Infinitesimally Locked Self-Touching Linkages with Applications to Locked Trees

Recently there has been much interest in linkages (bar-and-joint frameworks) that are locked or stuck in the sense that they cannot be moved into some other configuration while preserving the bar lengths and not crossing any bars. We propose a new algorithmic approach for analyzing whether planar linkages are locked in many cases of interest. The idea is to examine self-touching or degenerate frameworks in which multiple edges converge to geometrically overlapping configurations. We show how to study whether such frameworks are locked using techniques from rigidity theory, in particular first-order rigidity and equilibrium stresses. Then we show how to relate locked self-touching frameworks to locked frameworks that closely approximate the self-touching frameworks. Our motivation is that most existing approaches to locked linkages are based on approximations to self-touching frameworks. In particular, we show that a previously proposed locked tree in the plane [BDD+02] can be easily proved locked using our techniques, instead of the tedious arguments required by standard analysis. We also present a new locked tree in the plane with only one degree-3 vertex and all other vertices degree 1 or 2. This tree can also be easily proved locked with our methods, and implies that the result about opening polygonal arcs and cycles [CDR02] is the best possible. 1. Linkages A linkage is a graph together with an assignment of lengths to edges; each edge is called a rigid bar. We highlight three linkages of common study: a polygonal arc, polygonal cycle, or polygonal tree is a linkage whose graph is a single path, cycle, or tree, respectively. A configuration of a linkage in R is a mapping of the vertices to points in R that satisfies the bar-length constraints. A configuration is (strongly) simple if only incident bars intersect, and then only at the common endpoint. A motion is a continuum of configurations, that is, a continuous function mapping the time interval [0, 1] to configurations; often, each configuration is required to be simple. The configuration space of a given subset of configurations (e.g., simple configurations) is the space in which points correspond to configurations and paths correspond to motions. We focus here on planar linkages embedded in R. In this case, the linkage also specifies the combinatorial planar embedding because this cannot change by a motion that avoids crossings. It is known that the configuration space of simple 2000 Mathematics Subject Classification. Primary: 52C25. c ©2002 American Mathematical Society 1 2 ROBERT CONNELLY, ERIK D. DEMAINE, AND GÜNTER ROTE H2:01 H2:02 H2:03 H2:04 H2:05 H2:06 H2:07 H2:08 H2:09 H2:10 H2:11 (a) From [BDD+02] (b) Basis for new tree (c) New tree 3 3 1 3 1 2 1