Ileana Streinu

A combinatorial approach to planar non-colliding robot arm motion planning

We propose a combinatorial approach to plan noncolliding motions for a polygonal bar-and-joint framework. Our approach yields very efficient deterministic algorithms for a category of robot arm motion planning problems with many degrees of freedom, where the known general roadmap techniques would give exponential complexity. It is based on a novel class of one-degree-of-freedom mechanisms induced by pseudo triangulations of planar point sets, for which we provide several equivalent characterization and exhibit rich combinatorial and rigidity theoretic properties. The main application is an efficient algorithm for the Carpenters rule problem: convexify a simple bar-and-joint planar polygonal linkage using only non self-intersecting planar motions. A step in the convexification motion consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At that point, a local alteration restores the pseudo triangulation. The motion continues for O(n/sup 2/) steps until all the points are in convex position.

Planar minimally rigid graphs and pseudo-triangulations

Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than p). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide---to the best of our knowledge---the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.

Expansive Motions and the Polytope of Pointed Pseudo-Triangulations

We introduce the polytope of pointed pseudo-triangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1-skeleton is the graph whose vertices are the pointed pseudo-triangulations of the point set and whose edges are flips of interior pseudo-triangulation edges.

Periodic frameworks and flexibility

We formulate a concise deformation theory for periodic bar-and-joint frameworks in Rd and illustrate our algebraic–geometric approach on frameworks related to various crystalline structures. Particular attention is given to periodic frameworks modelled on silica, zeolites and perovskites. For frameworks akin to tectosilicates, which are made of one-skeleta of d-dimensional simplices, with each vertex common to exactly two simplices, we prove the existence of a space of periodicity-preserving infinitesimal flexes of dimension at least . However, these infinitesimal flexes need not come from genuine flexibility, as shown by rigid examples. The changes implicated in passing from a given lattice of periods to a sublattice of periods are illustrated with frameworks modelled on perovskites.

The Number of Embeddings of Minimally Rigid Graphs

Abstract Rigid frameworks in some Euclidean space are embedded graphs having a unique local realization (up to Euclidean motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with

KINARI-Web: a server for protein rigidity analysis

n

Efficient computation of location depth contours by methods of computational geometry

vertices. We show that, modulo planar rigid motions, this number is at most

Acute Triangulations of Polygons

{{2n-4}\choose {n-2}} \approx 4^n

Single-Vertex origami and spherical expansive motions

. We also exhibit several families which realize lower bounds of the order of

Using rigidity analysis to probe mutation-induced structural changes in proteins.

2^n