Combined Expansion and Orthogonalization of Experimental Modeshapes

2004 ◽  
Vol 127 (2) ◽  
pp. 188-196 ◽  
Author(s):  
Y. Halevi ◽  
C. A. Morales ◽  
D. J. Inman
Keyword(s):  
Closed Form ◽  
Degrees Of Freedom ◽  
Optimization Problem ◽  
Model Updating ◽  
Mass Matrix ◽  
Closed Form Solution ◽  
Form Solution ◽  
Entire Structure ◽  
Error Localization ◽  
Procrustes Problem

The paper describes a method of combined expansion and orthogonalization (CEO) of experimental modeshapes. Most model updating and error localization methods require a set of full length, orthogonal with respect to the mass matrix, eigenvectors. In practically every modal experiment, the number of measurements is less than the order of the model, and hence modeshape expansion, i.e., adding the unmeasured degrees of freedom, is required. This step is then followed by orthogonalization with respect to the mass matrix. Most current methods use two separate steps for expansion and orthogonalization, each one optimal by itself, but their combination is not optimal. The suggested method combines the two steps into one optimization problem for both steps, and minimizes a quadratic criterion. In the case of an equal number of analytical and experimental modeshapes, the problem coincides with the Procrustes problem and has a closed form solution. Otherwise the solution involves nonlinear equations. Several examples show the advantage of CEO, especially in cases where the measurements are limited either in number or in space, i.e., not spanned through the entire structure.

Structure design and kinematics of a robot manipulator

Robotica ◽  
1988 ◽  
Vol 6 (4) ◽  
pp. 299-309 ◽  
Author(s):  
Kesheng Wang ◽  
Terje K. Lien
Keyword(s):  
Numerical Solution ◽  
Closed Form ◽  
Inverse Kinematics ◽  
Degrees Of Freedom ◽  
Robot Manipulator ◽  
Closed Form Solution ◽  
Form Solution ◽  
Structure Design ◽  
General Algorithm ◽  
6 Degrees Of Freedom

SUMMARYIn this paper we show that a robot manipulator with 6 degrees of freedom can be separated into two parts: arm with the first three joints for major positioning and wrist with the last three joints for major orienting. We propose 5 arms and 2 wrists as basic construction for commercially robot manipulators. This kind of simplification can lead to a general algorithm of inverse kinematics for the corresponding configuration of different combinations of arm and wrist. The approaches for numerical solution and closed form solution presented in this paper are very efficient and easy for calculating the inverse kinematics of robot manipulator.

scholarly journals Closed form solution in buckling optimization problem of twisted rod

2021 ◽  
Author(s):  
Vladimir Kobelev
Keyword(s):  
Closed Form ◽  
Cross Section ◽  
Cross Sections ◽  
Optimization Problem ◽  
Closed Form Solution ◽  
Form Solution ◽  
Optimal Shape ◽  
Actual Problem ◽  
The Cross ◽  
Simply Connected

Abstract The applications of this method for stability problems are illustrated in this manuscript. In the context of twisted rods, the counterpart for Euler’s buckling problem is Greenhill's problem, which studies the forming of a loop in an elastic bar under torsion (Greenhill, 1883). We search the optimal shape of the rod along its axis. A priori form of the cross-section remains unknown. For the solution of the actual problem the stability equations take into account all possible convex, simply connected shapes of the cross-section. Thus, we drop the assumption about the equality of principle moments of inertia for the cross-section. The cross-sections are similar geometric figures related by a homothetic transformation with respect to a homothetic center on the axis of the rod and vary along its axis. The distribution of material along the length of a twisted rod is optimized so that the rod is of the constant volume T and will support the maximal moment without spatial buckling. The cross section that delivers the maximum or the minimum for the critical eigenvalue must be determined among all convex, simply connected domains. We demonstrate at the beginning the validity of static Euler’s approach for simply supported rod (hinged), twisted by the conservative moment. The applied method for integration of the optimization criteria delivers different length and volumes of the optimal twisted rods. Instead of the seeking for the twisted rods of the fixed length and volume, we directly compare the twisted rods with the different lengths and cross-sections using the invariant factors. The solution of optimization problem for twisted rod is stated in closed form in terms of the higher transcendental functions. In the torsion stability problem, the optimal shape of cross-section is the equilateral triangle.

3D Modeling and Closed-Form Inverse Kinematics Solution for 6dof Surgical Robot

2013 ◽  
Vol 455 ◽  
pp. 533-538
Author(s):  
Edris Farah ◽  
Shao Gang Liu
Keyword(s):  
Closed Form ◽  
Inverse Kinematics ◽  
Degrees Of Freedom ◽  
Closed Form Solution ◽  
Inverse Kinematic ◽  
Form Solution ◽  
Surgical Robot ◽  
Decoupling Method ◽  
Six Degrees Of Freedom ◽  
Inverse Kinematics Problem

Since robots began to inter the medical fields, more research efforts and more attention have been given to this kind of robots. In this paper six degrees of freedom surgical robot was studied. The Denavit-Hartenberg parameters of the robot have been computed and 3D model has been built by using open source robotics toolbox. The paper also discussed a closed form solution for the inverse kinematics problem by using inverse kinematic decoupling method.

scholarly journals Existence Conditions and General Solutions of Closed-form Inverse Kinematics for Revolute Serial Robots

2019 ◽  
Vol 9 (20) ◽  
pp. 4365 ◽  
Author(s):  
Wang Shanda ◽  
Luo Xiao ◽  
Luo Qingsheng ◽  
Han Baoling
Keyword(s):  
Closed Form ◽  
Inverse Kinematics ◽  
Degrees Of Freedom ◽  
Closed Form Solution ◽  
Form Solution ◽  
Algebraic Equations ◽  
Closed Form Solutions ◽  
Serial Robots ◽  
Existence Conditions ◽  
Low Degrees

This study proposes a method for judging the existence of closed-form inverse kinematics solutions based on the Denavit–Hartenberg (DH) model. In this method, serial robots with closed-form solutions are described using three types of sub-problems from the viewpoint of solving algebraic equations. If a serial robot can be described using these three types of sub-problems, i.e., if the inverse kinematics problems can be solved by several basic problems, then there is a closed-form solution. Based on the above method, we design a set of universal closed-form inverse kinematics solving algorithms. Since there is a definite formula solution for the three types of sub-problems, the joint angles can be rapidly determined. In addition, because the DH parameters can directly reflect the linkage of the robot, the judgment of the sub-problems is also quick and accurate. More importantly, the algorithm can be applied to serial robots with low degrees of freedom. This enables the algorithm to not only quickly and accurately solve inverse kinematics problems but also to exhibit high universality. This proposed theory improves the existence conditions for closed-form reverse solutions and further promotes the development of motion control techniques for serial robots.

scholarly journals Closed-Form Solutions to Differential Equations via Differential Evolution

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
L. Mex ◽  
Carlos A. Cruz-Villar ◽  
F. Peñuñuri
Keyword(s):  
Differential Equations ◽  
Differential Evolution ◽  
Closed Form ◽  
Optimization Problem ◽  
Closed Form Solution ◽  
Form Solution ◽  
Closed Form Solutions ◽  
Independent Variables ◽  
Effectiveness And Efficiency ◽  
Derivatives Of

We focus on solving ordinary differential equations using the evolutionary algorithm known as differential evolution (DE). The main purpose is to obtain a closed-form solution to differential equations. To solve the problem at hand, three steps are proposed. First, the problem is stated as an optimization problem where the independent variables areelementaryfunctions. Second, as the domain of DE is real numbers, we propose a grammar that assigns numbers to functions. Third, to avoid truncation and subtractive cancellation errors, to increase the efficiency of the calculation of derivatives, the dual numbers are used to obtain derivatives of functions. Some examples validating the effectiveness and efficiency of our method are presented.

scholarly journals Closed-form solution for column buckling optimization

2021 ◽  
Author(s):  
Vladimir Kobelev
Keyword(s):  
Closed Form ◽  
Dimensional Analysis ◽  
Optimization Problem ◽  
Closed Form Solution ◽  
Elliptic Functions ◽  
Isoperimetric Inequalities ◽  
Form Solution ◽  
Algebraic Equations ◽  
Closed Form Solutions ◽  
Axial Forces

Abstract An optimization problem for a column, loaded by axial forces, whose direction and value remain constant, is studied in this article. The dimensional analysis introduces the dimensionless mass and rigidity factors, which simplicities the mathematical technique for the optimization problem. With the method of dimensional analysis, the solution of the nonlinear algebraic equations for the Lagrange multiplier is superfluous. The closed-form solutions for Sturm-Liouville and mixed types boundary conditions are derived. The solutions are expressed in terms of the higher transcendental function. The principal results are the closed form solution in terms of the hypergeometric and elliptic functions, the analysis of single- and bimodal regimes, and the exact bounds for the masses of the optimal columns. The proof of isoperimetric inequalities exploits the variational method and the Hölder inequality. The isoperimetric inequalities for Euler’s column are rigorously verified.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations
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