Computation of Spatial Displacements From Redundant Geometric Features

1992 ◽  
Author(s):  
Q. J. Ge ◽  
B. Ravani
Keyword(s):  
Symmetric Matrix ◽  
Automatic Generation ◽  
Geometric Features ◽  
Least Eigenvalue ◽  
Special Cases ◽  
Minimum Number ◽  
Feature Information ◽  
Spatial Displacements ◽  
Simple Features ◽  
Cubic Function

Abstract This paper follows a previous one on the computation of spatial displacements (Ravani and Ge, 1992). The first paper dealt with the problem of computing spatial displacements from a minimum number of simple features of points, lines, planes, and their combinations. The present paper deals with the same problem using a redundant set of the simple geometric features. The problem for redundant information is formulated as a least squares problem which includes all simple features. A Clifford algebra is used to unify the handling of various feature information. An algorithm for determining the best orientation is developed which involves finding the eigenvector associated with the least eigenvalue of a 4 × 4 symmetric matrix. The best translation is found to be a rational cubic function of the best orientation. Special cases are discussed which yield the best orientation in closed form. In addition, simple algorithms are provided for automatic generation of body-fixed coordinate frames from various feature information. The results have applications in robot and world model calibration for off-line programming and computer vision.

Computation of Spatial Displacements from Redundant Geometric Features

1994 ◽  
Vol 116 (4) ◽  
pp. 1073-1080 ◽  
Author(s):  
Q. J. Ge ◽  
B. Ravani
Keyword(s):  
Symmetric Matrix ◽  
Automatic Generation ◽  
Geometric Features ◽  
Least Eigenvalue ◽  
Special Cases ◽  
Minimum Number ◽  
Feature Information ◽  
Spatial Displacements ◽  
Simple Features ◽  
Cubic Function

This paper follows a previous one on the computation of spatial displacements (Ravani and Ge, 1993). The first paper dealt with the problem of computing spatial displacements from a minimum number of simple features of points, lines, planes, and their combinations. The present paper deals with the same problem using a redundant set of the simple geometric features. The problem for redundant information is formulated as a least squares problem which includes all simple features. A Clifford algebra is used to unify the handling of various feature information. An algorithm for determining the best orientation is developed which involves finding the eigenvector associated with the least eigenvalue of a 4 × 4 symmetric matrix. The best translation is found to be a rational cubic function of the best orientation. Special cases are discussed which yield the best orientation in closed form. In addition, simple algorithms are provided for automatic generation of body-fixed coordinate frames from various feature information. The results have applications in robot and world model calibration for off-line programming and computer vision.

Computation of Spatial Displacements From Geometric Features

1993 ◽  
Vol 115 (1) ◽  
pp. 95-102 ◽  
Author(s):  
B. Ravani ◽  
Q. J. Ge
Keyword(s):  
Computational Geometry ◽  
Clifford Algebra ◽  
Theoretical Foundation ◽  
Geometric Features ◽  
Theoretical Interest ◽  
Robot Calibration ◽  
Minimum Number ◽  
The Common ◽  
Spatial Displacements ◽  
Three Space

This paper develops the theoretical foundation for computations of spatial displacements from the simple geometric features of points, lines, planes, and their combinations. Using an oriented projective three space with a Clifford Algebra, all these three features are handled in a similar fashion. Furthermore, issues related to uniqueness of computations and minimum number of required features are discussed. It is shown that contrary to the common intuition, specification of a minimum of four points (planes) or three lines are necessary for computation of a unique displacement. Only when the sense of the orientations of these features are specified then the minimum number of required features reduces to three for points and planes and two for lines. The results, in addition to their theoretical interest in computational geometry of motion, have application in robot calibration.

Fast Bilinear Algorithms for Symmetric Tensor Contractions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Edgar Solomonik ◽  
James Demmel
Keyword(s):  
Symmetric Matrix ◽  
Matrix Multiplication ◽  
Computational Cost ◽  
Symmetric Tensor ◽  
Partial Sums ◽  
Symmetric Tensors ◽  
Outer Product ◽  
Bilinear Complexity ◽  
Special Cases ◽  
Tensor Contractions

AbstractIn matrix-vector multiplication, matrix symmetry does not permit a straightforward reduction in computational cost. More generally, in contractions of symmetric tensors, the symmetries are not preserved in the usual algebraic form of contraction algorithms. We introduce an algorithm that reduces the bilinear complexity (number of computed elementwise products) for most types of symmetric tensor contractions. In particular, it lowers the bilinear complexity of symmetrized contractions of symmetric tensors of order {s+v} and {v+t} by a factor of {\frac{(s+t+v)!}{s!t!v!}} to leading order. The algorithm computes a symmetric tensor of bilinear products, then subtracts unwanted parts of its partial sums. Special cases of this algorithm provide improvements to the bilinear complexity of the multiplication of a symmetric matrix and a vector, the symmetrized vector outer product, and the symmetrized product of symmetric matrices. While the algorithm requires more additions for each elementwise product, the total number of operations is in some cases less than classical algorithms, for tensors of any size. We provide a round-off error analysis of the algorithm and demonstrate that the error is not too large in practice. Finally, we provide an optimized implementation for one variant of the symmetry-preserving algorithm, which achieves speedups of up to 4.58\times for a particular tensor contraction, relative to a classical approach that casts the problem as a matrix-matrix multiplication.

Automatic Generation of Anisotropic Patterns of Geometric Features for Industrial Design

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Diego Andrade ◽  
Ved Vyas ◽  
Kenji Shimada
Keyword(s):  
Boundary Conditions ◽  
Computer Aided Design ◽  
Industrial Design ◽  
Automatic Generation ◽  
Free Form ◽  
Geometric Features ◽  
Target Domain ◽  
Metric Tensor ◽  
Target Region ◽  
Free Form Surfaces

While modern computer aided design (CAD) systems currently offer tools for generating simple patterns, such as uniformly spaced rectangular or radial patterns, these tools are limited in several ways: (1) They cannot be applied to free-form geometries used in industrial design, (2) patterning of these features happens within a single working plane and is not applicable to highly curved surfaces, and (3) created features lack anisotropy and spatial variations, such as changes in the size and orientation of geometric features within a given region. In this paper, we introduce a novel approach for creating anisotropic patterns of geometric features on free-form surfaces. Complex patterns are generated automatically, such that they conform to the boundary of any specified target region. Furthermore, user input of a small number of geometric features (called “seed features”) of desired size and orientation in preferred locations could be specified within the target domain. These geometric seed features are then transformed into tensors and used as boundary conditions to generate a Riemannian metric tensor field. A form of Laplace's heat equation is used to produce the field over the target domain, subject to specified boundary conditions. The field represents the anisotropic pattern of geometric features. This procedure is implemented as an add-on for a commercial CAD package to add geometric features to a target region of a three-dimensional model using two set operations: union and subtraction. This method facilitates the creation of a complex pattern of hundreds of geometric features in less than 5 min. All the features are accessible from the CAD system, and if required, they are manipulable individually by the user.

scholarly journals On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture

10.37236/6516 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Megumi Asada ◽  
Ryan Chen ◽  
Florian Frick ◽  
Frederick Huang ◽  
Maxwell Polevy ◽  
...  
Keyword(s):  
Geometric Property ◽  
Convex Sets ◽  
Convex Hulls ◽  
Open Problems ◽  
Higher Dimensional ◽  
Special Cases ◽  
Minimum Number ◽  
Point Set ◽  
Colored Version ◽  
Special Case

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.

Computation of Spatial Displacements From Geometric Features

1991 ◽  
Author(s):  
B. Ravani ◽  
Q. J. Ge
Keyword(s):  
Computational Geometry ◽  
Clifford Algebra ◽  
Theoretical Foundation ◽  
Geometric Features ◽  
Theoretical Interest ◽  
Robot Calibration ◽  
The Common ◽  
Spatial Displacements ◽  
Three Space

Abstract This paper develops the theoretical foundation for computations of spatial displacements from the simple geometric features of points, lines, planes and their combinations. Using an oriented projective three space with a Clifford Algebra, all these three features are handled in a similar fashion. Furthermore, issues related to uniqueness of computations and minimal number of required features are discussed. It is shown that contrary to the common intuition, specification of a minimum of four points (planes) or three lines (each pair being non-planar) are necessary for computation of a unique displacement. Only when the sense of the orientations of these features are specified then the minimal number of required features reduces to three for points and planes and two for lines. The results, in addition to their theoretical interest in computational geometry of motion, have application in robot calibration.

Scalable Algorithms for Server Allocation in Infostations

2010 ◽  
pp. 645-656
Author(s):  
Alan A. Bertossi ◽  
M. Cristina Pinotti ◽  
Phalguni Gupta
Keyword(s):  
Bin Packing ◽  
Multiprocessor Scheduling ◽  
Interval Graph ◽  
Scalable Algorithms ◽  
Coverage Area ◽  
Special Cases ◽  
Minimum Number ◽  
Server Allocation ◽  
On Line ◽  
Web Surfing

The server allocation problem arises in isolated infostations, where mobile users going through the coverage area require immediate high-bit rate communications such as web surfing, file transferring, voice messaging, email and fax. Given a set of service requests, each characterized by a temporal interval and a category, an integer k, and an integer hc for each category c, the problem consists in assigning a server to each request in such a way that at most k mutually simultaneous requests are assigned to the same server at the same time, out of which at most hc are of category c, and the minimum number of servers is used. Since this problem is computationally intractable, a scalable 2-approximation online algorithm is exhibited. Generalizations of the problem are considered, which contain bin-packing, multiprocessor scheduling, and interval graph coloring as special cases, and admit scalable on-line algorithms providing constant approximations.

scholarly journals Pinning Decision in Interconnected Systems with Communication Disruptions under Multi-Agent Distributed Control Topology

2021 ◽  
Vol 2 (1) ◽  
pp. 18-36
Author(s):  
Samson S. Yu ◽  
Tat Kei Chau
Keyword(s):  
Decision Making ◽  
Transient Stability ◽  
Frequency Control ◽  
Automatic Generation ◽  
Distributed Generators ◽  
Number Of Generators ◽  
Minimum Number ◽  
Multi Agent ◽  
Active Communication ◽  
System Topology

In this study, we propose a decision-making strategy for pinning-based distributed multi-agent (PDMA) automatic generation control (AGC) in islanded microgrids against stochastic communication disruptions. The target microgrid is construed as a cyber-physical system, wherein the physical microgrid is modeled as an inverter-interfaced autonomous grid with detailed system dynamic formulation, and the communication network topology is regarded as a cyber-system independent of its physical connection. The primal goal of the proposed method is to decide the minimum number of generators to be pinned and their identities amongst all distributed generators (DGs). The pinning-decisions are made based on complex network theories using the genetic algorithm (GA), for the purpose of synchronizing and regulating the frequencies and voltages of all generator bus-bars in a PDMA control structure, i.e., without resorting to a central AGC agent. Thereafter, the mapping of cyber-system topology and the pinning decision is constructed using deep-learning (DL) technique, so that the pinning-decision can be made nearly instantly upon detecting a new cyber-system topology after stochastic communication disruptions. The proposed decision-making approach is verified using a 10-generator, 38-bus microgrid through time-domain simulation for transient stability analysis. Simulations show that the proposed pinning decision making method can achieve robust frequency control with minimum number of active communication channels.

scholarly journals A MBD Model-driven Automatic Generation Method of On-machine Measuring Program for CNC Machining

2021 ◽  
Vol 2095 (1) ◽  
pp. 012045
Author(s):  
Wenhao Zhong ◽  
Cheng Zhai ◽  
Yansheng Cao ◽  
Meiqing Wang
Keyword(s):  
Automatic Generation ◽  
Cnc Machining ◽  
Machining Process ◽  
Geometric Features ◽  
Probe Position ◽  
Model Driven ◽  
Tool Magazine ◽  
Manufacturing Information ◽  
Product Manufacturing ◽  
The Relationship

Abstract Aiming at the problem that the programming of on-machine measurement in CNC machining process is heavily dependent on operators and the efficiency is lower, a MBD model-driven automatic generation method of on-machine measuring program is proposed. The characteristics of measuring program is analysed and the relationship between the parameters of measuring cycle and the features of MBD model is built. The approach of PMI (product manufacturing information) data identifying and extracting is presented. The scheme for transforming the PMI data and geometric features to measuring cycle parameters is constructed. In order to make the measuring program adapt to the change of probe position which stored in different tool magazine, the way of collecting the position automatically by OPC DA protocol is provided. On the basis of the proposed methods and approaches, an on-machine measuring program generating software system has been developed. The running result of the system is given, and the feasibility and effectiveness of the method were demonstrated.

scholarly journals Covering Partial Cubes with Zones

10.37236/5076 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Jean Cardinal ◽  
Stefan Felsner
Keyword(s):  
Isometric Embedding ◽  
Linear Extension ◽  
Upper And Lower Bounds ◽  
Minimum Size ◽  
Worst Case ◽  
Line Arrangement ◽  
Minimal Dimension ◽  
Special Cases ◽  
Minimum Number ◽  
Partial Cube

A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called zones. The number of zones determines the minimal dimension of a hypercube in which the graph can be embedded. We consider the problem of covering the vertices of a partial cube with the minimum number of zones. The problem admits several special cases, among which are the following:cover the cells of a line arrangement with a minimum number of lines,select a smallest subset of edges in a graph such that for every acyclic orientation, there exists a selected edge that can be flipped without creating a cycle,find a smallest set of incomparable pairs of elements in a poset such that in every linear extension, at least one such pair is consecutive,find a minimum-size fibre in a bipartite poset.We give upper and lower bounds on the worst-case minimum size of a covering by zones in several of those cases. We also consider the computational complexity of those problems, and establish some hardness results.

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