Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond

Since the landmark work of R. E. Kalman in the 1960s, considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular, parametric filters that seek analytical estimates based on a closed-form Markov–Bayes recursion, e.g., recursion from a Gaussian or Gaussian mixture (GM) prior to a Gaussian/GM posterior (termed ‘Gaussian conjugacy’ in this paper), form the backbone for a general time series filter design. Due to challenges arising from nonlinearity, multimodality (including target maneuver), intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities), etc., new theories, algorithms, and technologies have been developed continuously to maintain such a conjugacy, or to approximate it as close as possible. They had contributed in large part to the prospective developments of time series parametric filters in the last six decades. In this paper, we review the state of the art in distinctive categories and highlight some insights that may otherwise be easily overlooked. In particular, specific attention is paid to nonlinear systems with an informative observation, multimodal systems including Gaussian mixture posterior and maneuvers, and intractable unknown inputs and constraints, to fill some gaps in existing reviews and surveys. In addition, we provide some new thoughts on alternatives to the first-order Markov transition model and on filter evaluation with regard to computing complexity

Performance Analysis of Bearings-only Tracking Problems for Maneuvering Target and Heterogeneous Sensor Applications

State estimation, i.e. determining the trajectory, of a maneuvering target from noisy measurements collected by a single or multiple passive sensors (e.g. passive sonar and radar) has wide civil and military applications, for example underwater surveillance, air defence, wireless communications, and self-protection of military vehicles. These passive sensors are listening to target emitted signals without emitting signals themselves which give them concealing properties. Tactical scenarios exists where the own position shall not be revealed, e.g. for tracking submarines with passive sonar or tracking an aerial target by means of electro-optic image sensors like infrared sensors. This estimation process is widely known as bearings-only tracking. On the one hand, a challenge is the high degree of nonlinearity in the estimation process caused by the nonlinear relation of angular measurements to the Cartesian state. On the other hand, passive sensors cannot provide direct target location measurements, so bearings-only tracking suffers from poor target trajectory estimation accuracy due to marginal observability from sensor measurements. In order to achieve observability, that means to be able to estimate the complete target state, multiple passive sensor measurements must be fused. The measurements can be recorded spatially distributed by multiple dislocated sensor platforms or temporally distributed by a single, moving sensor platform. Furthermore, an extended case of bearings-only tracking is given if heterogeneous measurements from targets emitting different types of signals, are involved. With this, observability can also be achieved on a single, not necessarily moving platform. In this work, a performance bound for complex motion models, i.e. piecewisely maneuvering targets with unknown maneuver change times, by means of bearings-only measurements from a single, moving sensor platform is derived and an efficient estimator is implemented and analyzed. Furthermore, an observability analysis is carried out for targets emitting acoustic and electromagnetic signals. Here, the different signal propagation velocities can be exploited to ensure observability on a single, not necessarily moving platform. Based on the theoretical performance and observability analyses a distributed fusion system has been realized by means of heterogeneous sensors, which shall detect an event and localize a threat. This is performed by a microphone array to detect sound waves emitted by the threat as well as a radar detector that detects electromagnetic emissions from the threat. Since multiple platforms are involved to provide increased observability and also redundancy against possible breakdowns, a WiFi mobile ad hoc network is used for communications. In order to keep up the network in a breakdown OLSR (optimized link state routing) routing approach is employed

Localizability Optimization for Multi Robot Systems and Applications to Ultra-Wide Band Positioning

RÉSUMÉ: RÉSUMÉ Les Systèmes Multi-Robots (SMR) permettent d’effectuer des missions de manière efficace et robuste du fait de leur redondance. Cependant, les robots étant des véhicules autonomes, ils nécessitent un positionnement précis en temps réel. Les techniques de localisation qui utilisent des Mesures Relatives (MR) entre les robots, pouvant être des distances ou des angles, sont particulièrement adaptées puisqu’elles peuvent bénéficier d’algorithmes coopératifs au sein du SMR afin d’améliorer la précision pour l’ensemble des robots. Dans cette thèse, nous proposons des stratégies pour améliorer la localisabilité des SMR, qui est fonction de deux facteurs. Premièrement, la géométrie du SMR influence fondamentalement la qualité de son positionnement pour des MR bruitées. Deuxièmement, les erreurs de mesures dépendent fortement de la technologie utilisée. Dans nos expériences, nous nous focalisons sur la technologie UWB (Ultra-Wide Band), qui est populaire pour le positionnement des robots en environnement intérieur en raison de son coût modéré et sa haute précision. Par conséquent, une partie de notre travail est consacrée à la correction des erreurs de mesure UWB afin de fournir un système de navigation opérationnel. En particulier, nous proposons une méthode de calibration des biais systématiques et un algorithme d’atténuation des trajets multiples pour les mesures de distance en milieu intérieur. Ensuite, nous proposons des Fonctions de Coût de Localisabilité (FCL) pour caractériser la géométrie du SMR, et sa capacité à se localiser. Pour cela, nous utilisons la Borne Inférieure de Cramér-Rao (BICR) en vue de quantifier les incertitudes de positionnement. Par la suite, nous fournissons des schémas d’optimisation décentralisés pour les FCL sous l’hypothèse de MR gaussiennes ou log-normales. En effet, puisque le SMR peut se déplacer, certains de ses robots peuvent être déployés afin de minimiser la FCL. Cependant, l’optimisation de la localisabilité doit être décentralisée pour être adaptée à des SMRs à grande échelle. Nous proposons également des extensions des FCL à des scénarios où les robots embarquent plusieurs capteurs, où les mesures se dégradent avec la distance, ou encore où des informations préalables sur la localisation des robots sont disponibles, permettant d’utiliser la BICR bayésienne. Ce dernier résultat est appliqué au placement d’ancres statiques connaissant la distribution statistique des MR et au maintien de la localisabilité des robots qui se localisent par filtrage de Kalman. Les contributions théoriques de notre travail ont été validées à la fois par des simulations à grande échelle et des expériences utilisant des SMR terrestres. Ce manuscrit est rédigé par publication, il est constitué de quatre articles évalués par des pairs et d’un chapitre supplémentaire. ABSTRACT: ABSTRACT Multi-Robot Systems (MRS) are increasingly interesting to perform tasks eÿciently and robustly. However, since the robots are autonomous vehicles, they require accurate real-time positioning. Localization techniques that use relative measurements (RMs), i.e., distances or angles, between the robots are particularly suitable because they can take advantage of cooperative schemes within the MRS in order to enhance the precision of its positioning. In this thesis, we propose strategies to improve the localizability of the SMR, which is a function of two factors. First, the geometry of the MRS fundamentally influences the quality of its positioning under noisy RMs. Second, the measurement errors are strongly influenced by the technology chosen to gather the RMs. In our experiments, we focus on the Ultra-Wide Band (UWB) technology, which is popular for indoor robot positioning because of its mod-erate cost and high accuracy. Therefore, one part of our work is dedicated to correcting the UWB measurement errors in order to provide an operable navigation system. In particular, we propose a calibration method for systematic biases and a multi-path mitigation algorithm for indoor distance measurements. Then, we propose Localizability Cost Functions (LCF) to characterize the MRS’s geometry, using the Cramér-Rao Lower Bound (CRLB) as a proxy to quantify the positioning uncertainties. Subsequently, we provide decentralized optimization schemes for the LCF under an assumption of Gaussian or Log-Normal RMs. Indeed, since the MRS can move, some of its robots can be deployed in order to decrease the LCF. However, the optimization of the localizability must be decentralized for large-scale MRS. We also propose extensions of LCFs to scenarios where robots carry multiple sensors, where the RMs deteriorate with distance, and finally, where prior information on the robots’ localization is available, allowing the use of the Bayesian CRLB. The latter result is applied to static anchor placement knowing the statistical distribution of the MRS and localizability maintenance of robots using Kalman filtering. The theoretical contributions of our work have been validated both through large-scale simulations and experiments using ground MRS. This manuscript is written by publication, it contains four peer-reviewed articles and an additional chapter

Approximate Gaussian Conjugacy: Parametric Recursive Filtering Under Nonlinearity, Multimodal, Uncertainty, and Constraint, and Beyond

This is a post-peer-review, pre-copyedit version of an article published in Frontiers of Information Technology & Electronic Engineering. The final authenticated version is available online at: https://doi.org/10.1631/FITEE.1700379Since the landmark work of R. E. Kalman in the 1960s, considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular, parametric filters that seek analytical estimates based on a closed-form Markov–Bayes recursion, e.g., recursion from a Gaussian or Gaussian mixture (GM) prior to a Gaussian/GM posterior (termed ‘Gaussian conjugacy’ in this paper), form the backbone for a general time series filter design. Due to challenges arising from nonlinearity, multimodality (including target maneuver), intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities), etc., new theories, algorithms, and technologies have been developed continuously to maintain such a conjugacy, or to approximate it as close as possible. They had contributed in large part to the prospective developments of time series parametric filters in the last six decades. In this paper, we review the state of the art in distinctive categories and highlight some insights that may otherwise be easily overlooked. In particular, specific attention is paid to nonlinear systems with an informative observation, multimodal systems including Gaussian mixture posterior and maneuvers, and intractable unknown inputs and constraints, to fill some gaps in existing reviews and surveys. In addition, we provide some new thoughts on alternatives to the first-order Markov transition model and on filter evaluation with regard to computing complexity

Stochastic Real-time Optimal Control for Bearing-only Trajectory Planning

A method is presented to simultaneously solve the optimal control problem and the optimal estimation problem for a bearing-only sensor. For bearing-only systems that require a minimum level of certainty in position relative to a source for mission accomplishment, some amount of maneuver is required to measure range. Traditional methods of trajectory optimization and optimal estimation minimize an information metric. This paper proposes constraining the final value of the information states with known time propagation dynamics relative to a given trajectory which allows for attainment of the required level of information with minimal deviation from a general performance index that can be tailored to a specific vehicle. The proposed method does not suffer from compression of the information metric into a scalar, and provides a route that will attain a particular target estimate quality while maneuvering to a desired relative point or set. An algorithm is created to apply the method in real-time, iteratively estimating target position with an Unscented Kalman Filter and updating the trajectory with an efficient pseudospectral method. Methods and tools required for hardware implementation are presented that apply to any real-time optimal control (RTOC) system. The algorithm is validated with both simulation and flight test, autonomously landing a quadrotor on a wire

Source localization within a uniform circular sensor array

Traditional source localization problems have been considered with linear and planar antenna arrays. In this research work, we assume that the sources are located within a uniformly spaced circular sensor array. Using a modified Metropolis algorithm and Polak-Ribière conjugate gradients, a hybrid optimization algorithm is proposed to localize sources within a two dimensional uniform circular sensor array, which suffers from far field attenuation. The developed algorithm is capable of accurately locating the position of a single, stationary source within 1% of a wavelength and 1° of angular displacement. In the single stationary source case, the simulated Cramer-Rao Lower Bound has also shown low noise susceptibility for a reasonable signal to noise ratio. Additionally, the localization of multiple stationary sources within the array is presented and tracking capabilities for a slowly moving non-stationary source is also demonstrated. In each case, results are presented, analyzed and discussed. Furthermore, the proposed algorithm has also been validated through hardware experimentation. The design and construction of four microstrip patch antennas and a wire antenna have been completed to emulate a circular sensor array and the enclosed source, respectively. Within this array, data has been collected at the four sensors from several fixed source positions and fitted into the proposed algorithm for source localization. The convergence of the algorithm with both simulated data and data collected from hardware are compared and sources of error and potential improvements are proposed

Exploiting Sparse Structures in Source Localization and Tracking

This thesis deals with the modeling of structured signals under different sparsity constraints. Many phenomena exhibit an inherent structure that may be exploited when setting up models, examples include audio waves, radar, sonar, and image objects. These structures allow us to model, identify, and classify the processes, enabling parameter estimation for, e.g., identification, localisation, and tracking.In this work, such structures are exploited, with the goal to achieve efficient localisation and tracking of a structured source signal. Specifically, two scenarios are considered. In papers A and B, the aim is to find a sparse subset of a structured signal such that the signal parameters and source locations maybe estimated in an optimal way. For the sparse subset selection, a combinatorial optimization problem is approximately solved by means of convex relaxation, with the results of allowing for different types of a priori information to be incorporated in the optimization. In paper C, a sparse subset of data is provided, and a generative model is used to find the location of an unknown number of jammers in a wireless network, with the jammers’ movement in the network being tracked as additional observations become available

MmWave V2V Localization in MU-MIMO Hybrid Beamforming

Recent trends for vehicular localization in millimetre-wave (mmWave) channels include employing a combination of parameters such as angle of arrival (AOA), angle of departure (AOD), and time of arrival (TOA) of the transmitted/received signals. These parameters are challenging to estimate, which along with the scattering and random nature of mmWave channels, and vehicle mobility lead to errors in localization. To circumvent these challenges, this paper proposes mmWave vehicular localization employing difference of arrival for time and frequency, with multiuser (MU) multiple-input-multiple-output (MIMO) hybrid beamforming; rather than relying on AOD/AOA/TOA estimates. The vehicular localization can exploit the number of vehicles present, as an increase in a number of vehicles reduces the Cramr-Rao bound (CRB) of error estimation. At 10 dB signal-to-noise ratio (SNR) both spatial multiplexing and beamforming result in comparable localization errors. At lower SNR values, spatial multiplexing leads to larger errors compared to beamforming due to formation of spurious peaks in the cross ambiguity function. Accuracy of the estimated parameters is improved by employing an extended Kalman filter leading to a root mean square (RMS) localization error of approximately 6.3 meters