2D position error in the Global Positioning System (GPS) depends on the Horizontal Dilution of Precision (HDOP) and User Equivalent Range Error UERE. The non-dimensional HDOP coefficient, determining the influence of satellite distribution on the positioning accuracy, can be calculated exactly for a given moment in time. However, the UERE value is a magnitude variable in time, especially due to errors in radio propagation (ionosphere and troposphere effects) and it cannot be precisely predicted. The variability of the UERE causes the actual measurements (despite an exact theoretical mathematical correlation between the HDOP value and the position error) to indicate that position errors differ for the same HDOP value.
The aim of this article is to determine the relation between the GPS position error and the HDOP value. It is possible only statistically, based on an analysis of an exceptionally large measurement sample. To this end, measurement results of a 10-day GPS measurement campaign (900,000 fixes) have been used. For HDOP values (in the range of 0.6–1.8), position errors were recorded and analysed to determine the statistical distribution of GPS position errors corresponding to various HDOP values.
The experimental study and statistical analyses showed that the most common HDOP values in the GPS system are magnitudes of: 0.7 (�� = 0•353) and 0.8 (�� = 0•432). Only 2.77% of fixes indicated an HDOP value larger than 1. Moreover, 95% of measurements featured a geometric coefficient of 0.973 – this is why it can be assumed that in optimal conditions (without local terrain obstacles), the GPS system is capable of providing values of HDOP ≤ 1, with a probability greater than 95% (2��). Obtaining a low HDOP value, which results in a low GPS position error value, calls for providing a high mean number of satellites (12 or more) and low variability in their number.

In order to guarantee the accuracy of turntable angle measurement, a real-time compensation method for turntable positioning precision based on harmonic analysis is proposed in this paper. Firstly, the principle and feasibility of the real-time compensation method are analysed, and a detailed description of harmonic compensation is provided herein. Secondly, we analyse the relationships between the surface number of the polygon with the compensation order of the harmonic function and its corresponding compensation accuracy. The effects of the iterations number and the data width on calculation accuracy in the coordinate rotation digital computer (CORDIC) algorithm are analysed and the quantization models of the approximation error and rounding error of the CORDIC algorithm are established. Then, the calculation of the harmonic error function and real-time compensation processes are implemented on a field programmable gate array (FPGA) chip. The resource occupation and time delay of the phase angle calculation and the harmonic component calculation are discussed separately. Finally, the validity of the harmonic compensation method is proven through comparing the compensation effect with that of linear interpolation and the polynomial compensation method. The influences of the compensation order, the iterations number and the data width on the compensation results are demonstrated by simulation. A test platform with a laboratory-made FPGA circuit is built to evaluate the effect of real-time compensation with the harmonic function and the positioning error compensation can be performed within 760 ns. The results confirmed the effectiveness of the harmonic compensation method, revealing an improvement of the positioning precision from 54.21″ to 1.63″, equivalent to 96.99% reduction in positioning error.

Angle calibrations are widely used in various fields of science and technology, while in the high-precision angle calibrations, a complete closure method which is complex and time-consuming is common. Therefore, in order to improve the measurement efficiency and maintain the accuracy of the complete closure method, an improved calibration method was proposed and verified by the calibration of a high-precision angle comparator with sub-arc-second level. Firstly, a basic principle and algorithm of angle calibration based on complete closure and symmetry connection theory was studied. Then, depending on the pre-established calibration system, the comparator was respectively calibrated by two calibration methods. Finally, by comparing En values of two calibration results, the effectiveness of the improved method was verified. The calibration results show that the angle comparator has a stable angle position error of 0:1700 and a measurement uncertainty of 0:0500 (k = 2). Through method comparisons, it was shown that the improved calibration method can greatly reduce calibration time and improve the calibration efficiency while ensuring the calibration accuracy, and with the decrease of measurement interval, the improvement of calibration efficiency was more obvious.