Two and Three Finger Caging of Polygons and Polyhedra

Multi-finger caging offers a rigorous and robust approach to robot grasping. This thesis provides several novel algorithms for caging polygons and polyhedra in two and three dimensions. Caging refers to a robotic grasp that does not necessarily immobilize an object, but prevents it from escaping to infinity. The first algorithm considers caging a polygon in two dimensions using two point fingers. The second algorithm extends the first to three dimensions. The third algorithm considers caging a convex polygon in two dimensions using three point fingers, and considers robustness of this cage to variations in the relative positions of the fingers.

This thesis describes an algorithm for finding all two-finger cage formations of planar polygonal objects based on a contact-space formulation. It shows that two-finger cages have several useful properties in contact space. First, the critical points of the cage representation in the hand’s configuration space appear as critical points of the inter-finger distance function in contact space. Second, these critical points can be graphically characterized directly on the object’s boundary. Third, contact space admits a natural rectangular decomposition such that all critical points lie on the rectangle boundaries, and the sublevel sets of contact space and free space are topologically equivalent. These properties lead to a caging graph that can be readily constructed in contact space. Starting from a desired immobilizing grasp of a polygonal object, the caging graph is searched for the minimal, intermediate, and maximal caging regions surrounding the immobilizing grasp. An example constructed from real-world data illustrates and validates the method.

A second algorithm is developed for finding caging formations of a 3D polyhedron for two point fingers using a lower dimensional contact-space formulation. Results from the two-dimensional algorithm are extended to three dimension. Critical points of the inter-finger distance function are shown to be identical to the critical points of the cage. A decomposition of contact space into 4D regions having useful properties is demonstrated. A geometric analysis of the critical points of the inter-finger distance function results in a catalog of grasps in which the cages change topology, leading to a simple test to classify critical points. With these properties established, the search algorithm from the two-dimensional case may be applied to the three-dimensional problem. An implemented example demonstrates the method.

This thesis also presents a study of cages of convex polygonal objects using three point fingers. It considers a three-parameter model of the relative position of the fingers, which gives complete generality for three point fingers in the plane. It analyzes robustness of caging grasps to variations in the relative position of the fingers without breaking the cage. Using a simple decomposition of free space around the polygon, we present an algorithm which gives all caging placements of the fingers and a characterization of the robustness of these cages.

Wrapping the cube and other polyhedra

An infinite series of twofold, two-way weavings of the cube, corresponding to 'wrappings', or double covers of the cube, is described with the aid of the two-parameter Goldberg- Coxeter construction. The strands of all such wrappings correspond to the central circuits (CCs) of octahedrites (four-regular polyhedral graphs with square and triangular faces), which for the cube necessarily have octahedral symmetry. Removing the symmetry constraint leads to wrappings of other eight-vertex convex polyhedra. Moreover, wrappings of convex polyhedra with fewer vertices can be generated by generalizing from octahedrites to i-hedrites, which additionally include digonal faces. When the strands of a wrapping correspond to the CCs of a four-regular graph that includes faces of size greater than 4, non-convex 'crinkled' wrappings are generated. The various generalizations have implications for activities as diverse as the construction of woven-closed baskets and the manufacture of advanced composite components of complex geometry. © 2012 The Royal Society.

Bi-Lipschitz G-triviality and Newton polyhedra, G = R, C, K, R-V, C-V, K-V

In this work we provide estimates for the bi-Lipschitz G-triviality, G = C or K, for a family of map germs satisfying a Lojasiewicz condition. We work with two cases: the class of weighted homogeneous map germs and the class of non-degenerate map germs with respect to some Newton polyhedron. We also consider the bi-Lipschitz triviality for families of map germs defined on an analytic variety V. We give estimates for the bi-Lipschitz G(V)-triviality where G = R,C or K in the weighted homogeneous case. Here we assume that the map germ and the analytic variety are both weighted homogeneous with respect to the same weights. The method applied in this paper is based in the construction of controlled vector fields in the presence of a suitable Lojasiewicz condition. In the last section of this work we compare our results with other results related to this work showing tables with all estimates that we know, including ours.

hp-DGFEM for second-order mixed elliptic problems polyhedra

We prove exponential rates of convergence of hp-version discontinuous Galerkin (dG) interior penalty finite element methods for second-order elliptic problems with mixed Dirichlet-Neumann boundary conditions in axiparallel polyhedra. The dG discretizations are based on axiparallel, σ-geometric anisotropic meshes of mapped hexahedra and anisotropic polynomial degree distributions of μ-bounded variation. We consider piecewise analytic solutions which belong to a larger analytic class than those for the pure Dirichlet problem considered in [11, 12]. For such solutions, we establish the exponential convergence of a nonconforming dG interpolant given by local L 2 -projections on elements away from corners and edges, and by suitable local low-order quasi-interpolants on elements at corners and edges. Due to the appearance of non-homogeneous, weighted norms in the analytic regularity class, new arguments are introduced to bound the dG consistency errors in elements abutting on Neumann edges. The non-homogeneous norms also entail some crucial modifications of the stability and quasi-optimality proofs, as well as of the analysis for the anisotropic interpolation operators. The exponential convergence bounds for the dG interpolant constructed in this paper generalize the results of [11, 12] for the pure Dirichlet case.

Fast accumulation of few polyhedra mutants during passage of a Spodoptera frugiperda multicapsid nucleopolyhedrovirus (Baculoviridae) in Sf9 cell cultures

Baculoviruses are a group of viruses that infect invertebrates and that have been used worldwide as a biopesticide against several insect pests of the Order Lepidoptera. In Brazil, the baculovirus Spodoptera frugiperda multicapsid nucleopolyhedrovirus (SfMNPV, Baculoviridae) has been used experimentally to control S. frugiperda (Lepidoptera: Noctuidae), an important insect pest of corn (maize) fields and other crops. Baculoviruses can be produced either in insect larvae or in cell culture bioreactors. A major limitation to the in vitro production of baculoviruses is the rapid generation of mutants when the virus undergoes passages in cell culture. In order to evaluate the potential of in vitro methods of producing SfMNPV on a large-scale, we have multiplied a Brazilian isolate of this virus in cell culture. Extensive formation of few polyhedra mutants was observed after only two passages in Sf9 cells.

Properties of a unique mutant of Helicoverpa armigera single-nucleocapsid nucleopolyhedrovirus that exhibits a partial many polyhedra and few polyhedra phenotype on extended serial passaging in suspension cell cultures

Serial passaging of wild-type Helicoverpa armigera, single-nucleocapsid (HaSNPV) in H. zea (HzAMI) illsect Cell Cultures results ill rapid selection for the few polyhedra (FP) phenotype. A unique HaSNPV mutant (ppC19) was isolated through plaque purification that exhibited a partial many polyhedra (MP) and FP phenotype. Oil serial passaging in suspension cell cultures, ppC19 produced fivefold more polyhedra than a typical FP mutant (FP8AS) but threefold less polyhedra than the wild-type virus. Most importantly, the polyhedra of ppC19 exhibited MP-like virion occlusion. Furthermore, ppC19 produced the same amount of budded virus (BV) as the FP mutant, which was fivefold higher than that of the wild-type virus. This selective advantage was likely to explain its relative stability in polyhedra production for six passages when compared with the wild-type Virus. However, subsequent passaging of ppC19 resulted in a steel) decline in both BV and polyhedra yields, which was also experienced by FP8AS and the wild-type virus Lit high passage numbers. Genomic deoxyribonueleic Licid profiling of the latter suggested that defective interfering particles (DIPS) were implicated in this phenomenon and represented another Undesirable mutation during serial passaging of HaSNPV Hence, a strategy to isolate HaSNPV Clones that exhibited MP-like polyhedra production but FP-like BV production, coupled with low multiplicities of infection during scale-up to avoid accumulation of DIPS, could prove commerically invaluable.

Experimental transformation of kaolinite to halloysite

A well-characterized kaolinite has been hydrated in order to test the hypothesis that platey kaolinite will roll upon hydration. Kaolinite hydrates are prepared by repeated intercalation of kaolinite with potassium acetate and subsequent washing with water. On hydration, kaolinite plates roll along the major crystallographic directions to form tubes identical to proper tubular halloysite. Most tubes are elongated along the b crystallographic axis, while some are elongated along the a axis. Overall, the tubes exhibit a range of crystallinity. Well-ordered examples show a 2-layer structure, while poorly ordered tubes show little or no 3-dimensional order. Cross-sectional views of the formed tubes show both smoothly curved layers and planar faces. These characteristics of the experimentally formed tubes are shared by natural halloysites. Therefore, it is proposed that planar kaolinite can transform to tubular halloysite.

Vibrational spectroscopy of the borate mineral henmilite Ca2Cu[B(OH)4]2(OH)4

Henmilite is a triclinic mineral with the crystal structure consisting of isolated B(OH)4 tetrahedra, planar Cu(OH)4 groups and Ca(OH)3 polyhedra. The structure can also be viewed as having dimers of Ca polyhedra connected to each other through 2B(OH) tetrahedra to form chains parallel to the C axis. The structure of the mineral has been assessed by the combination of Raman and infrared spectra. Raman bands at 902, 922, 951, and 984 cm−1 and infrared bands at 912, 955 and 998 cm−1 are assigned to stretching vibrations of tetragonal boron. The Raman band at 758 cm−1 is assigned to the symmetric stretching mode of tetrahedral boron. The series of bands in the 400–600 cm−1 region are due to the out-of-plane bending modes of tetrahedral boron. Two very sharp Raman bands are observed at 3559 and 3609 cm−1. Two infrared bands are found at 3558 and 3607 cm−1. These bands are assigned to the OH stretching vibrations of the OH units in henmilite. A series of Raman bands are observed at 3195, 3269, 3328, 3396, 3424 and 3501 cm−1 are assigned to water stretching modes. Infrared spectroscopy also identified water and OH units in the henmilite structure. It is proposed that water is involved in the structure of henmilite. Hydrogen bond distances based upon the OH stretching vibrations using a Libowitzky equation were calculated. The number and variation of water hydrogen bond distances are important for the stability off the mineral.