Standard Levenberg-Marquardt Algorithm Research Articles

Efficient Industrial Robot Calibration via a Novel Unscented Kalman Filter-Incorporated Variable Step-Size Levenberg–Marquardt Algorithm

Robots facilitate a critical category of equipments to implement intelligent production. However, due to extensively inevitable factors like structural errors and gear tolerances, the positioning error of an industrial robot is several millimeters, therefore failing in fulfilling the high-precision manufacture requirements. To address the critical problem, this work develops a novel calibration algorithm incorporates an Unscented Kalman Filter and a Variable Step-size Levenberg-Marquardt (UKF-VSLM) algorithm for efficient industrial robot calibration with the following two-fold ideas: a) developing a novel Variable Step-size Levenberg-Marquardt algorithm to address the local optimum issues encountered by a standard Levenberg-Marquardt algorithm; and b) incorporating an unscented Kalman filter into the proposed Variable Step-size Levenberg-Marquardt algorithm to suppressing the measurement noises during the calibration process. Empirical studies on an HSR JR680 industrial robot demonstrate that compared with state-of-the-art calibration algorithms, the calibration accuracy of the developed UKF-VSLM is 19.51% higher than that of the most accurate Levenberg-Marquardt algorithm measured by the maximum error. The empirical results strongly support the superior performance of the proposed algorithm in addressing robot calibration issues.

A Robust Magnetic Tracking Approach Based on Graph Optimization

Magnetic tracking approach (MTA) is mainly based on the observation of the magnetic field produced by a magnetic tracking target. A permanent magnet is usually employed as the tracking target, and a magnetic dipole model is used to estimate the magnetic field. In this article, we introduce an intuitive method to solve the magnetic tracking problem based on graph optimization. A graph is constructed to formulate the tracking problem, whose nodes correspond to the poses of the permanent magnet at different times and whose edges represent constraints derived by sensor measurements. The accuracy and robustness of MTA play a vital role in biomedical and industrial applications. However, a potential abnormal edge will appear in the graph when a magnetic sensor is disturbed by an interference source like a small magnet, which will deteriorate the localization accuracy of MTA. To improve its robustness, Huber cost function is used to reduce the weight of the abnormal edge. Finally, experiments were carried out to verify the performance of the proposed approach with and without magnetic field interference. Comparing the localization errors with the MTA based on the standard Levenberg–Marquardt algorithm, the results illustrate that the proposed approach could achieve superior localization accuracy and anti-interference ability.

Phasor-based widefield FLIM using a gated 512×512 single-photon SPAD imager.

Single-photon avalanche diode (SPAD) imagers can perform fast time-resolved imaging in a compact form factor, by exploiting the processing capability and speed of integrated CMOS electronics. Developments in SPAD imagers have recently made them compatible with widefield microscopy, thanks to array formats approaching one megapixel and sensitivity and noise levels approaching those of established technologies. In this paper, phasor-based FLIM is demonstrated with a gated binary 512×512 SPAD imager, which can operate with a gate length as short as 5.75 ns, a minimum gate step of 17.9 ps, and up to 98 kfps readout rate (1-bit frames). Lifetimes of ATTO 550 and Rhodamine 6G (R6G) solutions were measured across a 472×256 sub-array using phasor analysis, acquiring data by shifting a 13.1 ns gate window across the 50 ns laser period. The measurement accuracy obtained when employing such a scheme based on long, overlapping gates was validated by comparison with TCSPC measurements and fitting analysis results based on a standard Levenberg-Marquardt algorithm (>90% accuracy for the lifetime of R6G and ATTO 550). This demonstrates the ability of the proposed method to measure short lifetimes without minimum gate length requirements. The FLIM frame rate of the overall system can be increased up to a few fps for phasor-based widefield FLIM (moving closer to real-time operation) by FPGA-based parallel computation with continuous acquisition at the full speed of 98 kfps.

Power forecasting for a photovoltaic system based on the multi-agent adaptive fuzzy neuronet

This article presents a multi-agent adaptive fuzzy neuronet for a two days ahead forecasting of the hourly power from a photovoltaic system under random perturbations. In this research we consider a 5 KW Solar Power Plant for a residential building (model SA-5000M). The main objective of this research is to fulfil the multi-agent adaptive fuzzy neurone for hourly power forecasting for a photovoltaic system. The agents of the multi-agent adaptive fuzzy neuronet are fulfilled as two-layered recurrent networks. The standard Levenberg-Marquardt algorithm is described. The analysis of the evolving errors shows the potential of the multi-agent adaptive fuzzy neuronet in the hourly power forecasting for a photovoltaic system.

Localization of Multiple Underwater Objects With Gravity Field and Gravity Gradient Tensor

We present a novel algorithm to locate multiple underwater objects in real time using gravity field vector and gravity gradient tensor signals. This algorithm formulates the task of localization of multiple underwater objects into a regularized nonlinear problem, which is solved with the standard Levenberg-Marquardt algorithm. The regularization parameters are estimated by cross validation. The initial coordinates and masses of these underwater objects are automatically determined by solving a single-object localization problem. A synthetic navigation model with two underwater objects was adopted to validate the proposed algorithm. The results show that it has good stability and antinoise ability for multiple underwater objects localizations.

Characterization of periodic extreme ultraviolet multilayer based on multi-objective evolutionary algorithm

In order to refine the layered structure of extreme ultraviolet multilayers, a multi-objective evolutionary algorithm which is post-hybridized with the standard Levenberg-Marquardt algorithm is applied to analyze the grazing incidence X-ray reflectivity (GIXR) and the normal incidence extreme ultraviolet reflectance (EUVR). In this procedure, the GIXR data and EUVR data are simultaneously fitted as two objectives, and the high sensitivities of these two sets of data to layer thicknesses and densities are combined. This set of mathematical procedures is conducive to obtain a more correct model of periodic multilayers which can simultaneously describe both GIXR and EUVR measurements. As a result, the layered structure of Mo/Si multilayers with a period of about 7.0 nm is obtained.

Improvement of Levenberg-Marquardt algorithm during history fitting for reservoir simulation

Abstract In order to estimate reservoir parameters more effectively by history fitting, DE (Differential Evolution) was proposed to estimate the optimum damping factor so that the standard Levenberg-Marquardt algorithm was improved, and the improved algorithm was validated by analysis of examples. The standard LM algorithm uses trial-and-error method to estimate the damping factor and is less reliable for large scale inverse problems. DE can solve this problem and eliminate the use of line search for an appropriate step length. The improved Levenberg-Marquardt algorithm was applied to match the histories of two synthetic reservoir models with different scales, and compared with other algorithms. The results show that: DE speeds up the convergence rate of the LM algorithm and reduces the residual errors, making the algorithm suitable for not only small and medium scale inverse problems, but also large scale inverse problems; if the iteration termination criteria of LM algorithm is preset, the improved algorithm will save the number of iterations and reduce the total time greatly needed for the LM algorithm, leading to higher efficiency of history matching.

Unified approach to engine cylinder pressure reconstruction using time-delay neural networks with crank kinematics or block vibration measurements

Closed-loop combustion control in gasoline engines can improve efficiency, calibration effort, and performance using different fuels. Knowledge of in-cylinder pressures is a key requirement for closed-loop combustion control. Adaptive cylinder pressure reconstruction offers a realistic alternative to direct sensing, which is otherwise necessary as legislation requires continued reductions in CO2 and exhaust emissions. Direct sensing, however, is expensive and may not prove adequately robust. A new approach is developed for in-cylinder pressure reconstruction on gasoline engines. The approach uses time-delay feedforward artificial neural networks trained with the standard Levenberg–Marquardt algorithm. The same approach can be applied to reconstruction via measured crank kinematics obtained from a shaft encoder or measured engine cylinder block vibrations obtained from a production knock sensor. The basis of the procedure is initially justified by examination of the information content within measured data, which is considered to be equally important as the network architecture and training methodology. Key hypotheses are constructed and tested using data taken from a three-cylinder (direct injected spark ignition) engine to reveal the influence of the data information content on reconstruction potential. The findings of these hypotheses tests are then used to develop the methodology. The approach is tested by reconstructing cylinder pressure across a wide range of steady-state engine operation using both measured crank kinematics and block accelerations. The results obtained show a very marked improvement over previously published reconstruction accuracy for both crank kinematics and cylinder block vibration–based reconstruction using measurements obtained from a multi-cylinder engine. This article shows that by careful processing of measured engine data, a standard neural network architecture and a standard training algorithm can be used to very accurately reconstruct engine cylinder pressure with high levels of robustness and efficiency.

Design of broad angular phase retarders for the complete polarization analysis of extreme ultraviolet radiation**Project supported by the Fundamental Research Funds for the Central Universities, China (Grant Nos. JD1517, ZY1349, and 2652014012).

A method of designing broad angular phase retarders in the extreme ultraviolet (EUV) region is presented. The design is based on a standard Levenberg–Marquardt algorithm combined with a common merit function. Using this method, a series of broad angular EUV phase retarders were designed using aperiodic Mo/Si multilayers. At photon energy of 90 eV, broad angular phase retarders with 30°, 60°, and 90° phase retardations have been realized in the angular range of 39°–51°. By analyzing and comparing the performances of the designed broad angular phase retarders, we found that the Mo/Si multilayer with more layers could obtain higher phase retardation in broader angular range when used to design the broad angular phase retarder. Broad angular phase retarders possess lower sensitivity toward changing incident angle compared with the traditional phase retarders designed with transmission periodic multilayers, and can be used for the polarization control of broad angular EUV sources.

Design of broadband transmission quarter-wave plates for polarization control of isolated attosecond pulses

Using a standard Levenberg–Marquardt algorithm, broadband quarter-wave plates (QWPs) with bandwidth from 3 to 18 eV in the extreme ultraviolet (EUV) region were designed using aperiodic Mo/Si multilayers. By analyzing the design results of the Mo/Si multiayers with different bilayer numbers, we found that a Mo/Si multilayer with more bilayers can achieve broader phase control, but suffers from lower total throughput and a degree of circular polarization. In addition, the pulse broadenings caused by the group delay dispersions of the designed broadband QWPs were studied, and their layer distributions were investigated. The oscillating distribution of bilayer thickness in optimized multilayers was observed, which is considered to be the reason for forming the broadband phase control. Such broadband QWPs can be applied to generate a circularly polarized broadband EUV source, such as isolated attosecond pulse, directly from a linearly polarized source.

Simultaneous determination of dispersion model parameters and local thickness of thin films by imaging spectrophotometry

A least-squares data fitting procedure is developed for the analysis of measurements of thin films non-uniform in thickness by imaging spectroscopic reflectometry. It solves the problem of simultaneous least-squares fitting of film thicknesses in all image pixels together with shared dispersion model parameters. Since the huge number of mutually correlated fitting parameters prevents a straightforward application of the standard Levenberg–Marquardt algorithm, the presented procedure exploits the special structure of the specific least-squares problem. The free parameters are split into thicknesses and dispersion model parameters. Both groups of parameters are fitted alternately, utilising an unmodified Levenberg–Marquardt algorithm, correcting however the thicknesses during the dispersion model fitting step to preserve effective optical thicknesses. The behaviour of the algorithm is studied using experimental data of two highly non-uniform thin films of different materials, SiOxCyHz and CNx:H, and by numerical simulations using artificial data. It is found that the optical thickness correction enables the procedure to converge rapidly, permitting the analysis of large imaging spectroscopic reflectometry data sets with reasonable computational resources.

Maximum-Likelihood Fits to Histograms for Improved Parameter Estimation

Straightforward methods for adapting the familiar chi^2 statistic to histograms of discrete events and other Poisson distributed data generally yield biased estimates of the parameters of a model. The bias can be important even when the total number of events is large. For the case of estimating a microcalorimeter's energy resolution at 6 keV from the observed shape of the Mn K-alpha fluorescence spectrum, a poor choice of chi^2 can lead to biases of at least 10% in the estimated resolution when up to thousands of photons are observed. The best remedy is a Poisson maximum-likelihood fit, through a simple modification of the standard Levenberg-Marquardt algorithm for chi^2 minimization. Where the modification is not possible, another approach allows iterative approximation of the maximum-likelihood fit.

Multiparametric response surface construction by means of proper generalized decomposition: An extension of the PARAFAC procedure

Multidimensional problems are found in almost every scientific field. In particular, this is standard in parametric design, inverse analysis, in optimization and in metamodeling analysis, whereby statistical or deterministic approximations of multiparametric solutions are built from the results of experimental campaigns or computer simulations. Multidimensional fitting or approximation of response functions exponentially increase their complexity and computational cost with the number of dimensions responding to the well-known “curse of dimensionality”. To reduce the order of complexity and make the solution of many-parameter problems affordable, we propose to combine the model reduction technique known as Proper Generalized Decomposition (PGD) and the response surface (RSM) methodology. As a proof of concept we have used a simple fitting procedure as it is least squares, although other more complex fitting procedures may be easily included. The combined algorithm is presented and its capabilities discussed in a set of multidimensional examples. The data samples to be fit in each of these examples are obtained by means of appropriate discretizing the interval of interest for each design factor and then generating the output values (exact or stochastically modified) by means of virtual experiments. To have an idea of the number of discretization points needed along each direction, the Taguchi’s design of experiments is used. The obtained results show evident improvements in computer time and accuracy when compared to other traditional multiparametric approximation techniques based on polynomial functions and the standard Levenberg–Marquardt algorithm, especially in problems with non-linear behavior and with high number of design parameters. Further comparison was done with the PARAFAC-ALS algorithm. The combination of PGD and RSM seems to be an appealing tool for accurately modelling multiparametric problems in almost real-time if a sufficient set of previous off-line results is available despite the intrinsic complexity of the problem and of the number of parameters involved.

Reply: Shortened Dynamic 18F-FDG PET

REPLY: Dr. Disselhorst and colleagues note that most protocols use at least 45 or 60 min for data acquisition. This is absolutely correct if the standard Levenberg–Marquardt algorithm is used to calculate the compartment parameters. In our work ([1][1]) we use a predictive algorithm, which

Precise parameter estimation for chemical batch reactions in heterogeneous medium

A sequential experimental strategy for precise parameter estimation has been used in the case of liquid–liquid dispersions in batch-stirred tank reactors where slow chemical reactions take place. The mathematical model for a batch reaction in a stirred tank reactor is formulated as a system of non-linear differential equations standing for the mass balance of each component. Physical kinetic parameters and chemical kinetic parameters which arise from this model are estimated simultaneously. The estimation problem is posed as a weighted least squares problem and solved by using a standard Levenberg–Marquardt algorithm. In this work, we intend to show how it is possible to develop efficient experimental design strategies that lead to an accurate estimation of the parameters involved in phenomenological models and most particularly in kinetic models. Three design criteria for designing the experiments have been employed in order to increase the precision on the parameter estimates of the model. A standard non-linear sequential quadratic programming method ensures the determination of the operating conditions which define the experimental design. The well-known alkaline hydrolysis of esters in aqueous phase has been treated as a numerical application example.

Nonlinear-Least-Squares Analysis of Slow-Motion EPR Spectra in One and Two Dimensions Using a Modified Levenberg–Marquardt Algorithm

Abstract The application of the “model trust region” modification of the Levenberg–Marquardt minimization algorithm to the analysis of one-dimensional CW EPR and multidimensional Fourier-transform (FT) EPR spectra especially in the slow-motion regime is described. The dynamic parameters describing the slow motion are obtained from least-squares fitting of model calculations based on the stochastic Liouville equation (SLE) to experimental spectra. The trust-region approach is inherently more efficient than the standard Levenberg–Marquardt algorithm, and the efficiency of the procedure may be further increased by a separation-of-variables method in which a subset of fitting parameters is independently minimized at each iteration, thus reducing the number of parameters to be fitted by nonlinear least squares. A particularly useful application of this method occurs in the fitting of multicomponent spectra, for which it is possible to obtain the relative population of each component by the separation-of-variables method. These advantages, combined with recent improvements in the computational methods used to solve the SLE, have led to an order-of-magnitude reduction in computing time, and have made it possible to carry out interactive, real-time fitting on a laboratory workstation with a graphical interface. Examples of fits to experimental data will be given, including multicomponent CW EPR spectra as well as two- and three-dimensional FT EPR spectra. Emphasis is placed on the analytic information available from the partial derivatives utilized in the algorithm, and how it may be used to estimate the condition and uniqueness of the fit, as well as to estimate confidence limits for the parameters in certain cases.