In this chapter, we discuss aspects of the movements of linkage mechanisms embedded in the plane, permitting links to pass through one another, without obstacles. Our goal is to explain Kempe's universality theorem (Theorem 3.2.1).

STRAIGHT-LINE LINKAGES

In preparation for presenting this universality result, we discuss the fascinating history of the design of linkages to draw straight lines, which led directly to Kempe's work.

Degrees of Freedom

Linkages that have a point that follows a particular curve have 1 degree of freedom, that is, the configuration space is one-dimensional. We consider two types of joints: pinned and unpinned or free. Both are universal joints, but a pinned joint is fixed to a particular point on the plane. Let a linkage have j free joints and r rods (or links). Pinned joints add no freedom, a free joint adds 2 degrees of freedom, and a rod removes 1 degree of freedom if it is not redundant (Graver 2001, p. 135). Thus, if all rods are “nonredundant” (a common situation), a linkage has 2 j — r degrees of freedom. Thus, to follow a curve, a linkage should have r = 2 j — 1 rods; in particular, it must have an odd number of rods.

Watt Parallel Motion

It was a question of considerable practical importance to design a linkage so that one point moves along a straight line, a motion needed in many machines, for example, to drive the piston rod of a steam engine.