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In kinematics, cognate linkages are linkages that ensure the same input-output relationship or coupler curve geometry, while being dimensionally dissimilar. In case of four-bar linkage coupler cognates, the Roberts–Chebyschev Theorem, after Samuel Roberts and Pafnuty Chebyshev, states that each coupler curve can be generated by three different four-bar linkages. These four-bar linkages can be constructed using similar triangles and parallelograms, and the Cayley diagram (named after Arthur Cayley).

## Contents

Overconstrained mechanisms can be obtained by connecting two or more cognate linkages together.

## Roberts–Chebyschev theorem

The theorem states for a given coupler-curve there exist three four-bar linkages, three geared five-bar linkages, and more six-bar linkages which will generate the same path. The method for generating the additional two four bar linkages from a single four-bar mechanism is described below, using the Cayley diagram.

## Cayley diagram

From original triangle, ΔA1,D,B1

1. Sketch Cayley diagram
2. Using parallelograms, find A2 and B3 //OA,A1,D,A2 and //OB,B1,D,B3
3. Using similar triangles, find C2 and C3 ΔA2,C2,D and ΔD,C3,B3
4. Using a parallelogram, find OC //OC,C2,D,C3
5. Check similar triangles ΔOA,OC,OB
6. Separate left and right cognate
7. Put dimensions on Cayley diagram

## Dimensional relationships

The lengths of the four members can be found by using the law of sines. Both KL and KR are found as follows.

K L = sin ( α ) sin ( β ) K R = sin ( γ ) sin ( β )

## Conclusions

• If and only if the original is a Class I chain ( + s ) < ( P + q ) Both 4-bar cognates will be class I chains.
• If the original is a drag-link (double crank), both cognates will be drag links.
• If the original is a crank-rocker, one cognate will be a crank-rocker, and the second will be a double-rocker.
• If the original is a double-rocker, the cognates will be crank-rockers.