Sensor Position Errors Research Articles

A Study of the Influence of Calibration Uncertainty on the Global Uncertainty for Digital Image Correlation Using a Monte Carlo Approach

Stereo digital image correlation (DIC) is now a standard measurement technique. It is, therefore, important to quantify the measurement uncertainties when using it for experiments. Because of the complexity of the DIC measurement process, a Monte Carlo approach is presented as a method to discover the magnitude of the stereo-DIC calibration uncertainty. Then, the calibration errors, along with an assumed sensor position error, are propagated through the stereo-triangulation process to find the uncertainty in three-dimensional position and object motion. Details on the statistical results of the calibration parameters are presented, with estimated errors for different calibration targets and calibration image quality. A sensitivity study was done to look at the influence of the different calibration error sources. Details on the best approach for propagating the errors from a statistical perspective are discussed, including the importance of using a “boot-strap” approach for error propagation because of the covariance of many of the calibration parameters. The calibration and error propagation results are then interpreted to provide some best-practices guidelines for DIC.

Mapping Implementation for Multi-robot System with Glyph Localisation

Abstract An important prerequisite for creation of autonomous robot is the ability to create the map of the environment. Robots need to be able to construct a map of the environment and to use it for navigation. In this paper a multirobot mapping approach is proposed, where artificial landmarks (glyphs) are used for robot localisation and only simple sensors - bumper sensors - are available for obstacle detection. The proposed mapping system creates the map of the environment and allows changing the map rapidly in case of inaccurate robot position measurements or sensor errors. The system has a central server and robots use wireless network to communicate with the server. The mapping implementation takes into account the fact that the network has limited bandwidth and only limited amount of data can be sent.

A Semidefinite Relaxation Method for Source Localization Using TDOA and FDOA Measurements

Localization by a sensor network has been extensively studied. In this paper, we address the source localization problem by using time-difference-of-arrival (TDOA) and frequency-difference-of-arrival (FDOA) measurements. Owing to the nonconvex nature of the maximum-likelihood (ML) estimation problem, it is difficult to obtain its globally optimal solution without a good initial estimate. Thus, we reformulate the localization problem as a weighted least squares (WLS) problem and perform semidefinite relaxation (SDR) to obtain a convex semidefinite programming (SDP) problem. Although SDP is a relaxation of the original WLS problem, it facilitates accurate estimate without postprocessing. Moreover, this method is extended to solve the localization problem when there are errors in sensor positions and velocities. Simulation results show that the proposed method achieves a significant performance improvement over existing methods.

A Robust Algorithm for Multiple Disjoint Moving Sources Localization with Erroneous Sensor Locations

This paper considers the problem of multiple disjoint sources localization using time difference of arrivals (TDOA) and frequency difference of arrival (FDOA) with sensor position and velocity errors. By exploiting the observation that the multiple disjoint sources exhibit the same sensor location errors, the influence of sensor location errors can be greatly alleviated. The CTLS technique, as a natural extension of LS when noise occurs in all coefficients and the noise components of the coefficients are algebraically related, is more appropriate for the above problem, and a localization algorithm based on constrained total least-squares (CTLS) for estimating the positions and velocities of multiple disjoint moving sources with sensor location errors is proposed. Simulation results show that the proposed estimator achieves remarkably better performance than the two-step weighted least squares (WLS) approach, which makes the Crame r-Rao lower bound (CRLB) at a sufficiently high noise level before the threshold effect occurs.  Index Terms—first term, second term, third term, fourth term, fifth term, sixth term

Design and Implementation of a Nine-Axis Inertial Measurement Unit

A nine-axis inertial measurement unit (IMU) that utilizes three-axis angular velocity measurements from rate gyroscopes and six-axis linear acceleration measurements from three two-axis accelerometers is reported. This system can derive linear acceleration, angular acceleration, and angular velocity via simple memoryless matrix operations, and eliminates the requirement for accelerometer installation at the center of mass as in the traditional IMU. An optimal configuration of the system is proposed based on the analysis of rigid body dynamics and matrix theory. In this configuration, the computed angular acceleration is free of the gravity effect as well. Analyses of sensor position and orientation errors are reported. Experimental validation was executed to evaluate the performance of the system.

An Asymptotically Efficient Estimator for TDOA and FDOA Positioning of Multiple Disjoint Sources in the Presence of Sensor Location Uncertainties

This correspondence considers the problem of locating multiple disjoint sources using time differences of arrival (TDOAs) and frequency differences of arrival (FDOAs) in the presence of sensor position and velocity errors. Previous work applies to one source only with suboptimum performance for near source, or requires joint estimation of the source and sensor locations that could be computation demanding. Through nonlinearly transforming the measurements and converting them with respect to the inaccurate sensor locations, we have developed an algebraic solution to this problem. The solution is shown analytically to achieve the Cramer-Rao lower bound (CRLB) performance over small noise region and does not require joint estimation with sensor locations.

基于辅助阵元的阵列误差有源校正算法

The array mutual coupling, gain-phase errors and sensor position errors would significantly degrade the performance of high-resolution direction of arrival (DOA) estimation algorithms. Aiming at the combined influences of the above three array errors, a kind of active calibration algorithm is presented with the help of instrumental sensors in this paper. Firstly, the integrated effects of the three array errors are shown to be equivalent to angularly dependent gain-phase errors. Then, a non-linear least square (LS) optimization model for the three array errors is established through an auto-calibration algorithm using auxiliary sensors, and the corresponding resolving algorithms of the LS optimization problem are given according to the different models of array errors matrix. Next, the parameter estimation uniqueness of the proposed algorithms is analyzed in detail and an improvement strategy is given. Finally, the effectiveness of the novel algorithm is validated by simulations.

Array errors active calibration algorithm based on instrumental sensors

The array mutual coupling, gain-phase errors and sensor position errors would significantly degrade the performance of high-resolution direction of arrival (DOA) estimation algorithms. Aiming at the combined influences of the above three array errors, a kind of active calibration algorithm is presented with the help of instrumental sensors in this paper. Firstly, the integrated effects of the three array errors are shown to be equivalent to angularly dependent gain-phase errors. Then, a non-linear least square (LS) optimization model for the three array errors is established through an auto-calibration algorithm using auxiliary sensors, and the corresponding resolving algorithms of the LS optimization problem are given according to the different models of array errors matrix. Next, the parameter estimation uniqueness of the proposed algorithms is analyzed in detail and an improvement strategy is given. Finally, the effectiveness of the novel algorithm is validated by simulations.

Array errors active calibration algorithm and its improvement

The combined effects of mutual coupling, gain and phase errors, and sensor position errors have great negative impact on the direction-finding performance of the MUSIC algorithm. This paper investigates the calibration of the array errors induced by the three errors. Based on the array errors calibration algorithm (algorithm 1) presented by See, two improved array error-calibrating algorithms are presented. One (algorithm 2) uses the sparseness of mutual coupling matrix, and the other (algorithm 3) utilizes more special structures of mutual coupling matrix. Although the three algorithms have the same computation mode and theoretical framework, the latter two algorithms have much higher precision than the former one, as more properties of mutual coupling matrix are utilized in the two algorithms, and if the array manifold is uniform linear array or uniform circular array, algorithm 3 could have more obvious advantages. On the other hand, the three algorithms can be generalized to the situation when the calibration source azimuths have deviations. The three extended algorithms can not only calibrate the array errors, but also offset the deviations of calibration source azimuths simultaneously. Finally, the performances of the three algorithms are compared both in the absence and presence of the deviations of calibration source azimuths. Many simulation experiments demonstrate that the estimation precision of array errors can be obviously improved if more properties of mutual coupling matrix are effectively used.

Alleviating Sensor Position Error in Source Localization Using Calibration Emitters at Inaccurate Locations

A previous study shows that the use of a calibration emitter whose position is known exactly can significantly reduce the loss in time differences of arrival (TDOA) based source localization accuracy when the available sensor positions have random errors. This paper extends the previous work to a more practical scenario where the exact position of a calibration emitter is not known. By modeling the calibration position error as additive Gaussian noise, the amount of reduction in localization accuracy due to calibration position error is derived through Cramer-Rao lower bound (CRLB) analysis. In addition, the analysis also affirms the previous studies on Bayesian sensor network localization that it remains possible to improve the localization accuracy even if the calibration position is completely unknown. Next, a performance analysis illustrates that the penalty could be very high if one simply pretends the calibration position is accurate and ignores its error. A closed-form solution is then developed by accounting for the calibration position error and it is proved analytically to reach the CRLB accuracy when the sensor and calibration position errors are small relative to the distance between the calibration emitter and the sensor. Finally, the results are generalized to the case where multiple calibration emitters are available. When deploying multiple calibration emitters, although their positions may not be known exactly, we show that it is possible to completely eliminate the sensor position error and recover the best localization accuracy that is limited by the measurement noise in TDOAs only. All the theoretical developments are corroborated by simulations.

An Approximately Efficient TDOA Localization Algorithm in Closed-Form for Locating Multiple Disjoint Sources With Erroneous Sensor Positions

This paper considers the problem of time difference-of-arrival (TDOA) source localization when the TDOA measurements from multiple disjoint sources are subject to the same sensor position displacements from the available sensor positions. This is a challenging problem and closed-form solution with good localization accuracy has yet to be found. This paper proposes an estimator that can achieve this purpose. The proposed algorithm jointly estimates the unknown source and sensor positions to take the advantage that the TDOAs from different sources have the same sensor position displacements. The joint estimation is a highly nonlinear problem due to the coupling of source and sensor positions in the measurement equations. We introduce the novel idea of hypothesized source locations in the algorithm development to enable the formulation of psuedolinear equations, thereby leading to the establishment of closed-form solution for source location estimates. Besides the advantage of closed-form, the newly developed algorithm is shown analytically, under the condition that the TDOA measurement noise and the sensor position errors are sufficiently small, to reach the CRLB accuracy. For clarity, the localization of two disjoint sources is used in the algorithm development. The developed algorithm is then examined under the special case of a single source and extended to the more general case of more than two unknown sources. The theoretical developments are supported by simulations.

A Generic Method for RPC Refinement Using Ground Control Information

Geometric sensor models are used in image processing to model the relationship between object space and image space and to transform image data to conform to a map projection. An Rational Polynomial Coefficient (RCP) is a generic sensor model that can be used to transform images from a variety of different high resolution satellite and airborne remote sensing systems. To date, numerous researchers have published papers about RPC refinement, aimed at improving the accuracy of the results. So far, the Bias Compensation method is the one that is the most accepted and widely used, but this method has rigorous conditions that limit its use; namely, it can only be used to improve the RPC of images obtained from cameras with a narrow field of view and small attitude errors, such as those used on Ikonos or QuickBird satellites. In many cases, these rigorous conditions may not be satisfied (e.g., cameras with a wide field of view and some satellites with large ephemeris and attitude errors). Therefore, a more robust method that can be used to refine the RPC under a wider range of conditions is desirable. In this paper, a generic method for RPC refinement is proposed. The method first restores the sensor's pseudo position and attitude, then adjusts these parameters using ground control points. Finally a new RPC is generated based on the sensor's adjusted position and attitude. We commence our paper with a review of the previous ten years of research directed toward RPC refinement, and compare the characteristics of different refinement methods that have been proposed to date. We then present a methodology for a proposed generic method for RPC refinement and describe the results of two sets of experiments that compare the proposed Generic method with the Bias Compensation method. The results confirm that the Bias Compensation method works well only when the aforementioned rigorous conditions are met. The accuracy of the RPC refined by the Bias Compensation method decreased rapidly with the sensor's position error and attitude error. In contrast to this, the Generic method proposed in this paper was found to yield highly accurate results under a variety of different sensor positions and attitudes.

On the Use of a Calibration Emitter for Source Localization in the Presence of Sensor Position Uncertainty

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Sensor position uncertainty is known to degrade significantly the source localization accuracy. This paper investigates the use of a single calibration emitter, whose position is known to the sensor array, to reduce the loss in localization accuracy due to sensor position errors that are random. Using a Gaussian noise model, we first derive the CramÉr–Rao lower bound (CRLB) for a time difference of arrival (TDOA)-based source location estimate with the use of a calibration source. The differential calibration technique that is commonly used in Global Positioning System through the use of a calibration source to mitigate the inaccuracy in satellite ephemeris data is analyzed. The analysis indicates that differential calibration in most cases cannot reach the CRLB accuracy. The paper then proceeds to propose an algebraic closed-form solution for the source location estimate using both TDOA measurements from the unknown and the calibration source. The proposed algorithm is shown analytically, under high signal-to-noise ratio (SNR) and small sensor position noise, or under moderate level of SNR and sensor position noise together with distant unknown and calibration sources, to reach the CRLB accuracy. Simulations are used to corroborate and support the theoretical development. </para>

Ultra-tight GPS/INS/PL integration: a system concept and performance analysis

The architecture of the ultra-tight GPS/INS/PL integration is the key to its successful performance; the main feature of this architecture is the Doppler feedback to the GPS receiver tracking loops. This Doppler derived from INS, when integrated with the carrier tracking loops, removes the Doppler due to vehicle dynamics from the GPS/PL signal thereby achieving a significant reduction in the carrier tracking loop bandwidth. The bandwidth reduction provides several advantages such as: improvement in anti-jamming performance, and increase in post correlated signal strength which in turn increases the dynamic range and accuracy of measurements. Therefore, any degradation in the derived Doppler estimates will directly affect the tracking loop bandwidth and hence its performance. The quadrature signals from the receiver correlator, I (in-phase) and Q (quadrature), form the measurements, whereas the inertial sensor errors, position, velocity and attitude errors form the states of the complementary Kalman filter. To specify a reliable measurement model of the filter for this type of integrated system, a good understanding of GPS/PL signal characteristics is essential. It is shown in this paper that phase and frequency errors are the variables that relate the measurements and the states in the Kalman filter. The main focus of this paper is to establish the fundamental mathematical relationships that form the measurement model, and to show explicitly how the system error states are related to the GPS/PL signals. The derived mathematical relationships encapsulated in a Kalman filter, are tested by simulation and shown to be valid.

Bayesian Mitigation of Sensor Position Errors to Improve Unexploded Ordnance Detection

Phenomenological modeling coupled with statistical signal processing has been shown to significantly improve capabilities for discriminating unexploded ordnance (UXO) from benign clutter using electromagnetic induction (EMI) sensor data. The general premise underlying the majority of these coupled approaches is that a phenomenological model is fit to the measured data, and the parameters estimated from this model inversion, which characterize the interrogated target, are utilized in subsequent statistical signal processing algorithms to classify the target as either UXO or clutter. A potential limitation of this coupled approach is that the inversion has been shown to be sensitive to uncertainty associated with the sensor positions. When the measurement positions are uncertain, the inversion results are more variable, and consequently, discrimination performance degrades. In this letter, a Bayesian methodology is applied to estimate the desired features from the measured data. This method explicitly acknowledges that uncertainty in the sensor positions exists and incorporates this knowledge to find the maximum-likelihood feature estimates by integrating over the uncertain measurement positions. Due to the high dimensionality of the integration, Monte Carlo integration, a statistical technique to estimate the value of an integral, is employed. Simulation results show that this Bayesian approach in mitigating sensor position uncertainty produces features with lower variability and, therefore, provides improved discrimination performance.

Performance Capabilities of Long-Range UWB-IR TDOA Localization Systems

The theoretical and practical performance limits of a 2D ultra-wideband impulse-radio localization system operating in the far field are studied under the assumption that estimates of location are based on time-difference-of-arrival (TDOA) measurements. Performance is evaluated in the presence of errors in both the TDOA measurements and the sensor locations. The performance of both optimal (maximum-likelihood) and suboptimal location estimation algorithms is studied and compared with the theoretical performance limit defined by the Cramer-Rao lower bound on the variance of unbiased TDOA location estimates. A novel weighted total-least-squares algorithm is introduced that compensates somewhat for errors in sensor positions and reduces the bias in location estimation compared with a widely used weighted least-squares approach. In addition, although target tracking per se is not considered in this paper, performance is evaluated both under the assumption that sequential location estimates are not aggregated as well as under the assumption that some sort of tracker is available to aggregate a sequence of estimates.

Adaptive Beamforming Using Frequency Invariant Uniform Concentric Circular Arrays

This paper proposes new adaptive beamforming algorithms for a class of uniform concentric circular arrays (UCCAs) having near-frequency invariant characteristics. The basic principle of the UCCA frequency invariant beamformer (FIB) is to transform the received signals to the phase mode representation and remove the frequency dependence of individual phase modes through the use of a digital beamforming or compensation network. As a result, the far field pattern of the array is electronic steerable and is approximately invariant over a wider range of frequencies than the uniform circular arrays (UCAs). The beampattern is governed by a small set of variable beamformer weights. Based on the minimum variance distortionless response (MVDR) and generalized sidelobe canceller (GSC) methods, new recursive adaptive beamforming algorithms for UCCA-FIB are proposed. In addition, robust versions of these adaptive beamforming algorithms for mitigating direction-of-arrival (DOA) and sensor position errors are developed. Simulation results show that the proposed adaptive UCCA-FIBs converge much faster and reach a considerable lower steady-state error than conventional broadband UCCA beamformers without using the compensation network. Since fewer variable multipliers are required in the proposed algorithms, it also leads to lower arithmetic complexity and faster tracking performance than conventional methods.

Sound Field Analysis Based on Analytical Beamforming

The plane wave decomposition is an efficient analysis tool for multidimensional fields, particularly well fitted to the description of sound fields, whether these ones are continuous or discrete, obtained by a microphone array. In this article, a beamforming algorithm is presented in order to estimate the plane wave decomposition of the initial sound field. Our algorithm aims at deriving a spatial filter which preserves only the sound field component coming from a single direction and rejects the others. The originality of our approach is that the criterion uses a continuous instead of a discrete set of incidence directions to derive the tap vector. Then, a spatial filter bank is used to perform a global analysis of sound fields. The efficiency of our approach and its robustness to sensor noise and position errors are demonstrated through simulations. Finally, the influence of microphone directivity characteristics is also investigated.

Adaptive array beamforming with robust capabilities under random sensor position errors

The problem of adaptive array beamforming with multiple-beam constraints in the presence of steering error caused by random sensor position errors is considered. First the statistical relationship between the random sensor position errors and the induced random phase perturbation is derived. Based on the result, a cost function consisting of terms which utilise a posteriori information owing to the received array data and a priori information owing to the probabilistic distribution of the resulting phase perturbation, respectively, is constructed. Then, an appropriate estimate of the actual phase angle vector associated with each of the desired signals can be obtained by performing a nonlinear optimisation problem. An implementation algorithm is further presented to solve iteratively the problem. Theoretical analysis regarding the convergence property of the iterative procedure is also investigated. Finally, several computer simulation examples are provided for demonstrating the effectiveness of the proposed technique.

Array calibration of angularly dependent gain and phase uncertainties with carry-on instrumental sensors

Array calibration with angularly dependent gain and phase uncertainties has long been a difficult problem. Although many array calibration methods have been reported extensively in the literature, they almost all assumed an angularly independent model for array uncertainties. Few calibration methods have been developed for the angularly dependent array uncertainties. A novel and efficient auto-calibration method for angularly dependent gain and phase uncertainties is proposed in this paper, which is called ISM (Instrumental Sensors Method). With the help of a few well-calibrated instrumental sensors, the ISM is able to achieve favorable and unambiguous direction-of-arrivals (DOAs) estimate and the corresponding angularly dependent gain and phase estimate simultaneously, even in the case of multiple non-disjoint sources. Since the mutual coupling and sensor position errors can all be described as angularly dependent gain/phase uncertainties, the ISM proposed still works in the presence of a combination of all these array perturbations. The ISM can be applied to arbitrary array geometries including linear arrays. The ISM is computationally efficient and requires only one-dimensional search, with no high-dimensional nonlinear search and convergence burden involved. Besides, no small error assumption is made, which is always an essential prerequisite for many existing array calibration techniques. The estimation performance of the ISM is analyzed theoretically and simulation results are provided to demonstrate the effectiveness and behavior of the proposed ISM.