NAVIGATION:
Journal
of The Institute of Navigation
Vol. 43, No. 2, Summer
1996
Printed m U.S.A.
Development and Testing of a Mobile
Pseudolite Concept for Precise
Positioning
CAPT. J. RAQUET,*
C. PELLETIER
G. LACHAPELLE,
W. QIU, and
The University of Calgary, Calgary, Alberta, Canada
CAPT. T. NASH and CAPT. F. B. SNODGRASS
746th Test Squadron,
Holloman AFB, New Mexico
P. FENTON
NovAtel Communications
Ltd., Calgary, Alberta, Canada
T. HOLDEN
Stanford Telecom,
Sunnyvale,
California
Received December 1995
Revised March 1996
ABSTRACT
The concept of using a pseudolite in the reverse mode where the pseudolite is
positioned with respect to receivers deployed at known points is presented. Two
types of double-difference positioning approaches for eliminating clock and other
code and carrier-phase biases are described and analyzed-pseudolite
positioning
with satellite reference (PPSR) and pseudolite positioning with pseudolite reference
(PPPR). Results are given for an initial test conducted in a land vehicle at Holloman
APB, New Mexico. The test range consisted of a 600 m course surrounded by six
receivers deployed in a noncoplanar configuration at distances ranging from 100
to 1,500 m from the vehicle. The pseudolite position was calculated using both
integer and floating-point carrier-phase ambiguities, and the resulting trajectory
was analyzed to assess the differential positioning performance of this inverted
GPS system.
INTRODUCTION
A novel flight reference system concept has been developed by the 746th
Test Squadron in anticipation
of future testing requirements.
Originally conceived by [ll, the purpose of this system is to provide a true reference trajectory
for aircraft against which other navigation systems can be tested and analyzed.
At the top level, there are three driving requirements for this system: (1) very
*Capt Raquet is a Ph.D. candidate at The University of Calgary under sponsorship from The Air
Force Institute of Technology.
149
Summer 1996
Navigation
150
high position and velocity accuracy (on the order of 0.1 m and 0.005 m/s,
respectively), (2) ability to install easily on all aircraft in the Air Force inventory, and (3) ability to operate in the presence of jamming at GPS frequencies.
The reference system concept that is expected to meet these requirements is
shown in Figure 1. This system represents an inverse of the procedure normally
used to perform precise carrier-phase positioning. Normally, relative positioning is performed between two GPS receivers tracking many common GPS
satellites, and optionally from one or more ground-based fixed pseudolites. In
the new reference system concept, relative positioning is performed between
two pseudolites transmitting GPS-like signals, which are collected by many
common GPS receivers at precise 3-D coordinates on the ground. If one of the
pseudolites is at a fixed, known point on the ground, then the position of the
mobile pseudolite (which is on the aircraft) can be determined. The precise
carrier-phase measurements from the satellites can be used to determine the
fixed receiver positions at the centimeter accuracy level in batch mode. Once
the receiver positions are known, signals received from GPS satellites are used
only to time-synchronize the pseudolite measurements. The satellite geometry
then becomes of secondary importance, but the relative geometry of the mobile
pseudolite with respect to the fixed receivers becomes of primary importance.
This configuration boasts two distinct advantages over the traditional concept
of placing the GPS receiver in the aircraft. First, immunity to intentional GPS
jamming is gained by placing the receivers on the ground, away from the focus
of the jamming beam on the aircraft. Any residual jamming signals on the
receivers can be shielded locally at each receiver. This concept is currently
being tested by the 746th Test Squadron. Second, flexibility is gained and cost
is reduced by placing most of the hardware and software infrastructure on the
Mobile Pseudolite
(on aircraft)
Reference
A, B, C, D, E - GPS Receivers
Fig. l-Inverted
Central Processing
Station
Pseudolite Reference System Concept
Vol. 43, No. 2
Raquet, et al.: Mobile Pseudolite
Concept
151
ground, where power, size, and computational load constraints can be more
easily accommodated. The only equipment required on board the aircraft is a
GPS pseudolite and antenna, which are relatively small and could be easily
placed on most aircraft using several placement options (such as an empty
missile pod, empty fuel tank, or wing/fuselage mount).
In the past, pseudolites have been shown to improve GPS signal geometry
for mobile receivers 121.Pseudolites have also been used to help with ambiguity
resolution and autonomous integrity monitoring during aircraft precision
approach and landing [3-51. This is the first time the pseudolite itself
has been positioned using carrier-phase data in a double-difference mode. This
paper presents test results from a proof-of-concept test that took place at the
746th Test Squadron in April 1995. The results demonstrate the feasibility
of generating an accurate trajectory of a transmitting pseudolite on a mobile
platform.
TEST DESCRIPTION
The 746th Test Squadron mobile test van was selected as the mobile platform
for this proof-of-concept test. The van is a modified touring bus with on-board
power supplies, antenna mounts, and equipment racks that make it ideally
suited for this type of mission. Finding an adequate test site for this test was
a challenge. To obtain a reasonable 3-D solution, it was important for the
ground receivers to be placed in a configuration that was as noncoplanar as
possible to avoid a critical configuration singularity [61.This meant the receivers needed to be placed at varying altitudes. Another constraint was that each
receiver had to maintain signal lock with the pseudolite for the entire test
route, so line-of-sight was to be maintained between the ground receivers and
the moving vehicle. Additionally, the range between each receiver and the
pseudolites was constrained by the dynamic range of the pseudolites
(-100-1000 m). After 2 days of initial system checkout and testing, the test
route shown in Figure 2 was selected. This route covered a segment of straight,
level road. A large hill was located to the northeast of the road, which permitted
placement of the receivers at varying altitudes (see Figure 2) while maintaining
line-of-sight to the vehicle. The total length of the test route was 600 m.
A total of six stationary receivers were used for the test. The positions of
the receivers relative to the test route are shown in Figure 2. The nominal
height difference between the test route (which was nearly level) and each
receiver is given in parentheses. A vehicle was placed at each receiver location
to provide power. Receivers 1 through 5 were 12-channel NovAtel model 3151R
receivers in PowerpackTMenclosures. Data was transferred in real time through
the RS-232 port for storage on separate laptop computers. Receiver 6 was a
NovAtel PC card model 3951R that was mounted directly in a laptop computer.
All of the receivers used NovAtel’s Narrow CorrelatorTMtechnology 171.
NovAtel
model 501 dome antennas were used with each receiver. The antennas for
receivers 2-6 were mounted directly on the roof of the vehicles with a magnetic
mount. To improve reception along the test trajectory, the fixed receiver antennas were placed at a lo-20 deg inclination toward the trajectory. The antennas
for receiver 1 and the fixed pseudolite were mounted on surveying tripods in
152
Navigation
-400-600-400-200
Summer 1996
0
200 400 600
Easting (m)
Fig. 2-Positions
(nominal altitude
of Six Stationary
Receivers and Fixed Pseudolite
above test route shown in parentheses)
with Respect
to Test Route
pickup truck beds. All receivers and the fixed pseudolite remained stationary
during the entire test.
The coordinates of the fixed GPS receivers were determined from GPS measurement data collected by the receivers themselves. First, a floating ambiguity
survey was performed between receiver 4 and the 746th Test Squadron reference receiver, located 16 km away from the test site. From that point on, the
position of receiver 4 was considered the true reference point against which all
other positioning was performed. Next, each fixed GPS receiver was positioned
using an integer ambiguity solution relative to each of the other GPS receivers,
using about 2 h of data and the SEMIKIN’” software 181.A network adjustment
algorithm was then used to combine this redundant set of relative surveys to
generate the final positions of each of the fixed receiver antennas. Using this
method, the estimated relative positioning accuracy of the fixed receivers is
l-2 cm.
The pseudolite transmitters were Stanford Telecom Model 7201 Wideband
Signal Generators. The data rate for the broadcast was set at 50 bps, and
a prerecorded GPS message was transmitted. Two Mini-Circuits broadband
amplifiers were used to amplify the RF output from its standard level of
-50 dBm to an output of between - 15 and - 10 dBm in order to obtain the
desired transmission range. Each unit was controlled by a laptop computer
running an in-house Stanford Telecom software product called Synchronicity’“,
which was designed to simulate GPS, WAAS, and other pseudolite environments.
Vol. 43, No. 2
Raquet, et al.: Mobile Pseudolife
Concept
153
The fixed and mobile pseudolites transmitted on Ll using the PRN 10 and
PRN 8 Gold codes, respectively. Each of the receivers had the first two channels
manually locked onto PRN 10 and PRN 8, and the remaining 10 channels
remained free to receive as many GPS satellites as possible. No other modifications were made to the pseudolites or the receivers. Single-frequency (Ll)
pseudorange, carrier-phase, and Doppler measurements were collected at a
2 Hz rate from the pseudolites and each tracked satellite. The fixed pseudolite’s
antenna was mounted on a tripod in the back of a pickup truck in a position
visible to all of the fixed receivers. The position of the pseudolite antenna was
determined by attaching one of the NovAtel GPS receivers to the antenna for
a GPS survey (relative to the fixed GPS receivers) after the pseudolite testing
was complete.
Two phases of testing were performed on April 7,1995. Each phase consisted
of several passes along the test route at speeds between 5 and 20 m/s. During
the first phase, pseudolite 8 was mounted on the roof of the test van, and
pseudolite 10 was placed at the fixed location shown in Figure 2. The goal
during this phase was to use the data collected by the six stationary receivers
to determine the position of the test van’s pseudolite antenna. Two different
double-differencing methods were used to generate the results shown below.
The first, shown on the left of Figure 3, is referred to as pseudolite positioning
with satellite reference (PPSR). This type of double differencing is between
two receivers-the mobile pseudolite on the test van and a GPS satellite as
the reference transmitter (analogous to the reference receiver in a typical
case of relative positioning between two receivers). The other type of double
differencing is referred to as pseudolite positioning with pseudolite reference
(PPPR). This is the same as PPSR, except that the satellite used as the reference
transmitter is replaced with a fixed pseudolite (see Figure 3). Both types of
double differencing (PPSR and PPPR) were performed using the same set of
data collected during the first phase of testing.
During the second test phase, receiver 1 was mounted on the bus, using the
same antenna that had been used earlier by the mobile pseudolite. During this
Pseudolite Positioning with
Satellite Reference
(PPSR)
Pseudolite Positioning with
Pseudolite Reference
(PPPR)
GPS Satellite
0
Receiver A
Fixed
Pseudolite
Mobile
Mobile
Pseudolite
Receiver B
Fig. 3-Double-Differencing
Receiver A
Receiver B
Methods for Pseudolite Positioning
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154
Summer 1996
phase, the receiver collected measurements from the fixed pseudolite and all
visible GPS satellites. The trajectory of the antenna was calculated using
standard carrier-phase relative positioning techniques between one of the fixed
receivers and the mobile receiver. This type of positioning is referred to as
receiver positioning with receiver reference (RPRR). The measurements from
the fixed pseudolite were not used during this phase. As described later, this
RPRR positioning data has proven helpful in characterizing the performance
of the mobile pseudolite positioning algorithms (PPSR and PPPR).
MULTIPATH
ANALYSIS
Before attempting to generate a position solution, the raw measurements
were analyzed to determine basic measurement quality. Of primary interest
was the effect of multipath on the code and carrier measurements. High levels
of multipath were expected because both the receivers and pseudolites were
ground-based, and the GPS signal would have many potential reflected paths
in addition to the straight-line path. The placement of the pseudolite antenna
on the test van was also conducive to multipath. To increase reception at low
elevations and maximize visibility to the fixed receivers, the pseudolite antenna
was mounted on a wooden post 25 cm above a flat 1 m x 1 m metal plate,
which was itself mounted 21 cm above the rounded metal roof of the test van.
While this configuration
was required to accommodate
the necessary constraints in a ground-based test, it did create ideal conditions for multipath
(a worst-case scenario). Additionally, significant multipath was anticipated
because the signal-to-noise ratio (C/N,,) on each of the receivers tended to drop
noticeably whenever there was relative motion between the pseudolite and
receiver antennas. This effect may have been caused by uncharacteristically
high, rapidly changing levels of multipath due to the large potential for reflected
signals described above.
Analysis was performed on the data collected during all phases of the test,
but the results in this paper are shown for a 1673 s time period when one
pseudolite was mounted on the van, and the other was at a fixed site on the
ground. This time period was the only large segment in which all six receivers
maintained continuous carrier lock on both of the pseudolites with no cycle
slips (cycle slip detection and correction were not of primary importance in
this proof-of-concept demonstration).
The measurement data from this time
period are representative of the entire test.
Code/carrier difference plots are shown in Figure 4. The motion of the mobile
pseudolite can be inferred from the bottom plot, which shows latitude as a
function of time. Because carrier-phase multipath is very small relative to code
multipath, the code minus carrier difference is a good measure of code multipath
(plus measurement noise). As expected, there is a relatively high level of multipath variation in the signal from the mobile platform. This multipath correlates
strongly with vehicle motion. In contrast, the signal from the stationary pseudolite is relatively flat (note the difference in scale). This does not mean there is
no multipath, but that the multipath (if any) is nearly constant. The apparent
multipath from the satellite is also relatively small in magnitude as compared
with the moving pseudolite. Note that the receivers used in this test feature
Vol. 43, No. 2
Raquet, et al.: Mobile Pseudolite
-2
Concept
155
Test Van Latitude
33.025
9
s 33.020%
-I 33.019
492500
09:48
492800
09:53
493100
09:58
4f;M;O
41903;O 494000
IO:13
494300
IO:18
GPS Week Seconds/Local Time
Fig. 4-Code
Minus Carrier for Receiver 1
NovAtel’s Narrow CorrelatorTMspacing technology, which reduces code multipath effects [9].
While there is not a direct correlation between code and carrier-phase multipath, the high level of code multipath would imply that there may be high
levels of carrier-phase multipath as well. When the mobile pseudolite is actually
used on an airborne platform, code and carrier-phase multipath is expected to
decrease to the same level as that normally encountered when observing a
satellite. The accuracy is therefore expected to be better in the actual application than in the current demonstration.
POSITION
DOMAIN ANALYSIS
The primary goal for this test was to demonstrate the ability to position
a mobile pseudolite using carrier-phase processing techniques. This case is
analogous to the more typical case of positioning a GPS receiver (differentially)
using GPS satellites. The two pseudolites in this test correspond to the two
receivers in the typical case, and similarly the fixed receivers on the ground
correspond to the satellites. By considering the problem this way, standard
methods for ambiguity resolution and positioning can be used to generate the
desired trajectory. The two types of double differencing shown in Figure 3 were
used in the position domain analysis.
Pseudolite
Positioning
with Satellite Reference
(PPSR)
With PPSR double differencing, all clock errors are eliminated, and differential atmospheric errors are insignificant for the short baseline of this test case.
Navigation
156
Summer 1996
The positioning geometry (DOP) is a function of the relative location of the
mobile pseudolite and the receivers on the ground. The GPS SV is used only
in the double-differencing process to remove some of the errors, and it has no
bearing on the DOP values. DOP values are relevant in this case because no
weighting is used in the least-squares solution shown below. Plots of the HDOP
(horizontal) and VDOP (vertical) values are shown in Figure 5 as a function
of time. The latitude is also given to demonstrate that the DOP values are a
function of the position of the mobile pseudolite. Note the relatively high values
for VDOP (between 10 and 13), caused by having all of the receivers and the
moving pseudolite in nearly the same plane. In the real airborne case, this
will be less of a problem.
For a fixed set of double-difference integer ambiguities VAN and carrierphase measurements VA@, the measurement residual is defined as
(1)
v = VA+ + VAN - VAR,~~(P,,bile,Pref)Precl,Preez,...PreeG)
"measuTL?a
due"
“expected
value”
where VARal, is the calculated double difference of the ranges, based on the
position of the mobile PL (pm&ile),
the reference SV (pref), and the positions of
each of the six receivers (preel,prec2,...
pm,). The error in this residual as a function
of errors in the assumed positions of the PL, SV, and receivers is expressed by
sv = J%mbile~Pmobile
+ Hdpref + Hre$~rec~
+ i Hreej~preej
j=l
2.5
8
2
?
1.51.
I
I
1-q
I
IJU
Horizontal
J
+J
J
1
1,,1’_i
492500
492800
493100
493400
493700
494000
494300
09:48
0953
09:58
IO:03
IO:08
IO:13
IO:18
GPS Week Seconds/Local Time
Fig. 5-Dilution-of-Precision
(DOP) Plots
Vol. 43, No. 2
Raquet, et al.: Mobile Pseudolite
Concept
157
et - e,6
,Hm, =
eb
6 -
e6
-
e6
m
(3)
6
e,
where ek is the unit line-of-sight vector from the mobile PL to receiver k, et:
is the unit line-of-sight vector from the reference SV to receiver k, and Hrecjis
a 5 x 3 design matrix in which the jthrow is t$, - db, and all other rows are 0.
In this case, receiver 6 was chosen as the common receiver to minimize the
VDOP term. Because the receivers and the roving PL are very close to each
other relative to their distance from the SV, e;: = ei for all combinations x and
y. This means Href- 0, and the residual error equation becomes
6V = Hmobi&pmobile
+ J%,,~P,, + 9 H,,,~P,,
(4)
j=l
To determine the position of the roving PL (p m&i,e)ya nominal position is
chosen. The error in this position is calculated using the least-squares method
under the assumption that the errors in receiver positions are zero:
apmobile
=
6)
(~,bileHrnobile)-lH~,bileSV
The roving pseudolite position is then corrected by 8pm&ile,
a new set of
residuals is calculated, and the process is repeated until the solution has
sufficiently converged.
Pseudolite
Positioning
with Pseudolite
Reference
(PPPR)
With PPPR double differencing, all of the sensitivity (H) matrices in equation (2) are nonzero and cannot be eliminated. The roving pseudolite position
is calculated in the same manner as in PPSR double differencing using
equation (5), only the errors in the reference pseudolite position are also
assumed to be zero. The VDOP and HDOP values for this case are the same
as those shown in Figure 5. Measurements from the GPS SVs were used to
provide a common time base for each of the receivers, but they are not used
in the double-differencing process.
Double-Difference
integer Ambiguity
Resolution
In the initial attempts to generate a position solution for the mobile pseudolite, standard ambiguity resolution techniques were employed, which included
generating an initial estimate of the ambiguities (using code minus carrier
double differences), generating an integer search space about the initial estimate, calculating the sum of squares of the residuals CXr2)for each of the
candidate ambiguity sets, and selecting the ambiguity set with the minimum
29 1101.For the cases at hand, the residuals over the entire trajectory were
calculated for each candidate integer ambiguity set.
For PPPR double differencing, the above procedure seemed to work very
well, generating a trajectory that closely matched the known trajectory (as
described below) and had low measurement residuals throughout. However,
158
Navigation
Summer 1996
in the PPSR case, the correct set of integer ambiguities was not positively
identified. The ambiguity set with the minimum residuals did not generate a
valid solution (the altitude was several meters below the known altitude). One
cause of this difficulty may have been the high multipath levels present in this
test setup, as described earlier. Multipath is a systematic error that adds
noise to the residuals, creating difficulties for the integer ambiguity resolution
algorithms. A second possible cause involves the faulty assumption that the
errors in receiver position (Sp,,,) were zero. The relative positioning accuracy
of the receivers is actually around 1 cm, which is about the same magnitude
as the residuals themselves (for the correct set of ambiguities). The residuals
are therefore dependent not only on the position of the roving receiver, but also
on the position errors in the fixed receivers. Equation (41, however, attempts to
minimize the residuals by varying only the roving receiver’s position. In effect,
the receiver positioning errors are adding “noise” to the residuals because they
have not been properly accounted for from an estimation point of view. A
Monte-Carlo analysis showed that when the receiver and pseudolite positions
were artificially and randomly moved 1 cm (1 a) in each axis, the residual
could increase as much as 300 percent or decrease as much as 54 percent. The
average change was an increase of 42 percent. This demonstrated the high
sensitivity of the double-difference residuals to small positioning errors of the
fixed PL and receivers. A final cause of this ambiguity resolution problem may
have been some additional error that is common to both of the pseudolites and
is canceled out in the PPPR case, but not in the PPSR case.
To arrive at a good set of integer ambiguities for PPSR double differencing,
an altitude criterion was also added, based on a priori knowledge of the true
altitude. This hand-picked integer ambiguity set chosen for the PPSR double
differencing represents the minimum residual case of those that generate a
solution near the correct altitude (which was the 19th best residual overall).
While providing a reasonable solution, this approach is unacceptable in the
general case, since a priori knowledge of the trajectory is usually not available.
Double-Difference
Floating Ambiguity
Resolution
To present a more general positioning method for this demonstration,
a
different method was used which allowed the ambiguities VAN to be any constant floating-point number, rather than constraining them to integers as
before. Note that in keeping with commonly used terminology, these ambiguities are referred to as floating ambiguities, meaning only that they are constant
floating-point numbers. It does not mean that the values change or “float” over
time in this case because postmission batch solutions were used.
An iterative batch least-squares algorithm was used to calculate the floating
ambiguities. First, a subset of between 10 and 50 measurements was chosen
to ensure that the least-squares solutions were stable. These measurements
were selected so that they were evenly distributed over the entire trajectory
in order to maximize the observability ofthe least-squares parameters. Starting
values for the double-difference
carrier-phase ambiguities were estimated by
using double-differenced code measurements. Initially, a nominal trajectory of
the mobile pseudolite was calculated using the initial ambiguity values. Each
iteration then included the following steps:
Vol. 43, No. 2
Raquet, et al.: Mobile Pseudolite
159
Concept
1) Perform a batch least-squares estimation of the floating ambiguity errors
(total of five in this case) and the 3-D position error at each epoch. If n
represents the number of measurement epochs, then the total number of
unknowns is three dimensions times the number of epochs, plus five
ambiguities (3n + 5). Note that there were five double-difference measurements at each epoch, for a total of 5n measurements.
2) Correct the floating ambiguities and the trajectory by the estimated errors.
3) If the estimated floating ambiguity errors are sufficiently small, the process is complete. Otherwise, go back to step 1.
Four to eight iterations were typically required to converge on a solution.
This floating ambiguity algorithm generated surprisingly consistent sets of
ambiguities over a wide range of conditions. It was tested using seven different
reference satellites (PPSR) and the fixed pseudolite (PPPR), for a total of eight
different reference transmitters. The algorithm generated ambiguities for each
of the reference transmitters using between 10 and 50 data points. Figure 6
shows the typical convergence of floating ambiguities as more and more points
are used in the batch least-squares estimation. There were very few measurement epochs in common between the data used to generate each point in these
plots. The resulting ambiguities are relatively constant, especially when more
than 15 points are used, indicating that the estimation algorithm is stable.
Many of the floating ambiguities are noninteger values because of the multipath
and other error effects described above.
).
.t:
-1306
I
.s
DE3
I
1
-1306.5 - o
1
%
.ZZ!
00
-1307() 0
0
0 ooo
0
00 0 00
0 00 0
O O0 ~oOOoo
00
0
-1307.5
10
15
-19
I
0
20
25
30
35
40
45
I
I
I
I
1
I
50
Ret 5IRec 6/PL 8/SV 29 Double Difference
.G
O
$3
-19.5-20 -o”
5”
“E 0
-20.5 0-
‘F;
IEi
I
0
O
“E ‘Z=
3
I
00
0
6a,
3% 0
I
Ret 3IRec 6/PL 8/SV 29 Double Difference
O
-2110
non-integer due to multipath and other errors
00
Ooo”o
15
Oooo 0 000 OO
20
25
00O ~o”oooOooooo
30
35
40
0 0045
50
Number of Points for Least Squares Estimation of Ambiguities
Fig. 6-Typical
Convergence
Estimation Technique
of Floating Ambiguities
Using Least-Squares
160
Navigation
DEMONSTRATION
Summer 1996
RESULTS
To allow comparison of trajectories at different times, all results were transformed into a new coordinate frame. The origin of this new coordinate frame
is near the southeast end of the test trajectory, the X-axis points northwest
directly along the trajectory, the Y-axis points horizontally southwest normal
to the test trajectory, and the Z-axis points upward. If put in terms of the path
or “track” of the vehicle, the X-axis corresponds to down-track position (with
the origin near the southeast end of the trajectory and increasing to the northwest), the Y-axis to cross-track position, and the Z-axis to above-track position.
Once this coordinate transformation has been performed, the trajectory can
be correlated by down-track (X) position. This is especially useful for the vertical
axis since the road is relatively flat, and the altitude of the antenna is sufficiently determined by the down-track position on the road. The cross-track
component is also a function of down-track position, but is less repeatable
because it depends on the van driver’s ability to follow the same path each time.
Comparison Between PPSR and PPPR Double Differencing
Integer Ambiguities
Using
The differences between the trajectories generated by PPSR and PPPR double
differencing using integer ambiguities are shown on the left side of Figure 7.
The PPSR double differencing used satellite 18 as the reference transmitter.
Two different PPSR solutions are differenced with the PPPR solution. The first
(dashed line) is from the set of integers that yielded the lowest residuals (but
a poor position solution). The second (solid line) is from the hand-picked set
of integers chosen on the basis of a priori knowledge of the altitude. In both
cases there are significant differences between the PPSR and PPPR solutions,
due largely to the multipath and receiver positioning errors as described above.
The errors appear to be primarily a function of geometry, which is evident by
comparing the shapes of the plots in Figure 7 with the trajectory in the bottom
plot of Figure 5.
Floating Ambiguities
Difference Between 1 PPPR
and 7 PPSR Solutions
(Sk 2, 18, 19,27,28,29
31 as
Reference for PPSR Solutions)
Integer Ambiguities
Difference Between 1 PPPR
and 2 PPSR Solutions
(SV 18 as Reference
~
492500
w:48
492900
0953
493103
0958
49YW
,0:03
$“y
-&i&i
4937(10
10:09
>
494ooO
to:13
GPS Weak Seconds/Local Time
Fig. 7-Comparison
494300
lo:19
4;;go
09 48
493100
09 58
493400
10 03
493700
1008
494000
10 13
GPS Week SecondslLocal Tune
of PPPR and PPSR Position Solutions
494300
10 I9
Vol. 43, No. 2
Residuals
of Figure 8.
magnitudes
significantly
quate.
Raquet, et al.: Mobile Pseudolite
Concept
161
for the two types of double differencing are shown on the left
The residuals for PPPR double differencing look reasonable,
on the order of l-2 cm, while both of the PPSR residuals
larger, indicating that the PPSR integer solutions are not
Comparison Between PPSR and PPPR Double Differencing
Floating Ambiguities
side
with
are
ade-
Using
The difference between one trajectory generated using PPPR double differencing and seven trajectories generated using PPSR double differencing is shown
on the right side of Figure 7. The seven PPSR trajectories were generated using
seven different satellites as reference transmitters. All of the trajectories were
calculated using floating ambiguities. These plots show that all of the various
trajectories agree within about 6 cm horizontally and 1 m vertically. While the
vertical direction appears to be much worse, this is not surprising given the
poor vertical geometry of the test, as represented by high VDOP values between
10 and 13. These results are very encouraging because they demonstrate the
ability of the floating-ambiguity algorithm to calculate ambiguities that provide
consistent trajectories across many reference transmitters.
Typical double-difference residuals using floating ambiguities are shown on
the right side of Figure 8. For both PPSR and PPPR solutions, the residuals
are typically 1 cm or better, which is an improvement over the fixed integer
residuals. This is not surprising since the floating ambiguities are able to
absorb more of the errors, rather than forcing them into the residuals.
Stationary Positioning
Stability
As shown in the bottom plot of Figure 5, there were several time periods
when the van was stationary. During each of these stationary periods, the
position of the antenna was calculated using both PPSR and PPPR double
differencing. The standard deviation of the position during these periods pro-
Integer Ambiguity
Floating Ambiguity
Residuals
PPSR (Mfnimum Residual Solution)
Residuals
;;;t,
/ I,;
PPPR
-0 05
492500 492800 493100 493400 493700 494000 494300
0958
1003
,008
1013
1018
09:4e
0953
GPS Week Seconds/Local Time
Fig. &-Measurement
Residual
Comparison
GPS Week Seconds/Local
Among Different
Solution
Time
Types
162
Summer 1996
Navigation
vides a good measure of the basic stability of the pseudolite positioning methods.
For both PPSR and PPPR methods over several stops, the standard deviation
of the position varied between 0.9 and 2.7 mm horizontally and 16.7 and
18.4 mm vertically. Once again, the vertical results are poor as compared with
the horizontal results. If the vertical results are normalized by the VDOP,
however, they are approximately the same as the horizontal results (as they
should be). The van also stopped during the final phase of the testing, when
the NovAtel receiver was attached to the antenna previously used by the
pseudolite. The standard deviation of position from the RPRR (standard GPS
receiver) positioning over several stops varied between 2.3 and 6.7 mm in all
three axes. Based on these results, the stability of the PPSR and PPPR position
solutions is comparable to that of the standard RPRR.
Comparison with Receiver Positioning
(RPRR) Trajectory
with Receiver Reference
The data collected by the mobile GPS receiver during the final phase of
testing was processed using The University of Calgary’s SEMIKIN’” software,
which determined the correct set of integer ambiguities and generated a time
history of position [81. Residuals and other quality measurements were monitored to ensure that the trajectory would be correct. As a result, the GPS
receiver trajectory is considered a “truth” reference against which the pseudolite positioning can be compared. The estimated accuracy of the GPS trajectory
is l-3 cm, depending primarily on carrier-phase multipath. Rather than compare the pseudolite and GPS receiver trajectories verses time (which is impossible since they occurred at different times), they are compared spatially, using
the down-track distance (X) as the common reference variable.
Altitude comparisons between the pseudolite and GPS receiver trajectories
are given in Figure 9. The plots on the left show the altitude from the PPSR
and PPPR integer ambiguity pseudolite trajectories, and those on the right
show the altitude from the PPSR and PPPR floating-ambiguity
pseudolite
trajectories. These.plots depict a “side view” of each of the trajectories. On all
four plots, the gray lines are pseudolite trajectories, and the black lines are
GPS receiver trajectories. Both the GPS receiver and the pseudolite traversed
Floating Ambiguity Solutions
Integer Ambiguity Solutions
2
g
1
NI-E
5
100
150
PPPR
200
250
Solution
300
350
400
O
-1
0
/\
0
100
Along Track(X)
Fig. g-Altitude
Comparison
and RPRR Truth Solution
Between
Pseudolite
150
200
250
Along Track (X)
Positioning
Solutions
300
350
400
Vol. 43, No. 2
Raquet, et al.: Mobile Pseudolite
Concept
163
the same section of road several times (5 for the receiver and 4 for the PL),
and the plots give a measure of the repeatability of the solution. The pseudolitegenerated altitudes exhibit about 0.5 m of noise, which is good considering the
high VDOP values during this test. In the PPSR (integer) plot, one of the
pseudolite trajectories shows a 3-5 m bias in altitude. This trajectory was
generated using the minimum residual set of integer ambiguities. The better
pseudolite altitude shown on the same plot (referred to as “best altitude solution”) was generated by the integer ambiguities which were hand-picked to
obtain a good altitude. This is an excellent example showing that minimum
residual integer ambiguity resolution techniques may not be optimal in the
presence of significant biases due to multipath or other types of errors. It is
also apparent that the floating-ambiguity
trajectories in general exhibit a bias
of about 0.8 m. This is most likely a result of the amplification of remaining
survey or multipath errors due to the poor vertical geometry, and it is not
overly worrisome after DOP normalization is taken into account.
Horizontally, the pseudolite (both PPSR and PPPR) and RPRR solutions all
agree within about 0.5 m, which is as good as could be expected since they
represent different trips for the van, and the driver could not be expected to
repeat the same trajectory to an accuracy better than 0.5 m under normal
conditions. The horizontal accuracies are assumed to be better than the vertical
accuracies because of the relatively strong horizontal measurement geometry.
Comparison
with Tape Measure Positioning
One final analysis technique was used in an effort to provide an independent
measure of the van’s antenna position against which the pseudolite and GPS
solutions could be compared. On each southward pass of the trajectory, the
van stopped for a period of time near a fixed survey point on the middle of the
road. (These are the same points that were used to analyze position stability
above). Three measurements were made with a tape measure at each stop
point to determine the position of the edge of the van tires with respect to the
centerline of the road and the survey marker. These measurements were later
converted into measures of the horizontal position and heading of the van at
each stop point. After making an assumption about the roll and pitch angles,
the exact position of the antenna was calculated. An error analysis determined
that approximate 1 u error values for this method are 10 cm for the downtrack (X> axis, 20 cm for the cross-track (Y) axis, and 8 cm for the vertical (Z>
axis. The high Y-axis uncertainty is due to the uncertainty in the roll angle.
A comparison between the tape-measure and PIJGPS receiver positioning
methods is given in Table 1. While not subdecimeter in all cases, these results
do show agreement between the independent
measurements
to a level of
lo-20 cm horizontally and about 1 m vertically, which is reasonable considering
the accuracy of the tape measure method and the poor vertical geometry in
the pseudolite case.
CONCLUSIONS
The results from this demonstration clearly show that it is possible to generate an accurate position of a transmitting mobile pseudolite using an array of
GPS receivers fixed on the ground. Accuracies on the order of lo-30 cm were
Summer 1996
Navigation
164
Table 1-Difference
Between Tape-Measured and PUGPS Receiver Positions at
Selected Survey Points
Position Error vs. Tape Measure Method
Calculation Type
Ref. SVlpL
PRN
Along-Track
GPS Time 6s)
(X)
Cross-Track
Cy)
Above-Track
(Z)
PPPR
PPPR
PPPR
PPPR
Integer
Integer
Floating
Floating
PL
PL
PL
PL
10
10
10
10
493083-493333
493739-494089
493083-493333
493739-494089
8.2
8.8
14.5
15.2
cm
cm
cm
cm
- 7.0
- 8.5
-3.1
-4.5
cm
cm
cm
cm
35.1
41.1
92.7
98.7
cm
cm
cm
cm
PPSR
PPSR
PPSR
PPSR
PPSR
PPSR
PPSR
PPSR
PPSR
PPSR
Integer (low resid)
Integer (low resid)
Integer (best ah)
Integer (best ah)
Floating
Floating
Floating
Floating
Floating
Floating
SV 18
SV 18
SV 18
SV 18
sv2
sv2
SV 18
SV 18
SV 28
SV 28
493083-493333
493739-494089
2.3
3.0
18.3
19.0
16.2
17.2
14.8
15.5
14.5
15.5
cm
cm
cm
cm
cm
cm
cm
cm
cm
cm
- 26.4
-28.2
- 12.8
- 14.5
- 1.5
- 1.7
- 2.6
-4.3
- 3.8
- 6.2
cm
cm
cm
cm
cm
cm
cm
cm
cm
cm
-355.1
- 352.9
70.7
73.0
141.7
164.5
97.5
99.8
48.6
29.9
cm
cm
cm
cm
cm
cm
cm
cm
cm
cm
n/a
497812-498013
498389-498539
17.7 cm
25.2 cm
RPRR (Truth Solution)
RPRR (Truth Solution)
493739-494089
493083-493333
493739-494089
493083-493333
493739-494089
493083-493333
493739-494089
493083-493333
0.9 cm
4.8 cm
- 2.8 cm
1.1 cm
demonstrated.
Given the somewhat severe conditions that existed in this demonstration relative to an operational system (e.g., very poor vertical geometry
and high multipath potential), these results are encouraging, suggesting that
such a system may be a feasible means of meeting the reference system requirements for high accuracy (better than 10 cm). The analysis of the data also
pointed out areas that require attention if a reliable system is to be developed.
In particular, it is important to determine the position of the fixed receivers
and fixed pseudolite to a very high degree of accuracy (through better surveys
and/or error modeling techniques) because of the close proximity of the pseudolites and receivers.
REFERENCES
1. Snodgrass, F. B., Internal Memorandum (a sketch of the system), Holloman
AFB, Research and Development Section, 746th Test Squadron, December 1994.
2. Klein, D. and Parkinson, B. W., The Use of Pseudo-Satellites for Improved GPS
Performance, NAVIGATION, Journal of The Institute of Navigation, Vol. 31,
No. 4, Winter 1985, pp. 303-15.
3. Cohen, C., Lawrence, D., Pervan, B., Cobb, S., Barrows, A., Powell, D.,
Parkinson, B., Wullschleger, V., and Kalinowski, S., Flight Test Results of
Autocoupled Approaches Using GPS Integrity Beacons, Proceedings of the
Seventh International Technical Meeting of the Satellite Division of The
Institute of Navigation (ION GPS-941, Salt Lake City, UT, September 1994,
pp. 1145-53.
4. Cobb, H. S., Cohen, C. E., and Parkinson, B. W., Theory and Design of
Pseudolites, Proceedings of the 1994 National Technical Meeting of The Institute
of Navigation, San Diego, CA, January 1994, pp. 69-75.
5. Pervan, B. S., Cohen, C. E., and Parkinson, B. W., Zntegrity Monitoring for
Precision Approach Using Kinematic GPS and a Ground-Based Pseudolite,
Vol. 43, No. 2
6.
7.
8.
9.
10.
Raquet, et al.: Mobile Pseudolite
Concept
165
NAVIGATION, Journal of The Institute of Navigation, Vol. 41, No. 2, Summer
1994.
Blaha, G., Znvestigations
of Critical Configurations
for Fundamental
Range
Networks, Reports of the Department of Geodetic Science number 150, Ohio
State University, 1971.
Van Direndonck, A. J., Fenton, P., and Ford, T., Theory and Performance of
Narrow Correlator Spacing in a GPS Receiver, NAVIGATION, Journal of The
Institute of Navigation, Vol. 39, No. 3, Fall 1992, pp. 265-83.
Cannon, M. E., High-Accuracy
GPS Semikinematic
Positioning: Modeling and
Results, NAVIGATION, Journal of The Institute of Navigation, Vol. 37, No. 1,
Spring 1990, pp. 53-64.
Townsend, B. and Fenton, P., A Practical Approach to the Reduction of
Pseudorange Multipath
Errors in a Ll GPS Receiver, Proceedings of the Seventh
International Technical Meeting of the Satellite Division of The Institute of
Navigation (ION GPS-94), Salt Lake City, UT, September 1994, pp. 143-48.
Lachapelle, G., Cannon, M. E., and Lu, G., High Precision GPS Navigation
with
Emphasis on Carrier Phase Ambiguity Resolution,
Marine Geodesy, Vol. 15,
No. 4, 1992, pp. 253-69.