Plantigrade machine

Since James Watt invented the steam engine there has been a prob­lem to build a hinge mech­a­nism that trans­form cir­cu­lar motion to lin­ear.

A great Russ­ian math­e­mati­cian Pafnu­tiy Lvovich Tcheby­shev couldn't solve the ini­tial prob­lem, but inves­ti­gat­ing it he devel­oped a the­ory of approx­i­ma­tion and of mech­a­nism syn­the­sis. Apply­ing the lat­ter he could choose the para­me­ters of a lambda-mech­a­nism so that… Well, we'll talk about it below.

Two fixed red hinges, three edges of the same length. As it looks like the greek let­ter lambda, this mech­a­nism was named after it. The loose gray hinge of the lit­tle dri­ving edge turns around form­ing a cir­cle while the slave blue hinge has a tra­jec­tory that looks like a mush­room's cap.

Put on the cir­cle of the dri­ving hinge marks at reg­u­lar inter­vals and the cor­re­spond­ing marks on the tra­jec­tory of the slave one.

The lower part of the «cap» cor­re­sponds exactly to a half of the period of the dri­ving hinge's motion. At the same time, the lower part of the blue curve doesn't dif­fer much from a straight lin­ear motion (the dif­fer­ence is less that a per­cent of the short dri­ving edge length).

What else does this blue tra­jec­tory look like? Pafnu­tiy Lvovich could see sim­i­lar­ity with the horse hoof's motion tra­jec­tory!

Let's attach a mir­ror copy of the two-leg part we've already made. Addi­tional links coor­di­nate the phases of rota­tion and a com­mon plat­form con­nects the axes. As they say in mechan­ics, we've got a kine­mat­i­cal scheme of the first walk­ing machine in the world.

Being a pro­fes­sor in Saint-Peters­burg uni­ver­sity, Pafnu­tiy Lvovich spent most part of his salary on con­struc­tion of the mech­a­nisms he invented. He built the described one «in wood and steel» and called it a «Planti­grade machine». The first walk­ing machine in the world invented by a Russ­ian math­e­mati­cian was greatly approved dur­ing the Wold exhi­bi­tion in Paris, 1878.

Thanks to the Poly­tech­ni­cal museum of Moscow that pre­served the Tcheby­shev's orig­i­nal and let «Math­e­mat­i­cal etudes» mea­sure it, we have an oppor­tu­nity to see in action an exact 3D-model of a planti­grade machine of Pafnu­tiy Lvovich Tcheby­shev.