Advances in the Theory of Planar Curve Cognates

By offering alternative mechanism dimensions that achieve the same motion, knowledge of cognates can be useful in mechanical design. Roberts’s proof [1] that every planar four-bar coupler curve is generated by three distinct mechanisms was the first result in cognate theory. Much later, cognates were found for all the inversions of planar six-bar linkages, with Table 4 of Dijksman’s book [2], based on Refs. [3,4], purporting to be a complete list. For a review of all the cognates known before the publication of that table, we refer the reader to Ref. [5] and the references in Ref. [6].

Dijksman’s method of finding cognates included the following assumption [3]:

For succinctness, we introduce the following terminology.

Definition A permutation cognate is one for which the rotations of its links are a permutation of the rotations of the original linkage, both measured in the world reference frame.

Dijksman’s Conjecture All cognates for planar linkages with rotational links are permutation cognates.

This poses the question as to whether Dijksman’s table is really complete. In short, three issues remain unresolved:

1. Does Dijksman’s Conjecture hold for four-bar and six-bar linkages?

2. If it does hold, has Dijksman found all permissible permutations?

3. For each permissible permutation, is there only one cognate (or, in one case, only one two-dimensional family of cognates)?

Given these open issues, only two rows in Dijksman’s table can be considered complete:

• for four-bars, Roberts [1] settled the matter using arguments concerning the singular foci and nodal points of the coupler curve.

• Roth [7] gave a rigorous treatment of Stephenson-3 six-bars (and certain related geared linkages) by close examination of the coefficient equations of the coupler curve.

In addition to the above questions, we add:

1. Can we find all permutation cognates more efficiently than exhaustive testing?

2. Does Dijksman’s Conjecture hold for higher-order linkages (eight-bars, ten-bars, etc.)?

In Refs. [8,9], we have previously shown how to answer Question 3 using a formulation wherein planar vectors are written as complex numbers. Given a linkage type and a permutation of the angles, these papers reduce the construction of the cognates for that permutation to solving a linear system of equations. This system either has no solutions (meaning that the permutation is invalid for generating cognates), has a unique solution, or has a linear set of solutions. By automating this method, one could also answer Question 2 by exhaustively testing every possible permutation. However, Question 1 remains open.

The present paper answers Question 1 definitively in a way that also lets us easily answer Question 2 without resorting to exhaustive testing. Moreover, our approach gives a highly effective answer to Question 4, thus facilitating the construction of all permutation cognates for higher-order linkages. The approach we use to settle Dijksman’s conjecture for six-bars extends in principle to eight-bars and beyond, but the computations are currently beyond our capabilities, so Question 5 remains open. Nevertheless, we believe our method is a big step towards ultimately tackling that question.

Like Roberts [1] did for four-bars, for each linkage type, we start by determining the singular foci of the coupler curve. To do so, we use the methods of Refs. [10,11], supplemented by techniques from numerical algebraic geometry [12] as implemented in the software package bertini [13,14]. Since the singular foci belong to the coupler curve, not a specific linkage, all cognates must have the same singular foci with the same signatures. However, the singular foci alone do not completely determine the curve—recall that for four-bars, Roberts also considered the nodal points. To finish the job, our approach selects a few general points on the curve and solves the associated curve interpolation problem using numerical algebraic geometry. In this way, we find all curve cognates not just the permutation cognates. A side benefit is that the focal conditions already strongly limit the possible angle permutations, which is how we address Question 4.

To save space, we refer to the companion paper [9] for loop equations for all the six-bar inversions. Before treating these, we review the definition of a singular focus and describe how to compute the singular foci of a linkage. After first treating the Stephenson-2B linkage in detail, we summarize our results for all six-bars, omitting Watt-2 linkages because their coupler curves are the same as four-bars.

A caveat: throughout this paper, we restrict our attention to general linkages of the types we study. We do not rule out the possibility that additional curve cognates could appear in exceptional cases.

The rest of the paper is organized as follows. Section 2 reviews the definition of a singular foci and describes how to compute them, using the Stephenson-2B as an example. Section 3 summarizes the focal signatures for the four-bar, all six-bars, and one example each of an eight-bar and a ten-bar. It also details which permutations of the link rotations are compatible with these signatures, which is a necessary condition for a permutation cognate. In Sec. 4, to complete the analysis for six-bars, that is, to definitively answer Question 1, we find all cognates by appending an appropriate number of precision points to the focal point conditions and solve the system using numerical algebraic geometry. We summarize these results in Sec. 5.

The problem of finding coupler curve cognates is closely connected to precision-point path synthesis problems. In the case of curve cognates, we are not given a finite set of points on the curve, but rather we wish to find all linkages that exactly reproduce the whole curve.

In principle, one way to solve the curve cognate problem is to use a complete solution to the maximal precision-point path synthesis problem. For example, only a finite number of distinct four-bars interpolate nine general points. Given a four-bar curve, we could select nine general points on it and find all four-bars interpolating them. These four-bars will separate into groups of cognates. In Ref. [15], complete solutions of the nine-point problem for four-bars were computed using numerical algebraic geometry, resulting in 1442 coupler curves that interpolate the points, each curve generated by three cognate linkages. This accords to Roberts’s result that four-bars are triply generated. For six-bar linkages, such an approach becomes untenable since the precision-point problems are currently too big for current approaches.

For finding cognates, instead of choosing general points on the coupler curve, we can choose distinguished points associated with the coupler curve in order to make the synthesis problem easier. An advantageous choice is to select the points where the coupler curve approaches infinity, which is equivalent to finding the curve’s singular foci. This approach is motivated by recalling that the ground pivots of a four-bar linkage coincide with two singular foci of its coupler curve. The maximal number of general precision points that can be specified for the path synthesis of a four-bar with specified ground pivots is just five, and the problem can be solved ab initio with a 96-path homotopy [16]. Furthermore, since we know the location of the third singular focus, that constraint reduces the maximal number of general precision points to just three, which allows one to find all cognates even more simply. Thus, this suggests that analyzing the singular foci of the coupler curves of six-bars and higher-order linkages might provide significant leverage in solving for cognates. Indeed, this will turn out to be the case, but first we must review the definition of singular foci and how to compute them.

The notion that conics (parabolas, ellipses, and hyperbolas) have focal points is a familiar one. So too might one remember that as an ellipse is morphed into a circle, its two foci coalesce in to a single focus. This double focal point at the circle’s center is an example of a singular focus, and it clearly tells us something fundamental about the circle. Similarly, the singular foci of coupler curves hold important information that can help us solve the riddle of curve cognates.

A general definition of singular foci can be found in Ref. [17], and examples of their application to linkages appear in Ref. [18]. We will make use of the treatment found in Refs. [10,11] which greatly simplifies computations through the use of isotropic coordinates. From a geometric viewpoint, both treatments involve analyzing how a curve meets infinity, whereas from an algebraic viewpoint, they involve analyzing the highest order terms of the coupler curve equation. This makes sense because near infinity, when variables get large, the highest order terms dominate the lower order ones. A fundamental difference of Refs. [17,18] as compared to Refs. [10,11] comes in how “infinity” is defined.

If one confines their analysis to the reals, the question of how a coupler curve reaches infinity is vacuous: links are finite, so the whole coupler curve must remain finite. For example, this is true of a circle. However, if we allow the coordinates of the curve to take on complex values, this picture completely changes. Fundamentally, this is because in the reals, the polynomial for squared distance from the origin, x² + y² has only a single zero, (x, y) = (0, 0), whereas in complex space, any point with $y=±ix$ is a zero. Because of this, there are points that stretch out to infinity in the complex plane while still satisfying a circle’s equation. The same is true for any algebraic curve.

Let us call the two lines of points that are zero distance from the origin the isotropic lines. Their linear equations are $x+iy=0$ and $x−iy=0$⁠. Since the fundamental property of a rigid link is that it preserves distance, it should not be too surprising that these lines hold a special significance. The classical treatment of singular foci takes a one-homogeneous approach wherein the open Euclidean plane is closed up by adding a line at infinity, having one point on that line for each slope a finite line can have. In a sense that can be made precise, two distinct parallel lines intersect at the point at infinity associated with their slope. This implies that all lines parallel to one of the isotropic lines meets it in the same point at infinity. These two isotropic points play a major role in the classical theory.

A more facile way of treating singular foci, and in many ways, a more convenient way of treating nearly all problems in planar linkages, is to use isotropic coordinates. This is almost identical to a complex-vector formulation of kinematics that treats any point (x, y) in the Euclidean plane as the point $p=x+iy$ in the complex plane. To turn this into isotropic coordinates, we form a second coordinate $p¯=x−iy$ so that $(p,p¯)$ is just a linear change of coordinates from (x, y). When x and y are real, $p¯$ is the complex conjugate of p, but this is no longer true when x or y has a nonzero imaginary part.

If we close up the Euclidean plane one-homogeneously, i.e., with a single line at infinity, all the lines parallel to the p-axis meet in one isotropic point, and all lines parallel to the $p¯$-axis meet in the other one. But there is another way of closing up the plane that works to our advantage: add two lines at infinity, one parallel to each coordinate direction. This two-homogenization is done using the substitutions $p=P/w$ and $p¯=P¯/w^$⁠, and clearing denominators. The new coordinates are written $([P,w],[P¯,w^])$ and we do not allow $[P,w]=[0,0]$ nor $[P¯,w^]=[0,0]$⁠. The bracket notation means that only ratios matter: [a, b] and [λa, λb] for λ ≠ 0 are considered the same point.

The bottom line (see Ref. [10] for more) is that after making this transformation, the coordinates of the singular foci are formed as the values of p where $p¯$ hits infinity (i.e., where $w^=0$⁠) and vice versa for the values of $p¯$⁠. For the real curves we study here, if a term $αpjp¯k$ appears, so does its conjugate $α*pkp¯j$⁠. This means that $p¯$ hits infinity in the complex conjugates of where p hits infinity, so in fact we just need to compute the p-coordinates. To do this for plane curve given by $f(p,p¯)=0$⁠, make the substitution $p¯=1/w^$⁠, clear denominators, set $w^=0$⁠, and solve for p. Or put more simply, just cross out all terms of f except those with the highest power of $p¯$ and then set $p¯=1$⁠. The result is a simpler polynomial g(p) = 0 whose roots are the p-coordinates of the curve’s singular foci. As a simple example, consider the equation $(p−c)(p¯−c*)−r2=0$⁠, which is a real circle with center c and radius r. Omitting lower-order terms in $p¯$⁠, we are left with $(p−c)p¯=0$⁠, and setting $p¯=1$⁠, we find that the singular focus of the circle is its center p = c. (More properly speaking, it is $(p,p¯)=(c,c*)$⁠.)

To find singular foci following the prescription earlier, we need a curve’s coupler curve equation in the form $f(p,p¯)=0$⁠. In Ref. [10], it is shown how to carry this through using the Dixon determinant formulation from Ref. [19] to eliminate rotation angles from the kinematic loop equations. In Ref. [11], the singular foci are found by direct computations on the loop equations without elimination, which yields additional information about the rotations associated with the singular foci. To automate this, we follow the paradigm of numerical elimination theory [14,20] in numerical algebraic geometry and use the software package bertini [13,14]. This approach avoids any trouble with extraneous roots, which is a possibility when using the Dixon determinant. Furthermore, the information about rotations further constrains the possible cognates, making our search for cognates more efficient.

This homotopy-based approach computes the singular foci as numerical values that are dependent on the input coefficients a_kj. In some cases, symbolic formulas for each singular focus can be trivially observed from the numerical values such as when a singular focus is a ground pivot. Explicit symbolic expressions of the singular foci are not required to find permutation cognates but to settle the existence of any other cognates, they become essential. One approach to deriving is to employ exactness recovery methods [21] to determine symbolic expressions directly from numerical computations. Alternatively, one can use a computer algebra package, taking advantage of the knowledge of which rotations vanish at each singular focus to simplify expressions.

For linkage type, the term focal signature consists of the number of singular foci, the distinct patterns of vanishing rotations at each singular focus, and the multiplicity of each pattern. The computational approach in Sec. 2.2 yields all this information. For the Stephenson-2B, this is the information in Table 1 except the explicit symbolic formulas. It tells us that there are five singular foci: two with multiplicity 3 and three each having multiplicity one. Furthermore, one of the multiplicity 3 singular foci comes from a multiplicity 3 vanishing pattern, whereas the other one comes from three distinct vanishing patterns each of multiplicity 1. Table 2 lists the focal signature information for the four-bar and all six-bar types. For easier reading, it only shows the vanishing rotations in θ_j for j = 1, …, N − 1 since one immediately knows from this list that the complementary $θ^k$ vanish. As can be observed in that table, it is possible for a singular focus to have different multiplicities with respect to different vanishing rotations. In particular, this occurs for singular focus b₀ of the Watt-1A and Stephenson-2A linkages.

If a cognate is a permutation cognate, the permutation of the angles will permute the patterns of rotations that vanish where the coupler curve hits infinity. But this focal signature is a generic property of the linkage, so it cannot change. Only angle permutations that preserve the focal signature are candidates for permutation cognates. Applying any two signature-preserving permutations in succession still preserves the signature, so the collection of all such permutations forms a group.

Preservation of the focal signature is a necessary but not sufficient condition for an angle permutation to give a valid cognate. The final check is to apply the methods from Refs. [8,9] to see if the signature-preserving permutations are consistent with preserving the system of loop and coupler point equations. We will find that except in the case of the four-bar, the signature-preserving permutations are a small subset of full set of permutations among the moving links, i.e., for n ≥ 6, each n-link mechanism type has many fewer than (n − 1)! signature-preserving permutations. We will give the details in the following subsections, starting with our running example, the Stephenson-2B.

A three-way symmetry arises from two transpositions that have an element in common, which is denoted S₃ and has order 3! = 6. For example, the transpositions θ₁ ↔ θ₂ and θ₁ ↔ θ₃ have one element in common, namely θ₁, so this corresponds with the three-way symmetry group S₃. Finally, the product $S3×Z2$ has order 3! · 2 = 12 and arises from three permutations, two which have one element in common with the third one having distinct elements, e.g., θ₁ ↔ θ₂, θ₁ ↔ θ₃, and θ₄ ↔ θ₅.

Every permutation group includes the trivial permutation where nothing is changed. We will refer to any other permutation as nontrivial.

Using Table 1, we can determine which permutations are valid. As noted in Sec. 2.3, there are three types of singular foci: ground pivot a₀, ground pivot b₀, and the three singular foci of multiplicity 1. Hence, all valid permutations must keep the two ground pivots fixed, but the three singular foci of multiplicity 1 may permute among themselves.

From b₀, one sees that θ₁ cannot permute with any other rotation as $θ^1$ is the unique $θ^j$ vanishing at b₀. Since θ₁ cannot permute and θ₂ is the only other rotation that appears in each of the three sets of rotations for a₀, θ₂ also cannot permute. Therefore, the two ground pivots have shown that the only possible permutations are among θ₃, θ₄, and θ₅. Checking transpositions θ₃ ↔ θ₄ and θ₄ ↔ θ₅ separately shows that the vanishing structure for the ground pivots and one of the other singular foci of multiplicity 1 are maintained while transposing the other two singular foci of multiplicity 1. Hence, this yields a three-way symmetry among θ₃, θ₄, and θ₅ showing that the group action on the Stephenson-2B cognates corresponds with S₃. Note that since links 3 and 5 are both binary links connected between links 2 and 4, they are topologically equivalent, so the transposition θ₃ ↔ θ₅ corresponds with a relabeling of the mechanism rather than a new cognate. Accordingly, only three of the six permutations in S₃ have the potential to yield unique mechanisms. As found in Ref. [9], these are in fact valid permutation cognates.

As Roberts [1] identified, the three singular foci of the four-bar can be permuted in any way which corresponds to any permutation of the three rotations. Hence, the three-way symmetry shows that the group action on the rotations of the four-bar corresponds with S₃. Considering Fig. 2(a), the transposition θ₁ ↔ θ₃ corresponds with relabeling so only three of the six permutations in S₃ can produce unique mechanisms. These are the permutation cognates found by Roberts [1].

The only valid nontrivial permutation of rotations for the Stephenson-1 is the transposition θ₁ ↔ θ₂. Hence, the group action corresponds with $Z2$⁠. This action gives a permutation cognate.

The group of signature-preserving permutations for Stephenson-2A is $Z2×Z2$ generated by the two transpositions $θ2↔θ3$ and $θ4↔θ5$ yielding permutation cognates (Fig. 3).

Considering Fig. 4, Roth [7] determined that the cognates of the Stephenson-3 mechanism are generated by applying Roberts cognates to the four-bar formed by links 0-1-2-3 and a skew pantograph transformation to links 10-4-5 as discussed in Ref. [8]. The results of the computation from Sec. 2.2 match those previously known results as follows. First, the group of valid permutations of the Stephenson-3 mechanism corresponds with $S3×Z2$ where the S₃ arises from a three-way symmetry of the rotations θ₁, θ₂, and θ₃ while the $Z2$ arises from the transposition θ₄ ↔ θ₅. Finally, as in the case of the four-bar, the group S₃ can only produce three unique mechanisms due to relabeling.

For the Watt-1A, the only signature-preserving permutation is the trivial one. However, the Watt-1A is unique among the six-bars as it has a two-dimensional family of cognates (Fig. 5).

The group of signature-preserving permutations for Watt-1B is $Z2×Z2$ generated by the transpositions θ₂ ↔ θ₃ and θ₄ ↔ θ₅. These yield permutation cognates.

Whereas our analysis of the six-bars confirms that Dijksman found all permutation cognates, the equivalent question for higher-order linkages has never been addressed. An exhaustive check of all possible permutations for an eight-bar would have 7! = 5040 cases to check. (The ground link is fixed, so an n-link mechanism has (n − 1) link angles that might permute.) The necessity of preserving the focal signature greatly reduces the task. We illustrate this fact by treating the eight-bar shown in Fig. 6 followed by a ten-bar in the next section.

By computing the focal signature, we have reduced the complete analysis of permutation cognates from 5040 cases to just 2. The trivial permutation and the transposition θ₁ ↔ θ₂ each give a unique mechanism: the original one in Fig. 6 and its cognate shown in Ref. [8].

We have shown how to find all permutation cognates. It is much harder to show that no other type of cognate exists. To attack this problem, we solve a special type of precision-point path synthesis problem. For four-bars, one could settle the question by sampling ten general points from the coupler curve of a general four-bar, solving a path synthesis problem for nine of these points, and checking which of these solutions also interpolates the tenth point. This would give the original linkage and all its cognates. For four-bars, this is feasible, but for a six-bar, this would require solving a path synthesis problem having 15 precision points which is beyond current computational capability.

For six-bars, we proceed similarly. The first step is to check how many independent conditions are imposed by specifying the singular foci. Let x be the set of link parameters, x = (a₀, b₀, …), and suppose that S(x) is map that gives the singular foci. Choose a general (i.e., random) set of parameters x*. Then, all cognates to x* must satisfy S(x) − S(x*) = 0. Let JS(x*) be the Jacobian matrix of S evaluated at x*. By Ref. [20], assigning the singular foci imposes rank $JS(x*)$ conditions on the link parameters. Accordingly, we only need to specify $k=15−rankJS(x*)$ additional precision points to obtain an exactly constrained path synthesis problem. The k points are randomly sampled from the original coupler curve to ensure genericity. The second column in Table 5 lists k, the remaining degrees-of-freedom, for the four-bar and six-bar linkages.

When forming the path synthesis problems, for the four-bars and six-bars, we may write the j^th entry of S(x) form S_j(x) = p_j(x)/q_j(x). Clearing the denominator yields a polynomial of the form p_j(x) − S_j(x*) · q_j(x) = 0 for each singular focus.

Methods in numerical algebraic geometry [12,14] can be used to both sample random points from the coupler curve and solve the resulting path synthesis problem for four-bar and six-bar linkages. In particular, our experiments utilized regeneration [23,24] to solve the path synthesis problems with the resulting finite number of nondegenerate linkages summarized in Table 5.

Due to degrees-of-freedom counting, each nondegenerate linkage is a cognate of the original linkage if its coupler curve also passes through one additional randomly selected point on the coupler curve. If this additional check results in a unique cognate linkage, then the completeness test verifies that there is indeed a unique cognate per valid assignment of singular foci. This was indeed the case for the four-bar (confirming Roberts’s result [1]) and the Stephenson-3 (confirming Roth’s result [7]). The same was true for the Watt-1B, Stephenson-1, Stephenson-2A, and Stephenson-2B six-bar linkages which proves that Dijksman’s corresponding list of cognates is complete.

For the Watt-1A, Sec. 3.7 states that there is only the trivial assignment of the singular foci and there is a two-dimensional family of cognates. One natural way to parameterize this two-dimensional family is by the ground pivot a₀ which, in isotropic coordinates, corresponds with the two variables $(a0,a¯0)$⁠. Hence, one first fixes this ground pivot and then applies the completeness test to verify that there indeed is a unique cognate proving that Dijksman’s corresponding list of cognates is complete. We note that this computation first required the filtering of degenerate components before checking at an additional random point on the coupler curve.

Permutation cognates are those that arise by permuting link rotation angles. Dijksman conjectured that all planar cognates are permutation cognates and compiled a list of them for all six-bar planar mechanisms. He did not show how to make sure the list is exhaustive or how to generalize his approach to higher-order linkages. We have shown that analyzing the vanishing rotations corresponding to the singular foci of a mechanism type produces a short list of permutations that are compatible with preserving these focal signatures. This reproduces the result known for four-bars and also reproduces Dijksman’s list for six-bars, confirming that he found all permutation cognates. Testing an eight-bar example, our method reveals that just one nontrivial permutation preserves the focal signatures, and indeed, it yields a path cognate. For a ten-bar example, our method reveals that three independent transpositions of rotations preserve the focal signatures, but the further step of deriving cognates using these transpositions shows that one of them is inconsistent. Thus, this linkage type is shown to have exactly four permutation cognates (counting the original linkage).

Since its ground link is fixed, an n-bar linkage has (n − 1)! possible ways to permute its rotations. Restricting one’s search to just permutation cognates, our method provides an extreme reduction in the number of permutations to check compared to a naive approach of checking all possible permutations. For the eight-bar we studied, the reduction is from 7! = 5040 to just one transposition. For the ten-bar, it reduces the cases from 9! = 362,880 to just three transpositions.

For six-bars, we were able to check Dijksman’s Conjecture by computing all cognates that match a given assignment of singular foci by adding to the focal conditions a minimally sufficient number of precision points sampled generically from the coupler curve and solving the resulting path synthesis problem. Without the focal conditions, one would be faced with a 15-point path synthesis problem, which is currently impossible to solve completely. With the focal conditions, one needs to add only two or five precision points (depending on the linkage type). These smaller problems are feasible, and we find in every case that only permutation cognates result. This confirms Dijkman’s Conjecture for general six-bars, confirming that the list of permutation cognates is in fact the complete list of all possible path cognates.

While our methodology has proven effective for finding all permutation cognates for higher-order linkages, we are presently not able to solve the path synthesis problems that would confirm Dijksman’s Conjecture for eight-bars or beyond due to computational limitations.

SNS and JDH were partially supported by NSF CCF-181274. SNS was also supported by the Schmitt Leadership Fellowship in Science and Engineering.

There are no conflicts of interest.