Integer leastsquares theory for the GNSS compass
 P. J. G. Teunissen
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Abstract
Global navigation satellite system (GNSS) carrier phase integer ambiguity resolution is the key to highprecision positioning and attitude determination. In this contribution, we develop new integer leastsquares (ILS) theory for the GNSS compass model, together with efficient integer search strategies. It extends current unconstrained ILS theory to the nonlinearly constrained case, an extension that is particularly suited for precise attitude determination. As opposed to current practice, our method does proper justice to the a priori given information. The nonlinear baseline constraint is fully integrated into the ambiguity objective function, thereby receiving a proper weighting in its minimization and providing guidance for the integer search. Different search strategies are developed to compute exact and approximate solutions of the nonlinear constrained ILS problem. Their applicability depends on the strength of the GNSS model and on the length of the baseline. Two of the presented search strategies, a global and a local one, are based on the use of an ellipsoidal search space. This has the advantage that standard methods can be applied. The global ellipsoidal search strategy is applicable to GNSS models of sufficient strength, while the local ellipsoidal search strategy is applicable to models for which the baseline lengths are not too small. We also develop search strategies for the most challenging case, namely when the curvature of the nonellipsoidal ambiguity search space needs to be taken into account. Two such strategies are presented, an approximate one and a rigorous, somewhat more complex, one. The approximate one is applicable when the fixed baseline variance matrix is close to diagonal. Both methods make use of a search and shrink strategy. The rigorous solution is efficiently obtained by means of a search and shrink strategy that uses nonquadratic, but easytoevaluate, bounding functions of the ambiguity objective function. The theory presented is generally valid and it is not restricted to any particular GNSS or combination of GNSSs. Its general applicability also applies to the measurement scenarios (e.g. singleepoch vs. multiepoch, or singlefrequency vs. multifrequency). In particular it is applicable to the most challenging case of unaided, single frequency, single epoch GNSS attitude determination. The success rate performance of the different methods is also illustrated.
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 Title
 Integer leastsquares theory for the GNSS compass
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Journal of Geodesy
Volume 84, Issue 7 , pp 433447
 Cover Date
 20100701
 DOI
 10.1007/s0019001003808
 Print ISSN
 09497714
 Online ISSN
 14321394
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 GNSS
 Attitude determination
 Integer ambiguity resolution
 Constrained integer least squares
 Industry Sectors
 Authors

 P. J. G. Teunissen ^{(1)} ^{(2)}
 Author Affiliations

 1. Department of Spatial Sciences, Curtin University of Technology, Perth, Australia
 2. Delft Institute for Earth Observation and Space Systems (DEOS), Delft University of Technology, Delft, The Netherlands