Methods of computing weighted pseudoinverse matrices with singular weights

Three iterative processes are constructed and investigated for computing weighted pseudoinverse matrices with singular weights and ML-weighted pseudoinverse matrices. Two of them are based on the decompositions of the weighted pseudoinverse matrix with singular weights into matrix power series, and the third is a generalization of the Schulz method to nonsingular square matrices. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 150–169, September–October, 1999.     

Representations and expansions of weighted pseudoinverse matrices, iterative methods, and problem regularization. I. Positive definite weights

The paper reviews studies on the representations and expansions of weighted pseudoinverse matrices with positive definite weights and on iterative methods and regularized problems for calculation of weighted pseudoinverse matrices and weighted normal pseudosolutions. The use of these methods to solve constrained least-squares problems is examined. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 47–73, January–February 2008.     

Series expansion of weighted pseudoinverse matrices and iterative methods for calculating weighted pseudoinverse matrices and weighted normal pseudosolutions

Weighted pseudoinverse matrices are expanded into matrix power series with negative exponents and arbitrary positive parameters. Based on this expansion, iterative methods for evaluating weighted pseudoinverse matrices and weighted normal pseudosolutions are designed and analyzed. The iterative methods for weighted normal pseudosolutions are extended to solving constrained least-squares problems. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 32–62, January–February 2006.     

Expansion of weighted pseudoinverse matrices with positive definite weights into matrix power products. Iterative methods

Weighted pseudoinverse matrices with positive definite weights are expanded into matrix power products with negative exponents and arbitrary positive parameters. These expansions are used to develop and analyze iterative methods for evaluating weighted pseudoinverse matrices and weighted normal pseudosolutions and solving constrained least-squares problems. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 45–64, January–February 2007.     

Synthesis of systems of neurofunctional transformations in classification problems

Optimal synthesis of linear and nonlinear transformations is used to synthesize pattern recognition systems. The necessary and sufficient conditions for the existence of robust dichotomous linear separability of sets in feature space are represented in terms of pseudoinverse operations. The synthesis of classification systems is reduced to searching for the best nonlinear transformations of the components of the feature vector or optimal linear combinations of its components. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 47–56, May–June 2007.     

Representations and expansions of weighted pseudoinverse matrices, iterative methods, and problem regularization. II. Singular weights

The paper reviews studies on the representations and expansions of weighted pseudoinverse matrices with positive semidefinite weights and on the construction of iterative methods and regularized problems for the calculation of weighted pseudoinverses and weighted normal pseudosolutions based on these representations and expansions. The use of these methods to solve constrained least squares problems is examined. Continued from Cybernetics and Systems Analysis, 44, No. 1, 36–55 (2008). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 75–102, May–June 2008.     

Optimization of the synthesis of hyperplane clusters and neurofunctional transforms in signal classification systems

The paper continues the analysis of clustering and signal classification problems. For problems of point-set clustering in a feature space, an optimal algorithm to synthesize hyperplane clusters and an algorithm to solve a problem on linear separability of a finite point set are obtained, criteria for linear strip separability of points in a feature space into two classes are formulated, and methods of finding optimal nonlinear transform of a coordinate of the feature vector in given classes of functions and its index are developed. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 50–58, November–December 2008.     

Limiting Representations of Weighted Pseudoinverse Matrices with Positive Definite Weights. Problem Regularization

Limiting representations for weighted pseudoinverse matrices with positive definite weights are derived. It is shown that regularized problems can be constructed based on such limiting representations intended for evaluation of weighted pseudoinverse matrices and weighted normal pseudosolutions with positive definite weights. The results obtained, concerning regularization of problems on evaluation of weighted normal pseudosolutions, are employed for regularization of least-squares problems with constraints.     

Analysis and synthesis of signal classification systems by perturbing pseudoinverse and projection operations

The paper considers the synthesis of signal classification systems using the theory of perturbation of pseudoreverse and projection matrices and proposes an algorithm to select (filter) separable subsets on a finite point set from R ⁿ . Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 124–134, July–August 2009.     

Reliability analysis of computer solutions of systems of linear algebraic equations with approximate initial data

A weighted least squares problem {ie863-01} with positive definite weights M and N is considered, where A ∈ R^m×n is a rank-deficient matrix, b ∈ R^m. The hereditary, computational, and global errors of a weighted normal pseudosolution are estimated for perturbed initial data, including the case where the rank of the perturbed matrix varies. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 83–95, November–December 2008.     

机器人运动学反解中的奇异点处理

本文研究机器人运动学反解中的奇异点处理问题，给出了机器人微分运动Ｊａｃｏｂｉａｎ矩阵Ｊ（ｑ）条件数的一个上界，并在此基础上提出机器人关节速度阻尼伪逆解方法中阻尼系数的一种自适应调整方法，该方法可以保证在奇异点附近伪逆解的稳定性。     

Pseudoinverse and Projection Matrices in Problems of Synthesis of Functional Transformers

An approach is proposed to derive dependences in the form of functional transformers. Conditions of existence and the method of constructing such functional transformers as a superposition of linear and nonlinear functions from a specified class are presented. The principle of optimal synthesis of functional transformers is formulated based on the theory of disturbance of projective and pseudoinverse operators. Methods are proposed for constructing complex functional transformers of different topologies.     

广义逆在嵌入式大气数据传感系统中的应用

应用广义逆矩阵对嵌入式大气数据传感系统中的算法进行简化,提出了以Moore-Penrose广义逆矩阵为基础的动压、静压和修正参数的改进算法,并对其进行了收敛性分析,最后,应用M atlab软件进行了数字计算验证。计算结果表明:改进算法具有确定的收敛性,由于避免了迭代过程中求解逆矩阵,达到同样的精度所需要的计算时间只相当于原有算法的70%,改进算法可以满足系统的精度、可靠性和实时性要求。     

Iterative methods with different rates of convergence for calculating weighted pseudoinverse matrices and weighted normal pseudosolutions with positive definite weights

The authors develop and analyze iterative methods with different (linear, quadratic, or of p (p2) order) rates of convergence. The methods are used to calculate weighted pseudoinverse matrices with positive defined weights. To find weighted normal pseudosolutions with positive defined weights, iterative methods with a quadratic rate of convergence are developed and analyzed. The iterative methods for calculation of weighted normal pseudosolutions are used to solve least-square problems with constraints.Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 20–44, September–October 2004.     

用阻尼伪逆法控制冗余度机器人的一种新方案

提出一种带次级性能指标优化的阻尼伪逆控制方案,它是通常的阻尼伪逆解的扩展形 式.使用文中给出的方案和阻尼因子、优化因子选择法,冗余度机器人的灵活性有望得到更为有效 地利用.基于奇异值分解(SVD)方法,还提供一种高效并行算法,以利于在实际的机器人控制系统 中实施.仿真结果表明了所提方法比其它相关方法所具有的优点.     

Calculation of repeatable control strategies for kinematically redundant manipulators

A kinematically redundant manipulator is a robotic system that has more than the minimum number of degrees of freedom that are required for a specified task. Due to this additional freedom, control strategies may yield solutions which are not repeatable in the sense that the manipulator may not return to its initial joint configuration for closed end-effector paths. This paper compares two methods for choosing repeatable control strategies which minimize their distance from a nonrepeatable inverse with desirable properties. The first method minimizes the integral norm of the difference of the desired inverse and a repeatable inverse while the second method minimizes the distance of the null vectors associated with the desired and the repeatable inverses. It is then shown how the two techniques can be combined in order to obtain the advantages of both methods. As an illustrative example the pseudoinverse is approximated in a region of the joint space for a seven-degree-of-freedom manipulator.     

Application of Pseudoinverse and Projective Matrices to Studying Control, Observation, and Identification Problems

The method of perturbation of pseudoinverse matrices, developed earlier by the authors based on splitting matrices, is extended to projective matrices for their further application to identification, nonlinear regressive analysis, function approximation, and prediction problems.     

A simple derivation of the interactor matrix and its applications

An interactor matrix plays several important roles in the control system theory. In this article, we present a simple method to derive a special interactor matrix using Moore–Penrose pseudoinverse. The interactor by the proposed method has all its zeros at the origin, and has the all-pass property in the discrete-time. A systematic procedure to obtain an identity interactor, which has an lower triangular structure or has arbitrarily prespecified zeros, is also shown.     

A Fast Approach for Robot Motion Planning

This paper describes a new approach to robot motion planning that combines the end-point motion planning with joint trajectory planning for collision avoidance of the links. Local and global methods are proposed for end-point motion planning. The joint trajectory planning is achieved through a pseudoinverse kinematic formulation of the problem. This approach enables collision avoidance of the links by a fast null-space vector computation. The power of the proposed planner derives from: its speed; the good properties of the potential function for end-point motion planning; and from the simultaneous avoidance of the links collision, kinematic singularities, and local minima of the potential function. The planner is not defined over computationally expensive configuration space and can be applied for real-time applications. The planner shows to be faster than many previous planners and can be applied to robots with many degrees of freedom. The effectiveness of the proposed local and global planning methods as well as the general robot motion planning approach have been experimented using the computer-simulated robots. Some of the simulation results are included in this paper.     

基于广义马氏距离的缺损数据补值算法

在数据收集过程中数据缺损是不可避免的。如何还原这些缺损数据,成为数据挖掘研究的热点问题之一。与许多现有算法一样,基于马氏距离的缺损数据补值算法充分利用了实际数据之间的相关性,具有较好的补值效果,但它要求数据的相关性协方差矩阵可逆,使其应用范围受到了极大的限制。在改进传统主成分分析方法的基础上,利用矩阵的奇异值分解理论和Moors Pcnrosc广义逆性质,提出了广义马氏距离的概念,并运用于SOFM神经网络,结合信息嫡理论设计了基于广义马氏距离的缺损数据补值算法—GS算法。理论分析和数值仿真结果表明,广义马氏距离完全继承了马氏距离在处理相关性数据上的性能优势,新算法不仅在补值的精确度和稳定性上有很好的效果,而且适用于任意数据集合。