Geometric Dilution Of Precision Research Articles

On the exact solutions of pseudorange equations

Three formulations of exact solution algorithms to the system of determined pseudorange equations are derived. It is demonstrated that pseudorange equations are hyperbolic in nature and may have two solutions, even when the emitter configuration is nonsingular. Conditions for uniqueness and for the existence of multiple solutions are derived in terms of the Lorentz inner product. The bifurcation parameter for systems of pseudorange equations is also expressed in term of the Lorentz functional. The solution is expressed as a product of the geometric dilution of precision (GDOP) matrix, representing the linear part of the solution, and a vector of nonlinear term. Using this formulation stability of solutions is discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

SIMULATION OF BALLISTIC MISSILE MOTION IN FREE-FLIGHT (BALLISTIC) PHASE USING NAVSTAR (GPS) NAVIGATION MODEL

The accuracy of Ballistic Missile (BM) guidance is mainly dependent on the accuracy of the navigation system used. Majority of BM's apply the inertial navigation system (INS) where the inaccuracies of its sensors (accelerometers and gyros) induce errors which may accumulate to intolerable values due to increased flight time. The guidance is restricted only to the powered flight phase of the whole BM trajectory. In the present paper we navigate the BM motion in the free-flight phase (ballistic phase) using the NAVSTAR/GPS model and an implementation of the BM motion in the ballistic phase. The simulation includes several cases of ballistic trajectories applied on the GPS model that includes 24 satellites constellation. pseudo-range and ephemeries data are simulated, and the application of the principle of minimum geometric dilution of precision (GDOP) is performed. The implementation of the BM motion in the free-flight phase is applied on the optimum trajectories using the Keplerian orbit (elliptical section) in which Kepler's equation is solved at each instant on the trajectory by Newton Raphson method.The results of the different algorithms of implementation are evaluated. The solution of GPS equations is executed for BM motion in the ballistic trajectory using the iterative method. Different algorithms for selection of GPS model are given.

Calculation of geometric dilution of precision

A very simple closed-form formula for the calculation of the geometric dilution of precision (GDOP) in Global Positioning System (GPS) navigation and in Global Navigation Satellite System (GLONASS) navigation is presented, which requires less than 40 multiplications.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

IMPLEMENTING THE NAVSTAR/GPS IN EGYPT

Tire global Positioning System (NAVSTAR/GPS) should reach full 3 D capability in about 1992. Due to budget constraints, funds were cut back from a level which would provide a 24 satellite constellation to the present level which will maintain only 18 satellites plus three fully functioning on-orbit spares. As a consequence, there will be continuously some areas of temporarily reduced accuracy, requiring those who are using such data to periodically adjust their operations as consequence. One way of doing that is to calculate the changes in system measurements accuracy factors, mainly the geometric dilution of precision (GDOP) which is a measure of how satellite geometry degrades accuracy. This is very important for users relying only on four satellites GPS navigation such as our case. The used algorithm gives the user at Cairo city the changes of the GDOP with that of the satellites'' geometry and consequently with time.

Additional results on 'reducing geometric dilution of precision using ridge regression'

R.J. Kelly (1990) presented preliminary results on the feasibility of using ridge regression to reduce the effects of geometric dilution of precision (GDOP) error inflation in position-fix navigation systems. Results indicate that the ridge technique will not reduce bias inflation due to the effects of GDOP in applications where bias-like measurement errors persist for time periods which greatly exceed the response time of an aircraft's guidance loop. This conclusion will preclude the use of ridge regression on navigation systems whose dominant error sources are bias-like. This applies in particular to the Global Positioning System (GPS) selective availability error source. All the simulation results given in Kelly's work are, however, valid for the conditions defined. Although ridge regression has not yielded a satisfactory solution to the general GDOP problem, it has illuminated the role that multicollinearity plays in navigation signal processors such as the Kalman filter. After a background discussion, bias inflation, initial position guess errors, ridge parameter selection methodology, and the recursive ridge filter, are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Reducing geometric dilution of precision using ridge regression

A signal processing technique is proposed for improving position-fix navigation system accuracy performance when the geometry of the navigation landmarks (e.g. sensors) are nearly collinear. In the navigation literature, the accuracy degradation associated with a nearly collinear measure geometry is termed the geometric dilution of precision (GDOP). Its presence causes not only the variance of the position estimates to be highly inflated but also any bias terms which may be present in the model. Since a nearly collinear predictor matrix is mathematically equivalent to GDOP, it is proposed to use the ridge regression technique in a navigation signal processor. A position-fix algorithm based on ridge regression reduces the bias and variance inflation caused by GDOP and the overall mean-squared position error as well. Ridge regression contains the GDOP-sensitive least-mean-square (LMS) estimator as a special case. Even with a matched model, GDOP can inflate the mean-square error (MSE) of the ordinary least-squares estimator, whereas the ridge regression technique chooses a suitable biased estimator that will reduce the MSE, which is the main goal. The ridge concept is extended to include GDOP-amplified bias errors. A simple range/range navigation system is analyzed to illustrate the underlying principles of ridge regression.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Kinematic Survey of Clinton Lake Dam

Kinematic survey phase data are reduced using a standard double difference formulation when the initial integer values of the biases are known or determined. Two experimental results are presented. A byproduct of the reduction was the generation of a measure of success similar to the Geometrical Dilution of Precision (GDOP) familiar to those using measurements of pseudo ranges. This new measure is called RDOP for Relative Dilution of Precision.

測位システムにおける観測方向と推定精度に関する固有値的考察

測位システムにおける観測方向と推定精度に関する固有値的考察

Optimal Locations of Pseudolites for Differential GPS

Because of the announced DOD policy to degrade GPS accuracy, Differential GPS has assumed increasing importance. Furthermore, the delay in operational deployment of the NAVSTAR satellites has focused increased attention on the use of pseudo-satellites (pseudolites or PLs) to enhance accuracy and provide earlier operational capability. Pseudolites perform a double function: they both calibrate the local ranging errors and significantly improve the user's geometry as measured by GDOP (Geometric Dilution of Precision). This paper examines the following problem: What is the best location for a PL when one considers the worst possible satellite outage in twenty-four hours? What is the resulting worst case GDOP? Both single and dual PL systems are considered, and the results are extrapolated to expected user receiver performance (horizontal and vertical errors). The results indicate that a dual PL system will outperform the satellite-only GPS system, even with a worst case satellite outage. In addition, there should be negligible degradation during periods when the military intentionally reduces ephemeris or clock accuracy. Furthermore, the accuracy of such a system would provide aircraft positioning very close to Category I FAA landing requirements.

西豪州航路におけるオメガの使用結果について

From January to March in 1984, the phase data of Omega 10.2kHz signals are measured onboard, by a receiver with rubidium atomic standard, along the route between Japan and Western Australia for the purpose of analyzing the signal behavior and evaluating the relative error of LOPs by means of comparing with nominal LOPs (PPC included) by NNSS and Loran-C. Accordingly, recommended LOP combinations along the above route are selected in consideration of the relative accuracies of LOPs and Geometric Dilution of Precision (GDOP). As a result, signal behavior (10.2kHz) of A, G and H are confirmed to be almost the same as described in the Omega Coverage Diagrams. Accuracies of the Omega System are estimated 2〜3 N.M. (RMS) as a whole on the route concerned, but navigators should notice that it is difficult to make good Position Fix in the nighttime at the southern and the eastern parts of the South China Sea because of modal interference on H and G.

An Algebraic Solution of the GPS Equations

The global positioning system (GPS) equations are usually solved with an application of Newton's method or a variant thereof: Xn+1 = xn + H-1(t - f(xn)). (1) Here x is a vector comprising the user position coordinates together with clock offset, t is a vector of tour pseudorange measurements, and H is a measurement matrix of partial derivatives H = fx· In fact the first fix of a Kalman filter provides a solution of this type. If more than four pseudoranges are available for extended batch processing, H-1 may be replaced by a generalized inverse (HTWH)-1HTW, where W is a positive definite weighting matrix (usually taken to be the inverse of the measurement covariance matrix). This paper introduces a new method of solution that is algebraic and noniterative in nature, computationally efficient and numerically stable, admits extended batch processing, improves accuracy in bad geometric dilution of precision (GDOP) situations, and allows a "cold start" in deep space applications.

Operational Benefits and Design Approaches for Combining JTIDS and GPS Navigation

This paper describes the Joint Tactical Information Distribution System (JTIDS) Navigation Function and the Global Positioning System (GPS), emphasizing their technical similarities and differences. Based on these, the paper discusses the operational benefits, performance, and design approaches for combining the navigational and time data from these two systems. Both JTIDS and GPS are time synchronous, spread spectrum systems which provide high accuracy navigation data through the use of time-of-arrival measurements and pseudo-ranging techniques. Both are subject to Geometric Dilution of Precision (GDOP) effects, though under different conditions; and both are highly resistent to interference, though to different degrees. JTIDS is more a tactical or local system, while GPS is global in nature. Exchanging data between the two systems, under certain constraints, therefore has the potential of great performance and operational benefits with respect to system availability, under a variety of operational scenarios and interference conditions. These benefits and proposed methods for combining the data from the two systems are described in the paper.

Statistical Theory of Passive Location Systems

A derivation of the principal algorithms and an analysis of the performance of the two most important passive location systems for stationary transmitters, hyperbolic location systems and directionfinding location systems, are presented. The concentration ellipse, the circular error probability, and the geometric dilution of precision are defined and related to the location-system and received-signal characteristics. Doppler and other passive location systems are briefly discussed.

Comment on "Geometric Dilution of Precision in Global Positioning

Comment on "Geometric Dilution of Precision in Global Positioning

Geometric dilution of precision in Global Positioning System navigation

The evaluation of the GPS navigation performance is essentially equivalent to the computation of the diagonal terms of the geometric dilution of precision (GDOP) matrix. Certain theoretical results concerning the general properties of the GDOP matrix are obtained. An efficient algorithm for the computation of the GDOP matrix and the navigation performance index is given; and applications of the results are illustrated by numerical examples.

Geometric Dilution of Precision

A graph is developed which gives geometric dilution for fixes obtained hyperbolically using information received from three stations. Only bearings from the user to the stations are needed. The approach can be characterized as oriented to the user rather than to system development and is particularly valuable in assessing station choices in redundant systems such as Omega. Dilution is shown to be minor even if all bearings are in the same semi-circle provided the central station is on a nearly central bearing.

Maximum likelihood and optimum location of observation sites in geodetic and spatial surveying

In geodetic and spatial surveying, accurate statistical es-timates of the Cartesian coordinates of a stationary target depend, in general, on such factors as target position, number and type of instruments, their location, their accuracy, and their mix. Direct measurements are assumed to be made in elevation, azimuth, and range with errors independently and normally distributed. For a mix of instruments, theoretical arguments are established using maximum likelihood estimation to determine the optimal siting of the instruments which minimizes (GDOP)2 or the square of the geo-metric dilution of precision = sum of the variances of the estimates in the three orthogonal directions.