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GPS based navigation performance analysis within and beyond the Space Service
Volume for different transmitters' antenna patterns
Conference Paper · April 2017
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GPS based navigation performance analysis within and beyond
the Space Service Volume for different transmitters’ antenna
patterns
Endrit Shehaj*1, Vincenzo Capuano† , Cyril Botteron‡ , Paul Blunt§, Pierre-André Farine¶
École Nationale de l’Aviation Civile, Toulouse, 31055, France
École Polytechnique Fédérale de Lausanne (ESPLAB), Neuchâtel, 2002, Switzerland
In the last years, Global Navigation Satellite System (GNSS) based navigation in high earth
orbits (HEOs) has become a research field of interest, since it can increase the spacecraft’s
autonomy, reducing the operating costs. However, the GNSS availability and the GNSS-based
navigation performance for a spacecraft orbiting above the GNSS constellation, is strongly
constrained by the signals’ power levels at the receiver position and by its sensitivity. The
simulated level of signal power at the receiver’s position may considerably increase or decrease
when assuming different gain values of the transmitter antenna for a certain azimuth and
elevation. Assuming a slightly different pattern, therefore may significantly change the
simulated signal’s availability results and accordingly the simulated navigation accuracy,
leading to an inexact requirements identification for the receiver. This problem concerns
particularly orbital trajectories above the GNSS constellation, where most of the received
signals are radiated from the side lobes of the transmitters’ antennas, for which typically very
little information is known. At the time of this study, it was possible to model quite accurately
the GPS L1 antenna patterns for IIR and IIR-M Blocks because of the precise information
available. No accurate information was available for the GPS L1 antenna patterns of IIF Block.
Even less accurate information was available on the GPS L5 antenna patterns. In this context,
this paper aims at investigating the effect of different antenna pattern assumptions on the
simulated signal availability and on the consequent simulated navigation performance of a
spaceborne receiver orbiting in a very highly elliptical orbit from the Earth to the Moon.
Initially the impact of averaging the transmitter’s antenna gain over the azimuth, typical
assumption in many studies, is analyzed. Afterwards, we consider three different L5 antenna
patterns assumed in the literature (the precise pattern are unfortunately not yet fully
available). For each of the considered antenna pattern assumptions, we simulate received
signal power level, availability, GDOP and navigation accuracy, in order to evaluate their
effect. Finally, once identified the most conservative assumptions for the transmitters’ antenna
patterns, for each elevation of the receiver antenna, we also compute the number of available
GNSS observations and we analyze their distribution.
Nomenclature
𝑃𝑜
𝑃𝐼𝐶𝐷
𝑂𝐺
Received signal power
Minimum signal power on Earth
Global offset
𝑎𝑖
c𝛿𝑡𝑢 , 𝑐𝛿𝑡 𝑖
𝐴𝑖 , 𝐷 𝑖 , 𝐾 𝑖 , 𝑛 𝑖
𝐿𝐹𝑆
𝐿 𝑇𝑋 , 𝐿𝑅𝑋
𝜑, 𝜙
𝑥𝑠𝑎𝑡 𝑖 , 𝑦𝑠𝑎𝑡 𝑖 , 𝑧𝑠𝑎𝑡 𝑖
𝑥𝑢 , 𝑦𝑢 , 𝑧𝑢
𝑣𝑠𝑎𝑡 𝑖 , 𝑣𝑢
Free space loss
Transmitter and receiver antenna gain
Elevation, azimuth
ith satellite coordinates
Receiver coordinates
ith satellite and receiver velocity vectors
σDLL
σf
𝐵𝑛 , 𝐷, 𝑇, 𝐵𝑓𝑒 , 𝑅𝑐
𝐶/𝑁0
2
𝜎𝑖𝑓
𝜔𝐿1,𝐿5
Line of sight (LOS) unit vector
Receiver and ith satellite clock offsets
Atmospheric, Multipath, signal in space,
receiver delays
Thermal noise code tracking jitter
Doppler tracking jitter
Tracking loop parameters
Signal to noise ratio
Iono-free measurement variance
L1, L5 frequencies (𝑓𝐿1 , 𝑓𝐿5 ) square ratio
*Graduate student, MSc. GNSS, ENAC, Toulouse, and intern EPFL, Neuchatel.
†
Postdoc researcher, ESPLAB (EPFL), Neuchâtel.
‡
GNSS Team leader, ESPLAB (EPFL), Neuchâtel.
§
Researcher, ESPLAB (EPFL), Neuchâtel.
¶
Professor, ESPLAB (EPFL), Neuchâtel.
1
American Institute of Aeronautics and Astronautics
I.
Introduction
GNSS in space applications started to be investigated in the mid to late 70s while in 1982 the first spacecraft flew
with an onboard spaceborne receiver (1). Since then, the use of GNSS at LEO has been successfully adopted, favored
by the strong signal powers and the good relative geometry that a receiver at these altitudes experiences. Moreover,
the great benefits of GNSS based navigation over conventional ground based methods, such as increased spacecraft’s
autonomy and reduced operational costs, as well as, the will to improve the achieved accuracy, motivated the
investigation of GPS at higher altitudes, leading to the first recordings beyond LEO in 1997. The eager to exploit
GNSS signals at higher orbits has highly increased in recent years, with the perspective of new innovations such as
improved weather prediction, exploration moon missions and beyond, formation flying, etc.
Although space users showed great enthusiasm in GNSS based navigation, they had to face important difficulties.
Indeed, primarily intended for earth navigation, initial GPS system specifications (i.e. minimum power level and
availability) covered only Earth users or up to 14° half angle of the transmitters’ antenna, (1). Due to the great utility
for space users above 14° and the significant changes among different blocks, space users wanted formal specifications
to be included. Efforts to include Space Service Volume (SSV) did not start until only the 90s and it was in February
2000 when space requirements were officially included. However, this was not enough, and further improvements
were needed. GPS III included new requirements: 1) SSV was subdivided in two regions, MEO SSV (3000-8000 km)
and GEO/HEO SSV (8000-36000 km) and 2) specified worst case signal strength and availability. Even though it is
clear that processing side lobes signals can significantly improve the availability (and the receiver/transmitters
geometry) in GEO/HEO, so far, for many GNSS signals and GNSS satellites blocks, detailed specifications of the
transmitters’ side lobes have not been yet explicitly stated. As stated in (2), “Protection of GPS Side Lobes through
Specification is critically important to ‘Do No Harm’ to current and future users of SSV”. Under these circumstances,
a GPS receiver in GEO/HEO should be able to process signals transmitted from the side lobes of the GPS antenna
patterns. The lower power levels that these signals have, compared to the ones transmitted from the main lobe, affect
the range accuracy with higher receiver noise. Moreover, if the receiver’s sensitivity is not high enough, the availability
of GNSS signals can be drastically lower, with a consequent worsening of the GDOP (geometric dilution of position),
or in the worst case preventing navigation solution computation, for which at least four observations are needed.
In this context, in order to simulate correctly the GNSS based navigation performance of a spaceborne receiver,
the assumptions on the GNSS antenna patterns are very important and delicate, especially on the side lobes. The main
lobe is well known and no assumptions are needed. Figures 1 and 2 show respectively the elevation at which the signals
are transmitted by each GPS satellite and the number of signals transmitted from the main or the side lobes, for a direct
Moon Transfer Orbit (MTO), presented later in section II. As displayed, space users operating in all SSV and up to the
Moon, acquire and track most of the signals transmitted from the side lobes. Evidently, modelling correctly the side
lobes is very important as it would affect directly the navigation performance at high altitudes obtained in simulation.
Figure 1 Elevation angle of the GPS antenna pattern from which the signals are received, for the assumed MTO in section II.A.
In case of GPS L1 and L2, antenna gain data have been released for Blocks IIR and IIR-M, (3). On the other side,
Boeing, which is responsible for building GPS Block IIF satellites, has not released any similar document yet. Even
more obscure is the information regarding GPS L5 antenna directivity data. Existing information used by researchers,
is usually an approximation of the expected patterns using the information in (4). No definite data have been released
so far, causing possible inaccuracies in the results of the published researches. In (5), for instance, only an example of
L5 pattern for only PRN 5 is reported and without detailing how it was approximated from the available data.
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Figure 2 Number of GPS signals received from the main and side lobes of the transmitters’ antenna pattern, for the assumed
MTO in section II.A.
In our previous research (6), we have investigated the feasibility of using GNSS as navigation system to reach the
Moon; in (7) we described the implementation of an orbital filter in order to improve the navigation accuracy; in (8) we
detailed how such a filter can be used for aiding purposes; in (9) we demonstrated the GNSS accuracy at GEO altitude
achieved with the developed orbital filter considering that the receiver is able to process different signal combinations.
However, the lack of information about the transmitters’ antenna patterns has always limited the level of accuracy in
the results of our studies. Considering the importance of the antenna pattern assumptions, in this research we aim to
evaluate the effects of different antenna pattern assumptions when simulating the performance of a GNSS based
navigation system for space missions, to identify the most conservative assumptions to be used in our future
simulations and tests. Elements like experienced signal levels, availability, GDOP and ranging error were computed
for different antenna patterns assumptions. Moreover the single-point least squares solution was also computed and
compared. Initially, we evaluated the effect of averaging the gain pattern in azimuth for the GPS L1 signals and then
we explored the impact of assuming different GPS L5 patterns available from the literature.
These evaluations were carried out considering the characteristics of the “SANAG” receiver, a proof-of-concept
GPS receiver under development in our laboratory, specifically for lunar missions. The considered MTO is an
interesting trajectory case, as it includes the signals’ characteristics of the SSV and beyond up to the Moon.
The rest of the document is organized as follows. Section II describes the different models and assumptions used
in the course of this work. In sections III and IV, the assumption of averaging the pattern in azimuth for L1, as well as
the assumption of using different patterns for L5, are investigated. Once identified the most conservative assumptions
for the GPS antenna patterns, section V presents a short study about the number of available GNSS observations for
each elevation of the receiver antenna. Finally, section VI summarizes the conclusions obtained during this work.
II.
Models and Assumptions
In our simulations, we used the highly accurate multi-GNSS simulator “Spirent GSS8000” to simulate realistic
GPS signals for different antenna patterns (10). Basically, we defined the trajectory, the antenna patterns, the
transmitters’ ephemeris, the signal level characteristics of all simulated carrier frequencies at the RF output, and Spirent
provided simulated signals power levels at the receiver position. In our tests, we defined a GPS constellation of 29
satellites according to the almanac of 1st July 2005. The other assumptions are presented in the following paragraphs.
Receiver’s dynamics and kinematics
For this work, an accurate MTO trajectory of the receiver was generated using the Astrogator Tool of STK (11),
and after provided as reference trajectory in our Spirent GSS8000 simulator, as UMT (user motion) file.
The considered MTO is illustrated in Figure 3, where is identified by the yellow curve in LEO and the blue curve
in HEO with perigee at the Moon altitude. In this trajectory, a GNSS receiver experiences all signal characteristics that
it would experience orbiting at different altitudes in space (from LEO to the Moon).
The initial time of investigation is 18:00:00 of 1st July 2005 with initial ECI coordinates [-6188.944987; 1742.455236;
1155.838277] km and [2.484942; -6.510775; 3.563661] km/s, for position and velocity respectively; the launch of the vehicle
was performed earlier. The investigated part of this study is propagated until 05:11:12 of 6 th July 2005, which
corresponds to the end of the blue curve (Moon altitude) and it experiences an impulsive acceleration of 3.14217 km/s
in the x axis, at time 18:41:37:154 of 1st July 2005, to change the orbit from LEO to HEO with perigee at Moon altitude.
A.
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Figure 3 Assumed MTO for the receiver.
B.
Assumed receiver characteristics
As mentioned, in this study we assumed the characteristics of the spaceborne receiver “SANAG” (12) under
development in our laboratory, which is designed to acquire L1 C/A signals with a sensitivity of 15 dB-Hz (when aided
by an orbital filter) and to track both L1 C/A and L5Q signals, respectively with a tracking sensitivity of 15 dB-Hz and
12 dB-Hz. In particular, note that the tracking strategy adopted includes the use of ionosphere-free observations
combinations for some of the signals crossing the ionosphere, in order to mitigate their ionospheric delay. More
detailed information about the tracking strategy adopted in the SANAG receiver can be found in (12) .
The signal power received at the receiver position is modelled as follows, taking into account the gain patterns of
both transmitter and receiver antennas and the free space signal propagation losses:
𝑃𝑜 = 𝑃𝐼𝐶𝐷 + 𝑂𝐺 + 𝐿𝐹𝑆 − 𝐿𝑇𝑋 (𝜑, 𝜙) − 𝐿𝑅𝑋 (𝑑𝐵𝑚)
(1)
with 𝑃𝐼𝐶𝐷 the guaranteed minimum signal level for a receiver on Earth (as specified in signal-in-space specifications
document), 𝑂𝐺 the global offset (set to match the same level of signals as when real ones are received), 𝐿FS the free
space loss, 𝐿𝑇𝑋 and 𝐿𝑅𝑋 the loss due to the transmitter’s antenna pattern and receiver’s antenna pattern respectively,
𝜑, 𝜙 represent respectively the elevation and azimuth of the transmitter antenna pattern.
The 𝑃𝐼𝐶𝐷 for GPS L1 CA and GPS L5Q was assumed to be respectively -128.5 dBm and -127 dBm according to
the ICD documents (4) and (13); in addition, a global offset of 3 dB for both signals was assumed to account for the
difference between the minimum power transmitted and the real one (6). It should also be mentioned that the signal
power level range that Spirent simulator can generated is [𝑃𝐼𝐶𝐷 + 𝑂𝐺 + 20; 𝑃𝐼𝐶𝐷 + 𝑂𝐺 − 49.9] dBm. Thus, the powers
levels belong to the interval [-175.4, -105.5] dBm and [-173.9000, -104] dBm for GPS L1 and GPS L5 respectively.
The GPS observations were modelled as already done in our previous study (9); a summary is provided in Table 1.
Pseudorange
𝟐
𝟐
𝟐
𝝆𝒊 = √(𝒙𝒔𝒂𝒕 𝒊 − 𝒙𝒖 ) + (𝒚𝒔𝒂𝒕 𝒊 − 𝒚𝒖 ) + (𝒛𝒔𝒂𝒕 𝒊 − 𝒛𝒖 ) + 𝒄𝜹𝒕𝒖 − 𝒄𝜹𝒕𝒊 + 𝑨𝒊 + 𝑫𝒊 + 𝑲𝒊 + 𝒏𝒊
𝑥𝑠𝑎𝑡 𝑖 , 𝑦𝑠𝑎𝑡 𝑖 , 𝑧𝑠𝑎𝑡 𝑖 : ith satellite coordinates at time of transmission
𝑥𝑢 , 𝑦𝑢 , 𝑧𝑢 : the receiver coordinates at time of reception
𝑐𝛿𝑡𝑢 : receiver clock offset at signal reception time
𝑐𝛿𝑡 𝑖 : transmitter clock offset at signal transmission time
𝐴𝑖 : atmospheric delay (m)
𝐷𝑖 : multipath delay (m)
𝐾 𝑖 : signal-in-space ranging error (SISRE) (m)
𝑛𝑖 : receiver noise (m)
Pseudorange
rate
initial value :10 km
𝜎𝐴 = 7m (when the receiver is
below the ionosphere upper bound)
𝜎𝐷 = 0.2m
𝜎𝐾 = 0.5m
2
𝜎𝑛 = √𝜎𝐷𝐿𝐿
+ 0.12
𝝆̇ 𝒊 = (𝒗𝒔𝒂𝒕 𝒊 − 𝒗𝒖 ) ⋅ 𝒂𝒊 + 𝒃̇ + 𝝈𝒇
𝑣𝑠𝑎𝑡 𝑖 : satellite’s velocity vector
𝑣𝑢 : receiver’s velocity vector
𝑎𝑖 : the line of sight (LOS) unit vector from the receiver to the ith GNSS satellite
initial value :100 m/s
𝑏̇: receiver clock drift
𝜎𝑓 : Doppler tracking jitter (m/s)
ith
Table 1 Observations assumptions.
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C.
Different Antenna patterns assumptions analyzed
L1 Antenna patterns
We used the published data in (3) for GPS L1 Blocks IIR and IIR-M to model these blocks satellites’ antenna
patterns. The measured antenna gain data for each transmitter of Block IIR and IIR-M are provided for L1 and L2
signals by Lockheed Martin in APPENDIX B of (3). According to (14), some of the IIR satellites use the legacy antenna
pattern, while the remaining part and all IIR-M satellites use the improved antenna pattern. For the benefit of terrestrial
and space GPS users, the improved pattern includes a new design of the L-band elements mounted on the spacecraft’s
Earth-facing structure. Meanwhile, for the Block IIF satellites (built by Boeing) there have not been released similar
data. Logically the antenna patterns used for IIF Block should be at least as good as the improved antenna pattern.
Thus, we decided to use one single 3D pattern for the satellites of Block IIF, where for each azimuth and each elevation
the gain is the mean of the gains at the same azimuth and elevation of all the transmitters’ patterns with the improved
antenna pattern in (14). This pattern is illustrated in Figure 4; in APPENDIX A of (3), the antenna patterns for each
satellite of IIR and IIR-M Blocks are shown. In Figure 4 is also shown the assumed 2D pattern for Block IIF satellites,
which is the mean of the gains for different azimuth (in the same figure). In the same way the 2D patterns were formed
for each satellite of Block IIR and IIR-M from their respective patterns, shown in APPENDIX A of (3).
L5 Antenna patterns
In this study three different options for the L5 pattern, approximated from the L1 patterns. More in details, we
carried out a signal characteristics analysis for three different assumptions on the L5 antenna gain patterns:
1. L5 antenna pattern 1. In the first case we modelled a different L5 antenna gain pattern for each satellite of IIR and
IIR-M Blocks and only one L5 pattern for all the satellites of IIF Block, following the principle explained in (15).
For each IIR satellite and IIR-M satellite we derived the L5 patterns from the GPS L1 assumed patterns as follows:
i. First, for each satellite, we computed a 2D pattern from the L1 detailed 3D pattern, where the gain for each
elevation is the mean value of the gains for L1 for the same elevation and for all azimuths.
ii. Second, maintaining unchanged the gain, we multiplied each elevation angle by a 1.1 factor. Thus, essentially
the 2D gain pattern of L5 differs from the one of L1 only in elevation.
To form the single L5 pattern used for all the IIF satellites from the L1 pattern, the same procedure was adopted.
2. L5 antenna pattern 2. In this case we used the same pattern as the one defined for IIF satellites in the first case,
for all GPS L5 satellites.
3. L5 antenna pattern 3. In the third case we used the pattern provided in (5).
In Figure 5 is displayed in blue the assumed L5 pattern for the IIF satellites (in the first case) approximated from
the L1 pattern, used also in the second case for all GPS satellites. In red is shown the L5 antenna pattern 3.
Figure 4 Block IIF GPS L1 antenna Gain pattern.
III.
Figure 5 GPS L5 antenna Gain pattern.
GPS L1 receiver: 2D pattern vs 3D pattern
As confirmed by the published data in (3), the transmitters’ gain patterns vary not only among the elevation, but
also among the azimuth angles. On one side the gain variation for the elevations used for terrestrial services is less
than 1.5 dB, but, on the other side, it becomes significant for the remaining part of the main lobe; moreover it reaches
values larger than 10 dB for the side lobes. Thus, the experienced power levels for a signal tracked at the same elevation
but at different azimuth would be different, especially for signals processed from the side lobes. However, due to lack
of precise information on the antenna directivity data, in many research studies, this element is neglected. In most
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studies, a 2D antenna pattern is assumed, where the gain is function of the elevation, but not of the azimuth, for which
an average value is assumed. In this case, the term LTX in equation (1) will be only function of the elevation φ.
In this context, this section aims at comparing quantitatively and qualitatively the GNSS performance obtained
when assuming 3D antenna patterns with the one when 2D antenna patterns are considered.
A.
Received Signal power levels
As a minimum of four signals are required for a standalone GPS navigation solution, Figure 6 displays the fourth
highest received power levels of the GPS L1 CA signals at the receiver position as a function of the altitude, during
the full considered trajectory, assuming 0 dBi receiver antenna gain, when the GPS L1 transmitter antenna is assumed
to have a 3D pattern and a 2D pattern. What we can notice from this figure is that in the case of 3D pattern the fourth
highest received signal power is sometimes lower than -170 dBm, which is never the case for the 2D pattern.
Observing carefully Figure 6 we can notice that the received powers are different; however it is hard to have clear
evidence in which case the received powers are higher. To have a better understanding on the effect of the two
considered patterns, in the next section, we will investigate the signals’ availability.
Figure 6 Fourth highest received power level of GPS L1 signals, during the considered MTO, for a 0 dBi receiver gain pattern.
B.
Availability and GDOP
In this study the i-th GPS observation is considered available if the corresponding i-th GPS L1 C/A signal is acquired
and tracked. According to the capability of the SANAG receiver, the i-th GPS L1 C/A signal,
- is acquired if the corresponding signal power is equal to or stronger than 20 dB-Hz without any interruption
in the time interval [𝑡𝑘 − 5 𝑚𝑖𝑛, 𝑡𝑘 ], or if it is weaker than 20 dB-Hz but stronger than 15 dB-Hz without any
interruption in the time interval [𝑡𝑘 − 10 𝑚𝑖𝑛, 𝑡𝑘 ].
- Is tracked if the corresponding signal power is equal to or stronger than 15 dB-Hz.
Note that once the receiver is orbiting above the ionosphere upper bound, the few L1 signals that cross the ionosphere
are discarded, since they may be affected by large ionospheric delay due to the double crossing of the ionosphere.
Figure 7 illustrates the availability of satellites as a function of the altitude, during the full considered trajectory,
for both considered antenna patterns, assuming 10 dBi receiver antenna gain. From this plot we cannot conclude for
which pattern the availability is higher, however we can see that when assuming a 3D pattern it happens much more
often to lose a satellite for a short time and track it again shortly after, compared to the case when we are using a 2D
pattern. This is expected as in the case of a 3D pattern the gain might change a lot for different azimuth, causing more
often the sudden lose or track of a satellite. Another way to distinguish the impact of averaging the antenna pattern on
the availability is to plot the total number of observations available, as shown in Figure 8. From this plot we can see
that depending on the portion the availability might be higher for either 3D or 2D pattern. However, it is interesting to
notice that in case of 2D pattern after 200 000 km altitude, the number of satellites available has a lower standard
deviation compared to when using a 3D pattern. Hence, in case of 3D pattern there are more outages, as well as, there
are more peaks in the number of satellites. The precise statistics on the availability are reported in Table 3, where more
concretely the two assumed patterns are compared. The statistics are computed for the entire MTO as well as for three
representative portions, as defined in Table 2.
Portion Definition
Time interval (min)
Distance from Earth interval (km)
Portion 1
Portion 2
5-505
1446-1946
6530-100506
200000-237891
Table 2 Definition of representative portions.
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Portion 3
6421-6921
391219-398317
Figure 7 Available GPS L1 signals, during the assumed MTO, for a 10 dBi receiver gain pattern.
In Figure 9, the GDOP is shown. From this figure, it is difficult to assess which between the 3D or the 2D pattern
leads to the worst GDOP; in some portion the 3D pattern has higher GDOP and in some other portions lower GDOP.
However, at the end of the MTO for the 2D pattern the GDOP reaches its maximal value of 2.5e6, while, when we
assume a 3D pattern, there are much more important peaks with a maximum of about 4.7e7, as shown in Figure 9.
Figure 8 Number of available observations.
Figure 9 GDOP in log scale, during the assumed MTO.
Another important information related to the availability is the continuity of signal si as time interval duration when
the signal si is continuously available. This information is very useful for the acquisition module design. The continuity
can be expressed in terms of percentage of continuous time intervals with a duration equal to or longer than a time
interval T. This is shown in Figure 10 for both assumed patterns, for the full trajectory and the defined portions.
Figure 10 Percentage of continuous time intervals, a point of each curve identifies a time interval duration T (on the x-
axis) and the corresponding percentage (on the y-axis) of continuous time intervals equal or longer than T.
In Table 3 the statistics of the availability are reported for the three representative 500 min portions (as defined in
Table 2) of the trajectory and during the full trajectory, for both assumed patterns. As we can see in the first portion,
the availability is slightly higher when assuming a 2D pattern, however in the second and third portion assuming 3D
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pattern would result in slightly higher availability. Finally, as expected, averaging the pattern results in different
availability, although not significant; for instance considering the full trajectory the difference is less than 1% or less
than 0.2 satellites observed in average per epoch (1 second).
Portion1
Average
availability (%)
Mean
Max
Min
L1 antenna pattern 3D
Portion2 Portion3
60.21
25.19
16.58
All
MTO
24.61
18.67
26
11
7.81
11
5
5.14
9
1
7.63
26
1
L1 antenna pattern 2D
Portion2 Portion3
Portion1
62.63
22.49
14.06
All
MTO
24.01
19.42
26
11
6.97
9
4
4.36
7
2
7.44
26
1
Table 3 Statistics for the number of available observations.
C.
Ranging error
This section focuses on the impact of the two different assumed antenna patterns on the pseudorange error. The
pseudorange error is the sum of different errors affecting the measurement as we can see from Table 1. In this study,
all the pseudorange errors were modelled as in our previous study (9). However, it is relevant to mention here that,
assuming the use of code based observations (as for the SANAG receiver), the code tracking thermal noise error is the
main error affecting the pseudorange whose standard deviation can change over time due to a different signal power
level at the receiver position, and then due to a different transmitter’s antenna pattern model. Hence, for the two
considered antenna patterns, for simplicity, we only report the simulated code tracking thermal noise.
According to (16), for binary phase shift keying (BPSK) modulations, the thermal noise code tracking jitter is:
σDLL ≅ √
𝐵𝑛
2
𝜋𝑅𝑐
𝐷 [1 +
],𝐷 ≥
𝐶
𝐶
𝐵𝑓𝑒
(2 − 𝐷)
2
𝑇
𝑁0
𝑁0
(2)
with 𝐵𝑛 the code loop noise bandwidth (in Hz), 𝐷 the early-to-late correlator spacing (in chips), 𝑇 the coherent
integration time (in s), 𝐵𝑓𝑒 the double-sided front-end bandwidth (in Hz), 𝑅𝑐 = 1/𝑇𝑐 the chipping rate (in chips/s) and
𝐶/𝑁0 the carrier to noise ratio (in dB-Hz). In this work we assumed: 𝐵𝑛 = 0.1 Hz and 𝑇 = 20 ms for both signals; 𝐷 =
0.25 chips for L1 and 𝐷 = 0.8 chips for L5; 𝐵𝑓𝑒 = 13 MHz for L1 and 𝐵𝑓𝑒 = 40 MHz for L5.
Figure 11 shows the mean 𝜎𝑡𝐷𝐿𝐿 , assuming 10 dBi receiver antenna gain. The mean code tracking thermal noise
error is computed among the signals available at the same instant. Looking at this plot is difficult to state which pattern
(3D or 2D) would result in higher thermal noise, and thus ranging error. For example, at the beginning of the MTO,
up to 60 000 km, using the 3D pattern brings higher thermal noise, but, from 75 000 km to 160 000 km using the 2D
pattern results in higher thermal noise. Finally, due to different experienced 𝐶/𝑁0, using 2D or 3D pattern would result
in different 𝜎𝑡𝐷𝐿𝐿 for each pseudorange; however when computing the mean 𝜎𝑡𝐷𝐿𝐿 the difference becomes small.
Figure 11 Mean code tracking thermal noise range error, for the assumed MTO. The thermal noise range error is computed for
each available PRN and the mean is plotted.
D.
Navigation solution
In Table 4 the navigation solution statistics are shown, while Figure 12 illustrates the zoomed navigation solution
for the two assumed patterns. Looking at Figure 12 we can see that the performance is quite similar for both cases,
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however, according to Table 4 using a 2D pattern would result in better navigation solution. This is only true because
of few points where the GDOP affects considerably the navigation solution in case of 3D pattern. Indeed, as already
mentioned in section III.B the available number of observations when using a 3D pattern, has a higher standard
deviation, leading to larger number of outages (as well as more peaks in number of satellites), which causes a very bad
relative geometry between the transmitters and the receiver. Indeed, the maximal GDOP for the 2D pattern is 2.5×106
while for 3D pattern 4.7×107 and the GDOP has more peaks in case of 3D pattern compared to 2D one.
Figure 12 Single-point least squares estimation error, for the assumed MTO.
3D pattern
2D pattern
Position error (km)
Velocity error (km/s)
Mean
Std
Max
Mean
Std
Max
134.74
6445.39
381775.90
2.60
205.99
69692.35
4.25
40.84
19249.06
0.20
13.74
7249.07
Table 4 Statistics of the single-point least squares estimation error.
IV.
GPS L1/L5 receiver: L5 pattern assumption
Lack of information on the L5 antenna pattern has always been a big struggle for engineers, willing to design a
space-borne receiver. The only information is that the main lobe is wider than the one of L1 pattern, (2). In this sense
it can be approximated from the L1 pattern using the ICD document (4) and (13), but, on the other side, the side lobes
approximation is less accurate. Indeed, while the main lobe has almost the same gain values for different azimuth of
the same satellite, and furthermore for different satellites, the side lobes do not. Hence, this makes the L5 pattern
assumption, especially for side lobes very important, as it might, therefore, affect the results. In this context, here, we
want to evaluate the effect of assuming the different L5 antenna patterns defined in section II.C on the GNSS
availability and the GNSS-based navigation performance.
A.
Received Signal power levels
Figure 13 shows the fourth highest received power levels of the GPS L5Q signals at the receiver position as a
function of the altitude, during the full considered trajectory, when assuming a 0 dBi receiver antenna gain.
Figure 13 Fourth highest received power level of GPS L5 signals, during the considered MTO, for a 0 dBi receiver gain pattern.
From this figure we can see that when assuming L5 pattern 1 the signal power level is never smaller than -169
dBm; when assuming L5 pattern 2 it goes below -169 dBm at a very few points at the very end of the trajectory; and
when assuming L5 pattern 3 the received signal power level is below -169 dBm for a considerable amount of time
after 310 000 km altitude. Moreover, most of the time, the highest fourth signal power has higher values in case of L5
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pattern 1, while it has lower values in case of L5 pattern 3 for most of the time. In the next session we will investigate
the signals’ availability, in order see the impact of the three assumed L5 patterns.
B.
Availability and GDOP
In this case, according to the dual frequency L1/L5 capabilities of the SANAG receiver, the i-th GPS observation is
considered available at time 𝑡𝑘 if:
1.
The corresponding i-th GPS L1 C/A signal is acquired (according to the conditions specified in section III.B).
2.
In case the i-th GPS signal does not cross the ionosphere: the corresponding i-th GPS L5Q signal is tracked (i.e.
the corresponding signal power is equal to or stronger than 12 dB-Hz).
3.
If the i-th GPS signal crosses the ionosphere: the corresponding i-th GPS L5Q and GPS L1 C/A signals are both
tracked (i.e. the corresponding power is at least 12 dB-Hz for GPS L5Q and 15 dB-Hz for GPS L1 C/A).
In Figure 14 is illustrated the availability of satellites as a function of the altitude, for the three considered L5
antenna patterns, assuming 10 dBi receiver antenna gain. The total number of available observations at each altitude
is shown in Figure 15, where we can see more clearly that at the end of the trajectory the availability is lower when
assuming L5 pattern 3. Indeed, in this case at the end of the MTO the number of available observations is often less
than 4, which is never the case for L5 pattern 1 and L5 pattern 2). Table 5 reports the precise information on the
availability, such as percentage of availability, mean, max and min number of available satellites, and percentage of
time when more than four satellites are available. The GDOP for the three considered L5 patterns is shown in Figure
16. What is interesting to notice is that up to 250 000 km the GDOP has similar values for the three considered cases.
As displayed in Figure 16 in case of L5 pattern 3 the GDOP reaches very often larger values than 10 000.
Figure 14 Available GPS L5 signals, during the assumed MTO, for a 10 dBi receiver gain pattern.
Figure 15 Number of available observations.
Figure 16 GDOP in log scale, during the assumed MTO.
Meanwhile, the continuity in terms of percentage of continuous time intervals with a duration equal to or longer
than a time interval T, is shown in Figure 17 for the full MTO and the three considered portions. We can see that for
the first portion the continuity curves are almost the same for the three considered L5 patterns, while for the two other
portions and the entire trajectory they are different.
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Figure 17 Percentage of continuous time intervals, a point of each curve identifies a time interval duration T (on the x-
axis) and the corresponding percentage (on the y-axis) of continuous time intervals equal or longer than T.
From Table 5 we can conclude that the third assumed L5 (i.e. L5 pattern 3) patterns leads to worse availability
mainly in the third considered portion compared to L5 pattern 1 and L5 pattern 2. Indeed, in this portion, the availability
for L5 pattern 3 is almost two times less than the one for the other two patterns, which is reflected in the mean number
of satellites, respectively 7.13, 6.21 and 3.74 for the first, second and third assumed L5 patterns. For the other two
portions the statistics are quite similar; for instance less than one satellites is the difference of the mean number of
observations among the three considered assumptions. The smaller availability during portion 3 is probably related
with the big drop in the gain pattern of L5 pattern 3 between 35° and 50° elevation, as shown in Figure 5.
Portion
Average
availability (%)
Mean
Max
Min
>= 4 sats (%)
L5 antenna pattern 1
1
2
3
MTO
63.42 33.70 22.99 32.17
19.66
26
11
100
L5 antenna pattern 2
1
2
3
MTO
63.42 32.00 20.05 30.16
L5 antenna pattern 3
1
2
3
MTO
63.35 35.03 12.06 27.65
10.45
7.13
9.97
19.66
9.92
6.21
9.35
19.64
13
10
26
26
12
9
26
26
8
5
5
11
8
4
4
11
100
100
100
100
100
100
100
100
Table 5 Statistics for the number of available observations.
10.86
13
8
100
3.74
7
1
62.52
8.57
26
1
88.15
C.
Ranging error
When the line of sight vector between the transmitter and the receiver crosses the ionosphere layer, the observation
is available if both L1 and L5 signals are tracked. In this case we will have two measurements 𝜌𝐿1 and 𝜌𝐿5 affected by
different errors (see Table 1). The ionospheric delay can be mitigated using an ionosphere-free combination, (17):
1
(3 )
2
2
𝜎𝑖𝑓
=
[𝜔2
𝜎 2 + 𝜎𝐿5
]
(𝜔𝐿1,𝐿5 − 1)2 𝐿1,𝐿5 𝐿1
where 𝜔𝐿1,𝐿5 = 𝑓𝐿1 2 /𝑓𝐿5 2 , and 𝜎2𝐿1 and 𝜎2𝐿5 are respectively the variances of 𝜌𝐿1 and 𝜌𝐿5 . Hence, the variance of the
2
ionosphere-free pseudorange error 𝜎𝑖𝑓
is larger than 𝜎2𝐿1 and 𝜎2𝐿5 (without considering the ionospheric variance). Thus,
if the ranging error due to the ionosphere delay is reduced using equation (3), the other error sources are amplified.
Figure 18 shows the mean 𝜎𝑡𝐷𝐿𝐿 for the three considered L5 patterns, assuming 10 dBi receiver antenna gain.
Figure 18 Mean code tracking thermal noise range error. The 𝜎𝑡𝐷𝐿𝐿 is computed for each available PRN and the mean is plotted.
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We can see that the 𝜎𝑡𝐷𝐿𝐿 in case we assume L5 pattern 1 is generally smaller than the other two cases. Indeed, the
mean over all trajectory of the mean 𝜎𝑡𝐷𝐿𝐿 is 0.57 m, 0.68 m and 0.66 m when assuming L5 pattern 1, L5 pattern 2
and L5 pattern 3, respectively. Moreover, we can see that at the end of the trajectory the 𝜎𝑡𝐷𝐿𝐿 in case we assume L5
pattern 3 has a bigger standard deviation. As shown, at the end of the trajectory (from 330 000 km until the end), it is
when we assume the third L5 pattern that results in higher and smaller values of mean σtDLL .
D.
Navigation solution
Figure 19 shows the zoomed navigation solution for the three assumed L5 patterns. As we can see, when we assume
L5 pattern 1 the navigation solution has smaller error, whilst the biggest error happens when we assume L5 pattern 3.
Indeed, this was expected as the number of satellites is bigger for L5 pattern 1 leading to better GDOP values. In case
of L5 pattern 3 the relative geometry between the receiver and the transmitters is quite bad, reaching quite often values
above 10 000 at the end of trajectory (after 330 000 km), as explained in IV.B. Moreover, also the GDOP of L5 pattern
2 has more peaks compared to the one of L5 pattern 1, hence leading to worse positioning accuracy. In Table 6 are
shown the quantitative statistics of the navigation solution. Again these statistics confirm what is shown in Figure 19.
Figure 19 Single-point least squares estimation error, for the assumed MTO.
L5 pattern 1
L5 pattern 2
L5 pattern 3
V.
Position error (km)
Velocity error (km/s)
Mean
Std
Max
Mean
Std
Max
0.355
0.461
6.615
0.102
0.131
2.192
0.587
0.928
22.163
0.202
0.333
9.775
1.016
7.725
2123.291
0.269
7.430
4183.441
Table 6 Statistics of the single-point least squares estimation error.
Number of available GNSS observations for each receiver antenna elevation
In our previous studies, as well as here, we assumed to be able to always have 10 dB antenna gain at the receiver,
which could be provided by one steerable antenna or more than one antenna placed on different faces of the spacecraft.
In this section, assuming the L5 pattern 3 (which is the most conservative among the considered antenna patterns
defined in section II.C), we analyzed the elevations, at which the available signals are received by the receiver’s
antenna. Figure 21 displays the number of available GNSS observations for each receiver’s antenna elevation. Figure
22 displays the elevation values at which the four strongest signals are available at the receiver’s antenna. While Figure
23 shows the PRNs available for each elevation of the receiver’s antenna and for each altitude of the considered
trajectory. Figure 24 finally reports a histogram of the available observations for each receiver’s antenna elevation.
These information can be very useful in order to design an antenna system capable of providing a certain required gain
(e.g. 10 dB in our assumptions), at least at the receiver elevations where a minimum of four signals can be processed.
Figure 20 Earth pointing attitude configuration, assumed for the receiver.
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It is important to highlight that in our simulations, as illustrated in Figure 20, the spacecraft that hosts the receiver
is assumed to be earth pointing, with the same body axis always directed towards the earth’s center. Also note that here
the elevation is expressed as the semi aperture of a cone, thus always positive.
Figure 22 Elevation values at which the four strongest
signals are available at the receiver’s antenna.
Figure 21 Number of available observations for each receiver
antenna elevation, for each altitude of the considered MTO.
Figure 23 PRNs available for each receiver’s antenna elevation, for each altitude of the considered MTO.
Figure 24 Histogram of the available observations for each receiver’s antenna elevation, during the whole considered MTO.
From Figure 21, we can see that above 300 000 km all the signals are received between 0° and 5°, while between
150 000 and approximately 300 000 km there is no available signal at elevations higher than 10°, which means that
above 150 000 km, a high directivity antenna pointing towards the earth’s center, may be sufficient to process most of
the signals. From Figure 22, we see that above 100 000 km, the four strongest signals are always received within 10°
elevation and at the same time below the same altitude their powers never go below -159 dBm, which is the SANAG
receiver sensitivity. Therefore, in principle 10 dB gain from 0° to 10° elevation would already allow the processing of
at least four signals. Between the GPS constellation altitude and 150 000 km, most of the signals are received within
an elevation interval of 10°-30°; however as their power is already on average significantly higher than the signals
received at higher altitudes (see for instance Figure 6), lower gain for the receiver’s antenna may still allow the tracking
of a minimum number of signals. Finally we also see that, as expected, only for altitudes below the GPS constellation
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there are signals reaching the receiver between 90° and 180°. From Figure 22 and Figure 24 we can observe that in
general, most of the available signals are received between 1.5° and 7°; indeed most of the time of the mission is spent
above 200 000 km, where the spacecraft’s velocity decreases progressively until it reaches the minimum at the apogee.
VI.
Conclusions
In this paper, we investigated the effects of different GNSS transmitters’ antenna patterns assumptions on the
transmitted signals’ characteristics, in a trajectory that starts from LEO, crossing the SSV and ends at the Moon
altitude. In particular the analysis was carried out by simulating signal power levels, availability, GDOP, code tracking
jitter and finally the single-epoch least squares position and velocity solutions, assuming the characteristics of the
SANAG spaceborne receiver proof-of-concept, under development in our laboratory.
For the GPS antenna patterns of Block IIR and IIR-M L1 we used detailed information, available in the literature,
while we were forced to make an assumption (although conservative) on the L1 IIF Block transmitters’ patterns. We
analyzed the GPS L1 availability in case the assumed patterns vary as function of elevation and azimuth (3D pattern)
and in case they vary only as function of the elevation (2D pattern). The results show that the number of satellites
variation is smaller when a 2D pattern is used. The higher variation in number of observations when using a pattern
changing in both elevation and azimuth causes very big GDOP peaks, reflected in the navigation solution. When the
spacecraft is orbiting along the SSV regions only 0.23 satellites per epoch (1s) in average is the difference in availability
(with a maximum of 2 satellites), while above 200 000 km it is 1.24 satellites per epoch (maximum 5 satellites). Hence,
the approximation of the antenna gain pattern does not impact the SSV, but mainly very high altitudes.
We investigated the effect on the SANAG receiver navigation performance for different assumptions on the L5
transmitters’ patterns, for which little information is currently available. From our results we can state that the
assumption on the L5 antenna pattern has a small influence on the availability (consequently on the GDOP) for the
two regions of the SSV where there are always more than 7 satellites. However, it is obvious that at very high altitudes,
such as the last part of the assumed MTO, the number of available observations is more sensitive to the L5 pattern
assumptions; the same happens to the GDOP which worsens a lot for the third assumed pattern resulting in a big least
squares solution error. We can conclude that it is important to model the transmitters‘ antenna patterns dependency on
the azimuth (since using a constant average value is not conservative) and also that for the GPS L5 signal, as expected,
assuming a pattern with lower gain values for the side lobes is conservative.
Finally, we see that, during the full mission, 62.8% of the signals are received between 1.5° and 7° elevation and
above 150 000 km only within 10° elevation. Below 150 000 km many signals reach the receiver’s antenna between
10° and 30° elevation, but fortunately with stronger power and without the need of a strong receiver’s antenna gain.
Acknowledgments
The authors would like to sincerely thank the support from the Swiss Space Office of the State Secretariat for
Education, Research and innovation of the Swiss Confederation (SERI/SSO) who is financing this development.
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