Application of Distance Geometry to Tracing Coupler Curves

Nicolas Rojas2

e-mail: [email protected]

Federico Thomase-mail: [email protected]

Institut de Robotica i Informatica

Industrial (CSIC-UPC),

Llorens Artigas 4-6,

Barcelona 08028, Spain

Application of DistanceGeometry to Tracing CouplerCurves of Pin-Jointed Linkages1

In general, high-order coupler curves of single-degree-of-freedom plane linkages cannotbe properly traced by standard predictor–corrector algorithms due to drifting problemsand the presence of singularities. Instead of focusing on finding better algorithms fortracing curves, a simple method that first traces the configuration space of planar link-ages in a distance space and then maps it onto the mechanism workspace, to obtained thedesired coupler curves, is proposed. Tracing the configuration space of a linkage in theproposed distance space is simple because the equation that implicitly defines this spacecan be straightforwardly obtained from a sequence of bilaterations, and the configurationspace embedded in this distance space naturally decomposes into components corre-sponding to different combinations of signs for the oriented areas of the trianglesinvolved in the bilaterations. The advantages of this two-step method are exemplified bytracing the coupler curves of a double butterfly linkage. [DOI: 10.1115/1.4023515]

Keywords: linkages, configuration spaces, coupler curves, distance geometry, doublebutterfly linkage

1 Introduction

The mechanisms considered under the title linkwork, linkages,or articulated systems are plane mechanisms involving turningpairs only. That is, sets of plane links articulated through pins. Formechanisms of this type, the equation of the curve generated byan arbitrary point on the mechanism—the tracer point—can beobtained by solving a finite number of simultaneous equationsexpressing constancy of distance between pin centers whichinclude the tracer point. Then, the coordinates of all moving pinpoints, other than those of the tracer, can be seen as unknowns. Ifwe only considered single-degree-of-freedom mechanisms, thenumber of independent quadratic equations will be one fewer thanthe number of unknowns. The curve generated by the tracerpoint—usually known as a coupler curve—is, therefore, the elimi-nant of these equations. This reasoning permits to conclude thatthe curve generated by any point on a plane pin-jointed mecha-nism possessing a finite number of links of finite size is necessar-ily algebraic [1]. The same result can be attained, in a morecompact way, by computing the eliminant of the set of independ-ent loop equations [2–4]. All coupler curves can be seen as agroup of manifold curves joined through singular points usuallyclassified in kinematics as crunodes and cusps [5,6].

The problem of tracing a coupler curve is essentially that ofconnecting sampled points to give rise to its graph. Sampling acoupler curve is not a difficult task compared to that of connectingthe samples, mainly for high-order coupler curves. Continuationmethods are one of the major approaches reported in the literatureto solve this problem [7]. Since, in our case, the curves to betraced are algebraic, polynomial continuation can be used [8].These methods are global, that is, they are able to trace all theconnected components of a coupler curve but, depending on theapplication, one does not need to trace all components, but rather

one of them starting from a given point. Actually, this is theencountered problem when simulating the motion of a planemechanism [9]. In this case, a very popular method is the so-called predictor–corrector method [10]. It consists of two majorstages. In the first stage, called the predictor step, a point in thetangent line to the curve at the current given point is estimated(Fig. 1(a)). In the second stage, the corrector step, the predictedpoint is adjusted onto the curve, using typically a Newton-likemethod, to get a new point of the curve (Fig. 1(b)). In the case ofclosed curves, a third stage, called the filling step, is implementedto avoid overlaps. The predictor–corrector algorithm is simple toimplement and hence its popularity. Unfortunately, it exhibits, ingeneral, the undesirable phenomenon known as drifting in whichthe procedure fails to keep moving along a given branch of thecurve and drift to another (Fig. 1(c)). In most dramatic cases thismight even lead to cycling (Fig. 1(d)). This drawback can beresolved using more sophisticated mathematical tools, than thefirst order approximations used in standard predictor–correctoralgorithms, such as Runge–Kutta or Adam’s method [11]. Anotherimportant drawback of predictor–corrector algorithms ariseswhen the plane curve to be traced has singular points because thetangent is undefined at them. An approach to overcome thisissue is by first computing the singular points with symbolic proc-essing techniques for then use them as starting points of apredictor–corrector algorithm. Unfortunately, all these methods,generally based on independent loop equations, have only beenpresented for particular families of mechanisms [12–15]. A sim-pler and more elegant alternative, valid only for plane curves, isthe use of derivative-free methods such as the Morgado–Gomesfalse position numerical method [16].

The possibility of drifting, cycling, and having problems withsingular points, increases dramatically with the number of inde-pendent kinematic loops of the linkage because the order of thecoupler curves to be traced grows exponentially with it. Forexample, while the coupler curves of the well-known single-loopfour-bar linkage are of the sixth order [17], that of the three-loopdouble butterfly linkage can be up to the forty-eighth order [18].

In this paper, instead of focusing on a better algorithm for trac-ing coupler curves able to deal with all mentioned problems in theworkspace of the mechanism, a distance geometry approach thatfirst traces the configuration space of the mechanism in a distance

2Corresponding author.

1Some of the ideas contained in this paper were already presented at the ASME2011 International Design Engineering Technical Conferences and Computers andInformation in Engineering Conference IDETC/CIE 2011, August 28–31, 2011,Washington, DC.

Contributed by the Mechanisms and Robotics Committee of ASME forpublication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received July25, 2011; final manuscript received January 23, 2013; published online March 26,2013. Assoc. Editor: Qiaode Jeffrey Ge.

Journal of Mechanisms and Robotics MAY 2013, Vol. 5 / 021001-1Copyright VC 2013 by ASME

Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 03/28/2013 Terms of Use: http://asme.org/terms

space and then maps it onto the workspace to obtain the desiredcoupler curves is proposed. To get an intuitive idea of thisapproach and its advantages, Sec. 2 presents the main clueswithout going into mathematical details. Then, Sec. 3 presents thebasic mathematical tools to formalize them, and Sec. 4 concen-trates on the case study of tracing the coupler curves of the doublebutterfly linkage. An example is then presented in Sec. 5. Section6 discusses how the presented approach can be applied to othersingle-degree-of-freedom linkages and, finally, Sec. 7 summarizesthe main contributions.

2 Overview of the Proposed Approach

A linkage configuration is given by a set of parameters uniquelyspecifying the position of each of its links. The configurationspace of a linkage is thus simply the set of all its configurations.Then, since all points of a single-degree-of-freedom linkage traceplane curves which can readily be expressed in terms of the con-figuration parameters of the linkage itself, an alternative approach,other than directly tracing coupler curves in the linkage work-space, naturally arises: first trace the configuration space of themechanism and then compute the desired curves from it. Toexemplify this idea, let us consider the four-bar linkage inFig. 2(a). The coupler curve traced by any point linked to one ofits bars, while taking the opposite bar as fixed, are algebraiccurves of the sixth order; i.e., a straight line will cut it in not morethan six points [17] (Fig. 2(b)). The configuration space of thislinkage can be easily derived by expressing the location of theline segments P1P3 and P2P4 as a function of h1 and h2, respec-tively, and imposing the distance between P3 and P4 as closurecondition [19, pp. 26–27]. Thus, all possible configurations ofthis linkage defines a curve in the space defined by h1 and h2.Unfortunately, this idea cannot be applied, in general, to multilooplinkages. Actually, the valid configurations of a multiloop linkageare usually represented by the solution set of an independent setof its vector loop equations [2–4]. This requires introducing avariable for each link representing its orientation with respect tothe fixed link. For the case of the four-bar linkage, its vector loopequation defines a one-dimensional variety in the space defined byfh1; h2; h3g which seems quite complicated for such a simple link-age (Fig. 2(c)). Alternatively, the configuration space of a four-barlinkage can be represented by a single distance variable, for exam-ple the distance between P2 and P3, provided that the sign of theoriented areas of the triangles DP1P2P3 and DP2P4P3 are given(Fig. 2(d)). Besides an important reduction in the dimensionalityof the problem, the configuration space of the linkage is thusdecomposed into up to four components, one for each combina-tion of signs for the two oriented areas. Most importantly, thisidea of using distances and signs of oriented areas can be appliedto characterize the configuration spaces of arbitrary multilooplinkage. For example, let us consider the three-loop linkage, com-monly known as double butterfly linkage, depicted in Fig. 3(a).

Using the standard formulation based in vector loop equations, itsconfiguration space is determined by the solution set of a systemof six scalar equations in the space defined by fh1;…; h7g. Usingthe distance geometry approach proposed in this paper, we willshow how this configuration space can be characterized by a planecurve in the space defined by the lengths of P1P6 and P2P4, andhow this curve is decomposed into 16 components, one for eachcombination of signs of the oriented areas of the trianglesDP2P4P10, DP1P3P6;DP1P6P5, and DP4P9P7. This decomposi-tion, together with the reduction of the dimensionality of theambient space from 7 to 2, greatly simplifies the process of tracingthe configuration space of this linkage while retaining, at the sametime, the geometric flavor of the problem contrarily to what hap-pens to the fully algebraic current approaches.

It can be argued that the proposed approach belongs to the setof methods based on the idea of dividing the linkage’s circuits—

Fig. 1 (a) In the predictor step, a point p� in the tangent line to the curve at the current point pi is estimated. (b) In the correctorstep, the predicted point p� is adjusted onto the curve producing a new point piþ1. (c) The drifting problem. (d) The cyclingproblem.

Fig. 2 (a) A four-bar linkage and (b) the coupler curve tracedby a point affixed to one of its bars while taking the oppositeone as fixed. (c) Any coupler curve generated by this linkagecan be expressed in terms of its configuration space which canbe represented by a one-dimensional variety in the spacedefined by fh1; h2; h3g, or by fh1; h2g if the distance constraintbetween P3 and P4 is used as closure condition instead of thestandard loop equation. (d) Alternatively, this configurationspace can be represented by value ranges of a single variable,s2;3, one range for each combination of signs of the orientedareas of the triangles DP1P2P3 and DP2P4P3.

021001-2 / Vol. 5, MAY 2013 Transactions of the ASME


i.e., subsets of the configuration space in which a configurationcan be continuously transformed to another one—into segmentsconnected by special points because the linkage’s circuits are alsodivided into components. However, in the proposed technique thespecial points—those configuration space points where a givenoriented area vanishes—have not to be calculated beforehand.

The next section presents the necessary mathematical tools toformalize the proposed approach.

3 Distances in Strips of Triangles

The bilateration problem consists in finding the feasible loca-tions of a point, say Pk, given its distances to two other points, sayPi and Pj, whose locations are known. Then, according to Fig. 4(top), the result to this problem, in matrix form, can be expressedas:

pi;k ¼ Zi;j;kpi;j (1)

where pi;j ¼ PiPj��!

and

Zi;j;k ¼1

2si;j

si;j þ si;k � sj;k �4Ai;j;k

4Ai;j;k si;j þ si;k � sj;k

� �

is called a bilateration matrix, with si;j ¼ jjpi;jjj2, the squared dis-

tance between Pi and Pj, and

Ai;j;k ¼ 61

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisi;j þ si;k þ sj;k

� �2�2 s2i;j þ s2

i;k þ s2j;k

� r(2)

the oriented area of DPiPjPk which is defined as positive if Pk isto the left of vector pi;j, and negative otherwise. The interestedreader is addressed to Ref. [20] for a derivation of (1).

Given the triangle in Fig. 4 (top), it is possible to compute sixdifferent bilaterations. By algebraically manipulating the obtainedresults, it is possible to prove that:

Zi;j;k ¼ I� Zj;i;k (3)

Zi;j;k ¼ ðZi;k;jÞ�1(4)

Zi;j;k ¼ �Zk;j;iZj;i;k (5)

Moreover, it can be observed that the product of two bilatera-tion matrices is commutative. Then, it is easy to prove that the set

of bilateration matrices, i.e., matrices of the forma �bb a

�,

constitute a commutative group under the product and additionoperations.

Another important property, that will be useful later, comesfrom the fact that if v¼Zw, where Z is a bilateration matrix, thenjjvjj2 ¼ detðZÞjjwjj2.

Now, let us consider the two triangles sharing one edgedepicted in Fig. 4 (bottom). Then, pi;l can be expressed in terms ofpi;j by applying two consecutive bilaterations, as

Fig. 3 (a) Using the standard vector loop formulation, the configuration space of a doublebutterfly linkage can be represented by a one-dimensional variety in the space defined byfh1; . . . ; h7g. (b) Alternatively, using the proposed approach, this configuration space can berepresented by a one-dimensional variety in the space defined by fs1;6; s2;4g which can bedecomposed into 16 components, one for each combination of signs of the oriented areas ofthe triangles DP2P4P10;DP1P3P6;DP1P6P5, and DP4P9P7.

Fig. 4 Top: The bilateration problem. Bottom: Concatenationof two bilaterations.

Journal of Mechanisms and Robotics MAY 2013, Vol. 5 / 021001-3


pi;l ¼ Zi;k;lpi;k ¼ Zi;k;lZi;j;kpi;j (6)

Actually, a vector involving any two different points in the setfPi;Pj;Pk;Plg can be expressed in function of pi;j using bilatera-tion matrices. For example

pj;l ¼ pi;l � pi;j ¼ ðZi;k;lZi;j;k � IÞpi;j (7)

Therefore, the squared distance between Pj and Pl can be obtainedas

sj;l ¼ detðZi;k;lZi;j;k � IÞsi;j (8)

If this result is compared to the one presented for example inRef. [21, pp. 65–69], one starts to appreciate the ability of bilater-ation matrices to represent the solution of complex problems in avery compact form. This result can be extended to a strip of trian-gles—i.e., a series of connected triangles that share one edge withone neighbor and another with the next—to obtain the squareddistance between any couple of its vertices. As an example, let ussuppose that we are interested in finding p1;3 as a function of p2;4

for the strip of three triangles fDP1P10P2;DP2P10P4;DP10P3P4gappearing in Fig. 5(a). In this case,

p1;3 ¼ �p2;1 þ p2;4 þ p4;3

¼ �Z2;10;1p2;10 þ p2;4 þ Z4;10;3p4;10

¼ �Z2;10;1Z2;4;10p2;4 þ p2;4 þ Z4;10;3p4;10

¼ �Z2;10;1Z2;4;10 þ I� Z4;10;3Z4;2;10

� �p2;4

Therefore, the squared distance s1;3 can be expressed as

s1;3 ¼ detðX1Þs2;4 (9)

where X1 ¼ �Z2;10;1Z2;4;10 þ I� Z4;10;3Z4;2;10.The possibility of computing squared distances that involve

arbitrary couples of vertices, using sequences of bilaterations, isnot limited to strips of triangles. This can also be applied, forexample, to two strips sharing two arbitrary vertices. To exem-plify this, let us suppose that we are interested in finding p4;9 as afunction of p2;4 after attaching the strip of triangles defined byfDP1P6P3;DP1P5P6;DP6P5P9g to the previous strip so that theyshare vertices P1 and P3 (see Fig. 5(b)). Then

p4;9 ¼ �p2;4 þ p2;1 þ p1;6 þ p6;9

¼ �p2;4 þ Z2;10;1p2;10 þ p1;6 þ Z6;5;9p6;5

¼ ð�Iþ Z2;10;1Z2;4;10Þp2;4 þ ðI� Z6;5;9Z6;1;5Þp1;6

¼ ð�Iþ Z2;10;1Z2;4;10Þp2;4 þ ðI� Z6;5;9Z6;1;5ÞZ1;3;6p1;3

¼ ð�Iþ Z2;10;1Z2;4;10Þp2;4 þ ðI� Z6;5;9Z6;1;5ÞZ1;3;6X1p2;4

¼ ð�Iþ Z2;10;1Z2;4;10 þ ðI� Z6;5;9Z6;1;5ÞZ1;3;6X1Þp2;4

Therefore

s4;9 ¼ detðX2Þs2;4 (10)

where

X2 ¼ �Iþ Z2;10;1Z2;4;10 þ ðI� Z6;5;9Z6;1;5ÞZ1;3;6X1

The process of adding a strip of triangles sharing two arbitraryvertices with the obtained structure can be iterated further. For

Fig. 5 (a) In the strip of triangles fDP1P10P2;DP2P10P4;DP10P3P4g; s1;3 can be obtained frombilaterations. (b) After affixing the strip of triangles fDP1P6P3;DP1P5P6;DP6P5P9g to the previ-ous one, s4;9 can also be obtained using bilaterations. (c) Likewise, s2;6 can be obtained afteraffixing the strip of triangles fDP4P9P7;DP7P9P8g. (d) A double butterfly linkage. (e) If thelengths of dotted segments were known, this double butterfly linkage would be equivalent tothe obtained structure resulting from attaching three strips of triangles.



example, we can now add the strip of triangles defined byfDP4P9P7;DP7P9P8g, as shown in Fig. 5(c). In this case, wemight be interested in obtaining s2;8 as a function of s2;4. To thisend, we could compute

p2;8 ¼ p2;4 þ p4;9 þ p9;8

¼ p2;4 þ I� Z9;7;8Z9;4;7

� �p4;9

¼ Iþ I� Z9;7;8Z9;4;7

� �X2

� �p2;4

Therefore

s2;8 ¼ detðX3Þs2;4; (11)

where

X3 ¼ Iþ I� Z9;7;8Z9;4;7

� �X2

Then, if s2;8 is fixed, Eq. (11) can be seen as a closure condi-tion: it is satisfied if, and only if, the resulting structure can beassembled with the specified distances. This idea can be appliedto solve the position analysis problem of linkages, as shown nextfor the double butterfly linkage.

4 Tracing the Double Butterfly Linkage Configuration

Space

The double butterfly linkage has one of the sixteen topologiesavailable for eight-bar single-degree-of-freedom linkages [19].In the context of classical kinematics of mechanisms, the input–output problem for this linkage leads to either sixteenth order oreighteenth order polynomials depending on the selected fix andinput links [2,4,22]. This input–output problem, that was solvedusing continuation in Ref. [23], is equivalent to the position analy-sis problem of the seven-link Baranov trusses of types II and III[20,24]. A polynomial equation for the path of a point located ina coupler link of the double butterfly linkage was presented inRef. [18] for the first time. The resulting polynomial was shown tobe, at most, of forty-eighth order. A sampled plot of this curve ispresented in Ref. [25]. The interested reader is referred toRef. [26] for more details on this mechanism.

Figure 5(d) shows a double butterfly linkage. It consists of fourbinary links and four ternary links with three independent loops.The centers of the revolute joints of the binary links define theline segments P1P5;P3P6;P4P7, and P2P8, and those for the ter-nary links define the triangles DP1P10P2;DP10P3P4;DP6P5P9,and DP7P9P8. Instead of computing its configuration space interms of joint angles through loop-closure equations, we will usesquared distances in strips of triangles to compute the set of valuesof s2;4 and s1;6 compatible with all binary and ternary link sidelengths.

It can be verified that, if the distances s1;3; s1;6; s4;9, and s2;4 ofthe double butterfly linkage in Fig. 5(d) were fixed, the structurein Fig. 5(c) would be obtained (Fig. 5(e)). Then, if we rewriteEqs. (9)–(11), leaving these distances as variables, we get the fol-lowing system of equations:

s1;3 ¼ f1ðs2;4Þ ¼ detðX1Þs2;4 (12a)

s4;9 ¼ f2ðs2;4; s1;6; s1;3Þ ¼ detðX2Þs2;4 (12b)

s2;8 ¼ f3ðs2;4; s1;6; s1;3; s4;9Þ ¼ detðX3Þs2;4 (12c)

Computing a resultant from the above triangular systembecomes a trivial task that yields a scalar radical equation in twovariables: s2;4 and s1;6.

By expanding Eq. (12c), we get

s2;8 ¼1

s1;6s1;3s4;9W (13)

where

W ¼ W1 þW2A2;4;10 þW3A1;3;6 þW4A1;6;5 þW5A4;9;7

þW6A2;4;10A1;3;6 þW7A2;4;10A1;6;5 þW8A2;4;10A4;9;7

þW9A1;3;6A1;6;5 þW10A1;3;6A4;9;7 þW11A1;6;5A4;9;7

þW12A2;4;10A1;3;6A1;6;5 þW13A2;4;10A1;3;6A4;9;7

þW14A2;4;10A1;6;5A4;9;7 þW15A1;3;6A1;6;5A4;9;7

þW16A2;4;10A1;3;6A1;6;5A4;9;7

with Wi; i ¼ 1;…; 16, being polynomials in s2;4; s1;6; s1;3, and s4;9,and A2;4;10;A1;3;6;A1;6;5, and A4;9;7, the oriented areas ofDP2P4P10; DP1P6P3; DP1P6P5, and DP4P9P7, respectively, andsubstituting Eqs. (12a) and (12b) which account for the unknownsquared distances s1;3 and s4;9.

Equation (13) is the closure condition for the double butterflylinkage. This equation contains four variable oriented areas,namely, A2;4;10;A1;3;6;A1;6;5, and A4;9;7. Each of these orientedareas involves a radical. By iteratively isolating one radical onone side of the equation and then squaring both sides of the equa-tion, it is possible to transform Eq. (13) to polynomial form.Nevertheless, it is advantageous to keep it in its current form, notonly because of its compactness compared to the polynomialform, but because, and most importantly, it provides a decomposi-tion of the configuration space of the double butterfly linkage intosixteen varieties, one for each combination of signs of the orientedareas A2;4;10;A1;3;6;A1;6;5, and A4;9;7. Therefore, the configurationspace of the double butterfly linkage can be decomposed into six-teen components in the space defined by fs1;6; s2;4g.

The advantage of this decomposition is that the resultingcomponents can be easily traced. In the example presented in thefollowing section, it is shown how all the crunodes appear asintersections of two different components, and how all the cuspsappear when mapping these traced components onto the work-space. Starting from a given configuration in a component, thetracing process would proceed till the starting point is againreached or when it arrives at a point where one of the defining ori-ented areas vanishes. Then, the tracing process would have to skipto another component: the one defined by the same oriented areasigns except for the area that vanishes whose sign has to bechanged.

5 Example

According to the notation used in Fig. 5(d), let us sets1;2¼ 169;s1;5¼ 145; s1;10¼ 65;s2;8¼ 200; s2;10¼ 52;s3;4¼ 5;s3;6¼ 50;s3;10¼ 5;s4;7¼ 36;s4;10¼ 10, s5;6¼ 5;s5;9¼ 53;s6;9¼ 34;s7;8¼ 10;s7;9¼ 5, and s8;9¼ 25. Using triangular inequalities,s2;4 can be bound to lie in the interval ½62�4

ffiffiffiffiffi13p ffiffiffiffiffi

10p

;62þ4

ffiffiffiffiffi13p ffiffiffiffiffi

10p�. Fig. 6 (top) shows the root locus of Eq. (13).

This root locus has been obtained by clearing the radicals inEq. (13) to obtain a polynomial equation in s2;4 and s1;6 and thensolving this equation for sampled values of s2;4 at increments ofð1=100Þ. The result contains no information on the connectivity ofeach sample to its neighbors. Actually, finding this connectivity isthe difficult point but any of the obtained samples can be used as astarting point for a tracing algorithm. The sampled curve has beenincluded here for comparison purposes with the results obtainedby tracing as shown next.

Let us suppose that we are interested in tracing the configura-tion space followed by the linkage from the following three initialconfigurations:

(1) s2;4 ¼ 74; s1;6 ¼ 188:68, and A2;4;10 > 0; A1;3;6 < 0;A1;6;5 < 0, and A4;9;7 < 0,

(2) s2;4 ¼ 74; s1;6 ¼ 122, and A2;4;10 > 0;A1;3;6 > 0;A1;6;5 > 0,and A4;9;7 < 0,

(3) s2;4 ¼ 74; s1;6 ¼ 98:92, and A2;4;10 < 0; A1;3;6 < 0;A1;6;5 < 0, and A4;9;7 < 0.



The results using the procedure discussed in the previous sec-tion appear in Fig. 6 (bottom), from left to right, respectively. Inthe three plots, colors indicate the signs of the oriented areasaccording to Table 1.

In order to determine the coupler curve of a selected tracer ofthe linkage using the computed configuration space, we proceedto calculate the position of the linkage’s revolute pair centersusing bilateration. Let us suppose that DP1P2P10 is the fixed link.Then, for example, we can set p1 ¼ ð4; 0Þ

T ; p2 ¼ ð17; 0ÞT , andp10 ¼ ð11; 4ÞT , and the path traced by P3, P4, P5, P6, P7, P8, andP9 can be obtained by replacing each previously computed config-uration space point in the sequence of bilaterations given by:

p2;4 ¼ Z2;10;4p2;10;

p10;3 ¼ Z10;4;3p10;4;

p1;6 ¼ Z1;3;6p1;3;

p1;5 ¼ Z1;6;5p1;6;

p5;9 ¼ Z5;6;9p5;6;

p4;7 ¼ Z4;9;7p4;9;

p7;8 ¼ Z7;9;8p7;9

Figure 7 (top) shows the path followed by P9 from the first ini-tial configuration. It can be observed how the mapping from con-figuration space to workspace is surjective (two points of theconfiguration space can be mapped onto the same point in theworkspace) and how this fact is actually the underlying reasonthat makes coupler curves so difficult to be traced directly in thelinkage workspace. The zoomed-in areas in Fig. 7 (top) presentthis effect by showing how two overlapping branches next to aramphoid cusp, which leads to a reciprocating motion of the link-age, and a near-quadruple point are generated. Similar situationsarise for the path followed by P9 from the second initial configura-tion. In this case a cusp and a tacnode can be identified (Fig. 7(center)). The curve traced when starting from the third initialconfiguration appears in (Fig. 7 (bottom)). Observe how in thiscase a smooth simple curve in the configuration space maps ontothe linkage workspace as a curve with several singular points in areduced area which would be very difficult to be directly tracedusing a standard predictor–corrector procedure without highlyincreasing its resolution.

Fig. 6 Top: The real solution set of Eq. (13) in the plane defined by s2;4 and s1;6 for sampled values of s2;4. Bottom: From left toright, the connected components of the configuration space traced when starting from the initial configurations s2;4 5 74,s1;6 5 188:68, and A2;4;10 > 0;A1;3;6 < 0;A1;6;5 < 0, and A4;9;7 < 0; s2;4 5 74; s1;6 5 122, and A2;4;10 > 0;A1;3;6 > 0;A1;6;5 > 0, and A4;9;7 < 0,and s2;4 5 74;s1;6 5 98:92, and A2;4;10 < 0;A1;3;6 < 0;A1;6;5 < 0, and A4;9;7 < 0, respectively.

Table 1 Code of colors used in Figs. 6 and 7 for the signs ofA2;4;10;A1;3;6;A1;6;5, and A4;9;7.

Color A2;4;10 A1;3;6 A1;6;5 A4;9;7

� � � �� þ� þ � �� þ � þ� þ þ þþ � � �þ � � þþ þ � �þ þ � þþ þ þ �þ þ þ þ



If we were interested in the curve traced by a coupler point dif-ferent from the revolute pair centers, we could compute its loca-tion by introducing one extra bilateration with reference to therevolute pair centers of the corresponding coupler link.

For the curves traced in a different kinematic inversion of thelinkage, we simply have to calculate the Euclidean transformationbetween the constant values of the corresponding fixed link andthe values computed with the above set of bilaterations, and use itto recompute the values for the other revolute pair centers. Withthis simple procedure, the coupler curves of any kinematic inver-sion of the double butterfly linkage can be computed.

As it has been shown in this example, the main advantage ofthe proposed method for tracing the coupler curves is that theconfiguration space of single-degree-of-freedom linkages can bedecomposed into easy-to-trace branches. Actually, in the pre-sented example all singularities arise when mapping thesebranches onto the linkage workspace.

6 Other Single-Degree-of-Freedom Linkages

The proposed method for tracing coupler curves can be easilyapplied to any single-degree-of-freedom planar linkage. It can be

Fig. 7 The paths followed by the revolute pair center P9 from different initial configurations.Top: For the curve traced from the initial configuration s2;4 5 74, s1;6 5 188:68, andA2;4;10 > 0;A1;3;6 < 0;A1;6;5 < 0, and A4;9;7 < 0, zoomed-in areas show how, after mapping theconfiguration space onto the workspace, a ramphoid cusp, and near-quadruple point aregenerated. Center: For the curve traced from the initial configuration s2;4 5 74, s1;6 5 122, andA2;4;10 > 0;A1;3;6 > 0;A1;6;5 > 0, and A4;9;7 < 0, a cusp and a tacnode can be identified. Bottom:For the curve traced from the initial configuration s2;4 5 74; s1;6 5 98:92, and A2;4;10

< 0;A1;3;6 < 0;A1;6;5 < 0, and A4;9;7 < 0, zoomed-in areas show how, after mapping the configurationspace onto the workspace, several singular points are generated in a reduced region.



verified that tracing the coupler curves of the four-bar linkage andthe two six-bar linkages—the Watt and the Stephenson linkages—becomes trivial because their configuration spaces correspondto ranges of a single distance, one range for each combinationof sign of two or three oriented areas, depending on the case. Asimilar situation occurs for twelve of the sixteen possible topolo-gies for single-degree-of-freedom eight-bar linkages. In thesecases, four oriented areas are needed. For the remaining fourtopologies—in which the double butterfly linkage is included—the one-dimensional configuration space is embedded in a two-dimensional distance space. Since the sign of four oriented areasare needed in all these four cases to uniquely identify a configura-tion, the configuration space is naturally decomposed into 16components. Table 2 presents the equations representing the cor-responding configuration spaces for these four cases.

7 Conclusion

The current approaches for tracing the coupler curves of planemechanisms provide a rapid algebraization of the problem thusbecoming blind to the underlying geometry. A new distance ge-ometry approach that first computes the linkage configuration

space embedded in a space of squared distances and then maps itonto the linkage workspace has been presented. The used formu-lation involves products, additions, and square roots. The pres-ence of square roots permits a more compact representationthan the standard techniques based on polynomials. Square rootsactually play a fundamental role in the presented approachbecause their sign represent the orientation of triangles formedby sets of three joints of the linkage thus retaining importantgeometric information. Configuration spaces are thus decom-posed into components for which the signs of the oriented areasof the involved triangles remain invariant. This decomposition,besides providing a new insight in the analysis of coupler curves,avoids most of the possible drifts that could arise when using astandard predictor– corrector method directly in the linkageworkspace. In all the experiments we have carried out with theproposed method, all individual components, defined by constantsign areas, of the configuration space are free from singularities.As a result, tracing them is an easy task. We actually conjecturethat, in general, these components cannot self-intersect. Unfortu-nately, a formal proof or a counterexample remains elusivedespite our best efforts. This is a point that deserves furtherattention.

Table 2 The four eight-bar linkages whose configuration space can be embedded in a two-dimensional distance space. Since fouroriented areas are needed in the four cases to uniquely identify a configuration, the corresponding configuration spaces aredecomposed into 16 components. The equations representing the corresponding configuration spaces in implicit form are givenon the right column.

Distance spaceLinkage and oriented areas Closure conditions

ðs1;6; s2;7Þ

A1;6;3;A1;6;5;

A2;4;7;A2;7;8

s9;10 ¼ f ðs1;6; s2;7Þ¼ detð�Iþ Z6;5;9Z6;1;5 þ Z1;3;4Z1;6;3 � Z4;1;2Z1;3;4Z1;6;3

� I� Z7;8;10Z7;2;8

� �Z4;2;7Z4;1;2Z1;3;4Z1;6;3Þs1;6

ðs1;4; s6;9Þ

A1;4;3;A1;4;2;

A5;6;9;A6;9;8

s7;10 ¼ f ðs1;4; s6;9; s5;6Þ¼ detðZ4;2;7Z4;1;2 � Z4;3;6Z4;1;3 � I� Z9;8;10Z9;6;8

� �Z6;5;9X1Þs1;4

s5;6 ¼ f ðs1;4Þ ¼ detðX1Þs1;4

¼ detð�Z1;3;5Z1;4;3 þ I� Z4;3;6Z4;1;3Þs1;4

ðs1;7; s4;10Þ

A1;7;6;A1;7;2;

A3;10;4;A4;10;8

s5;9 ¼ f ðs1;7; s4;10; s3;10Þ¼ detð I� Z4;3;5Z4;10;3 � Z10;8;9Z10;4;8

� �Z10;3;4X1Þs1;7


¼ detð�Z1;2;3Z1;7;2 þ I� Z7;6;10Z7;1;6Þs1;7

ðs2;4; s1;6Þ

A2;4;10;A1;3;6;

A1;6;5;A4;9;7

s2;8 ¼ f ðs2;4; s1;6; s1;3; s4;9Þ¼ detðIþ I� Z9;7;8Z9;4;7

� �X2Þs2;4

s4;9 ¼ f ðs2;4; s1;6Þ ¼ detðX2Þs2;4

¼ detð�Iþ Z2;10;1Z2;4;10 þ I� Z6;5;9Z6;1;5

� �Z1;3;6X1Þs2;4


¼ detð�Z2;10;1Z2;4;10 þ I� Z4;10;3Z4;2;10Þs2;4



Acknowledgment

We gratefully acknowledge the financial support of the SpanishMinistry of Economy and Competitiveness, under the IþD ProjectNo. DPI2011-13208-E, and the Colombian Ministry of Communi-cations and Colfuturo through the ICT National Plan of Colombia.

References[1] Freudenstein, F., 1962, “On the Variety of Motions Generated by Mechanisms,”

J. Eng. Ind., 39, pp. 156–160.[2] Nielsen, J., and Roth, B., 1999, “Solving the Input/Output Problem for Planar

Mechanisms,” J. Mech. Des., 121(2), pp. 206–211.[3] Wampler, C., 1999, “Solving the Kinematics of Planar Mechanisms,” J. Mech.

Des., 121(3), pp. 387–391.[4] Dhingra, A., Almadi, A., and Kohli, D., 2000, “Closed-Form Displacement

Analysis of 8, 9, and 10-Link Mechanisms: Part I: 8-Link 1-DOF Mechanisms,”Mech. Mach. Theory, 35(6), pp. 821–850.

[5] Lan, Z., Huijun, Z., and Liuming, L., 2002, “Kinematic Decomposition of Cou-pler Plane and the Study on the Formation and Distribution of Coupler Curves,”Mech. Mach. Theory, 37(1), pp. 115–126.

[6] Shirazi, K., 2006, “Symmetrical Coupler Curve and Singular Point Classifica-tion in Planar and Spherical Swinging-Block Linkages,” J. Mech. Des., 128(2),pp. 436–443.

[7] Allgower, E., and Georg, K., 2003, Introduction to Numerical ContinuationMethods, SIAM, Philadelphia, PA.

[8] Sommese, A., and Wampler, C., 2005, The Numerical Solution of Systems ofPolynomials Arising in Engineering and Science, World Scientific, Singapore.

[9] Hernandez, A., and Petuya, V., 2004, “Position Analysis of Planar MechanismsWith R-Pairs Using a Geometrical-Iterative Method,” Mech. Mach. Theory,39(2), pp. 133–152.

[10] Gomes, A., Voiculescu, I., Jorge, J., Wyvill, B., and Galbraith, C., 2009,Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms,1st ed., Springer Publishing Company, Inc., New York.

[11] Lambert, J., 1991, Numerical Methods for Ordinary Differential Systems: TheInitial Value Problem, John Wiley and Sons, West Sussex, UK.

[12] Ting, K., Xue, C., Wang, J., and Currie, K., 2009, “Stretch Rotation and Com-plete Mobility Identification of Watt Six-Bar Chains,” Mech. Mach. Theory,44(10), pp. 1877–1886.

[13] Watanabe, K., and Katoh, H., 2004, “Identification of Motion Domains of Pla-nar Six-Link Mechanisms of the Stephenson-Type,” Mech. Mach. Theory,39(10), pp. 1081–1099.

[14] Ting, K., and Dou, X., 1996, “Classification and Branch Identification ofStephenson Six-Bar Chains,” Mech. Mach. Theory, 31(3), pp. 283–295.

[15] Wang, J., Ting, K., and Xue, C., 2010, “Discriminant Method for the MobilityIdentification of Single Degree-of-Freedom Double-Loop Linkages,” Mech.Mach. Theory, 45(5), pp. 740–755.

[16] Morgado, J., and Gomes, A., 2004, “A Derivative-Free Tracking Algorithm forImplicit Curves With Singularities,” International Conference on Computa-tional Science, pp. 221–228.

[17] Hunt, K., 1978, Kinematic Geometry of Mechanisms, Clarendon Press, Oxford.[18] Pennock, G., and Hasan, A., 2002, “A Polynomial Equation for a

Coupler Curve of the Double Butterfly Linkage,” J. Mech. Des., 124(1), pp.39–46.

[19] McCarthy, J., and Soh, G., 2011, Geometric Design of Linkages, Springer, NewYork.

[20] Rojas, N., and Thomas, F., 2011, “Distance-Based Position Analysis of theThree Seven-Link Assur Kinematic Chains,” Mech. Mach. Theory, 46(2), pp.112–126.

[21] Graver, J., 2001, Counting on Frameworks: Mathematics to Aid the Design of RigidStructures, The Mathematical Association of America, Oberlin, OH.

[22] Wampler, C., 2001, “Solving the Kinematics of Planar Mechanisms by DixonDeterminant and a Complex-Plane Formulation,” J. Mech. Des., 123(3), pp.382–387.

[23] Waldron, K., and Sreenivasan, S., 1996, “A Study of the Solvability of the Posi-tion Problem for Multi-Circuit Mechanisms by Way of Example of the DoubleButterfly Linkage,” J. Mech. Des., 118(3), pp. 390–395.

[24] Rojas, N., and Thomas, F., 2012, “On Closed-Form Solutions to the PositionAnalysis of Baranov Trusses,” Mech. Mach. Theory, 50, pp. 179–196.

[25] Porta, J., Ros, L., Creemers, T., and Thomas, F., 2007, “Box Approximations ofPlanar Linkage Configuration Spaces,” J. Mech. Des., 129(4), pp. 397–405.

[26] Pennock, G., and Raje, N., 2004, “Curvature Theory for the Double Flier Eight-Bar Linkage,” Mech. Mach. Theory, 39(7), pp. 665–679.



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Application of Distance Geometry to Tracing Coupler Curves