Modern global navigation satellite system receivers can access signals from several satellite constellations (including GPS, GLONASS, Galileo, BeiDou). Once these constellations are all fully operational, a typical receiver can expect to have on the order of 40-50 satellites in view. Motivated by that observation, this paper presents an asymptotic analysis of positioning algorithms in the large-constellation regime. We determine the exact asymptotic behavior for both pseudo-range and carrier-phase positioning. One interesting insight from our analysis is that the standard carrier-phase positioning approach based on resolving the carrier-phase integer ambiguities fails for large satellite constellations. Instead, we adopt a Bayesian approach, in which the ambiguities are treated as noise terms and not explicitly estimated.
The traditional method used for carrier-phase positioning is to resolve integer ambiguities prior to estimating the baseline coordinates. Multiple methods have been developed along these lines, including the works of Teunissen @cite_9 and @cite_7 (already mentioned above), Hatch @cite_11 , Remondi @cite_20 , Al- @cite_6 , and Hassibi and Boyd @cite_17 . Recent interest in positioning with multiple satellite constellations has spurred the development of integer ambiguity resolution algorithms that scale more favorably with the number of satellites @cite_10 .
@cite_7 @cite_9 @cite_17 @cite_6 @cite_10 @cite_20 @cite_11
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The least-squares ambiguity Decorrelation (LAMBDA) method has been widely used in GNSS for fixing integer ambiguities. It can also solve any integer least squares (ILS) problem arising from other applications. For real time applications with high dimensions, the computational speed is crucial. A modified LAMBDA (MLAMBDA) method is presented. Several strategies are proposed to reduce the computational complexity of the LAMBDA method. Numerical simulations show that MLAMBDA is (much) faster than LAMBDA. The relations between the LAMBDA method and some relevant methods in the information theory literature are pointed out when we introduce its main procedures. The GPS double difference carrier phase measurements are ambiguous by an unknown integer number of cycles. High precision relative GPS positioning based on short observational timespan data, is possible, when reliable estimates of the integer double difference ambiguities can be determined in an efficient manner. In this contribution a new method is introduced that enables very fast integer least-squares estimation of the ambiguities. The method makes use of an ambiguity transformation that allows one to reformulate the original ambiguity estimation problem as a new problem that is much easier to solve. The transformation aims at decorrelating the least-squares ambiguities and is based on an integer approximation of the conditional least-squares transformation. This least-squares ambiguity decorrelation approach, flattens the typical discontinuity in the GPS-spectrum of ambiguity conditional variances and returns new ambiguities that show a dramatic improvement in correlation and precision. As a result, the search for the transformed integer least-squares ambiguities can be performed in a highly efficient manner. We consider parameter estimation in linear models when some of the parameters are known to be integers. Such problems arise, for example, in positioning using carrier phase measurements in the global positioning system (GPS), where the unknown integers enter the equations as the number of carrier signal cycles between the receiver and the satellites when the carrier signal is initially phase locked. Given a linear model, we address two problems: (1) the problem of estimating the parameters and (2) the problem of verifying the parameter estimates. We show that with additive Gaussian measurement noise the maximum likelihood estimates of the parameters are given by solving an integer least-squares problem. Theoretically, this problem is very difficult computationally (NP-hard); verifying the parameter estimates (computing the probability of estimating the integer parameters correctly) requires computing the integral of a Gaussian probability density function over the Voronoi cell of a lattice. This problem is also very difficult computationally. However, by using a polynomial-time algorithm due to Lenstra, Lenstra, and Lovasz (1982), the LLL algorithm, the integer least-squares problem associated with estimating the parameters can be solved efficiently in practice; sharp upper and lower bounds can be found on the probability of correct integer parameter estimation. We conclude the paper with simulation results that are based on a synthetic GPS setup. The paper presents a new method of searching for a fixed solution in precise positioning based on carrier phase data. The main idea is to search for a fixed solution in the coordinate domain instead of in an ambiguity domain. The dimension of the ambiguity domain depends on the number of satellites. The dimension of the coordinate domain is always three, and it does not depend on the number of satellites. It considerably reduces computational load. Pseudo-kinematic (PK) positioning with GPS is a survey technique employed by the National Geodetic Survey. This technique, which could also be called broken static surveying (or intermittent static surveying), must not be confused with kinematic surveying. The kinematic survey method requires continuous carrier phase tracking while moving between survey monuments and places strong restrictions on the number of satellites without cycle slips. The PK method is a static technique whereby a geodetic monument is occupied for, typically, 1 to 5 min for two or more occupations separated substantially in time (e.g., 1 h). Tracking during the transitions between monuments is categorically not required, but is often done for practical reasons. PK promises substantial productivity gains over classical static surveys and can be employed where the regular kinematic method is impractical. Furthermore, PK is safer than the regular static mode in that a constant antenna height is normally involved. These advantages are discussed. The paper defines the PK method; provides a brief history; considers alternative possible names; presents the mathematical and physical basis for the method, with emphasis on what to do about all the cycle slips between occupations separated by 1 h or more; and provides a number of examples. With high quality dual-frequency receivers, it is possible to instantaneously resolve the whole-cycle carrier phase ambiguities in a kinematic scenario.