Dual Numbers and Invariant Theory of the Euclidean Group with Applications to Robotics

<p>In this thesis we study the special Euclidean group SE(3) from two points of view, algebraic and geometric. From the algebraic point of view we introduce a dualisation procedure for SO(3;ℝ) invariants and obtain vector invariants of the adjoint action of SE(3) acting on multiple screws. In the case of three screws there are 14 basic vector invariants related by two basic syzygies. Moreover, we prove that any invariant of the same group under the same action can be expressed as a rational function evaluated on those 14 vector invariants. From the geometric point of view, we study the Denavit-Hartenberg parameters used in robotics, and calculate formulae for link lengths and offsets in terms of vector invariants of the adjoint action of SE(3). Moreover, we obtain a geometrical duality between the offsets and the link lengths, where the geometrical dual of an offset is a link length and vice versa.</p>

Dual Numbers and Invariant Theory of the Euclidean Group with Applications to Robotics

<p>In this thesis we study the special Euclidean group SE(3) from two points of view, algebraic and geometric. From the algebraic point of view we introduce a dualisation procedure for SO(3;ℝ) invariants and obtain vector invariants of the adjoint action of SE(3) acting on multiple screws. In the case of three screws there are 14 basic vector invariants related by two basic syzygies. Moreover, we prove that any invariant of the same group under the same action can be expressed as a rational function evaluated on those 14 vector invariants. From the geometric point of view, we study the Denavit-Hartenberg parameters used in robotics, and calculate formulae for link lengths and offsets in terms of vector invariants of the adjoint action of SE(3). Moreover, we obtain a geometrical duality between the offsets and the link lengths, where the geometrical dual of an offset is a link length and vice versa.</p>

MACHINE LEARNING FOR MOBILE LIDAR DATA CLASSIFICATION OF 3D ROAD ENVIRONMENT

3D road mapping is essential for intelligent transportation system in smart cities. Road features can be utilized for road maintenance, autonomous driving vehicles, and providing regulations to drivers. Currently, 3D road environment receives its data from Mobile Laser Scanning (MLS) systems. MLS systems are capable of rapidly acquiring dense and accurate 3D point clouds, which allow for effective surveying of long road corridors. They produce huge amount of point clouds, which requires automatic features classification algorithms with acceptable processing time. Road features have variant geometric regular or irregular shapes. Therefore, most researches focus on classification of one road feature such as road surface, curbs, building facades, etc. Machine learning (ML) algorithms are widely used for predicting the future or classifying information to help policymakers in making necessary decisions. This prediction comes from a pre-trained model on a given data consisting of inputs and their corresponding outputs of the same characteristics. This research uses ML algorithms for mobile LiDAR data classification. First, cylindrical neighbourhood selection method was used to define point’s surroundings. Second, geometric point features including geometric, moment and height features were derived. Finally, three ML algorithms, Random Forest (RF), Gaussian Naïve Bayes (GNB), and Quadratic Discriminant Analysis (QDA) were applied. The ML algorithms were used to classify a part of Paris-Lille-3D benchmark of about 1.5 km long road in Lille with more than 98 million points into nine classes. The results demonstrated an overall accuracy of 92.39%, 78.5%, and 78.1% for RF, GNB, and QDA, respectively.

On the Convergence of Stochastic Process Convergence Proofs

Convergence of a stochastic process is an intrinsic property quite relevant for its successful practical for example for the function optimization problem. Lyapunov functions are widely used as tools to prove convergence of optimization procedures. However, identifying a Lyapunov function for a specific stochastic process is a difficult and creative task. This work aims to provide a geometric explanation to convergence results and to state and identify conditions for the convergence of not exclusively optimization methods but any stochastic process. Basically, we relate the expected directions set of a stochastic process with the half-space of a conservative vector field, concepts defined along the text. After some reasonable conditions, it is possible to assure convergence when the expected direction resembles enough to some vector field. We translate two existent and useful convergence results into convergence of processes that resemble to particular conservative vector fields. This geometric point of view could make it easier to identify Lyapunov functions for new stochastic processes which we would like to prove its convergence.

GeoGraph

In many applications of graph processing, the input data is often generated from an underlying geometric point data set. However, existing high-performance graph processing frameworks assume that the input data is given as a graph. Therefore, to use these frameworks, the user must write or use external programs based on computational geometry algorithms to convert their point data set to a graph, which requires more programming effort and can also lead to performance degradation. In this paper, we present our ongoing work on the Geo- Graph framework for shared-memory multicore machines, which seamlessly supports routines for parallel geometric graph construction and parallel graph processing within the same environment. GeoGraph supports graph construction based on k-nearest neighbors, Delaunay triangulation, and b-skeleton graphs. It can then pass these generated graphs to over 25 graph algorithms. GeoGraph contains highperformance parallel primitives and algorithms implemented in C++, and includes a Python interface. We present four examples of using GeoGraph, and some experimental results showing good parallel speedups and improvements over the Higra library. We conclude with a vision of future directions for research in bridging graph and geometric data processing.

Optimization of Airfoil Blend Limits with As-manufactured Geometry Finite Element Models

Conventional airfoil blend repair limits are established using nominal, design intent geometry. This convention does not explicitly consider inherent blade-to-blade structural response variations associated with geometric manufacturing deviations. In this work, we explore whether accounting for these variations leads to significant differences in blend depths and develop a novel approach to effectively predict blade-specific blend allowables. These blade-specific values maximize part repairability according to their proximity to defined structural integrity constraints. The methodology is demonstrated on as-manufactured geometry of an compressor rotor. Geometric point cloud data of this rotor is used to construct as-built finite element models of each airfoil. The effect of two large blends on these airfoils demonstrates the opportunity of blade-specific blend limits. A new approach to determine each airfoil's blend repair capacity is developed that uses sequential least squares quadratic programming and a parametric blended blade FEM that accounts for manufacturing geometry variations and variable blend geometry. A mesh morphing algorithm modifies a nominal geometry model to match the as-built airfoil surface and blend geometry. Numerical optimization maximizes blend depth values within frequency, mode shape, and high cycle fatigue (HCF) constraint boundaries. Large variations in blend depth allowables between blades are found and competing structural integrity criteria are responsible for their limits. It is also shown that, despite complex modal behavior caused by eigenvalue veering, the proposed optimization approach converges. The developed methodologies may be used to extend blend limits, enable continued operations, and reduce sustainment costs.

Cell Surface Area to Volume Relationship During Apoptosis and Apoptotic Body Formation.

Apoptosis is a programmed form of cell death culminating in packing cell content and corpse dismantling into membrane sealed vesicles called apoptotic bodies (ABs). Apoptotic bodies are engulfed and disposed by neighboring and immune system cells without eliciting a noxious inflammatory response, thus preventing sterile tissue damage. AB formation requires a total surface area larger than the apparent, initial cell's surface area. Apoptotic volume decrease (AVD) is a two-stage process leading to an isotonic, osmotic reduction in cell water content driven by net K+ and Cl- extrusion. In this article, the role of AVD is presented from a geometric point of view through the process of AB formation. AVD decisively contributes to (i) endowing the cell with the appropriate electrolytic environment for apoptotic execution; (ii) increasing the membrane surface area-to-volume ratio, along with the mobilization of membrane reservoirs (cell rounding, membrane folds and endosomal membranes), so that the cell corpse can be dismantled into ABs; and (iii) reducing plasmalemmal stretch, tension and stiffness, thus facilitating membrane bulging, blebbing and vesicle expansion ultimately leading to separation and release.

Study on Scanning Forming Methods of Machining Work-Piece Surface

In the process of machine tool cutting, there are strict geometric relations among the cutting edge curve / tool surface, machine tool movement and workpiece surface, and the machine tool movement is also related to the type of tool. Firstly, the forming methods of cutting workpiece surface are analyzed and summarized from the geometric point of view, and the scanning forming method and its geometric expression are studied, and the research technical route of forming turning scanning forming is put forward. Then, the mathematical modeling and Simulation of forming turning are carried out according to the proposed technical route. Finally, taking the groove of the inner ring of the formed turning ball bearing as an example, the mathematical modeling of the design surface of the workpiece and the machined surface of the workpiece is carried out. The radial dimension changes of the workpiece caused by the cutting force and tool wear are analyzed, and the simulation of the machined surface of the workpiece is carried out.