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1
Alwardat, Mohammad Y. "GEOMETRIC JACOBIANS DERIVATION AND KINEMATIC SINGULARITY ANALYSIS FOR 6-DOF ROBOTIC MANIPULATOR." international journal of advanced research in computer science 16, no. 1 (2025): 6–20. https://doi.org/10.26483/ijarcs.v16i1.7178.
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This article investigates the kinematic singularities and geometric Jacobians of a 6-DOF robotic manipulator, incorporating a prismatic joint, from the perspective of singularity theory. The study begins by deriving the forward kinematics using the Denavit-Hartenberg (D-H) convention and examines the Jacobian matrices to identify configurations where the Jacobian matrix becomes rank-deficient, signaling the presence of kinematic singularities. These singularities pose critical challenges, such as restricting end-effector mobility and leading to infinite solutions in inverse kinematics. The determinant of the Jacobian matrix is employed to detect singular configurations, and the implications for motion control and trajectory planning are discussed. Through a detailed analysis and MATLAB simulations, the article highlights the importance of singularity avoidance and provides a deeper understanding of the manipulator's kinematic behavior. The findings emphasize the need for strategic design and motion planning to ensure optimal performance and stability in robotic manipulation tasks
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Efimov, Danil, Anatoliy Petravchuk, and Maksym Sydorov. "Centralizers of Jacobian derivations." Algebra and Discrete Mathematics 36, no. 1 (2023): 22–31. http://dx.doi.org/10.12958/adm2186.
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Let K be an algebraically closed field of characte-ristic zero, K[x,y] the polynomial ring in variables x, y and let W2(K) be the Lie algebra of all K-derivations on K[x,y]. A derivation D∈W2(K) is called a Jacobian derivation if there exists f∈K[x,y] such that D(h)=det J(f,h) for any h∈K[x,y] (hereJ(f,h) is the Jacobian matrix for f and h). Such a derivation is denoted by Df. The kernel of Df in K[x,y] is a subalgebra K[p] where p=p(x,y) is a polynomial of smallest degree such that f(x,y)=φ(p(x,y) for some φ(t)∈K[t]. Let C=CW2(K)(Df) be the centralizer of Df in W2(K). We prove that C is the free K[p]-module of rank 1 or 2 over K[p] and point out a criterion of being a module of rank 2. These results are used to obtain a classof integrable autonomous systems of differential equations.
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Butuk, N., and J. P. Pemba. "Computing CHEMKIN Sensitivities Using Complex Variables." Journal of Engineering for Gas Turbines and Power 125, no. 3 (2003): 854–58. http://dx.doi.org/10.1115/1.1469006.
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This paper discusses an accurate numerical approach based on complex variables for the computation of the Jacobian matrix of complex chemical reaction mechanisms. The Jacobian matrix is required in the calculation of low dimensional manifolds during kinetic chemical mechanism reduction. The approach is suitable for numerical computations of large-scale problems and is more accurate than the finite difference approach of computing Jacobians. The method is demonstrated via a nonlinear reaction mechanism for the synthesis of Bromide acid and a H2/Air mechanism using a modified CHEMKIN package. The Bromide mechanism consisted of five species participating in six elementary chemical reactions and the H2/Air mechanism consisted of 11 species and 23 reactions. In both cases it is shown that the method is superior to the finite difference approach of computing derivatives with an arbitrary computational step size h.
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Yan, Yu, and Gong Ping Wu. "The Jacobian Matrix Structure of the Barrier Arm of Inspection Robot." Advanced Materials Research 706-708 (June 2013): 1183–86. http://dx.doi.org/10.4028/www.scientific.net/amr.706-708.1183.
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The Jacobian matrix represents the relationship of the linear mapping between the space velocity of robot operation and joint space velocity, which is the important link in the process of robot control. According to analyzing the gesture character of the inspection robot when it gets over the obstacles. This paper sets up the dynamitic model of robot by utilizing D-H. Based on that, it builds the Jacobian matrix by adopting differential transformation method, which establishes the foundation of the robots motion planning and real-time control.
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Rong, Xue Wen, Yi Bin Li, Jiu Hong Ruan, and Hong Jun Song. "Kinematics Analysis and Simulation of a Quadruped Robot." Applied Mechanics and Materials 26-28 (June 2010): 517–22. http://dx.doi.org/10.4028/www.scientific.net/amm.26-28.517.
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This paper presents the mechanical configuration of a quadruped robot with 4 rotary joints per leg firstly. All joints of the robot are activated by 16 identical hydraulic servo cylinders. Then it gives the forward and inverse kinematic equations of one leg with D-H transformation matrix. Furthermore, it also gives the jacobian matrix of the legs. The jacobian matrix establishes the relationships between the angle velocities of the 4 joints of the supporting leg and the moving velocity of the robot trunk. Lastly, a simple trot gait simulation is given based on MSC.ADAMS under the assumption that there are no slippages between the supporting feet and the ground.
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Yang, Fang, Lihui Yang, and Xikui Ma. "Fast-Scale Bifurcation Analysis in One-Cycle Controlled H-Bridge Inverter." International Journal of Bifurcation and Chaos 26, no. 12 (2016): 1650199. http://dx.doi.org/10.1142/s0218127416501996.
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This paper discusses the fast-scale instability in the one-cycle controlled H-bridge inverter which is widely used in practical applications. Simulations show that the fast-scale instability which occurs around the peak value of the inductive current is a type of local instability. The discrete-time model is derived to analyze the fast-scale instability. According to the Jacobian matrix derived from the discrete-time model, the theoretical analysis shows that the fast-scale instability manifests itself as a period-doubling bifurcation occurs, which reveals the intrinsic mechanism of the instability of one-cycle controlled H-bridge inverter. Furthermore, with the approximation of the matrix exponential in the discrete-time model and based on the Jacobian matrix, the stability boundary is obtained by the analytical expression which is helpful to select the proper values of parameters for avoiding the fast-scale instability. The experiments are carried out to validate the results of the simulation and theoretical analysis. It is shown that the analysis method is able to facilitate the design of the inverter.
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DE BONDT, MICHIEL, and XIAOSONG SUN. "CLASSIFICATION OF CUBIC HOMOGENEOUS POLYNOMIAL MAPS WITH JACOBIAN MATRICES OF RANK TWO." Bulletin of the Australian Mathematical Society 98, no. 1 (2018): 89–101. http://dx.doi.org/10.1017/s0004972718000345.
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Let $K$ be any field with $\text{char}\,K\neq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ whose Jacobian matrix, ${\mathcal{J}}H$, has $\text{rk}\,{\mathcal{J}}H\leq 2$. In particular, we show that, for such an $H$, if $F=x+H$ is a Keller map, then $F$ is invertible and furthermore $F$ is tame if the dimension $n\neq 4$.
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Lu, Jian, Guan Bin Gao, and Hui Ping Yang. "Structure Parameter Identification of Articulated Arm Coordinate Measuring Machines Based on the Least Squares Method." Applied Mechanics and Materials 303-306 (February 2013): 638–41. http://dx.doi.org/10.4028/www.scientific.net/amm.303-306.638.
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The AACMM is a kind of new coordinate measuring instrument based on non-Cartesian system. The errors of structure parameters are the main factors which affect the measuring accuracy, and the structural parameters identification can effectively improve the measuring accuracy of AACMM. The measuring model of an AACMM was established using the D-H method. Then the error model of the AACMM was established with total differential transforming method, from which the Jacobian matrix was obtained. The structure parameters’ errors which need to be identified through singular value decomposition of Jacobian matrix and the decomposition of orthogonal matrix elementary row transform can be determined. Finally, the least squares method was used in identifying the structure parameters’ errors of the AACMM, the results shows that the method proposed in this paper can identify the structure parameters efficiently.
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Cvetković, Dragana, Đorđe Vukelić, and Ksenija Doroslovački. "A New Subclass of H-Matrices with Applications." Mathematics 12, no. 15 (2024): 2322. http://dx.doi.org/10.3390/math12152322.
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The diagonal dominance property has been applied in many different ways and has proven to be very useful in various research areas. Its generalization, also known under the name H-matrix property, can be applied and produce significant benefits in economic theory, environmental sciences, epidemiology, neurology, engineering, etc. For example, it is known that the (local) stability of a (nonlinear) dynamic system is ensured if the (Jacobian) matrix belongs to the H-matrix class, and all its diagonal elements are negative. However, checking the H-matrix property itself is too expensive (from a computational point of view), so it is always worth investing effort in finding new subclasses of H-matrices, defined by relatively simple and practical conditions. Here, we will define a new subclass, which is closely related to the Euclidean vector norm, give some possible applications of this new class, and consider its relationship to some known subclasses.
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Meng, Guang Zhu, Ling Yu Sun, Ping Peng, Xian Chun Meng, Hong Mei Wang, and Jian Wei Zhang. "Jacobian Matrix of a Novel Continuum Robot for Search and Rescue." Applied Mechanics and Materials 303-306 (February 2013): 1695–701. http://dx.doi.org/10.4028/www.scientific.net/amm.303-306.1695.
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In this paper, a novel continuum robot for search and rescue is presented. A forward kinematic model is derived by product of exponentials formula, compare with conventional D-H method, this method is concise and simplicity. Finally, based on the differential kinematics using the chain rule, the overall Jacobian of the robot is established. This approach can be generally applied to various continuum robots, regardless of the specific actuation system used.
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Kozachok, O. Ya, and A. P. Petravchuk. "Action of derivations on polynomials and on Jacobian derivations." Researches in Mathematics 32, no. 1 (2024): 93. http://dx.doi.org/10.15421/242408.
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Let $\mathbb K$ be a field of characteristic zero, $A := \mathbb K[x_{1}, x_{2}]$ the polynomial ring and $W_2(\mathbb K)$ the Lie algebra of all $\mathbb K$-derivations on $A$. Every polynomial $f \in A$ defines a Jacobian derivation $D_f\in W_2(\mathbb K)$ by the rule $D_f(h)=\det J(f, h)$ for any $h\in A$, where $J(f, h)$ is the Jacobi matrix for $f, h$. The Lie algebra $W_2(\mathbb K)$ acts naturally on $A$ and on itself (by multiplication). We study relations between such actions from the viewpoint of Darboux polynomials of derivations from $W_2(\mathbb K)$. It is proved that for a Jordan chain $T(f_1)=\lambda f_1+f_2$, ..., $T(f_{k-1})=\lambda f_{k-1}+f_k$, $T(f_k)=\lambda f_k$ for a derivation $T\in W_2(\mathbb K)$ on $A$ there exists an analogous chain $[T,D_{f_1}]=(\lambda -\mathop{\mathrm{div}} T)D_{f_1} + D_{f_2}$, ..., $[T,D_{f_{k}}]=(\lambda -\mathop{\mathrm{div}} T)D_{f_{k}}$ in $W_2(\mathbb K)$. In case $A:=\mathbb K[x_1, \ldots , x_n]$, the action of normalizers of elements $f$ from $A$ in $W_n(\mathbb K)$ on the principal ideals $(f)$ is considered.
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Aboelnasr, Mohamed, Hussein M. Bahaa, and Ossama Mokhiamar. "Novel use of the Monte-Carlo methods to visualize singularity configurations in serial manipulators." Journal of Mechanical Engineering and Sciences 15, no. 2 (2021): 7948–63. http://dx.doi.org/10.15282/jmes.15.2.2021.02.0627.
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This paper analyses the problem of the kinematic singularity of 6 DOF serial robots by extending the use of Monte-Carlo numerical methods to visualize singularity configurations. To achieve this goal, first, forward kinematics and D-H parameters have been derived for the manipulator. Second, the derived equations are used to generate and visualize a workspace that gives a good intuition of the motion shape of the manipulator. Third, the Jacobian matrix is computed using graphical methods, aiming to locate positions that cause singularity. Finally, the data obtained are processed in order to visualize the singularity and to design a trajectory free of singularity. MATLAB robotics toolbox, Symbolic toolbox, and curve fitting toolbox are the MATLAB toolboxes used in the calculations. The results of the surface and contour plots of the determinate of the Jacobian matrix behavior lead to design a manipulator’s trajectory free of singularity and show the parameters that affect the manipulator’s singularity and its behavior in the workspace.
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Cheah, C. C., S. Kawamura, S. Arimoto, and K. Lee. "H/sub ∞/ tuning for task-space feedback control of robot with uncertain Jacobian matrix." IEEE Transactions on Automatic Control 46, no. 8 (2001): 1313–18. http://dx.doi.org/10.1109/9.940941.
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Doroslovački, Ksenija, and Dragana Cvetković. "On Matrices with Only One Non-SDD Row." Mathematics 11, no. 10 (2023): 2382. http://dx.doi.org/10.3390/math11102382.
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The class of H-matrices, also known as Generalized Diagonally Dominant (GDD) matrices, plays an important role in many areas of applied linear algebra, as well as in a wide range of applications, such as in dynamical analysis of complex networks that arise in ecology, epidemiology, infectology, neurology, engineering, economy, opinion dynamics, and many other fields. To conclude that the particular dynamical system is (locally) stable, it is sufficient to prove that the corresponding (Jacobian) matrix is an H-matrix with negative diagonal entries. In practice, however, it is very difficult to determine whether a matrix is a non-singular H-matrix or not, so it is valuable to investigate subclasses of H-matrices which are defined by relatively simple and practical criteria. Many subclasses of H-matrices have recently been discussed in detail demonstrating the many benefits they can provide, though one particular subclass has not been fully exploited until now. The aim of this paper is to attract attention to this class and discuss its relation with other more investigated classes, while showing its main advantage, based on its simplicity and elegance. This new approach, which we are presenting in this paper, will be compared with the existing ones, in three possible areas of applications, spectrum localization; maximum norm estimation of the inverse matrix in the point, as well as the block case; and error estimation for LCP problems. The main conclusion is that the importance of our approach grows with the matrix dimension.
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Zhao, Xin, Qi Fen Li, Jing Jing Xu, et al. "Hydraulic Calculation Principle and Todini Formulation Algorithm Analysis." Applied Mechanics and Materials 675-677 (October 2014): 927–33. http://dx.doi.org/10.4028/www.scientific.net/amm.675-677.927.
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The Todini formulation is effectively a type of Q–H equation formulation and it has an extremely efficient approach to the inversion of the Jacobian matrix by partitioning the governing equations in a smart way. The advantage of the Todini formulation is that the Jacobian matrices are symmetric. The aim of this paper is to study the hydraulic calculation principle and analyze the relationship between physical parameters and hydraulic calculation for each step. The comprehensive resistance factor (r) is affected by the loss coefficient along the tracks (λ), and the loss coefficient along the tracks (λ) is affected by the Reynolds number (Re), the Reynolds number (Re) is affected by the kinematic viscosity (), and the kinematic viscosity () is affected by the temperature (T). The coupling of hydraulic characteristics and thermal properties of the fluid can be got by modifying the physical parameters, and this article provided evidences for the further study.
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Yu, Xiaodong, Rui Ding, Jingbo Shao, and Xiaohui Li. "Hyperspectral Remote Sensing Image Feature Representation Method Based on CAE-H with Nuclear Norm Constraint." Electronics 10, no. 21 (2021): 2667. http://dx.doi.org/10.3390/electronics10212667.
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Due to the high dimensionality and high data redundancy of hyperspectral remote sensing images, it is difficult to maintain the nonlinear structural relationship in the dimensionality reduction representation of hyperspectral data. In this paper, a feature representation method based on high order contractive auto-encoder with nuclear norm constraint (CAE-HNC) is proposed. By introducing Jacobian matrix in the CAE of the nuclear norm constraint, the nuclear norm has better sparsity than the Frobenius norm and can better describe the local low dimension of the data manifold. At the same time, a second-order penalty term is added, which is the Frobenius norm of the Hessian matrix expressed in the hidden layer of the input, encouraging a smoother low-dimensional manifold geometry of the data. The experiment of hyperspectral remote sensing image shows that CAE-HNC proposed in this paper is a compact and robust feature representation method, which provides effective help for the ground object classification and target recognition of hyperspectral remote sensing image.
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Lu, Jian, Guan Bin Gao, and Hui Ping Yang. "Error Modeling and Analysis of Articulated Arm Coordinate Measuring Machines." Advanced Materials Research 798-799 (September 2013): 464–67. http://dx.doi.org/10.4028/www.scientific.net/amr.798-799.464.
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The Articulated Arm Coordinate Measuring Machine (AACMM) is a new type of non-orthogonal system precision instrument with the advantages of large measuring range, small volume, low weight and portability. To improve the measurement accuracy of AACMMs, an error analysis and calibration method for AACMMs is proposed. The kinematic model of the AACMM was established with D-H model, and then the error model of the AACMM was established on the basis of kinematic model with total differential transforming method and the singular value decomposition of Jacobian matrix and the decomposition of orthogonal matrix elementary row transform. Finally, the error model was validated by position error residual calculation. The error model provides a theoretical foundation for calibration and compensation of the AACMM.
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Sulaeman, Erwin, S. M. Afzal Hoq, Abdurahim Okhunov, and Marwan Badran. "Trilinear Finite Element Solution of Three Dimensional Heat Conduction Partial Differential Equations." International Journal of Engineering & Technology 7, no. 4.36 (2018): 379. http://dx.doi.org/10.14419/ijet.v7i4.36.28146.
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Solution of partial differential equations (PDEs) of three dimensional steady state heat conduction and its error analysis are elaborated in the present paper by using a Trilinear Galerkin Finite Element method (TGFEM). An eight-node hexahedron element model is developed for the TGFEM based on a trilinear basis function where physical domain is meshed by structured grid. The stiffness matrix of the hexahedron element is formulated by using a direct integration scheme without the necessity to use the Jacobian matrix. To check the accuracy of the established scheme, comparisons of the results using error analysis between the present TGFEM and exact solution is conducted for various number of the elements. For this purpose, analytical solution is derived in detailed for a particular heat conduction problem. The comparison shows promising result where its convergence is approximately O(h²) for matrix norms L1, L2 and L¥.
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Lee, Po-Chih, and Jyh-Jone Lee. "On the kinematics of a new parallel mechanism with Schoenflies motion." Robotica 34, no. 9 (2015): 2056–70. http://dx.doi.org/10.1017/s0263574714002732.
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SUMMARYThis paper investigates the kinematics of one new isoconstrained parallel manipulator with Schoenflies motion. This new manipulator has four degrees of freedom and two identical limbs, each having the topology of Cylindrical–Revolute–Prismatic–Helical (C–R–P–H). The kinematic equations are derived in closed-form using matrix algebra. The Jacobian matrix is then established and the singularities of the robot are investigated. The reachable workspaces and condition number of the manipulator are further studied. From the kinematic analysis, it can be shown that the manipulator is simple not only for its construction but also for its control. It is hoped that the results of the evaluation of the two-limb parallel mechanism can be useful for possible applications in industry where a pick-and-place motion is required.
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Wang, Peng, Jixiang Li, and Yuan Zhang. "The Nonfragile Controller with Covariance Constraint for Stable Motion of Quadruped Search-Rescue Robot." Advances in Mechanical Engineering 6 (January 1, 2014): 917381. http://dx.doi.org/10.1155/2014/917381.
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The problem of a stable motion for the quadruped search-rescue robots is described as a variance constrained uncertainty in the discrete systems. According to the model structure of the quadruped search-rescue robot, the kinematics of the robot is analyzed on the basis of the D- H parameter. Each joint of the robot angular velocity is planned using the Jacobian matrix, because the angular velocity is directly related to the stability of walking based on the ADAMS simulation. The nonfragile control method with the covariance constraint is proposed for the gait motion control of the quadruped search-rescue robot. The motion state feedback controller and the covariance upper bounds can be given by the solutions of the linear matrix inequalities (LMI), which makes the system satisfy the covariance constrain theory. The results given by LMI indicate that the proposed control method is correct and effective.
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Sabatini, Luca. "Volume Comparison in the presence of a Gromov-Hausdorff ε−approximation II". Annals of West University of Timisoara - Mathematics and Computer Science 56, № 1 (2018): 99–135. http://dx.doi.org/10.2478/awutm-2018-0008.
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Abstract Let (M, g) be any compact, connected, Riemannian manifold of dimension n. We use a transport of measures and the barycentre to construct a map from (M, g) onto a Hyperbolic manifold (ℍn/Λ, g0) (Λ is a torsionless subgroup of Isom(ℍn,g0)), in such a way that its jacobian is sharply bounded from above. We make no assumptions on the topology of (M, g) and on its curvature and geometry, but we only assume the existence of a measurable Gromov-Hausdorff ε-approximation between (ℍn/Λ, g0) and (M, g). When the Hausdorff approximation is continuous with non vanishing degree, this leads to a sharp volume comparison, if $\varepsilon < {1 \over {64\,{n^2}}}\min \left( {in{j_{\left( {{{\Bbb H}^n}/\Lambda ,{g_0}} \right)}},1} \right)$ , then $$\matrix{{Vol\left( {{M^n},g} \right) \ge }\cr {{{\left( {1 + 160n\left( {n + 1} \right)\sqrt {{\varepsilon \over {\min \left( {in{j_{\left( {{{\Bbb H}^n}/\Lambda ,{g_0}} \right)}},1} \right)}}} } \right)}^{{n \over 2}}}\left| {\deg \,h} \right| \cdot Vol\left( {{X^n},{g_0}} \right).} \cr }$$
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Yang, Xing, Bin Fu, Xiaochuan Ma, Yu Liu, Dongyu Yuan, and Xintong Wu. "Asynchronously H∞ Tracking Control and Optimization for Switched Flight Vehicles with Time-Varying Delay." Aerospace 11, no. 2 (2024): 107. http://dx.doi.org/10.3390/aerospace11020107.
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The current paper verifies the asynchronous H∞ control and optimization problem for flight vehicles with a time-varying delay. The nonlinear dynamic model and Jacobian linearization establish the flight vehicle’s switched model. An asynchronous H∞ tracking controller is designed, considering the existing asynchronous switching between the controllers and corresponding subsystems. In order to promote transient efficiency, the tracking controller comprises the model-based part and the learning-based part. The model-based part guarantees stability and prescribed efficiency, and the learning-based part compensates for undesirable uncertainties. The multiple Lyapunov function (MLF) and mode-dependent average dwell time (MDADT) methods are utilized to guarantee stability and the specified attenuation efficiency. The existing conditions and the solutions of model-based sub-controllers are represented by linear matrix inequalities (LMIs). The deep Q learning (DQL) algorithm provides the learning-based part. Different from the conventional method, the controller parameters are scheduled online. Therefore, robustness, stability, and dynamic efficiency can be met simultaneously. A numerical example illustrates the efficiency and advantage of the presented approach.
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Guti�rrez J�imez, Ram�n, and Jos� A. D�az-Garc�a. "Proof of the conjectures of H. Uhlig on the singular multivariate beta and the Jacobian of a certain matrix transformation." Annals of Statistics 25, no. 5 (1997): 2018–23. http://dx.doi.org/10.1214/aos/1069362383.
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Wang, Zhiming, Lizhen Cui, Zhenglong Cai, and Changfu Pang. "Modeling Analysis and Structural Design of Human Lower Limb Rehabilitation Robot." MATEC Web of Conferences 232 (2018): 02032. http://dx.doi.org/10.1051/matecconf/201823202032.
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With the rapid development of science and technology, robots are widely used in rehabilitation training. According to the physiological structure of human lower limbs and gait characteristics of walking, a lower limb rehabilitation robot is designed in this paper. We design the structure in a form of exoskeleton with three degrees of freedom in which kinematics analysis is carried out by the D-H coordinate transformation method. And then we obtain the relationship between the end effector and the angle of each joint. In addition, the relationship between end effector speed and joint speed is obtained through Jacobian matrix and Lagrange equilibrium method is used for dynamic analysis. The joint torque is calculated through the joint speed and three dimensional modeling of lower limb rehabilitation robot was reconstructed by Pro-e. Finally, the driving mode is selected and calculated.
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Yang, Lingyu, Xiaoke Feng, Jing Zhang, and Xiangqian Shu. "Multi-Ray Modeling of Ultrasonic Sensors and Application for Micro-UAV Localization in Indoor Environments." Sensors 19, no. 8 (2019): 1770. http://dx.doi.org/10.3390/s19081770.
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Due to its payload, size and computational limits, localizing a micro air vehicle (MAV) using only its onboard sensors in an indoor environment is a challenging problem in practice. This paper introduces an indoor localization approach that relies on only the inertial measurement unit (IMU) and four ultrasonic sensors. Specifically, a novel multi-ray ultrasonic sensor model is proposed to provide a rapid and accurate approximation of the complex beam pattern of the ultrasonic sensors. A fast algorithm for calculating the Jacobian matrix of the measurement function is presented, and then an extended Kalman filter (EKF) is used to fuse the information from the ultrasonic sensors and the IMU. A test based on a MaxSonar MB1222 sensor demonstrates the accuracy of the model, and a simulation and experiment based on the T h a l e s I I MAV platform are conducted. The results indicate good localization performance and robustness against measurement noises.
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Gong, Renxi, Tao Liu, Yan Qin, Jiawei Xu, and Zhihuan Wei. "A Novel Chaos Control Strategy for a Single-Phase Photovoltaic Energy Storage Inverter." Electronics 13, no. 14 (2024): 2854. http://dx.doi.org/10.3390/electronics13142854.
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The single-phase photovoltaic energy storage inverter represents a pivotal component within photovoltaic energy storage systems. Its operational dynamics are often intricate due to its inherent characteristics and the prevalent usage of nonlinear switching elements, leading to nonlinear characteristic bifurcation such as bifurcation and chaos. In this paper, a deep investigation of a single-phase H-bridge photovoltaic energy storage inverter under proportional–integral (PI) control is made, and a sinusoidal delayed feedback control (SDFC) strategy to mitigate the nonlinear characteristics is proposed. A frequency domain mapping model of the system is established, then, by analyzing the Jacobian matrix and equilibrium points, the bifurcation diagram is formed, and finally, the stable operational domains are determined under double and triple bifurcation parameters. Through simulation experiments, the efficacy of this strategy is validated. The findings show that through the control strategy, the stable operational envelope of the inverter can be greatly expanded and the nonlinear dynamic phenomena can be notably suppressed.
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Kim, Yeoun-Jae, Joon-Yong Lee, and Ju-Jang Lee. "A Torso-Moving Balance Control Strategy for a Walking Biped Robot Subject to External Continuous Forces." International Journal of Humanoid Robotics 12, no. 01 (2015): 1550003. http://dx.doi.org/10.1142/s0219843615500036.
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Moving the torso laterally in a walking biped robot can be mechanically more torque-efficient than not moving the torso according to recent research. Motivated by this observation, a torque-efficient torso-moving balance control strategy of a walking biped robot subject to a persistent continuous external force is suggested and verified in this paper. The torso-moving balance control strategy consists of a preliminary step and two additional steps. The preliminary step (disturbance detection) is to perceive the application of an external force by a safety boundary of zero moment point, detected approximately from cheap pressure sensors. Step 1 utilizes center of gravity (COG) Jacobian, centroidal momentum matrix and linear quadratic problem calculation to shift the zero moment point to the center of the support polygon. Step 2 makes use of H∞ controllers for a more stable state shift from single support phase to double support phase. By comparing the suggested torso moving control strategy to the original control strategy that we suggested previously, a mixed balance control strategy is suggested. The strategy is verified through numerical simulation results.
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Wang, Luyao, and Guolin Liu. "A Method for Solving Ill-Conditioned Nonlinear Least Squares Problems and Its Application in Image Distortion Correction Using Self-Calibration." Axioms 13, no. 3 (2024): 209. http://dx.doi.org/10.3390/axioms13030209.
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In this study, the ill-conditioning of the iterative method for nonlinear models is discussed. Due to the effectiveness of ridge estimation for ill-conditioned problems and the lack of a combination of the H-K formula with the iterative method, the improvement of the LM algorithm is studied in this paper. Considering the LM algorithm for ill-conditioned nonlinear least squares, an improved LM algorithm based on the H-K formula is proposed for image distortion correction using self-calibration. Three finite difference methods are used to approximate the Jacobian matrix, and the H-K formula is used to calculate the damping factor in each iteration. The Brown model, quadratic polynomial model and Fourier model are applied to the self-calibration, and the improved LM algorithm is used to solve the model parameters. In the simulation experiment of space resection of a single image, we evaluate the performance of the LM algorithm based on the gain ratio (LMh) and the improved LM algorithm based on the H-K formula (LMHK), and the accuracy of different models and algorithms is compared. A ridge trace analysis is carried out on the damping factor to illustrate the effects of the improved algorithm in handling ill-conditioning. In the second experiment, the improved algorithm is applied to measure the diameter of a coin using a single camera. The experimental results show that the improved LM algorithm can reach the same or higher accuracy as the LMh algorithm, and it can weaken the ill-conditioning to a certain extent and enhance the stability of the solution. Meanwhile, the applicability of the improved LM algorithm in self-calibration is verified.
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Liu, Yiqun, Xuanxia Fan, Liang Ding, Jianfeng Wang, Tao Liu, and Haibo Gao. "Fault-Tolerant Tripod Gait Planning and Verification of a Hexapod Robot." Applied Sciences 10, no. 8 (2020): 2959. http://dx.doi.org/10.3390/app10082959.
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In some hazardous or inaccessible applications, such as earthquake rescue, as a substitute for mankind, robots are expected to perform missions reliably. Unfortunately, the failure of components is difficult to avoid due to the complexity of robot composition and the interference of the environment. Thus, improving the reliability of robots is a crucial problem. The hexapod robot has redundant degrees of freedom due to its multiple joints, making it possible to tolerate the failure of one leg. In this paper, the Fault-Tolerant Tripod (F-TT) gait dealing with the failure of one leg is researched. The Denavit–Hartenberg (D-H) method is exploited to establish a kinematic model for the hexapod robot, the Jacobian matrix is analyzed, and it is proved that the body can be controlled when three legs are supported. Then, an F-TT gait phase sequence planning method based on a stability margin is established, and a method to improve stability is proposed. The trajectory for the center of gravity (COG) and foot is studied. Finally, a simulation model and prototype robot experiments are developed, and the effectiveness of the proposed method is verified.
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Chen, Lin, and Dragomir Ž. Ðoković. "Length filtration of the separable states." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2195 (2016): 20160350. http://dx.doi.org/10.1098/rspa.2016.0350.
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We investigate the separable states ρ of an arbitrary multi-partite quantum system with Hilbert space H of dimension d . The length L ( ρ ) of ρ is defined as the smallest number of pure product states having ρ as their mixture. The length filtration of the set of separable states, S , is the increasing chain ∅ ⊊ S 1 ′ ⊆ S 2 ′ ⊆ ⋯ , where S i ′ = { ρ ∈ S : L ( ρ ) ≤ i } . We define the maximum length, L max = max ρ ∈ S L ( ρ ) , critical length, L crit , and yet another special length, L c , which was defined by a simple formula in one of our previous papers. The critical length indicates the first term in the length filtration whose dimension is equal to Dim S . We show that in general d ≤ L c ≤ L crit ≤ L max ≤ d 2 . We conjecture that the equality L crit = L c holds for all finite-dimensional multi-partite quantum systems. Our main result is that L crit = L c for the bipartite systems having a single qubit as one of the parties. This is accomplished by computing the rank of the Jacobian matrix of a suitable map having S as its range.
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Yu, Hanwen, Qun Sun, Chong Wang, and Yongjun Zhao. "Frequency response analysis of heavy-load palletizing robot considering elastic deformation." Science Progress 103, no. 1 (2019): 003685041989385. http://dx.doi.org/10.1177/0036850419893856.
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For the palletizing robot’s operating characteristics of high speed, high acceleration, and heavy load, it is necessary to make a research on the structure optimization focusing on the vibration characteristics according to the mechanical and dynamic performance analysis. This article first introduces the mechanical feature and working principle of high-speed and heavy-load robot palletizer. Kinematics analysis is carried out by using D-H parameter method, which obtains positive kinematics solution and workspace. Jacobian matrix is deduced, and the relationship between joint space and Cartesian space is established. Second, for the reason that joint flexibility has a great influence on the vibration performance of the robot, a rigid–flexible coupling dynamic model is established based on the simplified model of the flexible reducer and Lagrange’s second equation to describe the joint flexibility of high-speed and heavy-load palletizing robot, and the vibration modes of the robot are analyzed. The influence of different joint stiffness on the frequency response of the system reveals the inherent properties of the heavy-load palletizing robot, which provides a theoretical basis for the optimal design and control of the heavy-load palletizing robot.
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Rigatos, Gerasimos, Dimitrios Serpanos, Vasilios Siadimas, et al. "Condition monitoring for the quadruple water tank system using H-infinity Kalman Filtering." MATEC Web of Conferences 188 (2018): 05008. http://dx.doi.org/10.1051/matecconf/201818805008.
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The problem of statistical fault diagnosis for the quadruple watertanks system is examined. The solution of the fault diagnosis problem for the dynamic model of the four-water tanks system is a non-trivial case, due to nonlinearities and the system’s multivariable structure. In the article’s approach, the system’s dynamic model undergoes first approximate linearization around a temporary operating point which is recomputed at each sampling period. The linearization procedure relies on Taylor series expansion and on the computation of the Jacobian matrices of the state-space description of the system. The H-infinity Kalman Filter is used as a robust state estimator for the approximately linearized model of the quadruple water tanks system. By comparing the outputs of the H-infinity Kalman Filter against the outputs measured from the real water tanks system the residuals sequence is generated. It is concluded that the sum of the squares of the residuals’ vectors, being weighted by the inverse of the associated covariance matrix, stands for a stochastic variable that follows the χ2 distribution. As a consequence, a statistical method for condition monitoring of the quadruple water tanks system is drawn, by using the properties of the χ2 distribution and the related confidence intervals. Actually, normal functioning can be ensured as long as the value of the aforementioned stochastic variable stays within the previously noted confidence intervals. On the other side, one can infer the malfunctioning of the quadruple water tanks system with a high level of certainty (e.g. of the order of 96% to 98%), when these confidence intervals are exceeded. The article’s method allows also for fault isolation, that is for identifying the specific component of the quadruple water tanks system that has been subject to fault or cyber-attack.
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Idà, Edoardo, Federico Nanetti, and Giovanni Mottola. "An Alternative Parallel Mechanism for Horizontal Positioning of a Nozzle in an FDM 3D Printer." Machines 10, no. 7 (2022): 542. http://dx.doi.org/10.3390/machines10070542.
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In 3D printer design, special care must be taken when choosing the print head positioning mechanism. Indeed, this choice has a significant influence on the manufacturing accuracy, printing speed, workspace characteristics, and total cost of the printer. Considering 3D printers with layer-based processes, many designs include two stages: a planar mechanism for positioning the nozzle on a horizontal plane and a linear mechanism for the vertical build–plate motion. From the literature, two designs are usually applied for horizontal motion, commercially known as “CoreXY” and “H-bot”. Their load distribution characteristics are compared here: it is found that both have significant drawbacks. Therefore, an alternative architecture, called “CoreH-bot,” is introduced to overcome such limitations; this mechanism is both fully planar, which greatly simplifies its design and assembly phases while increasing part life, and has low unbalanced torques during motion, which increases the maximum speed for the given accuracy. The CoreH-bot kinematic equations are analyzed to define the Jacobian matrix and the corresponding workspace. The static and dynamic analyses are also performed. A prototype with this architecture has been designed that shows interesting capabilities in terms of print speed, while being both simple and cost-effective to assemble.
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Singh, Amanpreet, Ekta Singla, Sanjeev Soni, and Ashish Singla. "Kinematic modeling of a 7-degree of freedom spatial hybrid manipulator for medical surgery." Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine 232, no. 1 (2017): 12–23. http://dx.doi.org/10.1177/0954411917741331.
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The prime objective of this work is to deal with the kinematics of spatial hybrid manipulators. In this direction, in 1955, Denavit and Hartenberg proposed a consistent and concise method, known as D-H parameters method, to deal with kinematics of open serial chains. From literature review, it is found that D-H parameter method is widely used to model manipulators consisting of lower pairs. However, the method leads to ambiguities when applied to closed-loop, tree-like and hybrid manipulators. Furthermore, in the dearth of any direct method to model closed-loop, tree-like and hybrid manipulators, revisions of this method have been proposed from time-to-time by different researchers. One such kind of revision using the concept of dummy frames has successfully been proposed and implemented by the authors on spatial hybrid manipulators. In that work, authors have addressed the orientational inconsistency of the D-H parameter method, restricted to body-attached frames only. In the current work, the condition of body-attached frames is relaxed and spatial frame attachment is considered to derive the kinematic model of a 7-degree of freedom spatial hybrid robotic arm, along with the development of closed-loop constraints. The validation of the new kinematic model has been performed with the help of a prototype of this 7-degree of freedom arm, which is being developed at Council of Scientific & Industrial Research–Central Scientific Instruments Organisation Chandigarh to aid the surgeon during a medical surgical task. Furthermore, the developed kinematic model is used to develop the first column of the Jacobian matrix, which helps in providing the estimate of the tip velocity of the 7-degree of freedom manipulator when the first joint velocity is known.
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Wang, Wenjie, Qing Tao, Xiaohua Wang, Yuting Cao, and Congcong Chen. "Denim-fabric-polishing robot size optimization based on global spatial dexterity." Mechanical Sciences 12, no. 1 (2021): 649–60. http://dx.doi.org/10.5194/ms-12-649-2021.
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Abstract. This paper presents a novel method to make denim-fabric-polishing robots perform their primary task flexibly and efficiently within a limited workspace. Link lengths are optimized based on an adaptive fireworks algorithm to improve the comprehensive dexterity index. A forward kinematics analysis of the denim-fabric-polishing robot is conducted via the D–H method; the workspace is analyzed according to the needs at hand to determine the range of motion of each joint. To solve the movement condition number of the Jacobian matrix, the concept of low-condition-number probability is established, and a comprehensive dexterity indicator is constructed. The influence of the robot's size on the condition number and comprehensive dexterity index is determined. Finally, the adaptive fireworks algorithm is used to establish the objective optimization function by integrating the dexterity index and other performance indicators. The optimization results show that when the comprehensive dexterity index is taken as the optimization objective, the dexterity comprehensive index and other performance indices of the robot are the lowest; that is, the robot is more flexible. Compared with the traditional genetic algorithm and particle swarm algorithm, the adaptive fireworks algorithm proposed in this paper has better solving speed and solving precision. The optimized workspace of the robot meets the requirements of the polishing task. The design also yields a sufficiently flexible, efficient, and effective robot.
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Mampi Saha. "On Fractional Epidemic Model Order Shigella." Communications on Applied Nonlinear Analysis 32, no. 9s (2025): 2478–86. https://doi.org/10.52783/cana.v32.4541.
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Introduction: We have adapted the continuous mathematical framework developed to examine the dynamics of a Shigella epidemic at a constant recruitment rate. They divided the population into seven groups in their model: susceptible (S), vaccination(V), exposé(E), infected (I), isolated (G), hospitalized(H) and recovered (R), each with its own set of parameters. We examined a mathematical model of a Shigella outbreak in a community with a constant population using the SVEIGHR compartmental nonlinear deterministic approach. The model was subjected to analytical investigations utilizing the linearized stability method. The greatest eigenvalue of the next-generation matrix yields the fundamental reproductive number , which controls the spread of the disease. The generalized Routh-Hurwitz and Jacobian criterion are used to determine the threshold value . The model is unstable if, , and stable if, . Additionally, we determine the endemic and disease-free equilibrium points, which are helpful for a faster recovery. We have created graphs in Matlab to provide a more accurate model representation. Objectives: To describe the disease Shigella with the help of SVEIGHER compartmental nonlinear deterministic approach through factorial differential method and compare it to classical differential method and see the results. Results: From the above studies we conclude that fractional differential method is more effective than classical differential method. Conclusion : In this paper, we discuss SVEIGHR epidemic model for disease Shigella . This SVEIGHR model controlss the spreading of the disease in Human population. By using Routh-Hurwitz Criteria we find all the eigen values for endemic point are negative which shows that the above model SVEIGHR (Susceptible Vacation Infectious Isolated Hospitalization Recovered) is stable. Extending our work, we can also use harmonic mean type incidence rate for better stability and control the disease.
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Ding, Yuxuan, Kefeng Li, Guangyuan Zhang, et al. "Multi-step depth enhancement refine network with multi-view stereo." PLOS ONE 20, no. 2 (2025): e0314418. https://doi.org/10.1371/journal.pone.0314418.
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This paper introduces an innovative multi-view stereo matching network—the Multi-Step Depth Enhancement Refine Network (MSDER-MVS), aimed at improving the accuracy and computational efficiency of high-resolution 3D reconstruction. The MSDER-MVS network leverages the potent capabilities of modern deep learning in conjunction with the geometric intuition of traditional 3D reconstruction techniques, with a particular focus on optimizing the quality of the depth map and the efficiency of the reconstruction process.Our key innovations include a dual-branch fusion structure and a Feature Pyramid Network (FPN) to effectively extract and integrate multi-scale features. With this approach, we construct depth maps progressively from coarse to fine, continuously improving depth prediction accuracy at each refinement stage. For cost volume construction, we employ a variance-based metric to integrate information from multiple perspectives, optimizing the consistency of the estimates. Moreover, we introduce a differentiable depth optimization process that iteratively enhances the quality of depth estimation using residuals and the Jacobian matrix, without the need for additional learnable parameters. This innovation significantly increases the network’s convergence rate and the fineness of depth prediction.Extensive experiments on the standard DTU dataset (Aanas H, 2016) show that MSDER-MVS surpasses current advanced methods in accuracy, completeness, and overall performance metrics. Particularly in scenarios rich in detail, our method more precisely recovers surface details and textures, demonstrating its effectiveness and superiority for practical applications.Overall, the MSDER-MVS network offers a robust solution for precise and efficient 3D scene reconstruction. Looking forward, we aim to extend this approach to more complex environments and larger-scale datasets, further enhancing the model’s generalization and real-time processing capabilities, and promoting the widespread deployment of multi-view stereo matching technology in practical applications.
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Deilami, Sara. "Online Coordination of Plug-In Electric Vehicles Considering Grid Congestion and Smart Grid Power Quality." Energies 11, no. 9 (2018): 2187. http://dx.doi.org/10.3390/en11092187.
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This paper first introduces the impacts of battery charger and nonlinear load harmonics on smart grids considering random plug-in of electric vehicles (PEVs) without any coordination. Then, a new centralized nonlinear online maximum sensitivity selection-based charging algorithm (NOL-MSSCA) is proposed for coordinating PEVs that minimizes the costs associated with generation and losses considering network and bus total harmonic distortion (THD). The aim is to first attend the high priority customers and charge their vehicles as quickly as possible while postponing the service to medium and low priority consumers to the off-peak hours, considering network, battery and power quality constraints and harmonics. The vehicles were randomly plugged at different locations during a period of 24 h. The proposed PEV coordination is based on the maximum sensitivity selection (MSS), which is the sensitivity of losses (including fundamental and harmonic losses) with respect to the PEV location (PEV bus). The proposed algorithm uses the decoupled harmonic power flow (DHPF) to model the nonlinear loads (including the PEV chargers) as current harmonic sources and computes the harmonic power losses, harmonic voltages and THD of the smart grid. The MSS vectors are easily determined using the entries of the Jacobian matrix of the DHPF program, which includes the spectrums of all injected harmonics by nonlinear electric vehicle (EV) chargers and nonlinear industrial loads. The sensitivity of the objective function (fundamental and harmonic power losses) to the PEVs were then used to schedule PEVs accordingly. The algorithm successfully controls the network THDv level within the standard limit of 5% for low and moderate PEV penetrations by delaying PEV charging activities. For high PEV penetrations, the installation of passive power filters (PPFs) is suggested to reduce the THDv and manage to fully charge the PEVs. Detailed simulations considering random and coordinated charging were performed on the modified IEEE 23 kV distribution system with 22 low voltage residential networks populated with PEVs that have nonlinear battery chargers. Simulation results are provided without/with filters for different penetration levels of PEVs.
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Chatters, A. W. "Representation of tiled matrix rings as full matrix rings." Mathematical Proceedings of the Cambridge Philosophical Society 105, no. 1 (1989): 67–72. http://dx.doi.org/10.1017/s0305004100001365.
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It can be very difficult to determine whether or not certain rings are really full matrix rings. For example, let p be an odd prime, let H be the ring of quaternions over the integers localized at p, and setThen T is not presented as a full matrix ring, but there is a subring W of H such that T ≅ M2(W). On the other hand, if we take H to be the ring of quaternions over the integers and form T as above, then it is not known whether T ≅ M2(W) for some ring W. The significance of p being an odd prime is that H/pH is a full 2 x 2 matrix ring, whereas H/2H is commutative. Whether or not a tiled matrix ring such as T above can be re-written as a full matrix ring depends on the sizes of the matrices involved in T and H/pH. To be precise, let H be a local integral domain with unique maximal ideal M and suppose that every one-sided ideal of H is principal. Then H/M ≅ Mk(D) for some positive integer k and division ring D. Given a positive integer n. let T be the tiled matrix ring consisting of all n x n matrices with elements of H on and below the diagonal and elements of M above the diagonal. We shall show in Theorem 2.5 that there is a ring W such that T ≅ Mn(W) if and only if n divides k. An important step in the proof is to show that certain idempotents in T/J(T) can be lifted to idempotents in T, where J(T) is the Jacobson radical of T. This technique for lifting idempotents also makes it possible to show that there are (k + n − 1)!/ k!(n−1)! isomorphism types of finitely generated indecomposable projective right T-modules (Theorem 2·10).
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Nursyarizal, Mohd Nor, Jegatheesan Ramiah, and Nallagownden Perumal. "Newton-Raphson State Estimation Solution Employing Systematically Constructed Jacobian Matrix." June 22, 2008. https://doi.org/10.5281/zenodo.1080213.
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Newton-Raphson State Estimation method using bus admittance matrix remains as an efficient and most popular method to estimate the state variables. Elements of Jacobian matrix are computed from standard expressions which lack physical significance. In this paper, elements of the state estimation Jacobian matrix are obtained considering the power flow measurements in the network elements. These elements are processed one-by-one and the Jacobian matrix H is updated suitably in a simple manner. The constructed Jacobian matrix H is integrated with Weight Least Square method to estimate the state variables. The suggested procedure is successfully tested on IEEE standard systems.
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Hu, Hao, Xinxin Li, Haesol Im, and Henry Wolkowicz. "A Semismooth Newton-Type Method for the Nearest Doubly Stochastic Matrix Problem." Mathematics of Operations Research, May 22, 2023. http://dx.doi.org/10.1287/moor.2023.1382.
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We study a semismooth Newton-type method for the nearest doubly stochastic matrix problem where the nonsingularity of the Jacobian can fail. The optimality conditions for this problem are formulated as a system of strongly semismooth functions. We show that the nonsingularity of the Jacobian does not hold for this system. By exploiting the problem structure, we construct a modified two step semismooth Newton method that guarantees a nonsingular Jacobian matrix at each iteration, and that converges to the nearest doubly stochastic matrix quadratically. Funding: This work was supported by Canadian Network for Research and Innovation in Machining Technology. The research of H. Hu, H. Im, and H. Wolkowocz was supported by The Natural Sciences and Engineering Research Council of Canada. The research of X. Li was supported by the National Natural Science Foundation of China [No. 11601183] and Natural Science Foundation for Young Scientist of Jilin Province [No. 20180520212JH].
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Hajiabadi, Mohammad Ebrahim, Mahdi Samadi, Mohammad Hassan Nikkhah, Hossein Lotfi, and Li Li. "Determining the optimal bid direction of a generation company using the gradient vector of the profit function in the network constraints of the electricity market." IET Generation, Transmission & Distribution, October 9, 2024. http://dx.doi.org/10.1049/gtd2.13280.
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AbstractOne of the primary challenges faced by generation companies (GenCos), which operate multiple generation units within the electricity market, is the determination of the optimal bid price for these units to maximize profit. This paper proposes a novel approach to ascertain the optimal bid price direction for GenCos by leveraging the gradient vector of the profit function within the constraints of the electricity market. First, the Jacobian matrix of unit profits is computed using the electricity market structural decomposition method. This matrix highlights how the profit of generation units is affected by market input parameters, including the bid prices of the units. Then, the gradient vector of the GenCos' profit function and the optimal bid price direction are derived from the Jacobian matrix. The methodology is applied to a 24‐bus IEEE network, with results validated against those from a simulation method to confirm the efficacy of the proposed approach. The simulation results show that the highest and lowest profit changes with a step increase of 0.1$/MWh are observed for GenCo 4 and GenCo 6 with values of 60.28 and 2.20 $/h, respectively. The proposed approach can be effective in the changes of bid direction of the units of a GenCo to achieve the highest possible profit.
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Bauchau, Olivier A., Alexander Epple, and Carlo L. Bottasso. "Scaling of Constraints and Augmented Lagrangian Formulations in Multibody Dynamics Simulations." Journal of Computational and Nonlinear Dynamics 4, no. 2 (2009). http://dx.doi.org/10.1115/1.3079826.
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This paper addresses practical issues associated with the numerical enforcement of constraints in flexible multibody systems, which are characterized by index-3 differential algebraic equations (DAEs). The need to scale the equations of motion is emphasized; in the proposed approach, they are scaled based on simple physical arguments, and an augmented Lagrangian term is added to the formulation. Time discretization followed by a linearization of the resulting equations leads to a Jacobian matrix that is independent of the time step size, h; hence, the condition number of the Jacobian and error propagation are both O(h0): the numerical solution of index-3 DAEs behaves as in the case of regular ordinary differential equations (ODEs). Since the scaling factor depends on the physical properties of the system, the proposed scaling decreases the dependency of this Jacobian on physical properties, further improving the numerical conditioning of the resulting linearized equations. Because the scaling of the equations is performed before the time and space discretizations, its benefits are reaped for all time integration schemes. The augmented Lagrangian term is shown to be indispensable if the solution of the linearized system of equations is to be performed without pivoting, a requirement for the efficient solution of the sparse system of linear equations. Finally, a number of numerical examples demonstrate the efficiency of the proposed approach to scaling.
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Wu, Ronghua, Xiaohong Zhang, and Wei Jiang. "Bifurcation Analysis and Chaos Control of Carrier Phase-Shifted Cascaded H-Bridge Inverter." International Journal of Bifurcation and Chaos, June 15, 2024. http://dx.doi.org/10.1142/s0218127424500949.
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Cascaded H-bridge inverters, which belong to strongly nonlinear time-varying system with complex dynamics, have been widely used in high-voltage and high-power engineering fields. In this paper, a carrier phase-shifted sinusoidal pulse-width modulation cascaded H-bridge inverter is taken as an example with piecewise analysis to establish the discrete iterative mathematical model. Furthermore, the nonlinear dynamic behavior of this inverter system is investigated using folding diagrams, spectrum diagrams, bifurcation diagrams, Lyapunov exponents, etc. We find that this inverter can exhibit multiperiod bifurcation, tangential bifurcation and paroxysmal chaos within certain ranges of control coefficients and other circuit parameters, which will influence the system’s stability seriously. On the basis of this discovery, the sinusoidal time-delay feedback control method is proposed. The difference between the load current of the controlled object and its own delay is used to form a feedback term, which is then fed into the sine function and proportional link to obtain the control term. Moreover, the Jacobian matrix of this feedback inverter system is derived, and the constraints on the feedback control coefficient are obtained based on the Jury criterion. Simulation results have verified the effectiveness of such a sinusoidal time-delay feedback control method, which can improve the stability and anti-interference of the system. The research can be used to provide some reference and guidance for the circuit design, debugging and control of cascaded H-bridge inverters.
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Chen, Ziming, Xuechan Chen, Min Gao, Chen Zhao, Kun Zhao, and Yanwen Li. "Motion Characteristics Analysis of a Novel Spherical Two-degree-of-freedom Parallel Mechanism." Chinese Journal of Mechanical Engineering 35, no. 1 (2022). http://dx.doi.org/10.1186/s10033-022-00702-7.
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AbstractCurrent research on spherical parallel mechanisms (SPMs) mainly focus on surgical robots, exoskeleton robots, entertainment equipment, and other fields. However, compared with the SPM, the structure types and research contents of the SPM are not abundant enough. In this paper, a novel two-degree-of-freedom (2DOF) SPM with symmetrical structure is proposed and analyzed. First, the models of forward kinematics and inverse kinematics are established based on D-H parameters, and the Jacobian matrix of the mechanism is obtained and verified. Second, the workspace of the mechanism is obtained according to inverse kinematics and link interference conditions. Next, rotational characteristics analysis shows that the end effector can achieve continuous rotation about an axis located in the mid-plane and passing through the rotation center of the mechanism. Moreover, the rotational characteristics of the mechanism are proved, and motion planning is carried out. A numerical example is given to verify the kinematics analysis and motion planning. Finally, some variant mechanisms can be synthesized. This work lays the foundation for the motion control and practical application of this 2DOF SPM.
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Zhao, Jing-Shan, and Han-Lin Sun. "Kinetostatics of a serial robot in screw form." Advances in Mechanical Engineering 15, no. 6 (2023). http://dx.doi.org/10.1177/16878132231180794.
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Kinetostatics of a robot is an essential component of robotic analysis that includes actuation distribution and torque control. The extant methods mainly include the static transfer formulae and force Jacobian matrix algorithms based on the Denavit-Hartenberg (D-H) method. They can calculate the joint driving torque, but there are some limitations. This paper proposes a kinetostatics approach for analyzing the driving torque of serial manipulators. The instantaneous work done by the external load is expressed with reciprocal screw. It is the key idea of the approach to derive the kinetostatics equation in screw form and determine the required torque at each joint. This methodology is straightforward for programming. Through kinetostatic analysis of some typical serial manipulators, this paper shows the process of establishing the kinetostatics of every joint to calculate the torque in Plücker coordinates. The general spatial manipulator was analyzed and the solution demonstrates the universality of the method presented in this paper. In addition, the kinetostatic analysis method applies to all kinds of serial manipulators.
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Arramy, D., J. de la Cruz Rodr'iguez, and J. Leenaarts. "Jacobian-free Newton-Krylov method for multilevel nonlocal thermal equilibrium radiative transfer problems." Astronomy & Astrophysics, August 26, 2024. http://dx.doi.org/10.1051/0004-6361/202449963.
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The calculation of the emerging radiation from a model atmosphere requires knowledge of the emissivity and absorption coefficients, which are proportional to the atomic level population densities of the levels involved in each transition. Due to the intricate inter dependence of the radiation field and the physical state of the atoms, iterative methods are required in order to calculate the atomic level population densities . A variety of different methods have been proposed to solve this problem, which is known as the n onlocal t hermodynamical e quilibrium (NLTE) problem. Our goal is to develop an efficient and rapidly converging method to solve the NLTE problem under the assumption of statistical equilibrium. In particular, we explore whether the Jacobian- F ree Newton -Krylov (JFNK) method can be used. This method does not require an explicit construction of the Jacobian matrix because it estimates the new correction with the Krylov-subspace method. We implemented an NLTE radiative transfer code with overlapping bound-bound and bound-free transitions. This solved the statistical equilibrium equations using a JFNK method, assuming a depth-stratified plane-parallel atmosphere. As a reference, we also implemented the 1992A&A...262..209R method based on linearization and operator splitting. Our tests with the Fontenla, Avrett and Loeser C model atmosphere (FAL-C) and two different six-level Ca II and H I atoms show that the JFNK method can converge faster than our reference case by up to a factor $2$. This number is evaluated in terms of the total number of evaluations of the formal solution of the radiative transfer equation for all frequencies and directions. This method can also reach a lower residual error compared to the reference case. The JFNK method we developed poses a new alternative to solving the NLTE problem. Because it is not based on operator splitting with a local approximate operator, it can improve the convergence of the NLTE problem in highly scattering cases. One major advantage of this method is that it is expected to allow for a direct implementation of more complex problems, such as overlapping transitions from different active atoms, charge conservation, or a more efficient treatment of partial redistribution, without having to explicitly linearize the equations.
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Meng, Xiangyi, and Yan Xu. "Adaptive local discontinuous Galerkin methods with semi-implicit time discretizations for the Navier-Stokes equations." Advances in Aerodynamics 4, no. 1 (2022). http://dx.doi.org/10.1186/s42774-022-00110-4.
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AbstractIn this paper, we present a mesh adaptation algorithm for the unsteady compressible Navier-Stokes equations under the framework of local discontinuous Galerkin methods coupled with implicit-explicit Runge-Kutta or spectral deferred correction time discretization methods. In both of the two high order semi-implicit time integration methods, the convective flux is treated explicitly and the viscous and heat fluxes are treated implicitly. The remarkable benefits of such semi-implicit temporal discretizations are that they can not only overcome the stringent time step restriction compared with time explicit methods, but also avoid the construction of the large Jacobian matrix as is done for fully implicit methods, thus are relatively easy to implement. To save computing time as well as capture the flow structures of interest accurately, a local mesh refinement (h-adaptive) technique, in which we present detailed criteria for selecting candidate elements and complete strategies to refine and coarsen them, is also applied for the Navier-Stokes equations. Numerical experiments are provided to illustrate the high order accuracy, efficiency and capabilities of the semi-implicit schemes in combination with adaptive local discontinuous Galerkin methods for the Navier-Stokes equations.
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Azizi, Nima, and Wolfgang Dornisch. "A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition." Computational Mechanics, February 20, 2024. http://dx.doi.org/10.1007/s00466-024-02444-w.
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AbstractA curved non-isoparametric Reissner–Mindlin shell element is developed for analyzing thin-walled structures. The standard kinematic description of the element requires the calculation of the director vector. To address this demand accurately, similar to isogeometric analysis (IGA), the geometry is defined by utilization of the non-uniform rational B-splines (NURBS) imported directly from computer-aided design (CAD) files. Then, shape functions of the Legendre spectral element method (SEM) are used to interpolate the displacements. Consequently, the shell director vector and Jacobian of the transformation are calculated properly according to the presented formulation. On the other hand, in Legendre SEM combined with Gauss–Lobatto–Legendre quadrature, the integration points and the element nodes coincide. Thus, the easily computable local coordinate systems at the integration points can be used directly as nodal basis systems. A separate calculation of nodal basis systems at control points, which is the source of either complexity or error in IGA shells, is not required. Given the condition number of the stiffness matrix in the developed method, super high-order elements can also be used. Very high order p-refined elements are used in addition to h-refinement of the mesh to show the capability of higher order elements to analyze problems without mesh refinement. The validity and convergence rate of the method are investigated and verified through various cases of h- and p-refinement in challenging obstacle course problems.
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Futai, Kouta, Niklas Kolbe, Hirofumi Notsu, and Tasuku Suzuki. "A Mass-Preserving Two-Step Lagrange–Galerkin Scheme for Convection-Diffusion Problems." Journal of Scientific Computing 92, no. 2 (2022). http://dx.doi.org/10.1007/s10915-022-01885-w.
Full textAbstract:
AbstractA mass-preserving two-step Lagrange–Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of $$L^2$$ L 2 -theory. The introduced scheme maintains the advantages of the Lagrange–Galerkin method, i.e., CFL-free robustness for convection-dominated problems and a symmetric and positive coefficient matrix resulting from the discretization. In addition, the scheme conserves the mass on the discrete level if the involved integrals are computed exactly. Unconditional stability and error estimates of second order in time are proved by employing two new key lemmas on the truncation error of the material derivative in conservative form and on a discrete Gronwall inequality for multistep methods. The mass-preserving property is achieved by the Jacobian multiplication technique introduced by Rui and Tabata in 2010, and the accuracy of second order in time is obtained based on the idea of the multistep Galerkin method along characteristics originally introduced by Ewing and Russel in 1981. For the first time step, the mass-preserving scheme of first order in time by Rui and Tabata in 2010 is employed, which is efficient and does not cause any loss of convergence order in the $$\ell ^\infty (L^2)$$ ℓ ∞ ( L 2 ) - and $$\ell ^2(H^1_0)$$ ℓ 2 ( H 0 1 ) -norms. For the time increment $$\Delta t$$ Δ t , the mesh size h and a conforming finite element space of polynomial degree $$k \in {\mathbb {N}}$$ k ∈ N , the convergence order is of $$O(\Delta t^2 + h^k)$$ O ( Δ t 2 + h k ) in the $$\ell ^\infty (L^2)\cap \ell ^2(H^1_0)$$ ℓ ∞ ( L 2 ) ∩ ℓ 2 ( H 0 1 ) -norm and of $$O(\Delta t^2 + h^{k+1})$$ O ( Δ t 2 + h k + 1 ) in the $$\ell ^\infty (L^2)$$ ℓ ∞ ( L 2 ) -norm if the duality argument can be employed. Error estimates of $$O(\Delta t^{3/2}+h^k)$$ O ( Δ t 3 / 2 + h k ) in discrete versions of the $$L^\infty (H^1_0)$$ L ∞ ( H 0 1 ) - and $$H^1(L^2)$$ H 1 ( L 2 ) -norm are additionally proved. Numerical results confirm the theoretical convergence orders in one, two and three dimensions.