scholarly journals A topological extension of movement primitives for curvature modulation and sampling of robot motion

2021 ◽  
Author(s):  
Adrià Colomé ◽  
Carme Torras
Keyword(s):  
Closed Curve ◽  
Robot Motion ◽  
Topological Feature ◽  
Closed Curves ◽  
Motion Data ◽  
Linking Number ◽  
Robot Trajectory ◽  
Topological Features ◽  
Curvature Information ◽  
Do So

AbstractThis paper proposes to enrich robot motion data with trajectory curvature information. To do so, we use an approximate implementation of a topological feature named writhe, which measures the curling of a closed curve around itself, and its analog feature for two closed curves, namely the linking number. Despite these features have been established for closed curves, their definition allows for a discrete calculation that is well-defined for non-closed curves and can thus provide information about how much a robot trajectory is curling around a line in space. Such lines can be predefined by a user, observed by vision or, in our case, inferred as virtual lines in space around which the robot motion is curling. We use these topological features to augment the data of a trajectory encapsulated as a Movement Primitive (MP). We propose a method to determine how many virtual segments best characterize a trajectory and then find such segments. This results in a generative model that permits modulating curvature to generate new samples, while still staying within the dataset distribution and being able to adapt to contextual variables.

Link prediction based on local weighted paths for complex networks

2017 ◽  
Vol 28 (04) ◽  
pp. 1750053
Author(s):  
Yabing Yao ◽  
Ruisheng Zhang ◽  
Fan Yang ◽  
Yongna Yuan ◽  
Rongjing Hu ◽  
...  
Keyword(s):  
Complex Networks ◽  
Real World ◽  
Link Prediction ◽  
Structural Similarity ◽  
Prediction Performance ◽  
Topological Feature ◽  
Topological Features ◽  
Node Similarity ◽  
Weighted Paths ◽  
Path Dependent

As a significant problem in complex networks, link prediction aims to find the missing and future links between two unconnected nodes by estimating the existence likelihood of potential links. It plays an important role in understanding the evolution mechanism of networks and has broad applications in practice. In order to improve prediction performance, a variety of structural similarity-based methods that rely on different topological features have been put forward. As one topological feature, the path information between node pairs is utilized to calculate the node similarity. However, many path-dependent methods neglect the different contributions of paths for a pair of nodes. In this paper, a local weighted path (LWP) index is proposed to differentiate the contributions between paths. The LWP index considers the effect of the link degrees of intermediate links and the connectivity influence of intermediate nodes on paths to quantify the path weight in the prediction procedure. The experimental results on 12 real-world networks show that the LWP index outperforms other seven prediction baselines.

On the Maximum Curvature of Closed Curves in Negatively Curved Manifolds

2015 ◽  
Vol 58 (4) ◽  
pp. 713-722
Author(s):  
Simon Brendle ◽  
Otis Chodosh
Keyword(s):  
Maximum Principle ◽  
Geodesic Curvature ◽  
Sharp Inequality ◽  
Closed Curve ◽  
Closed Curves ◽  
The Maximum Principle ◽  
Curved Manifolds ◽  
Maximum Curvature ◽  
Negatively Curved ◽  
Simply Connected

AbstractMotivated by Almgren’s work on the isoperimetric inequality, we prove a sharp inequality relating the length and maximum curvature of a closed curve in a complete, simply connected manifold of sectional curvature at most −1. Moreover, if equality holds, then the norm of the geodesic curvature is constant and the torsion vanishes. The proof involves an application of the maximum principle to a function defined on pairs of points.

Are Circular Shaped Masks Adequate in Adaptive Mask Overlay Topology Synthesis Method?

2010 ◽  
Vol 133 (1) ◽  
Author(s):  
Anupam Saxena
Keyword(s):  
Resource Constraints ◽  
Synthesis Method ◽  
Design Procedure ◽  
Topology Design ◽  
Two Dimensional ◽  
Closed Curves ◽  
Compliance Minimization ◽  
Topological Features ◽  
Mean Compliance ◽  
Material Layout

Previous versions of the material mask overlay strategy (MMOS) for topology synthesis primarily employ circular masks to simulate voids within the design region. MMOS operates on the photolithographic principle by appropriately positioning and sizing a group of negative masks and thus iteratively improves the material layout to meet the desired objective. The fundamental notion is that a group of circular masks can represent a local void of any shape. The question whether masks of more general shapes (e.g., any two-dimensional closed, nonself intersecting polygon) would offer significant enhancements in efficiently attaining the appropriate topological features in a continuum remains. This paper investigates the performance of two other mask types; elliptical and rectangular masks are compared with that of the circular ones. These are the respective modest representatives of closed curves and their polygonal approximations. First, two mean compliance minimization examples under resource constraints are solved. Thereafter, compliant pliers are synthesized using the three mask types. It is observed that the use of elliptical or rectangular masks do not offer significant advantages over the use of circular ones. On the contrary, the examples suggest that less number of circular masks are adequate to model the topology design procedure more efficiently. Thus, it is postulated that using generic simple closed curves or polygonal masks will not introduce significant benefits over circular ones in the MMOS based topology design algorithms.

scholarly journals Monotone homotopies and contracting discs on Riemannian surfaces

2018 ◽  
Vol 10 (02) ◽  
pp. 323-354 ◽  
Author(s):  
Gregory R. Chambers ◽  
Regina Rotman
Keyword(s):  
Finite Volume ◽  
Minimal Surfaces ◽  
Analogous Result ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Riemannian Surface ◽  
Closed Curves ◽  
Minimal Hypersurfaces ◽  
Riemannian Surfaces ◽  
Complete Manifolds

A monotone homotopy is a homotopy composed of simple closed curves which are also pairwise disjoint. In this paper, we prove a “gluing” theorem for monotone homotopies; we show that two monotone homotopies which have appropriate overlap can be replaced by a single monotone homotopy. The ideas used to prove this theorem are used in [G. R. Chambers and Y. Liokumovich, Existence of minimal hypersurfaces in complete manifolds of finite volume, arXiv:1609.04058] to prove an analogous result for cycles, which forms a critical step in their proof of the existence of minimal surfaces in complete non-compact manifolds of finite volume. We also show that, if monotone homotopies exist, then fixed point contractions through short curves exist. In particular, suppose that [Formula: see text] is a simple closed curve of a Riemannian surface, and that there exists a monotone contraction which covers a disc which [Formula: see text] bounds consisting of curves of length [Formula: see text]. If [Formula: see text] and [Formula: see text], then there exists a homotopy that contracts [Formula: see text] to [Formula: see text] over loops that are based at [Formula: see text] and have length bounded by [Formula: see text], where [Formula: see text] is the diameter of the surface. If the surface is a disc, and if [Formula: see text] is the boundary of this disc, then this bound can be improved to [Formula: see text].

Taming Wild Simple Closed Curves with Monotone Maps

1972 ◽  
Vol 24 (5) ◽  
pp. 768-788
Author(s):  
W. S. Boyd ◽  
A. H. Wright
Keyword(s):  
Simple Closed Curve ◽  
Closed Curve ◽  
Closed Curves ◽  
Monotone Map ◽  
Monotone Maps

Hempel [6, Theorem 2] proved that if S is a tame 2-sphere in E3 and f is a map of E3 onto itself such that f|S is a homeomorphism and f(E3 - S) = E3- f(S), then f(S) is tame. Boyd [4] has shown that the converse is false; in fact, if S is any 2-sphere in E3, then there is a monotone map f of E3 onto itself such that f |S is a homeomorphism, f(E3 — S) = E3 — f(S), and f(S) is tame.It is the purpose of this paper to prove that the corresponding converse for simple closed curves in E3 is also false. We show in Theorem 4 that if J is any simple closed curve in a closed orientable 3-manifold M3, then there is a monotone map f : M3 → S3 such that f |J is a homeomorphism, f(J) is tame and unknotted, and f(M3 - J) = S3 - f(J).In Theorem 1 of § 2, we construct a cube-with-handles neighbourhood of a simple closed curve in an orientable 3-manifold.

THE MELLIN CENTRAL PROJECTION TRANSFORM

2017 ◽  
Vol 58 (3-4) ◽  
pp. 256-264
Author(s):  
JIANWEI YANG ◽  
LIANG ZHANG ◽  
ZHENGDA LU
Keyword(s):  
Radial Direction ◽  
Mellin Transform ◽  
Closed Curve ◽  
Chinese Characters ◽  
Central Projection ◽  
Affine Invariant ◽  
Closed Curves ◽  
Affine Invariants ◽  
Invariant Features ◽  
Special Case

The central projection transform can be employed to extract invariant features by combining contour-based and region-based methods. However, the central projection transform only considers the accumulation of the pixels along the radial direction. Consequently, information along the radial direction is inevitably lost. In this paper, we propose the Mellin central projection transform to extract affine invariant features. The radial factor introduced by the Mellin transform, makes up for the loss of information along the radial direction by the central projection transform. The Mellin central projection transform can convert any object into a closed curve as a central projection transform, so the central projection transform is only a special case of the Mellin central projection transform. We prove that closed curves extracted from the original image and the affine transformed image by the Mellin central projection transform satisfy the same affine transform relationship. A method is provided for the extraction of affine invariants by employing the area of closed curves derived by the Mellin central projection transform. Experiments have been conducted on some printed Chinese characters and the results establish the invariance and robustness of the extracted features.

CONSTRUCTION OF INVARIANTS OF CURVES AND FRONTS USING WORD THEORY

2010 ◽  
Vol 19 (09) ◽  
pp. 1205-1245
Author(s):  
NOBORU ITO
Keyword(s):  
Infinite Sequence ◽  
Closed Curve ◽  
Closed Curves

We construct an infinite sequence of invariants for surface curves by using the word theory introduced by Turaev. We modify these invariants to various classes of curves: plane closed curves, long curves and fronts. For a plane closed curve, for example, these modified invariants contain Arnold's basic invariants. We also discuss how curves are classified by these modified invariants.

Quadrarcs, St. Peter's, and the Colosseum

1970 ◽  
Vol 63 (3) ◽  
pp. 209-215
Author(s):  
N. T. Gridgeman
Keyword(s):  
Abrupt Change ◽  
Applied Mathematics ◽  
Closed Curve ◽  
Closed Curves ◽  
The Everyday ◽  
Isolated Points ◽  
Axis Of Symmetry ◽  
Regular Closed

1. Ovals. An oval is a regular closed curve in the plane, having at least one axis of symmetry and continuons curvature. However, if this last condition is re laxed, the definition will admit a class of ovals (in the everyday sense of the word— perhaps they should formally be called pseudoövals) that are important in applied mathematics. These are the composite closed curves made up of symmetrically disposed circular ares joined “normally,” that is, where tangent and normal are common, so that the abrupt change of eurvature is not apparent. We may regard such an oval as having an evolute that is not itself a curve but a set of isolated points-discrete centers of curvature. Ovals of this kind have long been familiar to architects and engineers. We shall now discuss the commonest type, the quadrarc, and afterwards look at two celebrated architectural examples.

A New Class of Two-Dimensional Chaotic Maps with Closed Curve Fixed Points

2019 ◽  
Vol 29 (07) ◽  
pp. 1950094 ◽  
Author(s):  
Haibo Jiang ◽  
Yang Liu ◽  
Zhouchao Wei ◽  
Liping Zhang
Keyword(s):  
Fixed Points ◽  
Nonlinear Function ◽  
Closed Curve ◽  
Computer Search ◽  
Two Dimensional ◽  
Closed Curves ◽  
New Class ◽  
Dynamical Behaviors ◽  
The Mathematical Model ◽  
Search Program

This paper constructs a new class of two-dimensional maps with closed curve fixed points. Firstly, the mathematical model of these maps is formulated by introducing a nonlinear function. Different types of fixed points which form a closed curve are shown by choosing proper parameters of the nonlinear function. The stabilities of these fixed points are studied to show that these fixed points are all nonhyperbolic. Then a computer search program is employed to explore the chaotic attractors in these maps, and several simple maps whose fixed points form different shapes of closed curves are presented. Complex dynamical behaviors of these maps are investigated by using the phase-basin portrait, Lyapunov exponents, and bifurcation diagrams.

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