Constrained Total Least Squares Research Articles

An AOA Optimal Positioning Method Incorporating Station Error and Sensor Deployment

In order to improve the computational accuracy of the AOA (angle of arrival) location, an AOA location method based on CTLS (constrained total least squares) and incorporating the effect of station errors is investigated. Its approximate closed-form solution is derived by first calculating the least squares solution, then substituting the least squares solution into the approximate closed-form solution and iterating to calculate the final result. Subsequently, since the result of the Fisher information matrix is minimized when the stations are uniformly distributed, the final accuracy of the AOA target location is improved by finding the optimal relative position deployment strategy for the sensor-carrying station in 3D space, under the assumption that the sensor carriers are uniformly distributed. Simulations show that, in the presence of station errors, the accuracy of AOA location is improved by approximately 10% using this paper’s calculation method, compared to the common least squares location method. In the process of target location, the Fisher information matrix is used to calculate the lower bound of the location result for the CRLB (Cramer–Rao lower bound), which is optimal when the azimuth is uniformly distributed and all on the same horizontal plane, if its pitch angle is 35.56°.

Efficient and Robust Solution to Universal Symmetric Transformation for 3-D Point Sets

Recently, the 3D symmetric transformation, which can simultaneously consider the measurement errors of the source and target coordinates, has become a hot research topic especially in geodetic datum transformation and point clouds registration. However, there are rare studies on outlier resisting for 3D symmetric transformation and most of them only focused on similarity transformation. In this study, to unify affine, similarity, and rigid transformations, a universal expression of 3D symmetric transformation was abstracted as a partial EIV model with equality constraints. Then the equality constrained partial EIV model was solved by a proposed constrained total least squares (CTLS) algorithm and the corresponding efficient formulae were derived. Finally, a robust CTLS (RCTLS) algorithm of universal 3D symmetric transformation was presented by introducing the equivalent weight principle of robust estimation. Unlike the existing algorithms, the RCTLS can efficiently deal with all kinds of 3D symmetric transformation problems influenced by outliers with a unified formula framework. To verify the performance of the RCTLS, comparative experiments were employed in two simulated scenarios of similarity and affine transformation and a real-world rigid transformation scenario for point clouds registration, respectively. The proposed RCTLS was compared with three non-symmetric transformation algorithms (dual quaternion-based algorithm, linear least squares and singular value decomposition) and two symmetric transformation algorithms (outlier-detected TLS (OD-TLS) and CTLS) in terms of accuracy, robustness and computational efficiency. The results indicated both OD-TLS and RCTLS are robust to outliers with higher transformation accuracy, but the proposed RCTLS is more universal and efficient than OD-TLS.

Constrained total least squares method using TDOA measurements for jointly estimating acoustic emission source and wave velocity

The measurements of time-difference-of-arrival (TDOA) and wave velocity have significant influence on localization accuracy. To reduce the loss in localization accuracy induced by TDOA noises and velocity errors, the constrained total least squares (CTLS) method using the TDOA measurements for jointly estimating AE source and wave velocity is proposed in this paper. The proposed method exploits the CTLS principle of unknown wave velocity, and the Newton's iterative method is combined to give the optimal solution. The simulation tests are implemented to quantitatively analyze the performance of the proposed method, and the results show that the location accuracy and velocity accuracy of the proposed method are optimal and can attain the Cramér–Rao lower bound (CRLB). In addition, the experiment results of pencil-lead breaks further indicate that the proposed method can achieve higher positioning accuracy than traditional methods with a relatively low computational cost.

A total least squares solution to a 3D coordinate transformation parameters of large Euler angles with closure constrain

A new method for calculating three-dimensional (3D) coordinate transformation parameters of large Euler angles with closure constrain is presented. This method obtains the initial value by traditional seven-parameter model. Then we derive the closure constraint condition and linearize it. The constrained total least-squares (CTLS) approach is adopted to obtain corrected values under the closure constraint condition. A practical case study is presented to illustrate the effectiveness of this new model.

The Constrained Total Least Squares Solution for Virtual Reference Feedback Tuning

Abstract Virtual Reference Feedback Tuning (VRFT) is a direct data-driven control design method employed to tune a controller’s parameters aiming to achieve a prescribed closed-loop performance. Its primary formulation leads to a biased estimate in the presence of noise, so an instrumental variable (IV) alternative has been proposed and this alternative has been favoured whenever the noise level is significant. Even though VRFT thus formulated has been very successful, the bias reduction through the IV approach comes at the cost of an important increase in the variance of the parameters’ estimate. In this work we propose a different solution for the parameters estimation in VRFT which reduces bias without increasing the variance — the Constrained Total Least Squares (CTLS). The effectiveness of the proposed solution is illustrated by three case studies, showing that the mean square error of the parameters’ estimate is smaller when compared to previously proposed solutions and, most importantly, that the closed-loop performance is significantly better.

Closed form algorithm of double-satellite TDOA + AOA localization based on WGS-84 model

In this paper, we consider the double-satellite localization under the earth ellipsoid model of the Wideband Geodetic System (WGS-84) using the Time Difference of Arrival (TDOA) and the Angle-of-Arrival (AOA). Several closed-form solution algorithms via the pseudo-linearization of the measurement equations are presented to efficiently estimate the location. These algorithms include the Weighted Least Squares (WLS), the Constrained Total Least Squares (CTLS), and the Taylor-Series Iteration (TSI). Performance comparison of the proposed methods with the Cramér-Rao Lower Bound (CRLB) in the simulation is shown to demonstrate that the proposed algorithms are feasible and have stable performance.

Algorithms used in restraining random noise in measurements from a geomagnetic navigation magnetometer

Random error compensation is a key to high-precision geomagnetism measurements. A parameterized magnetometer measurement error model was established, where the error parameters are calibrated using different calibration algorithms. The standard least squares (LS) method supposes that random error compensation has a Gaussian distribution; however, random errors in a magnetometer observation equation have a non-Gaussian distribution. The robust total least squares (RTLS) method considers errors in both the coefficient matrix and vector data, with gross errors in the coefficient matrix. The constrained total least squares (CTLS) method calculates the constraint relationships of all error items synthetically, which is suitable for solving the linear optimization problems of associated errors. Owing to the fact that a geomagnetic measurement random error model is seriously sick, the truncated total least squares (TTLS) method was proposed to deal with ill-posed problems, as well as random errors existing on both sides of the observation equation. The simulation results demonstrate that the TTLS can inhibit the pathological effects on the magnetometer measurements well, and that the residual between the real data and the data after TTLS calibration is less than in other calibration results.

A Constrained-Total-Least-Squares Method for Joint Estimation of Source and Sensor Locations: A General Framework

It is well known that sensor location uncertainties can significantly deteriorate the source positioning accuracy. Therefore, improving the sensor locations is necessary in order to achieve better localization performance. In this paper, a constrained-total-least-squares (CTLS) method for simultaneously locating multiple disjoint sources and refining the erroneous sensor positions is presented. Unlike conventional localization techniques, an important feature of the proposed method is that it establishes a general framework that is suitable for many different location measurements. First, a modified CTLS optimization problem is formulated after some algebraic manipulations and then the corresponding Newton iterative algorithm is derived to give the numerical solution. Subsequently, by using the first-order perturbation analysis, the explicit expression for the covariance matrix of the proposed CTLS estimator is deduced under the Gaussian assumption. Moreover, the estimation accuracy of the CTLS method is shown to achieve the Cramér-Rao bound (CRB) before the thresholding effect occurs by a rigorous proof. Finally, two kinds of numerical examples are given to corroborate the theoretical development in this paper. One uses the TDOAs/GROAs measurements and the other is based on the TOAs/FOAs parameters.

Total Least-Squares Estimation for 2D Affine Coordinate Transformation with Constraints on Physical Parameters

AbstractThe error-in-variable (EIV) model takes the error of all variables into account and has been widely applied to many practical problems arising in environmental study, geology, geographic information science (GIS), and geodesy. Coordinate transformations are among the most frequently encountered problems in spatial data processing, and the EIV model can be built based on two sets of coordinates. In some applications, physical parameters—such as the shift, rotation angle, and scale factor—have constraints. Current implementations of the constrained EIV (CEIV) model do not consider physical constraints explicitly. The purpose of this paper is to convert physical constraints into mathematical forms and to use constrained total least squares (CTLS) to solve the CEIV problem of two-dimensional (2D) affine transformation. The effectiveness of this method is illustrated through a numerical example.

Theory and Algorithm for Planar Datum Establishment Using Constrained Total Least-squares

First, we present an efficient algorithm for establishing planar datums that is based on a constrained minimization search based on the L2 norm after forming a convex surface from sampled points. Visualized by Gauss maps, we prove that the problem reduces to a minimization search where the global minimum is localized about the minimizing facet. Second, we highlight advantages of this planar datum, including the major advantage that the datum planes have full mechanical contact with the datum features in stable cases yet are automatically balanced for rocking conditions. These advantages make this definition appealing for standardization.

基于外辐射源的约束总体最小二乘定位算法及其理论性能分析

A passive location and tracking system that locates “silent” targets using an external illuminator is of great importance in a variety of applications. Based on the spatial-, time-, and frequency-domain measures of this system, a constrained total least squares (CTLS) localization theory and a method that takes into account observer location and velocity errors are proposed. First, four kinds of joint location mechanisms are summarized, and the algebraic processing method that transforms nonlinear measurement equations into pseudo-linear equalities was considered. Then, the CTLS optimization model was constructed for locating the target, and the Newton iteration algorithm to solve the optimization problem was derived. Subsequently, an analytical performance analysis was performed for the CTLS location algorithm including observer location and velocity errors; thus, a performance analysis frame was built. In addition, the CTLS location variance was mathematically proved to be able to reach the corresponding Cramer-Rao lower bound (CRLB). Finally, some numerical experiments were devised to corroborate the advantage of the presented algorithm and the effectiveness of the theoretical analysis.

The constrained-total-least-squares localization algorithm and performance analysis based on an external illuminator

A passive location and tracking system that locates silent targets using an external illuminator is of great importance in a variety of applications. Based on the spatial-, time-, and frequency-domain measures of this system, a constrained total least squares(CTLS) localization theory and a method that takes into account observer location and velocity errors are proposed. First, four kinds of joint location mechanisms are summarized, and the algebraic processing method that transforms nonlinear measurement equations into pseudo-linear equalities was considered. Then, the CTLS optimization model was constructed for locating the target, and the Newton iteration algorithm to solve the optimization problem was derived. Subsequently, an analytical performance analysis was performed for the CTLS location algorithm including observer location and velocity errors;thus, a performance analysis frame was built. In addition, the CTLS location variance was mathematically proved to be able to reach the corresponding Cram′er-Rao lower bound(CRLB). Finally, some numerical experiments were devised to corroborate the advantage of the presented algorithm and the effectiveness of the theoretical analysis.

Accurate single-observer passive coherent location estimation based on TDOA and DOA

This paper investigates the problem of target position estimation with a single-observer passive coherent location (PCL) system. An approach that combines angle with time difference of arrival (ATDOA) is used to estimate the location of a target. Compared with the TDOA-only method which needs two steps, the proposed method estimates the target position more directly. The constrained total least squares (CTLS) technique is applied in this approach. It achieves the Cramer–Rao lower bound (CRLB) when the parameter measurements are subject to small Gaussian-distributed errors. Performance analysis and the CRLB of this approach are also studied. Theory verifies that the ATDOA method gets a lower CRLB than the TDOA-only method with the same TDOA measuring error. It can also be seen that the position of the target affects estimating precision. At the same time, the locations of transmitters affect the precision and its gradient direction. Compared with the TDOA, the ATDOA method can obtain more precise target position estimation. Furthermore, the proposed method accomplishes target position estimation with a single transmitter, while the TDOA-only method needs at least four transmitters to get the target position. Furthermore, the transmitters’ position errors also affect precision of estimation regularly.

Passive Location Using TDOA Measurements from Compass Satellite Illuminators

AbstractIn recent years, the passive radar detection system based on satellite borne transmitters has attracted widespread attention. In this paper, by taking the Compass satellite signal as the research object, the practicality of the Compass signal as an external illuminator of the passive radar is discussed with respect to ambiguity function. Then, a four satellite transmitters and one receiver location model based on the time difference of arrival (TDOA) is proposed and the corresponding location algorithm of the constrained total least squares (CTLS) in earth centered earth fixed coordinates is derived. Finally, the location accuracy of different algorithms is discussed and the geometric dilution of precision (GDOP) of the location system is analyzed. The comparative simulation results demonstrate that the CTLS‐based location algorithm can improve the location accuracy significantly, as the CTLS algorithm can reduce nearly 69% and 39% of the positioning error over LS and TLS algorithms, respectively.

Constrained total least-squares calibration of three-axis magnetometer for vehicular applications

Strapdown three-axis magnetometer is widely used to determine the heading angle of a vehicle to which it is attached. Unfortunately, the outputs of three-axis magnetometer are often distorted by various magnetic field sources. Therefore, a calibration procedure must be performed before it is used. This paper presents a new calibration algorithm to determine calibration parameters based on a constrained total least-squares (CTLS) technique. A Newton iteration method is utilized to determine the CTLS solution. Unlike existing calibration algorithms, the proposed algorithm yields efficient estimates without assuming a priori knowledge of the noise distribution. Thus, it is easily implemented and suitable for practical applications. The results of simulations and experiments reveal the superiority of the proposed algorithm to the compared algorithms.

Constrained Total Least Squares Algorithm Applied to Three-Axis Magnetometer Calibration

Strapdown three-axis magnetometer is becoming increasingly used in navigation systems and in biomedical applications. A calibration must be performed before the magnetometer can be used, because its outputs are often corrupted by various magnetic field sources. This paper presents a new calibration algorithm based on constrained total least squares (CTLS) to determine different calibration parameters such as sensor non-orthogonality, scale factors, offsets, hard iron and soft iron errors. The basic idea is to convert the calibration problem into an unconstrained nonlinear optimization problem. An iterative technique based on Newton’s method is applied to give a numerical solution. Simulation results show that the proposed CTLS algorithm gives more accurate solutions than compared algorithms.

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

Constrained total least-squares localisation algorithm using time difference of arrival and frequency difference of arrival measurements with sensor location uncertainties

In this study, a constrained total least-squares (CTLS) algorithm for estimating the position and velocity of a moving source with sensor location uncertainties that uses the time difference of arrival and frequency difference of arrival measurements of a signal received at a number of sensors is proposed. The CTLS method, as a natural extension of LS when noise occurs in all the data and the noise components of the coefficients are linearly dependent, is more appropriate than the LS method for the above problem. By utilising the Lagrange multipliers technique, the known relation between the intermediate variables and the source localisation coordinates has been exploited to constrain the solution. In addition, the Lagrange multipliers can be obtained efficiently and robustly, which can allow real-time implementation as well as ensure global convergence. After a perturbation analysis, the bias and covariance of the proposed CTLS algorithm are also derived, indicating that the proposed CTLS algorithm is an unbiased estimator, and it could achieve the Cramer Rao lower bound (CRLB) when the measurement noise and the sensor location errors are sufficiently small. The simulation results show that the proposed estimator achieves remarkably better performance than the TLS and two-step weighted least squares approach, which makes it possible that the CRLB is attained at a sufficiently high noise level before the threshold effect occurs.

On weighted total least-squares with linear and quadratic constraints

A weighted total least-squares (WTLS) approach with linear and quadratic constraints is developed. This method is according to the traditional Lagrange approach to optimize the target function of this problem. The WTLS and constrained total least-squares (CTLS) approach had been distinctively investigated, however, these two problems have not been simultaneously considered yet; furthermore, among the contributions on the CTLS problem, only Schaffrin and Felus considered linear and quadratic constraints together; nevertheless, in many practical examples, some elements of the design/coefficient matrix are fixed and should not be modified and this approach cannot deal with these cases. The main necessity of this research appears after the desirable property of the WTLS approach in preserving the structure of the design matrix was proven by Mahboub. In other words, currently, the WTLS approach is one of the most efficient methods for solving the so-called errors-in-variables model and an attempt for equipping it with constraints seems necessary. Also it is demonstrated that the additional constraints have a ’regularization role’ for ill-conditioned problems and the unconstrained solution suffers from ill-conditioning effects which give it an added advantage over the unconstrained WTLS algorithm. Four geodetic applications indicate the significant of this problem in the presence of colored and white noise in the data.

Reverse Engineering Partially-Known Interaction Networks from Noisy Data

AbstractOne of the most difficult challenges associated with the problem of inferring functional interaction networks from experimental data is that of dealing with the effects of measurement noise in the data used for reverse engineering. A second important challenge is that of taking full advantage of prior knowledge about some elements of the network to improve the results of the reconstruction process. This paper introduces a new inference algorithm, PACTLS, which addresses both of the above issues. The algorithm combines methods to exploit mechanisms underpinning scale–free networks generation, i.e. network growth and preferential attachment (PA), with a technique to optimally reduce the effects of measurement noise in the data on the reliability of the inference results, i.e. the Constrained Total Least Squares (CTLS) algorithm. The technique is assessed through numerical tests on in silico random networks and is shown to consistently outperform approaches based on Bayesian networks.