Instantaneous Ambiguity Resolution in Global-Navigation- Satellite-System-Based Attitude Determination Applications: A Multivariate Constrained Approach Gabriele Giorgi ∗ Delft University of Technology, 2629 HS Delft, The Netherlands Peter J. G. Teunissen † Curtin University of Technology, Perth, Western Australia 6102, Australia and Sandra Verhagen ‡ and Peter J. Buist § Delft University of Technology, 2629 HS Delft, The Netherlands DOI: 10.2514/1.54069 Carrier phase integer ambiguity resolution is the key to high-precision Global Navigation Satellite System (GNSS) positioning, navigation, and attitude determination. It is the process of resolving the unknown cycle ambiguities of the carrier phase data as integers. After ambiguity resolution, precise baseline estimates become available, which can be used to derive the attitude of a multi-antenna platform. The purpose of this contribution is to present and test a rigorous GNSS-based attitude determination method, optimally exploiting the complete set of geometrical constraints. The key to this new method is an extension of the popular LAMBDA method: the multivariate constrained LAMBDA. The method estimates the integer ambiguities and the platform’s attitude in an integral manner, fully exploiting the known body geometry of the multi-antenna configuration. As a result, the ambiguity resolution performance is greatly improved. The method is extensively tested addressing the most challenging scenario: single-epoch single-frequency GNSS observations are processed without any filtering, external aid, or dynamic modeling. I. Introduction G LOBAL-NAVIGATION-SATELLITE-SYSTEM (GNSS)- based attitude determination employs multiple antennas firmly mounted on a body in order to estimate its orientation with respect to a given reference frame. Many studies have already been carried out to investigate the feasibility and performance of multiantenna GNSS receivers as attitude sensors [1–16]. Attitude estimation via GNSS observations is demonstrated to be a viable technique with a wide spectrum of challenging applications, ranging from terrestrial to maritime (guidance of land vehicles, precise docking of vessels, and precision farming), and from air to space (landing assistance, unmanned air vehicles, and space platforms guidance and control) [1–3,11,12,17–22]. High-precision GNSS-attitude determination requires the use of the carrier phase observations, which provide higher range mea- surement accuracy than the code observables but are ambiguous by an unknown integer number of cycles. Integer ambiguity resolution is the process of resolving these unknown ambiguities as integers. Integer ambiguity resolution is a nontrivial problem, especially if one aims at developing reliable and fast techniques ultimately capable of instantaneous correct fixing. Various ambiguity resolution methods have been developed, differing in the way the problem is approached and solved. The earliest strategies for attitude ambiguity resolution were the so-called motion-based methods [4,23–25]. These methods take advantage of the change in receiver-satellite geometry partially induced by the platform motion. However, these methods are not applicable on an epoch-by-epoch basis, since motion of the platform is a prerequisite. Another class of methods are the search-based methods. These methods do not depend on motion and can be applied instantaneously (for further discussion, see [9,26] and references therein). Several search-based methods make use of the integer least-squares (ILS) estimators [27], which extend the least- squares theory to linear models where a subset of the unknowns is integer valued. A widely used ILS implementation is the least- squares ambiguity decorrelation adjustment (LAMBDA) method [28], which is currently the standard method for solving un- constrained GNSS ambiguity resolution problems [29–33]. For unconstrained and linearly constrained GNSS models, the method is known to be optimal in the sense that it provides the highest possible success rate in a numerically efficient way [34–36]. Although the standard LAMBDA method has been used for attitude determination [6,10,14,37–39], it is not designed for rigorously solving such nonlinearly constrained problems. Incorporating the known body frame geometry of the GNSS antennas into the integer estimation process results in a nonlinear mixed estimation problem. The non- linear constraints originate from the a priori knowledge of the length and orientation of the baselines. It is the purpose of this work to provide and test the rigorous solution of the nonlinearly constrained ILS problem. First, the inclusion of a baseline-length constraint for two-antenna configurations was analyzed and tested: observations from two antennas separated by a known distance were processed including the baseline-length constraint, resulting in a large increase in success rate [40,41]. In case several antennas are installed aboard a platform, one may benefit from the additional a priori constraints on the mutual baseline orientations in the body frame. Instead of separately accounting for all the different constraints, the problem can be Presented at the IEEE/AIAA Aerospace Conference, Big Sky, MT, 6–13 March 2010; received 22 February 2011; revision received 21 April 2011; accepted for publication 21 April 2011. Copyright © 2011 by Gabriele Giorgi. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0731-5090/12 and $10.00 in correspondence with the CCC. ∗ Researcher, Delft Institute of Earth Observation and Space Systems, Kluyverweg 1; G.Giorgi@TUDelft.nl. † Professor, ARC Federation Fellow, Head of GNSS Research Laboratory, Department of Spatial Sciences, Kent Street, Bentley; P.Teunissen@ Curtin.edu.au. ‡ Professor, Delft Institute of Earth Observation and Space Systems, Kluyverweg 1; A.A.Verhagen@TUDelft.nl. § Researcher, Delft Institute of Earth Observation and Space Systems, Kluyverweg 1; P.J.Buist@TUDelft.nl. JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 35, No. 1, January–February 2012 51