The key to precise global navigation satellite system (GNSS) positioning is carrier phase integer ambiguity resolution with a high success rate. On the other hand when the success rate is too low, the user will normally prefer the float solution. The alternative can be to use the best integer equivariant (BIE) estimator, since it is optimal in the minimum mean squared error (MMSE) sense. Low-cost receiver real-time kinematic precise positioning has become possible through the many signals that can be obtained by combining several GNSSs, such as BDS, Galileo, QZSS and GPS. In this contribution, we will use both simulations and such low-cost multi-GNSS data to compare the performance of the BIE and integer least squares (ILS) estimator, based on full ambiguity resolution. The GNSS data are evaluated in Dunedin, New Zealand, with a short- (670 m) and long-baseline (112.9 km) where the relative atmospheric delays can be neglected and need to be estimated, respectively. We compare the BIE and ILS results by using both single-frequency and dual-frequency (DF) low-cost and survey-grade receivers and antennas. We demonstrate, for the first time, the distributional properties of BIE positioning, where it will be shown that a ‘star-like’ pattern reveals itself once the model gets stronger and the ILS success rate increases. It will further be shown that the DF low-cost receivers give a very good positioning performance, but still not yet competitive to the survey-grade counterparts for the long-baseline. We will also demonstrate that the positioning performance of the BIE estimator will always equal or be better than that of the float solutions. It will finally be shown that BIE will always be better in the MMSE sense than the ILS solution when the success rate is at low to medium levels, whereas for high success rates we get a similar performance to ILS.

中文翻译：

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最佳整数等变估计：使用低成本、单频和双频、多 GNSS 接收器收集的真实数据进行性能分析，用于短到长基线 RTK 定位

精确全球导航卫星系统（GNSS）定位的关键是具有高成功率的载波相位整数模糊度分辨率。另一方面，当成功率太低时，用户通常会更喜欢浮动解决方案。另一种方法是使用最佳整数等变 (BIE) 估计量，因为它在最小均方误差 (MMSE) 意义上是最佳的。通过组合多种GNSS（如BDS、Galileo、QZSS和GPS）可以获得的多种信号，低成本接收机实时运动学精确定位已成为可能。在这篇文章中，我们将使用模拟和这种低成本的多 GNSS 数据来比较 BIE 和整数最小二乘 (ILS) 估计器的性能，基于全模糊度分辨率。GNSS 数据在新西兰但尼丁进行评估，具有短基线（670 m）和长基线（112.9 km），其中相对大气延迟可以忽略并需要分别估计。我们通过使用单频和双频 (DF) 低成本和调查级接收机和天线来比较 BIE 和 ILS 结果。我们首次展示了 BIE 定位的分布特性，其中将表明，一旦模型变得更强大并且 ILS 成功率增加，“星状”模式就会显现出来。将进一步表明，DF 低成本接收机提供了非常好的定位性能，但仍无法与长基线的测量级同行竞争。我们还将证明 BIE 估计器的定位性能将始终等于或优于浮动解决方案的定位性能。