The Chebyshev linkage is a mechanical linkage that converts rotational motion to approximate straight-line motion.

It was invented by the nineteenth-century mathematician Pafnuty Chebyshev, who studied theoretical problems in kinematic mechanisms. One of the problems was the construction of a linkage that converts a rotary motion into an approximate straight-line motion. This was also studied by James Watt in his improvements to the steam engine.[1]

The straight-line linkage confines the point P – the midpoint on the link L3 – on a straight line at the two extremes and at the center of travel. (L1, L2, L3, and L4 are as shown in the illustration.) Between those points, point P deviates slightly from a perfect straight line. The proportions between the links are

${\displaystyle L_{1}:L_{2}:L_{3}=2:2.5:1=4:5:2.\,}$

Point P is in the middle of L3. This relationship assures that the link L3 lies vertically when it is at one of the extremes of its travel.[2]

The lengths are related mathematically as follows:

${\displaystyle L_{4}=L_{3}+{\sqrt {L_{2}^{2}-L_{1}^{2}}}.\,}$

It can be shown that if the base proportions described above are taken as lengths, then for all cases,

${\displaystyle L_{4}=L_{2}.\,}$

and this contributes to the perceived straight-line motion of point P.

## Equations of motion

The motion of the linkage can be constrained to an input angle that may be changed through velocities, forces, etc. The input angles can be either link L2 with the horizontal or link L4 with the horizontal. Regardless of the input angle, it is possible to compute the motion of two end-points for link L3 that we will name A and B, and the middle point P.

${\displaystyle x_{A}=L_{2}\cos(\varphi _{1})\,}$
${\displaystyle y_{A}=L_{2}\sin(\varphi _{1})\,}$

while the motion of point B will be computed with the other angle,

${\displaystyle x_{B}=L_{1}-L_{4}\cos(\varphi _{2})\,}$
${\displaystyle y_{B}=L_{4}\sin(\varphi _{2})\,}$

And ultimately, we will write the output angle in terms of the input angle,

${\displaystyle \varphi _{2}=\arcsin \left[{\frac {L_{2}\,\sin(\varphi _{1})}{\overline {AO_{2}}}}\right]-\arccos \left({\frac {L_{4}^{2}+{\overline {AO_{2}}}^{2}-L_{3}^{2}}{2\,L_{4}\,{\overline {AO_{2}}}}}\right)\,}$

Consequently, we can write the motion of point P, using the two points defined above and the definition of the middle point.

${\displaystyle x_{P}={\frac {x_{A}+x_{B}}{2}}\,}$
${\displaystyle y_{P}={\frac {y_{A}+y_{B}}{2}}\,}$

### Input angles

Illustration of the limits

The limits to the input angles, in both cases, are:

${\displaystyle \varphi _{\text{min}}=\arccos \left({\frac {4}{5}}\right)\approx 36.8699^{\circ }.\,}$
${\displaystyle \varphi _{\text{max}}=\arccos \left({\frac {-1}{5}}\right)\approx 101.537^{\circ }.\,}$