After seeing that many fundamental linkage reconfiguration problems are intractable, we specialize in this chapter to the simplest situations, where polynomial-time—sometimes even linear-time—algorithms are possible.

We start in Section 5.1 by exploring from Table 1.1 the simpler end of each spectrum: the simplest linkage graph stucture, an open chain; the simplest intersection constraints, none; the simplest dimension, 2D; and the simplest problem, reachability. In Section 5.2 we continue to permit the chain to cross itself, but not a surrounding boundary. Finally, we initiate the exploration of reconfigurations of chains that avoid self-intersection in Section 5.3, saving the richest topic (locked chains) for Chapter 6.

RECONFIGURATION PERMITTING INTERSECTION

Chain Reachability

Let C be an open chain of n edges/links ei, with lengths |ei| = ℓi i = 1, 2,…, n, and with n + 1 vertices/joints v0,…, vn so that ei = vi−1vi (see Figure 5.1). Such a pinned chain is often called an arm (or “robot arm”), with v0 the shoulder joint and vn the hand. Fix v0 at the origin of a coordinate system.

Connectivity of Configuration Space

The configuration space of this arm C is certainly connected. From any configuration with n ≥ 2links, we can rotate the last link en to become a collinear extension of the next to-last link en−1. Then the arm can be viewed as an arm with one fewer link, treating the last two links en−1 and en as fused together into a single “virtual link” of length ℓn−1 + ℓn.