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27 pages, 11130 KiB

Open AccessArticle

A Dual-Modal Robot Welding Trajectory Generation Scheme for Motion Based on Stereo Vision and Deep Learning

by Xinlei Li, Jiawei Ma, Shida Yao, Guanxin Chi and Guangjun Zhang

Materials 2025, 18(11), 2593; https://doi.org/10.3390/ma18112593 - 1 Jun 2025

Viewed by 695

To address the challenges of redundant point cloud processing and insufficient robustness under complex working conditions in existing teaching-free methods, this study proposes a dual-modal perception framework termed “2D image autonomous recognition and 3D point cloud precise planning”, which integrates stereo vision and [...] Read more.

To address the challenges of redundant point cloud processing and insufficient robustness under complex working conditions in existing teaching-free methods, this study proposes a dual-modal perception framework termed “2D image autonomous recognition and 3D point cloud precise planning”, which integrates stereo vision and deep learning. First, an improved U-Net deep learning model is developed, where VGG16 serves as the backbone network and a dual-channel attention module (DAM) is incorporated, achieving robust weld segmentation with a mean intersection over union (mIoU) of 0.887 and an F1-Score of 0.940. Next, the weld centerline is extracted using the Zhang–Suen skeleton refinement algorithm, and weld feature points are obtained through polynomial fitting optimization to establish cross-modal mapping between 2D pixels and 3D point clouds. Finally, a groove feature point extraction algorithm based on improved RANSAC combined with an equal-area weld bead filling strategy is designed to enable multi-layer and multi-bead robot trajectory planning, achieving a mean absolute error (MAE) of 0.238 mm in feature point positioning. Experimental results demonstrate that the method maintains high accuracy under complex working conditions such as noise interference and groove deformation, achieving a system accuracy of 0.208 mm and weld width fluctuation within ±0.15 mm, thereby significantly improving the autonomy and robustness of robot trajectory planning. Full article

(This article belongs to the Section Materials Simulation and Design)

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20 pages, 2292 KiB

Open AccessArticle

Kekulé Counts, Clar Numbers, and ZZ Polynomials for All Isomers of (5,6)-Fullerenes C₅₂–C₇₀

by Henryk A. Witek and Rafał Podeszwa

Molecules 2024, 29(17), 4013; https://doi.org/10.3390/molecules29174013 - 24 Aug 2024

Viewed by 1663

Abstract

We report an extensive tabulation of several important topological invariants for all the isomers of carbon

(5, 6)

-fullerenes

C_{n}

with n = 52–70. The topological invariants (including Kekulé count, Clar count, and Clar number) are computed and reported [...] Read more.

We report an extensive tabulation of several important topological invariants for all the isomers of carbon

(5, 6)

-fullerenes

C_{n}

with n = 52–70. The topological invariants (including Kekulé count, Clar count, and Clar number) are computed and reported in the form of the corresponding Zhang–Zhang (ZZ) polynomials. The ZZ polynomials appear to be distinct for each isomer cage, providing a unique label that allows for differentiation between various isomers. Several chemical applications of the computed invariants are reported. The results suggest rather weak correlation between the Kekulé count, Clar count, Clar number invariants, and isomer stability, calling into doubt the predictive power of these topological invariants in discriminating the most stable isomer of a given fullerene. The only exception is the Clar count/Kekulé count ratio, which seems to be the most important diagnostic discovered from our analysis. Stronger correlations are detected between Pauling bond orders computed from Kekulé structures (or Clar covers) and the corresponding equilibrium bond lengths determined from the optimized DFTB geometries of all 30,579 isomers of C₂₀–C₇₀. Full article

(This article belongs to the Special Issue Multiconfigurational and DFT Methods Applied to Chemical Systems—2nd Edition)

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18 pages, 713 KiB

Open AccessArticle

The Multivariable Zhang–Zhang Polynomial of Phenylenes

by Niko Tratnik

Axioms 2023, 12(11), 1053; https://doi.org/10.3390/axioms12111053 - 15 Nov 2023

Cited by 5 | Viewed by 1502

Abstract

The Zhang–Zhang polynomial of a benzenoid system is a well-known counting polynomial that was introduced in 1996. It was designed to enumerate Clar covers, which are spanning subgraphs with only hexagons and edges as connected components. In 2018, the generalized Zhang–Zhang polynomial of [...] Read more.

The Zhang–Zhang polynomial of a benzenoid system is a well-known counting polynomial that was introduced in 1996. It was designed to enumerate Clar covers, which are spanning subgraphs with only hexagons and edges as connected components. In 2018, the generalized Zhang–Zhang polynomial of two variables was defined such that it also takes into account 10-cycles of a benzenoid system. The aim of this paper is to introduce and study a new variation of the Zhang–Zhang polynomial for phenylenes, which are important molecular graphs composed of 6-membered and 4-membered rings. In our case, Clar covers can contain 4-cycles, 6-cycles, 8-cycles, and edges. Since this new polynomial has three variables, we call it the multivariable Zhang–Zhang (MZZ) polynomial. In the main part of the paper, some recursive formulas for calculating the MZZ polynomial from subgraphs of a given phenylene are developed and an algorithm for phenylene chains is deduced. Interestingly, computing the MZZ polynomial of a phenylene chain requires some techniques that are different to those used to calculate the (generalized) Zhang–Zhang polynomial of benzenoid chains. Finally, we prove a result that enables us to find the MZZ polynomial of a phenylene with branched hexagons. Full article

(This article belongs to the Special Issue Spectral Graph Theory, Molecular Graph Theory and Their Applications)

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16 pages, 5528 KiB

Open AccessArticle

Comparison of Camera Calibration and Measurement Accuracy Techniques for Phase Measuring Deflectometry

by Renhao Ge, Dahai Li, Xinwei Zhang, Ruiyang Wang, Wanxing Zheng, Xiaowei Li and Wuxiang Zhao

Appl. Sci. 2021, 11(21), 10300; https://doi.org/10.3390/app112110300 - 2 Nov 2021

Cited by 3 | Viewed by 2943

Abstract

Phase measuring deflectometry (PMD) is a competitive method for specular surface measurement that offers the advantages of a high dynamic range, non-contact process, and full field measurement; furthermore, it can also achieve high accuracy. Camera calibration is a crucial step for PMD. As [...] Read more.

Phase measuring deflectometry (PMD) is a competitive method for specular surface measurement that offers the advantages of a high dynamic range, non-contact process, and full field measurement; furthermore, it can also achieve high accuracy. Camera calibration is a crucial step for PMD. As a result, a method based on the calibration of the entrance pupil center is introduced in this paper. Then, our proposed approach is compared with the most popular photogrammetric method based on Zhang’s technique (PM) and Huang’s modal phase measuring deflectometry (MPMD). The calibration procedures of these three methods are described, and the measurement errors introduced by the perturbations of degrees of freedom in the PMD system are analyzed using a ray tracing technique. In the experiment, a planar window glass and an optical planar element are separately measured, and the measurement results of the use of the three methods are compared. The experimental results for the optical planar element (removing the first 6 terms of the Zernike polynomial) show that our method’s measurement accuracy reached 13.71 nm RMS and 80.50 nm PV, which is comparable to accuracy values for the interferometer. Full article

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27 pages, 3539 KiB

Open AccessArticle

Zhang–Zhang Polynomials of Multiple Zigzag Chains Revisited: A Connection with the John–Sachs Theorem

by Henryk A. Witek

Molecules 2021, 26(9), 2524; https://doi.org/10.3390/molecules26092524 - 26 Apr 2021

Cited by 8 | Viewed by 2807

Abstract

Multiple zigzag chains

Z (m, n)

of length n and width m constitute an important class of regular graphene flakes of rectangular shape. The physical and chemical properties of these basic pericondensed benzenoids can be related to their various topological invariants, conveniently [...] Read more.

Multiple zigzag chains

Z (m, n)

of length n and width m constitute an important class of regular graphene flakes of rectangular shape. The physical and chemical properties of these basic pericondensed benzenoids can be related to their various topological invariants, conveniently encoded as the coefficients of a combinatorial polynomial, usually referred to as the ZZ polynomial of multiple zigzag chains

Z (m, n)

. The current study reports a novel method for determination of these ZZ polynomials based on a hypothesized extension to John–Sachs theorem, used previously to enumerate Kekulé structures of various benzenoid hydrocarbons. We show that the ZZ polynomial of the

Z (m, n)

multiple zigzag chain can be conveniently expressed as a determinant of a Toeplitz (or almost Toeplitz) matrix of size

⌈\frac{m}{2}⌉ \times ⌈\frac{m}{2}⌉

consisting of simple hypergeometric polynomials. The presented analysis can be extended to generalized multiple zigzag chains

Z_{k} (m, n)

, i.e., derivatives of

Z (m, n)

with a single attached polyacene chain of length k. All presented formulas are accompanied by formal proofs. The developed theoretical machinery is applied for predicting aromaticity distribution patterns in large and infinite multiple zigzag chains

Z (m, n)

and for computing the distribution of spin densities in biradical states of finite multiple zigzag chains

Z (m, n)

. Full article

(This article belongs to the Special Issue Molecular Modeling: Advancements and Applications)

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14 pages, 1056 KiB

Open AccessFeature PaperArticle

Zhang–Zhang Polynomials of Ribbons

by Bing-Hau He, Chien-Pin Chou, Johanna Langner and Henryk A. Witek

Symmetry 2020, 12(12), 2060; https://doi.org/10.3390/sym12122060 - 11 Dec 2020

Cited by 12 | Viewed by 1849

Abstract

We report a closed-form formula for the Zhang–Zhang polynomial (also known as ZZ polynomial or Clar covering polynomial) of an important class of elementary peri-condensed benzenoids

R b (n_{1}, n_{2}, m_{1}, m_{2})

, usually referred to [...] Read more.

We report a closed-form formula for the Zhang–Zhang polynomial (also known as ZZ polynomial or Clar covering polynomial) of an important class of elementary peri-condensed benzenoids

R b (n_{1}, n_{2}, m_{1}, m_{2})

, usually referred to as ribbons. A straightforward derivation is based on the recently developed interface theory of benzenoids [Langner and Witek, MATCH Commun. Math. Comput. Chem.2020, 84, 143–176]. The discovered formula provides compact expressions for various topological invariants of

R b (n_{1}, n_{2}, m_{1}, m_{2})

: the number of Kekulé structures, the number of Clar covers, its Clar number, and the number of Clar structures. The last two classes of elementary benzenoids, for which closed-form ZZ polynomial formulas remain to be found, are hexagonal flakes

O (k, m, n)

and oblate rectangles

O r (m, n)

. Full article

(This article belongs to the Special Issue Symmetry on the Genealogy of Conjugated Acyclic Polyenes ‑ Dedicated to the Two Active Mathematical Chemists, Diudea and Aihara)

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47 pages, 6519 KiB

Open AccessArticle

ZZ Polynomials for Isomers of (5,6)-Fullerenes C_n with n = 20–50

by Henryk A. Witek and Jin-Su Kang

Symmetry 2020, 12(9), 1483; https://doi.org/10.3390/sym12091483 - 9 Sep 2020

Cited by 13 | Viewed by 2960

Abstract

A compilation of ZZ polynomials (aka Zhang–Zhang polynomials or Clar covering polynomials) for all isomers of small (5,6)-fullerenes C

_{n}

with n = 20–50 is presented. The ZZ polynomials concisely summarize the most important topological invariants of the fullerene isomers: the number of [...] Read more.

A compilation of ZZ polynomials (aka Zhang–Zhang polynomials or Clar covering polynomials) for all isomers of small (5,6)-fullerenes C

_{n}

with n = 20–50 is presented. The ZZ polynomials concisely summarize the most important topological invariants of the fullerene isomers: the number of Kekulé structures K, the Clar number

C l

, the first Herndon number

h_{1}

, the total number of Clar covers C, and the number of Clar structures. The presented results should be useful as benchmark data for designing algorithms and computer programs aiming at topological analysis of fullerenes and at generation of resonance structures for valence-bond quantum-chemical calculations. Full article

(This article belongs to the Section Chemistry: Symmetry/Asymmetry)

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13 pages, 288 KiB

Open AccessArticle

Four Constructions of Asymptotically Optimal Codebooks via Additive Characters and Multiplicative Characters

by Xia Wu and Wei Lu

Mathematics 2019, 7(12), 1144; https://doi.org/10.3390/math7121144 - 23 Nov 2019

Cited by 1 | Viewed by 2169

Abstract

In this paper, we present four new constructions of complex codebooks with multiplicative characters, additive characters, and quadratic irreducible polynomials and determine the maximal cross-correlation amplitude of these codebooks. We prove that the codebooks we constructed are asymptotically optimal with respect to the [...] Read more.

In this paper, we present four new constructions of complex codebooks with multiplicative characters, additive characters, and quadratic irreducible polynomials and determine the maximal cross-correlation amplitude of these codebooks. We prove that the codebooks we constructed are asymptotically optimal with respect to the Welch bound. Moreover, we generalize the result obtained by Zhang and Feng and contain theirs as a special case. The parameters of these codebooks are new. Full article

(This article belongs to the Special Issue Information Theory, Cryptography, Randomness and Statistical Modeling)

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