Our focus in this first part is on one-dimensional (1D) linkages, and mostly on especially simple linkages we call “chains.” Linkages are useful models for robot arms and for folding proteins; these and other applications will be detailed in Section 1.2. After defining linkages and setting some terminology, we quickly review the contents of this first part.

Linkage definitions. A linkage is a collection of fixed-length 1D segments joined at their endpoints to form a graph. A segment endpoint is also called a vertex. The segments are often called links or bars, and the shared endpoints are called joints or vertices. The bars correspond to graph edges and the joints to graph nodes. Some joints may be pinned to be fixed to specific locations. Although telescoping links and sliding joints are of considerable interest in mechanics, we only explore fixed-length links and joints fixed at endpoints. (We'll use the term mechanism to loosely indicate any collection of rigid bodies connected by joints, hinges, sliders, etc.) An example of a linkage is shown in Figure 1.1.

Overview. After classifying problems in this chapter, we turn to presenting some of the basic upper and lower complexity bounds obtained in the past 20 years in Chapter 2.We then explore in Chapter 3 classical mechanisms, particularly the pursuit of straight-line linkage motion. In contrast to these linkages, whose whole purpose is motion, we next study in Chapter 4 when a linkage is rigid, that is, can move at all. Most of the remainder of Part I concentrates on chains, starting with reconfiguring chains under various constraints in Chapter 5.