A Computer Simulation of Dilution of Precision in the Global Positioning System Using Matlab
ABSTRACT The aim of this study is to design a computer simulation of the Dilution of Precision (DOP) which affects the GPS accuracy. DOP simulator is a usefull interface which provides a fast and practical analysis of both the DOP parameters and satellite geometry. The simulator was designed as a graphical user interface in MATLAB.
A COMPUTER SIMULATION OF DILUTION OF PRECISION IN THE
GLOBAL POSITIONING SYSTEM USING MATLAB
Fevzi Aytaç Kaya
Gazi University, Faculty of Engineering & Architecture
Department of Electrical&Electronics Engineering
06570 - Maltepe, Ankara, Turkey
Key words: Dilution of Precision, satellite geometry, simulation, user interface
The aim of this study is to design a computer simulation of
the Dilution of Precision (DOP) which affects the GPS
accuracy. DOP simulator is a usefull interface which
provides a fast and practical analysis of both the DOP
parameters and satellite geometry. The simulator was
designed as a graphical user interface in MATLAB.
GPS devices are designed to send any positional data
they calculate, regardless of its accuracy. Indeed, the
latitude, longitude and altitude reported by GPS devices
are frequently inaccurate by three hundred meters. This
inaccuracy can lead to poor business intelligence, or,
worse still, get an end-user into a precarious situation.
Developers who wish to write location-based services
must get smart about GPS precision. Fortunately, GPS
devices provide information which can be used to make
300m inaccuracies a thing of the past. This information
is known as "Dilution of Precision," and is the key to
writing commercial GPS applications . The Dilution
of Precision (DOP) is a value of probability for the
geometrical effect on
GPS devices calculate the position using a technique
called “3-D multilateration,” which is the process of
figuring out where a number of spheres intersect. In the
case of GPS, each sphere has a satellite at its center. The
radius of the sphere is the calculated distance from the
satellite to the GPS device. Ideally, these spheres would
intersect at exactly one point, causing there to be only
one possible solution to the current location, but in
reality, the intersection forms more of an oddly shaped
area. The device could be located within any point in the
area, forcing devices to choose from many possibilities.
Figure 1 shows such an area created from three satellites.
The current location could be any point within the gray-
colored area. Precision is said to be “diluted” when the
area grows larger, which leads to this article’s focus:
dilution of precision. The monitoring and control of
dilution of precision is the key to writing high-precision
Figure 1: GPS devices must choose one of several
possible solutions to the current location .
Several external sources introduce errors into a GPS
position estimated by a GPS receiver. One important
factor in determining positional accuracy is the
constellation, or geometry, of the group of satellites from
which signals are being received. DOP only depends on
the position of the satellites: how many satellites you can
see, how high they are in the sky, and the bearing towards
them. This is often refered to as the geometry . The
computed position can vary depending on which satellites
are used for the measurement. Different satellite
geometries can magnify or lessen the position error. A
greater angle between the satellites lowers the DOP, and
provides a better measurement. A higher DOP indicates
poor satellite geometry, and an inferior measurement
cofiguration, or in other words: the lower the value the
greater the confidence in the solution.
II. THEORETICAL BACKGROUND
Satellite navigation depends on accurate range
measurements in order to determine the position of the
user. Because the receiver clock is generally not
synchronized with GPS system time, the range
measurements are erroneous and are therefore called
pseudoranges . A pseudorange equation is thus:
where: PRi is the pseudorange to the ith satellite (m)
(X, Y, Z) is the unknown three dimensional user
(Xi , Yi , Zi ) is the known three dimensional satellite
tB is the unknown receiver clock offset (s),
c is the speed of light (m/s).
Because there are four unknowns, at least four
measurements are required to solve for X, Y, Z and tB.
Because the pseudorange equations are nonlinear, the
system must be linearized in order to derive a linear
relationship between pseudorange errors and position
errors. If we linearize the GPS pseudo-ranges by using a
first order Taylor Series expantion, we obtain a linear
equation of the form:
where the hmn elements represent direction cosines to
each of the satellites.
In general, this equation can be written as:
∂Y = H ∂β ( 3 )
Thus, a receiver modifies an initial estimate of β using
∂β and iterates until convergence is achieved.
The column of ones in H shows that the receiver clock
offset (ctB) biases each pseudorange measurement by
exactly the same amount . In practice, there may be
interchannel biases that affect each measurement
differently. However, great care is taken by receiver
manufacturers to calibrate these effects.
The GPS satellite geometry relates position errors to
range measurement errors. Considering equation 1 with
m pseudorange measurements. Then ∂Y is an m x 1
vector, H is an m x 4 matrix, and as usual ∂β is a 4 x 1
vector. If we think of ∂β as a zero-mean vector containing
the errors in the estimated user state, then we are
interested in the statistics of ∂β because that will
characterize the expected position errors. Using the
generalized inverse of H, we find the covariance of ∂β
Now we have the covariance of ∂Y, the pseudorange
errors. These are assumed to be uncorrelated, Gaussian
random variables. As such, they are statistically
independent which results in a diagonal covariance
matrix. Furthermore, the range measurement errors are
assumed to have the same variance ( σr ) for each satellite.
So, we have:
cov (∂Y) = σr
2 I ( 5 )
which results in :
It can easily be shown that HTH is symmetric, so the
transpose is unnecessary:
cov (∂β) = σr
Now the relationship between range measurement errors
and position errors becomes clear. Let G=(HTH )-1 so that
cov (∂β) = σr
( 8 )
The elements of G give a measure of the satellite
geometry called the dilution of precision (DOP) .
Various DOP values can be calculated from the diagonal
elements of G. For example:
2 (HTH )-1 ( 7 )
2 G. Using B = ctB and expanding Eq. 7,
where GDOP is the Geometric Dilution of Precision
Other DOPs are as follows:
( 11 )
PDOP is the Position Dilution of Precision
HDOP is the Horizontal Dilution of Precision
VDOP is the Vertical Dilution of Precision
TDOP is the Time Dilution of Precision
While each of these GDOP terms can be individually
computed, they are formed from covariances and so are
not independent of each other. A high TDOP (time
dilution of precision), for example, will cause receiver
clock errors which will eventually result in increased
position errors. Good GDOP, a small value representing
a large unit-vector-volume, results when angles from
receiver to satellites are different. Wheras poor GDOP, a
large value representing a small unit vector-volume,
results when angles from receiver to the set of satellites
Figure 2 : Good satellite geometry .
Figure 3 : Poor satellite geometry .
used are similar. Clustered satellites give poor GDOP
values (Figures 2,3). The higher the DOP, the weaker the
ECEF COORDINATE REFERENCE FRAME
The Cartesian coordinate frame of reference used in GPS
is called Earth-Centered, Earth-Fixed (ECEF). ECEF uses
three-dimensional XYZ coordinates (in meters) to
describe the location of a GPS user or satellite. The term
"Earth-Centered" comes from the fact that the origin of
the axis (0,0,0) is located at the mass center of gravity.
The term "Earth-Fixed" implies that the axes are fixed
with respect to the earth (that is, they rotate with the
earth). The Z-axis pierces the North Pole, and the XY-
axis defines the equatorial plane .
III. COMPUTER SIMULATION
The fundamental aim in designing the Dilution of
Precision computer simulator, namely the DOP Simulator,
is to obtain a useful interface which provides a fast and
practical analysis of both the DOP parameters and the
satellite geometry. The simulator was designed as a
graphical user interface in MATLAB .
The DOP Simulator, when started, asks the user to enter
the ECEF coordinates of the four satellites and the GPS
receiver with respect to the ECEF coordinate reference
frame in meters. The user enters the coordinates to be
simulated and presses the Analyze button. The program
gets the coordinates as inputs and calculates the DOP
parameters according to the equations which were
mentioned above (Equations 1 to 11). The results are
displayed in the corresponding windows. All the
parameters, namely GDOP, PDOP, TDOP, HDOP, and
VDOP are calculated and displayed. Later the results are
rated according to levels between 1 and 50 (Table 1).
Table 1: DOP Ratings
Finally the description of the rating associated with each
parameter is stated in an EXPLANATIONS window. To
see the explanation associated with each parameter, the
user should click the EXPLAIN button which is located
under the rating of each parameter (Figure 4). Finally, the
program shows the satellite geometry (on the right top of
the interface) associated with the coordinates that the user
has entered. This provides the opportunity to see both the
DOP values and the satellite geometry at the same time so
that a better imagination is created in the users
Figure 4: The Dilution of Precision computer simulation user interface.
mind. The satellites are simulated with four spheres that
have equal volumes. The GPS device is simulated with a
sphere having a bigger volume than the four satellites.
From another point of view; the user can clear the
coordinates using the Clear Coordinates button and
enter different new coordinates to see how the DOP
values change with the satellite geometry and also how
the precision of the positioning is diluted. The user can
also decide on which parameter mostly causes the
dilution of precision by looking both at the RESULTs
and the RATINGs. That is; the user can consider
whether the dilution of precision occurred vertically,
horizontally, positional or a timing error.
The DOP Simulator has some other properties. One of
these is; the program detects invalid inputs and warns the
user to correct the mistake. That is, when a user enters
for example a letter between the numbers in one of the
coordinates by mistake, the program detects the invalid
input and warns the user with a message saying which of
the coordinates he has entered wrong and asks to correct
it. Another property is the additional explanation
shortcuts to each of the ratings. If the user wants to
compare the ratings, he can simply click the ? button
near the rating names which are located at the right
bottom of the interface. These shortcuts directly give the
explanations of the DOP ratings.
IV. RESULTS AND CONCLUSION
The graphical user interface, which is named as the DOP
Simulator was operated and various different coordinates
were tried. The results proved that the DOP parameters
change according to the satellite geometry. The DOP
values were decreased, as the angles between the satellites
were increased, as a result the accuracy was increased.
However the simulator is considered as a sufficient
interface, the user can plot the coordinates by writing the
necessary code in the MATLAB command window and
do detailed analysis using the camera view option. This
option is not used in the simulation since it will be a time
consuming process which is a contradiction to the design
criteria of the simulation.
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