Statistical approach to GPS positioning of mobile robot

Abstract

This article analyzes the present state of GPS system, possibilities and constraints of the mobile robot localization under environment influences. Real measured data from GPS receiver are statistically evaluated.

Full text

CEAI, Vol.12, No.2, pp. 44-51, 2010 Printed in Romania
Statistical approach to GPS positioning of mobile robot
Ladislav Jurišica, Anton Vitko, František Duchoň, Dušan Kaštan
Faculty of Electrical Engineering and Information Technology, Institute of Control and Industrial Informatics, Ilkovičova 3,
Bratislava, ladislav.jurisica@stuba.sk, anton.vitko@stuba.sk, frantisek.duchon@stuba.sk, dusan.kastan@gmail.com
Abstract: This article analyzes the present state of GPS system, possibilities and constraints of the
mobile robot localization under environment influences. Real measured data from GPS receiver are
statistically evaluated.
Keywords: GPS, Kalman filter, moving average
1. INTRODUCTION
Navigation of a mobile robot in an unknown and uncertain
environment requires knowledge about the robot’s current
position. A natural approach to manage the task is using
GPS (Global Positioning System) [1]. Because the GPS’s
signals are commonly corrupted by various disturbances like
signal bounces, inaccurate receivers etc. the disturbances
should be adequately mitigated at the highest possible rate.
The most effective means of doing this seems to be statistical
ones.
2. THE GPS SYSTEM
GPS serves for position of an object on Earth independently
of actual meteorological conditions. Position of a measured
point is given by intersection point of spherical surface of the
radius given by distances between satellites and measured
point. From geometric point of view, for determination of
the position it is needed to know positions of three satellites
minimally. Because the distance between satellite and
measured point must be defined at the same time, for exact
position identification of measured point it is needed to
know positions of four satellites. To achieve the high
accuracy of position determination, it is important to use the
maximum possible number of visible satellites that should be
properly distributed on the sphere. Applications based on the
GPS technology are almost unlimited. They cover virtually
all the areas of humane activities. Application area of GPS is
continually widening. For instance, pilots can utilize GPS
for searching airports, marines for searching ports, tourists
can orient themselves in an unknown country, fish men can
find out suitable time for fishing, land surveyors can identify
position of a point with high accuracy and mobile robots can
localize themselves in the environment. [2].
The GPS in comparison with conventional measuring
methods has several advantages:
• among particular measuring points may not be
immediate visibility
• it has high accuracy
• the measurement is quick and provides results in
the united world coordinate system WGS 84
• it provides three-dimensional position data
• it works regardless of weather, daily or night time
[4].
But it suffers from several disadvantages:
• it can't be used for measuring in the underground
• it provides worse results in dense vegetation (forest
etc.)
• it is required for satellite to be immediately visible
and the sky should be visible from measurement
point within 15° above the horizon and more in all
other directions
• problematic measurement in heavy inhabited areas
(town with narrow streets)
• problematic measurement in valleys
2.1 The system navstar gps
The system consists of three segments [1]:
1. Cosmic segment
Nominal constellation of GPS consists of 24 satellites
deployed above the ground at the height of 22 200 km. It is
required minimal number of 24 operating satellites.
Satellites are deployed on 6 orbits (4 satellites on orbit
minimally), that are equally spaced (60 degrees from each
other) and inclined 55 degrees with regard to equatorial
plane. This constellation ensures that from an optional point
on Earth a user is visible from 5 to 8 satellites. Orbit time of
one satellite from GPS system is 12 h of star time what
means that satellites have identical configuration during 11
h 58 min of solar day.
2. The control segment
Tasks of the control segment of GPS are: non-stop
monitoring and control of satellite system, definition of the
system time of GPS, prediction satellites’ paths and clock
operation on satellites and regular regeneration of navigation
report of each satellite [3]. The segment consists of five
monitoring stations (Hawaii, Kwajalein, Ascension Island,
Diego Garcia, Colorado Springs), four ground antennas

CONTROL ENGINEERING AND APPLIED INFORMATICS 45
(Ascension Island, Diego Garcia, Kwajalein, Cape
Canaveral) and so called Master Control Station on Falcon
Air Force Base (Colorado) [5].
Ground monitoring stations accept signals of all visible
satellites. Data are sent into Master Control Station where
orbital elements of satellites are defined, the correction of
atomic clocks is performed and navigation report is arranged
[3]. Navigation report is then sent on particular satellites by
ground transmitting antennas. Satellites then send back their
orbital elements and exact time to the Earth. Transmitting
antennas are deployed in such a way that the connection
with each satellite is possible at least three times per day.
[2].
3. User segment
This segment is composed of the GPS receivers. By way of
utilization they can be divided into navigational (ground,
naval, aerial and others navigation), geodetic (single-
frequency and two-frequency devices, RTK systems etc.) and
receivers for time synchronization (astronomic
measurements and telecommunications).
2.2 GPS systems with higher accuracy
1. Inertial Navigation System (INS)
By addition of the INS device to the GPS receiver it is
possible to achieve accuracy of position determination up to
one meter. In this configuration the GPS provides a short-
term accuracy as far as the INS provides long-lasting
stability. Outputs of both systems are compared and properly
filtered, whereby corrections on both systems are executed.
Corrections are done by using Kalman filter, which
combines two estimates and provides the most probable
estimation.
2. Differential GPS (DGPS)
By using DGPS the accuracy of position determination can
be improved up to one meter. DGPS includes second receiver
on a fixed position. The second receiver is set to calculate
correction for GPS data. Many free services based on DGPS
corrections exist, though paid services have better accuracy.
3. Wide Area Augmentation System (WAAS)
WAAS is extremely precise navigation system developed for
civilian navigation. System WAAS allows horizontal and
vertical navigation for exact operations (e.g. aircraft
landing). WAAS is available in USA and in some Pacific
areas. In Europe it is known as the system EGNOS [10]. It
improves GPS accuracy of position determination up to three
meters. From the above mentioned systems the WAAS is
most suitable system for mobile navigation. DGPS isn't
accessible everywhere and on the top of this it is a paid
service. INS sensors and their implementations are
financially demanding. WAAS technology is in the newest
models of GPS receivers integrated as a standard. It depends
only on the user whether he utilizes this possibility. Ideal
solution is usage of GPS with WAAS, whereby position data
are further improved with Kalman filter and combined with
data about local robot position, e.g. information from
incremental sensors or digital compass [6].
2.3 GPS standards
GPS receivers and their large proliferation represent
enormous source of data. But there can be problem with data
transmission from GPS receiver to the program of
measurement evaluation, which was manufactured by a
different producer. Receivers often use their own (often
undocumented) formats. For this reason the communication
standards with GPS receivers such as NMEA 0183, RTCM
SC-104 or RINEX were created.
2.4 GPS errors
1. Selective Availability (SA) error
SA error is the main factor, which influences accuracy of
GPS. It is an intentionally transmitted error with the aim to
decrease accuracy of civilian GPS receivers. An objective is
to make utilization of GPS system for enemy armies and
terroristic organizations impossible. [1]. This error can
lower accuracy up to cca. 100 m. The calculated falling-off
GPS accuracy for civilian sector ended 2nd May 2000. At
present the GPS receivers from civilian sector reach
accuracy around five meters.
2. Satellite geometry
Satellite geometry specifies satellites positions relatively to
view of GPS receiver. If GPS receiver sees four satellites
situated to the north and west it is possible that it will not be
able to identify position, because all measurements are
incoming from the same direction. Triangulation is
becoming imprecise and resulting plane on which the
measurements intersect is very large. Following this error,
the variance of a real position can be from 100 to 150 m. If
each of assuming visible satellites is on other world's side
(90° angle between satellites) the measurement accuracy is
advanced because measurements come from different world's
sides and the resultant plane is much smaller. Influence of
satellite geometry make the measurement worse for cars near
high buildings or in uneven terrains. High-end GPS receiver
then does not show only the satellites from which it is able to
accept signals, but also their positions on sky (azimuth and
elevation).
Quality of the satellite geometry can be evaluated by the
parameter “Dilution of Precision – DOP”, which is an
explicit indicator of position determination quality.
Calculation of DOP rests in determination of the relative
position of each visible satellite with respect to other visible
satellites. Smaller value of DOP corresponds to higher
accuracy. There are several types of DOP. They indicate
influences of various parameters on accuracy. To the family
of DOPs belong the following ones: the relative (RDOP,
relative position error), the positional (PDOP, horizontal and
vertical measurement), the horizontal (HDOP, horizontal
measurement), the vertical (VDOP, height measuring) and
the time DOP (TDOP, time shift).

46 CONTROL ENGINEERING AND APPLIED INFORMATICS
3. Atmospheric effects
Earth atmosphere is a two-part environment (troposphere
and ionosphere) [2][10] with essentially different effects on
the broadcasted high-frequency signals. The troposphere is a
neutral part without electrically charged elements. For waves
of frequency up to 15 GHz the troposphere is a non-
dispersive environment. Both types of GPS measurements
(phased and encoded) are equally affected by tropospheric
refraction, the size of which depends on meteorological
parameters of atmosphere (especially temperature, pressure
and humidity). The ionosphere is characterized by high
capacity of free electrons and ions. Accordingly, it is
electrically active. Properties of ionosphere are changing in
time, sun activity and changes of Earth magnetic field. As a
result, the reflexive properties of particular layers are
changing. For radio waves the ionosphere has character of
non-dispersive environment. In ionized layers of the
ionosphere the radio waves are not only bounced back, but
they are also absorbed. Besides the electrons and ions, these
layers contain also electrically neutral molecules that do not
oscillate. However, electrons and ions vibrating by action of
electromagnetic field bump on these molecules. This causes
the energy losses exhibited as radio wave dumping. Losses
are the greater the more no-ionized molecules are in the
layer, hence the layer is closer to Earth's surface [2]
4. Multipath effect
The multipath effect of signal propagation means that the
transmitted radio signal is reflected from some objects. GPS
satellite does not transmit signals just in the direction toward
a certain receiver but into the wide cone [2]. Provided that
the reflexive plane exists in the surrounding of GPS receiver
the signal may approach receiver's antenna also indirectly.
The path that signal is required to pass becomes longer and
the same goes for the time needed. Due to this the GPS
receiver measures longer distance to satellite. As a result, the
overall error at position determination is approximately 5 m
greater. [7]. There are many ways to reduce influence of
multipath effect. The one of the most used consists in a
construction modification of GPS antenna, technological
improvement of receiver and modification of received signal
processing. From among the construction modifications the
protective plate below antenna is mainly used, which
prevents receiving the signals reflected from Earth or water
level and other reflexive planes.
5. Relativistic effects
Relativistic effects manifest themselves especially at high
speed of surveyed objects and in presence of non-
homogeneous gravitation field. Another important factor is
accuracy with which it is required to sense given events. In
essence those are reasons due to which the relativistic effects
are needed to be included into operative equations of GPS
measurements. These are mainly the non neglect-able
changes of frequency due to quickly moving GPS satellites
and changes of signal propagation from quickly moving
satellites w.r.t. the rotating Earth as well as their great
distance from the Earth. There is also need to include the
relativistic disturbance effects on the satellites paths, which
are caused by non-homogeneous gravity field of the Earth, as
follows from general theory of relativity.
6. Other GPS errors
From among the additional measurement errors one can
mention the clock inaccuracy of the GPS receiver (receiver
can't contain atomic clock and the error removal is solved by
using signals from several satellites), inaccurate
determination of parameters of the satellite path (so-called
ephemeris error) and number of visible satellites (for higher
accuracy of measurement is needed to have higher number of
visible satellites).
3. KALMAN FILTER
The term “measurement” is historically related to
determination of angles, lengths and elevations by technical
means. As early as at the beginning of 19th century German
mathematician C. F. Gauss developed method [11] known as
the Least Squares Method (LSM). It allows for parameter
estimation by minimization of the sum of squares of the
differences between individual realizations of the
measurement.
At present, thanks to technological progress, we commonly
process an enormous quantity of data. Primarily with the
arrival and development global satellite systems for
navigation and position identification (GNS), the interval of
repeated measurements of three-dimensional position has
shorten from days to hours, from minutes to seconds or even
fragments of the second. Such quantity of data is no longer
possible effectively evaluate using LSM, because the LMS
necessitates processing large matrices. The problem is solved
with recursion filters, which use a part of their outputs as
input for next calculation. One of them is Kalman filter [9].
Kalman filter is able to process dynamic data with minimal
delay even if the fast dynamic changes occour. The result of
Kalman filtering is the system state estimated from
measurements corrupted with errors. Let a discrete process
of subsequent mobile robot positions is described by the state
equation:
111 −−− ++= kkkk wBuAxx (1)
with measurements:
(
)
kkk fvxHxz +== . (2)
Random variables
w
and
v
represent process noise and
measurement noise respectively. It is supposed that these
variables are independent of each other and have normal
Gauss distributions with covariance matrices Qand
R
. The
matrix
A
represents relations between the system state at
time steps k and 1
−
k without influence of control
function u and the signal noise. The matrix
B
is a control
matrix, which joins the known control vector
u
with the
system state
x
. Matrix
H
represents relations between the

CONTROL ENGINEERING AND APPLIED INFORMATICS 47
system state
x
and the measurement
z
.
Algorithm of Kalman filter is recursive, thus all previous
input data are included in the last estimation and also create
(except new measurement) the input into the new cycle. In
comparison with LSM method there is not need to calculate
inversions of large matrices.
Principle of discrete Kalman filter (Fig. 1) consists of two
basic steps: prediction and correction (sometimes called
update or actualization) [8].
Fig.1 Principle of discrete Kalman filter
In the prediction step the system state and its covariance
matrix at time step k on basis of system state estimation at
time 1
−
k is predicted in accordance with eq. (3)
QAPAP
uBxAx
+= +=
−
−−−
−
T
kk
kkk
1
11
.
.
ˆ
.
ˆ, (3)
where −
k
x
ˆstands for the state estimation (position of mobile
robot) at time step k. Matrices k
P and 1−k
P can be
expressed:
(
)
( )
11 ˆ
cov
ˆ
cov
−− −= −=
kkk
kkk xxP xxP (4)
Correction step represents improvement of the state
estimation based on actual measurements and it is defined
as:
(
)
( )
( )
−
−−
−
−−
−= −+= +=
kkk
kkkkk
T
k
T
kk
PHKIP
xHzKxx
RHPHHPK
..
ˆ
.
ˆˆ
... 1
, (5)
where k
K is so-called Kalman gain. It expresses a weight
of actual measurement concerning estimated variable. By a
simple analysis it becomes clear that with more precise
measurement (i.e. with decreasing covariance matrix of
measurement noise -
R
) its weight rises:
1
0
lim −
→=HK
Rk, (6)
On the contrary, if covariance matrix −
k
P approaches zero
the weight of actual measurement decreases [4]:
0K
P=
→
−k
k0
lim , (7)
The part
(
)
−
−kk xHz ˆ
. in (5), is often defined as a-priori
residuum
e
and represents difference between real value of
actual measurement and expected measurement, assigned
from last estimation of the state (thus position of mobile
robot).
The described procedure requires an initialization. It is a
process, which at the beginning of calculation defines values
of basic parameters. Vector 0
x is designed from initial
measurement or it has zero initial value. Covariance matrix
0
P at initialization is mostly defined as a diagonal matrix
with sufficiently large members, which will have almost no
weight in next iteration [9].
4. MOVING AVERAGE
Moving average [12] is in technical applications frequently
used because of its simplicity and possibility to combine
various moving averages together. Moving averages
smoothes data and simplify identification of trends. Many
types of moving averages exist but in technical analysis the
simple and exponential moving averages are mostly used.
The simple moving average is established by calculation of
average value in specific number of periods (
n
):
n
PP
SMA n
++
=...
1, (8)
where n
PP ,...,
1 generally stand for measured values (in
our case positions of mobile robot). The exponential moving
average eliminates delays, which appear within simple
moving average. With usage of exponential moving average
the delay is reduced by application of a bigger weight (
K
)
on recent values relatively to older ones:
(
)
(
)
[
]
1
.1. −
−+= nn EMAKKPEMA (9)
Selection of small calculation period causes that moving
averages are more sensitive and generate more signals.
Longer calculation period increase reliability but the
calculated value may exhibit non-permissible delay
(undesirable at estimation of position of mobile robot).
5. EXPERIMENTAL RESULTS
Statistical methods were tested by a cheap I-Tec Bluetooth
GPS receiver utilizing encoded way of measuring. The
receiver works with recording period from 0.1s and uses
standard NMEA-0183 at 57 600 b/s. Measurements were
carried out by method of the absolute position determination
of the area smaller than 30 km2. Therefore, measurements
were equally loaded by global errors. The only one
measuring equipment was used, consisting of sensor I-Tec

48 CONTROL ENGINEERING AND APPLIED INFORMATICS
GPS and software EvEgps developed for data acquisition
from GPS device with standard NMEA 0183.
5.1 Stationary point measurement
The stationary point measurement identifies position of
stationary point and was realized without external antenna.
A measurement file contains 90 measurements.
Measurement is corrupted with errors caused by multipath
effect and smaller number of visible satellites. This follows
from the selection of placement of stationary point in urban
area and also from partial visibility of the sky.
The measurement data were processed by statistical
methods. In case of moving average the set up time period
was equal to the time of measurement (Fig. 2 b). With
Kalman filter the outputs were almost identical with input
(Fig. 2 a). As far as the need of repeatability isconcerned,
after application of moving average with period equal to the
time of measurement, it results into different values of
position. In the case of using Kalman filter the results are
close to the measured values.
a.)
b.)
Fig.2. Stationary point measurement
a.) Application of Kalman filter (red colour) on file of
measured data (blue colour)
b.) Application of moving average (red colour) on file of
measured data (blue colour)
5.2 Geomtric shape measurement
The second measurement consisted in the measurement of a
circular path and it was also realized without application of
external antenna. Measuring data file contained 63 points.
Fig. 3. Comparison of real path (blue colour) and measured
path (red colour) at geometric shape measurement
By comparison of the measured and real path can be
concluded that measurements were influenced by some
errors (Fig. 3). The space where measurements were realized
was localized in a dense built-up urban area. Errors were
caused by insufficient number of visible satellites (during
entire measurements the average number of visible satellites
were equal to 3), the multipath effect of signal propagation
and satellites’ geometry.
From observations of the graph comparing the real and
measured path are follows that measurements were
influenced with constant (global) error. After application of
statistical methods the similar outputs were obtained (Fig. 4)
and uncertainties caused by variable (local) errors were
eliminated. Global and also local errors of measurement
seem to be constant, hence, to discriminate local from global
errors it was realized next measurement.
a.)

CONTROL ENGINEERING AND APPLIED INFORMATICS 49
b.)
Fig. 4. Geometric shape measurement
a.) Application of Kalman filter (red colour) on file of
measured data (blue colour)
b.) Application of moving average (red colour) on file of
measured data (blue colour)
5.3 Measurement of road following a cusped line
A road of the shape of a cusped line was measured without
application of external antenna. Measured data file consisted
of 200 points.
Fig. 5. Comparison of real path (yellow colour) and
measured path (red colour) at road with shape of cusped line
measurement
By comparison of the real and measured path (Fig. 5) it can
be pointed out that the measurement was influenced by
errors. Measurement was realized in a dense urban area. In
the area situated between high-rise buildings were visible at
most four satellites. The visibility became better after leaving
this area (white dot in Fig. 5) and thereby number of visible
satellites increased to seven. This fact becomes more evident
when position determination is improved. In case that
greater number of satellites was visible the real and
measured path was almost equal. On the top of this, no
multipath effect was noticed, because there were much less
reflexive planes.
In this case the measurement was influenced by errors just in
a certain period. These errors can be considered as non-
constant and can be removed by application of statistical
methods (Fig. 6).
a.)
b.)
Fig. 6. Road with shape of cusped line measurement
a.) Application of Kalman filter (red colour) on file of
measured data (blue colour)
b.) Application of moving average (red colour) on file of
measured data (blue colour)
5.4 Road way measurement
The measurement consisted in passing a long road way by
car without application of external antenna. It was measured
3010 points.
In this case, measured path was almost equal to the real one
(Fig. 7). During measurement there were seven visible
satellites on average. With a very long path and sufficient
number of visible satellites properly distributed on the sphere
the deviations from real path were minimal. Applications of

50 CONTROL ENGINEERING AND APPLIED INFORMATICS
statistical methods lead to minimal improvements only (Fig.
8).
Fig. 7. Measured path of road way (red colour)
a.)
b.)
Fig. 8. Road way measurement
a.) Application of Kalman filter (red colour) on file of
measured data (blue colour)
b.) Application of moving average (red colour) on file of
measured data (blue colour)
6. CONCLUSIONS
Statistical methods were applied on GPS data in order to
improve precision of a commonly available GPS receiver. In
the realm of mobile robotics they are mainly used for robot
localization in the environment. Both applied methods
(Kalman filter and moving average) are characterized by a
small delay.
These methods filtered errors caused by non-constant local
errors (insufficient number of satellites, multipath effect of
signal propagation and satellites geometry). If measurements
were influenced also with global errors (atmospheric and
relativistic effects) then measurements were corrupted by
constant errors. To eliminate global errors it is necessary to
use DGPS or WAAS system or employ additional sensors.
These systems can eliminate global errors; hence in
comparison with standard GPS receivers they are more
useful in robot localization in external environments.
7. ACKNOWLEDGEMENTS
This work was supported by MS SR under the contract
VEGA 1/0690/09.
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