A closed-form of Newton method for solving over-determined pseudo-distance equations

Ning

Mathematical Problems in Engineering

et al. 2020

Ultrawideband (UWB) is considered as a promising technology in indoor positioning due to its accurate time resolution and good penetration through objects. Since the functional model of UWB positioning is nonlinear, iterative algorithms are often considered for solving the nonlinear problem. With a rough initial value, we can obtain the optimal solution by the way of continuous iteration. As an iterative descent method of high efficiency, the Gauss–Newton method is widely used to estimate the position. However, in the UWB indoor positioning, since the location information of the user is rather limited, it is not easy to get a good initial value for iteration. Besides, the positioning system is prone to become ill-posed. These factors make the Gauss–Newton method not easy to converge to a global optimal solution or even diverge, especially under ill-conditioned positioning configuration. Furthermore, the linearization of the positioning functional model results in biased least-squares estimation. The Gauss–Newton method only includes the first-order Taylor expansion of distance equations. The bias comes from neglected higher order terms. In this study, the closed-form Newton method which considers the high order partial derivatives is introduced and proposed. The simulation and measurement experiments are implemented to analyze the performance of the closed-form Newton method in UWB positioning. Both the initial value factor and geometric configuration factor are discussed. Experimental results are given to demonstrate that the closed-form Newton method can converge to the global optimal solution with better convergence and higher efficiency regardless of whether the initial iteration value is reasonable or not. Meanwhile, the closed-form Newton method can improve the accuracy of positioning results, particularly when the anchors are not uniformly and symmetrically distributed, or the ranging error is relatively large. The study shows that closed-form of the Newton method has better convergence and positioning performance than the Gauss–Newton method, especially under ill-conditioned positioning configuration or relatively low measurement precision, even though it adds a little computational cost.

Section: The Closed-form Newton Methodsmentioning

confidence: 99%

“…e closed-form of the Newton method by which the Newton method and Gauss-Newton method are connected has been obtained for solving the distance equations [26,27]. To minimize the objective function for the optimal position estimation, the performance of Gauss-Newton type and the quasi-Newton methods is analyzed and discussed [28].…”

Section: Introductionmentioning

confidence: 99%

Performance Analysis of Parameter Estimator Based on Closed-Form Newton Method for Ultrawideband Positioning

Ning

Mathematical Problems in Engineering

et al. 2020

“…, } denotes stations, x is geocenter, and m is the dimension of the configuration (m = 3) [10]. For the configuration comprising n ground stations, if and only if [15,18,19]…”

Section: Station-geocenter Dop Measuring the Uniformity Of The Stationsmentioning

confidence: 99%

Random Optimization Algorithm on GNSS Monitoring Stations Selection for Ultra‐Rapid Orbit Determination and Real‐Time Satellite Clock Offset Estimation

Yang

Mathematical Problems in Engineering

Xue

2019

Self Cite

Geographical distribution of global navigation satellite system (GNSS) ground monitoring stations affects the accuracy of satellite orbit, earth rotation parameters (ERP), and real-time satellite clock offset determination. The geometric dilution of precision (GDOP) is an important metric used to measure the uniformity of the stations distribution. However, it is difficult to find the optimal configuration with the lowest GDOP when taking the 71% ocean limitation into account, because the ground stations are hardly uniformly distributed on the whole of the Earth surface. The station distribution geometry needs to be optimized and besides the stability and observational quality of the stations should also be taken into account. Based on these considerations, a method of configuring global station tracking networks based on grid control probabilities is proposed to generate optimal configurations that approximately have the minimum GDOP. A random optimization algorithm method is proposed to perform the station selection. It is shown that an optimal subset of the total stations can be obtained in limited iterations by assigning selecting probabilities for the global stations and performing a Monte Carlo sampling. By applying the proposed algorithm for observation data of 201 International GNSS Service (IGS) stations for 3 consecutive days, an experiment of ultra-rapid orbit determination and real-time clock offset estimation is conducted. The distribution effects of stations on the products accuracy are analyzed. It shows that (1) the accuracies of GNSS ultra-rapid observed and predicted orbits and real-time clock offset achieved using the proposed algorithm are higher than those achieved with the traditional method having the drawbacks of lacking evaluation indicators and being time-consuming, corresponding to the improvements 17.15%, 19.30%, and 31.55%, respectively. Only using 30 stations selected by the proposed method, the accuracies achieved reach 2.01 cm (RMS), 4.93 cm (RMS), and 0.20 ns (STD), respectively. Using 60 stations, the accuracies are 1.47 cm, 3.50 cm, and 0.17 ns, respectively. (2) With the increasing number of stations, the accuracies of the Global Positioning System (GPS) orbit and clock offset improve continuously, but more than 60 stations, the improvement on the orbit determination becomes more gradual, while for more than 30 stations, there is no appreciable increase in the accuracy of the real-time clock offset.

Advances in Space Research

“…For a positioning configuration composed of n points with known coordinates, the unknown coordinates can be determined by the distances or pseudo-distances from the known stations to the unknown stations with the positioning model (Xue et al, 2014a)…”

Section: Gdops For Positioning and Their Minimummentioning

confidence: 99%

Understanding GDOP minimization in GNSS positioning: Infinite solutions, finite solutions and no solution

Xue

Yang²

2017

Self Cite

Although purely using the GNSS (Global Navigation Satellite System) users cannot obtain the theoretical GDOP minimum unless the GNSS positioning is aided by a certain number of pseudolites, discussing this problem is still meaningful in understanding the issues about the positioning geometry, such as the PDOP minimization. Many literatures have pointed that the GDOP (Geometric Dilution of Precision) minimum in 3-D positioning is the root square of 10/n where n is the total number of GNSS satellites or ground-based beacons with known coordinates. As the case with five known points concerned in this paper, the current knowledge indicates that the GDOP can reach the minimum the root square of 2, but our discussion shows that the GDOP minimum with five known points cannot get the theoretical minimum the root square of 2, although there are infinite positioning configurations with the lowest PDOP. Fortunately, we can find a positioning configuration with the GDOP 1.428 which is very close to the theoretical minimum 1.414. The PDOP can always reach the theoretical minimum the root square of 9/n, and there are infinite solutions for n>4. However for GDOP minimization, only when n>5, infinite solutions can be obtained. The configurations with the lowest GDOPs can be given by solving a set of nonlinear algebraic equations.