Deterministic Error Research Articles

A Novel Positioning Module and Fusion Algorithm for Unmanned Aerial Vehicle Monitoring

Aiming to solve the monitoring problem of unmanned aerial vehicles (UAVs), a novel positioning module is designed and implemented. The hardware design meets the prerequisites of miniaturization and low-cost. Then, a calibration procedure based on six-position calibration and ellipsoid fitting is executed to eliminate the bad influence on the system’s accuracy and stability caused by deterministic errors. Also, a stochastic error modeling procedure based on Allan variance is performed to identify stochastic errors’ model type and model parameters. Finally, data from inertial measurement units (IMUs), global navigation satellite system (GNSS), magnetometers, and a barometer are fused by an extended Kalman filter (EKF) based sensor fusion algorithm. This algorithm is applied to a UAV flight test and comparison result using calibrated data and uncalibrated data respectively are shown. Besides, the proposed EKF algorithm is compared with unscented Kalman filter (UKF) and unscented quaternion estimator (USQUE) to show its practicability in the application. The main contribution of this research is the design of the positioning module and the validation of the sensor fusion algorithm in this hardware. It is found that, under the constraint of low-cost sensors, the accuracy of positioning can achieve around 7.5m horizontally, and 2m vertically. The frequency of positioning increases to 50Hz after sensor fusion, a large improvement over the previous 1Hz GNSS positioning frequency.

Deterministic error bounds for kernel-based learning techniques under bounded noise

We consider the problem of reconstructing a function from a finite set of noise-corrupted samples. Two kernel algorithms are analyzed, namely kernel ridge regression and ɛ-support vector regression. By assuming the ground-truth function belongs to the reproducing kernel Hilbert space of the chosen kernel, and the measurement noise affecting the dataset is bounded, we adopt an approximation theory viewpoint to establish deterministic, finite-sample error bounds for the two models. Finally, we discuss their connection with Gaussian processes and two numerical examples are provided. In establishing our inequalities, we hope to help bring the fields of non-parametric kernel learning and system identification for robust control closer to each other.

The calibration method for accelerometers in the redundant MEMS inertial navigation system

Compared with the non-redundant inertial navigation system (INS), the redundant INS (RINS) has more error parameters and the system is more complicated. The calibration methods for non-redundant INS are unable to be adopted by RINS directly. Meanwhile, the inner lever arm effect of accelerometers is more severe in RINS, which is not supposed to be ignored in the error compensation. To solve the problems above, this paper proposes a novel calibration method for accelerometers in RINS. First, the paper analyses and models the bias, scale factor error, installation angle error and lever arm error. Based on the error model, two Kalman filters are designed to estimate the error parameters. The calibration is divided into two steps: the bias, scale factor error and installation angle error are calibrated by the static multi-position experiment first, and then the lever arm error is calibrated by the rotation experiment. Experiments prove that the proposed method can effectively calibrate the deterministic error of the accelerometers, that the estimation errors are controlled at 1 × 10−4 level. Further, the paper studies and optimizes the turntable rotation scheme in the lever arm error calibration experiment, and proposes the design principles for the rotation scheme. Comparing with the casually designed scheme, the optimized procedure can improve the accuracy by almost an order of magnitude as well as the time-consumption being shortened by 40%. The design principles are applicable to the other inertial navigation systems to improve the calibration accuracy and reduces the time cost.

Guaranteed a posteriori estimation of unknown right-hand sides of linear periodic systems of ODEs

We consider the problem of recovery of unknown right-hand sides of the first-order linear systems of ordinary differential equations with periodic coefficients from indirect observations of their solutions on a finite system of points and intervals. Under the assumption that right-hand sides and deterministic errors in observations are subjected to some quadratic restrictions, we obtain a posteriori estimates of right-hand sides which are compatible with observations data.

Deterministic error compensation for slow tool servo-driven diamond turning of freeform surface with nanometric form accuracy

• A deterministic process flow was proposed for error compensation in free from surface turning. • All the main error factors were modeled, simulated and compensated before machining. • The method enabled achieving nanometer-level form accuracy without trial-and-error. • A two-dimensional sine wave grid was generated on single-crystal silicon by a single cut. • The form error was reduced to 8 nm peak-to-valley with surface roughness of ∼ 1 nm Sa. Slow tool servo (STS) turning takes an important role in fabrication of freeform optical elements. However, in conventional STS turning, it is technologically difficult to obtain nanometer-level form accuracy due to multiple error factors such as tool trajectory errors, tool alignment errors and dynamic follow-up errors of machine tools used. In this study, a deterministic process flow was proposed where all the main error factors were comprehensively analyzed, simulated and then compensated before machining based on the feedforward method, and the workpiece form error after compensation was predicted accurately. The proposed process flow enabled achieving nanometer-level form accuracy by a single cut without the necessity of repetitive trial-and-error. To demonstrate the effectiveness of the proposed method, cutting tests of a two-dimensional sine wave grid were attempted on single-crystal silicon and the proposed error compensation was applied. As a result, the form error was reduced to 8 nm P–V (peak to valley) with a surface roughness of 1 nm Sa by a single cut.

Visualizing Uncertainty for Non-Expert End Users: The Challenge of the Deterministic Construal Error

There is a growing body of evidence that numerical uncertainty expressions can be used by non-experts to improve decision quality. Moreover, there is some evidence that similar advantages extend to graphic expressions of uncertainty. However, visualizing uncertainty introduces challenges as well. Here, we discuss key misunderstandings that may arise from uncertainty visualizations, in particular the evidence that users sometimes fail to realize that the graphic depicts uncertainty. Instead they have a tendency to interpret the image as representing some deterministic quantity. We refer to this as thedeterministic construal error. Although there is now growing evidence for thedeterministic construal error, few studies are designed to detect it directly because they inform participants upfront that the visualization expresses uncertainty. In a natural setting such cues would be absent, perhaps making the deterministic assumption more likely. Here we discuss the psychological roots of this key but underappreciated misunderstanding as well as possible solutions. This is a critical question because it is now clear that members of the public understand that predictions involve uncertainty and have greater trust when uncertainty is included. Moreover, they can understand and use uncertainty predictions to tailor decisions to their own risk tolerance, as long as they are carefully expressed, taking into account the cognitive processes involved.

Multi-Signal Approaches for Repeated Sampling Schemes in Inertial Sensor Calibration

Inertial sensor calibration plays a progressively important role in many areas of research among which navigation engineering. By performing this task accurately, it is possible to significantly increase general navigation performance by correctly filtering out the deterministic and stochastic measurement errors that characterize such devices. While different techniques are available to model and remove the deterministic errors, there has been considerable research over the past years with respect to modelling the stochastic errors which have complex structures. In order to do the latter, different replicates of these error signals are collected and a model is identified and estimated based on one of these replicates. While this procedure has allowed to improve navigation performance, it has not yet taken advantage of the information coming from all the other replicates collected on the same sensor. However, it has been observed that there is often a change of error behaviour between replicates which can also be explained by different (constant) external conditions under which each replicate was taken. Whatever the reason for the difference between replicates, it appears that the model structure remains the same between replicates but the parameter values vary. In this work we therefore consider and study the properties of different approaches that allow to combine the information from all replicates considering this phenomenon, confirming their validity both in simulation settings and also when applied to real inertial sensor error signals. By taking into account parameter variation between replicates, this work highlights how these approaches can improve the average navigation precision as well as obtain reliable estimates of the uncertainty of the navigation solution.

Training of Mixed-Signal Optical Convolutional Neural Networks With Reduced Quantization Levels

Mixed-signal artificial neural networks (ANNs) that employ analog matrix-multiplication accelerators can achieve higher speed and improved power efficiency. Though analog computing is known to be susceptible to noise and device imperfections, various analog computing paradigms have been considered as promising solutions to address the growing computing demand in machine learning applications, thanks to the robustness of ANNs. This robustness has been explored in low-precision, fixed-point ANN models, which have proven successful on compressing ANN model size on digital computers. However, these promising results and network training algorithms cannot be easily migrated to analog accelerators. The reason is that digital computers typically carry intermediate results with higher bit width, though the inputs and weights of each ANN layers are of low bit width; while the analog intermediate results have low precision, analogous to digital signals with a reduced quantization level. Here we report a training method for mixed-signal ANN with two types of errors in its analog signals, random noise, and deterministic errors (distortions). The results showed that mixed-signal ANNs trained with our proposed method can achieve an equivalent classification accuracy with noise level up to 50% of the ideal quantization step size. We have demonstrated this training method on a mixed-signal optical convolutional neural network based on diffractive optics.

Analytical Assessment of the Propagation of Colored Sensor Noise in Strapdown Inertial Navigation.

Knowledge of the propagation of sensor errors in strapdown inertial navigation is crucial for the design of inertial and integrated navigation systems. The propagation of initialization errors and deterministic sensor errors is well covered in the literature. If considered at all, the propagation of inertial sensor noise has typically been assessed for un-correlated (white) Gaussian noise. Real inertial sensor noise, however, is time-correlated (colored) and best described by a combination of different stochastic processes. In this paper, we demonstrate how a navigation system’s response to colored noise input differs from the response to bias-like or white noise inputs. We present a method for assessing the navigation error from various inertial sensor noise processes without the need for time-consuming Monte Carlo simulations and demonstrate its application and validity with real sensor data. The proposed method is used to determine in which scenarios the sensor’s real noise can be approximated by simple white Gaussian noise. The results indicate that neglecting colored sensor noise is justified for many applications, but should be checked individually for each sensor configuration and mission.

A Field Calibration Method for Low-Cost MEMS Accelerometer Based on the Generalized Nonlinear Least Square Method

This paper proposes a field calibration method for an accelerometer without the need of having any external devices for calibration. In the proposed calibration method, generalized nonlinear least-square (GNLS) is used to estimate deterministic errors. A novel sensor’s data collection procedure is developed to collect data of an accelerometer along all three axes and all possible orientations where the expectation of influence of all possible errors is very high. The proposed calibration method is verified by applying it to two different accelerometers. The proposed calibration method achieved an accurate estimation of calibration parameters. The results of the proposed GNLS based calibration method are compared with two other commonly used algorithms, such as Levenberg–Marquardt (LM) and Gauss–Newton (GN). Simulation and experimental results show that the proposed GNLS-based calibration method is slightly more accurate than the LM and GN. The GNLS convergence rate for estimating the calibration parameters is also faster than the LM and GN.

A Review of Innovation-Based Methods to Jointly Estimate Model and Observation Error Covariance Matrices in Ensemble Data Assimilation

AbstractData assimilation combines forecasts from a numerical model with observations. Most of the current data assimilation algorithms consider the model and observation error terms as additive Gaussian noise, specified by their covariance matrices and , respectively. These error covariances, and specifically their respective amplitudes, determine the weights given to the background (i.e., the model forecasts) and to the observations in the solution of data assimilation algorithms (i.e., the analysis). Consequently, and matrices significantly impact the accuracy of the analysis. This review aims to present and to discuss, with a unified framework, different methods to jointly estimate the and matrices using ensemble-based data assimilation techniques. Most of the methods developed to date use the innovations, defined as differences between the observations and the projection of the forecasts onto the observation space. These methods are based on two main statistical criteria: 1) the method of moments, in which the theoretical and empirical moments of the innovations are assumed to be equal, and 2) methods that use the likelihood of the observations, themselves contained in the innovations. The reviewed methods assume that innovations are Gaussian random variables, although extension to other distributions is possible for likelihood-based methods. The methods also show some differences in terms of levels of complexity and applicability to high-dimensional systems. The conclusion of the review discusses the key challenges to further develop estimation methods for and . These challenges include taking into account time-varying error covariances, using limited observational coverage, estimating additional deterministic error terms, or accounting for correlated noise.

Convergence analysis of quasi-Monte Carlo sampling for quantile and expected shortfall

Quantiles and expected shortfalls are usually used to measure risks of stochastic systems, which are often estimated by Monte Carlo methods. This paper focuses on the use of the quasi-Monte Carlo (QMC) method, whose convergence rate is asymptotically better than Monte Carlo in the numerical integration. We first prove the convergence of QMC-based quantile estimates under very mild conditions, and then establish a deterministic error bound of O ( N − 1 / d ) O(N^{-1/d}) for the quantile estimates, where d d is the dimension of the QMC point sets used in the simulation and N N is the sample size. Under certain conditions, we show that the mean squared error (MSE) of the randomized QMC estimate for expected shortfall is o ( N − 1 ) o(N^{-1}) . Moreover, under stronger conditions the MSE can be improved to O ( N − 1 − 1 / ( 2 d − 1 ) + ϵ ) O(N^{-1-1/(2d-1)+\epsilon }) for arbitrarily small ϵ > 0 \epsilon >0 .

An Ultrasonic Heading Goniometer Intrinsically Robust to Magnetic Interference

An ultrasonic heading measurement method, working under magnetic interference prohibitive for magnetometers, was prototyped, validated, and metrologically characterized. Two capacitive ultrasonic transducers convert the mechanical rotation in two correspondingly time-delayed electrical sine waves. Then, the time delay is estimated using the standard tree parameter sine-fitting algorithm. The prototyped goniometer achieves the same repeatability level (<36 mrad) of a magnetometer-based heading in the range [−437 mrad, 437 mrad]. Simultaneously, a throughput of 505 Sa/s is proven on an STM32F303xC Arm Cortex −M4 32-bit microcontroller. An interference analysis revealed the experimental deterministic error well explained by the combined effect of beacon and receiver directivity, as well as by the relative position of beacon, receiver, and reflective surfaces. Noise robustness was assessed in case of SNR decayed to 9.1 dB from the initial value of 36.6 dB; the maximum deterministic error in the range [−437 mrad, 437 mrad] increased less than 10% (from 21 to 23 mrad in absolute value).

Energy Fluctuations in the Homogenized Hyper-Elastic Particulate Composites with Stochastic Interface Defects

The principle aim of this study is to analyze deformation energy of hyper-elastic particulate composites, which is the basis for their further probabilistic homogenization. These composites have some uncertain interface defects, which are modelled as small semi-spheres with random radius and with bases positioned on the particle-matrix interface. These defects are smeared into thin layer of the interphase surrounding the reinforcing particle introduced as the third component of this composite. Matrix properties are determined from the experimental tests of Laripur LPR 5020 High Density Polyurethane (HDPU). It is strengthened with the Carbon Black particles of spherical shape. The Arruda–Boyce potential has been selected for numerical experiments as fitting the best stress-strain curves for the matrix behavior. A homogenization procedure is numerically implemented using the cubic Representative Volume Element (RVE). Spherical particle is located centrally, and computations of deformation energy probabilistic characteristics are carried out using the Iterative Stochastic Finite Element Method (ISFEM). This ISFEM is implemented in the algebra system MAPLE 2019 as dual approach based upon the stochastic perturbation method and, independently, upon a classical Monte-Carlo simulation, and uniform uniaxial deformations of this RVE are determined in the system ABAQUS and its 20-noded solid hexahedral finite elements. Computational experiments include initial deterministic numerical error analysis and the basic probabilistic characteristics, i.e., expectations, deviations, skewness and kurtosis of the deformation energy. They are performed for various expected values of the defects volume fraction. We analyze numerically (1) if randomness of homogenized deformation energy can correspond to the normal distribution, (2) how variability of the interface defects volume fraction affects the deterministic and stochastic characteristics of composite deformation energy and (3) whether the stochastic perturbation method is efficient in deformation energy computations (and in FEM analysis) of hyper-elastic media.

Multi-Antenna Aided Secrecy Beamforming Optimization for Wirelessly Powered HetNets

The new paradigm of wirelessly powered two-tier heterogeneous networks (HetNets) is considered in this paper. Specifically, the femtocell base station (FBS) is powered by a power beacon (PB) and transmits confidential information to a legitimate femtocell user (FU) in the presence of a potential eavesdropper (EVE) and a macro base station (MBS). In this scenario, we investigate the secrecy beamforming design under three different levels of FBS-EVE channel state information (CSI), namely, the perfect, imperfect and completely unknown FBS-EVE CSI. Firstly, given the perfect global CSI at the FBS, the PB energy covariance matrix, the FBS information covariance matrix and the time splitting factor are jointly optimized aiming for perfect secrecy rate maximization. Upon assuming the imperfect FBS-EVE CSI, the worst-case and outage-constrained SRM problems corresponding to deterministic and statistical CSI errors are investigated, respectively. Furthermore, considering the more realistic case of unknown FBS-EVE CSI, the artificial noise (AN) aided secrecy beamforming design is studied. Our analysis reveals that for all above cases both the optimal PB energy and FBS information secrecy beamformings are of rank-1. Moreover, for all considered cases of FBS-EVE CSI, the closed-form PB energy beamforming solutions are available when the cross-tier interference constraint is inactive. Numerical simulation results demonstrate the secrecy performance advantages of all proposed secrecy beamforming designs compared to the adopted baseline algorithms.

On the use of a calibration emitter for direct position determination with single moving array in the presence of sensor gain and phase errors

The problem of direct position determination (DPD) with single moving array in the presence of deterministic sensor gain and phase errors is addressed. To reduce the loss in localization accuracy due to these errors, a calibration emitter with known position is introduced. The Cramér-Rao lower bound (CRLB) for estimating these errors using this calibration emitter are first derived in two situations that the calibration emitter signals are exactly known or unknown. An interesting property of the CRLB is concluded that its value is independent with the sensor phase errors. The paper then proceeds to propose a two step procedure where the considered errors are estimated in the first step and the positions of other emitters are determined by DPD given the estimated errors in the second step. In the first step, the two situations introduced above are respectively considered. For the former, the errors are jointly estimated based on Maximum Likelihood with analytical solution. For the latter, a method that estimating the sensor gain and phase errors in sequence is proposed, which performs independently with the phase errors as the CRLB does. Furthermore, analytical solutions of the errors without estimation accuracy degradation can be provided by this method. Besides, the theoretical localization bias caused by these errors is also provided. Numerical simulations are used to examine the performance of the proposed method and corroborate the theoretical results at last.

Reliable Bounding Zones and Inconsistency Measures for GPS Positioning using Geometrical Constraints

Reliable confidence domains for Global Navigation Satellite System (GNSS) positioning and inconsistency measures of the observations are of great importance for any navigation system, especially for safety critical applications. In this work, deterministic error bounds are introduced in form of intervals to assess remaining observation errors. The intervals can be computed based on expert knowledge or based on a sensitivity analysis of the measurement correction process. Using convex optimization, bounding zones are computed for GPS positioning using the geometrical constraints imposed by the observation intervals. The bounding zone is a convex polytope, where exploiting only the navigation geometry, confidence domain is computed in form of zonotope. We show that the relative volume between the polytopes and the zonotope is an inconsistency measures. Small polytope volume indicates bad consistency of the observations. In extreme cases empty sets are obtained which indicates large outliers. We determine the observation intervals via sensitivity analysis of the Klobuchar ionospheric model and Saastamoinen tropospheric model. The remaining errors are treated as white noise. We explain how the shape and the volume of the polytope are related to the positioning geometry. We show that this assignment has to be interpreted with care. Furthermore, we propose a new concept of Minimum Detectable Biases (MDB). Taking GPS data from simulations and real experiments, a comparison analysis between the proposed deterministic bounding method and the classical least-squares adjustment has been conduct in terms of accuracy and reliability. This helps validating that our proposed deterministic bound methods shows high internal and external reliability compared to the probabilistic approaches and that it provides rigorous inconsistency measures.

Self-Calibration direct position determination using a single moving array with sensor gain and phase errors

The problem of direct position determination (DPD) using a single moving array in the presence of deterministic sensor gain and phase errors is considered. To eliminate the localization bias caused by these errors, an eigenstructure based self-calibrating DPD method is first introduced, in which the sensor gain and phase errors and the emitter positions are jointly estimated by an iterative process. Considering the performance deterioration of eigenstructure methods when the signal to noise ratio or the number of samples is not sufficiently large, a maximum likelihood (ML) based two-step self-calibration approach for DPD is subsequently proposed. The sensor gain errors are provided using the diagonal of the covariance matrix of the array output by a closed form solution at the first step. Then, the phase errors and the emitter positions are jointly estimated by an iterative scheme based on ML, in which the phase errors are also determined by a closed form solution in each iteration. Besides, detailed analyses and discussions about the differences between the introduced eigenstructure based and the proposed ML based self-calibration DPD methods are also provided. At last, numerical simulations are involved to examine their performance.

Outliers in the spectrum for products of independent random matrices

Pour un $m\geq 1$ fixé, nous considérons le produit de $m$ matrices aléatoires indépendantes de taille $n\times n$, à coefficients i.i.d., lorsque $n\to \infty $. Sous certaines hypothèses sur les coefficients de chaque matrice, il est connu que la loi empirique limite des valeurs propres est décrite par la puissance $m$-ième de la loi circulaire. De plus, cette même loi limite apparaît toujours si chacune des matrices i.i.d. est perturbée additivement par une erreur déterministe de rang borné. Néanmoins, les perturbations de rang borné peuvent créer quelques valeurs propres atypiques (outliers). Nous décrivons la localisation asymptotique de ces valeurs propres atypiques, ce qui généralise un résultat de Tao (Probab. Theory Related Fields 155 (2013) 231–263) dans la cas d’une seule matrice i.i.d. Nos méthodes nous permettent également de considérer d’autres types de perturbations, dont des perturbations multiplicatives.

Plato

Plato provides fast approximate analytics on time series, by precomputing and storing compressed time series. Plato's key novelty is the delivery of tight deterministic error guarantees for the linear algebra operators over vectors/time series, the inner product operator and arithmetic operators. Composing them allows for evaluating common statistics, such as correlation and cross-correlation. In the offline processing phase, Plato (i) segments each time series into several disjoint segmentations using known fixed-length or variable-length segmentation algorithms; (ii) compresses each segment by a compression function that is coming from a user-chosen compression function family; and (iii) associates to each segment 1 to 3 precomputed error measures. In the online query processing phase, Plato uses the error measures to compute the error guarantees. Importantly, we identify certain compression function families that lead to theoretically and experimentally higher quality guarantees.