What do a car's windshield wiper, a robot arm, some lamps (including desk lamps and dental office lamps), a human finger and a carpenter's ruler have in common? All of these items consist of sticks or rods that are hinged to enable rotation with respect to one another. Where is the mathematics?
On its face, the idea of a linkage is natural and straightforward: one is given a collection of (straight) sticks which are pinned (hinged) to one another at their ends. In the simplest case the hinged rods form a path. We see something like a linkage in our car (its windshield wiper, though the rods are not connected at the ends here but at an interior point of one of the rods), on our desk, and on the arm used in helping to assemble the International Space Station. The study of linkages, in part for applications reasons, was initiated in the 19th century. The work was in response to problems which involved the machinery that was being developed as part of the Industrial Revolution. As a simple example, contemplate the question of how one would guarantee that a point move along a straight line? When I want to draw a straight line segment I use a ruler, but how does one manufacture a ruler with a straight edge? At first glance it may not be apparent how this is related to linkages, but there is a connection.