A Satellite Selection Algorithm for Achieving High Reliability of Ambiguity Resolution with GPS and Beidou Constellations

Reliability of carrier phase ambiguity resolution (AR) of an integer least-squares (ILS) problem depends on ambiguity success rate (ASR), which in practice can be well approximated by the success probability of integer bootstrapping solutions. With the current GPS constellation, sufficiently high ASR of geometry-based model can only be achievable at certain percentage of time. As a result, high reliability of AR cannot be assured by the single constellation. In the event of dual constellations system (DCS), for example, GPS and Beidou, which provide more satellites in view, users can expect significant performance benefits such as AR reliability and high precision positioning solutions. Simply using all the satellites in view for AR and positioning is a straightforward solution, but does not necessarily lead to high reliability as it is hoped. The paper presents an alternative approach that selects a subset of the visible satellites to achieve a higher reliability performance of the AR solutions in a multi-GNSS environment, instead of using all the satellites. Traditionally, satellite selection algorithms are mostly based on the position dilution of precision (PDOP) in order to meet accuracy requirements. In this contribution, some reliability criteria are introduced for GNSS satellite selection, and a novel satellite selection algorithm for reliable ambiguity resolution (SARA) is developed. The SARA algorithm allows receivers to select a subset of satellites for achieving high ASR such as above 0.99. Numerical results from a simulated dual constellation cases show that with the SARA procedure, the percentages of ASR values in excess of 0.99 and the percentages of ratio-test values passing the threshold 3 are both higher than those directly using all satellites in view, particularly in the case of dual-constellation, the percentages of ASRs (>0.99) and ratio-test values (>3) could be as high as 98.0 and 98.5 % respectively, compared to 18.1 and 25.0 % without satellite selection process. It is also worth noting that the implementation of SARA is simple and the computation time is low, which can be applied in most real-time data processing applications.

Ambiguity Acceptance Testing : A Comparison of the Ratio Test and Difference Test

Integer ambiguity resolution is an indispensable procedure for all high precision GNSS applications. The correctness of the estimated integer ambiguities is the key to achieving highly reliable positioning, but the solution cannot be validated with classical hypothesis testing methods. The integer aperture estimation theory unifies all existing ambiguity validation tests and provides a new prospective to review existing methods, which enables us to have a better understanding on the ambiguity validation problem. This contribution analyses two simple but efficient ambiguity validation test methods, ratio test and difference test, from three aspects: acceptance region, probability basis and numerical results. The major contribution of this paper can be summarized as: (1) The ratio test acceptance region is overlap of ellipsoids while the difference test acceptance region is overlap of half-spaces. (2) The probability basis of these two popular tests is firstly analyzed. The difference test is an approximation to optimal integer aperture, while the ratio test follows an exponential relationship in probability. (3) The limitations of the two tests are firstly identified. The two tests may under-evaluate the failure risk if the model is not strong enough or the float ambiguities fall in particular region. (4) Extensive numerical results are used to compare the performance of these two tests. The simulation results show the ratio test outperforms the difference test in some models while difference test performs better in other models. Particularly in the medium baseline kinematic model, the difference tests outperforms the ratio test, the superiority is independent on frequency number, observation noise, satellite geometry, while it depends on success rate and failure rate tolerance. Smaller failure rate leads to larger performance discrepancy.

Assessment of code bias variations of BDS triple-frequency signals and their impacts on ambiguity resolution for long baselines

Carrier phase ambiguity resolution over long baselines is challenging in BDS data processing. This is partially due to the variations of the hardware biases in BDS code signals and its dependence on elevation angles. We present an assessment of satellite-induced code bias variations in BDS triple-frequency signals and the ambiguity resolutions procedures involving both geometry-free and geometry-based models. First, since the elevation of a GEO satellite remains unchanged, we propose to model the single-differenced fractional cycle bias with widespread ground stations. Second, the effects of code bias variations induced by GEO, IGSO and MEO satellites on ambiguity resolution of extra-wide-lane, wide-lane and narrow-lane combinations are analyzed. Third, together with the IGSO and MEO code bias variations models, the effects of code bias variations on ambiguity resolution are examined using 30-day data collected over the baselines ranging from 500 to 2600 km in 2014. The results suggest that although the effect of code bias variations on the extra-wide-lane integer solution is almost ignorable due to its long wavelength, the wide-lane integer solutions are rather sensitive to the code bias variations. Wide-lane ambiguity resolution success rates are evidently improved when code bias variations are corrected. However, the improvement of narrow-lane ambiguity resolution is not obvious since it is based on geometry-based model and there is only an indirect impact on the narrow-lane ambiguity solutions.

An approximate method for the solution of an influence function foil rolling model

Fleck and Johnson (Int. J. Mech. Sci. 29 (1987) 507) and Fleck et al. (Proc. Inst. Mech. Eng. 206 (1992) 119) have developed foil rolling models which allow for large deformations in the roll profile, including the possibility that the rolls flatten completely. However, these models require computationally expensive iterative solution techniques. A new approach to the approximate solution of the Fleck et al. (1992) Influence Function Model has been developed using both analytic and approximation techniques. The numerical difficulties arising from solving an integral equation in the flattened region have been reduced by applying an Inverse Hilbert Transform to get an analytic expression for the pressure. The method described in this paper is applicable to cases where there is or there is not a flat region.

An Efficient and Verifiable Solution to the Millionaire Problem

A new solution to the millionaire problem is designed on the base of two new techniques: zero test and batch equation. Zero test is a technique used to test whether one or more ciphertext contains a zero without revealing other information. Batch equation is a technique used to test equality of multiple integers. Combination of these two techniques produces the only known solution to the millionaire problem that is correct, private, publicly verifiable and efficient at the same time.