Optimal Positioning of Pseudolites Augmented With

Optimal Positioning of Pseudolites Augmented With GPS Finding an Optimum Location for a Fixed Pseudolite In Support of GPS Navigation in Poor Visibility Conditions By Dr. M. Halis Saka Assistant Professor Department of Geodetic and Photogrammetric Engineering Gebze fnstitute of Technology Gebze-Kocaeli, Turkey

ver the last decade or so, global positioning system (GPS) position­ ıng has played an escalating role in nav­ ıgation, and it has become a primary tool for precise positioning. However, in some applıcations (e.g., urban

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canyons, valleys, deep open-cut mines and near-shore applıcationsl the num­ ber of visible satellites may not be suffi­ cient to reliably determine the user's position. In addition, GPS satellite avail­ ability tends to be a function of the obsel'ver latitudes, and some tech­ niques (e.g., phase smoothing) are sen­ sitive to cycle sı ips, hence they require continuous data acquisition.' Pseudo satellites, known as pseudolites (PLs), provide extra and continuous signals to GPS users so that they can significantly

improve the satellite geometry and positional accuracy. PLs typically trans­ mit pseudorange and carrier phase sig­ nals from the ground at Ll and L2 GPS frequency bands. Normally, standard GPS receivers with minor firmware modifications can track PL signals. The placement of a PL with respect to the user's location can be critical. Therefore, an optimization approach that quantifies the level of improvement with respect to the geometry of the GPS satellites together with PLs can provide

Typical configuration for' GPS satellites and a PL:' (Photo courtesy of Mustafa Danisman.)

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considerable benefits.' The philosophy behind the optimization approach is to obtain an optimal geometry, assuring the 2 minimum in terms of a cost function. Additionally, the effect of the signal mask­ ing due to the local horizon or man-made structures is a constraint in the placement of Pls. In this study, two optimization strategies 4 are deseribed and applied to a data set using Cıifferent geometrical dilution of pre­ 5 cision (GDOP) eriteria. The fundamental idea behind these strategies is to observe 6 the maximum GDOP anel total GDOP variations, then to detect the eritical e1irec­ tions producing optimal GPS-Pl configurations. The results of these considerations are compared to verify their effectiveness, especially for the case where no Pl was employed.

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A typical satellite-Pl configuration was consiel­ ered for the selected region in this study. Such a system is not reliable when the configuration of the satellites is weak due to the number of visible satel­ lites anel their geometrical configuration. In this case, one must support the system by adding a ground Pl to the system. At this point, an impartant question must be asked: Which locatian is the opti­ malone for the Pl? The linearized pseudorange equation can be expresseel as equation one (shown top of page); where p is a vector of pseudorange observa­

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tions, v is the vector of residuals, C is the design matrix that explains the satellite geometry relat­ ed to the user's position, x shows the user's posi­ tion and parameters of the receiver clock offset and pO is the approximate range between the user and the satellites. The design matrix, C, can be written using the azimuth and elevation of the satellites in the form of equation two. Although various cost functions are suggested in the literature, the most popular one is the GDOP, which is used as a precision indicator in GPS applications. J GDOP is estimated from the design matrix, C. The minimum of the GDOP corresponds to the best geometry of the observed satellites. Therefore, the optimization of new Pl locations should minimize the GDOP (i.e., GDOP=min). The Pl location with respect to the user is defined by its azimuth (Aeı) and eleva­

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(Second From Top) GDOP values for the satellites over e/evation angles of 23° without a PL, with the pessimalır positioned mobile PL (upper) and with the optimalır positioned mobile PL (lower).

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(Second From Bottom) Results of the optimization strategies for the data set.

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(Bottom) Solutions of the optimization strategies using the mini­ mum ofmaximum GDOPs (Ieft) and the minimum of the sam of GDOPs (right).

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tion angles (Epı). As a east function (f), COOP can be estimated as equation three. Note that the parameters of Epı and Apı are the unknowns, which are the design variables for the optimizatian. The remaining coefficients of C are known constants for given satellite con­ figurations. In order to minimize {(Aı'ı, Epl.), the necessary conditions are shown ın equations four and five. Solving the nonlinear equations four and five yields the solution of Apı and Epı that minimize the cost function in equation three. it should be pointed out that the solutions obtained from equations four and five belong to an instant of time. In practical applications, the solu­ tions for a period of time (e.g., 24 hours) are required. In this case, the neweast function converts in the form of equa­ tion six, where i refers to eachinstant of time in the simulation data and nT shows the number of sampling data for 24 hours. Obviously, obtaining the analytical solution using the east functions is computationally expensive due to the complexity of calculating the inverse of the term C{t)G(t) in equation six. In

order to overcome this problem, effi­ cient numerical approaches are required.

Numerical Optimization Methods The COOP differences between the optimal and pessimal positioning of the Pl can help find the most sensitive geometry of satellite constellation with the inclusion of the PL. Maximum improvement in COOP can be achieved by positioning the Pl for the instant when the geometry is most sen­ sitive. However, it should be pointed out that this does not guarantee the lowest COOP for all epochs. The opli­ mal and pessimal locations of the Pl in terms of the GOOP criterian were esti­ mated for the data set. It was found that the use of a Pl can make significant improvements in terms of COOP with a good selectian of its location, even for the case where a maximum number of vısible satellites was used. This is valid for the positioning of a fully mobile Pl. However, ın real-world applications, a fixed locatian is required for Pls. Sametimes, though rarely, the use of a Pl may not provide any improvement when it is placed at the worst locatıon,

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as in the data set for GPS elevations greater than 23°. In order to select the best locatian of a Pl for a specified time span, robust approaches or strategies are needed. Two strategies were adopted in this study for this purpose. The first deter­ mines the Pl directian, giving the mini­ mum sum of GOOPs estimated for all epochs. Thus, it aims to find the satel­ lite-Pl configuration providing the low­ esI total GOOP value for the consiclered time span. This strategy can be better explained in the fallawıng steps for c1arity. The first is to estimate a GOOP value for each directian covering the time span; second, to estimate the sums of the GOOP values based on the direc­ tions; third, to find the lowest total and the corresponding directian; and fourth, to use this directian for position­ ing of the PL. The second strategy is based on the selectian of the lowest GOOP among maximum COOP values estimated for each direction. 4 This is carried out in two stages: GOOP values are estimated for each direction for a 24-hour period and their maximum values are selected,

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and then the lowest GDOP value of the selected GDOPs is taken into consideration. Thus, the corresponding direction is determined.

Results In order to validate the optimization approaches suggested in this study, one-day GPS satellite visibility data for Istanbul, Turkey (acquired above elevation angles of 23°, which was set considering the topography of the study area and the vısıbility condilions), were used. . . Visibi i ity esti mations were carried out at three-mınute i ntervals for a 24-hour period (totaling 480 epochs), for which the GDOP values were estimated. Without a 1055 of generality, it is assumed that the elevation of the PL is zero (i.e., Epı=O). The azimuth angle of the PL is then used to estimate the GDOP values for all epochs. As a result of this study, both straıegies suggest close directions (27r and 278°) for the PL location. When the estimated GDOP values were analyzed, it was observed that the second strategy outperforms the firsl in terms of the maximum GDOP value. The difference, however, is negligible. As a result, both methods can be successfully applied for the particular data set used in this study. Results of the optimization strategies illustrate the variation in GDOP values. When the PL is placed at the center of the project area, it is estimated that the azimuth directions for the borders of the study area range from 240° to 60°. It was also found that the GDOP values for this range do not exceed 7.5. Various experiments conducted in the studies have shown

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Conclusions PLs provide extra and continuous signals to GPS users, which improves the positional accuracy in GPS applications. PLs transmit pseudorange and carrier phase signals at one or two GPS frequency bands (L1 or L2). The use of PLs is particularly important for marine applications (e.g., hydrographic surveying, navigation in narrow waters and harbor entrances, indoor navigations and aircraft landing). Although they provide extra information and improve the quality of the results, their positioning is a crucıal problem that needs further investigation. This study ıs an attempt to determine an optimal location for a PL augmented with GPS. Two strategies based on different GDOP criteria were applied to data sets for Istanbul, Turkey. The first str~tegy ıs based on determining the direction of the PL by g~yıng the minimum GDOP total for a 24-hour period. It searches a satellite-PL configuration providing the lowest total GDOP value for the considered time span. The second strategy discussed, on the other hand, searches the directian yielding minimum of the maximum GDOPs estimated for the 24-hour data.' Both strategies were applied to the data set, and the results were thoroughly analyzed. It was observed that the methods produce comparable results and provide significant improvements over the case where no PL was employed. Under the light of present results, it is recommended the results of the two techniques be analyzed together in order to benefit from their partiCLılar advantages. •

References 1. Saka, M.H., T. Kavzoglu, C. Ozsamli and R.M. Alkan, "Sub-Meter Accuracy for Stand Alone GPS Positioning in Hydrographic Surveying," The Journalaf Navigation, vol. 571 no. 1, pp. 135-1441 2004. 2. Morley, T.G., "Augmentation of GPS with Pseudolites in a Marine Environment/' M.Sc. Thesisı Department of Geomatics Engineeringı University of Calgary, Canadaı pp. 5,26, 43, 50, 138, 1997. 3. McKay, ).B., and M. Pachter, "Geometry Optimization for GPS Navigation," Proceedings of the 36th Conrerence on Decision and Control, pp. 4,695-4,699, 1997. 4. Parkinson, B.W., and K.T. Fitzgibbon, "Optimal Locations of Pseudolites for Differential GPS Navigation," Journalaf the Institute of Navigation, vol. 33, no. 4, pp. 259-283, 1986. Visit our Web site at www.sea-technology.com. and click on the title of this article in the Table of Conteiıts to be linked to the respective company's Web site.

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that both methods may produce differenl results and suggest different directions for the positioning of a PL depending on the satellite geometry. The second method in particular, which is based on the analysis of the sum of the GDOP values , does not bauarantee the minimum for GDOP maxima for all cases.

Or. M. Halis Saka is an assistant professor, currently working in the Department of Geodetic and Photogrammetric Engineering at Gebze Institute of Technology in Turkey. He gives lectures on geodesy and global positioning system technology. His main research cavers satellite positioning, navigation and geodesy.

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