Geometric Dilution Of Precision Research Articles

Weighted geometric dilution of precision calculations with matrix multiplication.

To enhance the performance of location estimation in wireless positioning systems, the geometric dilution of precision (GDOP) is widely used as a criterion for selecting measurement units. Since GDOP represents the geometric effect on the relationship between measurement error and positioning determination error, the smallest GDOP of the measurement unit subset is usually chosen for positioning. The conventional GDOP calculation using matrix inversion method requires many operations. Because more and more measurement units can be chosen nowadays, an efficient calculation should be designed to decrease the complexity. Since the performance of each measurement unit is different, the weighted GDOP (WGDOP), instead of GDOP, is used to select the measurement units to improve the accuracy of location. To calculate WGDOP effectively and efficiently, the closed-form solution for WGDOP calculation is proposed when more than four measurements are available. In this paper, an efficient WGDOP calculation method applying matrix multiplication that is easy for hardware implementation is proposed. In addition, the proposed method can be used when more than exactly four measurements are available. Even when using all-in-view method for positioning, the proposed method still can reduce the computational overhead. The proposed WGDOP methods with less computation are compatible with global positioning system (GPS), wireless sensor networks (WSN) and cellular communication systems.

Seeking GDOP-optimal Flower Constellations for global coverage problems through evolutionary algorithms

In this paper, given a certain number of satellites (Nsat), which is limited due to the sort of mission or economical reasons, the Flower Constellation with Nsat satellites which has the best geometrical configuration for a certain global coverage problem is sought by using evolutionary algorithms. In particular, genetic algorithm and particle swarm optimization algorithm are used. As a measure of optimality, the Geometric Dilution Of Precision (GDOP) value over 30000 points randomly and uniformly distributed over the Earth surface during the propagation time is used. The GDOP function, which depends on the geometry of the satellites with respect to the 30000 points over the Earth surface (as ground stations), corresponds to the fitness function of the evolutionary algorithms used throughout this work. Two different techniques are shown in this paper to reduce the computational cost of the search process: one that reduces the search space and the other that reduces the propagation time. The GDOP-optimal Flower Constellations are obtained when the number of satellites varies between 18 and 40. These configurations are analyzed and compared. Owing to the Flower Constellation theory we find explicit examples where eccentric orbits outperform circular ones for a global positioning system.

Positioning configurations with the lowest GDOP and their classification

The positioning configuration optimization is a basic problem in surveying, and the geometric dilution of precision (GDOP) is a key index to handle this problem. Simplex graphs as regular polygons and regular polyhedrons are the well-known configurations with the lowest GDOP. However, it has been proved that there are at most five kinds of regular polyhedrons. We analytically solve the GDOP minimization problem with arbitrary observational freedom to extend the current knowledge. The configuration optimization framework established is composed of the algebraic and geometric operators (including combination, reflection, collinear mapping, projection and three kinds of equivalence relations), basic properties to GDOP minimization (including continuity, combination invariant, reflection invariant, rotation invariant and collinear invariant) and the lowest GDOP configurations (including cones, regular polygons, regular polyhedrons, Descartes configuration, helical configuration and generalized Walker configuration, and their reflections and combinations). GDOP minimization criterion and D-maximization criterion both reduce to the same criterion matrices that the optimization becomes the problem for solving an underdetermined quadratic equation system. Making use of the concepts for solving underdetermined linear equation system, the concepts of base configuration (single classification) and general configuration (combined classification) are applied to the GDOP minimization to analytically solve the quadratic equation system. Firstly, the problems are divided into two subproblems by two kinds of GDOP to reveal the impact of the clock-offset on the configuration optimization, and it shows that the symmetry and uniformity play a key role in identifying the systematic errors. Then, the solution of the GDOP minimization is classified by the number of symmetry axes, that the base configurations with at least one symmetry axis and the general configurations without symmetry axis are categorized to be two large classifications. Complex configurations can be then generated by the combination and the reflection of those base configurations with simplex structure, and this indicates that completely solving the GDOP minimization needs to solve the simplex classifications primarily. Ultimately, constrained or unconstrained configuration optimization examples including GDOP distribution analysis, single-global satellite navigation system (GNSS) or multi-GNSS constellation design, configuration optimization of pseudolites and configuration design of buoys for underwater positioning are performed by employing the properties, lemmas, theorems and corollaries proposed.

New Neural Network-based Approaches for GPS GDOP Classification based on Neuro-Fuzzy Inference System, Radial Basis Function, and Improved Bee Algorithm

Global positioning system (GPS) is the most widely used military and commercial positioning tool for real-time navigation and location. Geometric dilution of precision (GDOP) stands as a relevant measure of positioning accuracy and consequently, the performance quality of the GPS positioning algorithm. Since the calculation of GPS GDOP has a time and power burden that involves complicated transformation and inversion of measurement matrices, in this paper we propose hybrid intelligent methods, namely adaptive neuro-fuzzy inference system (ANFIS), improved ANFIS, and radial basis function (RBF), for GPS GDOP classification. Through investigation it is verified that the ANFIS is a high performance and valuable classifier. In the ANFIS training, the radius vector has very important role for its recognition accuracy. Therefore, in the optimization module, bee algorithm (BA) is proposed for finding the optimum vector of radius. In order to improve the performance of the proposed method, a new improvement for the BA is used. In addition, to enhance the accuracy of the method, principal component analysis (PCA) is utilized as a pre-processing step. Experimental results clearly indicate that the proposed intelligent methods have high classification accuracy rates comparing with conventional ones.

Characterisation of GNSS Space Service Volume

There is increasing demand for navigation capability for space vehicles. The idea to extend the application of Global Navigation Satellite Systems (GNSS) from terrestrial to space applications by the use of main beam and side lobe signals has been shown to be feasible. In order to understand the performance and the potential space applications GNSS can support, this paper characterises the Space Service Volume (SSV) in terms of the four parameters of minimum received power, satellite visibility, pseudorange accuracy and Geometric Dilution of Precision (GDOP). This new definition enables the position errors to be estimated. An analytical methodology is proposed to characterise minimum received power for the worst location. Satellite visibility and GDOP are assessed based on grid points at different height layers (to capture the relationship between height and visibility) for single and multiple GNSS constellations, the former represented by BeiDou III (BDS III) and the latter, BDS III in various combinations with GPS, GLONASS and GALILEO. Additional simulation shows that GNSS can potentially support lunar exploration spacecraft at the Earth phasing orbit. This initial assessment of SSV shows the potential of GNSS for space vehicle navigation.

New Characteristics of Geometric Dilution of Precision (GDOP) for Multi-GNSS Constellations

For multi-Global Navigation Satellite System (GNSS) constellations, the Geometric Dilution of Precision (GDOP) is an important parameter utilised for the selection of satellites. This paper has derived new formulae to describe the change of GDOP. The result shows that, for GNSS single point positioning solutions, if one more satellite belonging to the existing tracked multi-GNSS constellation used in the single point positioning solution is added, the GDOP always decreases with the number of the added satellites. On the other hand, when the constellation of the added satellite is not from the tracked existing constellations, the different numbers of the added satellites have different influences on the change of GDOP. Generally, adding one satellite from another constellation into the existing multi-GNSS constellations will increase the GDOP, but adding two satellites will decrease the GDOP compared with adding one from another constellation. Additionally, the GDOP also increases in the cases of adding two satellites from two different constellations into the tracked existing constellations.

Target Positioning with GDOP Assisted Nodes Selection Algorithm in Wireless Sensor Networks

In wireless sensor networks (WSN), the geometric distribution of anchor nodes has a significant influence on the positioning accuracy. Geometric dilution of precision (GDOP) can be used to measure the positioning precision of the localization system. In order to select the optimal node combination, traditional algorithms based on GDOP need to spend much time on calculating every possible combination of nodes. This paper proposes GDOP assisted nodes selection (GANS) algorithm to calculate GDOP value of the current geometric distribution. Sensor node's contribution to the overall GDOP value is adopted as the evaluation criteria. The nodes whose contribution value is greater than the threshold will be selected. The anchor nodes subset, which participates in the positioning, will be real-time determined. Simulation results show that the GANS algorithm can effectively reduce the energy consumption of the system, while the positioning accuracy has no obvious loss. Meanwhile, computational complexity is also obviously decreased.

Probabilistic Error Modeling and Topology-Based Smoothing of Indoor Localization and Tracking Data, Based on the IEEE 802.15.4a Chirp Spread Spectrum Specification

Location awareness is a core capability in many context-aware computing platforms. Multiple existing systems either provide inadequate accuracy or require extensive calibration or preexisting measurements in order to be functional. This work presents an extensive study of indoor tracking based on the chirp spread spectrum (CSS) specification and an associated analytical framework that allows comparisons to be made between different deployments. CSS provides resilience to fading, while being rapidly deployable. Wireless CSS modules are used to provide time of arrival measurements, necessary to infer the coordinates of a mobile user through trilateration. CSS resilience is tested in four deployments: an indoor space where line of sight (LoS) conditions are always satisfied, an indoor site that includes concrete, nonreflective obstructions, an industrial space with metallic, reflective obstacles, and a Tunnel. Empirical data are discussed in conjunction with the geometric dilution of precision (GDoP) metric, which depends on the system's deployment topology. The probabilistic modeling of the normalized localization error provides insight into the underlying distribution and is utilized in the context of a novel topology-based smoothing technique. Results indicate that CSS can provide accurate tracking. The application of the smoothing algorithm, however, further reduces the normalized error by a considerable amount.

Receiver autonomous integrity monitoring availability and fault detection capability comparison between BeiDou and GPS

This paper used the statistical methods of quality control to assess receiver autonomous integrity monitoring (RAIM) availability and fault detection (FD) capability of BeiDou14 (Phase II with 14 satellites), BeiDou (Phase III with 35 satellites) and GPS (with 31 satellites) for the first time. The three constellations are simulated and their RAIM performances are quantified by the global, Asia-Pacific region and temporal variations respectively. RAIM availability must be determined before RAIM detection. It is proposed that RAIM availability performances from satellites and constellation geometry configuration are evaluated by the number of visible satellites (NVS, NVS > 5) and geometric dilution of precision (GDOP, GDOP < 6) together. The minimal detectable bias (MDB) and minimal detectable effect (MDE) are considered as a measure of the minimum FD capability of RAIM in the measurement level and navigation position level respectively. The analyses of simulation results testify that the average global RAIM performances for BeiDou are better than that for GPS except global RAIM holes proportion. Moreover, the Asia-Pacific RAIM performances for BeiDou are much better than that for GPS in all indexes. RAIM availability from constellation geometry configuration and RAIM minimum FD capability for BeiDou14 are better than that for GPS in Asia-Pacific region in all cases, but the BeiDou14 RAIM availability from satellites are worse than GPS’s. The methods and conclusions can be used for RAIM prediction and real-time assessment of all kinds of Global Navigation Satellite Systems (GNSS) constellation.

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.

Minimum of PDOP and its applications in inter-satellite links (ISL) establishment of Walker-δ constellation

Within the next decade, there will be a number of GNSS (Global Navigation Satellite System) available, i.e. modernized GPS, Galileo, restored GLONASS, BeiDou and many other regional GNSS augmentation systems. Thus, measurement redundancies and geometry of the satellites can be improved. GDOP (Geometric Dilution of Precision) and PDOP (Position Dilution of Precision) are associated with the constellation geometry of satellites, and they are the geometrically determined factors that describe the effect of geometry on the relationship between measurement error and position error. GDOP and PDOP are often used as standards for selecting good satellites to meet the desired positioning precision. In this paper, the related conclusions of minimum of GDOP which was discussed are given, and it is used to study the minimum of PDOP for two cases that the receiver is on the earth’s surface and the receiver is on satellite. The corresponding theorem and constructive solutions of minimum of PDOP are given. Then, the rationality of the ISL (inter-satellite link) establishment criteria in Walker-δ constellation is discussed by using the theory of minimum of PDOP. Finally, the minimum of PDOP is calculated when the number of satellites is 4–10, and these results are verified by using Monte Carlo method.

A Practical Method of Observer Velocity Optimization with Airborne ESM Sensor

In bearings-only target motion analysis, the observer trajectory has an important effect on the accuracy of target localization and tracking. This paper presented a practical method of observer velocity optimization based on Monte-Carlo run for airborne ESM measurement system. For verifying the validity of our method, the geometric dilution of precision (GDOP) is taken as the performance index. Finally, three scenarios are given and the GDOP values of the ESM measurement system in scenarios are calculated to illustrate the observer velocity optimization strategy.

An analysis of height precision in applying single frequency static GPS to landslide monitoring

As we all know,GPS has brought great convenience to surveying and mapping work. Its horizontal accuracy has reached sub- millimeter. However,due to the influences of such factors as the effects of ionosphere,multipath effect,GDOP( geometric dilution of precision) and height of antenna,the vertical deformation measurements from GPS are seldomly used by precision survey workers. Based on the reasonable layout of deformation monitoring network of Shuping landslide in the Three Gorges Reservoir Area,the authors obtained a large number of leveling and GPS data from long- term monitoring,and used leveling data as the reference to analyze the height precision of single- frequency static GPS in landslide vertical deformation monitoring. The results show that the vertical deformation monitoring precision of single- frequency static GPS is ± 2cm,that the deformation monitoring accuracy should be greater than 1 /5 of deformation in the measurement period according to the monitoring code of landslide,and that single frequency static GPS is suitable for the landslide whose vertical deformation is above 10 cm in its monitoring period. For the Shuping landslide,GPS measurement can provide good monitoring data in the second to third period whose vertical deformation is up to 30 cm,but it is unsuitable for other monitoring periods in slow state.

Effect of Radio Frequency Interference (RFI) on the Precision of GPS Relative Positioning

The successful of GPS observations are dependent on several factors between satellite vehicles and GPS receivers, where low GPS power levels have led to the threat of radio frequency interference (RFI) on the GPS signals. This study was conducted to evaluate the effect of RFI on the precision of positions of single and dual frequency receivers through relative positioning technique by taking into consideration the radius of GPS receiver from interference source, length of baseline and response of rejection. Several tests were conducted in real environment by simulating the interference signal towards GPS receivers in the nominated GPS L1 and L2 bands. Calculations were made to indentify the distance and interference signal power between interference source and GPS receiver in order to investigate the level of effect. To be able to study this effect on the precision of GPS positions, the 3D residual positions and geometric dilution of precision (GDOP) have been used. The findings of this study have demonstrated that a sufficient time for the GPS receiver to respond in particular interference signal power level and the radius from the interference source were made as compared to previous work. It was also indicated that the residual positions and GDOPs were affected proportionally when nearly to interference source but not similar for both days due to GPS coverage and other probable errors. Therefore, a good investigation on RFI towards GPS signals should be conducted in secured environment which can control the various GPS error parameters in order to obtain a reliable result on this effect.

Rapid and Precise GLONASS GDOP Approximation using Neural Networks

Russian global navigation satellite system (GLONASS) provides civilian and military users three-dimensional position determination and navigation services as same as US global positioning system. Geometric dilution of precision (GDOP) provides a simple interpretation of positioning precision. Usual method for GLONASS GDOP calculation is matrix inversion. However this process imposes a huge calculation load on receiver, especially when large number of visible satellites exists. To overcome this problem, artificial neural network is used. Different configurations and training methods are simulated on a data base obtained by a GLONASS receiver. Then navigation precision and execution times are explored and compared. Results show that recurrent neural network has 0.00024 RMS error, which is the best against other focused tools including feed forward back propagation and radial basis function neural network with usual training and with genetic algorithm adopted weights and biases.

Ultrawideband-Based 3-D Localization Using Compact Base-Station Configurations

This letter presents theoretical and experimental investigations and analysis on three-dimensional ultrawideband (UWB) localization using compact base-station configurations. A comparative study is performed between the three proposed configurations (Y-shape, L-shape, and mirror-based) and the commonly used cuboid-shape configuration in terms of accuracy and compactness. Results show that the average localization accuracy of the Y- and L-shape configurations is in the range of 3-5 cm, whereas the cuboid- and mirror-based configurations give an accuracy of around 2-5 cm. The Geometric Dilution of Precision (GDOP) values have been calculated for all configurations, which shows a high level of accuracy (i.e., precision values are in the range of 2-4). In addition, a tradeoff regarding occupancy area and complexity is taken into account, and the results show that the proposed configurations have the advantage of compactness as compared to the cuboid configuration and also provide high accuracy in centimeter range, hence making it suitable for various applications such as motion capture in healthcare systems and entertainment.

Interoperability of SeaSondes and Wellen Radars in Mapping Radial Surface Currents

Abstract A dual-station high-frequency (HF) Wellen Radar (WERA) transmitting at 16 MHz has observed near-real-time surface currents over an approximate range of 100 km across the Florida Straits since July 2004. During a 10-day period in April 2005 (15–25 April), a pair of 12.6-MHz SeaSondes (SS) were deployed south of the WERAs sites by NOAA's Center for Operational Oceanographic Products and Services (CO-OPS). The resulting SS grid overlapped the southern portion of the WERA domain. During the same period of time, a bottom-mounted acoustic Doppler current profiler (ADCP) acquired subsurface current measurements within these HF radar grids starting at 14 m below the surface in water of 86-m depth. The interoperability of beam-forming (WERA) and direction-finding (SS) HF radar technologies was examined. Comparisons of radial and vector currents for an 8-day concurrent time series suggested good agreement in current direction over both domains, where the surface currents' magnitudes were a maximum of 1.2 m s−1. In the core of the radar domains consisting of 108 cells, hourly vector currents were obtained by combining WERA and SS radials. Generally, this can be done in a relatively straightforward manner, considering the geometric dilution of precision (GDOP). A second key issue is downscaling the SS measurements from a 3-km grid to a 1.1-km grid to match the WERA output. This enhanced grid spacing is important along the western flank of the Florida Current, where energetic, small-scale surface features have been observed.

GPS GDOP classification via improved neural network trainings and principal component analysis

Geometric dilution of precision (GDOP) is an engineering expression that denotes how well the constellation of global positioning system (GPS) satellites is organised geometrically. In the analysis of received signals, it is often essential to invert and transform the data matrices. This requires tremendous computational burden on the navigator’s processor. Since classification of GPS GDOP is a non-linear problem, neural networks (NNs) can be used as an acceptable solution. Since the back propagation (BP) does not have sufficient speed to train a feed-forward NN, in this paper several improved NN trainings, including Levenberg–Marquardt (LM), modified LM, and resilient BP (RBP), scaled conjugate gradient, one-step secant (OSS) and quasi-Newton methods are proposed to classify the GPS GDOP. In this study, in order to have uncorrelated and informative features of the GPS GDOP, principal component analysis (PCA) is used as a pre-processing step. The simulation results show that using the RBP and PCA leads to greater accuracy and lower calculation time comparing with other existing and proposed methods and it can improve the classification accuracy of GPS satellites to about 99.65%. Moreover, the modified LM is the fastest algorithm that requires only 10 iterations for training the NN and it can be used in online applications.

Study on UAV Path Planning Oriented to Optimization of Positioning Error

For many applications in wireless sensor network (WSN) node positioning by unmanned aerial vehicle (UAV) need a path planning algorithm optimizing positioning error. In this article, we present an algorithm combining Voronoi diagram and geometric dilution of precision (GDOP) to make track points, considering both the efficiency of flying and accuracy of positioning. Then, we connect all the track points using genetic algorithm and we get the minimized flight path.

Optimal MLAT Tracking Based on Improved Simulation for Airport Surveillance

MLAT (Multilateration) is becoming one of the leading techniques in the airport surveillance. It locates the positions of aircrafts by the TDOAs (Time Difference of Arrivals) of signal from the aircraft to the different remote stations. In views of the influence from the geometric relation and the clock error among different stations, an improved simulation is proposed to estimate the measurement error for the optimal tracking. The model of simulation takes both the GDOP (Geometric Dilution of Precision) and TOA (Time of Arrival) error into consideration. Experiment results show that compared with the traditional simulation method, the improved simulation is more accurate and is able to be used in the practical application.