GPS World - May 2008 - (Page 49)

Algorithms & Methods | INNOVATION pand the concept of differential GPS to the time domain. The use of time differences overcomes the need for a base receiver or any initialization process, while still providing decimeter and even sub-decimeter precision when postprocessing data from low-cost receivers. Solely using phase measurements, the method does not use any smoothing or filtering, which could cause hard-to-resolve effects in the resulting solution. However, these advantages do not come for free as the accuracy of a time-difference-based solution inevitably degrades with time (that is, the length of the data-collection time span). Nevertheless, the use of external correction data does allow for precise processing of parts of a platform' s trajectory for up to several minutes at a time. The time-difference GPS postprocessing technique is currently implemented at the Institute of Flight System Dynamics at the Technical University of Munich, where it is used for the precise evaluation of flight-path sections and individual maneuvers such as take-off, flare-out, landing, and dynamic soaring cycles, to gain insight into the flight-mechanics characteristics of the observed aircraft. Because our research is also focused on the flight characteristics of miniaturized aerial vehicles and even birds, the sensor used must be fully self-contained, small, and lightweight. The maximum size should be 150 × 100 × 20 millimeters and the weight must not exceed 100 grams, including the data logger and power supply, which should last for 70 hours. These specifications can be met using one of the recently developed single-frequency GPS receiver modules. As the nature of the intended applications requires high precision without having to worry about any kind of initialization pattern or operating a second nearby receiver, time differences are used. By postprocessing the kinematic phase measurements in time-difference mode, we can achieve the desired results. To validate our approach, we use RTK solutions generated by the Department of Earth Observation and Space Systems at Delft University of Technology (TU Delft). An RTK solution uses both L1 phase and code measurements and is based on resolving the integer ambiguities by means of the LAMBDA method. signal lock-on. Differencing across two epochs, and , yields . With denoting the time-difference operator, one obtains The Concept The basic problem when working with GPS phase data is the unknown cycle ambiguity in all range measurements. The benefit of any time-difference-based approach is the possibility of simply canceling these parameters while obviating the need for a second receiver and dedicated statistical methods for ambiguity estimation. The Observable. An ideal phase range measurement can be modeled as follows: with the (true) geometric range, , between receiver and satellite (in meters); the vacuum speed of light, (in meters per second); the receiver clock offset with respect to GPS System Time, (in seconds); the L1 carrier wavelength ( = 0.1903 meters); and the (non-integer) ambiguity (in cycles). Note that is not a function of time, , assuming that phase lock is maintained to the observed satellite; it has a constant value since www.gpsworld.com where the ambiguity has been canceled! Applications. The advantage of canceling ambiguities is the basis of various applications using time differences in different ways. For example, time-differenced double differences (so called triple differences across two receivers, two satellites, and two observation epochs) can be used for carrier-phase cycle-slip detection. Such observables can also be used for computing a precise baseline between a base or reference receiver and a roving receiver, provided that there are at least seven satellites in view (if only phase data is used). In tightly coupled GPS/INS systems, triple differences can support the dynamics estimation for attitude computation. In a similar context, carrier phases directly differenced between sequential epochs can be used instead of the noisier delta-range measurements to improve attitude information without the need for a base station. Time differences have been used in stand-alone GPS applications to process static data for gun-laying. This approach, enhanced by a loop-misclosure procedure, can also be applied to static measurements from civil receivers. Kinematic Time Differences. Using time differences for processing kinematic data is an unconventional approach, which emerged from the need for a high-quality but low-effort navigation , is a solution. The model of time-differenced phase ranges, function of receiver position and clock bias at the base epoch, , and the current time, . Assuming the position and time bias at the base epoch to be known, one can solve if measurements, , to at least four satellites continuously tracked between and are available. These equations are linearized and solved iteratively via a least-squares estimator in a similar manner as that for standard single-point positioning. The resulting solution is the baseline vector pointing from the position of the receiver at the base epoch, , to its location at the current time, . For transforming this basic approach into a tool for successful data processing, several real-world effects have to be accounted for. The measured phase ranges are inevitably afflicted with nonmodeled, remaining errors caused by the ª usual suspectsº in GPS navigation: receiver-independent atmospheric delays, satellite clock and ephemeris errors, and receiver-dependent multipath and measurement noise. The receiver-independent errors cancel completely in the very first moment of processing but start to grow with increasing time spans, . This slow drift directly degrades the solution and is the limiting factor when working with time differences. Increasing the temporal correlation of the nonmodeled errors is the only way to compensate for this effect. In addition, a ª geometricº error affecting the quality of the relative solution is caused by an offset of the base position at from the true location. To keep this error acceptably small, the base position has to be determined within an absolute accuracy of a few meters. For these reasons, a way to monitor the quality May 2008 | GPS World 49

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GPS World - May 2008 Contents Out in Front Letters to the Editor Expert Advice GPS III Contract Award Now a Reality? Near-Space Location Boost RTK Crops Up in Precision Ag Safety Afoot 50+ Leaders to Watch Making a Difference with GPS Product Showcase Advertisers Index & Company Directory Marketplace Classifieds Seen & Heard

GPS World - May 2008

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