## 1 Introduction

Cognate linkages are mechanisms that share the same motion. Curve cognates, in particular, are distinct mechanisms, each with one degree-of-freedom, whose respective coupler points draw the same curve. Since cognates may occupy different regions of space and have different transmission characteristics, they can be useful in finding a more suitable mechanical design for the same function. Knowledge of cognates can also be useful when solving mechanism synthesis problems, especially in confirming that a complete solution list has been found [1,2].

We will show how curve cognates for general planar linkages can be generated by a simple sequence of operations: form complex-vector loop equations, interchange certain link rotations, match coefficients in the kinematic equations, and then solve the resulting linear equations. An interchange of link rotations means to permute complex rotations applied to the links. For example, swapping rotations for two links means that one is aiming to construct a cognate in which the two links in the cognate linkage simply have rotational characteristics that are swapped between the corresponding two links in the original linkage. The matching of coefficients ensures that the kinematic equations hold for all input angles, and thus, the resulting linkage is indeed a curve cognate since it traces out the same coupler curve.

Given knowledge of which link rotations can be interchanged, the procedure is straightforward to carry out, and the results are easy to interpret graphically and ready for computer simulation. If one posits an interchange that does not correspond to a cognate, this is revealed as an inconsistency in the linear equations. The method can be used to reproduce all the known results for curve cognates of the four-bar and all the six-bar linkages. In addition, curve cognates are also constructed for an eight-bar and a ten-bar linkage.

Even if one considers all possible permutations for the interchange of rotations, a process whose complexity grows quickly with the number of links, this does not a priori mean that all possible cognates have been found. Until proven otherwise, there remains the possibility that some more subtle transformation of the linkage could leave the coupler curve invariant. A companion paper [3] completes the theory of cognates for general six-bar linkages by showing that the interchange of rotations does in fact produce all possible cognates. A complete cognate theory for eight-bars and beyond is still open, but already the approach of Ref. [3] sets limits on which interchanges have a chance to produce cognates. The method presented here produces the cognates for any valid permutation one specifies.

The most famous result in cognate theory is from 1875 in which Roberts [4] showed that every four-bar coupler curve is triply generated. That result is sometimes called the Roberts–Chebyshev theorem in recognition of Chebyshev’s independent discovery of it 3 years later [5]. To our knowledge, no results on cognates of six-bar linkages were found until the work of Hartenberg and Denavit [6], followed by Roth [7], and Soni [8]. See the study by Nolle [9] (with reference list in Ref. [10]) for a historical review as of 1974. Finally, nearly 100 years after Roberts, Dijksman [1113] compiled cognates for all the six-bar planar linkages. In the conference paper [14] upon which the present article expands, we showed how our method reproduces the skew pantograph, all known cognates for the four-bar, Watt six-bars, Stephenson-I and Stephenson-III six-bars, and a new eight-bar cognate. Soni also found cognates for certain eight-bars [15].

Dijksman provides the most comprehensive list of six-bar cognates by showing that cognates can be generated from permutations of link rotations and presented his results by means of intricate geometric constructions. Although correct, these drawings and their explanatory text can be rather difficult to decode, thereby presenting a barrier to understanding and using those results. The purpose of this article is to present a simple method of understanding by constructing planar cognates using a complex-vector approach. We are by no means the first to approach cognates from this direction; indeed, Nolle [9] states that Schor (1941), Schmid (1950), Meyer zur Capellen (1956), and Wunderlich (1958) all used some version of a complex plane formulation in treating Roberts cognates. No doubt there have been others as well since the complex plane formulation is arguably the most natural way to treat any planar linkage with all rotational joints. Rather than studying each mechanism type in isolation, our contribution is an approach that allows one to understand all the known results in terms of how a permutation of link rotations leads to a cognate and to construct new cognates for eight-bars, ten-bars, and beyond.

The rest of this article is organized as follows. Following a list of nomenclature in Sec. 2, Sec. 3 provides background information on types of cognates and complex-vector notation. Section 4 presents our method of constructing cognates. Section 5 illustrates the method by considering in succession Roberts cognates for four-bars, the Stephenson-2A six-bar, the Watt-1A six-bar, an eight-bar, and a ten-bar. After the summary in Sec. 6, an Appendix gives recipes for constructing all possible six-bar cognates.

## 2 Nomenclature

We use the following conventions and notations. Let N be the number of links and L = N/2 − 1 be the number of loops in the mechanism. One link is designated the ground and another one is designated the coupler. For example, the Watt six-bar has Watt-1 and Watt-2 inversions depending on which link is called ground, and then, a choice of coupler link gives curve types Watt-1A and Watt-1B.

• The ground link is always link 0.

• Rotation θi, i = 1, …, N − 1, is the rotation of link i relative to ground in a complex-vector form, i.e., $θi=e−1Θi$, where Θ i is the (real) rotation angle of link i.

• Link parameters ai, bi, ci, i = 0, 1, …, N − 1, are complex vectors fixed in link i. We choose one joint of each moving link as its origin, then ai, bi, ci specify the relative locations of the other joints. The origin for the ground link is chosen arbitrarily.

• θi′, i = 1, …, N − 1 and ai′, bi′, ci′, i = 0, …, N − 1 are the rotations and link parameters, respectively, for a cognate mechanism.

• p is the complex-vector from the ground origin to the coupler point. By definition, it is the same for both the original and cognate linkages.

• Complex number γj, j = 1, …, L is a nonzero scaling factor arising in the proof that loop equation j for the original mechanism is equivalent to a corresponding loop equation for its cognate.

## 3 Background

This section provides an introductory review to types of cognates and to complex-vector notation as applied to planar linkages. In this article, we restrict ourselves to planar mechanisms with rigid links connected by rotational (pin) joints. Each joint connects two links, thereby imposing one vector constraint, equivalent to two scalar constraints, requiring that the respective center points of the joint on the two links must coincide. For the purpose of classifying mechanisms with one degree-of-freedom, we consider only unexceptional mechanisms, being those whose number of freedoms does not change when the link dimensions are perturbed in a general fashion. For N links and J joints, these mechanisms obey the Grashof mobility criterion: M = 3(N − 1) − 2J. For M = 1 degree-of-freedom, this implies the mechanism must have N = 2L + 2 links and J = 3L + 1 joints, where L is the number of independent loops in the mechanism.

### 3.1 Types of Cognates.

The cognates under consideration here are curve cognates, that is, linkages that draw the same curve. We only treat cognates of the same curve type, ignoring the possibility that a six-bar might duplicate a four-bar curve or that two types of six-bars might draw the same curve.

Some curve cognates satisfy additional criteria that define subclasses of interest. A coupler cognate is a curve cognate where the coupler link maintains the same orientation as the original. Alternatively, after selecting an input link, a timed curve cognate is a curve cognate with the same functional relationship between the input rotation and the point on the coupler curve. Finally, a timed coupler cognate is both a coupler cognate and a timed curve cognate with respect to an input link. Once a curve cognate is found, it is straightforward to check these additional criteria and we will do so.

Another class of cognates that have been considered elsewhere are function cognates. These cognates maintain the functional relationship between an input crank and an output link. Some function cognates are not curve cognates, but those that are can be easily recognized here.

### 3.2 Complex-Vector Notation.

To simplify the mathematical formulas used to represent linkages and compute cognates, we use a complex-vector formulation. Thus, a vector [ab] in the plane is represented by a complex number a + bi, where $i=−1$. Any complex number can be cast in the form seiΘ, where s is a scalar and Θ is an angle in radians. Complex arithmetic facilitates geometric transformations. In particular, complex addition implements translation, while multiplication by $seiΘ$ corresponds to a stretch-rotation, which stretches by s and performs a complex rotation by angle Θ. Throughout this article, we use θ to abbreviate the complex rotation, $θ=eiΘ$, and more specifically, after numbering the links of a mechanism, θj is the complex rotation of link j. By convention, the ground link, which does not move, is always link 0.

To illustrate the complex-vector notation, we begin with the case of a four-bar linkage. Referring to Fig. 1, we have a loop closure equation:
(1)
and a coupler-point equation
(2)
Note that by subtracting one from the other, we have an alternate coupler-point equation:
(3)
This is just the sum of vectors going on a different path from the origin to the coupler point. Although equivalent, one of Eq. (2) or Eq. (3) will prove more convenient depending on which cognate of the initial linkage one wishes to pursue.
Fig. 1
Fig. 1
Close modal

The link dimensions and the placement of the ground pivots in the plane are given by a0, b0, a1, a2, b2, and a3. To compute the mechanism’s motion, consider that given one link rotation, say θ1, one can solve Eq. (1) for θ2 and θ3, keeping in mind that the complex loop equation is equivalent to two scalar equations (by taking its real and imaginary parts) and the rotations are each parameterized by a single scalar angle. Then, one can evaluate the coupler point position using Eq. (2) or Eq. (3). A more facile approach based on using the complex conjugate of the loop equation is presented in Refs. [16,17]. This article does not need to solve the loop and coupler-point equations; instead, we merely need to show that cognate mechanisms satisfy the same equations, i.e., trace the same coupler curve as the original mechanism.

## 4 Cognate Construction Procedure

The key steps in our method for constructing curve cognates for an L-loop mechanism are as follows. It can be seen that including the origin point in the ground link and the coupler point in the coupler link, an unexceptional, mobility-1, L-loop mechanism has 4L + 2 independent link parameters and N = 2L + 2 links. Let q = (q1, …, q4L+2) be the link parameters of the original linkage and q′ be those for the cognate. Similarly, let θ = (θ1, …, θ2L+1) be the link rotations for the original linkage and θ′ the rotations for the cognate.

1. Choose a permutation P for interchanging link rotations between the original and the cognate mechanism. Hence, we are seeking a cognate whose link rotations are θ′ = P(θ).

2. For the original mechanism, form L independent complex-vector loop equations,
and one coupler-point equation for a complex-vector path from the origin to the coupler point, p,
3. Similarly, for the cognate mechanism, write equations
4. Substitute θ′ = P(θ) into equations from step 3 yielding
5. If the corresponding equations in steps 3 and 4 do not contain the same set of link rotations, replace equations
with independent linear combinations to allow every link rotation to be properly matched in the final step.
6. Set each loop function for the cognate equal to a stretch-rotation of its corresponding function from the original:
7. Set the coupler points equal:
8. Solve for the cognate parameters q′ and stretch-rotations γ1, …, γL by matching the coefficients of the link rotations and constant term in the loop (step 6) and coupler-point (step 7) equations.

The final solution is a mechanism that satisfies the same set of loop equations and the same coupler-point equation as the original mechanism, and hence, it is a curve cognate. If the permutation chosen at step 1 does not alter the rotation of the input link, then we obtain a timed curve cognate. If the permutation does not alter the rotation of the coupler link, then we obtain a coupler cognate. It is sometimes possible to satisfy both of these and thus obtain a timed coupler cognate.

For the matching procedure to succeed in step 8, the equations produced in steps 6 and 7 must have the same link rotations appearing on both sides. For example, if the link rotation 1 is to be interchanged with link rotation 2, then every equation must involve either both links 1 and 2 or neither of them. Usually, but not always, with appropriate choices of loops and paths in steps 2 and 3, one is able to match rotations without needing rearrangement in step 5.

If rotations match properly, the process of solving for the cognate in step 8 is simple. As illustrated in Eqs. (1)(3), the equations in the complex-vector formulation are linear in the parameters, and the only unknowns on the right-hand side of the equations in steps 6 and 7 are the stretch-rotations, which also appear linearly. Hence, the equations to solve in step 8 are all linear, and so the system is straightforward to solve symbolically. We note that not all permutations one might consider at step 1 lead to cognates: invalid choices become apparent in step 8 as an inconsistent set of linear equations. This may happen if the number of coefficients to match exceeds the number of unknowns. Conversely, it may also happen that the number of coefficients to match is smaller than the number of unknowns, in which case there exists a positive-dimensional set of cognates. In particular, this happens for the Watt-1A mechanism, which has a two-dimensional family of curve cognates (see Sec. 5.3).

## 5 Examples

Our procedure becomes much clearer when applied to a specific mechanism. We will first illustrate it using the simplest example of interest, the four-bar linkage, followed by a Stephenson-2A six-bar, the Watt-1A six-bar, an eight-bar, and a ten-bar. The Appendix summarizes how to derive cognates for all the planar six-bar curve mechanisms.

### 5.1 Four-Bar: Roberts Cognates.

The existence of three curve cognates for every four-bar linkage was first proved by Roberts [4]. Figure 2 shows their geometric construction, as was found by Chebyshev [5] and Cayley [18]. This arrangement contains three similar triangles, links 2, 2′, and 2″, and three parallelograms. A proof that the four-bars labeled “swap 1-2” and “swap 2-3” are truly cognates of the original requires showing that point c0 stays fixed as the original four-bar moves. In a geometric approach, this fact follows from showing that the focal triangle a0, b0, c0 is also similar to the coupler triangle, link 2. Our approach proves this while at the same time generating the formulas for the cognate link parameters and for the location of point c0.

Fig. 2
Fig. 2
Close modal
Table 1

Four-bar linkage and cognate parameters

OriginalSwap 1-2Swap 2-3
a00.0 + 0.0i0.0000 + 0.0000i−0.6549 + 2.2196i
b03.0 + 0.8i−0.6549 + 2.2196i3.0000 + 0.8000i
a10.8 + 0.8i0.2000 + 0.9000i1.4118 + 0.2196i
a21.2 − 0.3i−0.6118 + 0.5804i1.2431 − 0.4392i
b20.2 + 0.9i0.8000 + 0.8000i0.2431 − 0.7392i
a31.0 + 0.3i−0.2431 + 0.7392i1.0000 − 1.2000i
OriginalSwap 1-2Swap 2-3
a00.0 + 0.0i0.0000 + 0.0000i−0.6549 + 2.2196i
b03.0 + 0.8i−0.6549 + 2.2196i3.0000 + 0.8000i
a10.8 + 0.8i0.2000 + 0.9000i1.4118 + 0.2196i
a21.2 − 0.3i−0.6118 + 0.5804i1.2431 − 0.4392i
b20.2 + 0.9i0.8000 + 0.8000i0.2431 − 0.7392i
a31.0 + 0.3i−0.2431 + 0.7392i1.0000 − 1.2000i

We begin by considering the “swap 1-2” cognate. The four-bar has one loop and four links, and its six-link parameters are denoted a0, b0, a1, a2, b2, and a3. In Step 1, we consider the permutation (θ1′, θ2′, θ3′) = (θ2, θ1, θ3). The loop equation Eq. (1) and the coupler-point equation Eq. (2) are used for the original mechanism in step 2 and for the cognate mechanism in step 3 with a0 in place of a0, θ1′ in place of θ1, and so on. After the substitution in step 4, step 5 is unnecessary since the same set of link rotations appear in f1, f1′ and in f2, f2′, respectively. Steps 6–7 yield
(4)
(5)
In step 8, equating the coefficients of the link rotations 1, θ1, θ2, and θ3 on both sides of these equations yields a set of seven linear equations in seven unknown parameters, these being (a0′, b0′, a1′, a2′, b2′, a3′) for the cognate linkage and the stretch-rotation γ1. Listing these out, the loop equation Eq. (4) gives
(6)
while the coupler-point equation Eq. (5) yields
(7)

Since a1′ = γ1a2 and a1′ = b2, one finds that γ1 = b2/a2, which is the stretch-rotation that transforms a2 into b2. With γ1 known, all of the cognate parameters are easily determined from Eqs. (6) and (7). In particular, one sees that ground pivot a0′ stays fixed, i.e., a0′ = a0, whereas ground pivot b0′ moves to a new location b0′ = c0 : = a0 + (b2/a2)(b0a0). We label this new ground pivot as c0 in Fig. 2. As is well known, (a0, b0, c0) are the singular foci of the four-bar’s coupler curve, and they form a triangle that is similar to the coupler triangle, (0, a2, b2) [19].

The simple linear relations of Eqs. (6) and (7) directly imply the parallelograms and similar triangles in the Chebyshev–Cayley geometric construction. In particular, b2′ = a1, a1′ = b2 in Eq. (7) imply that quadrilateral a1, b2, b2′, a1′ is a parallelogram, while b2′ = a1, a2′ = γ1a1, hence γ1 = a2′/b2′, along with γ1 = b2/a2 shows that triangle 2′ is similar to triangle 2. These geometric relations automatically appear using our approach.

A second cognate is found by swapping rotations between links 2 and 3. Call its parameters a0″, …, a3″ to distinguish them from those for the first cognate. The same procedure applies, but to successfully match terms in step 8, we can use the alternate coupler-point equation Eq. (3) at step 3. Since Eq. (3) was obtained by subtracting Eq. (1) from Eq. (2), this preliminary rearrangement avoids needing any additional rearrangement in step 5. This time, one finds that b0″ = b0 stays in place, while a0 moves to the third singular focus, a0″ = b0′ = c0. Matching all coefficients, we find the stretch-rotation factor, call it ζ1, to be
with cognate link parameters

One may also swap rotations between links 1 and 3. To carry through the matching procedure without needing step 5, one picks Eq. (2) at step 2 and Eq. (3) at step 3. Thus, after the substitution θ1′ = θ3 and θ3′ = θ1 at step 4, the same rotations, namely, θ1 and θ2 appear on both sides in step 7. The result computed in step 8 is valid, but it does not produce a new four-bar. Instead, this simply produces the original four-bar with its links renumbered 0-3-2-1 in place of 0-1-2-3. While this renumbering is not interesting from a mechanical standpoint, it is meaningful algebraically. As shown in Ref. [2], the system of path-synthesis equations for a four-bar coupler curve to pass through nine given precision points has 8652 solutions that arise in a six-way symmetry: three cognates which each allow a two-way renumbering. Hence, the 8652 solutions correspond with 1442 distinct coupler curves, each of which is generated by three four-bar curve cognates.

Finally, we observe that the first cognate (a0′, …, a3′) is a timed curve cognate if link 3 is the input, while the second cognate (a0″, …, a3″) is if link 1 is the input. Neither is a coupler cognate since they do not preserve the rotation of link 2.

Example 1

Table 1 lists the parameters for the four-bar linkage along with the parameters (to 4 decimal places) for the cognates derived above and drawn in Fig. 2.

### 5.2 Stephenson-2A.

The Stephenson-2 six-bar has two coupler curve types depending on which link is designated the coupler. Figure 3 shows the Stephenson-2A option, where link 5 carries the coupler point. (The alternative, Stephenson-2B, places the coupler point on link 2.) Cognates for the Stephenson-2A appear as a group of four: the original, swap rotations 2-3, swap rotations 4-5, and swap both 2-3 and 4-5. We will show how to derive all the Stephenson-2A curve cognates and also illustrate what goes wrong for an inadmissible permutation.

Fig. 3
Fig. 3
Close modal
Table 2

Stephenson-2A linkage and cognate parameters

OriginalSwap 2-3Swap 4-5Swap 2-3 and 4-5
a00.0 + 0.0i0.3204 + 0.6180i0.3139 + 0.3869i1.1198 + 1.1559i
b01.0 + 0.0i1.0000 + 0.0000i1.0000 + 0.0000i1.0000 + 0.0000i
a1−0.2 + 0.5i0.1731 + 0.4634i0.0562 + 0.4204i0.6019 + 0.1713i
a20.4 − 0.1i0.3460 − 0.4388i0.5535 − 0.3591i0.2957 − 0.8438i
b2−0.1 + 0.6i−0.3662 − 0.0965i−0.1542 + 0.5861i−0.8562 + 0.4630i
a3−0.6 + 0.1i−0.2100 + 0.3152i−0.6044 + 0.0442i0.1603 + 0.9601i
a4−0.2 + 0.3i0.0495 + 0.3275i0.5280 + 0.4040i0.8572 − 0.4767i
b4−0.4 − 0.3i−0.1505 − 0.2725i1.1280 − 0.2960i1.4572 − 1.1767i
a51.1 − 0.4i0.7268 + 0.0538i−0.3693 − 0.3343i−0.7612 − 0.2463i
b5−0.6 + 0.7i−0.6000 + 0.7000i0.2000 + 0.6000i0.2000 + 0.6000i
OriginalSwap 2-3Swap 4-5Swap 2-3 and 4-5
a00.0 + 0.0i0.3204 + 0.6180i0.3139 + 0.3869i1.1198 + 1.1559i
b01.0 + 0.0i1.0000 + 0.0000i1.0000 + 0.0000i1.0000 + 0.0000i
a1−0.2 + 0.5i0.1731 + 0.4634i0.0562 + 0.4204i0.6019 + 0.1713i
a20.4 − 0.1i0.3460 − 0.4388i0.5535 − 0.3591i0.2957 − 0.8438i
b2−0.1 + 0.6i−0.3662 − 0.0965i−0.1542 + 0.5861i−0.8562 + 0.4630i
a3−0.6 + 0.1i−0.2100 + 0.3152i−0.6044 + 0.0442i0.1603 + 0.9601i
a4−0.2 + 0.3i0.0495 + 0.3275i0.5280 + 0.4040i0.8572 − 0.4767i
b4−0.4 − 0.3i−0.1505 − 0.2725i1.1280 − 0.2960i1.4572 − 1.1767i
a51.1 − 0.4i0.7268 + 0.0538i−0.3693 − 0.3343i−0.7612 − 0.2463i
b5−0.6 + 0.7i−0.6000 + 0.7000i0.2000 + 0.6000i0.2000 + 0.6000i

#### 5.2.1 Swap Rotations 2-3.

To derive the cognate that swaps link rotations 2 and 3, we wish to form loop and coupler-point equations that either contain both or neither of links 2 and 3. Loops 0-1-2-3-4 and 2-3-4-5 suffice, as does the path to the coupler point using links 0-4-5. Accordingly, at step 2, we have
(8)
(9)
(10)
At step 3, we use the same equations with a0′ in place of a0, θ1′ in place of θ1, and so on. After substituting θ2′ = θ3, and θ3′ = θ2, steps 6–7 give
Matching terms on the left to those on the right of these equalities, one obtains 12 linear conditions in the 12 unknowns, these being the 10 cognate link parameters, a0′, …, b5′, and the two stretch-rotations γ1, γ2. The 12 conditions are quite simple and sparse, including
(11)
From the whole set of 12 equations, one finds the stretch-rotations to be
and the cognate link parameters as follows:

Links 1 and 4 adjacent to ground keep their original rotations as does the coupler link 5. Accordingly, this cognate is a timed coupler cognate for input at either link 1 or 4.

#### 5.2.2 Swap Rotations 4-5.

We will not write out the details for deriving the cognate obtained by swapping the rotations of links 4 and 5. One may find it similarly to the procedure in Sec. 5.2.1 by using loops 0-1-2-5-4 and 2-3-4-5 and the path 0-4-5 to the coupler point. Since links 4 and 5 appear in every equation, these satisfy the matching condition so step 5 is not utilized. Moreover, we again obtain 12 linear conditions in 12 unknowns, which are readily solved for the required stretch-rotations,
and the cognate mechanism’s parameters,
Since link 1 keeps its original rotation, this cognate is a timed curve cognate for input at link 1.

#### 5.2.3 Swap Both Rotations 2-3 and 4-5.

Once one has the formulas in hand for swapping 2-3 and for swapping 4-5, one can compute the result of swapping both by applying the two sets of formulas in sequence. The order of the sequence, 2-3 then 4-5 versus 4-5 then 2-3, does not matter: the same final cognate results.

Even though the sequential option is available, let us consider how to derive the cognate as one double swap. This will illustrate a case where step 5 comes into play. The trouble is that although both the loop 2-3-4-5 and the coupler path 0-4-5 satisfy the matching criterion, there is no second loop equation that does so. The possibilities for a second loop are 0-1-2-3-4, which swaps to 0-1-3-2-5, as well as the loop 0-1-2-5-4, which swaps to 0-1-3-4-5. For both, we see that the cognate loop has a link rotation that is not matched in the original. The way out of this bind is to use a linear combination of the loops to eliminate the unmatched rotation.

We already wrote the equations for loops 0-1-2-3-4 and 2-3-4-5 and for path 0-4-5 to the coupler point as f1, f2, f3 in Eqs. (8)(10). After the double swap, we get rotations 0-1-3-2-5 in f1′. To match these, consider the linear combination that eliminates θ4 between f1 and f2:
(12)
This combination contains rotations for links 0-1-2-3-5, which are the same ones that appear in loop 0′-1′-2′-3′-4′ after the double swap of rotations turns it into 0-1-3-2-5. Therefore, at step 5, f1 is replaced by g1 yielding at steps 6–7 the following equations:
where P(θ1, θ2, θ3, θ4, θ5) = (θ1, θ3, θ2, θ5, θ4).
After matching coefficients, we again end up with 12 linear equations in 12 unknowns and find the stretch-rotation factors to be
and the cognate parameters as follows:
Example 2

Table 2 lists the parameters for the Stephenson-2A six-bar mechanism along with the parameters (to four decimal places) for the three cognates derived above and drawn in Fig. 3.

#### 5.2.4 An Inadmissible Permutation: Swap Rotations 2-5.

One may wonder why other permutations besides the swapping of 2-3 and 4-5 do not give curve cognates for the Stephenson-2A. To illustrate, consider swapping the rotations of links 2 and 5. Loops 0-1-2-5-4 and 2-3-4-5 and vector path 0-1-2-5 all contain both 2 and 5, so the matching condition is satisfied. The trouble is that in contrast to the swaps previously considered, this time, the path to the coupler point traverses four instead of just three links. The consequence is that we have 13 coefficients to match, and we only have 12 unknowns. A row reduction procedure shows that these equations are in general incompatible, so rotations 2 and 5 cannot be interchanged.

### 5.3 Watt-1A Six-Bar.

Among the six-bar coupler curve mechanisms, only the Watt-1A, shown in Fig. 4, has a positive-dimensional set of curve cognates. Let us see how our method of constructing cognates deals with this.

Fig. 4
Fig. 4
Close modal
Table 3

Watt-1A linkage and cognate parameters

OriginalCognateOriginalCognate
a00.0 + 0.0i0.4000 + 0.1000ia30.1 − 0.1i0.0286 − 0.0571i
b00.7 + 0.0i0.7000 + 0.0000ib3−0.2 − 0.4i−0.1286 − 0.4429i
a1−0.1 + 0.3i0.0000 + 0.1429ia40.4 + 0.1i0.3871 + 0.1757i
a20.7 − 0.2i0.2714 − 0.1857ia50.1 − 0.2i0.1386 − 0.1843i
b2−0.3 + 0.5i−0.3971 + 0.4514ib50.2 + 0.2i0.2000 + 0.2000i
OriginalCognateOriginalCognate
a00.0 + 0.0i0.4000 + 0.1000ia30.1 − 0.1i0.0286 − 0.0571i
b00.7 + 0.0i0.7000 + 0.0000ib3−0.2 − 0.4i−0.1286 − 0.4429i
a1−0.1 + 0.3i0.0000 + 0.1429ia40.4 + 0.1i0.3871 + 0.1757i
a20.7 − 0.2i0.2714 − 0.1857ia50.1 − 0.2i0.1386 − 0.1843i
b2−0.3 + 0.5i−0.3971 + 0.4514ib50.2 + 0.2i0.2000 + 0.2000i

The Watt-1A cognates derive from the trivial permutation: no swaps at all. The loops are 0-1-2-3 and 2-3-5-4, and a path to the coupler point is 0-3-5. The number of coefficients to match in these equations is 4 + 4 + 3 = 11, and as is always the case for six-bars, we have 12 unknowns: 10 link parameters and 2 stretch-rotations. Accordingly, we can specify one link vector, say a0′, and still satisfy all the conditions imposed by matching coefficients of the rotations. Since a link vector has a real and an imaginary part, the set of curve cognates is two real dimensional.

Specifying a0′, we find the stretch-rotation factors to be
and the cognate mechanism link parameters

Since all the links of the cognate keep their original rotations, the result is a timed coupler cognate.

Example 3

Table 3 lists the parameters for the Watt-1A six-bar linkage along with the parameters (to four decimal places) for a cognate with a0′ = 0.4 + 0.1i. Both mechanisms are drawn in Fig. 4.

### 5.4 An Eight-Bar Cognate.

So far, we have used our approach to provide simple demonstrations of known results in the cognate theory. However, the beauty of the approach is that it easily generates cognates for more complex linkages. To show this, we generate a novel cognate of the eight-bar linkage in Fig. 5.

Fig. 5
Fig. 5
Close modal
Table 4

Eight-bar linkage and cognate parameters

OriginalCognateOriginalCognate
a0−4.0 + 0.0i−2.2665 + 1.2640ia42.1 + 0.5i0.7494 + 1.4184i
b00.0 + 0.0i0.0000 + 0.0000ib4−1.5 + 0.4i−1.4238 − 0.6638i
a10.8 + 1.0i1.4797 − 0.6767ia52.5 + 0.6i1.5503 + 2.0892i
b1−1.0 + 0.5i−1.1130 − 1.4949ib5−2.3 + 0.4i−1.3503 − 1.0892i
a22.5 + 0.2i0.7693 + 0.3138ia61.9 + 0.6i1.0847 + 1.6995i
a30.7 − 1.2i0.0174 − 0.9011ia72.1 − 0.4i1.8894 + 1.0534i
b30.9 + 0.4i0.2174 + 0.6989ib7−0.5 + 0.9i−0.5000 + 0.9000i
OriginalCognateOriginalCognate
a0−4.0 + 0.0i−2.2665 + 1.2640ia42.1 + 0.5i0.7494 + 1.4184i
b00.0 + 0.0i0.0000 + 0.0000ib4−1.5 + 0.4i−1.4238 − 0.6638i
a10.8 + 1.0i1.4797 − 0.6767ia52.5 + 0.6i1.5503 + 2.0892i
b1−1.0 + 0.5i−1.1130 − 1.4949ib5−2.3 + 0.4i−1.3503 − 1.0892i
a22.5 + 0.2i0.7693 + 0.3138ia61.9 + 0.6i1.0847 + 1.6995i
a30.7 − 1.2i0.0174 − 0.9011ia72.1 − 0.4i1.8894 + 1.0534i
b30.9 + 0.4i0.2174 + 0.6989ib7−0.5 + 0.9i−0.5000 + 0.9000i

We can generate a cognate by swapping the rotations of links 1 and 2. Three compatible loops are 0-1-2-3, 1-2-3-5-4, and 4-5-7-6. The vector path 0-3-5-7 to the coupler point is also compatible. There are 4 + 5 + 4 + 4 = 17 coefficients to match. The number of link parameters is 4L + 2 = 14, and there will be three stretch-rotations associated with matching the loops for a total of 17 unknowns. Therefore, we can expect that this swap will lead to a unique cognate.

The relevant loop and coupler-point equations are as follows:
(13)
(14)
(15)
(16)
The cognate uses these same equations except interchanging rotations 1 and 2. After introducing three stretch-rotation factors, one for each loop equation Eqs. (13)(15), and equating coefficients, one obtains stretch-rotations
and cognate mechanism link parameters

Since rotation 7 is preserved, this is a coupler cognate. When link 3 is the input crank, this is a timed coupler cognate. If instead link 1 is the input, timing is not preserved.

Example 4

Table 4 lists the parameters for the eight-bar linkage along with the parameters (to four decimal places) for the cognate derived earlier and drawn in Fig. 5.

### 5.5 A Ten-Bar Cognate.

As a final demonstration of the simplicity of the approach, we use the method to generate a novel cognate to the ten-bar linkage in Ref. [20, Fig. 15], which is shown in Fig. 6.

Fig. 6
Fig. 6
Close modal
Table 5

Ten-bar linkage and cognate parameters

OriginalCognateOriginalCognate
a00.0 + 0.0i0.0000 + 0.0000ia40.6 + 0.3i−0.5533 − 0.0067i
b01.0 + 0.0i2.3667 + 1.0000ib4−0.1 + 0.5i−0.2000 + 0.7000i
c02.2 + 0.1i2.2000 + 0.1000ia5−0.6 + 0.3i−0.6000 + 0.3000i
a10.5 + 0.3i0.8833 + 1.2100ib50.7 + 0.3i0.8941 − 0.3235i
b10.4 − 0.7i0.7833 + 0.2100ia60.2 + 0.0i0.4733 + 0.2000i
a20.3 − 0.3i1.0100 − 0.4100ia7−0.3 − 0.5i−0.3000 − 0.5000i
b20.4 + 0.5i0.2867 − 0.3933ia8−0.1 − 1.1i−1.0294 − 0.9176i
a30.2 − 0.7i0.1000 − 0.5000ia9−0.9 − 0.1i−0.9000 − 0.1000i
b3−0.5 + 0.3i0.3353 + 0.5412ib90.6 + 0.5i0.6000 + 0.5000i
OriginalCognateOriginalCognate
a00.0 + 0.0i0.0000 + 0.0000ia40.6 + 0.3i−0.5533 − 0.0067i
b01.0 + 0.0i2.3667 + 1.0000ib4−0.1 + 0.5i−0.2000 + 0.7000i
c02.2 + 0.1i2.2000 + 0.1000ia5−0.6 + 0.3i−0.6000 + 0.3000i
a10.5 + 0.3i0.8833 + 1.2100ib50.7 + 0.3i0.8941 − 0.3235i
b10.4 − 0.7i0.7833 + 0.2100ia60.2 + 0.0i0.4733 + 0.2000i
a20.3 − 0.3i1.0100 − 0.4100ia7−0.3 − 0.5i−0.3000 − 0.5000i
b20.4 + 0.5i0.2867 − 0.3933ia8−0.1 − 1.1i−1.0294 − 0.9176i
a30.2 − 0.7i0.1000 − 0.5000ia9−0.9 − 0.1i−0.9000 − 0.1000i
b3−0.5 + 0.3i0.3353 + 0.5412ib90.6 + 0.5i0.6000 + 0.5000i
The following loop and coupler-point equations are compatible with swapping link rotations 3 and 4:
(17)
(18)
(19)
(20)
(21)
A count of the conditions to be satisfied from matching coefficients is 4 + 4 + 4 + 7 + 4 = 23, while there are just 22 unknowns: 18 link parameters and 4 stretch-rotation factors. By that count, one might hastily conclude that there are too few freedoms to match all the conditions, but it turns out that the conditions are in fact compatible. In particular, since neither θ1 nor θ7 is involved in the swap, and their coefficients in Eqs. (20) and (21) lead to the following four equations:
It is easily seen that these four equations only place three conditions on the parameters. Carrying out the entire procedure, one finds a unique solution for the stretch-rotations:
and the link parameters:
This is a coupler cognate since the rotation of link 9 is preserved. When link 1 or 2 is the input, this is a timed coupler cognate. When link 3 is the input, timing is not preserved.
Example 5

Table 5 lists the parameters for the ten-bar linkage along with the parameters (to 4 decimal places) for the cognate derived earlier and drawn in Fig. 6.

## 6 Conclusion

We have presented a method of deriving planar curve cognates and illustrated its application to the four-bar, several six-bars, one eight-bar, and one ten-bar linkage. Cognates are found by interchanging link rotations in a complex-vector formulation of loop and coupler-point equations, resulting in formulas that are easy to apply, especially in a computer graphics environment. As we saw in Sec. 5.2.4, not all permutations lead to valid cognates. Thus, at the level of development presented here, some trial and error would be required to find all cognates of a given linkage type for linkages with N > 6. A companion paper [3] addresses the issue of finding all possible valid permutations.

The Appendix summarizes how to derive all known six-bar curve cognates. Beyond giving simple derivations for known cognates, the procedure also allows one to produce new cognates as demonstrated in Secs. 5.4 and 5.5 by finding a novel cognate of an eight-bar and ten-bar linkage.

## Acknowledgment

S.N.S. and J.D.H. were partially supported by NSF CCF-181274. S.N.S. was also supported by the Schmitt Leadership Fellowship in Science and Engineering.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.

### Appendix: Deriving All Six-Bar Cognates

For easy reference, we provide a quick summary of how to derive all planar four-bar and six-bar cognates. For each curve type, we provide

• the mechanism’s type graph (which represents links as nodes and joints as edges), where the ground link is always link 0 and the coupler is marked with an overbar,

• the link rotations that are to be swapped,

• the loops and the path to the coupler point used for the original mechanism,

• for one cognate of the Stephenson-2A, where a linear combination of loops is required, it is denoted as (0-1-2-$4^$-5)+(2-3-$4^$-5) to indicate that the combination eliminates θ4 as detailed in Eq. (12),

• for one cognate of the Watt-1B where a linear combination of the path to the coupler point and a loop is required, it is denoted as p = (0-$3^$-4-5)+(2-$3^$-4-5) to indicate that the combination eliminates θ3, which is obtained by simply rescaling the loop and adding it to the path to the coupler point,

• the loops and the path to the coupler point used for the cognate mechanism, given before and after the permutation of rotations, e.g., if rotations 2 and 3 are to be swapped, we might write 0′-1′-2′-3′ → 0-1-3-2.

Comparing the specifications in this appendix for the four-bar, Stephenson-2A, and Watt-1A to the corresponding derivations in the main body may help clarify the abbreviated notation. One can easily verify that the functions for the original and the cognate mechanism contain the same rotations, so that coefficient matching can be done and that the number of coefficients to be matched is less than or equal to the number of unknowns (six link parameters plus one stretch-rotation for the four-bar and ten link parameters plus two stretch-rotations for the six-bars). The “less than” case occurs only for the Watt-1A (as shown in Sec. 5.3). In all other cases, further analysis of these linear matching conditions shows that they are independent, and hence, each permutation corresponds with a unique cognate.

Four-bar. A cognate triple exists (original plus two cognates) (Table 6).

Table 6

Four-bar type-graph and swaps leading to cognates

SwapOriginalCognate
1′2′ →210-1-2-30′-1′-2′-3′ → 0-2-1-3
p = 0-1-2p = 0′-1′-2′ → 0-2-1
2′3′ → 320-1-2-30′-1′-2′-3 ′ → 0-1-3-2
p = 0-3-2p = 0′-3′-2′ → 0-2-3
SwapOriginalCognate
1′2′ →210-1-2-30′-1′-2′-3′ → 0-2-1-3
p = 0-1-2p = 0′-1′-2′ → 0-2-1
2′3′ → 320-1-2-30′-1′-2′-3 ′ → 0-1-3-2
p = 0-3-2p = 0′-3′-2′ → 0-2-3

Stephenson-1. A cognate pair exists (Table 7).

Table 7

Stephenson-1 type graph and swap leading to one cognate

SwapOriginalCognate
0-1-2-30′-1′-2′-3′ → 0-2-1-3
1′2′ → 211-4-5-3-21′-4′-5′-3′-2′ → 2-4-5-3-1
p = 0-3-5p = 0′-3′-5′ → 0-3-5
SwapOriginalCognate
0-1-2-30′-1′-2′-3′ → 0-2-1-3
1′2′ → 211-4-5-3-21′-4′-5′-3′-2′ → 2-4-5-3-1
p = 0-3-5p = 0′-3′-5′ → 0-3-5

Stephenson-2A. A cognate quadruple exists (Table 8).

Table 8

Stephenson-2A type graph and swaps leading to cognates

SwapOriginalCognate
0-1-2-3-40′-1′-2′-3′-4′ → 0-1-3-2-4
2′3′ → 322-3-4-52′-3′-4′-5′ → 3-2-4-5
p = 0-4-5p = 0′-4′-5′ → 0-4-5
0-1-2-5-40′-1′-2′-5′-4′ → 0-1-2-4-5
4′5′ → 542-3-4-52′-3′-4′-5′ → 2-3-5-4
p = 0-4-5p = 0′-4′-5′ → 0-5-4
2′3′ → 322-3-4-52′-3′-4′-5′ → 3-2-5-4
and(0-1-2-3-$4^$)+(2-3-$4^$-5)0′-1′-2′-3′-4′ → 0-1-3-2-5
4′5′ → 54p = 0-4-5p = 0 ′-4′-5′ → 0-5-4
SwapOriginalCognate
0-1-2-3-40′-1′-2′-3′-4′ → 0-1-3-2-4
2′3′ → 322-3-4-52′-3′-4′-5′ → 3-2-4-5
p = 0-4-5p = 0′-4′-5′ → 0-4-5
0-1-2-5-40′-1′-2′-5′-4′ → 0-1-2-4-5
4′5′ → 542-3-4-52′-3′-4′-5′ → 2-3-5-4
p = 0-4-5p = 0′-4′-5′ → 0-5-4
2′3′ → 322-3-4-52′-3′-4′-5′ → 3-2-5-4
and(0-1-2-3-$4^$)+(2-3-$4^$-5)0′-1′-2′-3′-4′ → 0-1-3-2-5
4′5′ → 54p = 0-4-5p = 0 ′-4′-5′ → 0-5-4

Stephenson-2B. A cognate triple exists (Table 9).

Table 9

Stephenson-2B type graph and swaps leading to cognates

SwapOriginalCognate
0-1-5-2-40′-1′-5′-2′-4′ → 0-1-5-4-2
2′4′ → 422-4-3-52′-4′-3′-5′ → 4-2-3-5
p = 0-1-5p = 0 ′-1′-5′ → 0-1-5
0-1-5-3-40′-1′-5′-3′-4′ → 0-1-5-4-3
3′4′ → 432-4-3-52′-4′-3′-5′ → 2-3-4-5
p = 0-1-5p = 0 ′-1′-5′ → 0-1-5
SwapOriginalCognate
0-1-5-2-40′-1′-5′-2′-4′ → 0-1-5-4-2
2′4′ → 422-4-3-52′-4′-3′-5′ → 4-2-3-5
p = 0-1-5p = 0 ′-1′-5′ → 0-1-5
0-1-5-3-40′-1′-5′-3′-4′ → 0-1-5-4-3
3′4′ → 432-4-3-52′-4′-3′-5′ → 2-3-4-5
p = 0-1-5p = 0 ′-1′-5′ → 0-1-5

Stephenson-III. There exists a group of six cognates. As found in Ref. [7] and discussed in Ref. [14], these can be generated by applying the three-way Roberts cognates to the four-bar 0-1-2-3 and a 2-way skew pantograph transformation to the dyad 0-4-5. Our method also applies, as summarized below (Table 10).

Table 10

Stephenson-III type graph and swaps leading to cognates

SwapOriginalCognate
0-1-2-30′-1′-2′-3′ → 0-2-1-3
1′2′ → 210-1-2-5-40′-1′-2′-5′-4′ → 0-2-1-5-4
p = 0-4-5p = 0′-4′-5′ → 0-4-5
0-1-2-30′-1′-2′-3′ → 0-1-3-2
2′3′ → 320-3-2-5-40′-3′-2′-5′-4′ → 0-2-3-5-4
p = 0-4-5p = 0′-4′-5′ → 0-4-5
0-1-2-30′-1′-2′-3′ → 0-1-2-3
4′5′ → 540-1-2-5-40′-1′-2′-5′-4′ → 0-1-2-4-5
p = 0-4-5p = 0′-4′-5′ → 0-5-4
1′2′ → 210-1-2-30′-1′-2′-3′ → 0-2-1-3
and0-1-2-5-40′-1′-2′-5′-4′ → 0-2-1-4-5
4′5′ → 54p = 0-4-5p = 0′-4′-5′ → 0-5-4
2′3′ → 320-1-2-30′-1′-2′-3′ → 0-1-3-2
and0-3-2-5-40′-3′-2′-5′-4′ → 0-2-3-4-5
4′5′ → 54p = 0-4-5p = 0′-4′-5′ → 0-5-4
SwapOriginalCognate
0-1-2-30′-1′-2′-3′ → 0-2-1-3
1′2′ → 210-1-2-5-40′-1′-2′-5′-4′ → 0-2-1-5-4
p = 0-4-5p = 0′-4′-5′ → 0-4-5
0-1-2-30′-1′-2′-3′ → 0-1-3-2
2′3′ → 320-3-2-5-40′-3′-2′-5′-4′ → 0-2-3-5-4
p = 0-4-5p = 0′-4′-5′ → 0-4-5
0-1-2-30′-1′-2′-3′ → 0-1-2-3
4′5′ → 540-1-2-5-40′-1′-2′-5′-4′ → 0-1-2-4-5
p = 0-4-5p = 0′-4′-5′ → 0-5-4
1′2′ → 210-1-2-30′-1′-2′-3′ → 0-2-1-3
and0-1-2-5-40′-1′-2′-5′-4′ → 0-2-1-4-5
4′5′ → 54p = 0-4-5p = 0′-4′-5′ → 0-5-4
2′3′ → 320-1-2-30′-1′-2′-3′ → 0-1-3-2
and0-3-2-5-40′-3′-2′-5′-4′ → 0-2-3-4-5
4′5′ → 54p = 0-4-5p = 0′-4′-5′ → 0-5-4

Watt-1A. A two-dimensional set of cognates exists (Table 11).

Table 11

Watt-1A type graph, the trivial swap leads to cognates

SwapOriginalCognate
0-1-2-30′-1′-2′-3′ → 0-1-2-3
None2-3-5-42′-3′-5′-4′ → 2-3-5-4
p=0-3-5p=0′-3′-5′ → 0-3-5
SwapOriginalCognate
0-1-2-30′-1′-2′-3′ → 0-1-2-3
None2-3-5-42′-3′-5′-4′ → 2-3-5-4
p=0-3-5p=0′-3′-5′ → 0-3-5

Watt-1B. A cognate quadruple exists (Table 12).

Table 12

Watt-1B type graph and swaps leading to cognates

SwapOriginalCognate
0-1-2-30′-1′-2′-3′ → 0-1-3-2
2′3′ → 322-3-4-52′-3′-4′-5′ → 3-2-4-5
p = 0-3-2-5p = 0 ′-3′-2′-5′ → 0-2-3-5
0-1-2-30′-1′-2′-3′ → 0-1-2-3
4′5′ → 542-3-4-52′-3′-4′-5′ → 2-3-5-4
p = 0-3-4-5p = 0 ′-3′-4′-5′ → 0-3-5-4
2′3′ → 320-1-2-30′-1′-2′-3′ → 0-1-3-2
and2-3-4-52′-3′-4′-5′ → 3-2-5-4
4′5′ → 54p =(0-$3^$-4-5)+(2-$3^$-4-5)p = 0 ′-3′-4′-5′ → 0-2-5-4
SwapOriginalCognate
0-1-2-30′-1′-2′-3′ → 0-1-3-2
2′3′ → 322-3-4-52′-3′-4′-5′ → 3-2-4-5
p = 0-3-2-5p = 0 ′-3′-2′-5′ → 0-2-3-5
0-1-2-30′-1′-2′-3′ → 0-1-2-3
4′5′ → 542-3-4-52′-3′-4′-5′ → 2-3-5-4
p = 0-3-4-5p = 0 ′-3′-4′-5′ → 0-3-5-4
2′3′ → 320-1-2-30′-1′-2′-3′ → 0-1-3-2
and2-3-4-52′-3′-4′-5′ → 3-2-5-4
4′5′ → 54p =(0-$3^$-4-5)+(2-$3^$-4-5)p = 0 ′-3′-4′-5′ → 0-2-5-4

Watt-2.

The Watt-2 linkage can only draw circles and four-bar curves. The formulas for the four-bar can be applied to loop 0-1-2-3.

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