Analysis of Three Ambiguity Resolution Methods for Real Time Static and Kinematic Positioning of a GPS Receiver

The objective of this work is to compare the performance of three methods of ambiguity resolution: LSAST, LAMBDA, and FASF. To do this evaluation, the methods were implemented in software and data were collected using two dual frequency geodetic quality Trimble R8 GPS receivers in static situation and two Ashtech Z12 GPS receivers in kinematic situation. The data were processed using LSAST, LAMBDA, and FASF methods for ambiguity resolution. During the processing, the float solution and ambiguity resolution process was reset once every 5 minutes. The ability of the methods to obtain a fixed solution was then analyzed in terms of time to fix and percentage of correct ambiguity fixes, where truth ambiguities were obtained by processing the entire data set using LAMBDA without any filter resets. As these methods use distinct search space reduction processes, differences in ambiguity resolution time and correct fix percentage were observed. INTRODUCTION The Global Positioning System (GPS) is a satellite-based navigation system which allows the user to determine position and time with high precision. The GPS signal is subject to several error sources in the measurements. The combined effects of these errors in the propagation signal cause a degradation in precision of positioning. However, using phase measurements, it is possible to reduce positioning error up to 100 times, if compared with positioning using code measurements (Misra and Enge, 2001). However, this type of measurement contains an inherent difficulty that is the determination of the ambiguity in the number of wavelengths of the corresponding signal. At the beginning of the positioning procedure, this ambiguity must be resolved (determined) in order to obtain an unambiguous phase pseudorange and then determine a position with the highest precision. Positioning by means of phase measurement is attained in three steps: float solution, ambiguity resolution and fixed solution. The float solution consists of estimating baseline values between the receivers and ambiguities as real values. The ambiguity resolution step consists of an integer estimate of the ambiguities, that is, to determine the value as an integer number. These integer values are used to correct the baseline value and provide the fixed solution. Three ambiguity resolution methods are tested: LAMBDA, FASF and LSAST. LAMBDA method (Least-squares AMBiguity Decorrelation Adjustment) is a procedure for integer ambiguity estimation in carrier phase measurements. This method executes the integer ambiguity estimation through a Z-transform, in which ambiguities are decorrelated before the integer values search process. Then, minimization problem is approached as a discrete search inside an ellipsoidal region defined by decorrelated float ambiguities, which is smaller than original ones. As a result, integer least-squares estimates for the ambiguities are obtained. This method was introduced in Teunissen (1993; 1994). De Jonge and Tiberius (1996) and de Jonge et al. (1996) show computational implementation aspects and ambiguity search space reducing. The FASF method (Fast Ambiguity Search Filter) has reduced both computational effort and required number of 2020 22nd International Meeting of the Satellite Division of The Institute of Navigation, Savannah, GA, September 22-25, 2009 observations to resolve the ambiguities. This enables this method to be tested in situations in which ambiguities should be resolved in real time (Chen, 1993; 1994). The search space of each ambiguity is determined recursively and sequentially updating the restrictions. To calculate the search space for an ambiguity, all previous ambiguities are assumed known. The number of potential solutions is used as an index to exit the ambiguity search. An attempt is made to fix the ambiguities if the total number of potential ambiguity sets from the search is less than a certain threshold, 20 candidates is used here. If the number is one, the ambiguity set is regarded as the correct one. Otherwise, the validation W-test is carried out on candidates. However, if the W-test fails, the ambiguities are estimated as float values. Since the full search of potential ambiguities is avoided with this method, only a relatively small amount of computation time is needed for ambiguity searching. LSAST method (Least-Squares Ambiguity Solution Technique) was proposed in Hatch (1990). This method involves a modified sequential least-squares technique, in which ambiguity parameters are divided into two groups: primary ambiguities (typically three double difference ambiguities), and the secondary ambiguities. Only the primary ambiguities are fully searched, in ±5 cycles around the corresponding float ambiguity, after rounded to the nearest integer. For each set of the primary ambiguities, there is a unique set of secondary ambiguities. Therefore, the search dimension is smaller and the computation time is significantly shorter than the full search approach. The choice of primary group measurements is based on GDOP value. Satellites with low GDOP will lead to a search with less potential solutions. However, GDOP cannot be very low, in order to avoid the position uncertainty including more than one solution for secondary group measurements. The procedure is to choose primary group of satellites which have a reasonable GDOP. The W-test, described in Wang et al. (1998), was used as a criterion for ambiguities to be considered resolved. If the value of W-test is greater than a certain threshold, ambiguities are considered resolved. If this threshold is not reached, the real valued ambiguities given by the Kalman filter are used. Algorithms were implemented for static positioning, in which both receivers (base and user) are kept fixed during the whole test, and kinematic, in which user receiver is moving. Results for the ambiguity resolution presented in both cases (static and kinematic) were obtained from the same estimation process using a Kalman filter, processing code and carrier phase measurements. Ambiguity resolution is achieved epoch by epoch, applying LAMBDA, FASF and LSAST methodologies over real valued estimated ambiguities obtained from the Kalman filter (float solution) to evaluate the ambiguity values and W-test behaviors, as data accumulates over time, processed by the filter. The position solution was obtained using an iterated least-squares method, processing carrier phase with ambiguities resolved measurements. This procedure was implemented for real time ambiguity resolution and positioning. However, in a practical situation, once ambiguities are validated and accepted, they are kept constant and the process of resolving ambiguity is not carried out until a signal interruption is found. This is a common procedure in carrier phase positioning. Test results were obtained using real dual frequency data, and were analyzed in terms of the percentage of resolved ambiguities (PR) and correctly resolved (PRC) and, if static, the error of position of user receiver with respect to the position of the landmark in which this receiver was located. Kinematic positioning algorithms were applied in a data set from an aircraft on a flight test. Results can be compared to a post-processed reference trajectory. The parameters used in result analysis are defined as: • Percentage of correctly resolved ambiguities (PRC): PRC is the percentage of ambiguities which are estimated to the correct integer value. PRC is calculated as the total number of ambiguities that are resolved to correct integer values divided by total number of ambiguities resolved to an integer value. • Percentage of resolved ambiguities (PR): PR is the total number of ambiguities resolved divided by the total number of ambiguities in the whole period. • Time to resolve ambiguity (TR): TR is the time required for the first ambiguity set validated as a solution.