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, ΔA _{1},D,B_{1}*

- Sketch Cayley diagram
- Using parallelograms, find
*A*_{2}and*B*_{3}//O_{A},*A*_{1},*D*,*A*_{2}and //O_{B},*B*_{1},*D*,*B*_{3} - Using similar triangles, find
*C*_{2}and*C*_{3}Δ*A*_{2},*C*_{2},*D*and Δ*D*,*C*_{3},*B*_{3} - Using a parallelogram, find O
_{C}//O_{C},*C*_{2},*D*,*C*_{3} - Check similar triangles ΔO
_{A},O_{C},O_{B} - Separate left and right cognate
- Put dimensions on Cayley diagram

## Dimensional relationships

The lengths of the four members can be found by using the law of sines. Both *K*_{L} and *K*_{R} are found as follows.

## Conclusions

**Class I**chain

**double crank**), both cognates will be drag links.

**crank-rocker**, one cognate will be a crank-rocker, and the second will be a double-rocker.