Analytical elastostatic stiffness modeling of parallel manipulators considering the compliance of the link and joint

This paper discusses the analytical elastostatic stiffness modeling of parallel manipulators (PMs) considering the compliance of the link and joint. The proposed modeling is implemented in three steps: (1) the limb constraint wrenches are formulated based on screw theory; (2) the strain energy of the link and the joint is formulated using material mechanics and a mapping matrix, respectively, and the concentrated limb stiffness matrix corresponding to the constraint wrenches is obtained by summing the strain energy of the links and joints in the limb; and (3) the overall stiffness matrix is assembled based on the deformation compatibility equations. The strain energy factor index (SEFI) is adopted to describe the influence of the elastic components on the stiffness performance of the mechanism. Matrix structural analysis (MSA) using Timoshenko beam elements is applied to obtain analytical expressions for the compliance matrices of different joints through a three-step process: (1) formulate the element stiffness equation for each element; (2) extend the element stiffness equation to obtain the element contribution matrix, allowing the extended overall stiffness matrix to be obtained by summing the element contribution matrices; and (3) determine the stiffness matrices of joints by extracting the node stiffness matrix from the extended overall stiffness matrix and then releasing the degrees of freedom of twist. A comparison with MSA using Euler–Bernoulli beam elements demonstrates the superiority of using Timoshenko beam elements. The 2PRU-UPR PM is presented to illustrate the effectiveness of the proposed approach. Finally, the global SEFI and scatter matrix are used to identify the elastic component with the weakest stiffness performance, providing a new approach for effectively improving the stiffness performance of the mechanism.     

有理平均逼近的新算法

有理平均逼近应比多项式逼近好,但至今研究很少.本文分析了此问题有奇性,并提出一种新算法:延拓 正则化,则以获得非常好的结果.数值实验也证实了此结论.     

A realization of constraint feasibility in a moving least squares response surface based approximate optimization

In the context of approximate optimization, the most extensively used tools are the response surface method (RSM) and the moving least squares method (MLSM). Since traditional RSMs and MLSMs are generally described by second-order polynomials, approximate optimal solutions can, at times, be infeasible in cases where highly nonlinear and/or nonconvex constraint functions are to be approximated. This paper explores the development of a new MLSM-based meta-model that ensures the constraint feasibility of an approximate optimal solution. A constraint-feasible MLSM, referred to as CF-MLSM, makes approximate optimization possible for all of the convergence processes, regardless of the multimodality/nonlinearity in the constraint function. The usefulness of the proposed approach is verified by examining various nonlinear function optimization problems.     

Global optimization of polynomial-expressed nonlinear optimal control problems with semidefinite programming relaxation

In this paper, we propose a new deterministic global optimization method for solving nonlinear optimal control problems in which the constraint conditions of differential equations and the performance index are expressed as polynomials of the state and control functions. The nonlinear optimal control problem is transformed into a relaxed optimal control problem with linear constraint conditions of differential equations, a linear performance index, and a matrix inequality condition with semidefinite programming relaxation. In the process of introducing the relaxed optimal control problem, we discuss the duality theory of optimal control problems, polynomial expression of the approximated value function, and sum-of-squares representation of a non-negative polynomial. By solving the relaxed optimal control problem, we can obtain the approximated global optimal solutions of the control and state functions based on the degree of relaxation. Finally, the proposed global optimization method is explained, and its efficacy is proved using an example of its application.     

Bernstein polynomial basis for numerical solution of boundary value problems

The purpose of this paper is to propose a computational method for the approximate solution of linear and nonlinear two-point boundary value problems. In order to approximate the solution, the expansions in terms of the Bernstein polynomial basis have been used. The properties of the Bernstein polynomial basis and the corresponding operational matrices of integration and product are utilized to reduce the given boundary value problem to a system of algebraic equations for the unknown expansion coefficients of the solution. On this approach, the problem can be solved as a system of algebraic equations. By considering a special case of the problem, an error analysis is given for the approximate solution obtained by the present method. At last, five examples are examined in order to illustrate the efficiency of the proposed method.     

Genetic Algorithm Optimization of a Six Degree of Freedom Micro Parallel Robot

In this paper a mono–objective optimum design procedure for a six–degree of freedom parallel micro robot is outlined by using optimality criterion of workspace and numerical aspects. A mono–objective optimization problem is formulated by referring to a basic performance of parallel robots. Additional objective functions can be used to extend the proposed design procedure to more general but specific design problems. A kinematic optimization was performed to maximize the workspace of the mini parallel robot. Optimization was performed using Genetic Algorithms. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variance of the spectral numbers and Newton polygons

Using the theory of the mixed Hodge structure one can define a notion of spectrum of a singularity or of a polynomial. Recently Claus Hertling proposed a conjecture about the variance of the spectrum of a singularity. Alexandru Dimca proposed a similar conjecture on polynomials. Here, we prove these two conjectures in the case of dimension 2 and when the singularity or the polynomial is Newton non-degenerated and commode.     

Optimal design of a 2 DOF parallel robot

This paper is aimed at presenting a study on the optimization of the Biglide mini parallel robot, which comprises two-degree-of-freedom (DOF) mini parallel robots with constant struts. The robot workspace is characterized and the inverse kinematics equation is obtained. In the paper, design optimization is implemented with Genetic Algorithms (GA) for optimization considering transmission quality index and workspace. Here, intended to show the advantages of using the GA, we applied it to a multicriteria optimization problem of 2 DOF mini parallel robot. Genetic algorithms (GA) are so far generally the best and most robust kind of evolutionary algorithms. A GA has a number of advantages. It can quickly scan a vast solution set. Bad proposals do not affect the end solution negatively as they are simply discarded. The obtained results have shown that the use of GA in such kind of optimization problem enhances the quality of the optimization outcome, providing a better and more realistic support for the decision maker. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

海洋表面波约化Hamilton方程的新发展:从小幅波到有限幅波的推广

海洋表面波的3-波至5-波约化Hamilton方程由于其对称多项式简化结构以及保能量等独特优点,得到广泛应用.但是,据相关近似假设,其适用范围局限于波陡很小的弱非线性波.于是进一步探讨下述推广问题: 对一定范围内的有限幅非线性波,在足够精确意义上是否也能获得具对称多项式简化结构的约化Hamilton方程？由于涉及复杂非线性强耦合,在该重要方面至今尚未取得进展.提出基于Chebyshev(切比雪夫)多项式逼近处理精确水波方程强非线性耦合的新简化途径,导出具对称多项式简化结构的新约化Hamilton方程.新结果将波数与波陡之积为小量的弱非线性情形拓广到该积直至1.035的非线性情形.分析表明,在该范围内新结果的误差不超过5%,特别,当前述积邻近于0.9时新结果给出精确结果.     

Convergence and conditioning of a Nyström method for Stokes flow in exterior three-dimensional domains

Convergence and conditioning results are presented for the lowest-order member of a family of Nyström methods for arbitrary, exterior, three-dimensional Stokes flow. The flow problem is formulated in terms of a recently introduced two-parameter, weakly singular boundary integral equation of the second kind. In contrast to methods based on product integration, coordinate transformation and singularity subtraction, the family of Nyström methods considered here is based on a local polynomial correction determined by an auxiliary system of moment equations. The polynomial correction is designed to remove the weak singularity in the integral equation and provide control over the approximation error. Here we focus attention on the lowest-order method of the family, whose implementation is especially simple. We outline a convergence theorem for this method and illustrate it with various numerical examples. Our examples show that well-conditioned, accurate approximations can be obtained with reasonable meshes for a range of different geometries.     

Multi-objective optimization of parallel manipulators using a game algorithm

In this paper, a multi-objective optimization game algorithm (MOOGA) for parallel manipulators (PMs) is proposed. The proposed algorithm considers the volume of the regular cylindrical workspace, motion/force transmission performance, and stiffness performance as objective functions. First, the distributions of the objective functions in the complete parameter space are calculated and sorted by importance. Second, game weighting factors and lower bound values are assigned to different objective functions according to the engineering requirements. Finally, after multiple rounds of gaming according to the weighting factors and lower bound values, the objective functions reach an optimal balance point and obtain a balance intersection subspace. In addition, a new comprehensive stiffness index (CSI) is proposed, that takes the coupling of the non-diagonal elements into consideration. This index decouples the linear and angular stiffness and has definite physical dimensions as well as a clear physical meaning. A Lagrangian function is used to obtain the maximum and minimum stiffnesses at a given position along with their corresponding directions. To compare the difference between the CSI and the principal diagonal stiffness index (PDSI), a divergence index κ is proposed. 2UPR–RPU and 2UPR–2RPU PMs are employed as examples to implement the proposed algorithm, where U, P and R denote a universal joint, prismatic pair and revolute pair, respectively. The corresponding slice distributions of the local CSI and κ in the regular cylindrical workspace are presented. Additionally, the distributions of the extreme linear stiffness indices and their corresponding directions are presented. The results show that the CSI is decreased by 99% relative to the PDSI. The numerical results demonstrate the effectiveness of the algorithm proposed in this paper.     

The Relationships between the Shape of the Workspace and GeometricalDimensions of Parallel Manipulators

In this paper, a new simulation program for the workspace analysis of parallel robots is presented. The system consists of modules, which enable a complex study of the parallel structures. A special method for generating the workspace of the parallel structures is used. The simulation program offers the possibility to generate automatically the virtual model of the parallel robot using the facilities of a 3D modeling software. The structure geometric parameters, which could be modified within the modeling software, influence the simulation environment. These facilities enable to establish the relationships between the workspace shape, elements links and geometrical parameters of the structure. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Studying a mathematical model of parallel computation by algebraic topology methods

Under study is one of the mathematical models of parallel computational processes, i.e., an asynchronous transition system. Some methods are proposed for calculating the homology groups and the Poincaré polynomial of a finite asynchronous transition system. Conditions are obtained for the decomposability of an asynchronous transition system as a parallel product.     

Volterra series approximation for rational nonlinear system

Rational nonlinear systems are widely used to model the phenomena in mechanics, biology, physics and engineering. However, there are no exact analytical solutions for rational nonlinear system. Hence, the approximate analytical solutions are good choices as they can give the estimation of the states for system analysis, controller design and reduction. In this paper, an approximate analytical solution for rational nonlinear system is derived in terms of the solution of a polynomial system by Volterra series theory. The rational nonlinear system is transformed to a singular polynomial system with finite terms by adding some algebraic constraints related to the rational terms. The analytical solution of singular polynomial system is approximated by the summation of the solutions of Volterra singular subsystems. Their analytical solutions are derived by a novel regularization algorithm. The first fourth Volterra subsystems are enough to approximate the analytical solution to guarantee the accuracy. Results of numerical experiments are reported to show the effectiveness of the proposed method.     

The parallel volume at large distances

In this paper we examine the asymptotic behavior of the parallel volume of planar non-convex bodies as the distance tends to infinity. We show that the difference between the parallel volume of the convex hull of a body and the parallel volume of the body itself tends to 0. This yields a new proof for the fact that a planar body can only have polynomial parallel volume if it is convex. Extensions to Minkowski spaces and random sets are also discussed.     

Isothermal flow behind a shock wave propagating in the solar wind

A Parker-type blast wave, which is headed by a strong shock, driven out by a propelling contact surface, moving into an ambient solar wind having a strictly inverse square law radial decay in density, is studied. Assuming the self-similar flow behind the shock to be isothermal, approximate analytical and exact numerical solutions are obtained. There is a good agreement between the approximate analytical and exact numerical solutions. It is observed that the mathematical singularity in density at the contact surface is removed for the isothermal flow.     

Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel

In this study, a new collocation method based on the Bernstein polynomials is introduced for the approximate solution of a class of linear Volterra integro-differential equations with weakly singular kernel. If the exact solution is polynomial, then the exact solution can be obtained. If the exact solution is not a polynomial, then an accurate solution can be obtained with a combination of choice in the number of nodes and the number of digits in the solver. In addition, the method is presented with error and stability analysis.     

A tearing-based hybrid parallel banded linear system solver

A new parallel algorithm for the solution of banded linear systems is proposed. The scheme tears the coefficient matrix into several overlapped independent blocks in which the size of the overlap is equal to the system’s bandwidth. A corresponding splitting of the right-hand side is also provided. The resulting independent, and smaller size, linear systems are solved under the constraint that the solutions corresponding to the overlap regions are identical. This results in a linear system whose size is proportional to the sum of the overlap regions which we refer to as the “balance” system. We propose a solution strategy that does not require obtaining this “balance” system explicitly. Once the balance system is solved, retrieving the rest of the solution can be realized with almost perfect parallelism. Our proposed algorithm is a hybrid scheme that combines direct and iterative methods for solving a single banded system of linear equations on parallel architectures. It has broad applications in finite-element analysis, particularly as a parallel solver of banded preconditioners that can be used in conjunction with outer Krylov iterative schemes.     

不等式约束最优化的非光滑精确罚函数的一个光滑近似

为不等式约束最优化问题提出一个连续可微近似罚函数并研究它的性质.在此基础上,提出了两个罚函数方法并证明这两个方法是全局收敛的.