Set Of Sample Points Research Articles

Unscented Kalman filter for spacecraft attitude estimation using Quaternions and Euler angles

A spacecraft attitude estimation approach based on the Unscented Kalman Filter is derived. For nonlinear systems the Unscented Kalman Filter uses a carefully selected set of sample points to map more accurately the probability distribution than the linearization of the standard Extended Kalman Filter, leading to faster convergence from inaccurate initial conditions in attitude estimation problems. The filter formulation is based on standard attitude- vector measurements using a gyro-based model for attitude propagation. This paper compares the performance of a new technique, the Unscented Kalman Filter, when two different mathematical constructs are used to represent the attitude: the Euler angles and quaternions. In this study, the attitude of satellite is estimated with real time algorithms using real data supplied by gyros, Earth sensors and Sun sensors that are on board of the CBERS-2 (China Brazil Earth Resources Satellite).

Study on Improved Multidiscipline Feasible Strategy for Complicated Turbine Component Optimization

A new improved multidiscipline feasible (MDF) strategy based on the updating approximate function constructed by Response Surface Method (RSM) is firstly pointed out and denoted as MDF-update-RSM which involves double cycles. Before the start of inner loop on MDF-update-RSM strategy, the approximate function is initialized by RSM on a set of sampling points which are generated from the Design of Experiment (DOE). With the process of outer loop, the coefficients and one high-order term of polynomial are updated and upgrade by the Least Square Regression. The ability of this new strategy promotes the efficiency and saves huge calculational cost incurred by the MDF strategy which always runs on the multidisciplinary analysis (MDA). The MDF-update-RSM is good procedure to ameliorate the accuracy that has been tested by the Rosenbrock function in this paper. Finally, the optimization strategy is employed on executing the low pressure turbine shrouded blade design and provides a true optimal result compared to the classical MDF strategy.

MSCTrees: a mean-shift based toolkit for cluster analysis of phylogenetic trees

Background Mean shift, an iterative technique for identifying the local maxima of a probability density function, has been successfully used as a clustering method in computer vision and image processing. We apply the mean shift technique to the high dimensional space of phylogeny trees. The basic idea behind this technique is to, given a set of sample points, shift each point in the direction of the gradient of the underlying density function in an iterative manner until the points concentrate at the local maxima of the density function and form natural clusters [1]. We have developed software named MSCTrees based on a variant of the mean shift method, called the adaptive mean shift [2], to perform cluster analysis on a set of multidimensional data points corresponding to phylogenetic trees.

SHA-Less Pipelined ADC With In Situ Background Clock-Skew Calibration

A 10-b, 100-MS/s pipelined analog-to-digital converter (ADC) without dedicated front-end sample-and-hold amplifier (SHA) converts from dc to the 12th Nyquist band with in situ, mostly digital background calibration for the clock skew in the 3.5-b front-end stage. The skew information is extracted from the first-stage residue output with two comparators sensing out-of-range errors; a gradient-descent algorithm is used to adaptively adjust the timing of the front-end sub-ADC to synchronize with that of the sample-and-hold (S/H) in the multiplying digital-to-analog converter (MDAC). The prototype ADC, implemented in a 90-nm CMOS process, digitizes inputs up to 610 MHz without skew errors in experiments; in contrast, the same ADC fails at 130 MHz with calibration disabled (with the default sub-ADC sample point set at the midpoint of the delay range). The prototype with calibration circuits fully integrated on chip consumes 12.2 mW and occupies 0.26-mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> silicon area, while the calibration circuits dissipate 0.9 mW and occupy 0.01 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . A 71-dB spurious-free dynamic range (SFDR) and a 55-dB signal-to-noise and distortion ratio (SNDR) were measured with a 20-MHz sine-wave input, and a larger than 55-dB SFDR was measured in the 10th Nyquist band.

Unscented Kalman Filtering for Single Camera Based Motion and Shape Estimation

Accurate estimation of the motion and shape of a moving object is a challenging task due to great variety of noises present from sources such as electronic components and the influence of the external environment, etc. To alleviate the noise, the filtering/estimation approach can be used to reduce it in streaming video to obtain better estimation accuracy in feature points on the moving objects. To deal with the filtering problem in the appropriate nonlinear system, the extended Kalman filter (EKF), which neglects higher-order derivatives in the linearization process, has been very popular. The unscented Kalman filter (UKF), which uses a deterministic sampling approach to capture the mean and covariance estimates with a minimal set of sample points, is able to achieve at least the second order accuracy without Jacobians’ computation involved. In this paper, the UKF is applied to the rigid body motion and shape dynamics to estimate feature points on moving objects. The performance evaluation is carried out through the numerical study. The results show that UKF demonstrates substantial improvement in accuracy estimation for implementing the estimation of motion and planar surface parameters of a single camera.

On using prototype reduction schemes to optimize locally linear reconstruction methods

This paper concerns the use of prototype reduction schemes (PRS) to optimize the computations involved in typical k-nearest neighbor (k-NN) rules. These rules have been successfully used for decades in statistical pattern recognition (PR) [1,15] applications and are particularly effective for density estimation, classification, and regression because of the known error bounds that they possess. For a given data point of unknown identity, the k-NN possesses the phenomenon that it combines the information about the samples from a priori target classes (values) of selected neighbors to predict the target class of the tested sample, or to estimate the density function value of the given queried sample. Recently, an implementation of the k-NN, named as the locally linear reconstruction (LLR) [2], has been proposed. The salient and brilliant feature of the latter is that by invoking a quadratic optimization process, it is capable of systematically setting model parameters, such as the number of neighbors (specified by the parameter, k) and the weights. However, the LLR takes more time than other conventional methods when it has to be applied to classification tasks. To overcome this problem, we propose a strategy of using a PRS to efficiently compute the optimization problem. In this paper, we demonstrate, first of all, that by completely discarding the points not included by the PRS, we can obtain a reduced set of sample points, using which, in turn, the quadratic optimization problem can be computed far more expediently. The values of the corresponding indices are comparable to those obtained with the original training set (i.e., the one which considers all the data points) even though the computations required to obtain the prototypes and the corresponding classification accuracies are noticeably less. The proposed method has been tested on artificial and real-life data sets, and the results obtained are very promising, and could have potential in PR applications.

Spatial autocorrelation in predictors reduces the impact of positional uncertainty in occurrence data on species distribution modelling

Aim To investigate the impact of positional uncertainty in species occurrences on the predictions of seven commonly used species distribution models (SDMs), and explore its interaction with spatial autocorrelation in predictors. Methods A series of artificial datasets covering 155 scenarios including different combinations of five positional uncertainty scenarios and 31 spatial autocorrelation scenarios were simulated. The level of positional uncertainty was defined by the standard deviation of a normally distributed zero-mean random variable. Each dataset included two environmental gradients (predictor variables) and one set of species occurrence sample points (response variable). Seven commonly used models were selected to develop SDMs: generalized linear models, generalized additive models, boosted regression trees, multivariate adaptive regression spline, random forests, genetic algorithm for rule-set production and maximum entropy. A probabilistic approach was employed to model and simulate five levels of error in the species locations. To analyse the propagation of positional uncertainty, Monte Carlo simulation was applied to each scenario for each SDM. The models were evaluated for performance using simulated independent test data with Cohen's Kappa and the area under the receiver operating characteristic curve. Results Positional uncertainty in species location led to a reduction in prediction accuracy for all SDMs, although the magnitude of the reduction varied between SDMs. In all cases the magnitude of this impact varied according to the degree of spatial autocorrelation in predictors and the levels of positional uncertainty. It was shown that when the range of spatial autocorrelation in the predictors was less than or equal to three times the standard deviation of the positional error, the models were less affected by error and, consequently, had smaller decreases in prediction accuracy. When the range of spatial autocorrelation in predictors was larger than three times the standard deviation of positional error, the prediction accuracy was low for all scenarios. Main conclusions The potential impact of positional uncertainty in species occurrences on the predictions of SDMs can be understood by comparing it with the spatial autocorrelation range in predictor variables.

B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness

Let ξ = { x j } j = 1 n be a set of n sample points in the d -cube I d ≔ [ 0 , 1 ] d , and Φ = { φ j } j = 1 n a family of n functions on I d . We define the linear sampling algorithm L n ( Φ , ξ , ⋅ ) for an approximate recovery of a continuous function f on I d from the sampled values f ( x 1 ) , … , f ( x n ) , by L n ( Φ , ξ , f ) ≔ ∑ j = 1 n f ( x j ) φ j . For the Besov class B p , θ α of mixed smoothness α , to study optimality of L n ( Φ , ξ , ⋅ ) in L q ( I d ) we use the quantity r n ( B p , θ α ) q ≔ inf ξ , Φ sup f ∈ B p , θ α ‖ f − L n ( Φ , ξ , f ) ‖ q , where the infimum is taken over all sets of n sample points ξ = { x j } j = 1 n and all families Φ = { φ j } j = 1 n in L q ( I d ) . We explicitly constructed linear sampling algorithms L n ( Φ , ξ , ⋅ ) on the set of sample points ξ = G d ( m ) ≔ { ( 2 − k 1 s 1 , … , 2 − k d s d ) ∈ I d : k 1 + ⋯ + k d ≤ m } , with Φ a family of linear combinations of mixed B-splines which are mixed tensor products of either integer or half integer translated dilations of the centered B-spline of order r . For various 0 < p , q , θ ≤ ∞ and 1 / p < α < r , we proved upper bounds for the worst case error sup f ∈ B p , θ α ‖ f − L n ( Φ , ξ , f ) ‖ q which coincide with the asymptotic order of r n ( B p , θ α ) q in some cases. A key role in constructing these linear sampling algorithms, plays a quasi-interpolant representation of functions f ∈ B p , θ α by mixed B-spline series with the coefficient functionals which are explicitly constructed as linear combinations of an absolute constant number of values of functions. Moreover, we proved that the quasi-norm of the Besov space M B p , θ α is equivalent to a discrete quasi-norm in terms of the coefficient functionals.

Weighted geometric discrepancies and numerical integration on reproducing kernel Hilbert spaces

We extend the notion of L 2 – B -discrepancy introduced in [E. Novak, H. Woźniakowski, L 2 discrepancy and multivariate integration, in: W.W.L. Chen, W.T. Gowers, H. Halberstam, W.M. Schmidt, and R.C. Vaughan (Eds.), Analytic Number Theory. Essays in Honour of Klaus Roth, Cambridge University Press, Cambridge, 2009, pp. 359–388] to what we shall call weighted geometric L 2 -discrepancy. This extension enables us to consider weights in order to moderate the importance of different groups of variables, as well as to consider volume measures different from the Lebesgue measure and classes of test sets different from measurable subsets of Euclidean spaces. We relate the weighted geometric L 2 -discrepancy to numerical integration defined over weighted reproducing kernel Hilbert spaces and settle in this way an open problem posed by Novak and Woźniakowski. Furthermore, we prove an upper bound for the numerical integration error for cubature formulas that use admissible sample points. The set of admissible sample points may actually be a subset of the integration domain of measure zero. We illustrate that particularly in infinite-dimensional numerical integration it is crucial to distinguish between the whole integration domain and the set of those sample points that actually can be used by the algorithms.

Low-Rank Unscented Kalman Filter for Freeway Traffic Estimation Problems

Data assimilation techniques are a process of using measurements combined with model predictions to (optimally) estimate the value of a certain set of variables. These techniques belong to a group of model-based estimation approaches. In traffic science, the Kalman filter and its family, namely, the extended Kalman filter (EKF), have been widely applied to problems of traffic state estimation. The EKF is successful for computation and provides some reasonably accurate results. However, traffic systems are generally so nonlinear that the linearization used in the EKF is not always straightforward. To avoid such linearization, another family of Kalman filter, the unscented Kalman filter (UKF), has been designed to account for the nonlinear system in which a set of deterministic sample points is chosen to capture the initial probability distribution. These sample points are then propagated through the nonlinear system, and the probability density function of the actual state is approximated by the ensemble of the estimates. In the UKF, the number of sample points is determined by the dimension of the states to be estimated so that it becomes computationally expensive when the size of traffic networks is increased. Thus an effort was made to approximate a UKF in which the required sample points were reduced. Numerical experiments were carried out to assess the relevance of the proposed approach with real traffic data, and comparisons with the UKF for the level of accuracy and computational time were made.

Multi-camera calibration based on an invariant pattern

In this paper we present a method for the calibration of multiple cameras based on the extraction and use of the physical characteristics of a one-dimensional invariant pattern which is defined by four collinear markers. The advantages of this kind of pattern stand out in two key steps of the calibration process. In the initial step of camera calibration methods, related to sample points capture, the proposed method takes advantage of using a new technique for the capture and recognition of a robust sample of projective invariant patterns, which allows to capture simultaneously more than one invariant pattern in the tracking area and recognize each pattern individually as well as each marker that composes them. This process is executed in real time while capturing our sample of calibration points in the cameras of our system. This new feature allows to capture a more numerous and robust set of sample points than other patterns used for multi-camera calibration methods. In the last step of the calibration process, related to camera parameters' optimization, we explore the collinearity feature of the invariant pattern and add this feature in the camera parameters optimization model. This approach obtains better results in the computation of camera parameters. We present the results obtained with the calibration of two multi-camera systems using the proposed method and compare them with other methods from the literature.

Finding Deterministic Solution From Underdetermined Equation: Large-Scale Performance Variability Modeling of Analog/RF Circuits

The aggressive scaling of integrated circuit technology results in high-dimensional, strongly-nonlinear performance variability that cannot be efficiently captured by traditional modeling techniques. In this paper, we adapt a novel L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -norm regularization method to address this modeling challenge. Our goal is to solve a large number of (e.g., 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> -10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</sup> ) model coefficients from a small set of (e.g., 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) sampling points without over-fitting. This is facilitated by exploiting the underlying sparsity of model coefficients. Namely, although numerous basis functions are needed to span the high-dimensional, strongly-nonlinear variation space, only a few of them play an important role for a given performance of interest. An efficient orthogonal matching pursuit (OMP) algorithm is applied to automatically select these important basis functions based on a limited number of simulation samples. Several circuit examples designed in a commercial 65 nm process demonstrate that OMP achieves up to 25× speedup compared to the traditional least-squares fitting method.

Comparing methods for measuring the rate of spread of invading populations

Measuring rates of spread during biological invasions is important for predicting where and when invading organisms will spread in the future as well as for quantifying the influence of environmental conditions on invasion speed. While several methods have been proposed in the literature to measure spread rates, a comprehensive comparison of their accuracy when applied to empirical data would be problematic because true rates of spread are never known. This study compares the performances of several spread rate measurement methods using a set of simulated invasions with known theoretical spread rates over a hypothetical region where a set of sampling points are distributed. We vary the density and distribution (aggregative, random, and regular) of the sampling points as well as the shape of the invaded area and then compare how different spread rate measurement methods accommodate these varying conditions. We find that the method of regressing distance to the point of origin of the invasion as a function of time of first detection provides the most reliable method over adverse conditions (low sampling density, aggregated distribution of sampling points, irregular invaded area). The boundary displacement method appears to be a useful complementary method when sampling density is sufficiently high, as it provides an instantaneous measure of spread rate, and does not require long time series of data.

CONDITION MONITORING IN A HYDRAULIC SYSTEM OF AN INDUSTRIAL MACHINE USING UNSCENTED KALMAN FILTER

This paper develops a model-based technique based on the Unscented Kalman Filter (UKF) for on-line condition monitoring, and applies it to the hydraulic system of an automated industrial fish processing machine. First an analytical model of the hydraulic system is developed and the system parameters are identified (determined). Then the developed UKF approach is implemented in the machine. The UKF employs an unscented transformation to select a minimal set of sample points around the mean, which are then propagated through nonlinear functions, from which the mean and covariance of the estimate are recovered. This approach is known to be more accurate for nonlinear systems. For experimental investigation of the performance of the approach, four common hydraulic faults are deliberately introduced into the machine. The four faults are external leakage in the two chambers of the hydraulic cylinder; internal leakage; and dry friction build-up at the moving support plate of the cutter carriage. Three levels of leakage are manually introduced to the system for each fault scenario using needle valves. The criteria that are considered in fault diagnosing are residual moving average of the errors, chamber pressures, and actuator characteristics. The experimental results show that the developed technique is able to accurately determine the fault conditions of the machine.

Kernel-Induced Sampling Theorem

A perfect reconstruction of functions in a reproducing kernel Hilbert space from a given set of sampling points is discussed. A necessary and sufficient condition for the corresponding reproducing kernel and the given set of sampling points to perfectly recover the functions is obtained in this paper. The key idea of our work is adopting the reproducing kernel Hilbert space corresponding to the Gramian matrix of the kernel and the given set of sampling points as the range space of a sampling operator and considering the orthogonal projector, defined via the range space, onto the closed linear subspace spanned by the kernel functions corresponding to the given sampling points. We also give an error analysis of a reconstructed function by incomplete sampling points.

Localized Delaunay Refinement for Sampling and Meshing

AbstractThe technique of Delaunay refinement has been recognized as a versatile tool to generate Delaunay meshes of a variety of geometries. Despite its usefulness, it suffers from one lacuna that limits its application. It does not scale well with the mesh size. As the sample point set grows, the Delaunay triangulation starts stressing the available memory space which ultimately stalls any effective progress. A natural solution to the problem is to maintain the point set in clusters and run the refinement on each individual cluster. However, this needs a careful point insertion strategy and a balanced coordination among the neighboring clusters to ensure consistency across individual meshes. We design an octtree based localized Delaunay refinement method for meshing surfaces in three dimensions which meets these goals. We prove that the algorithm terminates and provide guarantees about structural properties of the output mesh. Experimental results show that the method can avoid memory thrashing while computing large meshes and thus scales much better than the standard Delaunay refinement method.

Uncorrelated unscented filtering for spacecraft attitude determination

Abstract Spacecraft attitude estimation based on the nonlinear unscented filter is addressed to fully utilize capabilities of the unscented transformation. To overcome significant computational load, an efficient technique is proposed by appropriately eliminating correlation between random variables. This modification leads to a considerable reduction of computational burden in matrix square-root calculation for most nonlinear systems. The unscented filter makes use of a set of sample points to predict mean and covariance. For attitude estimation based on quaternions, an approach to computing quaternion means from sampled quaternions with guarantee of the normalization constraint is described by using a constrained optimization technique. Finally, the performance of the new approach is demonstrated by attitude determination using a star tracker and rate-gyro measurements.

Rate of Convergence in Approximating the Spectral Factor of Regular Stochastic Sequences

Common methods for the calculation of the spectral factorization rely on an approximation of the given spectral density by a polynomial and a subsequent factorization of this polynomial. It is known that the regularity of the stochastic sequence determines the achievable approximation rate of its spectrum. However, since the approximative polynomial should be factorized, it has to be positive. It is shown that this restriction on the approximation polynomial implies a limitation on the approximation rate for linear methods whereas for nonlinear methods the optimal approximation rate can still be achieved. This has also consequences for the rate of convergence of the spectral factor, which is investigated in the second part. There, a lower and an upper bound for the error in the spectral factor is derived, which shows the dependency on the approximation degree and on the regularity of the stochastic sequence. Finally, if the spectral density is given only on a finite set of sampling points, no linear approximation method exists such that the error in the spectral factor can be controlled by the approximation degree.

Teaching and Learning Guide for: Application of Virtual Globes in Education

Teaching and Learning Guide for: Application of Virtual Globes in Education

Injection Molding Warpage Optimization Based on a Mode-Pursuing Sampling Method

This paper presents a new optimization approach for minimizing the warpage defect of injection-molded plastic parts. Existing methods in warpage optimization are either computationally expensive or, when inexpensive surrogate models are employed with fixed set of sample points, the accuracy of the surrogate model will only be ensured by a large number of sample points, which in turn will increase the amount of required computation. To address this problem, this paper applies a mode-pursuing sampling (MPS) method for warpage optimization, by integrating injection molding simulation with MPS, and by proposing a reinforced convergence criterion for the optimization process, in an attempt to search for the optimal process parameters of injection molding for minimizing warpage defect both effectively and efficiently. The MPS method can systematically generate more sample points in the neighborhood of the current optimal solution while statistically covering the entire search space. A case study of a scanner frame, where injection time, melt temperature and mold temperature are selected as the design variables, demonstrates that the proposed optimization method can effectively decrease the warpage deflection of an injection-molded part with significantly less computation required. Based on the optimization results, the paper also studied the influences of different process parameters on the severity of the warpage defect, providing a guideline for the setting of the proper process parameters.