~ Pergamon Mech. Mach. Theory Vol. 30, No. 8, pp. 1295-1303, 1995

Copyright © 1995 Elsevier Science Ltd 0094-114X(95)00~5-3 Printed in Great Britain. All rights reserved

0094-114X/95 $9.50 + 0.00

POLYNOMIAL SOLUTION TO THE POSITION ANALYSIS OF THE 7-LINK ASSUR KINEMATIC CHAIN WITH ONE

QUATERNARY LINK

CARLO INNOCENTI Dipartimento di lngegneria delle Costruzioni Meccaniche, Facolt~ di Ingegneria, Universifft di Bologna

Viale Risorgimento, 2 - 40136 Bologna, Italy

(Received 9 July 1993; in rev&ed form 17 February 1994; received for publication 30 May 1995)

Abstract--Following a couple of previous papers devoted to the position analysis of two different 7-1ink Assur kinematic chains, this paper presents the algebraic-form position analysis of the remaining 7-1ink Assur kinematic chain, which can be distinguished from the others because it alone features one quaternary link. The position analysis, devoted at determining all possible assembly configurations of the considered Assur kinematic chain, is performed by first devising a system of two algebraic equations in two unknowns. After dialytic elimination, a final polynomial equation of eighteenth order is found whose solutions provide, in the complex field, 18 assembly configurations for the examined Assur kinematic chain. As a corollary, it can also be stated that two coupler point curves drawn by two distinct four-bar linkages intersect each other at 18 real points at most. Finally, a numerical example shows application of the new theoretical results.

1. I N T R O D U C T I O N

The position analysis of a linkage is aimed at finding the assembly configurations that correspond to a given choice of the variables of motion in the actuated kinematic pairs. The problem is usually transposed in mathematical terms by specifying a set of constraint equations whose unknowns are the parameters defining the position of all driven links of the mechanism. Consistently with the strategy pursued in selecting the constraint equations, either of two well-known approaches can be followed, namely, the global or the modular approach.

According to the global approach, a number of constraint equations are collected--no matter how involved--and subsequently solved via numerical techniques. Convergence is achieved only if a good estimate of the solution is a priori available [1, 2]: that is why this approach is generally unsuitable for computing all closure configurations.

The modular approach, on the other hand, offers a criterion for identifying the constraint equations in their simplest form. Such a preliminary step is of basic relevance for analytical solution of the position analysis or, which is the same, determination of all assembly configurations of a mechanism [3-5]. Being able to compute all assembly configurations is particularly valuable in dimensional synthesis of linkages if performed by iterative optimization techniques. Since the outcome of optimization depends on the initially selected assembly configuration, the ability to scan all possible initial assemblies considerably enhances the probability of finding a satisfactory solution to the synthesis problem.

The modular approach takes advantage of the observation that the position analysis of a linkage is equivalent to the position analysis of a structure, that is, the one obtained from the linkage by freezing--at the given specified posit ions--all actuated kinematic pairs. In facing the problem of determining the assembly configurations of a structure, the modular approach prescribes a search for substructures that can be assembled regardless of the remainder of the structure (a substructure is a set of links that can be ideally obtained from the original structure by suppressing other links in such a way that the resulting kinematic chain has zero mobility). I f a substructure with more than one link is spotted, it can be assembled on its own by solving a reduced number of constraint equations in a corresponding number of unknowns. Once assembled, the substructure is replaced by a rigid link in the original structure, and the process reiterated, thus substantially reducing the complexity of the position analysis of the linkage.

1295

1296 Carlo Innocenti

Obviously, adoption of the modular approach does not guarantee success in solving the position analysis of a given linkage, because any encountered substructure has eventually to be assembled. Although the constraint equations for a substructure could be numerically solved [1], the modular approach is mainly adopted for performing the position analysis of linkages in analytical form, which also calls for each of the involved substructures to be solved analytically.

Clearly, the modular approach proves fully efficient if substructures are singled out that cannot be further reduced. These special substructures, generally referred to as Assur kinematic chains (AKCs), represent the core elements of the modular approach, since their assemblage cannot be further eluded. If any AKC is solved in analytical form, a univariate polynomial equation is found whose roots correspond to the assembly configurations of the AKC. Although for degrees greater than four a polynomial equation must be numerically solved, this does not prevent any solution from being determined, since every root of an algebraic equation can be numerically computed by robust algorithms.

Although AKCs are infinite in number, the large majority of linkages deserving practical interest require consideration of a limited set of AKCs. If the analytical-form position analysis of these AKCs were available, they could be ideally filed in a library in order to be readily singled out as soon as needed. Unfortunately, few AKCs have so far been assembled in analytical form. If attention is focused on AKCs with exclusively revolute pairs (the prismatic pair is a special revolute pair having its center at infinity), the solved AKCs are only the four represented in Fig. 1, i.e., the triad, the pentad, and two 7-1ink AKCs. The twofold solution for the triad [see Fig. l(a)] has been reported by Suh and Radcliffe [3]. The pentad [see Fig. l(b)] was solved for the first time by Peysah [6], subsequently by Li and Matthew [7], and Gosselin et al. [8]; it admits six assembly configurations in the complex field.

Following triad and pentad are, in order of increasing complexity, the three AKCs with seven links. The analytical-form position analyses of the two 7-1ink AKCs represented in Figs l(c) and l(d), referred to as 7a- and 7b-AKC, have been presented in [9] and [10] where 14 and, respectively, 16 assembly configurations were declared possible in the complex field.

(a) (b)

(c) (d)

Fig. I. The solved Assur kinematic chains.

7-1ink Assur kinematic chain 1297

This paper presents the position analysis in analytical form of the last of the three existing 7-1ink AKCs, henceforth referred to as 7c-AKC and represented in Fig. 2. It features nine revolute pairs joining four binary links, two ternary links, and one quaternary link. The analysis is performed by first devising a suitable kinematic model that allows a set of two constraint equations in a corresponding number of unknowns to be written. After algebraic elimination, a final polynomial equation of eighteenth order is obtained whose 18 roots represent as many assembly configurations for the 7c-AKC in the complex field.

The reported results conclude the efforts devoted to finding a polynomial solution to the position analysis of the 7-1ink AKC family: only now, for instance, can the position analysis of all one-degree-of-freedom mechanisms derived from the 16 8-1ink kinematic chains reported by Hain [11] be solved exhaustively.

In addition, the presented results contribute to the characterization of a locus often considered in planar kinematics, namely, the four-bar coupler point curve. As a matter of fact, the position analysis of the 7c-AKC corresponds to finding all intersections of two coupler curves generated by two distinct four-bar linkages. Hence, it can also be stated that two four-bar linkages intersect each other at 18 real points at most.

Finally, a numerical example shows application of the proposed procedure to a case study.

2. THE KINEMATIC MODEL

With reference to Fig. 3, a 7c-AKC is considered as having a reference frame Oxy attached to the quaternary link A~B~ A2 B2. Dimensions of all links and, in particular, lengths ej, fj, mj, nj, angles ctj, and coordinates in reference frame Oxy of points A; = (x,j, yoj) and Bj = (Xbj, Ybj) (J = 1, 2) are to be considered as analysis data.

If the revolute pair placed at P is released, the structure becomes a two-degree-of-freedom mechanism (see Fig. 4). Indeed two four-bar linkages Aj B~ E~ F~ and A2B2E2F2 can be spotted both having the quaternary link A~ B~ A2B2 as frame. The former point P of the 7c-AKC can still be singled out on each four-bar linkage, and labeled as Pj (j = 1,2).

The locus of possible locations of point Pj (j = 1,2) with respect to Oxy is a four-bar coupler point curve. Each mutual intersection of the coupler curves traced out by P~ and P2 defines a possible location for point P of the 7c-AKC or, which is the same, an assembly configuration for the 7c-AKC. The converse is also true, i.e., each assembly configuration of the 7c-AKC places point P at an intersection of the coupler curves generated by P~ and P2. As a consequence, performing the position analysis of the 7c-AKC is equivalent to finding the intersections of the coupler curves described by points P~ and P2 of the four-bar linkages represented in Fig. 4.

The remainder of the present section will be devoted to determining the cartesian equations of the above mentioned coupler curves, as well as the constraint conditions for the 7c-AKC.

2. I. The coupler point curve equation

Deduction of the coupler point curve equation, although substantially similar to those traceable on planar kinematics treatises [12-14], is here reported for the sake Of completeness.

. '..... ,7..'..:-~ ...

"~..'...'/.~

,°,.,.*,%

~i.':~.'..:.

Fig. 2. The 7c-AKC.

1298 Carlo Innocenti

P

m E1 ~ F 2

O x Fig. 3. The geometric parameters of the 7c-AKC.

n 2

P1

E I ~F2

E 2

A 2

A 1

¢ £

Fig. 4. The two four-bar linkages resulting from the 7c- AKC.

B2

Figure 5 represents the coupler of either of the two four-bar linkages of Fig. 4 in a generic location relative to the reference frame Oxy on the quaternary link of the 7c-AKC. The coupler location can be parametrized by coordinates xj and yj of points Pj ( j = 1, 2) in Oxy, and by angle 0j measuring the counterclockwise rotation that superimposes the x-axis of Oxy on vector (Ej - Pj). In Fig. 5, the centers of revolute pairs Aj and Bj on the quaternary link of the 7c-AKC are also spotted.

Parameters x i, yj, and 0j together represent a consistent location for the coupler of the j - th four-bar linkage (j = 1, 2) if the congruence equations

2 (la) ( E j - A~) 2 = mj

Bj) = nj (lb)

are satisfied. On the left-hand side of equations (1), the square of a vector stands for the scalar product of the vector by itself.

In order to put equations (1) into explicit form, some positions are introduced:

(Ej - As) = (Ej - Pj) + (Pj - As,) (2a)

(F; - Bj) = (F; - Pj) + ( P j - B;) (2b)

yl O

B . ~ j (Xbj' Ybj) -(3

nj .. -- *~

Fj .. -* "* ~

J X ~' ~m J X

Pj = (x j, yj) x O

Aj = (Xaj, Yaj )

x

Fig. 5. The j-th (j = 1,2) four-bar coupler.

7-1ink Assur kinematic chain 1299

where:

(Ej - Pj) = ej(cj, sj) T (3a)

(~ - Pj) = f ( u j c j - vjsj, vjcj + ujsj) r (3b)

(Pj -- At) = (x r - x~r, yj - y~r) T (3c)

(Pr - Br) = (xr - Xbr, Yj -- Ybr) ~ (3d)

and T means transpose. In writing relations (3a) and (3b), the fo l lowing notat ion has been adopted:

c r = cos Oj; sj = sin 0 r (4a)

uj = cos cer; v r = sin ~j (4b)

By substituting posi t ions (2) in equat ions (1), and rearranging, the fo l lowing relations can be obtained:

[er(x r - x~j)]c r + [er(y r - y . r ) ] s j + [e 2 - m 2 + ( x j - Xaj) 2 "Jr- (Yr --Y~r)2]/2 = 0 (5a)

{ £[ur(xr -- Xbr) + vr(yj -- Ybr)]}Cj + {£[-- Vi(Xj -- Xbr) + ur(Yr - Ybr)]}sj

+[J~J -- n2 + (xr - Xbr) 2 + (Yr -- Ybr)2]/2 = 0 (5b)

which are linear in c r and sj. Eliminat ion o f u n k n o w n 0 r is accompl i shed by first solving system (5) for cj and s r via Cramer's

rule:

c r = O , j ( x j , y j ) /Doj(Xj , Yr); sr = D2j(xr, YJ)/Dor(XJ, YJ) (6)

In relations (6), Dor(Xy, y j ) , D~j(x r, y j ) , and D2j(xj , y j ) are polynomials of degree two, three, and three respectively in xj and yj; their coefficients depend exclusively on the geometry of the j th four-bar linkage. If the following congruence condition is imposed

2 2 1 (7) C r + Sj =

and expressions (6) substituted for cj and sj, one can obtain

[D, j (x j , yj)]2 + [D2j(xj, yr)]2 _ [Doj(Xj, yj)]2 = 0 (8)

Equation (8) can be given the arranged form

E h k Cjhk Xj yj = 0 (9) h,k = 0 , 6

h+k<~6

where the 28 coefficients Cyhk (h, k = 0 . . . . ,6; h + k ~< 6) depend on the geometry of the four-bar linkage only and are defined apart from an arbitrary factor. Equation (9) represents the cartesian expression for the coupler point sextic of a four-bar linkage.

2.2. The cons t ra in t equa t ion se t

The two four-bar linkages represented in Fig. 4 can be joined together at points P~ and P2--and reconstitute the 7c-AKC from which they derive--when P~ superimposes on P2.

Let x and y denote the common coordinates in reference frame Oxy of points P~ and P2 when they superimpose or, which is the same, they denote the position of revolute pair P of the 7c-AKC (see Fig. 3). From equation (9) it can be derived that all possible locations of point P are those satisfying the following conditions:

Clhkxhy k = 0 (10a) h . k = 0 ,6

h+k<~6

E C2hkxhy k = 0 (10b) h,k = 0 ,6

h+k<~6

1300 Carlo Innocenti

where quantities x and y are to be considered as unknown. Equations (10) represent the constraint equation set for the 7c-AKC.

3. S O L U T I O N P R O C E D U R E

Since equation set (10) is algebraic, one of its unknowns, say y, can be dialytically eliminated by the Sylvester procedure [15].

3.1. Elimination of unknown y If the following positions are adopted:

G, = ~ Cih," X h (k = 0 . . . . . 6) (1 la) h=O, 6 - k

Hk= ~" C2h,'x h (k=O . . . . . 6) ( l i b ) h=O, 6 k

equations (10) can be rewritten as:

where only unknown y Equations (12) admit

Go

0

0

0

0

0 M =

H0 0

0

0

0

0

satisfies the condition

~, Gk" yk = 0 (12a) k = 0 , 6

Hk" yk = 0 (12b) k = 0 , 6

appears explicitly. a common root for y

GI G2 G3 G4

Go Gi G2 G3 0 Go Gi G2

0 0 Go G1

0 0 0 G O

0 0 0 0

HI H2 H3 H,

Ho H1H2 H3

0 Ho Hi 1-12 0 0 Ho Hi o o o Ho 0 0 0 0

if the 12 x 12 matrix M:

G 5 G 6 0 0 0 0 0

G4 G5 G6 0 0 0 0

G3 G4 G5 G6 0 0 0

G2 G3 G4 G5 G6 0 0

Gi G2 G3 G4 G5 G6 0

Go GI G2 G3 G4 G5 G6 H5 //6 0 0 0 0 0

H. H~ H~ 0 0 0 0

n~ H. n~ H~ 0 0 0

H2 H3 H4 H5 H6 0 0 H~ H2 H3 Ha t15 H6 0

Ho H~ H2 H3 H4 H5 H6

(13)

det M = 0 (14)

Equation (14) contains unknown x only, and represents the result of eliminating unknown y from the constraint equation set (10).

3.2. The final constraint equation If G k and Ilk (k = 0 , . . . , 6) were arbitrary polynomials of degree 6-k in x, the left hand side

of equation (14) would generally result in a 36th order polynomial in x [15]. However, the 28 coefficients of each of equations (9) are interrelated because they depend on l0 parameters only [seven parameters defining a four-bar linkage with respect to reference frame Oxy, two parameters for selecting a point on the coupler, and one arbitrary factor for the coefficients of equation (9)].

7-1ink Assur kinematic chain 1301

Actually, direct computation shows that the degrees of equation (14) reduces to 18, and equation (14) itself can be written in the form

wixi= 0 (15) i=0.18

where coefficients wi (i = 0 . . . . . 18), depending on coefficients Cjhk ( j = 1, 2;

h,k = 0 . . . . . 6,

h + k ~< 6) of equations (10), are functions of the geometry of the 7c-AKC only. Equation (15) represents the sought polynomial solution for the Assur kinematic chain under

study. If solved in the complex field, it provides 18 roots for unknown x. Each root, as will be shown in the following subsection, corresponds to an assembly configuration for the 7c-AKC.

3.3. Back substitution Let xh (h = 1 . . . . ,18) be a generic root of the final algebraic equation (15). For x = xh, the left

hand sides of equations (12) are polynomials in y that have a first-order greatest common divisor (GCD). By equation this GCD to zero, the root Yh for y is linearly obtained. Hence the location of revolute pair P of the 7c-AKC with respect to reference frame Oxy is determined (see Fig. 3).

Table 1. Assembly configurations of the 7c-AKC in terms of coordinates of points P, E~, F~, E 2, and F 2 in reference frame Oxy

# P E, F 1 E 2 F 2

I 0.0520383420 - 1.7846754756 0.0989505315 - 1.3395454529 0.5233747628 1.9616734593 -0 . 8459068784 -0 . 42 1 8 6 4 9 2 8 6 0.0928785155 -0 .5718558929

2 0.0895355968 --1.6006471952 0.2587302394 - 1.3755029810 0.4594116571 4.0551145684 1.1569587531 1.6771260964 2.2433303744 1.5047968039

3 2.3919476515 0.4254381060 2.3268963480 1.7526865775 0.0930602413 0.7896131533 - 1.9286335624 - 1.5934991663 3.0302035396 1.9541068952

4 2.8499055219 0.8406465314 2.7471678350 1.8154215766 0.3781333210 0.8996423749 - 1.7871593128 - 1.4821428732 2.9874026501 1.6285258955

5 0.8593821885 - 1.0110941074 0.8775141379 0.4900186900 - 1.2853872731 1.0886414145 - 1.6965593729 - 1.2952896316 3.3891784401 2.5172396634

6 0.7138846184 - 1.1488629336 0.7386255930 - 1.3964503227 -0 .0699999329 1.1839078498 - 1.6064678521 - 1.1999637668 0.1962913066 - 1.2709758563

7 - 1.1366338717 1.8717860969 1.2238894810 - 1.1216843509 0.7657388516 2.1151830027 0.6301125356 2.4489276676 -0 .2147690380 0.3768196858

8 - 1.0958279256 - 1.5772508406 -0 .0481309672 - 1.1357033651 0.7651166589 2.1316229385 --1.1886567149 --0.0098221538 --0.1980358239 0.3489804467

9 --0.4319952273 1.8239890926 --0.1029139930 -- 1.4721313561 -2 .9057329859 - 1.7258334118 0.7574224647 0.6353445868 0.3591166676 - 1,0036488775

10 2.1808019423 - 1.0747336129 --0.0546365274 1.6239611210 --0,0739322679 0.8461050582 1.6569769646 0.0176903986 3.1085878315 2.0939442470

11 - 1.4418984644 --1.3533840583 --0.0462806048 -0 .9788019016 0,7594767823 1.9154384126 -- 1.4383937537 --0.0173592462 --0.3680767665 0.5757085804

12 --1.1701965771 1.8677924797 0.0423555131 1.1565772624 0,6773990831 --0.7817499359 0.6418537628 1.2708516885 --0.9043203237 1.0147241876

13 --1.3838813842 1.7563690347 --0.0141345851 - 1.4210250074 --3.3027271866 --2.0842511524 --0.9032124966 -0 .1330336591 0.2454527668 -0 .3640884314

14 1.6878886174 1.9554432049 0.5050497155 1.5655898523 --0.2925677593 - 3.0670657545 0.2772487556 --0.9971990603 --0.7402776224 - 1.4182111399

15 -- 1.2812421185 + 1.8712547564 0.1089484377 - 1.5569528403 --3.3662771549 --0.5162609557 0.6316887181 1.4204440341 1.7973690056 0.9981887454

16 1.7395638135 0.8696523513 --0.0691300453 2.6742210464 0.7094212916 -- 1.4670332878 1.7732258141 0.0860547732 0.6672853053 0.8951130634

17 --1.2391738068 1.8102206502 --0.0100735711 --1.3651700329 -3 .2222479601 --2.1887696157 --0.7897633807 -0 .1460340453 0.1378212273 -0 .5430643255

18 0.9119981675 1.8209350898 2.7392960416 -- 1.0337797524 0.4904725105 3.9942646804 0.7647355087 2.4631087515 2.7125282463 1.4519733674

1302 Carlo Innocenti

Now equations (5) are considered for both possible values of j ( j = 1, 2). Substituting in equations (5) xh and yj, for xj and yj respectively, equations (5) themselves linearly provide the values cj, and sjh for cj and sj, which means that also value 0jh for angle 0j ( j = 1, 2) can de determined. Since both ternary links of the 7c-AKC now have a definite position with respect to reference frame Oxy, coordinates of points Et, Fi, E2, and F2 with respect to Oxy can be computed [see equations (3a) and (3b)]. This completes the description of the h-th assembly configuration of the 7c-AKC.

Since each root of equation (15) allows an assembly configuration to be determined, 18 assembly configurations for the 7c-AKC are possible in the complex field.

4. N U M E R I C A L EX A MP LE

With reference to Fig. 3, a 7c-AKC is considered as having the following dimensions (lengths are expressed in arbitrary length unit, while angles are measured in radians):

(xal, Yat ) = (0., 0.) (xo2, Ya2) = (0.55, 1.2)

(Xbl, Ybl ) = (2., 0.489) (Xb2, Yb2) = (-- 1.342, 0.41)

el = 3.355 e2 = 2.330

f~ = 2.384 f2 = 2.577

~1 = 0.599 ~2 = 0.824

ml = 1.975 m2 = 2.190

nl = 2.108 n2 = 2.108

-2

-4 -4- - 2 @ 2 4

Fig. 6. Two four-bar coupler curves intersecting at 18 real points.

7-1ink Assur kinematic chain 1303

The position analysis of this 7c-AKC, solved by keeping to the proposed procedure, has provided 18 real assembly configurations, which are reported in Table 1 in terms of coordinates of points P, E~, F~, E2, and F2 in reference frame Oxy. It has been verified that all solutions satisfy equations (10).

In order to have a glance at the alternative interpretation of the 7c-AKC position analysis, Fig. 6 represents the 18 assembly configurations in terms of intersections of coupler point curves. Precisely, the coupler curve labeled by Pj(j = 1, 2) identifies the trajectory of point Pj belonging to the j - th four-bar linkage (see Fig. 4). Both coupler curves are unicursal because both the four-bar linkages resulting from the considered 7c-AKC are non-Grashof chains [13]. The 18 intersections of the two coupler curves are labeled consistently with the assembly configurations listed in Table I.

5. C O N C L U S I O N S

The position analysis of the 7-1ink Assur kinematic chain featuring one quaternary link is presented. The analysis is accomplished by first devising a set of two constraint equations in a corresponding number of unknowns. After algebraic elimination, a final polynomial equation of eighteenth order in only one unknown is obtained whose 18 roots correspond to as many assembly configurations of the Assur kinematic chain in the complex field.

The paper completes the position analysis of the existing 7-1ink Assur kinematic chains and, further on, expands the class of mechanisms whose position analysis can be exhaustively performed.

The new results can also be interpreted as proof that two coupler point curves generated by two different four-bar linkages intersect each other at 18 proper points, either real or complex.

Finally, a numerical example is reported which confirms the new theoretical results.

Acknowledgement--The financial support of MURST by 60% funds is gratefully acknowledged.

R E F E R E N C E S 1. C. U. Galleni, Meccanica (March), 6-10 (1969). 2. V. S. Karelin, Mech. Mach. Theory 27, 403414 (1992). 3. C. H. Suh and C. W. Radcliffe, Kinematics and Mechanism Design. Wiley, New York (1978). 4. G. L. Kinzel and C. Chang. Mech. Mach. Theory 19, 165-172 (1984). 5. C. U. Galletti, Mech. Mach. Theory 21, 385-391 (1986). 6. E. E. Peysah, Machinery, No. 5, pp. 55~51 (1985). 7. S. Li and G. K. Matthew, Proc. of the 7th Worm Congress on the Theory of Machines and Mechanisms, Sevilla, Spain,

17-22 Sept., Vol. 1, pp. 141-145 (1987). 8. C. M. Gosselin, J. Sefrioui and M. J. Richard, Mech. Mach. Theory 27, 107-119 (1992). 9. C. Innocenti, Mech. Mach. Theory, submitted (1996).

10. C. Innocenti, J. Mech. Design 116, 622-628 (1994). 11. K. Hain, Applied Kinematics. McGraw-Hill, New York (1967). 12. R. S. Hartenberg and J. Denavit, Kinematic Synthesis of Linkages. McGraw-Hill, New York (1964). 13. K. J. Hunt, Kinematic Geometry of Mechanisms. Clarendon Press, Oxford (1978). 14. O. Bottema and B. Roth, Theoretical Kinematics. North-Holland Amsterdam (1979). 15. G. Salmon, Modern Higher Algebra. Hodges, Figgs & Co., Dublin (1885).