Mobile sensing systems based on improved GDOP for target localization and tracking

Code phase GNSS receivers convert the measured satellite pseudoranges into estimates of the position and clock offset of the receiver, typically via an iterative, linearized least squares method. Since the pseudoranges themselves are noisy, the resulting estimates of position and time are random variables. To describe the accuracy of this solution, it is common to describe it statistically via the error covariance matrix. Rather than considering the individual elements of this covariance matrix, users frequently reduce it to a scalar performance indicator; the most common of these is the Geometric Dilution of Precision (GDOP). It is well known that the GDOP is a function of the satellite geometry; with only a few visible satellites in poor locations, the GDOP can become quite large. However, for a future with multiple, fully occupied GNSS constellations it is expected that receivers would select those satellites to track so as to achieve the best possible performance. Hence, an understanding of both how small the GDOP can be as a function of the number of satellites visible and the characteristics of the constellations that meet that bound are of value. Further, once identified, a receiver could exploit those constellation characteristics in selecting a subset of satellites. Investigating the best possible GNSS satellite constellation with respect to the GDOP is not a new problem. Recently, these authors developed achievable lower bounds to the GDOP as a function of the number of satellites; the bounds were also extended to non-zero mask angle and to multiple GNSS constellations. Further, using actual GPS satellite ephemeris data, it was shown by example that good GDOP performance resulted from constellations similar to the “best” constellations resulting from the bounds. This paper examines augmentation of the GNSS pseudoragnes with data from non-GNSS sensors; specifically, ranges. While integration of GNSS and non-GNSS sensors is not novel, the perspective in the paper is how such external sensors impact potential receiver performance (i.e. minimum GDOP) and what role they play in satellite selection. Specifically, tight lower bounds to GDOP when the GNSS is augmented by this additional measurement (barometric altimeter or a DME slant range) are presented; achievability of the bounds is also examined.

Geometric dilution of precision (GDOP) is defined as the ratio of the root-mean-square position determination error to the root-mean-square measurement error. Based on the Cramer-Rao lower bound (CRLB), new mathematical expressions of GDOP are presented in Time-of-Arrival (TOA) and Angle-of-Arrival (AOA) positioning systems in this paper. The expressions obviously reveal that the geometrical shape of the anchor nodes make a significant influence on GDOP. Then the influence of adding a new anchor node at different locations is analyzed. Simulations show that in both positioning systems, two optimal placement angles exist to make the GDOP smallest, when the original anchor nodes are no uniformly scattered. However, when uniformly scattered, the GDOP keeps constant no matter what angle the new anchor node is placed at. In AOA positioning systems, the GDOP is also affected by the distance between the newly added anchor node and the target, and increases when the distance increases. Meanwhile, the GDOP is always reduced when more anchor nodes are used.

In the zenith-horizon placement for achieving minimum geometric dilution of precision (GDOP), one access node or sensor is positioned along the z-axis, while the remaining nodes are placed symmetrically on a three-dimensional (3D) cone. This configuration yields the minimum GDOP at the cone’s tip, which we term the designated min-GDOP point. However, in practical localization applications, the unknown node is not necessarily located at this designated min-GDOP point; instead, it may be situated anywhere within an area. As a result, evaluating localization accuracy across the entire area, rather than at a single point, is more relevant. Averaged horizontal dilution of precision (HDOP) and GDOP across the region provide more meaningful metrics for system-wide performance than values computed only at a specific location. Although many recent positioning applications leverage multiple unmanned aerial vehicles (UAVs), many established fixed sensor deployments predate the widespread adoption of UAVs. This paper proposes a novel approach with a single UAV working in conjunction with existing fixed access nodes for positioning. This approach offers improved adaptability for fixed infrastructure while circumventing the expense of establishing entirely new UAV systems, thus providing a valuable compromise. We investigate the criteria of average HDOP and GDOP over the given area. The objective is to determine optimal UAV positions along the flight path that minimize the average HDOP and/or GDOP across the area. Due to the analytical complexity, we employ numerical methods. Our simulation results demonstrate that minimizing average HDOP and GDOP often requires different UAV positions, depending on the number of access nodes and the size of the area. Consequently, achieving simultaneous minimization of both metrics with a single UAV trajectory is generally infeasible. When minimizing the average HDOP with a small number of access nodes, aligning the UAV’s XY-plane angle with those of the stationary nodes, offset by , proves advantageous. This angular alignment becomes less critical as the number of access nodes increases. For scenarios where both HDOP and GDOP are important, UAV placement can be optimized by selecting appropriate trade-offs. Additionally, we quantify how increasing the number of access nodes improves the average HDOP and GDOP over the specified area.

Geometric Dilution of Precision (GDOP) represents the magnification positioning error in satellite navigation, smaller GDOP represents high positioning accuracy. The analytic results of relevant methods to derive the minimum of GDOP by geocentric coordinate system cannot be achieved in reality, because it does not think fully about the satellite visibility constrains. This article employs the elevation angle to express the satellite visibility, and on basis of the local Cartesian coordinates coordinate system, the essay derives the minimum of GDOP with constraint of four satellites visible. Based on the information track of Beidou satellite navigation system, the minimum of GDOP is verified by using the STK simulation software.

UWB (Ultra Wideband) TOF (Time of Flight) location system is a high accuracy real time indoor location system. It includes many anchors and positioning tags. GDOP (geometric dilution of precision) is an index that used to measures the accuracy of GPS (Global positioning system). In this paper we will study GDOP from the GPS and use GDOP into indoor location system. Researching whether GDOP is a big impact to the accuracy of UWB indoor location system. At the end of the paper, there are two-dimensional positioning GDOP simulation based on MATLAB and several groups of experimental data which calculated by hyperbolic positioning algorithm.

With the development of the Global Navigation Satellite System (GNSS), the integrated system including two or more individual constellations will be the main navigation mode in the future. For integrated system, the Geometric Dilution of Precision (GDOP) is widely used for the selection of satellites and the evaluation of positioning accuracy. The change of GDOP is related with the number of the satellites. In this paper, the impact of the number of satellites on the change of GDOP is theoretically derived. The result shows that, adding satellites from the existing tracked constellations into the integrated system, the GDOP always decreases when the number of satellites increases.

Passive source localization is an important research area with numerous applications in mobile communications and signal processing. This paper presents an analysis of energy-based localization performance with unknown transmit power using received signal strength (RSS) and RSS difference (RSSD) models. These models are widely used because of their low cost and simple implementation. Previous studies were based on the assumption that the source node transmit power is known, which is not practical in many situations. The Cramer–Rao lower bound for the RSS and RSSD models with correlated noise is derived as a performance benchmark for the mean squared error (MSE) of the location and transmit power estimation. It is shown that the MSE can be factored into two independent terms corresponding to the geometric distribution of the sensors and the channel parameters including the noise variance. The effect of the sensor and source node positions on the location accuracy and the MSE is derived via the geometric dilution of precision (GDOP). The GDOP is then used to evaluate the effect of the joint estimation of unknown power and source location on the performance. Lower bounds on the GDOP are derived to obtain the minimum MSE. The RSSD solution is obtained by formulating the nonlinear and nonconvex objective functions into a convex optimization problem through relaxation and semidefinite programming. Simulation results are presented with the source inside or outside the sensor network.

The workshop Measurement Positioning System (wMPS) is a large-scale measurement system that better copes with the current challenges of dimensional metrology. However, as a distributed measuring system with multiple transmitters forming a spatial measurement network, the network topology of transmitters relative to the receiver exerts a significant influence on the measurement accuracy albeit one that is difficult to quantify. An evaluation metric, termed the geometric dilution of precision (GDOP), is introduced to quantify the quality of the network topology of the wMPS. The GDOP is derived from the measurement error model of wMPS and its mathematical derivation is expounded. Two significant factors (density and layout of the transmitter) affecting the network topology are analyzed by simulations and experiments. The experimental results show that GDOP is approximately proportional to the measurement error. More transmitters, and a relatively good layout thereof, can decrease the value of GDOP and the measurement error.

This paper presents a new method to calculate the geometric dilution of precision (GDOP) of GPS by incorporating the concept of model predictive filtering in the training process of neural networks to learn the relationship between GDOP and the azimuth and elevation of satellite. This method overcomes the shortcomings of the traditional back propagation neural networks, such as the slow convergence speed and easily falling into local minimum. A model predictive filtering algorithm is developed by using network weights as system state variables to optimize the network weights based on the neural network’s error correction. During the training process, the neural network model error is corrected by compensating the deviation between the actual and target output via the model predictive filtering. Experimental results and comparison analysis demonstrate that the proposed method can effectively approximate GDOP with improved accuracy and reduced training time.

To evaluate the performance of real time kinematic (RTK) network algorithms without applying actual measurements, a new method called geometric precision evaluation methodology (GPEM) based on covariance analysis was presented. Three types of multiple reference station interpolation algorithms, including partial derivation algorithm (PDA), linear interpolation algorithms (LIA) and least squares condition (LSC) were discussed and analyzed. The geometric dilution of precision (GDOP) was defined to describe the influence of the network geometry on the interpolation precision, and the different GDOP expressions of above-mentioned algorithms were deduced. In order to compare geometric precision characteristics among different multiple reference station network algorithms, a simulation was conducted, and the GDOP contours of these algorithms were enumerated. Finally, to confirm the validation of GPEM, an experiment was conducted using data from Unite State Continuously Operating Reference Stations (US-CORS), and the precision performances were calculated according to the real test data and GPEM, respectively. The results show that GPEM generates very accurate estimation of the performance compared to the real data test.

Multi-constellation integrated navigation receiver will increase the number of visible satellites and improve positioning accuracy of the receiver. If all-in-view satellites are used for receiver positioning, the computational burden of the receiver will be increased. The traditional satellite selection algorithm is traversal algorithm; however, as the number of visible satellites increases, the traversal algorithm exists huge computation. In order to the problem, the improved particle swarm optimization (PSO) is given for satellite selection, in the proposed algorithm, each satellite subset is considered a particle without mass in search space, and the selected objective function is the geometric dilution of precision (GDOP). Particles update their position based on the proposed algorithm model. Moreover, the optimal satellite subset and the corresponding GDOP value are obtained. The performance of the algorithms is compared based on real navigation data. The simulation results show that proposed algorithm can improve satellite selection speed, and the satellite selection accuracy is better than that of the basic PSO satellite selection algorithm.

Reconfigurable intelligent surfaces (RISs) are anticipated to constitute a critical component of forthcoming communication networks due to their ability to establish controllable wireless environments. Furthermore, RIS can be leveraged to solve infeasible localization problems. This paper presents a novel adoption of the geometric dilution of precision (GDoP) analysis in the design of non-line of sight (NLoS) single anchor millimetre wave (mmWave) large RIS-aided localization in the near-field. Considering downlink transmission, the time difference of arrival (TDoA) is used to estimate the positioning by extracting the signal contribution of each RIS tile. Considering the enriched time of arrival (ToA) measurements provided by RIS, the RIS tiles involved in TDoA-based estimation should be selected properly. Therefore, the GDoP adopted in this context is to select RIS tile combinations that achieve minimum GDoP values. It has been shown via numerical simulation that the proposed design of the RIS-aided localization outperforms other state-of-the-art techniques that adopt the signal-to-noise ratio (SNR) for the selection of the RIS tiles. It is demonstrated that the proposed scheme can provide sufficient localization accuracy involving only 10% of RIS tiles, whereas the SNR method requires about (70–80) % of the tiles to approximately achieve the same accuracy.

Mobile sensor nodes are featured with node mobility along with standard sensor functions. They can move around after being deployed. In many situations placement of static sensor nodes might not be possible and human intervention is not feasible. Mobile sensor nodes are very useful in such hazardous and disastrous situations. When a group of mobile sensor nodes are deployed to cover an area for search operations, coordination among the deployed nodes is very important. A mobile sensor node (MSN) depletes more energy during the traversal. Whenever there arises a failure, in one of the deployed mobile sensor nodes, the remaining fault-free mobile sensor nodes should travel to cover the remaining uncovered area. In this work we propose a Mobile Traversal Algorithm (MTA), for mobile sensor nodes to cover a rectangular region of interest (ROI). It makes MSNs to travel shorter distances to extend effective operational duration of the network and also to provide fault tolerance mechanism.

The navigation performance of Iridium satellite is simulated and analyzed based on STK from two aspects—the number of visible satellites and geometric dilution of precision (GDOP). Meanwhile we compare it with GPS and GALILEO system to demonstrate its performance. Experiment shows that the Iridium satellite system has a good coverage merit at two poles, which is even better than that of GPS and GALILEO. However, its global coverage is unstable and GDOP changes sharply from high latitude area to low latitude area which means Iridium satellite cannot orientate or navigate around the equator well. Then we explore the application fields of the Iridium satellite system.

In order to improve the accuracy and engineering feasibility of four-Satellite localization system, the frequency difference measurement is introduced to the four-Satellite TDOA (Time Difference of Arrival) localization algorithm. The TDOA/FDOA (Frequency Difference of Arrival) localization algorithm is used to optimize the GDOP (geometric dilution of precision) of four-Satellite localization. The simulation results show that the absolute position measurement accuracy has little influence on TDOA/FDOA localization accuracy as compared with TDOA localization. Under the same conditions, TDOA/FDOA localization has better accuracy and its GDOP shows more uniform distribution in diamond configuration case. The localization accuracy of four-Satellite TDOA/FDOA is better than the localization accuracy of four-Satellite TDOA.