Condition Number Of Jacobian Matrix Research Articles

An analysis and solution of ill-conditioning in physics-informed neural networks

Physics-informed neural networks (PINNs) have recently emerged as a novel and popular approach for solving forward and inverse problems involving partial differential equations (PDEs). However, ensuring stable training and obtaining accurate results remain challenging in many scenarios, often attributed to the ill-conditioning of PINNs. Despite this, a deeper analysis is still lacking, which hampers progress and application of PINNs in complex engineering problems. Drawing inspiration from the ill-conditioning analysis in traditional numerical methods, we establish a strong connection between the ill-conditioning of PINNs and the Jacobian matrix of the PDE system. Specifically, for any given PDE system, we construct a controlled system that allows for the adjustment of the Jacobian matrix's condition number while retaining the same solution as the original system. Our numerical experiments show that as the condition number of the Jacobian matrix decreases, PINNs exhibit faster convergence and higher accuracy. Building upon this principle and the extension of controlled systems, we propose a general approach to mitigate the ill-conditioning in PINNs, leading to successful simulations of three-dimensional flow around the M6 wing at a Reynolds number of 5,000. To the best of our knowledge, this is the first time that PINNs have successfully simulated such complex systems, offering a promising new technique for addressing industrial complexity problems. Our findings also provide valuable insights to guide the future development of PINNs.

An adaptive singularity avoidance method for five-axis toolpath generation with invariant cutter-contact points and bounded tool orientation variation based on the reduction of Jacobian condition number

To enhance the five-axis machine-tool performance, it is significant to avoid the singular configuration where the kinematic and dynamic performances degrade dramatically. Most existing methods aim at solving this issue by avoiding the singularity posture in the geometric manner. Different from them, this paper aims at avoiding the singular phenomenon by reducing the Jacobian-matrix condition number, and proposes an adaptive singularity avoidance toolpath generation method. This method adds a damping diagonal matrix into the computation of the pseudo-inverse Jacobian matrix to reduce its ill-condition extent during axial path generation, thus solving the singularity phenomenon. However, the adding of damping matrix gives arise to the toolpath error. To ensure the accuracy meanwhile, an adaptive law for determination of the damping matrix is presented to balance the singularity-avoiding effect and the path error, under the constraints of condition number and tool-orientation error-tolerance thresholds, as well as invariant cutter-contact points. Simulation and experimental tests verify that the proposed method can not only realize the singular avoidance without preliminary adjustment of the original toolpath, thus can be realized in real-time NC system, but also has better singular-avoiding performance and lower toolpath error when comparing with typical methods.

Research on Power Flow Calculation of Distribution Network with Distributed Power Based on Optimal Multiplier

With a large number of distributed power connected to the grid, there are many new node types in the distribution network. The traditional power flow calculation method cannot be applied to the distribution network with multiple distributed generations. Three kinds of distributed power grid model such as synchronous machine, asynchronous machine and inverter are established, according to the characteristics of different distributed power. The P0 node model is proposed for nodes with reactive power output as active power output and terminal voltage function. To improve the condition number of Jacobian matrix included P0 node model, the optimal multiplier is introduced to correct Jacobian matrix. Through the simulation of a variety of operation mode in certain district grid, the results show that the introduction of distributed generation has effects on the process of power flow calculation. The optimal multiplier can effectively improve the ill-posed problem of power flow caused by P0 node model, and increase the fast convergence speed of power flow calculation.

Link Lengths Optimization Based on Multiple Performance Indexes of Anthropomorphic Manipulators

How to determine the optimal link length parameters according to task requirements and constraints is a key problem in the design of anthropomorphic manipulators. Considering multiple performance indexes such as dexterity and end stiffness, this article presents a method on how to maximally improve the comprehensive motion performance of anthropomorphic manipulators by optimizing the link lengths. Firstly, the joint motion ranges of human arm are obtained through analyzing the motion mechanism and the motion forms. Then the kinematics model of manipulators with the optimal configuration is established by using screw theory, and a comprehensive evaluation index for the motion performance is proposed by considering the Jacobian matrix condition number, manipulability, and end stiffness. Taking the evaluation index as the objective function, the corresponding constraints and the optimization model are established to optimize the link lengths of manipulators with both flexibility and stiffness. Through comparative analyses, it can be seen that the global comprehensive performance index value of the optimized anthropomorphic manipulators increases by about 23.97%, and the motion performance is significantly improved.

Physics-based preconditioning of Jacobian free Newton Krylov for Burgers’ equation using modified nodal integral method

The application of Nodal Integral Method (NIM) is mostly limited to the low Reynolds number problems; due to lack of suitable nonlinear solvers. This necessitates the development of advanced nonlinear solvers for higher nonlinearity. Newton’s method is a well-established solver for such equations. However, the rate of convergence in Newton methods is dependent on the initial guess. Additionally, the condition number of Jacobian matrix dictates the time required for matrix inversion. In this work, for the first time a preconditioner is developed for fluid flow problem using Modified-NIM (MNIM). This preconditioned Jacobian free Newton Krylov algorithm uses the linearized Modified-MNIM (M2NIM) as the preconditioner, resulting in better eigenvalue clustering, reducing Krylov iterations. The effectiveness of proposed method is demonstrated using 1-D Burgers' equation for larger time steps and higher Reynolds number up to 2500. The proposed preconditioner drastically reduces the spectral radius, CPU run-time and Krylov iterations.

Optimal design of kinematic performance for a novel 2R1T parallel mechanism with pantograph units

A new type of 2R1T 3-PzPxS parallel mechanism is presented in this paper to satisfy the needs of configuration innovation for multi-axis machining platform. The limb configuration with a planar pantograph unit and the 3-PzPxS mechanism configuration are proposed and optimized. The analytical solutions of the forward and inverse position for the mechanism are derived, which are convenient to subsequent motion planning and control. After obtaining the workspace of the mechanism by the boundary search algorithm, the influence of key scale parameters on the rotation capacity of the mechanism in the whole workspace is investigated, and the reasonable ranges of the scale parameters required for large rotation capacity of the mechanism are confirmed. Furthermore, based on the Jacobian matrix condition number, the global performance indices of the angular velocity and the angular acceleration of 3-PzPxS parallel mechanism are defined, and ultimately the optimal ranges of the scale parameters are determined according to these two performance indices. The above results lay a theoretical foundation for the further development of high-efficiency hybrid NC machining center.

Thumb Configuration and Performance Evaluation for Dexterous Robotic Hand Design

The force and/or motion transmissibility and the analyticity of inverse kinematics for a thumb mechanism depend on thumb configuration. This paper presents a general framework for the thumb configuration and performance evaluation in the design of dexterous robotic hand. The thumb configuration is described by the functional analysis of human thumb, and the thumb of robotic hand is generalized into 15 configurations. A performance evaluation method is proposed based on kinetostatic and dynamic dexterity as well as workspace. The kinetostatic dexterity is based on a Jacobian matrix condition number (JMCN). A dynamic dexterity measure is presented via acceleration analysis, which keeps a clear geometric meaning. The proposed method is applied to evaluate the performance of three examples, which cover thumb configurations of most existing dexterous hands. Performance evaluation results demonstrate the effectiveness of the proposed method. Using these results and the proposed performance evaluation method, meaningful design principles are presented to guide the design of the thumb configuration.

Improvement of Composite Load Modeling Based on Parameter Sensitivity and Dependency Analyses

This paper presents an effective optimization scheme for the measurement-based load modeling based on the sensitivity analysis of composite load model parameters. Each parameter of load model has different effects on its dynamic response. Moreover, some parameters are insensitive to the change of others. To estimate the dynamic interactions between parameters, their sensitivity is analyzed by using the eigenvalues of Hessian matrix used in the optimization algorithm. Also, the linear dependence between two load model parameters is then identified by examining the condition number of Jacobian matrix. With this parameter analysis, the performance of optimization process for measurement-based composite load modeling is improved by reducing the number of necessary parameters to consider. The performance of proposed method is verified with the practical data measured at a feeder in a real substation.

The Analysis on the Processing Dexterity of a 3-TPT Parallel Machine Tool

This paper took a 3-TPT Parallel Machine Tool (PMT) as the objective of research. First, the kinematics equation and the Jacobian matrix were obtained according to the kinematics analysis of PMT and the condition number of Jacobian matrix was obtained further. Then, the analysis on the processing dexterity of PKM was conducted in terms of the condition number of Jacobian matrix as measurable index. Therefore, its distribution characteristics within the workspace were extracted. The result shows that this PMT is in effective processing dexterity. The conclusions based on the analysis have an important theoretical significance to the structural design, workspace design and inspection as well as kinematics simulation of this PMT.

Algorithm for Simulator’s Road Spectrum Signal

In response to the impossibility of directly using data of the actual road show experiment to the motion simulation of simulators, due to the limited work space of simulators, this paper begins with an attempt to extract the main frequency components of original signals by spectrum analysis and allow the input signal to meet performance requirements by adjusting the phase offset corresponding to the various frequency components. The paper proceeds with the regeneration of the signals capable of reflecting the motion characteristics of the original data and meeting the performance requirements of motion simulators by selecting these phase offsets as the design variables, selecting simulator’s displacement, velocity, and acceleration as the optimization objectives and applying Genetic Algorithm toolbox of Matlab to develop the optimization model. The paper ends with the efforts to check mechanism singularity by calculating the Jacobian matrix condition number of the regenerating signals and thus obtain the regenerating signals compatible with the work space, capable of fulfilling the performance requirements of motion simulators and free of continuous region singularity.

Multi-Criteria Optimal Design of Parallel Manipulators Based on Natural Frequency

The Stewart manipulator has characteristics of low natural frequency, high cost and large size which make it difficult to obtain optimum performance with high dynamic response. The lowest natural frequency in the total workspace and average of six frequencies at home configuration of Stewart manipulator are introduced as indices to evaluate dynamic stability. Multi-criteria optimal design based on genetic algorithm (GA) was presented synthetically considering the workspace requirement, lowest natural frequency, average frequency and global dimensionally homogeneous Jacobian matrix condition number. An optimal result was obtained through standard GA using penalty function and the Pareto-optimal set was also obtained through parallel selection method.

Dimensional Synthesis of a 2UPS-UPR Parallel Machine Tool

This paper deals with the dimensional synthesis of a 2UPS-UPR parallel mechanism tool newly designed by Northeastern University. On the basis of establishing kinematics equations and obtaining Jacobian matrix, the performance index of dimensional synthesis is given which is the average of 729 values of the condition number of Jacobian matrix corresponding to 729 positions in the workspace. With MATLAB software, the effects are simulated which the structural parameters of parallel machine tools have on dimensional synthesis, their change laws are gained, and then dimensional synthesis of parallel machine tools is conducted based on these laws.

ANALYSIS OF PERFORMANCES OF PLANAR 2-DOF PARALLEL MANIPULATOR WITH ACTUATION REDUNDANCY

As the subject investigated, the planar 2-DOF parallel manipulator with actuation redundancy is treated as non-dimensional structure. The physical model of the solution space, which can reflect the scope in which the geometric parameters change, is presented. Kinematic equation is established, solution to inverse kinematics and velocity Jacobian matrix are given. The global performance indices of the manipulator are defined and calculated. The relationship between link lengths and global performance indices, such as condition number of Jacobian matrix, payload and stiffness are discussed; corresponding atlases are plotted in the physical model of the solution space. According to these atlases, the distribution rule of the three performance indices in the physical model of the solution space is concluded; the better dimension scope of each performance index is made out respectively. Finally, when the three performance indices are considered as the optimum indices simultaneously, the optimal dimension scope is obtained by synthesizing the three dimension scopes. These atlases give a possible path to optimize manipulator using kinematic performance indices. This method also provides a simple and effective way for designing parallel manipulator using computer-aided.

A real‐coded genetic optimal kinematic design of a Stewart fine tuning platform for a large radio telescope

AbstractA real‐coded genetic design methodology for an optimal kinematic Stewart platform is presented in this article. The Jacobian matrix connecting the dexterity performance of the parallel platform is first deduced, and then the condition number of Jacobian matrix is employed as the objective function to implement the optimal kinematic design of Stewart fine tuning platform for a large spherical radio telescope. A niched‐penalty approach is used to transform this optimal kinematic design problem to an unconstrained one, and a niching technique and the dynamic mutation operator are applied. A kinematic accuracy comparison of the genetically designed Stewart fine tuning platform with the quasi‐Newtonian designed platform is made. The comparison results have shown that the kinematic accuracy of the genetically designed Stewart fine tuning platform has a much higher accuracy and compact structure than that of the quasi‐Newtonian designed platform, which guarantees the implementation of high accuracy requirement of trajectory tracking and reduces the disturbance of random wind for large radio telescopes. © 2001 John Wiley & Sons, Inc.