HOW YOUR GPS WORKS

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Maintaining the fix means that we need to continuously recalculate the information based on the moving satellites. Once we have a number of fixes we can derive much more information than just location data. For example a gps can compute the travel direction (compass heading) by comparing current location to previous location. Similarly the gps can keep track of travel distance, compute speed, record travel time and other valuable data.

This view is simplified. In addition to the data already mentioned the unit uses Doppler data from the moving satellites, almanac data to figure out the approximate positions of all the satellites, and ephemeris data download directly from the satellite that can be used to compute its position in the sky. For a more detailed look at this information you should read the section on obtaining a fix. Similar to the geometry problem we had in the older system of taking bearings on fixed sites, the satellite geometry has a significant effect in the accuracy of our final position. A unitless number representing this geometry is called Dilution Of Position, DOP and is used by the gps in determining which of the satellites available represents the best ones to use. The smaller the number the better the geometry.

MATHEMATICAL VIEW

Another way to understand the operation of a gps system is to look at the math that goes into calculating a position. From Pythagoras we have:

Prs + T + Es = sqrt{(X - Xs)^2 + (Y - Ys)^2 + (Z - Zs)^2}

Where X, Y, Z are the positions we are trying to find and T is the time error at the receiver. The terms Xs, Ys, Zs are the satellite positions that can be calculated from ephemeris information sent from each satellite. The Es term is a lump sum of all the modeling errors considered by the gps. These include such things as troposphere and ionosphere errors, clock errors from the satellite and any other error the gps receivers thinks is significant enough to model. Prs is the approximate (pseudorange) distance from the receiver to the satellite. Since we can calculate the pseudorange and satellite positions independently and we can factor in modeling information from hardcoded data we are left with four unknowns, X, Y, Z, and T. Therefore we need 4 equations to solve for the 4 unknowns. Mathematically this is a standard least squares problem. One approach is to use guesses of our current position to calculate delta's from what we would expect and then iterate towards a converged solution. This is the reason that the unit requires an estimate of our current location to compute our position. Once we have the delta's down to an acceptable level we have a solution.

In actual practice a Garmin receiver calculates a set of equations with 7 unknowns. In addition to the 3 positions and time they have added the Doppler data dx, dy, and dz which represents the relative speed between the satellite and the receiver. These terms are needed because our solution is based on moving objects and dx and dy can be used as part of the receiver velocity calculation (dz is discarded). Four equations will compute a full 3D solution but new 12 channel Garmin units can use additional satellites to perform an overdetermined solution that will offer more accuracy. Older multiplex units pick the best 4 satellites based on their DOP. As satellites move out of view or get blocked from the receivers view by buildings, trees, and other objects the receiver will switch to other satellites to maintain a location fix. If the number of tracked satellites drops to three then a 3D solution is no longer possible and the receiver will use the last available altitude and compute a 2D fix for horizontal position.

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