Laman graph

Abstract: This article sets out the rudiments of a theory for analyzing and creating architectures appropriate to the control of formations of autonomous vehicles. The theory rests on ideas of rigid graph theory, some but not all of which are old. The theory, however, has some gaps in it, and their elimination would help in applications. Some of the gaps in the relevant graph theory are as follows. First, there is as yet no analogue for three-dimensional graphs of Laman's theorem, which provides a combinatorial criterion for rigidity in two-dimensional graphs. Second, for three-dimensional graphs there is no analogue of the two-dimensional Henneberg construction for growing or deconstructing minimally rigid graphs although there are conjectures. Third, global rigidity can easily be characterized for two-dimensional graphs, but not for three-dimensional graphs.

Abstract: 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 Carpenter's 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.

Abstract: Recent work has shown that if an isostatic bar-and-joint framework possesses nontrivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are "fixed" by various symmetry operations of the framework. For the group C-3 which describes 3-fold rotational symmetry in the plane, we verify the conjecture proposed by Connelly et al. (Int. J. Solids Struct. 46: 762-773, 2009) that these restrictions on the number of fixed structural components, together with the Laman conditions, are also sufficient for a framework with C-3 symmetry to be isostatic, provided that its joints are positioned as generically as possible subject to the given symmetry constraints. In addition, we establish symmetric versions of Henneberg's theorem and Crapo's theorem for C-3 which provide alternate characterizations of "generically" isostatic graphs with C-3 symmetry. As shown in (Schulze, Combinatorial and geometric rigidity with symmetry constraints, Ph.D. thesis, York University, Toronto, Canada, 2009; Schulze, Symmetrized Laman theorems for the groups C-2 and C-s, in preparation, 2009), our techniques can be extended to establish analogous results for the symmetry groups C-2 and C-s which are generated by a half-turn and a reflection in the plane, respectively.