Efficient DOP Calculation for GPS with and without Altimeter Aiding

Dilution of precision (DOP) for time-of-arrival (TOA) or time-difference-of-arrival (TDOA) positioning systems has been thoroughly investigated. This is not the case for angle-of-arrival (AOA) positioning systems, especially three dimensional AOA systems. However, the recent increase in deployment of antenna arrays and multipath mitigation techniques make AOA systems more attractive. In this paper, the DOPs for position systems using range measurements are reviewed and an expression for 3D AOA DOP is derived. The average values of DOPs for different deployments of base station geometries are examined. It is found that for a 3D AOA system, unlike TOA or TDOA systems, vertical DOP (VDOP) is generally smaller than horizontal DOP (HDOP), and the best VDOP can be achieved when the base stations are at the same height as the user. Moving the base stations away from the plane of the user may decrease the HDOP and Geometric DOP (GDOP) slightly, but it also increases the VDOP. For applications that the height of the user changes frequently, the base stations should be deployed with height close to half the height of the area of interest.

Global Positioning System (GPS) satellites signal processing to obtain all in view satellite measurements and to use them to find a solution and to do integrity monitoring forms a major component of the load on the receiver's processing element. If processing capability is limited there is restriction on the number of measurements which can be obtained and processed. Alternatively, the number of measurements can be restricted and the resulting saving in load on the processor can be used to offer more spare processing time which can be used for other user specific requirements. Thus if m visible satellites can provide measurements only n measurements can be used (n < m). The arrangement and the number of GPS satellites influence measurement accuracy. Dilution of Precision (DOP) is an index evaluating the arrangement of satellites. Geometric DOP (GDOP) is, in effect, the amplification factor of pseudo-range measurement errors into user errors due to the effect of satellite geometry. The GDOP approximation is an essential feature in determining the performance of a positioning system. In this paper, knowledge-based methods such as neural networks and evolutionary adaptive filters are presented for optimum approximation of GDOP. Without matrix inversion required, the knowledge-based approaches are capable of evaluating all subsets of satellites and hence reduce the computational burden. This would enable the use of a high-integrity navigation solution without the delay required for many matrix inversions. Models validity is verified with test data. The results are highly effective techniques for GDOP approximation.

Positioning of User Equipment (UE) in 5G Ultra-Dense Networks (UDN) promises significant improvement in location accuracy estimation due to densification of Access Points (AP) infrastructure. For range and bearing primary measurements network geometry plays a significant role in the resultant positioning accuracy, regardless of time (TOA) and angle (AOA) of arrival secondary processing technique. Existing approaches for positioning system topology search in terrestrial cellular networks utilize geometric dilution of precision (GDOP) minimization under certain conditions, satisfying application requirements. Drawback of existing approaches, yielding tractable analytical results, is that they predominantly have assumption of 2-dimensional (2D) AP distribution according to regular hexagonal grid, square lattice or Poisson point process (PPP). If for previous 2G-4G cellular networks it is justified, for 5G UDN these assumptions do not retain. According to latest 3GPP specifications, enhanced positioning service area should provide sub-meter 3D UE location estimation in horizontal and vertical planes. The contribution of current research is a simulation model for 5G UDN positioning system topology search using well known in Global Navigation Satellite Systems (GNSS) horizontal (HDOP), vertical (VDOP) and position (PDOP) dilution of precision (DOP) criterion. Simulation results reveal, that fixed AP number topology, achieving DOP criterion in horizontal plane, does not provide satisfying DOP in vertical one.

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.

The challenges of optical wireless positioning are addressed in this paper by a thorough investigation of angle of arrival (AOA) positioning characteristics. The overall positioning precision for AOA positioning is studied in terms of two contributing factors—being angular precision and geometric dilution of precision (DOP). Angular precision is characterized for an optical wireless receiver having an especially wide angular field-of-view (FOV). Geometric DOP is characterized for optical beacons deployed in the form of triangle, square, and hexagon cell geometries. The mean and standard deviation of the positioning errors are extracted from the positioning error distributions for each of the three cell geometries. It is found that the overarching goal to establish low and uniform positioning error distributions can be met by implementing an optical wireless receiver with a wide angular FOV and by implementing the optical beacon geometry with a correspondingly small height-to-side-length ratio. The prospects of these findings are discussed for future optical wireless positioning systems.

Dilution of Precision (DOP) is used as a parameter representing the geometrical position of visible satellites, and as a measure of position error of Global Navigation Satellite Systems (GNSSs). In previous studies, the Weight DOP (WDOP) was proposed by setting different weights based on the signal quality of the GPS satellite, and the signal quality of satellites was evaluated based on the elevation angle. In this study, Multi-Constellation Weighted DOP (MWDOP) is defined by expanding the weight matrix of GPS to the global navigation satellite systems (GPS, GLONASS, Galileo, and Beidou). The weight matrix is defined through actual data considering the elevation angle and pseudorange for each satellite system. Furthermore, the satellite selection algorithm is set up through sensitivity, which is calculated by dividing the difference between the DOP calculated for the entire set of satellites and the DOP calculated for a set excluding specific satellites divided by the DOP calculated for the entire set of satellites. If the calculated sensitivity is large, the excluded satellites are considered to have a large effect on the satellite positioning. This study provides more reliable criteria for selecting satellites by comparing DOP sensitivity and MWDOP sensitivity.

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) 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.

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.

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.

In mobile sensing for target localization and tracking, the geometric relationship between sensor nodes and a target may affect the positioning accuracy, of which the effect may be indicated by geometric dilution of precision (GDOP). In the paper, a target localization scheme based on GDOP for time-difference of arrival mobile sensing systems is presented. To attain the best sensor formation for maintaining good GDOP values, a time-varying function based on the GDOP curved surface is defined. Based on the derived function, path planning and relocation of mobile sensor nodes (MSNs) at each time cycle are discussed. Simulation results show that location accuracy improves when the MSNs steadily move toward the best deployment for tracking the target.

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.

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.

This paper introduces an indoor integrated navigation system for the auto-control of unmanned aerial vehicle (UAV) and robot that fuses inertial sensors and ultra wide band (UWB) networks. In this paper, we concentration on the problem of selecting the optimized UWB anchor node to obtain the distance measurement between the selected anchor node and the tag node for this indoor integrated navigation system. Taking the navigation accuracy and system power consumption into account according to the requirements and features of UWB system, we propose and analyze the performance of four kinds of anchor node selection algorithms, named alternation algorithm, powerfirst algorithm, accuracy-first algorithm and the optimized algorithm considering power and accuracy. In the fourth algorithm, we introduce dynamic local dilution of precision (DOP) values which reflect the impact of the geometric distribution of anchor nodes to guarantee precision and define a cost function of DOP values and UWB communication distance for a complementary consideration of power consumption. To fuse the inertial sensor data and the selected UWB distance measurement, an extended Kalman filter (EKF) is applied in this integrated system to correct the state errors. Simulations and experiments with low-cost Micro-electromechanical inertial sensors and Decawave UWB transceivers have been done to evaluate the performance of these four anchor node selection algorithms. The simulation and experiments results verify that the fourth algorithm proposed in this paper can achieve a good balance in system multiple demands of navigation accuracy and system power consumption.