Mobility and equilibrium stability analysis of pin-jointed mechanisms with equilibrium matrix SVD

Under certain load pattern, the geometrically indeterminate pin-jointed mechanisms will present certain shapes to keep static equalization. This paper proposes a matrix-based method to determine the mobility and equilibrium stability of mechanisms according to the effects of the external loads. The first and second variations of the potential energy function of mechanisms under conservative force field are analyzed. Based on the singular value decomposition (SVD) method, a new crite- rion for the mobility and equilibrium stability of mechanisms can be concluded by analyzing the equilibrium matrix. The mobility and stability of mechanisms can be classified by unified matrix formulae. A number of examples are given to demonstrate the proposed criterion. In the end, criteria are summarized in a table.

Algorithm for Detecting Volumetric Geometric Accuracy of NC Machine Tool by Laser Tracker

Quick and accurate detecting the error of NC machine tool and performing the error compensation are important to improve the machining accuracy of NC machine tool. Currently, there are many methods for detecting the geometric accuracy of NC machine tool. However, these methods have deficiencies in detection efficiency and accuracy as well as in versatility. In the paper, a method with laser tracker based on the multi-station and time-sharing measurement principle is proposed, and this method can rapidly and accurately detect the geometric accuracy of NC machine tool. The machine tool is controlled to move in the preset path in a 3D space or 2D plane, and a laser tracker is used to measure the same motion trajectory of the machine tool successively at different base stations. The original algorithm for multi-station and time-sharing measurement is improved. The space coordinates of the measuring point obtained by the laser tracker are taken as parameter values, and the initial position of each base point can be determined. The redundant equation concerning the base point calibration can be established by the distance information of the laser tracker, and the position of each base point is further determined by solving the equation with least squares method, then the space coordinates of each measuring point can be calibrated. The singular matrix does not occur in calculation with the improved algorithm, which overcomes the limitations of the original algorithm, that the motion trajectory of machine tool is in a 3D space and there exits height difference between the base stations. Adopting the improved algorithm can expand the application of multi-station and time-sharing measurement, and can meet the quick and accurate detecting requirements for different types of NC machine tool.

A local f-x Cadzow method for noise reduction of seismic data obtained in complex formations

A noise-reduction method with sliding called the local f-x Cadzow noise-reduction method, windows in the frequency-space （f-x） domain, is presented in this paper. This method is based on the assumption that the signal in each window is linearly predictable in the spatial direction while the random noise is not. For each Toeplitz matrix constructed by constant frequency slice, a singular value decomposition （SVD） is applied to separate signal from noise. To avoid edge artifacts caused by zero percent overlap between windows and to remove more noise, an appropriate overlap is adopted. Besides flat and dipping events, this method can enhance curved and conflicting events. However, it is not suitable for seismic data that contains big spikes or null traces. It is also compared with the SVD, f-x deconvolution, and Cadzow method without windows. The comparison results show that the local Cadzow method performs well in removing random noise and preserving signal. In addition, a real data example proves that it is a potential noise-reduction technique for seismic data obtained in areas of complex formations.

Research and Design of Coordinated Control Strategy for Smart Electromechanical Actuator System

In order to improve the frequency response and anti-interference characteristics of the smart electromechanical actuator(EMA)system,and aiming at the force fighting problem when multiple actuators work synchronously,a multi input multi output(MIMO)position difference cross coupling control coordinated strategy based on double‑closed-loop load feedforward control is proposed and designed.In this strategy,the singular value method of return difference matrix is used to design the parameter range that meets the requirements of system stability margin,and the sensitivity function and the H_(∞)norm theory are used to design and determine the optimal solution in the obtained parameter stability region,so that the multi actuator system has excellent synchronization,stability and anti-interference.At the same time,the mathematical model of the integrated smart EMA system is established.According to the requirements of point-to-point control,the controller of double-loop control and load feedforward compensation is determined and designed to improve the frequency response and anti-interference ability of single actuator.Finally,the 270 V high-voltage smart EMA system experimental platform is built,and the frequency response,load feedforward compensation and coordinated control experiments are carried out to verify the correctness of the position difference cross coupling control strategy and the rationality of the parameter design,so that the system can reach the servo control indexes of bandwidth 6 Hz,the maximum output force 20000 N and the synchronization error≤0.1 mm,which effectively solves the problem of force fighting.

Efficient Generalized Inverse for Solving Simultaneous Linear Equations

Solving large scale system of Simultaneous Linear Equations (SLE) has been (and continue to be) a major challenging problem for many real-world engineering and science applications. Solving SLE with singular coefficient matrices arises from various engineering and sciences applications [1]-[6]. In this paper, efficient numerical procedures for finding the generalized (or pseudo) inverse of a general (square/rectangle, symmetrical/unsymmetrical, non-singular/singular) matrix and solving systems of Simultaneous Linear Equations (SLE) are formulated and explained. The developed procedures and its associated computer software (under MATLAB [7] computer environment) have been based on “special Cholesky factorization schemes” (for a singular matrix). Test matrices from different fields of applications have been chosen, tested and compared with other existing algorithms. The results of the numerical tests have indicated that the developed procedures are far more efficient than the existing algorithms.

A Geometric View on Inner Transformation between the Variables of a Linear Regression Model

In the teaching and researching of linear regression analysis, it is interesting and enlightening to explore how the dependent variable vector can be inner-transformed into regression coefficient estimator vector from a visible geometrical view. As an example, the roadmap of such inner transformation is presented based on a simple multiple linear regression model in this work. By applying the matrix algorithms like singular value decomposition (SVD) and Moore-Penrose generalized matrix inverse, the dependent variable vector lands into the right space of the independent variable matrix and is metamorphosed into regression coefficient estimator vector through the three-step of inner transformation. This work explores the geometrical relationship between the dependent variable vector and regression coefficient estimator vector as well as presents a new approach for vector rotating.

Estimation of a Linear Model in Terms of Intra-Class Correlations of the Residual Error and the Regressors

Objectives: The objective is to analyze the interaction of the correlation structure and values of the regressor variables in the estimation of a linear model when there is a constant, possibly negative, intra-class correlation of residual errors and the group sizes are equal. Specifically: 1) How does the variance of the generalized least squares (GLS) estimator (GLSE) depend on the regressor values? 2) What is the bias in estimated variances when ordinary least squares (OLS) estimator is used? 3) In what cases are OLS and GLS equivalent. 4) How can the best linear unbiased estimator (BLUE) be constructed when the covariance matrix is singular? The purpose is to make general matrix results understandable. Results: The effects of the regressor values can be expressed in terms of the intra-class correlations of the regressors. If the intra-class correlation of residuals is large, then it is beneficial to have small intra-class correlations of the regressors, and vice versa. The algebraic presentation of GLS shows how the GLSE gives different weight to the between-group effects and the within-group effects, in what cases OLSE is equal to GLSE, and how BLUE can be constructed when the residual covariance matrix is singular. Different situations arise when the intra-class correlations of the regressors get their extreme values or intermediate values. The derivations lead to BLUE combining OLS and GLS weighting in an estimator, which can be obtained also using general matrix theory. It is indicated how the analysis can be generalized to non-equal group sizes. The analysis gives insight to models where between-group effects and within-group effects are used as separate regressors.

Discrete Time Mean-variance Analysis with Singular Second Moment Matrixes and an Exogenous Liability

We apply the dynamic programming methods to compute the analytical solution of the dynamic mean-variance optimization problem affected by an exogenous liability in a multi-periods market model with singular second moment matrixes of the return vector of assets. We use orthogonai transformations to overcome the difficulty produced by those singular matrixes, and the analytical form of the efficient frontier is obtained. As an application, the explicit form of the optimal mean-variance hedging strategy is also obtained for our model.

Construction of compressed sensing matrixes based on the singular pseudo-symplectic space over finite fields

Compressed sensing（CS） provides a new approach to acquire data as a sampling technique and makes it sure that a sparse signal can be reconstructed from few measurements. The construction of compressed matrixes is a central problem in compressed sensing. This paper provides a construction of deterministic CS matrixes, which are also disjunct and inclusive matrixes, from singular pseudo-symplectic space over finite fields of characteristic 2. Our construction is superior to De Vore＇s construction under some conditions and can be used to reconstruct sparse signals through an efficient algorithm.

ON SINGULAR MATRIX VARIATE BETA DISTRIBUTION

In this paper the density of the matrix variate beta distribution of rank lower than itsdimensionality is obtained with respect to a suitably defined differential form under the condi-tion that the difference between the identity and this matrix has full rank. As preliminaries,the Jacobian of a transformation related to decomposing a nonnegative-definite matrix into theproduct of a matrix of full column rank and its transpose and that of the transformation of anonnegative-definite matrix into its congruent matrix are established.