an iteration process, all higher order terms may be neglected. The matrix loop equation, for the nth position of the mechanism with all revolute pairs, becomes 7
7
7
7
6",., da, + ~] D,, dot, + ~ E,, d0,, + ~ Fi, as, = I - B, /=l
/=l
i=l
(7)
i=1
where
/3. = 5,,,.~i2 ..... AT.
D,, = AI,.A2 ..... A,,.Q,,.A,+, ...... .47, Ein = AIn.Az ..... Ai-l.n.Q,.min....mTn
F,. = .~,../i2 ..... ~,-,.,.Q,.~,,....5,7,. Replacing (6) in (4a) and neglecting all the higher order terms, for the nth position of the mechanism, we get 3
x0p.--
3
3
3
3
+
~X4
2F,.d,,,.o: e
+ Bn3(Qx " .X4P d x , P + O r . X 4 P d y , P + Oz" X 4 P dz4P)
(8)
where B. 3 = Y,o.A,°.A2..A3.
C~n = mo./6tl ..... min.Qa.Ai+l ...... m3n D~. = A o . a , ..... A,..Qo..4,+, ...... A 3 .
E ,3 = a o . A , , . . . . A H . , . O o . f t , ..... P,3. F,3. = Ao A , .... A,_,.,.Q.,.A, ..... fi~3..
Expanding eqn (7) and equating the matrix on the left side to that on the right side, element for element, we get 12 linear equations, 6 of which can be eliminated since they are either identities or restatements of the other six. To determine the equations which can be eliminated, we must make several observations about the transformation matrices A;[4--6], namely: the sizes characterizing the translation of the two coordinate axes systems, the a, and s, distances, are contained in the first column, and the 3 × 3 rotational submatrix is orthogonal. We can also notice that the differentiation matrix operators Q have zero elements in the top row, while the 3 × 3 submatrices, made up of the elements Q,-j (i, ./= 2, 3, 4), are antisymmetric. It is known that if H is a non-singular matrix, H t is its transpose, and Q is antisymmetric matrix, then the product Hj = H" Q . H '
295
is also an antisymmetric matrix. Making use of this, we can operate a series of changes on the Ci,, Di., E~,, F~, product matrices. Thus t i n = A1n.A2 ..... A,..Oa.,A,+I ...... .,,~7,(A,,.A2 ..... .4,,)-'-.'~,,.A2 ..... -4,,, or
Ci, = A1,.A2 ..... AI,.Oa(A1~.A2 ..... A i n ) - l A I n . A 2
. . . . . .~7n.
But (/~ln.A2 ..... Ain) -1 = (A~..A2 ..... A~)'
and A1,.A2 . . . . . A7n = I
if the initial estimates of the unknowns are exact. Therefore, the 3 × 3 rotational submatrix existing within the Ci, product matrix is antisymmetric. In a similar manner, it is possible to prove that the Di,, Ei., F~n product matrices have this remarkable characteristic, Using the above, it is obvious that from the matrix eqn (7), it is sufficient to keep only the six equations resulting from the equalization of the terms situated under the major diagonal. If, as it was assumed, the initial estimates are exact, the da, da, ds;, d0~, errors vanish, and the left side of eqn (7) also vanishes. The same must occur to the right side of the equation. The I - B~ difference vanishes if the B. product matrix converges to the unit matrix. We have already imposed that the elements existing under the major diagonal should vanish. In order that the elements in the major diagonal of the B, product matrix approach unity, it is necessary to add three additional equations to those already mentioned. These result from the last three elements of the major diagonal of the matrices contained in eqn (7). This means that for the n positions imposed on the P point on the (T) path, there results 12. n equations with 28 + 7n unknowns. Since the roots of this system are the approximation errors of the unknown parameters and because all higher order terms have been neglected in the calculations made so far, it is not possible to get an unique solution which mathematically exact for all these equations. However, applying the root mean square method, one can find a solution which should contain no significant errors. The roots of this system of errors improve the initial estimates of the unknown parameters of the mechanism and the entire process is repeated until the desired accuracy is obtained. If some of the kinematic pairs belonging to the mechanism are prismatic, the above formulae are changed accordingly with no difficulty. The process is not convergent if one of the mechanism positions is near a dead position of the mechanism. It is manifest that the synthesis problem has more solutions and obtaining one or another solution depends on the choice of the initial estimates of the unknown parameters of the mechanism. 5. The improvement of the path approximation, that is the increase of the number of points in which a given path is intersected by the P point curved trajectory, is obtained by increasing the number of links of the mechanism. In this manner it is possible to achieve double or poly-closed mechanisms, writing for each loop an additional matrix loop equation of type (2). A double closed mechanism formed by 12 links with constrained motion can approximate a given path by a maximum of 25 points specified by their coordinates in the reference coordinate axes system. The calculation of the exact dimensions of the links belonging to such a mechanism implies that we must solve a system of algebraic equations resulting from the development of two matrix loop equations A D,.A 2..A 3..A4,,.A ~n.A6..A 7,~ = [ A l,,.A z,,.A 3,,.A4..As,,.A9,,.A Io..A m,.A12..A13,, = I
as well as a matrix equation defining the position of the P point: XoP~ = Ao.AIn.A2..A3n.A4..Asn.A9..Alon.XllP.
296 I n a s i m i l a r m a n n e r , t h e p r o p o s e d s y n t h e s i s m e t h o d c a n b e e m p l o y e d f o r d e s i g n i n g all t y p e ~ of mechanisms.
6. Examples 6.1 T h e
7-1ink m e c h a n i s m
shown
in Fig.
1 is c h o s e n
as an example
to d e m o n s t r a t e
a p p l i c a t i o n o f t h e m e t h o d to a n o t h e r w i s e difficult p r o b l e m a n d its flexibility. A s t h e t r a j e c t o r y o f a P p o i n t in l i n k 3 a p p r o x i m a t e s
a c i r c l e g i v e n b y t h e f o l l o w i n g six p o i n t s
P~(XoPj =
-20.00
YoPl=
19.00
ZoPl =
16.00)
P2(XoP2
=
-
20.00
YoP2
4.28
ZoP2=
7.50)
P3(XoP3=
-
20.00
YoP3=
4.28
ZoP3=
- 9.50)
P4(XoP4=
- 20.00
YoP,=
19.00
ZoP4 = - 18.00)
Ps(XoP~=
- 20.00
YoPs=
33.72
ZoP~=
- 9.50)
P6(XoP6=
- 20.00
YoP6=
33.72
ZoP6=
7.50)
=
the following dimensions of the mechanism a2 = 3 2 . 5 7 4 0
a2 = 7 3 . 0 5 3 2 °
a3 = 10.7269
sl = 115.329
a6
=
23.1425
x4P =
- 78.541
a7
=
40.8621
Y4P =
- 17.869
znP =
17.5402.
a~ = 4 . 2 5 8 2 6 ° The mechanism
were calculated
p o s i t i o n r e l a t i v e to t h e r e f e r e n c e c o o r d i n a t e s y s t e m is d e f i n e d b y
a0 = - 49.6631
So = 20.7733.
The other dimensions of the mechanism
a r e i m p o s e d b y initial c o n d i t i o n s
al = 0 . 0
or3 = 0.0
a4 = 0.0
a4 = 90.0 °
as = 0.0
a5 = 90.0 °
o~6 = 0.0
$5 = 0.0
a7 = 0.0
s6 = 0.0
s2 = - 118.353
s7 = 0.0
s3 = 0.0
0o = 17.42 °
s4 = 0.0
ao = 32.42 °.
T h e first p o s i t i o n o f t h e m e c h a n i s m
( c o r r e s p o n d i n g to t h e P1 p o i n t ) is d e f i n e d b y
011 = 29.9103 °
0~1 = 2 5 0 . 3 8 4 °
021=
129.347 °
06t = 31.7453 °
031 = 1.20108 °
87r = 8 0 . 0 9 4 2 °
041 = 2 6 5 . 3 0 4 °.
297
Figure 2.
6.2 The synthesis of a 4--bar linkage which approximates a planar path (Fig. 2) implies the determination of eight dimensions, namely: aj, a2, a3, a4, x3P, y3P, ao, 0o. Therefore, the maximum number of given points through which the trajectory of the point P passes exactly is eight. If the trajectory of this point must approximate a curve given by ten points, namely
Pl(xoPl = 339.0
YoP1= 200.0)
P2(xoP~= 335.0
YoP~= 185.0)
P3(xoP3 = 298.0
YoP3 = 183.0)
P4(XoP4 255.0
YoP4= 183.0)
Ps(xoP5 = 215.0
YoP5 = 181.O)
P6(xoP6 = 179.0
YoP6 = 182.0)
P7(XoP7= 146.0
YoP7= 186.0)
Ps(xoP8 = 122.0
Y0P8 = 194.0)
P9(xoP9 = 132.0
YoP9 = 215.0)
Pio(xoPlo = 241.0
YoPlo= 239.0)
=
the accurate solution of this problem is not possible. A solution in the root mean square sense represents an optimum one for the synthesis problem, because the trajectory of the point P approaches almost all the given ten points. The dimensions of the mechanism calculated were: al = 162.366
x~P = - 243.088
as = 155.654
y3P = 50.9991
a3
=
58.4625
a4 = 116.540
ao = 222.581 0o = 0.143076°.
In the direction of the point P6, the deviation of the trajectory is maximum, namely d6 = 0.0141. For the beginning of the iterative process, we utilized the initial estimates: til= 60.0
~3P = - 250.0
298
6~ = 150.0
~P
= 30.0
~ = 150.0
¢-io= 100.0
64 = 120.0
~ = 0.0.
T h e e r r o r s da,, da2 . . . . etc. b e c a m e less than 0.05 m m (or deg) after five iterations.
References I. R. S. Hartenberg and J. Denavit, Kinematic Synthesis o[ Linkages. McGraw-Hill, New York (1964). 2. D. Mangeron and C. Drhgan, Asupra unei noi metode tensoriale de stidiu a mecanismelor. BuletinullnstitutuluiPolitehnic din la~i llI (VII) (1957). 3. Chr. Pelecudi, Teoria Mecanismelor Spatiale. Editura Academiei Republicii Socialiste Rom~inia, Bucure~ti tl972). 4. J. J. Uicker. Jr., J. Denavit and R. S. Hartenberg, An iterative method for the displacement analysis of spatial mechanisms. J. Appl. Mech., ASME Trans. 903-910 (1964). 5. J. J. Uicker, Jr., Displacement Analysis o[ Spatial Mechanisms by an lterative Method Based on 4 × 4 Matricesl MS. Thesis, Technological Institute, Evanston, Illinois (1963). 6. J. J. Uicker, Jr., Velocity and Acceleration Analysis o[ Spatial Mechanisms using 4 × 4 Matrices. Technological Institute, Evanston, Illinois (1963). "7. 1. Simionescu and C. Duca, Sinteza mecanismelor pentru realizarea poziliilor asociate cu ajutorul operatorilor matriceali. A IV-a Conferintd de Mecanic6. Bucure~ti (1975).
I. Simionescu und C. Duca E~u~z~asstu~ - Ziel de~ Arbelt ist es~ eine allgameine Methode £t~ die Syn~hese von
Getrleben, die elne dutch eine endllche Znzahl yon Punkten gegebene Ku/ve dutch die Ba.hnlcurve eines Gliedpunktes ann~hert, vorzustellen. DaB Verfahren berttht auf dez Ma~rizenalgebra und ist auf die Anwendung eines Computers abgestimmt. Die Anwendung der vorgelegten Synthesemethode wird am Beispiel eines slebenglied~igen zwangl~ufigen ~onokontu~-Raum6etriebes beschrleben. Sie kann abet auch f~r die Synthese ,Con Getrieben mit elnfach geschlossener odez Pol~kontur angewendet werden. Die Anwendung der Methode wird dutch zwel Belapiele erl~utert.
On a General Method for the Synthesis of the Path Approximation Mechanisms Simionescu Iont and Duca Cezar, Received 3 March 1978; in revised form 3 September 1978; for publication 5 March 1979 Abstract The purpose of this paper is to give a general method for the synthesis of the path approximation mechanisms. The method is based on matrix algebra and is adjusted for use with a digital computer. The procedure described for applying the proposed method of synthesis is made on the one-loop spatial mechanism with seven links having constrained motion, but it may be employed for the synthesis of any type of simple closed or poly-loop mechanism,
1. T,E USE of matrix algebra in the geometrical and kinematic analysis of mechanisms is advantageous because it presents the calculations in a concise form, which can easily be converted into a programming language. In this respect, several works (Hartenberg and Denavit[1], D. Mangeron and Drfigan[2], Pelecudi[3]) have laid the theoretical bases for using the transformation matrices in the kinematic study of 3-dimensional mechanisms. The development of the matrix method and its effectiveness in solving the problem of acceleration, velocity and displacement analysis of spatial mechanisms in a unique manner is owed to Uicker Jr. [4--6]. In his works, Uicker Jr. [4--6] presents a simple and concise numerical method, easily accessible to the design engineers, for the investigation of any mechanism type, with the aid of the digital computer. This paper gives an extension of the matrix method for the synthesis of mechanisms used to approximate a path described by given points. Any type of planar or spatial, simple closed or poly-loop mechanism can be designed using the method described in this paper. 2. The mechanism synthesis which theoretically fits a given path exactly, is a problem which can only be solved for a limited number of cases. In general, a defined path can be approximated by the curved trajectory of a point of a link in a mechanism if both the given path and the considered curved trajectory have a finite number of common points. The maximum number of common points which the given path may have with the curved trajectory of the considered point of the mechanism, follows from the condition of uniqueness and exactness of the synthesis solution and depends on the mechanism complexity. This condition is achieved only when the number of equations in the system whose solutions represent the unknown parameters of the mechanism is equal to the number of the unknowns. When the number of imposed common points is less than the maximum one calculated in the above manner, some of the dimensions of the mechanism links may be imposed initially in a convenient way. If, on the tResearch,The NationalInstitutefor ThermalEngines,SplaiBahluiStingnr.61 bis, lasi, Romania. :~Lecturer,PolytechnicInstitute,Calea23 Augustnr.l1, lasi, Romania. MMT VoL 14, No. 5--A
289
290
contrary, the number of imposed positions exceeds the calculated maximum, it is possible to find a solution for the problem which would approximately correspond to the requirements. 3. An affine transformation in the three-dimensional space is defined analytically by expressing x2, Y2, z2 as functions of x~, y~, z~. Expressing this by a homogeneous matrix equation, we get: 1 X2
=
Y2 , Z2
,~
x2
=A •
a22
a23
a32
a33
a42
a43
Xl 34
Yl
a
or
Z2 i
Xl
(la)
zli
Consider two consecutive links belonging to a mechanism connected by a revolute or prismatic pair. Two right-handed cartesian coordinate systems are considered to be attached to each link. Between the coordinates of a point belonging to the former link and the coordinates of the same point considered on the latter link a correspondence may be established, defined by eqn (1). The transformation matrix A contains six independent parameters, among which there is an established number of relations imposed by the kinematic pair and which are equal to the pair class. If the transformation equation is written around the entire chain formed by m links, the m + 1 coordinate system being identical with the 1 coordinate system, then the product of the transformation matrices is equal to the 4 × 4 unit matrix (2)
ApA2.A3....Ai....Am = L
This equation is known as the matrix loop equation. A convenient choice of location of the coordinate systems attached to each mechanism iink[1-5] reduces the number of independent parameters of the transformation matrix from six to four. Thus, if: The zt axis is the characteristic axis of the ith pair. The x; axis is formed by the common perpendicular from the zi-j to zt axes, then the four parameters are: st is the distance along the zt axis from the xt to xi+r axes, at is the length of the common perpendicular from the zt to Zt÷r, ai is the angle between the zt and zt+j axes, taken as positive along the xt+~ axis, Ot is the angle between the positive xt and positive xi+~ axes, as seen from positive z~. In this case, the transformation matrix At has the following form
A~=
at clos Ot cos0 Ot - sin Ot 0 cos c~t sin o~Osinat a;sinOf sinO; cosOicosat - cos Ot sin at si 0 sin at cos a;
(3)
The cylindric kinematic pair, the spherical pair or other kinematic pairs of class lower than five can be reduced to combinations of revolute and prismatic pairs, attached to conveniently placed links. The position of the P point belonging to the i - 1 link, which is defined in the system attached to this link by the xtP, ytP, ziP coordinates, can be determined in the system attached to the frame by calculating the x~P, y~P, z~P coordinates by means of the equation
= AbAp...Ai_l " YlP I z~P
1 x~P y~P ziP
(4)
291
For calculating the coordinates of the P point in a certain system of coordinate axes, x0, yo, Zo, eqn (4) becomes
YoP zoP
= Ao.A1.A2....Ai-I • xiP y~P ziP
(4a)
where: Ao represents the transformation matrix of the coordinates of a point in the system of coordinate axes attached to the frame from the system of axes chosen arbitrarily. If the position of these two systems of coordinate axes is so chosen that the xj axis is the common perpendicular from the zo to zl axes, then the A0 transformation matrix has the particular form expressed by formula (3). This system of coordinate axes, xoy0zo, will hereafter be called the reference coordinate axes system. 4. Consider a simple closed spatial mechanism (Fig. 1) formed by seven binary links with constrained motion. Each link is completely defined by three of the four parameters mentioned above, namely: Theai, si and al for the links whose ith pair is a revolute one, or The aj, a t and 0j for the links whose jth pair is a prismatic one. Such a mechanism is therefore completely defined in terms of its link dimensions by 3 x 7 = 21 sizes. The position of the P point belonging to the i - 1 link is given in the system of coordinate axes attached to the link by the three coordinates, xiP, yiP, ziP. The position of the mechanism frame with respect to the reference coordinate axes system is defined by the four parameters present in the Ao transformation matrix, i.e. a0, ao, so, 0o. In all, the number of sizes necessary to build a mechanism for approximating the path by the curved trajectory of the P point is 28. The possibilities of computing these parameters are as follows: for each position of the mechanism, corresponding to each given point on the (T) path to be approximated, we can write a matrix loop eqn (2) equivalent to six algebraic equations, and a matrix eqn (4a) equivalent to three algebraic equations which define the position of the P point. These nine algebraic equations introduce an additional seven unknowns, namely the variables of the kinematic pairs, 0i or st. Thus, for each defined position of the P point whose curved trajectory approximates the (T) path, we have nine algebraic equations containing 28 + 7 = 35 unknowns. For n positions of the mechanism, if the number of equations must be equal to the number of the unknowns: 9n = 28 + 7n, then the maximum number of points through which P passes is 14. These points are given by their coordinates in the reference coordinate axes system. This assertion is true only if the subsystem made up of the algebraic equations which follow from the matrix eqns (4a) is also compatible. It is necessary to take into consideration this ~'Z 3
Zo ~
l
j Xo
Figure 1.
~'x2
,%
292
matter in choosing the link the P point belongs to. The mechanism under consideration has six movable links. It is useless to choose for this purpose links 1 and 6, which are attached to the frame. The links 2 with 5, and 3 with 4 have a similar position in the mechanism from this point of view. If the P point is on link 3, then the subsystem of the (4a) eqns contains 3n eqns having 16+3n unknowns, therefore compatibility always exists. If the P point is on link 2, the subsystem of the (4a) eqns contains 3n algebraic eqns having only 13 + 2n unknowns and it is compatible only when n ~< 13, which, of course, constitutes a limitation of the possibilities to approximate a given path. From now on, we will consider the P point to be on link 3 of the mechanism. In solving this problem, an iterative method for possible use with a digital computer is appropriate. The Newton-Raphson method is suggested, since the transformation matrices are continuous functions and have differentials with respect to the unknowns a~, o~, S i, 0 i in a vicinity of solutions. Expanding in a Taylor's series one of the transformation matrices, for instance, the one corresponding to the ith revolute kinematic pair when the mechanism is in the nth position, we get
Ai=fjg+{OAil
"~dai+(c~Ail
\Oai, .....,~,1
(OAil
/OA, I
\d
\-~ail.~=a,) dai+ \ Os~,~,=~)dsi+t~i.[o~ =G) Oi.
(5)
where rig, ai, sg, ~. represent the initial estimates of the unknown parameters a~, a~, s~, 0~.. Noting that this problem is to be solved by employing a digital computer, the introduction of several differentiation matrix operators is very useful in performing the differentiation indicated. We make use of the following definition:
aA,
= ,~,Q,.
OOi ai=a i
Under this definition, the Qo matrix is found to be the following[5, 6]
0 0 0 0 1
Q"=
0
0 0
0 0 0 0 0 0 0 0
Likewise OOti c~i=~~
where
Q~=
0 0 0 0
oo! 0 0
0
0 00 I
aA'] = 0o ,, 00i. [ol.=~i. where
i 0 0 ~' Oo = 0 - 1
OAi [
1
0
0
0
= Qs,4i,
293
where 0 0 0 0 0 0 0 0
~ = o o o o 1 0 0 0 With these expressions, eqn (5) becomes [7]
A~ = .X.i + AiQ, dai + fi.~Q,, dal + Qofi.i dOi, + Q,,E,~ dsi. If we note
(5a)
//'
X4P = x4P Y4P z4P
then we get in the same manner
X4P = -~4P + ('gX'P \ Ox4P
"~dx4P
x4P=y4p]
+
( 8X4P I
dy4P
\~-~4P4PIy4p=~4p] aX4P
dz4P.
+ ( Oz4P Iz4P=24P)
In this case we can also introduce the differentiation matrix operator. Thus
OX, P = Q, " f(4P Ox4P x4P=x4P where
Q=
o o 0 0 0 0
'
OX4P [' = Qy " f(4P Oy4P I y4P=~,4P where
Q=
ioo i oo O0 O0
ax'P I'
=
az4P I z4p=~.4p
'
Q~ " ff4P
where
Qz =
i°°i o o 0 0
0 0
"
(6)
294 With these formulae, eqn (6) becomes: X 4 P = X4P +
Q, "
.'~4P dx4P + Q~ " X 4 P dy4P + Q~ . X 4 P dz, P.
(6al
Substituting the relation (5a) in the matrix loop eqn (2) and performing the multiplications, we get a very lengthy and complicated equation. However, since we are making use of an iteration process, all higher order terms may be neglected. The matrix loop equation, for the nth position of the mechanism with all revolute pairs, becomes 7
7
7
7
6",., da, + ~] D,, dot, + ~ E,, d0,, + ~ Fi, as, = I - B, /=l
/=l
i=l
(7)
i=1
where
/3. = 5,,,.~i2 ..... AT.
D,, = AI,.A2 ..... A,,.Q,,.A,+, ...... .47, Ein = AIn.Az ..... Ai-l.n.Q,.min....mTn
F,. = .~,../i2 ..... ~,-,.,.Q,.~,,....5,7,. Replacing (6) in (4a) and neglecting all the higher order terms, for the nth position of the mechanism, we get 3
x0p.--
3
3
3
3
+
~X4
2F,.d,,,.o: e
+ Bn3(Qx " .X4P d x , P + O r . X 4 P d y , P + Oz" X 4 P dz4P)
(8)
where B. 3 = Y,o.A,°.A2..A3.
C~n = mo./6tl ..... min.Qa.Ai+l ...... m3n D~. = A o . a , ..... A,..Qo..4,+, ...... A 3 .
E ,3 = a o . A , , . . . . A H . , . O o . f t , ..... P,3. F,3. = Ao A , .... A,_,.,.Q.,.A, ..... fi~3..
Expanding eqn (7) and equating the matrix on the left side to that on the right side, element for element, we get 12 linear equations, 6 of which can be eliminated since they are either identities or restatements of the other six. To determine the equations which can be eliminated, we must make several observations about the transformation matrices A;[4--6], namely: the sizes characterizing the translation of the two coordinate axes systems, the a, and s, distances, are contained in the first column, and the 3 × 3 rotational submatrix is orthogonal. We can also notice that the differentiation matrix operators Q have zero elements in the top row, while the 3 × 3 submatrices, made up of the elements Q,-j (i, ./= 2, 3, 4), are antisymmetric. It is known that if H is a non-singular matrix, H t is its transpose, and Q is antisymmetric matrix, then the product Hj = H" Q . H '
295
is also an antisymmetric matrix. Making use of this, we can operate a series of changes on the Ci,, Di., E~,, F~, product matrices. Thus t i n = A1n.A2 ..... A,..Oa.,A,+I ...... .,,~7,(A,,.A2 ..... .4,,)-'-.'~,,.A2 ..... -4,,, or
Ci, = A1,.A2 ..... AI,.Oa(A1~.A2 ..... A i n ) - l A I n . A 2
. . . . . .~7n.
But (/~ln.A2 ..... Ain) -1 = (A~..A2 ..... A~)'
and A1,.A2 . . . . . A7n = I
if the initial estimates of the unknowns are exact. Therefore, the 3 × 3 rotational submatrix existing within the Ci, product matrix is antisymmetric. In a similar manner, it is possible to prove that the Di,, Ei., F~n product matrices have this remarkable characteristic, Using the above, it is obvious that from the matrix eqn (7), it is sufficient to keep only the six equations resulting from the equalization of the terms situated under the major diagonal. If, as it was assumed, the initial estimates are exact, the da, da, ds;, d0~, errors vanish, and the left side of eqn (7) also vanishes. The same must occur to the right side of the equation. The I - B~ difference vanishes if the B. product matrix converges to the unit matrix. We have already imposed that the elements existing under the major diagonal should vanish. In order that the elements in the major diagonal of the B, product matrix approach unity, it is necessary to add three additional equations to those already mentioned. These result from the last three elements of the major diagonal of the matrices contained in eqn (7). This means that for the n positions imposed on the P point on the (T) path, there results 12. n equations with 28 + 7n unknowns. Since the roots of this system are the approximation errors of the unknown parameters and because all higher order terms have been neglected in the calculations made so far, it is not possible to get an unique solution which mathematically exact for all these equations. However, applying the root mean square method, one can find a solution which should contain no significant errors. The roots of this system of errors improve the initial estimates of the unknown parameters of the mechanism and the entire process is repeated until the desired accuracy is obtained. If some of the kinematic pairs belonging to the mechanism are prismatic, the above formulae are changed accordingly with no difficulty. The process is not convergent if one of the mechanism positions is near a dead position of the mechanism. It is manifest that the synthesis problem has more solutions and obtaining one or another solution depends on the choice of the initial estimates of the unknown parameters of the mechanism. 5. The improvement of the path approximation, that is the increase of the number of points in which a given path is intersected by the P point curved trajectory, is obtained by increasing the number of links of the mechanism. In this manner it is possible to achieve double or poly-closed mechanisms, writing for each loop an additional matrix loop equation of type (2). A double closed mechanism formed by 12 links with constrained motion can approximate a given path by a maximum of 25 points specified by their coordinates in the reference coordinate axes system. The calculation of the exact dimensions of the links belonging to such a mechanism implies that we must solve a system of algebraic equations resulting from the development of two matrix loop equations A D,.A 2..A 3..A4,,.A ~n.A6..A 7,~ = [ A l,,.A z,,.A 3,,.A4..As,,.A9,,.A Io..A m,.A12..A13,, = I
as well as a matrix equation defining the position of the P point: XoP~ = Ao.AIn.A2..A3n.A4..Asn.A9..Alon.XllP.
296 I n a s i m i l a r m a n n e r , t h e p r o p o s e d s y n t h e s i s m e t h o d c a n b e e m p l o y e d f o r d e s i g n i n g all t y p e ~ of mechanisms.
6. Examples 6.1 T h e
7-1ink m e c h a n i s m
shown
in Fig.
1 is c h o s e n
as an example
to d e m o n s t r a t e
a p p l i c a t i o n o f t h e m e t h o d to a n o t h e r w i s e difficult p r o b l e m a n d its flexibility. A s t h e t r a j e c t o r y o f a P p o i n t in l i n k 3 a p p r o x i m a t e s
a c i r c l e g i v e n b y t h e f o l l o w i n g six p o i n t s
P~(XoPj =
-20.00
YoPl=
19.00
ZoPl =
16.00)
P2(XoP2
=
-
20.00
YoP2
4.28
ZoP2=
7.50)
P3(XoP3=
-
20.00
YoP3=
4.28
ZoP3=
- 9.50)
P4(XoP4=
- 20.00
YoP,=
19.00
ZoP4 = - 18.00)
Ps(XoP~=
- 20.00
YoPs=
33.72
ZoP~=
- 9.50)
P6(XoP6=
- 20.00
YoP6=
33.72
ZoP6=
7.50)
=
the following dimensions of the mechanism a2 = 3 2 . 5 7 4 0
a2 = 7 3 . 0 5 3 2 °
a3 = 10.7269
sl = 115.329
a6
=
23.1425
x4P =
- 78.541
a7
=
40.8621
Y4P =
- 17.869
znP =
17.5402.
a~ = 4 . 2 5 8 2 6 ° The mechanism
were calculated
p o s i t i o n r e l a t i v e to t h e r e f e r e n c e c o o r d i n a t e s y s t e m is d e f i n e d b y
a0 = - 49.6631
So = 20.7733.
The other dimensions of the mechanism
a r e i m p o s e d b y initial c o n d i t i o n s
al = 0 . 0
or3 = 0.0
a4 = 0.0
a4 = 90.0 °
as = 0.0
a5 = 90.0 °
o~6 = 0.0
$5 = 0.0
a7 = 0.0
s6 = 0.0
s2 = - 118.353
s7 = 0.0
s3 = 0.0
0o = 17.42 °
s4 = 0.0
ao = 32.42 °.
T h e first p o s i t i o n o f t h e m e c h a n i s m
( c o r r e s p o n d i n g to t h e P1 p o i n t ) is d e f i n e d b y
011 = 29.9103 °
0~1 = 2 5 0 . 3 8 4 °
021=
129.347 °
06t = 31.7453 °
031 = 1.20108 °
87r = 8 0 . 0 9 4 2 °
041 = 2 6 5 . 3 0 4 °.
297
Figure 2.
6.2 The synthesis of a 4--bar linkage which approximates a planar path (Fig. 2) implies the determination of eight dimensions, namely: aj, a2, a3, a4, x3P, y3P, ao, 0o. Therefore, the maximum number of given points through which the trajectory of the point P passes exactly is eight. If the trajectory of this point must approximate a curve given by ten points, namely
Pl(xoPl = 339.0
YoP1= 200.0)
P2(xoP~= 335.0
YoP~= 185.0)
P3(xoP3 = 298.0
YoP3 = 183.0)
P4(XoP4 255.0
YoP4= 183.0)
Ps(xoP5 = 215.0
YoP5 = 181.O)
P6(xoP6 = 179.0
YoP6 = 182.0)
P7(XoP7= 146.0
YoP7= 186.0)
Ps(xoP8 = 122.0
Y0P8 = 194.0)
P9(xoP9 = 132.0
YoP9 = 215.0)
Pio(xoPlo = 241.0
YoPlo= 239.0)
=
the accurate solution of this problem is not possible. A solution in the root mean square sense represents an optimum one for the synthesis problem, because the trajectory of the point P approaches almost all the given ten points. The dimensions of the mechanism calculated were: al = 162.366
x~P = - 243.088
as = 155.654
y3P = 50.9991
a3
=
58.4625
a4 = 116.540
ao = 222.581 0o = 0.143076°.
In the direction of the point P6, the deviation of the trajectory is maximum, namely d6 = 0.0141. For the beginning of the iterative process, we utilized the initial estimates: til= 60.0
~3P = - 250.0
298
6~ = 150.0
~P
= 30.0
~ = 150.0
¢-io= 100.0
64 = 120.0
~ = 0.0.
T h e e r r o r s da,, da2 . . . . etc. b e c a m e less than 0.05 m m (or deg) after five iterations.
References I. R. S. Hartenberg and J. Denavit, Kinematic Synthesis o[ Linkages. McGraw-Hill, New York (1964). 2. D. Mangeron and C. Drhgan, Asupra unei noi metode tensoriale de stidiu a mecanismelor. BuletinullnstitutuluiPolitehnic din la~i llI (VII) (1957). 3. Chr. Pelecudi, Teoria Mecanismelor Spatiale. Editura Academiei Republicii Socialiste Rom~inia, Bucure~ti tl972). 4. J. J. Uicker. Jr., J. Denavit and R. S. Hartenberg, An iterative method for the displacement analysis of spatial mechanisms. J. Appl. Mech., ASME Trans. 903-910 (1964). 5. J. J. Uicker, Jr., Displacement Analysis o[ Spatial Mechanisms by an lterative Method Based on 4 × 4 Matricesl MS. Thesis, Technological Institute, Evanston, Illinois (1963). 6. J. J. Uicker, Jr., Velocity and Acceleration Analysis o[ Spatial Mechanisms using 4 × 4 Matrices. Technological Institute, Evanston, Illinois (1963). "7. 1. Simionescu and C. Duca, Sinteza mecanismelor pentru realizarea poziliilor asociate cu ajutorul operatorilor matriceali. A IV-a Conferintd de Mecanic6. Bucure~ti (1975).
I. Simionescu und C. Duca E~u~z~asstu~ - Ziel de~ Arbelt ist es~ eine allgameine Methode £t~ die Syn~hese von
Getrleben, die elne dutch eine endllche Znzahl yon Punkten gegebene Ku/ve dutch die Ba.hnlcurve eines Gliedpunktes ann~hert, vorzustellen. DaB Verfahren berttht auf dez Ma~rizenalgebra und ist auf die Anwendung eines Computers abgestimmt. Die Anwendung der vorgelegten Synthesemethode wird am Beispiel eines slebenglied~igen zwangl~ufigen ~onokontu~-Raum6etriebes beschrleben. Sie kann abet auch f~r die Synthese ,Con Getrieben mit elnfach geschlossener odez Pol~kontur angewendet werden. Die Anwendung der Methode wird dutch zwel Belapiele erl~utert.