A method for the identification and recognition of equivalence of kinematic chains

Mechanism and Machine Theory, 1975, Vol. 10, pp. 375-383. Pergamon Press. Printed in Great Britain

A Method for the Identification and Recognition of Equivalence of Kinematic Chains J. J. Uicker, Jr.t and A. Raicu$

Received on 23 May 1974 Abstract

This paper presents a method for determining a set of identification numbers for each possible kinematic chain. A theorem is also proven which shows that kinematically equivalent chains may be detected by the equality of these identifying numbers. An extension of the method allows testing the equivalence of chains inCluding different types of kinematic pairs. Another variation is also discussed which explicitly determines the kinematic loops and paths of different lengths. 1. Introduction OVER the past several years much work has been reported in the literature on the structural (topological) analysis and synthesis of mechanisms[I-32]. Motives behind these studies range from the desire for an orderly classification system, to studies of mechanism mobility, to the hope of identifying heretofore unknown mechanisms, to the need for the automated recognition of topology in generalized computer-aided design programs. A commonly recurring problem in many of the studies to date, however, particularly those utilizing the digital computer, is the lack of a conclusive and easily applied test for the possible isomorphism (equivalent topology) of two kinematic chains. Indeed, such a test has been sought for some time; even before Reuleaux[33], kinematicians have tried to devise classification systems which will allow a precise and distinctive terminology or symbolism for each possible linkage topology. Techniques for the recognition and identification of a given mechanism's kinematic structure can be divided into two main categories, graphical methods based on the visual inspection of various forms of simplified schematic diagrams[3,5,7,25,26], and numerical methods[4, 10, 16-20, 22-24] many of which are based on the theory of graphs. However, even with the current numerical methods, no efficient test is known which will determine whether two given kinematic chains are isomorphic without depending on the analyst to discover at least part of the answer by "inspection". Current tests tend to be heuristic and consist of applying a series of many basic tests to limit, but not conclusively exclude, the possibility of error. Also, because tests are applied in sequence, these algorithms tend to be inefficient, often resorting at the end to testing all possible permutations of element number arrangements. In this paper the authors present a theorem based on graph theory which leads to a numerical algorithm for testing for isomorphism of two kinematic chains. Although the theorem, as proven below, provides only a necessary (and not necessarily sufficient) test for uniqueness, the test is extremely selective and no two dissimilar kinematic chains are known to the authors which are not discovered by this test. tAssociateProfessor,Departmentof MechanicalEngineering,The Universityof Wisconsin,Madison,W153706,U.S.A. SFacultatea de Ma~ini ~i Utilaje, Institutul de Constructii Bucuresti, Bul. Republicii, No. 176, Sector 3, Bucuresti, Romania.

375

376

2. The Distance Matrix One possible symbolism for the topology or connectivity of a kinematic chain is the link-link form of the incidence matrix, more properly referred to as the distance matrix. Once the links of the chain have been numbered from 1 to n, the distance matrix D is defined as a square matrix of order n, each row and each column representing the link with the corresponding number. The elements of the matrix are then entered as either zero or unity, depending on the absence or presence of a direct kinematic connection (pair) between the links corresponding to that row and column+. 10 if link i is joined to link j by a pair, if link i is not directly joined to link j.

Dsj ~

(I)

For example, the distance matrix for the four-bar kinematic chain with links numbered as in Fig. l is

D=

1

0 1

1 0

o

1 t~

(2)

'

Note, however, that the form of the distance matrix is dependent on the order of numbering the links of the chain. For example, the four-bar chain, with links numbered as in Fig. 2 has the following distance matrix:

D*=

Iil

~ 01 01 (il

)

0 0 1 1

"

(3)

Yet, although the two chains are topologically equivalent, the distance matrices are not equal, the difference being that the numbering of links 3 and 4 has been reversed. The relationship between the two distance matrices (2) and (3) is through a permutation of the

C

A

Figure

I

D

1. Four-bar chain, open c o n f i g u r a t i o n .

I Figure

2. Four-bar chain, crossed c o n f i g u r a t i o n .

+The term distance matrix stems from the fact that the entries can be interpreted as the topologicaldistances (number of pairs) between links. Algorithmscan also be found to determinethe distancesbetween non-adjacentlinksby operatingon the entries of D (see for example [341).

377 r o w and column numbers. If E is the permutation matrixf which corresponds to the renumbering

of the links, L* = EL,

(4)

= 0 1 0 0 2 0 0 1 3 0 1 4 then D can be obtained from D* by the congruence transformation, D = E' D ' E ,

(5)

where E' denotes the transpose of E. Although this equation has only been demonstrated by example, it is true in general whenever the two matrices D and D* are the distance matrices of two isomorphic kinematic chains. Equations (4, 5) correspond to the definition of isomorphism for chains, i.e. that there exists a one to one correspondence between the numbering of links in the two chains (4) and that, when the links are consistently renumbered, the topology of the two chains is identical (5). Note the advantage of using the link-link (distance) form of the incidence matrix over the link-joint form, where two sets of permutations (link numbers and pair numbers) must both be dealt with in discovering isomorphism.

3. The Characteristic Polynomial The problem of testing for isomorphism has now been reduced to one of determining whether a permutation matrix E in fact exists which will make eq. (5) an identity for the D and D* matrices of the two chains in question. However, since there are n ! possible choices for this permutation matrix, trial and error is not an attractive alternative. Fortunately, a convenient numerical test does exist to determine the existence of such a permutation matrix. A well known theorem of matrix algebra (e.g. [35]) states that a congruence relationship such as eq. (5) can exist only if the characteristic polynomials of D and D* are identical:~, i.e. if [ x I - D I = J x I - D*[

(6)

for all x. For the above example, this becomes

[x 0olI:: !1 -1 0 -1

x -1 0

-1 x -1

=

x 0 -1

0 x -I

-

x4-4x2= X4--4X 2 which, since the polynomials are identical, demonstrates the isomorphism of the linkages of Figs. 1 and 2. From the above discussion, the following theorem can be stated. Theorem : Two kinematic chains which are isomorphic to each other have identical characteristic polynomials for their associated distance matrices. Its proof is clear from eqns (4-6). tA permutation matrix is derived from an identity matrix whosecolumnshave been reordered. Note that it has a positive or negative unit determinant. Also, it is a unitary matrix, i.e. its transpose is equal to its inverse. ~:Notethat this condition is also guaranteed if the eigenvalues of D and D* are identical. However, the characteristic polynomial coefficients will always be real integers, and are more easily found.

378

It should be pointed out that, although this test demonstrates a necessary condition for isomorphism, it has not been proven to be a sufficient condition. Although the authors believe the converse of this theorem to be true, it has not yet been conclusively proven. Nevertheless, this is an extremely selective test, and no counterexamples are known to the authors which disprove the converse theorem. More will be said about the converse theorem below.

4. Calculation Technique When working on a digital computer--one of the key goals of this test--the calculation of characteristic polynomial coefficients (or eigenvalues) is not a difficult chore. Most computer systems have such library programs available, e.g. [36]. However, when calculating by hand, it may be argued that the test is impractical because of the computation of (n × n) determinants. The Souriau-Frame algorithm, presented below, is quite easily applied to matrices of the form of the distance matrix, however, partly because the values encountered will always be small integers. Also, since most practical mechanisms have only a small number of links, the order of the distance matrix is not large and hand computation is often quite feasible. The Souriau-Frame algorithm[37] is suited to the numerical evaluation of the inverse and characteristic polynomial of the particular form of matrix required in this test. It is based on the fact that the inverse can be written as the ratio of the adjoint matrix and the determinant. [xl-

D ] -~ =

Adjoint [xI

- D]

M-DI

(7)

Since the determinant and the adjoint of a matrix of this form will both be polynomials in the parameter x, the coefficients of these polynomials will be expressed as A~ and dr. Adjoint[x I - I)] -

x n-, Ai

=x"lA,+x

Ixl- I)l =

"-2 A

2+ " ' + x A .

,+A.

(8)

Xn-ldj ] =o

= x"do+x"

Idl + x , 2d 2 + " ' + x d ,

t+d,.

(9)

Note that, since x appears only on the diagonal of ( x I - D), the adjoint is a matrix polynomial of order n - 1 while the determinant is a polynomial of order n. The Souriau-Frame algorithm is a recursive scheme which generates the coefficients Aj and dj. It is started by initializing do and A, to the values d,,= 1 and

A~ = I.

(10a,b)

The coefficients are then generated successively by the following relations: 1

d i = --~ Trace (AID), J

(11)

Ai+l = AiD + djl.

(12)

When these recursion relations are applied n times to the distance matrix, the coefficients of the desired characteristic polynomial (9) are found and can be used to test for isomorphism of one kinematic chain with another. Also, this algorithm has an additional advantage over most others, in that the term A,+, can be evaluated as a check on calculations. With no computation errors, A.+~ will be zero. As an example of the use of the Souriau-Frame algorithm, consider finding the characteristic

379 F

A

6

5

E

2

I

B

Figure 3. Watt six-bar chain. polynomial of the Watt six-bar chain (Fig. 3). The distance matrix is O 1 1 D= 0 0 1

l O 0 1 0 0

l 0 0 0 1 0

0 1 0 0 0 1

0 O 1 0 0 1

f 0 0 1 1 0

The polynomial coefficients are found as follows

~e=l, AI

=

A2 =

I, D,

dl = 0, d2 = -7,

--4 0 0 2 2 00 -5 1 0 0 2 0 1 -5 0 0 2 A3 = 2 0 0 -5 1 0 2 0 0 1 -5 0 0 2 2 0 0 -4

d3 ~ Os

0 0 00 -2 -2 0 -3 3 0 -2 0 0 3 -3 0 -2 0 3 0 0 -2 0 -3 0 0 -2 0 3 -3 0 -2 -2 0 0 0

d4 = 7s

A4 =

-3 0 0 A~ = - 2 -2 0

A6--

0 0 -2 -2 )7 1 0 0 -2 2 2 0 0 -2 1 0 0 2 1 ' 0 1 2 0 0 0 -2 -2 0

-1-]

00 -10 -10 -1 0 0 0 0 0 -1 0 1 1 0

o0iJ

Ii

I 0°0

I

0

'

d5 ~ O~

d6 = --1.

380 The characteristic polynomial of the Watt six-bar chain is, therefore, x 6 - 7x4 + 7x z - 1.

(13)

The characteristic polynomial of the Stephenson six-bar chain, shown in Fig. 4, is found in the same way and is x 6 - 7x" + 9 x 2 - 4 x .

(14)

6

I Figure 4. Stephenson six-bar chain.

5. Characteristic Coefficients A new classification scheme for mechanisms can be based on these polynomials by listing the coefficients of the corresponding polynomial in order. For example, the four-bar linkage would be identified as the 1/0/-4/0/0 chain, the Watt linkage as the 1/0/-7/0/7/0/-1 chain, and the Stephenson linkage as the 1/0/-7/0/9/--4/0 chain. This would provide a highly selective (believed unique) identification for each possible chain topology. It is also interesting to note that many of the coefficients can be given physical significance and may be found directly by inspection of the linkage itself. According to eq. (10a), the leading coefficient is always unity. Also, since the diagonal elements of the distance matrix must be zero, i.e. no kinematic pair can connect a link to itself, the second coefficient must be zero. This results from the fact that this coefficient is always equal to the trace of the D matrix[35]. The third coefficient is always the negative of the number of pairs in the chain. This is a result of the fact that the jth coefficient is equal to the summation of the determinants of all jth order principal minors of D [35]. Each non-zero second order principle minor of D contains the two unit entries indicating a pair, and one such principal minor occurs once for each pair. The fourth coefficient is always twice the number of three-pair loops (substructures) within the chain, and will be zero for mechanisms. This also results from the fact that all third order principal minors will be zero except for those with all off-diagonal terms of unity. The higher order coefficients have not been as carefully studied or explained by the authors. It is felt however, that more study will also show physical significance for these. It is also felt that the entries of the adjoint matrix coefficient Aj will show physical significance. This, however, has not been studied by the authors. 6. Comments on the Converse Theorem

As stated above, the converse theorem which guarantees that the characteristic polynomial is a sufficient test for isomorphism, i.e. that the characteristic coefficients are unique for a given topology, is believed to be true. However, the authors have not been able to conclusively prove this converse. Arguments which lead in this direction can be based on a study of the possible topologies implied by non-zero determinants of the principal minors. Each non-zero determinant of a jth order principal minor will imply one of only a few possible connectivities among the j links so connected. The possibility of equal coefficients for non-isomorphic topologies exists only if it is possible to find that the interchange of two (or more) of these implied connectivities can take place without affecting the remaining coefficients, and without defying the other constraints of the problem. For example, these equal coefficients must occur with a symmetric matrix of known order, having only zero and unit entries, with zeroes on the major diagonal, and having precisely the same number of unit entries as pair elements in the chain. The authors feel that this is

381

extremely unlikely. However, it is difficult to express and manipulate in conjunction with the known theory of the eigenvalue problem, particularly the conditions implied by only zero and unit entries. Another very strong argument can be made for the truth of the converse theorem. The first author has written a computer program to evaluate the characteristic coefficients of all the known four-, six-, eight- and ten-bar chains [3, 5, 24] and no contradictions have been found. The authors would be most happy to hear from others who may find further information on the proof of this converse theorem.

7. Variations of the Method Many interesting variations of the above test for isomorphism have been found, and each has its own special properties and advantages. Most of these stem from a redefinition of the form of the entries in the distance matrix (eq. 1). The unit entries of the distance matrix can each be replaced with a symbol indicating the type of kinematic pair. For example, the distance matrix for the slider-crank linkage shown in Fig. 5 can be written as

D= R 0 R0 R 0 i1 0 P

R 0

0 R

(15)

"

The characteristic polynomial will still be invariant with the renumbering of links, and is x 4 + (-3R ~- P:)x 2+ ( - R " + 2R 3p _ R2p2).

(16)

The disadvantage of this variation of the method is that it is more difficult to program for digital computer evaluation, and standard library programs cannot be used. However, the Souriau-Frame algorithm can be programmed for this additional complication, and this variation is very powerful if the test for isomorphism is to distinguish between chains which differ only in the location of a variety of pair types. Possibly the most interesting variation found by the authors involved defining an implied sense for each pair in the chain. Then, in the distance matrix, each non-zero entry was made as a literal symbol for the particular pair, with the symmetric entry (indicating the same pair, but oppositely directed) carrying a bar over the symbol. For example, the Stephenson chain of Fig. 4 would have the following entries in this form of the distance matrix O a c b O O ~ 0 0 0 0 f C 0 0 0 0 e 1)= b 0 0 0 d 0 0 00 d 0 g 0 f ~ 0 ~ 0

(17)

Next, in applying the Souriau-Frame algorithm with the symbolic entries, the barred symbols were given the interpretation of the logical NOT, i.e. t] was interpreted as NOT(a), and all products of the form at~ were set equal to zero. The resulting characteristic polynomial for the

I

I

L

Figure 5, Slider-crank mechanism.

MMT VoL 10, NO. 5---B

382 Stephenson chain example (17} is

x ~'+ (afO( + gffec )x ; + (aNdb + fifgdb + ce~db + OO,~db)x.

(181

Note that, with the non-symmetric entries, the above theorem is no longer valid since the symbol and sense of each pair is chosen arbitrarily, and the form of each coefficient will depend on these choices. However, if the logical A N D and OR operations are associated with multiplication and addition respectively, an entirely new property appears in the characteristic coefficients. The coefficients now show explicitly all possible choices for the kinematic loops of the chain! Also, if TRUE and F A L S E values are assigned to the pair symbols, the coefficients will reduce to only the independent kinematic loops! In addition, the coefficient matrices A~ of the adjoint expression (8), under the same conventions, will show all possible topological paths of length j between each combination of links! For example, the A~ coefficient matrix for the Stephenson chain is

0 fsf A, = ,j _ e N 0 Ig,/ci +gE( L 0

cef 0 O 0 rig:fdba ~¢a

af~ 0

0 fgd e~d 0

dg~ dbc 0 [8c .[ab+~eb

a[~ + ce~ 8bd (bd 0 0 0

0 ace e,t.f baf + b,'e 0 0

(19)

8. Conclusions The authors do not claim that this paper presents an exhaustive study of this new method of identification and structural analysis of kinematic chains. There is much work which can yet be done before a final, "best" form is chosen. However, it is felt that this paper presents a new concept on which a new classification system for mechanisms can be based. Such a new classification system would be extremely selective and would minimize, if not completely eliminate, the possibility of duplicate identifications for structurally different mechanisms. The areas which most deserve additional attention are: (a) the proof of the converse to the theorem of Section 3, (b) the search for other possible meaningful forms for the distance matrix and the resulting coefficients, (c) the development of general computer programs for the application of each variation found, and most important, (d) the continued search for the physical meaning behind each of the coefficients. Hopefully, it might be possible to develop a form for which all coefficients could be identified immediately by inspection. This would eliminate the need for a tedious evaluation process and provide a very powerful classification system.

References [ll CROSSLEY F. R. E.. A contribution to Gruebler's theory in the number synthesis of plane mechanisms. Trans. ASME, .1. Engng. fnd. 86B(1), 1-8 (1964). 12] MANOLESCU N. I., Une m~thode unitaire pour la formation des chaines cin6matiques el des m6canismes phms articul6s avec diff6rents degr6s de libert6 et mobilit6. M¢canique Appliqu~e 9(6), 1263-1313 (19641. [3] CROSSLEY F. R. E., The permutations of ten-link plane kinematic chains. Antriebstechnik 3(5), 181-185 (1964). [41 FREUDENSTEIN F. and DOBRJANSKYJ I,., On a theory of the type synthesis of mechanism>,. Proc. l lth hltern. Congr. of Applied Mechanics, pp. 420-428. Springer, Berlin (19641. 15] CROSSLEY F. R. E., Permutations of kinematic chains with eight members or less from the graph-theoretic viewpoint. Developments in Theoretical and Applied Mechanics Vol. 2, pp. 467-486. Pergamon Press, Oxford (19651. [6] CROSSLEY F. R. E., On an unpublished work of Air. J. Mechanisms !(2), 165-170 (19661. [71 DAVIES T. H. and CROSSLEY F. R. E., Strnctural analysis of plane linkages by Franke's condensed notation. J. Mechanisms 1(2). 171-184 (19661. [8] FREUDENSTEIN F., The basic concepts of Polya's Theory of enumerations with applications to the structural classification of mechanisms. J. Mechanisms 1(31, 275-290 (19671. [9] DEJONGE A. E. R., The structure and synthesis of plane kinematic chains. Annals oJ the New York Academy o.! Sciences 136(211, 575-654 (19671. [10l DOBRJANSKYJ L. and FREUDENSTEIN F., Some applications of graph theory to the structural analysis of mechanisms. Trans. ASME, J. Engng. h~d. 89B(11, 153-158 (19671. [111 HAIN K., Systematik sechsglieder kinematischer Kenen, Maschinenmarkt 74(381, (19681. [12] DAVIES T. H., An extension of Manolescu's classification of planar kinematic chains and mechanisms of mobility m/> I, using graph theory. J. Mechanisms 3(2), 87-100 (1968). 1131 MANOLESCU N. I., For a united point of view in the study of the structural analysis of kinematic chains and mechanisms. J. Mechanisms 3(3), 149-170 (19681.

383 [14] BUCHSBAUM F. and FREUDENSTEIN F., Synthesis of kinematic structure of geared kinematic chains and other mechanisms. J. Mechanisms 5(3), 357-392 (1970). [15] ROSSNER W., Zur struckturellen Ordnung der Getriebe. Wissenschaft. Tech. Univ. Dresden 10, 1101-1115 (1961). [16] ATANASIU M. C. and RO~CA I., The classification of plane mechanisms using the associated graphs. Studii si cercetdri de mecanic6 applicatd 30(2), 402 (1971) (in Romanian). [17] RAICU A., The use of numerical and literal symbols for Type Analysis and Synthesis of Mechanisms. Bulletimd Sciintific Inst. Constructii Bucuresti XIV(1-2), 34%367 (1971) (in Romanian). [18] RAICU A., Methode pour la determination des schemas cinematiques des mecanismes. Proc. Third World Congr. for the Theory of Machines and Mechanisms D(16), 223-267 (1971). [19] HUANG M., Application of linear and non-linear graphs in structural synthesis of kinematic chains Doctural dissertation, Oklahoma State Univ., Stillwater, Okla., U.S.A. (1972). [20] MANOLESCU N. I., A method based on Baranov trusses, and using graph theory to find the set of planar jointed kinematic chains and mechanisms. Mechanism and Machine Theory 8(1), 3-22 (1973). [21] VERHO A., An extension of the concept of the group. Mechanism and Machine Theory 8(2), 24%256 (1973). [22] RAICU A., Matrices associated with kinematic chains with from 3 to 5 members. Mechanism and Machine Theory 9(1 ), 123-130 (1974). [23] MANOLESCU N. I., On mechanisms composed of kinematic chains of different families (in Romanian). Stud. Res. appl. Mech. XII, 589-615 (1961) (in Romanian). [24] WOO L. S., Type synthesis of plane linkages. Trans. ASME, J. Engng. Ind. 89B(1), 15%172 (1967). [25] FRANKER., Vom Aufbau der Getriebe. VDI Verlag, Dusseldorf, Germany (1958). [26] ASSUR L. V., Collected Works on the St~dy of Plane Pivoted Mechanisms with Lower Pairs (edited by I. 1. Artobolevskii). Moscow Akad. Nauk (1952) (in Russian). [27] MOLIAN S., Storage and retrieval of descriptions of mechanisms and mechanical devices according to kinematic type. J. Mechanisms 4(4), 311-324 (1969). [28] KOLCHIN N. I., Experiment in the construction of an expanded structural classification of mechanisms and a structural table based on it. Trans. 2nd USSR Conf. Basic problems of the theory of machines and mechanisms L Moscow, pp. 85-97 (1958) (in Russian). [29] VOINEA R. P. and ATANASIU M. C., Contributions ~ I'~tude de la structure des chafnes cinematiques. Bull. Polytech. Inst. Bucharesti 22, 3-51 (1960). [30] BUGAEVSKI E., BOGDAN R. C. and PELECUDI C., Contributions to the classification of spatial mechanisms. Stud. Res. appl. mech. VIII, 8, 407 (1957) (in Romanian). [31] PELECUDI C., The method of contours in numerical and structural synthesis of kinematic chains. Stud. Res. appl. Mech. 25(34 573-599 (1967) (in Romanian). [32] HARRISBERGER L., Number synthesis survey of three-dimensional mechanisms. Trans. ASME, J. Engng. Ind. 87B, 213-220 (1965). [33] REULEAUX F., TheoretischeKinematische. Friedrich Vieweg, Braunschweig, Germany (1875); English translation by A. B. W. Kennedy, Reuleaux' Kinematics of Machinery. Macmillan, London (1876); reprinted by Dover, New York (1963). [34] SESHU S. and REED M. B., Linear Graphs and Electrical Networks. Addison-Wesley, Reading, Mass. (1961), [35] HILDEBRAND F. B., Methods of Applied Mathematics, 2rid Edn. Prentice-Hall, Englewood Cliffs, New Jersey (1965). [36] Scientific Subroutine Package, Version II. Programmer's Manual, IBM Corporation, White Plains, New York. U.S.A. (1967). [37] FRAME J. S., Matrix functions and applications--IV. Matrix functions and constituent matrices. IEEE Spectrum 1, 123-131 (1964). Ein Verfahren fGr die Identifizierun~ Gleichwertigkeit J. J. Uicker, Kurzfassun~

und Erkennun~ der

kinematisc~er Ketten

Jr. und A. Raicu

- Zur Bestimmung einer Gruppe yon Kennzahlen f ~

jede m ~ l i c h e

kinematische Kette wird eine Meti~ode geschil-

deft. Ein Lehrsatz wird entwickelt,

der zeigt, dab = gleich-

wertige kinematisc~e Ketten dutch die Gleichheit dieser Kennzahlen bewiesen ~ird. Diese Kennzahlen sind die Koeffizienten des eigenwertigen

Polynoms der topologischen Abstand-~iatrix

der Kette und deshalb unabh~ngig yon der Numerierung der einzelnen

Glieder.

Eine Erweiterung der MetLode erlaubt die Probe auf Gleichwertigkeit von Ketten, enthalten.

die verschiedene

Typen von Gelenken

Eine andere Variation wird diskutiert,

kinematisonen Maschen und Wege verschiedener bestimmen.

die die

Lange explizit