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.     

A closed-form formula for GPS GDOP computation

Geometric dilution of precision (GDOP) is often used for selecting good satellites to meet the desired positioning precision. An efficient closed-form formula for GDOP has been developed when exactly four satellites are used. It has been proved that increasing the number of satellites for positioning will always reduce the GDOP. Since most GPS receivers today can receive signals from more than four satellites, it is desirable to compute GDOP efficiently for the general case. Previous studies have partially solved this problem with artificial neural network (ANN). Though ANN is a powerful function approximation technique, it needs costly training and the trained model may not be applicable to data deviating too much from the training data. Using Newton’s identities from the theory of symmetric polynomials, this paper presents a simple closed-form formula for computing GDOP with the inputs used in previous studies. These inputs include traces of the measurement matrix and its second and third powers, and the determinant of the matrix.     

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。     

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。     

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。     

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。     

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.      

北斗／GPS双模定位中快速选星算法研究

采用双系统组合定位模式,导航接收机能在同一时刻接收到超过20颗可见星,大大增加了可见星的数量,能提供十分精准的定位。但可见星数量大幅度增加带来定位精度提升的同时也带来了大量的计算量,加上实际作业中的一些问题,都大大增加了选星时间。本文从GDOP与星座几何布局的关系出发,结合实际应用场景,提出了基于仰角和方位角排序的选星算法。该方法以满足定位要求为前提,根据可见星仰角排序初筛卫星,对不同仰角卫星进行合理的顶星选择,以卫星均匀分布为原则,将在理想情况和实际情况下寻找有效定位平衡点,以规避无效选星并减少计算量,从而大大减少选星时间,有效实现了在北斗和GPS双模定位中的快速选星。      

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。     

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/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。     

“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。      

GPS/BDS/GLONASS/Galileo多系统融合在中国区域的仿真分析

通过STK软件对GPS、BDS、GLONASS、Galileo四个系统的星座结构进行仿真,并选择单系统与多系统组合定位的方式对中国区域内的可见卫星数、GDOP值和定位精度进行覆盖分析。结果表明,GPS/BDS/GLONASS/Galileo四系统组合定位在我国的GDOP值可达0.7~0.8,定位精度可达3~4m,优于其他方式的组合定位;同时四系统组合定位下的GDOP值降低,定位精度更好,GDOP值与定位精度的波动异常得到了抑制,导航定位的性能与稳定性也得到了相应的提升。     

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.      

北斗系统及GNSS多星座组合导航性能研究

针对北斗、GPS、GLONASS和GALILEO等单星座系统定位中存在的定位精度不足、可见星不多、定位可靠性不强等问题,研究了一种利用北斗、GPS、GLONASS和GALI—LEO多星座信息在统一坐标系中采用最小二乘法进行组合导航定位的方法。仿真结果表明：北斗与GPS双系统的定位精度优于单纯的北斗系统精度,而采用北斗／GPS／GLONASS／GALILEO多星座组合导航定位能够有效提高用户的定位精度和可靠性,研究成果对北斗系统的精度验证和多星座接收机的实现具有参考意义。     

"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。     

GPS,Galileo, QZSS and IRNSS differential ISBs: estimation and application

Knowledge of inter-system biases (ISBs) is essential to combine observations of multiple global and regional navigation satellite systems (GNSS/RNSS) in an optimal way. Earlier studies based on GPS, Galileo, BDS and QZSS have demonstrated that the performance of multi-GNSS real-time kinematic positioning is improved when the differential ISBs (DISBs) corresponding to signals of different constellations but transmitted at identical frequencies can be calibrated, such that only one common pivot satellite is sufficient for inter-system ambiguity resolution at that particular frequency. Recently, many new GNSS satellites have been launched. At the beginning of 2016, there were 12 Galileo IOV/FOC satellites and 12 GPS Block IIF satellites in orbit, while the Indian Regional Navigation Satellite System (IRNSS) had five satellites launched of which four are operational. More launches are scheduled for the coming years. As a continuation of the earlier studies, we analyze the magnitude and stability of the DISBs corresponding to these new satellites. For IRNSS this article presents for the first time DISBs with respect to the L5/E5a signals of GPS, Galileo and QZSS for a mixed-receiver baseline. It is furthermore demonstrated that single-frequency (L5/E5a) ambiguity resolution is tremendously improved when the multi-GNSS observations are all differenced with respect to a common pivot satellite, compared to classical differencing for which a pivot satellite is selected for each constellation.     

三种典型低轨增强星座与北斗系统联合应用的RAIM性能分析

接收机自主完好性监测(receiver autonomous integrity monitoring,RAIM)是终端用户高可靠导航定位的保障,低轨卫星的发展为完好性监测带来新的机遇,然而不同低轨星座增强下的终端RAIM性能可能会存在显著差异。基于高轨道倾角(80颗)、中轨道倾角(120颗)和混合轨道倾角(168颗)3种典型的低轨星座,系统评估了低轨卫星增强下的北斗卫星导航系统（BeiDou navigation satellite system,BDS）RAIM可用性及故障检测效果。仿真计算结果表明：对于高纬区域,高轨道倾角增强下的RAIM可用性效果最好,而在中、低纬区域,中轨道倾角星座增强下的RAIM可用性效果最优;在全球范围内,高轨道倾角、中轨道倾角和混合轨道倾角星座增强下非精密进近阶段的RAIM可用性较BDS分别提升30.5%、29.0%和41.0%。由此可知,由不同轨道倾角组成的混合星座可较好地弥补可视卫星在空间覆盖上的缺陷,其全球RAIM可用性增强效果最优,增强下的RAIM可检测到的最小伪距偏差较之前平均减小33.3 m。     

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).