Damped Least-squares Solution Research Articles

Estimation of the Strain Rate Hardening of Aluminium Using an Inverse Method and Blast Loading

The dynamic behaviour of structures under blast wave loading is increasingly studied due to the rising risks of intentional and accidental explosions and the development of new protective technologies and methods. To ensure the efficiency of these protective measures, the identification of the material behaviour under blast loading is of utmost importance. This paper investigates the feasibility of the identification of the strain rate hardening of aluminium using an inverse method and blast loads. The results of a series of tests on aluminium plates loaded by means of an Explosive Driven Shock Tube are reported. A numerical study is performed to develop a model that is both in concordance with the experimental measurements and suitable for loop implementation. The 3D high-speed full-field measurements and finite element simulations are used to accurately assess the dynamic response of the plates. The identification is based on the Levenberg-Marquardt formulation for damped least-squares solution. The approach is virtually validated and its sensitivity to noise is studied using virtual measurements from finite element calculations. Then, experimental data are used for the identification. The results obtained are within the range of expected values based on literature.

Cartesian control of an advanced tractors rear hitch – damped least-squares solution

Three-point hitch is a mechanism used to connect an agricultural implement, e.g. a plow, rigidly to an agricultural tractor. Movement of a traditional three-point hitch provides only one degree of freedom. This limits the use of hitched implements and forces implement manufacturers to build additional hydraulics into the implement. In this research a tractor was instrumented to provide two rotational and two translational degrees of freedom for the rear hitch. The goal was to develop a control system which allows the operator to control the rear hitch in Cartesian coordinates. In this paper an iterative, damped least-squares (DLS) solution for the rear hitch inverse kinematic transform is presented. The method is tested on the instrumented tractor. The result is a functional control system allowing Cartesian control. The system also provides the rear hitch pose information via a virtual terminal (VT) interface.

Identification of the plastic behavior of aluminum plates under free air explosions using inverse methods and full-field measurements

This article describes an inverse method for the identification of the plastic behavior of aluminum plates subjected to sudden blast loads. The method uses full-field optical measurements taken during the first milliseconds of a free air explosion and the finite element method for the numerical prediction of the blast response. The identification is based on a damped least-squares solution according to the Levenberg–Marquardt formulation. Three different rate-dependent plasticity models are examined. First, a combined model based on linear strain hardening and the strain rate term of the Cowper–Symonds model, secondly, the Johnson–Cook model and finally, a combined model based on a bi-exponential relation for the strain hardening term and the strain rate term of the Cowper–Symonds model. A validation of the method and its sensitivity to measurement uncertainties is first provided according to virtual measurements generated with the finite element method. Next, the plastic behavior of aluminum is identified using measurements from real free air explosions obtained from a controlled detonation of C4. The results show that inverse methods can be successfully applied for the identification of the plastic behavior of metals subjected to blast waves. In addition, the material parameters identified with inverse methods enable the numerical prediction of the material’s response with increased accuracy.

Estimation of shear wave velocity profiles by the inversion of spatial autocorrelation coefficients

The subsurface shear-wave velocity (Vs) is considered to be a key parameter for site characterization and assessment of earthquake hazard because of its great influence on local ground-motion amplification. Array microtremor measurements are widely used for the estimation of shear-wave velocities. Compared to other methods such as frequency-wavenumber (f-k) methods, the spatial autocorrelation (SPAC) method requires fewer sensors and thus is relatively easier to implement and gives robust estimations of shear-wave velocity profiles for depths down to a few hundred meters. The quantity derived from observed data is the SPAC coefficient, which is a function of correlation distance, frequency and phase velocity. Generally, estimation of Vs profiles is a two stage process: Estimation of the dispersion data from the SPAC coefficients and inversion of the dispersion data for shear-wave velocity structure. In this study, instead of inverting dispersion curves, a more practical approach is used; that is, observed SPAC coefficients are directly inverted for the S-wave velocities. A synthetic case and a field data application are presented to test the potential of the inversion algorithm. We obtain an iterative damped least-squares solution with differential smoothing. The differential smoothing approach constrains the change in shear-wave velocities of the adjacent layers and thus stabilizes the inversion.

A revised relaxation-time spectrum for Fennoscandia

Summary The Fennoscandian relaxation-time spectrum (RTS), first derived by McConnell (1968), is a classic data set in studies of glacial isostatic adjustment (GIA). We outline a new method for estimating an RTS from a set of strandline data, which is based on a damped least-squares solution for spherical harmonic coefficients associated with the strandline heights. In contrast to the Hankel transform approach outlined by McConnell (1968), the method does not require interpolation or extrapolation of the data, nor does it use the assumption that peripheral deformations are zero. We begin by applying the new approach to a suite of synthetic strandlines. These synthetic calculations quantify the effect on the RTS estimates of the common assumptions of free-decay uplift, an axisymmetric Fennoscandian deformation field, and the uncertainty introduced by limited spatial and temporal sampling of this field. Recently, the accuracy of the Sauramo (1958) strandline data upon which the McConnell (1968) RTS was based has been questioned (Wolf 1996); accordingly, we apply our new approach to a set of more robust strandline data published by Donner (1964, 1969, 1980, 1995) to compute a revised RTS for Fennoscandia. At high harmonic degrees (above 50), our new RTS is characterized by weak constraints, whereas McConnell’s (1968) RTS suggests a significant reduction in relaxation times relative to the values at low degrees. This reduction was the basis for McConnell’s (1968) inference of an elastic lithosphere. In contrast to this, we conclude that the trend is an artefact of the observational data set he adopted. At lower degrees, McConnell’s (1968) relaxation-time estimates lie at the lower end of the range implied by the present analysis. To complete our study, we apply the techniques of linearized Bayesian inference to invert our newly derived RTS for mantle viscosity. We find that the RTS provides an estimate of ~ 5 × 1020 Pa s for the volumetric mean viscosity in a region extending from the base of the lithosphere to about 550 km depth. Target regions which extend from the transition zone to ~ 1200 km depth are less well constrained; however, the average viscosity from the base of the lithosphere to 1200 km depth is consistent with the classic Haskell (1935) value of 1021 Pa s for mantle viscosity. Finally, we demonstrate that viscosity models within the class inferred by Mitrovica & Forte (1997) from joint inversions of postglacial relaxation times associated with GIA and mantle convection observables simultaneously fit the revised Fennoscandian RTS.

Review of the damped least-squares inverse kinematics with experiments on an industrial robot manipulator

The goal of this paper is to present experimental results on the implementation of the damped least-squares method for the six-joint ABB IRb2000 industrial robot manipulator. A number of inverse kinematics schemes are reviewed which allow robot control through kinematic singularities. The basic scheme adopts a damped least-squares inverse of the manipulator Jacobian with a varying damping factor acting in the neighborhood of singularities. The effect of a weighted damped least-squares solution is investigated to provide user-defined accuracy capabilities along prescribed end-effector space directions. An online estimation algorithm is employed to measure closeness to singular configurations. A feedback correction error term is introduced to ensure algorithm tracking convergence and its effect on the joint velocity solution is discussed. Computational aspects are discussed in view of real-time implementation of the proposed schemes. Experimental case studies are developed to investigate manipulator performance in the case of critical end-effector trajectories passing through and near the shoulder and wrist singularities of the structure.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Symbolic Singular Value Decomposition for a PUMA Robot and Its Application to a Robot Operation Near Singularities

Singular value decomposition has received considerable atten tion in robot control during the past several years because of the fact that it may be used in solving the problem of robot operation in singular configurations. However, its main draw back is its high computational complexity when compared with that of the Gaussian elimination or analytical inverse kinematic solution. In this article we have derived symbolic expressions for matrices of the decomposition for both a 2-DOF planar manipulator and a 6-DOF PUMA robot. This reduces the com putational burden by an order of magnitude and gives us more insight into the nature of the singularities. The symbolic ex pressions have been applied to obtain the damped least-squares solution for joint velocities. Some modifications of the damped least-squares solution are proposed, and the simulation results are included. The method can be applied to other robots when position and orientation problems can be separated.

Resolving redundancy through a weighted damped least-squares solution

Resolving redundancy through a weighted damped least-squares solution

Resolving Redundancy Through a Weighted Damped Least-Squares Solution

Singularity robust redundancy resolution with task priority can be implemented using the extended Jacobian technique with weighted damped least-squares. The resulting scheme is simple to implement and involves less computation than the task priority scheme. The minimum singular value of the Jacobian can be estimated reliably and accurately with little computation, and this estimate was used to calculate an appropriate damping factor. A constant damping factor was also used with good results. The scheme was successfully implemented in a simulation study with a seven-joint manipulator with a kinematic design derived from the PUMA geometry.