This paper addresses the multistatic localization problem with the receiving sensors distributed on a rigid body (RB) whose position and orientation are not known. We propose to jointly estimate the unknown object position together with the RB position and orientation, simply called joint object and rigid body localization (JORBL), using time delay (TD) measurements from both the direct-propagation and indirect-reflection paths between the transmitters and sensors. Two scenarios are considered, one with synchronization between the sensors and transmitters and the other without. We begin by investigating the benefits of having the direct-path TD measurements compared to using the indirect-path TDs only for JORBL, in terms of estimation accuracy and minimum number of transmitters and sensors needed, through analyzing the Cramér-Rao lower bound (CRLB). Afterward, in the presence of synchronization, we propose a JORBL method by formulating a non-convex constrained weighted least squares problem and applying semidefinite relaxation to obtain the solution by semidefinite programming. The proposed method is then extended to the case without synchronization. The mean squared error analysis demonstrates that the proposed methods can achieve the CRLB accuracy for small Gaussian noise, which is further confirmed by numerical simulations.

Moving rigid body localization requires the estimation of not only the position vector and rotation angles but also its translational and angular velocities, which leads to a very challenging problem. This paper demonstrates the feasibility and develops a semidefinite relaxation (SDR) method for locating a moving rigid body with constant translational and angular velocities using time-of-arrival (TOA) measurements only collected at several successive instants over a time period. More specifically, we analyze the localizability and determine the minimum numbers of anchors, sensors, and observation samples required for this particular problem. Afterward, beginning with a proposed model by transforming the TOA relation with the unknowns, we construct a constrained weighted least squares (CWLS) problem by introducing auxiliary variables. The CWLS problem is nonlinear and complicated to tackle owing to the large number of variables and the special orthogonal group constraint on the rotation matrix. To solve this difficult problem, we use SDR and transform it into a convex semidefinite programming problem to obtain a preliminary estimate. Subsequently, a more accurate solution is achieved through orthogonalization and refinement to compensate for the performance loss caused by approximations and relaxation. Theoretical mean square error analysis shows that the proposed method is able to achieve the Cramér-Rao lower bound performance under small Gaussian noise. Simulation results also demonstrate the effectiveness and robustness of the method proposed.

When we try to locate an object in a crowded urban area or an obstacle-rich environment, direct line-of-sight (LOS) propagation is usually absent and the Global Navigation Satellite System (GNSS) is often denied. In such a difficult scenario, we employ a moving transceiver that exhibits LOS propagations to connect the object and each sensor to determine its location. On the other hand, the transceiver positions and velocities are not known due to unreliable GNSS signals, which creates an extra level of challenge to the problem. By collecting the time delay and angle-of-arrival (AOA) sensor measurements with the object, two methods are proposed to address this localization problem. The first solution is efficient in computation; it linearizes the measurement model and obtains the transceiver parameters and object position by two linear weighted least squares (LWLS) estimators. The second method is resilient against large measurement noise; it solves the problem using semidefinite programming optimization by applying semidefinite relaxation. Both solutions are suboptimal due to the linearization or relaxation errors and they are refined by a separate LWLS estimator to improve the accuracy. Additionally, the theoretical analyses include the derivation of the Cramér-Rao lower bound, the contribution of AOA to the localization performance, the effect of the number of transceivers on the positioning accuracy, and the optimality of the proposed methods after refinement. The algorithms developed and the analyses performed are supported by simulations.

This paper investigates the joint source and sensor localization (JSSL) problem using hybrid time-difference-of-arrival (TDOA) and angle-of-arrival (AOA) measurements in a multi-source-multi-sensor scenario. Without the knowledge of sensor positions, this problem jointly estimates the relative positions of the sources and sensors together. We first conduct the localizability analysis by examining the rank of the Jacobian matrix, from which we obtain the minimum number of sources and sensors required for TDOA-AOA JSSL and the special configurations where the JSSL problem is not solvable. Next, we construct a novel parametric model by integrating the TDOA and AOA measurements, and use it to formulate a non-convex constrained weighted least squares (CWLS) problem. JSSL is then accomplished by solving the CWLS problem through semidefinite relaxation and semidefinite programming. Finally, theoretical analyses are conducted. It shows that the proposed CWLS solution is able to achieve the Cramér-Rao lower bound performance under small Gaussian noise. Moreover, it reveals that increasing the number of sources or sensors can improve the overall JSSL performance. Simulation results validate the theory and also demonstrate the capability of the proposed method for TDOA-AOA JSSL.

Based on the exact spherical wavefront model, an under-determined estimation method for three-dimensional (3-D) parameters of near-field (NF) sources using L-shaped nested arrays is proposed, referred to as the cumulant algorithm. This algorithm leverages the temporal-spatial domain cumulants of NF sources by constructing virtual data through delayed fourth-order cumulant (FOC) calculations of the original received data. Subsequently, a spatial-spectrum-based subspace method is applied for 3-D NF localization, which involves a 3-D spectral search procedure. Additionally, the maximum number of identifiable NF sources of the proposed algorithm is analyzed. Simulation results demonstrate that, based on the exact spherical wavefront model, the proposed algorithm can achieve underdetermined 3-D parameter estimation of NF sources without any matching process, and it performs better in localization than existing methods.

In this paper, a high-precision three-dimensional (3-D) near-field (NF) localization method is proposed under an underdetermined case based on a symmetric enhanced nested array (SENA). Firstly, the symmetry of the array and the fourth-order cumulant (FOC) are utilized to construct the equivalent virtual far-field (FF) reception data. Then, a gridless sparse and parametric approach (SPA), combined with an l1-SVD based pairing procedure, is used to obtain estimates for two paired angles. Finally, a one-dimensional (1-D) spectral estimator is applied to obtain the estimate of range parameter. Simulation results show the effectiveness of the proposed method.

This paper investigates the rigid body localization (RBL) problem in 3-D using hybrid time-of-arrival (TOA) and angle-of-arrival (AOA) measurements without synchronization among the sensors on the rigid body and with the anchors, where relays at unknown positions are exploited to improve the RBL performance by generating additional measurements in the relay propagation paths and enhance the localization geometry by having the relays as pseudo receivers. The large number of unknown parameters in the localization system leads to a very challenging problem. We propose a two-step method that jointly estimates the rotation angles and position of the rigid body, the relay positions, and the synchronization offsets. In the first step, a non-convex constrained least squares problem (CWLS) is formulated based on the transformed measurement models, and it is solved by semidefinite programming (SDP) after semidefinite relaxation (SDR). The CWLS solution is suboptimal due to the loss of useful information incurred by the measurement model transformations. In the second step, a refinement procedure is followed to compensate for the performance loss by solving a linear weighted least squares problem. Apart from the solution method, the performance gain from the relays is confirmed by a theoretical analysis with the Cramér-Rao lower bound (CRLB). Moreover, the mean square error performance of the proposed two-step method is shown to reach the CRLB under the small Gaussian noise condition. Simulation results validate the theoretical studies and demonstrate accurate RBL of the proposed method.

In practical underwater environments, the uncertain and varying propagation speed of an acoustic signal makes time-based localization of an acoustic source challenging. In this paper, we utilize the isogradient sound speed profile (SSP) model with random model parameters to address the underwater source localization problem using time-of-arrival (TOA) measurements. To better reflect the real-world environments, the SSP we used is indeterminate where its model parameters have unknown random deviations from the nominal values. We propose a two-step method for locating an underwater acoustic source for this highly nonlinear challenging problem. In the first step, we propose an approximation to the TOA model that takes the sound speed dependence on the source depth into consideration, and formulate a non-convex constrained weighted least squares (CWLS) problem to obtain an initial source position estimate. In the second step, we expand the original TOA model by the second-order Taylor series at the initial estimate, and then create a different non-convex CWLS problem to obtain a refined solution. Both CWLS problems are solved by applying the semidefinite relaxation technique. Moreover, the proposed method is extended to the case where the source is not time synchronized with the sensors. Furthermore, we derive the Cramér-Rao lower bound (CRLB) for this particular localization problem and show by mean squared error (MSE) analysis that the proposed method is capable of achieving the CRLB performance under small Gaussian errors in the measurements and SSP model parameters. Simulation results confirm the effectiveness of the proposed method in achieving good localization performance.

Mobile edge computing (MEC) has emerged as a cutting-edge technique that brings computation and storage resources closer to the edge of the mobile network. However, MEC is vulnerable to be attacked by malicious users. To improve the security of computation tasks and enhance user connectivity, we design a deep reinforcement learning (DRL) network for reconfigurable intelligence surface (RIS)-aided MEC system. Specifically, we jointly optimize the phase shifts at the RIS, tasks offloaded by users and task assignment to maximize the secrecy offloading capacity and minimize energy consumption under different delay requirements of users. Furthermore, a multi-agent twin delayed deep deterministic policy gradient (TD3)-based algorithm is exploited to tackle the non-convex optimization problem. Numerical results validate the feasibility and applicability of our proposed scheme, demonstrating that the proposed scheme significantly improves the security and energy performance of the system compared to the baseline DRL algorithm.

This paper addresses the passive source localization problem using hybrid angle-of-arrival (AOA) and time-difference-of-arrival (TDOA) measurements observed by single stationary receiver at several time intervals, when the source is moving with a constant velocity trajectory. We first show that the localization problem cannot be achieved using a sequence of one kind of measurements (i.e., AOAs only or TDOAs only), but can be accomplished by using both AOA and TDOA observations together. Then we develop a parametric model by integrating AOA and TDOA, and then formulate a non-convex constrained weighted least squares (CWLS) problem that is applicable to the 2-D and 3-D scenarios for locating the source. Benefiting from the integrated model, the global solution of the CWLS problem is guaranteed without having local convergence or divergence issues. Furthermore, we conduct the mean square error analysis to validate that the proposed method is able to reach the Cramer-Rao lower bound accuracy under small Gaussian noise for the 2-D and 3-D cases. Simulation results also confirm the good performance of the proposed method.

Most existing direction of arrival (DOA) estimation methods are based on antenna arrays for line-of-sight (LOS) propagation. In this article, a different and challenging DOA estimation problem with a single receiving antenna in the non-line-of-sight (NLOS) scenario is addressed, where a reconfigurable intelligent surface (RIS) is combined with two robust array spatial covariance matrix (SCM) reconstruction schemes to solve the problem. In detail, a two-stage approach for high-efficiency RIS phase shifting is first designed, and then the Tikhonov regularization criterion and total least-squares (TLS) criterion are respectively exploited for SCM reconstruction with and without phase shift error (PSE), yielding an improved DOA estimation performance with reduced complexity. Theoretical analysis on the performance of SCM reconstruction is conducted, and simulation results are provided to show the effectiveness of the proposed solutions.

Based on the exact source-sensor spatial geometry, a three-dimensional (3-D) spatial-temporal localization algorithm for multiple near-field (NF) sources is proposed without adopting the Fresnel approximation, which simplifies the spatial phase difference by Taylors polynomial. In addition, considering the propagation attenuation which varies from different sensors, the spatial and temporal information can be exploited to construct a third-order parallel factor (PARAFAC) data model and the array manifold matrices can be extracted by trilinear decomposition; then, estimation of the unambiguous range and angle parameters of the NF sources is achieved from the spatial amplitude-phase factors by the least squares method. The obtained 3-D parameters associated with each source require no additional pairing process, as also demonstrated by simulation results.

In this paper, based on sum-difference co-array analysis, a new fourth-order sparse array called sum-difference-FODC (SD-FODC) is proposed, allowing the construction of a fourth-order DCA with long consecutive lags using the continuous segments in the second-order DCA and SCA of the original array. It has a closed-form expression for sensor positions that can be generated by two arbitrary nonuniform linear arrays (NLAs) called generators. If the second-order SCA and DCA of the generators have long consecutive segments, the designed fourth-order sparse array can achieve a large number of uDOFs. Numerical results are provided to demonstrate the superior performance of the proposed design.

In this paper, an underdetermined three-dimensional (3-D) near-field source localization method is proposed, based on a two-dimensional (2-D) symmetric nonuniform cross array. Firstly, the fourth-order cumulant of the near-field observations with multiple delay lags is exploited to construct virtual far-field pseudo-observations, leading to increased degrees of freedom (DOF); then, 2-D angles of the nearfield sources are jointly estimated by employing the recently proposed sparse and parametric approach (SPA) and Vandermonde decomposition technique, eliminating the need for parameter discretization. To estimate the range term, the one-dimensional (1-D) MUSIC algorithm is applied by resorting to the conjugate symmetry property of the signal's autocorrelation function. Numerical results are provided to demonstrate the superiority of our method.

We address the unified near-field and far-field time-difference-of-arrival source localization problem, in which the signal propagation speed is unknown, and the prior knowledge that whether the source is in the near-field or far-field is unavailable. Using the unified near-field and far-field model obtained by expressing the source position in the modified polar representation (MPR), two different methods, the semidefinite relaxation (SDR) method that achieves noise resilience and the two-step closed-from solution method that accomplishes low complexity, are proposed to jointly estimate the MPR coordinates and the propagation speed. For the SDR method, we express the propagation speed as the sum of a pre-selected constant and the residual to form a non-convex constrained weighted least squares problem, which is then relaxed into a convex semidefinite program by applying SDR. The two-step method obtains a coarse estimate in Step 1 by solving the quadratic programming problem without considering the relations among the variables, and then refines the coarse estimate in Step 2 by estimating the correction to compensate the error resulting from ignoring the relations. Theoretical mean square error analysis confirms that the proposed methods can reach the Cramer-Rao Lower Bound performance, which is also validated by using both simulated and real data.

This paper addresses the localization problem in unique coordinates of a far-field object. Using moving transceivers as relays for the sensor signals in reaching the object, we utilize the range measurements from the sensors through the transceivers to the object to determine its position. The transceivers have no self-localization capability such that their motion parameters are unknown. Moreover, neither transceiver relay times nor object reflection delay are known, causing additional unknown range offsets. We propose an effective three-step method for this localization problem. The first step eliminates the object position and range offsets by formulating a constrained weighted least squares (CWLS) problem and estimates only the transceiver motion parameters. Using the preliminary estimate of the motion parameters, the second step formulates another CWLS problem to obtain the object position estimate, which is used to determine the range offsets by a linear WLS estimator. Finally, a refinement procedure follows in the third step by formulating a different CWLS problem to compensate for the performance loss. To solve the non-convex CWLS problems, we introduce semidefinite relaxation to transform them into convex semidefinite programs. Both mean square error analysis and simulation results show that the refined CWLS solution is able to achieve the Cramer–Rao lower bound performance when the SNR is not very low.

This paper addresses the rigid body localization (RBL) problem using an unsynchronized sensor network, where clock offsets between sensors and anchors are present. We first investigate RBL using unsynchronized time-of-arrival (TOA) measurements to determine the necessary and sufficient conditions for this problem by analyzing the rank of the Fisher Information Matrix. When the condition is fulfilled, a constrained weighted least squares (CWLS) problem is formulated to jointly estimate the position and rotation angles of the rigid body together with the clock offsets. Besides the non-convex nature, the CWLS problem contains nonlinear constraints resulting from the orthogonality property of the rotation matrix. To solve this difficult problem, we propose to apply semidefinite relaxation (SDR) to relax it as a convex semidefinite program. Furthermore, we investigate the RBL problem using hybrid TOA and angle-of-arrival (AOA) measurements. The Cramer-Rao Lower Bound (CRLB) analysis shows better performance of RBL by hybrid TOA-AOA than by AOA only when more than one anchors are used. The CWLS problem is extended to include the AOAs and the associated SDR method is derived. Finally, mean square error analysis shows that the proposed methods are able to achieve the CRLB performance under small Gaussian noise. Numerical results confirm the theoretical analyses and validate the performance of the proposed methods.

In this paper, we address the multistatic localization problem when the transmitter position is not known and the synchronizations among the receivers and with the transmitter are not available. Aided by a calibration object, the problem is solvable where the unknown object position, the transmitter position and the synchronization errors can be estimated using the time delay measurements from the direct paths between the transmitter and receivers, and the indirect paths reflected by the calibration and unknown objects. To solve this challenging problem, we propose a three-step estimation method that obtains a different unknown in each step. Specifically, the transmitter position is estimated by formulating a constrained weighted least squares (CWLS) problem in the first step, and a linear weighted least squares estimator follows to determine the synchronization errors in the second step. Finally, the object position is estimated by formulating a different CWLS problem in the third step. The solutions are closed-form in all three steps, and thus, the proposed method is computationally efficient. Furthermore, the proposed method is extended to the more practical scenario in which the calibration object position error is present. The mean square error analysis validates that the object position estimate is able to achieve the Cramer-Rao lower bound performance under small Gaussian noise, which is further confirmed by numerical simulations.

Considering all or part of the sensors are at known positions, this paper addresses the problem of locating a 3-D moving object with a linear constant velocity trajectory using angle-of-arrival (AOA) observations. We start by analyzing the minimum number of sensors and show that at least two sensors with known positions are needed for this problem. For the case of all sensors at known positions, we formulate a weighted least squares (WLS) problem with the capability of bias reduction, called the bias-reduced WLS (BR-WLS) problem, to limit the bias caused by the transformation of the measurement models. The BR-WLS problem has a closed-form solution, and thus, solving it is very computationally efficient. For the case where only part of the sensors are at known positions, we first theoretically analyze the performance gain by including the sensors at unknown positions for localization, and then extend the proposed BR-WLS method to jointly estimate the object motion parameters and the unknown sensor positions. For both cases, we show that the localization mean square error (MSE) can reach the Cramér-Rao lower bound (CRLB) when the noise is small and Gaussian distributed. Moreover, we also derive the theoretical expressions of the residual biases for the object position and velocity estimation by the proposed solutions. Simulation results confirm the good performance of the proposed solutions and also validate the theoretical results.

This paper addresses the challenging problem of multistatic localization of a stationary object with a set of synchronized receivers, when the transmitter position is unknown and the synchronization with the transmitter is unavailable. Using the time delay measurements from the direct and indirect paths, we propose to jointly estimate the object and transmitter positions together with the clock offset. To accomplish the joint estimation, we first formulate a non-convex constrained weighted least squares (CWLS) minimization problem, where the approximations involved could introduce a large amount of estimation bias. We then extend the formulation to arrive at a bias-reduced CWLS (BR-CWLS) problem that has the ability of reducing the bias. The BR-CWLS problem, which is non-convex, is handled by applying the semidefinite relaxation technique to reach a convex semidefinite program that can be directly solved by a software package. Simulation results demonstrate the good performance of the proposed method in achieving the Cramer-Rao lower bound performance and reducing the bias.

The presence of radically irregular data points (RIDPs), which are referred to as the subset of measurements that represents no or little information, can significantly degrade the performance of ellipse fitting methods. We develop an ellipse fitting method that is robust to RIDPs based on the maximum correntropy criterion with variable center (MCC-VC), where an adaptable Laplacian kernel is used. For single ellipse fitting, we formulate a non-convex optimization problem and divide it into two subproblems, one to estimate the kernel bandwidth and the other the kernel center. We design sufficiently accurate convex approximation to each subproblem that will lead to computationally efficient closed-form solutions. The two subproblems are solved in an alternate manner until convergence is reached. We also investigate coupled ellipses fitting. While there exist multiple ellipses fitting methods in the literature, we develop a coupled ellipses fitting method by exploiting the underlying special structure, where the associations between the data points and ellipses are absent in the problem. The proposed method first introduces an association vector for each data point and then formulates a non-convex mixed-integer optimization problem to establish the data associations, which is approximately solved by relaxing it into a second-order cone program. Using the estimated data associations, we then extend the proposed single ellipse fitting method to accomplish the final coupled ellipses fitting. The proposed method is shown to perform significantly better than the existing methods using both simulated data and real images.

This paper investigates the elliptic localization for moving object problem from time delay (TD) and Doppler frequency shift (DFS) measurements, where the transmitter position and velocity are unknown. The transmitter is not perfectly time synchronized such that unknown offsets exist in the TD and DFS measurements. We propose to jointly estimate the object and transmitter positions and velocities and the offsets. Using the TD and DFS measurements from both the indirect and direct paths between the transmitter and the receivers, we formulate a non-convex weighted least squares (WLS) problem. Local convergence may occur when solving the non-convex WLS problem, implying that good estimate is not guaranteed. Thus, we relax the non-convex WLS problem into a convex semidefinite program by applying semidefinite relaxation (SDR). Moreover, we theoretically show that the performance can be improved by using multiple transmitters as compared to that using single transmitter, although more unknown parameters are introduced. We then extend the proposed SDR method to handle the multiple transmitters case. Finally, the mean square error analysis is provided to show that the proposed WLS method reaches the Cramer-Rao lower bound accuracy under small Gaussian noise condition. Simulation results validate the theoretical analysis and show the superior performance over the existing methods.

This paper addresses the localization of a constant velocity moving object in 3-D using angle-of-arrival (AOA) measurements. Compared with the maximum-likelihood estimator, pseudo-linear approach for AOA localization has a large amount of bias resulting from the model transformation to simplify the solution finding. This work aims at reducing the bias and developing bias-reduced semidefinite relaxation (SDR) methods for estimating the initial position and velocity of the object, in a batch or sequential manner. This is accomplished by first formulating a bias reduced constrained weighted least squares (BR-CWLS) problem from the transformed measurements, through introducing an auxiliary variable and adding a quadratic constraint. Such an intractable non-convex problem is tackled next by applying the SDR technique and relaxing it into a convex semidefinite program (SDP), which is shown to be capable of reaching the solution of the original BR-CWLS problem. For sequential estimation, we formulate a different BR-CWLS problem and utilize SDR for obtaining a sequential estimation method that updates the initial position and velocity estimates at each time step. We conduct the mean square error (MSE) and bias analyses for both estimation methods to assess their expected performance. Simulation results verify the ability of bias reduction and the good performance of the proposed methods.

A new near-field source localization method is proposed for two-dimensional (2-D) direction-of-arrival (DOA) and range estimation based on a symmetrical cross array. It first employs the conjugate symmetry property of the signal auto-correlation at different time delays to construct a conjugate augmented spatial-temporal cross correlation matrix, then the extended steering vector is decoupled to avoid the usual multiple-dimensional (M-D) search based on the properties of the Khatri-Rao product, and finally three one-dimensional (1-D) MUSIC type searches are employed to obtain the results. The proposed method can realize automatic pairing of multiple parameters associated with each source and it also works in the underdetermined case.

This paper addresses the multistatic localization of a moving object in position and velocity using time delay (TD) and Doppler frequency shift (DFS) measurements, where the position of the transmitter is unknown and has not yet been synchronized with the receivers. Based on the TD and DFS measurements from the direct and indirect paths, we formulate a non-convex weighted least squares (WLS) minimization problem, and then apply semidefinite relaxation (SDR) to relax the WLS problem into a convex semidefinite program. Simulation results show that the proposed SDR method is able to achieve the Cramer-Rao lower bound accuracy under mild Gaussian noise condition and outperforms the existing method.

Locating a device is a basic element for many Internet of Things (IoT) applications. In particular, it often demands an algorithm having low complexity to limit the energy consumption and most important, sufficient robustness without knowing the device in the near-field for point localization or in the far-field for direction of arrival (DOA) estimation. This article proposes a new localization algorithm that can achieve the two purposes, with the theoretical analysis to validate the optimal accuracy and the real data experiment to support the promising performance. The first objective is achieved by a closed-form solution and the second is accomplished by using the modified polar representation (MPR) of the source position, based on a new formulation for the localization problem. While the MPR localization method has been introduced before, it is not sufficiently robust for IoT application to handle the large equal radius (LER) scenario or the presence of sensor position errors. The proposed algorithm uses a different MPR formulation, which is able to handle the LER scenario, sensor position errors, and has low computational complexity.

In this letter, we address the multistatic object localization problem when the transmitter position and signal propagation speed are not known. By transforming the time delay measurement model, we first formulate a non-convex weighted least squares problem, which is then relaxed into a convex semidefinite programming (SDP) problem by applying semidefinite relaxation. It turns out that the relaxed SDP problem is too loose and fails to achieve the Cramer-Rao lower bound (CRLB) performance. To address this problem, we propose to tighten the relaxed SDP problem by adding a second-order cone constraint to reach better solutions. Simulation results show that the proposed method is able to achieve the CRLB.

The performance of ellipse fitting may significantly degrade in the presence of outliers, which can be caused by occlusion of the object, mirror reflection or other objects in the process of edge detection. In this paper, we propose an ellipse fitting method that is robust against the outliers, and thus maintaining stable performance when outliers can be present. We formulate an optimization problem for ellipse fitting based on the maximum entropy criterion (MCC), having the Laplacian as the kernel function from the well-known fact that the ℓ1-norm error measure is robust to outliers. The optimization problem is highly nonlinear and non-convex, and thus is very difficult to solve. To handle this difficulty, we divide it into two subproblems and solve the two subproblems in an alternate manner through iterations. The first subproblem has a closed-form solution and the second one is cast as a convex second-order cone program (SOCP) that can reach the global solution. By so doing, the alternate iterations always converge to an optimal solution, although it can be local instead of global. Furthermore, we propose a procedure to identify failed fitting of the algorithm caused by local convergence to a wrong solution, and thus, it reduces the probability of fitting failure by restarting the algorithm at a different initialization. The proposed robust ellipse fitting method is next extended to the coupled ellipses fitting problem. Both simulated and real data verify the superior performance of the proposed ellipse fitting method over the existing methods.

Elliptic localization where a transmitter actively sending out a signal to locate an object from its echo appears in many practical systems including multiple-input-multiple-output (MIMO) radars and wireless communications. In this article, the elliptic localization problem when the transmitter position is not known is addressed. We propose an effective method for joint estimation of the object and transmitter positions. Using the range measurements from the direct paths between the transmitter and the receivers together with the indirect paths through the object, we formulate a non-convex weighted least squares (WLS) minimization problem. The non-convex nature of this problem makes it difficult for an iterative search algorithm to reach the optimal solution, implying that good estimate is not guaranteed. To handle this difficulty, we propose to solve the formulated WLS localization problem by applying the semidefinite relaxation (SDR) technique, resulting in a convex semidefinite program (SDP). The proposed SDR method is then extended to the more general case when multiple transmitters at unknown positions are used and the receiver positions are subject to errors. Simulation results show that the proposed method can attain the Cramer-Rao lower bound (CRLB) accuracy under Gaussian noise and has superior performance over the existing method.

This paper addresses the rigid body localization problem using angle-of-arrival measurements. We formulate the problem as a constrained weighted least squares (CWLS) minimization problem with the rotation matrix and position vector as variables, which is a challenging non-convex problem. To approximately solve this problem, we first relax it as a convex semidefinite program (SDP), and then tighten the relaxed problem by adding some reasonable second-order cone constraints. Simulations show that the tightened SDP problem is able to reach the performance of the original CWLS problem, making its solution achieve the Cramer-Rao lower bound accuracy, when the noise level is not too high.

Research on received signal strength (RSS) based localization has received a lot of attention from both academia and industry because of its low complexity and high efficiency. In this letter, the RSS based localization problem when the transmit power is unknown, is addressed based on the least squared relative error (LSRE) estimation. By taking exponential transformation, the original log-normal RSS measurement model is transformed into a multiplicative model, which is used to formulate a non-convex LSRE estimation problem with the source location and the transmit power as variables. We then apply semidefinite relaxation (SDR) to the non-convex LSRE problem to obtain a convex semidefinite programming (SDP) problem. To facilitate SDR, we introduce two compound variables constructed by the source location and the transmit power, and then estimate the two compound variables instead of directly estimating the source location and the transmit power. The source location estimate is recovered according to its relation with the compound variable. Both simulations and real field test demonstrate the superior performance of the proposed method over several existing methods.

Accurate localization is critical to facilitate location services for Internet of Things (IoT). It is particular challenging to provision localization based on nonline-of-sight (NLOS) signals. Thus, we actualize source localization based on time difference of arrival (TDOA) derived from NLOS signal propagations. The existing robust least squares (RLS) method exhibits two shortcomings: 1) it is formulated using too large upper bounds on the NLOS errors, and 2) it suffers from the possible inexact triangle inequality problem. Aiming at circumventing the shortcomings of the existing RLS method, we propose two new RLS formulations. On one hand, to reduce the upper bounds on the NLOS errors, we propose to jointly estimate the source position and the NLOS error in the reference path. On the other hand, to avoid using the triangle inequality, we introduce a “balancing parameter” in the first formulation and develop the second formulation by transforming the measurement model. Both formulations are transformed via the S-lemma into optimization problems that are amendable to semidefinite relaxation. The proposed methods achieve superior performance over the existing methods, as validated by using both simulated and experimental data.

This paper addresses the rigid body localization problem using a convex optimization approach. We propose a semi-definite relaxation (SDR) method for locating a stationary rigid body using arrival time measurements, and extend it for moving rigid body using both arrival time and Doppler measurements. Localization of a stationary (moving) rigid body involves not only the position (and velocity) but also the rotation angles (and angular velocity), making it a challenging optimization problem with nonlinear constraints. We approximate the maximum likelihood problem with a constrained weighted least-squares (CWLS) minimization and apply SDR to obtain a coarse estimate. The orthogonalization and refinement procedures are followed next to recover the performance loss caused by relaxation and approximation. It is shown analytically that the CWLS solution can achieve the Cramér–Rao lower bound accuracy for Gaussian noise when the noise level is not significant. The simulations show that the proposed method achieves better accuracy than the previously developed methods.

This paper develops two convex relaxation solutions for the unified localization of a signal source using time difference of arrival measurements, regardless of whether the source is in the near field for coordinate positioning or in the far field for the direction of arrival estimation. The previous study on unified estimation only derived an iterative solution, which is sensitive to initialization. Albeit a coarse initialization was supplied to start the iteration, it may not be sufficient to ensure convergence to the global solution especially when the source is close to the sensors. The proposed solutions come from two novel formulations for optimization, one using the weighted least squares with the modified polar representation of the source position as variable and the other applying fractional programming with the Cartesian coordinate representation instead. Both the optimization problems are solved by first performing semidefinite relaxation and then tightening the relaxed problem by including a set of second-order cone constraints. The two formulations are created from different approaches. Nevertheless, we are able to prove that both the formulations reduce to solving exactly the same mixed semidefinite/second-order cone program and thus establish their equivalence. Furthermore, the proposed solution method is extended to the more practical scenario when sensor position errors are present. The results from both the simulated and real experiments show that the proposed method achieves almost the same performance of the iterative maximum likelihood estimator under ideal initialization.

In this paper, we investigate the cooperative localization problem in non-line-of-sight (NLOS) environments using time-of-arrival measurements. We propose a novel robust localization method that is robust against the effect of NLOS errors on the localization performance. The proposed method only requires knowledge of the upper bound on the magnitude of the NLOS errors rather than their statistics, which are more difficult to obtain in practice. Our method is shown to perform better than the existing non-robust method especially in severe NLOS environments.

Angles of Arrival (AOAs) are popular measurements to locate a signal source. They can yield a position if the source is near the sensors or a DOA if it is far away. Point positioning and DOA localization require different estimators and the knowledge if the source is near or far is needed. Nevertheless, such knowledge about the source range is often not available in practice. This paper first analyzes the consequences of point positioning of a distant source and DOA estimation of a near one. It next proposes a unified estimator that provides a position estimate if it is near or a DOA estimate if it is far, without requiring any prior knowledge about the range region of where it lies. The estimator is derived using the Maximum Likelihood criterion with the Gauss Newton implementation, using the modified polar coordinates to represent the source location. A preliminary solution using the semi-definite relaxation is developed to initialize the estimator. Simulations validate the performance of the proposed estimator in reaching the CRLB performance.

In this letter, we propose a method to improve the localization accuracy of a rigid body using the range measurements between the anchors and several sensors mounted on the body. Rigid body localization aims to estimate the rotation matrix and position vector of the sensor array on the body and it is a nonlinear and challenging problem to solve. Through the application of semidefinite relaxation to the original maximum likelihood formulation, we can obtain a coarse estimate of the rotation matrix and position vector that is robust to measurement noise. We next apply orthogonalization and refinement to the estimate and gain back the performance loss from relaxation and approximation. Simulation results show that the proposed method can reach the Cramér–Rao lower bound performance even at high noise levels where existing estimators cannot.

In this letter, we address the differential received signal strength based problem with model parameter errors. To deal with the model parameter errors, we adopt the robust weighted least squares criterion, leading to a minimax optimization problem. By assuming the model parameter errors lie in a ball, the minimax problem is transformed into a tractable reformulation via the S-Lemma. Confronted with the nonconvexity of the reformulated problem, we approximately solve it by applying the semidefinite relaxation. The proposed approach only requires the knowledge of the upper bounds of the model parameter errors, which are practically easy to acquire. Simulation results show that the proposed method is robust to the model parameter errors and outperforms the existing methods.

In this letter, the two-way time-of-arrival (TW-TOA)-based localization problem with clock imperfections in an asynchronous network is addressed. In the TW-TOA measurement model, the unknown turn-around times and clock skews may significantly degrade the localization performance. Under the assumption that the ranges of the turn-around times and the upper bound of the clock skews are known, we propose a robust least squares (RLS) formulation by taking the turn-around times and the clock skews as nuisance parameters. The RLS problem is approximately solved by employing the second-order cone relaxation technique. Simulation results illustrate the superior performance of the proposed method over the existing methods.

In this letter, the time-difference-of-arrival-based localization problem under mixed line-of-sight (LOS)/non-LOS (NLOS) conditions is addressed. Under the assumption that the path status information is known, we formulate a robust weighted-least-squares method to solve this problem. To fully utilize the more accurate LOS measurements, we impose a weight to the term with respect to the NLOS measurements, and the choice of the weight is derived and explicitly given. We then employ the second-order cone relaxation technique to relax the problem as a tractable second-order cone program. Simulation results show that the prior path status information significantly improves the localization performance.

In this paper, we address the time-of-arrival (TOA) based localization problem in an adverse environment, where line-of-sight (LOS) signal propagation between the source and the sensor is not readily available, in which case we have to resort to non-line-of-sight (NLOS) signals. Two convex relaxation methods, i.e., the semidefinite relaxation (SDR) and the second-order cone relaxation (SOCR) methods, are proposed to mitigate the effect of NLOS errors on the localization performance. We consider two separate cases in which the information of the NLOS status is totally unknown and perfectly known, respectively. The proposed methods can be applied without knowing the distribution of NLOS errors. Moreover, we propose a NLOS error mitigation method that is robust to detection errors, which are generated in the process of detecting NLOS paths. Simulation results show that the proposed convex relaxation methods outperform some existing state-of-the-art methods.

An unknown-position sensor can be localized if there are three or more anchors making time-of-arrival (TOA) measurements of a signal from it. However, the location errors can be very large due to the fact that some of the measurements are from non-line-of-sight (NLOS) paths. In this paper, a semi-definite programming (SDP) based node localization algorithm in NLOS environments is proposed for ultra-wideband (UWB) wireless sensor networks. The positions of sensors can be estimated using the distance estimates from location-aware anchors as well as other sensors. However, in the absence of line-of-sight (LOS) paths, e.g., in indoor networks, the NLOS range estimates can be significantly biased. As a result, the NLOS error can remarkably decrease the location accuracy, and it is not easy to accurately distinguish LOS from NLOS measurements. According to the information known about the prior probabilities and distributions of the NLOS errors, three different cases are introduced and the respective localization problems are addressed. Simulation results demonstrate that this algorithm achieves high location accuracy even for the case in which NLOS and LOS measurements are not identifiable.

We consider the source localization problem using time-difference-of-arrival (TDOA) measurements in sensor networks. The maximum likelihood (ML) estimation of the source location can be cast as a nonlinear/nonconvex optimization problem, and its global solution is hardly obtained. In this paper, we resort to the Monte Carlo importance sampling (MCIS) technique to find an approximate global solution to this problem. To obtain an efficient importance function that is used in the technique, we construct a Gaussian distribution and choose its probability density function (pdf) as the importance function. In this process, an initial estimate of the source location is required. We reformulate the problem as a nonlinear robust least squares (LS) problem, and relax it as a second-order cone programming (SOCP), the solution of which is used as the initial estimate. Simulation results show that the proposed method can achieve the Cramer-Rao bound (CRB) accuracy and outperforms several existing methods.

In this letter, we propose a new approach to the localization problem in wireless sensor networks using received-signal-strength (RSS) measurements. The problem is reformulated under the equivalent exponential transformation of the conventional path loss measurement model and the unscented transformation (UT), and is approximately approached by the maximum likelihood (ML) parameter estimation, which we refer to as the weighted least squares (WLS) approach. This formulation is used for sensor node localization in both noncooperative and cooperative scenarios. Simulation results confirm the effectiveness of the approach for both outdoor and indoor environments.

Recently, energy-based localization using acoustic energy measurements has received much attention in wireless sensor networks. Since the objective function of the energy-based maximum likelihood (ML) localization is non-convex, the global solutions are hardly obtained without good initial estimates. In this paper, we relax this non-convex problem as a convex semidefinite programming (SDP), based on which a good estimate can be obtained and be improved by a procedure called randomization. Simulation results show that the proposed method is effective and outperforms the existing methods.

In practice, the transition probability matrix (TPM) in the approach to track a maneuvering target is often unknown. We propose a new method to estimate the optimal TPM according to the maximum a posteriori (MAP) or maximum likelihood (ML) criterion via convex optimization. We apply the proposed method to the nonlinear/non- Gaussian cases, where the interacting multiple model (IMM) particle filter (IMMPF) is employed to estimate the corresponding base state. Simulation results of tracking a maneuvering target show the efficacy of the proposed method with improved performance.