2.
BASIC GPS CONCEPT
The position of a certain point in space can be found from distance measured from this point to some known
positions in space.
In a three-dimensional case four satellites and four distances are needed. The equal-distance trace to a fixed
point is a sphere in a three-dimensional case. Two spheres intersect to make a circle. This circle intersects
another sphere to produce two points. In order to determine which point is the user position, one more satellite is
needed.
In GPS, the position of the satellite is known from the ephemeris data transmitted by the satellite. One can
measure the distance from the receiver to the satellite. Therefore, the position of the receiver can be determined.
In the above discussion, the distance measured from the user to the satellite is assumed to be very accurate and
there is no bias error. However, the distance measured between the receiver and the satellite has a constant
unknown bias, because the user clock usually is different from the GPS clock. In order to resolve this bias error
one more satellite is required. Therefore, in order to find the user position five satellites are needed.
If one uses four satellites and the measured distance with bias error to measure a user position, two possible
solutions can be obtained. Theoretically, one cannot determine the user position. However, one of the solutions
is close to the earth’s surface and the other one is in space. Since the user position is usually close to the surface
of the earth, it can be uniquely determined. Therefore, the general statement is that four satellites can be used to
determine a user position, even though the distance measured has a bias error.
In Figure, there are three known points at locations r1 or (x1, y1, z1), r2 or (x2, y2, z2), and r3 or (x3, y3, z3), and an
unknown point at ru or (xu, yu, zu). If the distances between the three known points to
the unknown point can be measured as ρ1, ρ2, and ρ3, these distances can be written as:
where bu is the user clock bias error expressed in distance.
For ‘n’ satellites, the above equation can be written as:
where i=1,2,..n and xu, yu, zu, and bu are the unknowns. The pseudorange ρi and the positions of the satellites
xi , yi , zi are known. One common way to solve above equations is to linearize them.
Differentiating the above equation,
Linearize this equation, and the result is
The user position calculated from the above discussion is in a Cartesian coordinate system. It is usually
desirable to convert to a spherical system and label the position in latitude, longitude, and altitude as the every-
day maps use these notations. The latitude of the earth is from −90 to 90 degrees with the equator at 0 degree.
The longitude is from −180 to 180 degrees with the Greenwich meridian at 0 degree. The altitude is the height
above the earth’s surface. If the earth is a perfect sphere, the user position can be found easily from the figure as:
From this figure, the distance from the center of the earth to the user is
2.9 EARTH GEOMETRY(4–6)
The earth is not a perfect sphere but is an ellipsoid; thus, the latitude and altitude calculated must be modified.
However, the longitude l calculated also applies to the nonspherical earth. Therefore, this quantity does not need
modification.
For an ellipsoid, there are two latitudes. One is referred to as the geocentric latitude Lc, which is calculated from
the previous section. The other one is the geodetic latitude L and is the one often used in every-day maps.
Therefore, the geocentric latitude must be converted to the geodetic latitude.
For an ellipsoid, the following relations are obtained:
where ep is the eccentricity of earth.
Dilution of Precision
The dilution of precision (DOP) is often used to measure user position accuracy. There are several different
definitions of the DOP. All the different DOPs are a function of satellite geometry only. The positions of the
satellites determine the DOP value.
The geometrical dilution of precision (GDOP) is defined as:
where σ is the measured rms error of the pseudorange, which has a zero mean, σxσyσz are the measured rms
errors of the user position in the xyz directions, and σb is the measured rms user clock error expressed in
distance.
The position dilution of precision is defined as:
The horizontal dilution of precision is defined as:
The vertical dilution of precision is:
The time dilution of precision is:
The smallest DOP value means the best satellite geometry for calculating user position.
Program:
