New characteristics of weighted GDOP in multi-GNSS positioning

In positioning, navigation and timing applications of multi-GNSS (global navigation satellite system) constellations, the geometric dilution of precision (GDOP) offers an important index for selecting satellites and evaluating positioning accuracy. However, GDOP assumes that the measurement errors of all the tracked satellites are independent and have the same accuracy level, which is impossible in practice, especially when the tracked satellites are from various constellations. Through introducing a weighted matrix describing the measurement errors of different satellites into a common GDOP, we focus on new characteristics of weighted GDOP (WGDOP) in two aspects. First, we compare the sizes of WGDOP and the common GDOP based on the range of the weights of different satellites, i.e., the diagonal elements of the weighted matrix. In addition, when the weights of different satellites increase, the change of WGDOP with the weights is also derived. Moreover, a closed-form formula for calculating WGDOP is also presented. The theoretical derivations demonstrate that the closed-form can reduce the computation burden effectively. Furthermore, numerical tests verify these analyses.     

A closed-form formula to calculate geometric dilution of precision (GDOP) for multi-GNSS constellations

With the future global navigation satellite system (GNSS), the multi-GNSS constellations, which are composed of various single systems, will be the main navigation method in future. For the multi-GNSS constellations, the geometric dilution of precision (GDOP) is an important parameter used for satellite selection and the evaluation of positioning accuracy. However, the calculation of GDOP is a time-consuming and power-consuming task. Using Schur complement, we present a closed-form formula to calculate GDOP for multi-GNSS constellations. The formula can be applied to multi-GNSS constellations that include two, three or four different single systems. Furthermore, a closed-form formula for the case of exactly five satellites is also derived. Compared with the conventional numerical methods, the formula can reduce the amounts of multiplication and addition effectively. Numerical experiments validate the effectiveness and feasibility of the closed-form formula.     

基于伪卫星的改善GPS几何精度因子的研究

随着人们活动范围的日益扩大和周边环境的日益复杂,高精度GPS导航技术逐渐成为国内外研究的重点。GPS系统的定位精度在很大程度上取决于参与定位卫星的数目和几何布局,而几何精度因子(GDOP)正是衡量定位卫星几何布局优劣的量度。文章从几何精度因子着手,从理论上证明了伪卫星对GPS系统GDOP的改善,分析了伪卫星数量对GPS系统定位精度的影响。借助于仿真实验,结果表明,在GPS导航定位中,伪卫星能够显著增强卫星几何图形结构、提高测量精度、改善精度因子从而提高定位精度。     

Satellite selection with an end-to-end deep learning network

Benefiting from multi-constellation Global Navigation Satellite Systems (GNSS), more and more visible satellites can be used to improve user positioning performance. However, due to limited tracking receiver channels and power consumption, and other issues, it may be not possible, or desirable, to use all satellites in view for positioning. The optimal subset is generally selected from all possible satellite combinations to minimize either Geometric Dilution of Precision (GDOP) or weighted GDOP (WGDOP). However, this brute force approach is difficult to implement in real-time applications due to the time- and power-consuming calculation of the DOP values. As an alternative to a brute force satellite selection procedure, the authors propose an end-to-end deep learning network for satellite selection based on the PointNet and VoxelNet networks. The satellite selection is converted to a satellite segmentation problem, with specified input channel for each satellite and two class labels, one for selected satellites and the other for those not selected. The aim of the satellite segmentation is that a fixed number of satellites with the minimum GDOP/WGDOP value can be segmented from any feeding order of input satellites. To validate the proposed satellite segmentation network, training and test data from 220 IGS stations tracking GPS and GLONASS satellites were used. The segmentation performance using different architectures and representations of input channels, including receiver-to-satellite unit vector and elevation and azimuth, were compared. It was found that the input channel with elevation and azimuth can achieve better performance than using the receiver-to-satellite unit vector, and an architecture with stacked feature encoding (FE) layers has better satellite segmentation performance than one without stacked FE layers. In addition, the models with GDOP and WGDOP criteria for selecting 9 and 12 satellites were trained. It was demonstrated that the satellite segmentation network was about 90 times faster than using the brute force approach. Furthermore, all the trained models can effectively select the satellites making the most contribution to the desired GDOP/WGDOP value. Approximately 99% of the tests had GDOP and WGDOP value differences smaller than 0.03 and 0.2, respectively, between the predicted subset and the optimal subset.     

基于组合卫星导航系统的编队卫星分析

针对编队飞行中星间相对定位的任务需求,分析了卫星导航系统对编队卫星的动态观测几何问题,引入了相对定位精度衰减因子(RDOP)描述,并讨论了其性质。在对编队中单颗低轨卫星进行导航卫星GDOP分析的基础上,研究了不同编队宽度下编队集合的共视卫星和共视时段,仿真了一定场景下的编队卫星RDOP,并比较了与PDOP的大小关系。接收机的截止高度角对于导航卫星GDOP影响较大;编队宽度会影响到共视卫星的选择;而与采用单个GPS系统相比,采用GPS-Galileo组合卫星导航系统对编队卫星进行相对定位,RDOP数值明显减小,从而有利于高精度的星间位置确定。     

“一带一路”沿线及周边地区BDS定位性能动态分析

本文通过STK软件对BDS全球组网时至当前的星座结构进行了动态仿真,确定了“一带一路”沿线及周边地区最小可见卫星数、GDOP值以及定位误差的动态变化过程。结果表明,当前BDS已实现“一带一路”沿线大部分地区的覆盖,中国大陆、东南亚、南亚等地区的定位精度优于13 m,但北欧、东欧、西亚等小部分地区的定位精度仍有待提高;MEO卫星较GEO／IGSO卫星对“一带一路”沿线及周边地区定位精度的提升作用要大;同时发现,随着可见卫星数的增加,GDOP值逐渐减小,但当可见卫星数达到一定值时,随着可见卫星数的增加,GDOP值的减小幅度不明显,说明单纯增加可见卫星数有时并不完全能提高定位精度。      

GNSS/地磁组合的室内外无缝定位平滑过渡方法

针对室内外无缝定位在室内外过渡点精度低、不能平滑自动切换等问题,结合GNSS定位技术以及室内地磁指纹节点的组合方法来实现室内外无缝定位及导航。由于从室外至室内时接收机接收到的卫星星数减少、GDOP值逐渐增加、定位的误差增大,因此室内地磁定位精度逐渐优于GNSS定位精度,在两个定位精度临界点通过分析计算得出GDOP最优转化范围值,进行平稳切换。此次试验仿真结果表明,GDOP在3~3.5进行切换与单一GNSS或地磁方法定位的精度相比,分别提高85.7%和82.6%,从而达到了室内外的高精度无缝定位,填补了国内外在室内外无缝定位上没有合适的切换界定的空白。     

"北斗"系统低纬度区域定位精度增强方案探讨

针对我国“北斗”双星定位系统(BDS BeiDou System)在低纬度地区定位精度差的弱点，探讨了几种增强方案，并从精度因子(GDOP Geography Dilution of Precision)的角度比较各方案的优劣，研究表明，基于伪卫星增强双星的方案可改善卫星几何布局，从而有效提高该区域用户的定位精度。仿真说明改方案可提供百米级的定位结果，优于常规的单点定位精度。     

Neural network-based GPS GDOP approximation and classification

In this paper, the neural network (NN)-based navigation satellite subset selection is presented. The approach is based on approximation or classification of the satellite geometry dilution of precision (GDOP) factors utilizing the NN approach. Without matrix inversion required, the NN-based approach is capable of evaluating all subsets of satellites and hence reduces the computational burden. This would enable the use of a high-integrity navigation solution without the delay required for many matrix inversions. For overcoming the problem of slow learning in the BPNN, three other NNs that feature very fast learning speed, including the optimal interpolative (OI) Net, probabilistic neural network (PNN) and general regression neural network (GRNN), are employed. The network performance and computational expense on NN-based GDOP approximation and classification are explored. All the networks are able to provide sufficiently good accuracy, given enough time (for BPNN) or enough training data (for the other three networks).     

GPS/GLONASS组合单点定位中选星算法的研究

通过分析影响定位精度的因素,给出了一种基于高度角和GDOP的组合选星算法,研究了其算法在GPS/GLONASS组合单点定位中的应用,在提高定位效率的同时提高了定位精度。结果表明,在动态定位中能快速选出较少的卫星组合用于定位解算,能满足动态用户对实时性的需求。     

On the relation between GDOP and the volume described by the user-to-satellite unit vectors for GNSS positioning

GPS Solutions - The geometric dilution of precision (GDOP) is related to the satellite geometry. Thus far, research has focused on minimizing the GDOP by selecting a set of satellites that...     

BDS-3/GPS在遮挡环境下定位性能分析

针对在全球卫星导航系统(GNSS)信号易遮挡地区,单一系统可见卫星数较少,定位性能不理想甚至难以满足定位需求的问题,分析北斗三号(BDS-3)在不同区域遮挡环境下对定位性能的改善. 通过全球不同区域MGEX(Multi-GNSS Experiment)监测站的观测数据,采用GPS、BDS-3、BDS-3/GPS组合定位三种模式,在不同模拟遮挡环境下进行伪距单点定位,分析了各模式下可见卫星颗数、历元利用率、几何精度衰减因子(GDOP)值和定位精度. 结果表明:在北半球区域,相较于其他方向遮挡,GPS模式在低纬度地区南面遮挡的定位稳定性和精度最高,在中高纬度地区北面遮挡的定位稳定性和精度最高,BDS-3和BDS-3/GPS组合模式在低纬度地区各方向遮挡定位精度相当,在中纬度和中高纬度地区,北面遮挡的精度明显优于其他方向遮挡的定位精度. BDS-3/GPS组合定位模式,大大增加了可见卫星颗数,历元利用率提高,卫星空间几何结构改善,GDOP值降低,稳定性和定位精度明显优于单系统.      

天文定位中几何精度衰减因子最小值分析

天文定位是一种重要的导航定位方法,被广泛应用于大地天文测量、天文航海等领域。该方法中观测恒星的选择会影响最终的定位精度,目前缺少针对同时测定经纬度天文定位算法中最优选星问题的研究。随着观测仪器自动化水平的提高,观测数据的获取变得更加高效,这就要求研究最优的选星方案以达到最高的定位精度。本文借鉴卫星导航中几何精度衰减因子GDOP的概念,研究了天顶距法中恒星的数量以及分布对定位精度的影响,最后通过仿真试验和实测数据验证得到结论:在天顶距观测误差的统计特性一定时,GDOP能够用来描述恒星的分布对定位结果影响的优劣,且观测的恒星方位角均匀分布时定位误差最小。考虑到不同高度的恒星天顶距大气折射改正残差不同,在实际测量中应尽量采用等天顶距且方位角均匀分布的恒星。     

基于STK的典型机场BDS性能分析

为分析北斗卫星导航系统(BDS)在我国机场的导航性能,通过卫星仿真工具包(STK)建立了BDS星座,仿真并分析了11个典型机场的卫星可见星数和几何精度衰减因子(GDOP)的值．仿真结果表明,BDS在多数典型机场拥有较多的可见星数目与较低的GDOP值,通过对比GDOP值对应的定位精度分级表,得出BDS在国内大部分典型机场能够提供优级的导航服务,为利用BDS实施进近提供了良好的技术支撑.      

低轨卫星增强载波相位差分定位

针对低轨卫星快速空间几何变化和抗干扰能力强等特征,该文基于卫星工具包软件对全球导航定位系统和铱星系统星座进行了仿真,并假定铱星具有导航卫星的功能,分析铱星对GPS定位的增强作用。首先对GPS和铱星增强星座的可见卫星数量和几何精度因子值进行了分析,然后通过对不同的误差值建模,对GPS系统和铱星系统的观测值进行了仿真,分析了低轨卫星对双差定位浮点解和模糊度固定的增强作用,结果表明:低轨卫星的加入增加了可见卫星数量,几何精度因子也优于单GPS系统。单频双差模糊度浮点解的RMS值优于1周,双频双差模糊度浮点解的RMS值优于0.5周,与单GPS相比有了较明显的提高,同时,低轨卫星的加入更有利于单频短基线的模糊度固定。     

GPS/GALILEO组合系统可见卫星与GDOP的区域和时序分析

卫星星座方案的选择对导航定位精度具有很大影响。本文根据GALILEO系统的设计轨道参数模拟得到的GALILEO系统的卫星位置,分别计算了GPS系统、GALILEO系统和GPS/GALILEO组合系统在不同卫星截止高度角的情况下,全国范围内可见卫星和GDOP值的分布情况,并选择了北京、武汉和乌鲁木齐三个城市,连续观测24小时,分析了各城市的可见卫星和GDOP值随时间的变化规律。     

一种改进的组合导航系统选星算法

针对GPS,GLONASS,BDS组合导航系统定位中卫星的选择作了相关分析。首先用STK软件进行仿真,分析几何精度因子与卫星数的关系,得出组合导航系统最佳选星数;再根据卫星星座的空间几何分布,基于次优选星算法的成本函数模型,结合各导航系统卫星测量精度的差异性以及次优选星算法的峰值、不稳定特点,构建一种以卫星高度角和载噪比确定的加权成本函数模型,提出一种依据加权成本函数选星的分步次优加权选星算法。实验结果表明,该选星算法能近似到达最优选星算法的效果,计算负荷也相对较小,可满足导航定位解算精度和实时性要求。     

BDS-2/BDS-3伪距单点定位精度分析

北斗三号(BDS-3)基本系统于2018年底开始提供全球服务．通过处理37个全球GNSS服务组织(IGS)多模实验跟踪网(MGEX)观测站90天北斗数据,评估了北斗二号(BDS-2)和BDS-3在全球范围内的可见卫星数、几何精度衰减因子(GDOP)和单频伪距单点定位精度,分析了BDS-2／BDS-3组合对BDS-2、BDS-3单系统空间几何构型、伪距单点定位(SPP)精度的改善程度．结果表明,BDS-3的空间几何构型较BDS-2有明显的提升,定位精度在东方向、北方向和高程方向分别为1.490、2.610、5.238 m(RMS),相较于BDS-2分别提高了58％、1％、24％．BDS-2／BDS-3组合在东方向、北方向和高程方向分别为1.45、2.36、4.90 m(RMS),较BDS-2与BDS-3单系统分别提高了59％、11％、29％,以及3％、10％、6％．并且BDS-2／BDS-3组合明显削弱了BDS-2定位精度与地理经度相关的边缘效应.      

卫星导航中几何精度衰减因子最小值分析及应用

DOP是评估卫星导航定位性能的重要手段之一。在分析GDOP的数学意义和测量意义的基础上,提出两种GDOP最小值的求解方法,并对最小值进行理论分析。结果表明,传统场合中认为DOP为"精度衰减"因子是有局限性的;随着卫星与用户几何关系的改变,包括GDOP在内的所有DOP完全可以小于1,从而具有一定的精度增强作用。在此基础上,提出一种评价星座空间分布均匀程度的星座几何构型品质因数,并以GPS和Galileo星座为例进行试验,结果表明该因数可以较好地评价星座的空间分布均匀程度。基于GPS星座在近地空间内导航精度性能的试验结果很好地证明了DOP具有精度增强的属性。     

几何精度因子的算法研究及矩阵病态性探讨

针对GPS定位中的影响定位精度的几何精度因子（GDOP）的传统矩阵求逆算法和基于矩阵QR分解法进行了叙述,并对两种方法进行了比较分析,结果表明：后者具有数值稳定性好,计算效率高等优点。最后对其矩阵病态性进行了讨论。