Enumerating Constrained Non-Crossing Minimally Rigid Frameworks

In this paper we present an algorithm for enumerating without repetitions all the non-crossing generically minimally rigid bar-and-joint frameworks under edge constraints, which we call constrained non-crossing Laman frameworks, on a given set of n points in the plane. Our algorithm is based on the reverse search paradigm of Avis and Fukuda. It generates each output graph in O(n4) time and O(n) space, or, with a slightly different implementation, in O(n3) time and O(n2) space. In particular, we obtain that the set of all the constrained non-crossing Laman frameworks on a given point set is connected by flips which preserve the Laman property

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. For points in convex position we obtain a new realization of the associahedron, i.e., a geometric representation of the set of triangulations of an n-gon, or of the set of binary trees on n vertices, or of many other combinatorial objects that are counted by the Catalan numbers. By considering the 1-dimensional version of the polytope of constrained expansive motions we obtain a second distinct realization of the associahedron as a perturbation of the positive cell in a Coxeter arrangement. Our methods produce as a by-product a new proof that every simple polygon or polygonal arc in the plane has expansive motions, a key step in the proofs of the Carpenter's Rule Theorem by Connelly, Demaine and Rote (2000) and by Streinu (2000).Comment: 40 pages, 7 figures. Changes from v1: added some comments (specially to the "Further remarks" in Section 5) + changed to final book format. This version is to appear in "Discrete and Computational Geometry -- The Goodman-Pollack Festschrift" (B. Aronov, S. Basu, J. Pach, M. Sharir, eds), series "Algorithms and Combinatorics", Springer Verlag, Berli

Pseudo-Triangulations, Rigidity and Motion Planning

This paper proposes a combinatorial approach to planning non-colliding trajectories for a polygonal bar-and-joint framework with n vertices. It is based on a new class of simple motions induced by expansive one-degree-of-freedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointed pseudo-triangulations, a class of embedded planar graphs for which we give several equivalent characterizations and exhibit rich rigidity theoretic properties. The main application is an efficient algorithm for the Carpenter\u27s Rule Problem: convexify a simple bar-and-joint planar polygonal linkage using only non-self-intersecting planar motions. A step of the algorithm consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At the alignment event, a local alteration restores the pseudo-triangulation. The motion continues for O(n3) steps until all the points are in convex position. © 2005 Springer Science+Business Media, Inc

Energy-Economical Heuristically Based Control of Compass Gait Walking on Stochastically Varying Terrain

Investigation uses simulation to explore the inherent tradeoffs ofcontrolling high-speed and highly robust walking robots while minimizing energy consumption. Using a novel controller which optimizes robustness, energy economy, and speed of a simulated robot on rough terrain, the user can adjust their priorities between these three outcome measures and systematically generate a performance curveassessing the tradeoffs associated with these metrics

Energy-Economical Heuristically Based Control of Compass Gait Walking on Stochastically Varying Terrain

Investigation uses simulation to explore the inherent tradeoffs ofcontrolling high-speed and highly robust walking robots while minimizing energy consumption. Using a novel controller which optimizes robustness, energy economy, and speed of a simulated robot on rough terrain, the user can adjust their priorities between these three outcome measures and systematically generate a performance curveassessing the tradeoffs associated with these metrics

A Proposal for a Three Detector Short-Baseline Neutrino Oscillation Program in the Fermilab Booster Neutrino Beam

A Short-Baseline Neutrino (SBN) physics program of three LAr-TPC detectors located along the Booster Neutrino Beam (BNB) at Fermilab is presented. This new SBN Program will deliver a rich and compelling physics opportunity, including the ability to resolve a class of experimental anomalies in neutrino physics and to perform the most sensitive search to date for sterile neutrinos at the eV mass-scale through both appearance and disappearance oscillation channels. Using data sets of 6.6e20 protons on target (P.O.T.) in the LAr1-ND and ICARUS T600 detectors plus 13.2e20 P.O.T. in the MicroBooNE detector, we estimate that a search for muon neutrino to electron neutrino appearance can be performed with ~5 sigma sensitivity for the LSND allowed (99% C.L.) parameter region. In this proposal for the SBN Program, we describe the physics analysis, the conceptual design of the LAr1-ND detector, the design and refurbishment of the T600 detector, the necessary infrastructure required to execute the program, and a possible reconfiguration of the BNB target and horn system to improve its performance for oscillation searches.Comment: 209 pages, 129 figure