An algebraic formulation of static isotropy and design of statically isotropic 6-6 Stewart platform manipulators

In this paper, we present an algebraic method to study and design spatial parallel manipulators that demonstrate isotropy in the force and moment distributions. We use the force and moment transformation matrices separately, and derive conditions for their isotropy individually as well as in combination. The isotropy conditions are derived in closed-form in terms of the invariants of the quadratic forms associated with these matrices. The formulation is applied to a class of Stewart platform manipulator, and a multi-parameter family of isotropic manipulators is identified analytically. We show that it is impossible to obtain a spatially isotropic configuration within this family. We also compute the isotropic configurations of an existing manipulator and demonstrate a procedure for designing the manipulator for isotropy at a given configuration. (C) 2008 Elsevier Ltd. All rights reserved.

An algebraic formulation of static isotropy and design of statically isotropic 6–6 Stewart platform manipulators

In this paper, we present an algebraic method to study and design spatial parallel manipulators that demonstrate isotropy in the force and moment distributions. We use the force and moment transformation matrices separately, and derive conditions for their isotropy individually as well as in combination. The isotropy conditions are derived in closed-form in terms of the invariants of the quadratic forms associated with these matrices. The formulation is applied to a class of Stewart platform manipulator, and a multi-parameter family of isotropic manipulators is identified analytically. We show that it is impossible to obtain a spatially isotropic configuration within this family. We also compute the isotropic configurations of an existing manipulator and demonstrate a procedure for designing the manipulator for isotropy at a given configuration.

An algebraic formulation of exact force-, moment-isotropy in spatial parallel manipulators

In this paper, we present an algebraic method to study and design spatial parallel manipulators that demonstrate isotropy in the force and moment distributions.We use the force and moment transformation matrices separately,and derive conditions for their isotropy individually as well as in combination. The isotropy conditions are derived in closed-form in terms of the invariants of the quadratic forms associated with these matrices. The formulation has been applied to a class of Stewart platform manipulators. We obtain multi-parameter families of isotropic manipulator analytically. In addition to computing the isotropic configurations of an existing manipulator,we demonstrate a procedure for designing the manipulator for isotropy at a given configuration.

ENTANGLEMENT IN A 3-SPIN HEISENBERG-XY CHAIN WITH NEAREST-NEIGHBOR INTERACTIONS, SIMULATED IN AN NMR QUANTUM SIMULATOR

The evolution of entanglement in a 3-spin chain with nearest-neighbor Heisenberg-XY interactions for different initial states is investigated here. In an NMR experimental implementation, we generate multipartite entangled states starting from initial separable pseudo-pure states by simulating nearest-neighbor XY interactions in a 3-spin linear chain of nuclear spin qubits. For simulating XY interactions, we follow algebraic method of Zhang et al. Phys. Rev. A 72 (2005) 012331]. Bell state between end qubits has been generated by using only the unitary evolution of the XY Hamiltonian. For generating W-state and GHZ-state a single qubit rotation is applied on second and all the three qubits, respectively after the unitary evolution of the XY Hamiltonian.

The alpha method for solving differential algebraic inequality (DAI) systems

This paper describes an algorithm for ``direct numerical integration'' of the initial value Differential-Algebraic Inequalities (DAI) in a time stepping fashion using a sequential quadratic programming (SQP) method solver for detecting and satisfying active path constraints at each time step. The activation of a path constraint generally increases the condition number of the active discretized differential algebraic equation's (DAE) Jacobian and this difficulty is addressed by a regularization property of the alpha method. The algorithm is locally stable when index 1 and index 2 active path constraints and bounds are active. Subject to available regularization it is seen to be stable for active index 3 active path constraints in the numerical examples. For the high index active path constraints, the algorithm uses a user-selectable parameter to perturb the smaller singular values of the Jacobian with a view to reducing the condition number so that the simulation can proceed. The algorithm can be used as a relatively cheaper estimation tool for trajectory and control planning and in the context of model predictive control solutions. It can also be used to generate initial guess values of optimization variables used as input to inequality path constrained dynamic optimization problems. The method is illustrated with examples from space vehicle trajectory and robot path planning.

The Alpha-Method Direct Transcription In Path Constrained Dynamic Optimization

Numerically discretized dynamic optimization problems having active inequality and equality path constraints that along with the dynamics induce locally high index differential algebraic equations often cause the optimizer to fail in convergence or to produce degraded control solutions. In many applications, regularization of the numerically discretized problem in direct transcription schemes by perturbing the high index path constraints helps the optimizer to converge to usefulm control solutions. For complex engineering problems with many constraints it is often difficult to find effective nondegenerat perturbations that produce useful solutions in some neighborhood of the correct solution. In this paper we describe a numerical discretization that regularizes the numerically consistent discretized dynamics and does not perturb the path constraints. For all values of the regularization parameter the discretization remains numerically consistent with the dynamics and the path constraints specified in the, original problem. The regularization is quanti. able in terms of time step size in the mesh and the regularization parameter. For full regularized systems the scheme converges linearly in time step size.The method is illustrated with examples.

Application of partition method to vibration problems of plates

The “partition method” or “sub-domain method” consists of expressing the solution of a governing differential equation, partial or ordinary, in terms of functions which satisfy the boundary conditions and setting to zero the error in the differential equation integrated over each of the sub-domains into which the given domain is partitioned. In this paper, the use of this method in eigenvalue problems with particular reference to vibration of plates is investigated. The deflection of the plate is expressed in terms of polynomials satisfying the boundary conditions completely. Setting the integrated error in each of the subdomains to zero results in a set of simultaneous, linear, homogeneous, algebraic equations in the undetermined coefficients of the deflection series. The algebraic eigenvalue problem is then solved for eigenvalues and eigenvectors. Convergence is examined in a few typical cases and is found to be satisfactory. The results obtained are compared with existing results based on other methods and are found to be in very good agreement.

Application of partition method to vibration problems of plates

The “partition method” or “sub-domain method” consists of expressing the solution of a governing differential equation, partial or ordinary, in terms of functions which satisfy the boundary conditions and setting to zero the error in the differential equation integrated over each of the sub-domains into which the given domain is partitioned. In this paper, the use of this method in eigenvalue problems with particular reference to vibration of plates is investigated. The deflection of the plate is expressed in terms of polynomials satisfying the boundary conditions completely. Setting the integrated error in each of the subdomains to zero results in a set of simultaneous, linear, homogeneous, algebraic equations in the undetermined coefficients of the deflection series. The algebraic eigenvalue problem is then solved for eigenvalues and eigenvectors. Convergence is examined in a few typical cases and is found to be satisfactory. The results obtained are compared with existing results based on other methods and are found to be in very good agreement.

Regularized numerical integration of multibody dynamics with the generalized alpha method*

This paper discusses the consistent regularization property of the generalized α method when applied as an integrator to an initial value high index and singular differential-algebraic equation model of a multibody system. The regularization comes from within the discretization itself and the discretization remains consistent over the range of values the regularization parameter may take. The regularization involves increase of the smallest singular values of the ill-conditioned Jacobian of the discretization and is different from Baumgarte and similar techniques which tend to be inconsistent for poor choice of regularization parameter. This regularization also helps where pre-conditioning the Jacobian by scaling is of limited effect, for example, when the scleronomic constraints contain multiple closed loops or singular configuration or when high index path constraints are present. The feed-forward control in Kane's equation models is additionally considered in the numerical examples to illustrate the effect of regularization. The discretization presented in this work is adopted to the first order DAE system (unlike the original method which is intended for second order systems) for its A-stability and same order of accuracy for positions and velocities.

An FETI-preconditioned conjuerate gradient method for large-scale stochastic finite element problems

In the spectral stochastic finite element method for analyzing an uncertain system. the uncertainty is represented by a set of random variables, and a quantity of Interest such as the system response is considered as a function of these random variables Consequently, the underlying Galerkin projection yields a block system of deterministic equations where the blocks are sparse but coupled. The solution of this algebraic system of equations becomes rapidly challenging when the size of the physical system and/or the level of uncertainty is increased This paper addresses this challenge by presenting a preconditioned conjugate gradient method for such block systems where the preconditioning step is based on the dual-primal finite element tearing and interconnecting method equipped with a Krylov subspace reusage technique for accelerating the iterative solution of systems with multiple and repeated right-hand sides. Preliminary performance results on a Linux Cluster suggest that the proposed Solution method is numerically scalable and demonstrate its potential for making the uncertainty quantification Of realistic systems tractable.

Fractal boundaries of basin of attraction of Newton-Raphson method in helicopter trim

A key problem in helicopter aeroelastic analysis is the enormous computational time required for a numerical solution of the nonlinear system of algebraic equations required for trim, particularly when free wake models are used. Trim requires calculation of the main rotor and tail rotor controls and the vehicle attitude which leads to the six steady forces and moments about the helicopter center of gravity to be zero. An appropriate initial estimate of the trim state is needed for successful helicopter trim. This study aims to determine the control inputs that can have considerable effect on the convergence of trim solution in the aeroelastic analysis of helicopter rotors by investigating the basin of attraction of the nonlinear equations (set of initial guess points from which the nonlinear equations converge). It is illustrated that the three main rotor pitch controls of collective pitch, longitudinal cyclic pitch and lateral cyclic pitch have a significant contribution to the convergence of the trim solution. Trajectories of the Newton iterates are shown and some ideas for accelerating the convergence of a trim solution in the aeroelastic analysis of helicopters are proposed. It is found that the basins of attraction can have fractal boundaries. (C) 2010 Elsevier Ltd. All rights reserved.

An Algebraic Implicitization and Specialization of Minimum KL-Divergence Models

In this paper we study representation of KL-divergence minimization, in the cases where integer sufficient statistics exists, using tools from polynomial algebra. We show that the estimation of parametric statistical models in this case can be transformed to solving a system of polynomial equations. In particular, we also study the case of Kullback-Csiszar iteration scheme. We present implicit descriptions of these models and show that implicitization preserves specialization of prior distribution. This result leads us to a Grobner bases method to compute an implicit representation of minimum KL-divergence models.

Spherical aberration and its correction using Lie algebraic methods

We discuss three methods to correct spherical aberration for a point to point imaging system. First, results obtained using Fermat's principle and the ray tracing method are described briefly. Next, we obtain solutions using Lie algebraic techniques. Even though one cannot always obtain analytical results using this method, it is often more powerful than the first method. The result obtained with this approach is compared and found to agree with the exact result of the first method.

A simple numerical method for three-dimensional analysis of simple expansion chamber mufflers of rectangular as well as circular cross-section with a stationary medium

Three-dimensional effects are a primary source of discrepancy between the measured values of automotive muffler performance and those predicted by the plane wave theory at higher frequencies. The basically exact method of (truncated) eigenfunction expansions for simple expansion chambers involves very complicated algebra, and the numerical finite element method requires large computation time and core storage. A simple numerical method is presented in this paper. It makes use of compatibility conditions for acoustic pressure and particle velocity at a number of equally spaced points in the planes of the junctions (or area discontinuities) to generate the required number of algebraic equations for evaluation of the relative amplitudes of the various modes (eigenfunctions), the total number of which is proportional to the area ratio. The method is demonstrated for evaluation of the four-pole parameters of rigid-walled, simple expansion chambers of rectangular as well as circular cross-section for the case of a stationary medium. Computed values of transmission loss are compared with those computed by means of the plane wave theory, in order to highlight the onset (cutting-on) of various higher order modes and the effect thereof on transmission loss of the muffler. These are also compared with predictions of the finite element methods (FEM) and the exact methods involving eigenfunction expansions, in order to demonstrate the accuracy of the simple method presented here.

Hardware Accelerator for 3D Method of Moments based Parasitic Extraction

A Field Programmable Gate Array (FPGA) based hardware accelerator for multi-conductor parasitic capacitance extraction, using Method of Moments (MoM), is presented in this paper. Due to the prohibitive cost of solving a dense algebraic system formed by MoM, linear complexity fast solver algorithms have been developed in the past to expedite the matrix-vector product computation in a Krylov sub-space based iterative solver framework. However, as the number of conductors in a system increases leading to a corresponding increase in the number of right-hand-side (RHS) vectors, the computational cost for multiple matrix-vector products present a time bottleneck, especially for ill-conditioned system matrices. In this work, an FPGA based hardware implementation is proposed to parallelize the iterative matrix solution for multiple RHS vectors in a low-rank compression based fast solver scheme. The method is applied to accelerate electrostatic parasitic capacitance extraction of multiple conductors in a Ball Grid Array (BGA) package. Speed-ups up to 13x over equivalent software implementation on an Intel Core i5 processor for dense matrix-vector products and 12x for QR compressed matrix-vector products is achieved using a Virtex-6 XC6VLX240T FPGA on Xilinx's ML605 board.