Radionavigation Alternatives for US Army
Ground Forces in GPS Denied Environments
Mark S. Asher, Stephen J. Stafford, Robert J. Bamberger, Aaron Q. Rogers, David Scheidt, Robert Chalmers
The Johns Hopkins University Applied Physics Laboratory
BIOGRAPHIES
engineering and advanced concept solutions for
government sponsors, and is currently the Deputy Mission
System Engineer on the Multi-Mission Bus
Demonstration (MBD) flight program. Prior to joining
APL, Mr. Rogers held technical and management
positions with AeroAstro and Lockheed Martin. He has
authored multiple publications relating to small spacecraft
systems and enabling technologies. He has a B.S. in
Aeronautics & Astronautics from MIT.
Mark S. Asher is a member of the Johns Hopkins APL
Principal Staff working as a radionavigation concept
development engineer. He has worked for 25 years in
radio and inertial navigation analysis, including GPS and
inertial data fusion for analysis of Trident missile tests,
and Doppler-based geolocation methods. He designed the
algorithms for the orbit determination Kalman filter for
the TIMED spacecraft. Mr. Asher holds a B.S. in
mechanical engineering and an M.S. in engineering
mechanics, both from Virginia Tech.
David H. Scheidt is a member of the Principal Staff at the
Johns Hopkins University Applied Physics Laboratory.
He holds a B.S. in Computer Engineering from Case
Western Reserve University. He has been working with
distributed control and agent-based systems since 1988
and unmanned air systems since 2003.
Stephen J. Stafford is a member of the Senior Professional
Staff at the Johns Hopkins University Applied Physics
Laboratory. He holds B.S. and M.S. degrees in electrical
engineering from the University of Maryland and the
University of California, Berkeley, respectively. His
work includes the development of signal processing
algorithms for radionavigation, with a focus on weaksignal GPS detection.
Robert W. Chalmers leads the autonomy simulation and
analysis efforts for the Johns Hopkins University Applied
Physics Laboratory where he is a member of the Senior
Professional Staff. A member of the first class of
Electrical Engineering graduates at Florida State
University he also holds an M.S. in Applied Physics from
Johns Hopkins University.
Robert J. Bamberger is a member of the Principal Staff at
the Johns Hopkins University Applied Physics
Laboratory. He holds a B.S. in Electrical Engineering
from the University of Maryland, and an M.S. in Applied
Physics from the Johns Hopkins University.
Mr.
Bamberger has over 25 years experience in
communications systems development, aircraft systems
integration, and sensor payload development. He has been
working with autonomous unmanned aerial systems
(UASs) since 2001, leading a number of UAS-related
R&D projects, including development of architectures
and communications standards for small UASs, unique
autopilot control methodologies, RF sensing and
geolocation from small UASs, and acoustic-based sensing
for airspace deconfliction and gunshot detection. He has
authored multiple publications on technologies related to
small UASs.
ABSTRACT
This study, convened and funded by the U.S. Army,
analyzes the tradeoffs involved in designing a local or
theater GPS replacement system.
Our operating
assumption was that such a system would not be
constrained by legacy GPS architecture, equipment, or
waveforms. It could therefore employ more efficient Z4
cyclostationary ranging codes and reap the benefits of
two-way measurement and communication including:
elimination of User Device (UD) clock error, dramatic
reduction of the UD infill bandwidth by computing the
UD position on the Reference Device (RD), code
ambiguity elimination, and (perhaps) the use of cognitive
radio techniques to resist jamming. The existence of
robust cellular telephony with compact handsets in very
crowded spectrum shared by many users testifies to the
viability of this approach.
Aaron Q. Rogers is a Space Systems Engineer and
Supervisor of the Military & Intelligence Systems section
at The Johns Hopkins University Applied Physics
Laboratory. While with APL he has led several multidisciplinary teams focused on providing system
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
508
We studied a physical architecture in which the RDs were
carried by Unmanned Aircraft Systems (UASs). It was
shown that such architectures are robust to jamming, and
that the Horizontal Dilution Of Precision (HDOP)
benefits greatly from two-way ranging.
We also
described how the use of Mission Level Autonomy
(MLA) by a resource limited UAS constellation provides
robust location services to dynamically changing ground
operations in a near optimal way.
architecture tradeoff study between receive-only,
transmit-only, and two-way transponder PNT solutions
will show that a two-way system has very desirable
properties relative to a one-way system, including:
1. The UD clock is eliminated, thus a two-way system
requires one fewer RD to achieve a solution.
2. The location of the UD can be computed on the RD
and returned as a very small data payload. This
eliminates the need to transmit lengthy ephemeris
messages or Position, Velocity, and Time (PVT)
information from the RDs. PVT data can be rapidly
changing for UAS-borne RDs, requiring high data
rates, which makes the RD to UD link more vulnerable
to jamming.
3. The range ambiguity associated with a cyclostationary
code can be removed using data encoded as code shifts
in packets subsequent to the initial acquisition3.
4. A two-way system can use Cognitive Radio (CR)
techniques to resist jamming and spoofing. The
benefits gained from CR, if any, are highly scenario
dependent4.
5. Two-way data transmission is possible.
6. A satellite-borne RD in a two-way system is capable
of single-ball, single-ping geolocation of a static UD,
if two-way Doppler measurements are employed in
addition to two-way ranging. This is important
because the RF signal may be able to penetrate foliage
or structures for only a brief time during the satellite
pass.
For operations over very large areas of denied airspace,
we considered geolocation of static UDs from two-way
satellite links which are consistent with specifications
developed in the Operationally Responsive Satellite
(ORS) program. Two-way satellite links that can measure
Doppler as well as range enable single-satellite, singleepoch geolocation. This reduces the number of satellites
required in the constellation by about a factor of two. A
notional satellite design consistent with launch from an
SSBN submarine was presented.
I. INTRODUCTION
The U.S. Army ground forces are presented with
operational scenarios that present unique challenges in
accessing Position, Navigation, and Timing (PNT)
information in a GPS-denied environment. The most
stressing case is that of the infantryman, who must locate
himself with a small, low power receiver, which must
produce a solution quickly from a “cold start” and be very
robust to countermeasures, extreme Radio Frequency
(RF), and battlefield environments. Operation in urban
warfare environments will have to contend with high
multipath and signal blockage, but may be required to
determine position very precisely. Any infrastructure
required to support this capability must be rapidly
deployable. In the following we will speak of user
devices (UDs) and reference devices (RDs). The UD,
analogous to a GPS receiver, tells the user where he is.
The RD is analogous to a GPS satellite. It knows its
location by some means, and supports a one or two-way
RF ranging link to the UD. Stationary RDs can be at
surveyed positions, and Aerostat or Unmanned Aircraft
System (UAS)-borne RDs might use GPS or Precision
Terrain Aided Navigation1 (PTAN).
Our signal
architecture trade space included:
Two physical architectures will be studied: a UAS or
aerostat based approach and a rapidly deployable satellite
constellation. These physical architectures correspond to
ground operations in scenarios ranging from complete air
superiority to completely denied airspace.
II. GPS: THE INCUMBENT
The Global Positioning System5 has the desirable features
that:
1. It has global reach and requires no infrastructure that
must be deployed prior to operation.
2. It does not require receivers to have highly accurate
clocks, e.g. atomic clocks, !"#$%%&'()$*+&,*-!(%(+(."&
./&+0$&12345*.'$&(%&6$("7&,++$89+$'.
3. The modern GPS signal can be locally jammed by
friendly forces to deny service to the enemy.
4. Relatively low Doppler presented by the Medium
Earth Orbit (MEO) GPS constellation and the presence
of the civilian C/A code, make the signal easy to
acquire in an unjammed environment.
• RD transmits and UD receives (like GPS)
• UD transmits and RD receives (as in some tagging
systems)
• RD and UD both transmit and receive to produce
range measurements (as in the Nanotron2 system)
Architectures and concepts explored to solve this problem
were not constrained by the existing waveforms and
frequencies, the existing one-way space to ground link, or
compatibility with existing equipment. The signal
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
Regarding (1): The problem of providing global PNT
services is well studied by the designers of GPS. The fact
that all subsequent Global Navigation Satellite Systems
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(GNSS) generally resemble GPS shows that its basic
architecture is nearly optimal. We therefore focus on the
problem of rapidly deployable, less costly local or theater
systems that can be superior to GPS in accuracy, jamresistance or other properties in its restricted area, which
in the case of a Low Earth Orbit (LEO) constellation
might be a whole band of latitude.
contains information on the transmitter location and
velocity (APPENDIX B). Doppler measurements from a
LEO satellite are more sensitive to UD position than a
MEO (GPS) due to the faster line-of sight rotation rate.
The UD oscillator rate error must be estimated or
removed by using Frequency Difference Of Arrival
(FDOA) measurements. The RDs can measure the
jamming environment and instruct the UD to adapt its
ranging code transmission to mitigate the jamming using
the same communication link that serves the PNT
solution.
Regarding (2): The retention of this feature is highly
desirable, so we have focused on the use of
cyclostationary codes.
Regarding (4): If a LEO physical architecture is used, the
Doppler is much higher, as is the sensitivity of the
Doppler with respect to position. Although this presents a
difficulty for signal acquisition, it permits precise
geolocation from a single measurement if a two-way
system is employed with both range and Doppler
measurements.
One-way navigation signal: UD Receives
Inverting the direction of the transmission results in a
more traditional GPS pseudolite system in which the UDs
compute their position locally. The time-aligned airborne
or LEO RD transmitters can still monitor the jamming
environment and use “dirty paper” techniques6 to adapt
the transmission from an array of pre-planned
possibilities, each of which is tested by the receiver. Like
GPS, the ranging signal from the RD pseudolites would
contain information on the position and velocity of the
transmit platforms. For LEO RDs, this information could
be supplied at a low rate in a form similar to the GPS
message ephemeris.
For airborne pseudolites, the
position and velocity information would have to be served
out more rapidly because of unknown accelerations. The
presence of high rate modulation reduces jam resistance
because it limits the coherent integration time. The
coherent integration time is limited to symbol period, e.g.
20 ms in GPS.
III. SIGNAL ARCHITECTURES
One-way navigation signal: UD Transmits
The simplest UD solution is a device that transmits
ranging codes and places the majority of the signal
processing and navigation calculations on a set of
spatially distributed RDs. The RDs (1) receive the
ranging signals from each UD, (2) do the signal
processing to form a pseudorange (and optionally
Doppler) measurement, (3) share the measurements with
other RDs, (4) compute the PVT of the UD, and (5) serve
PVT back on a communication link. The ranging link
must be robust to jamming. The communication link
could be data riding as modulation on the ranging link or
a separate RF link which is itself robust to jamming. The
RDs can be fixed in surveyed locations, carried in aircraft,
or a LEO spacecraft. In each case, the RD must know its
position and velocity. Fixed RDs can be placed at
fiduciary locations determined by satellite imagery. RD
platforms on aircraft or UASs might use GPS in their
navigation solution since they can be much more immune
to jamming due to their height, use of CRPA antennas,
beam forming, and Inertial Measurement Units (IMUs).
Or they may derive their solution from other means such
as PTAN1. If the RD resides on a LEO, its orbit can be
determined out-of-theater by GPS or other means. The
RDs must be time and frequency aligned so that the
pseudoranges and Doppler measurements are formed with
respect to a common basis. The UD oscillator need not be
particularly stable. The RD solution must therefore
estimate the UD clock error, or remove it by using Time
Difference Of Arrival (TDOA) measurements. A heightconstrained geolocation will require a minimum of three
RDs in a non-degenerate spatial arrangement. More RDs
result in an overdetermined solution that allows for
measurement integrity monitoring. For RDs on aircraft
and LEO satellites, the frequency shift of the signal also
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
The minimum number of RDs required for a heightconstrained PNT solution in either of the one-way ranging
link cases is three. The PNT solution is obtained from the
pseudorange and Doppler measurements and contains two
position parameters (latitude and longitude) and their
rates, as well as epoch clock and frequency error. Thus
there are six equations in six unknowns, which are
solvable if the geometry is adequate.
Two-way navigation signal: UD, RD Transponders
The number of measurements, and hence reference
platforms, can be reduced to two if a two-way ranging
link is used. Moreover, the reference platforms no longer
require precise clock or frequency alignment. This can be
a huge advantage in reducing the cost of the systems.
Since the link is now bidirectional, the data payload for
realizing a cognitive radio solution can ride as modulation
on the ranging links. If the dynamic range of the link
losses between RD and UD transceivers is small relative
to the cross-correlation of the links (accomplished with a
combination of FDMA, CDMA, and TDMA), a single
radio can two-way multiplex a large number of UD
transceivers. One shortcoming of the two-way solution is
that since the links must operate in both directions, if
either direction is jammed, the system fails. For best
510
performance, the links must be balanced in the sense of
equal post-correlation SNR. If the RDs have a high
power transmission relative to the UDs, they must make
up for it in processing gain to achieve a balanced link.
The Karr patent3 describes techniques to acquire the
ranging signals, to balance the links, to resolve
ambiguities, to transfer data, and to have many UDs and
RDs (tags and locators) cohabitating the same spectrum.
Waveform Design
Signal bandwidth is required to enable ranging. GPS
accomplishes this with Direct Sequence Spread Spectrum
(DSSS) Biphase Shift Keyed (BPSK) signals. There are
two signals in phase-quadrature on the GPS L1 frequency:
the C/A code with a 1.023 MHz chipping rate and 1
millisecond code length (~300 km), and the encrypted
P(Y) code with a 10.23 MHz chipping rate and a 37 week
period, short cycled to one week. The advantages of a
high chipping rate are more precise ranging, better
multipath rejection, and higher jam resistance. The
advantages of a long code are a longer ambiguity
distance, and more simultaneous users. In addition, long
codes have spectral lines that are smaller and closer
together—in the limit they approach a continuous spectral
density with a sinc-squared envelope. The P(Y) code is
long enough that it can be regarded as having no
ambiguities since they are separated by an enormous
distance (one light-week), and the spectrum is essentially
continuous (the lines are separated by about two microHertz). The continuous spectrum makes the signal harder
to distinguish from thermal noise using a spectrum
analyzer.
Figure 1: Ambiguities for 1023 length code at 10.23 MHz
chipping rate
The obstacle with a very long code is acquisition, which
requires fairly precise time information to limit the lag
search space. Even if the code is in fact cyclostationary,
for code lengths much longer than ~106 it becomes
computationally infeasible to utilize frequency-domain
techniques that exploit the code repetition.
The
magnitude of the search space is largely driven by the
accuracy of the UD’s local oscillator. A 1 ppm UD clock
will drift about 1000 P(Y) chips in 100 seconds, which is
about equal to the entire C/A code space. Fortunately,
modern, highly optimized GPS receivers can test many
thousands of lag and Doppler hypotheses simultaneously.
These architectures make the long-code acquisition
problem much more tenable, except for unaided receivers
that must be dormant for long periods of time (hours).
The problem with using a short codes with a high
chipping rate is illustrated in Figure 1, which shows a
TDOA system in a planar arrangement of three RDs
(black circles) in an equilateral triangle with 100 km
edges and the UD (red circle) at the center. This is an
optimum precision geometry. The code is 1023 chips
long and clocked at 10.23 MHz, corresponding to the C/A
code length with the P(Y) code chipping rate. This results
in an ambiguity length of about 30 km. The dashed lines
have constant TDOA modulo 30 km, and every
intersection is a possible solution. This proliferation of
hypotheses is highly undesirable on the battlefield. As
noted earlier, these ambiguities can be completely
3
removed in a two-way system using the sequel packets .
Since the development of GPS in 1970s, research in the
field of information theory has led to the development of
new sequence families with improved properties for
ranging applications. For radionavigation, the most
pertinent parameters of the DSSS code are the length,
family size, and correlation properties. These parameters
are
explicitly
defined
as
follows:
the
set
! !! ! ! ! ! !! ! ! ! ! !! ! ! !! provides a family of M
sequences, each of each of length N. The autocorrelation
of the sequences is defined as
!
(1)
!!!! !
!! !!! !! ! ! ! ! ! ! !! ! ! !
!!!
where the overbar represents the complex conjugate and
the cross-correlation is
(2)& !!!!!! !
!
!!! !!
! !! ! ! ! ! !! !! ! ! !! ! ! !! ! ! !:&
And, it is required that
(3)&
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
511
!!!! ! !! ! ! !! ! ! !.
The cross-correlation strength of the sequence is
measured in terms of the maximum sidelobe amplitude,
!!"# , given as
The Welch7 lower bound states that, for all codes, the
maximum sidelobe amplitude must satisfy:
(4)&
(5)
!!"# ! !"#! !"# !!!!!! ! !"#!!! !!!! .
The effects of code length, N, were described in the
preceding paragraphs. Family size, M, determines the
number of simultaneous transmitters that can be used.
This parameter is particularly significant for two-way
ranging, in which each UD in a localized region needs a
unique code. The family size monotonically increases
with code length. The correlation strength of the code,
!!"# , is a measure of the interference between
transmitters. This interference is acutely consequential in
ad hoc networks because they are susceptible to the nearfar problem, in which a strong transmitter effectively jams
weaker transmitters.
Code families with improved
correlation sidelobes are able to reject nearby transmitters
more effectively, thus mitigating the near-far problem.
!!"# ! !
!"!!
!"!!
.
Gold codes8, which are binary, serve as the basis of the
C/A code. The C/A code has length 1023, but generally
Gold codes are of length ! ! !! ! ! and family size
! ! ! ! !, for odd ! ! !. The maximum sidelobe for
Gold codes is ! ! !! ! !. Asymptotically, the Gold
code maximum sidelobe power is 3 dB worse than the
Welch bound. Furthermore, there are no binary codes
with a family size on the same order as Gold codes that
also approach the Welch bound. For example, Kasami
codes9 are a binary code family that approaches the Welch
bound; however, their family size is only ! ! (see
Table 1):
Table 1: Comparison of Code Families [Gold8, Kumar9, Boztas10, Tang11, Jiang12]
Family
Gold
Kasami
!! !!Family!!
Length, N
!! ! !
odd p
!! ! !
even p
!! ! !
!
Alphabet
!!!!
Family Size, M
!!!
!!!!
!!!
!!"#
! ! !! ! !
!! !!!
!! !!! !!! ! !!
!!!
!! !!!
! ! ! !!
!! ! !!!!
!! !!!
!! !!!
!! !!Family!!! !
!! !!Family!!
! ! !!
! !! ! !
!! !!! !!! ! !!
!! !!! !!! ! !!
!! !!Family!!
!! ! !
!! !!! !!! ! !!
!!!
!! !!!
!! ! Family!!
! !! ! !
!! !!! !!! ! !!
!! ! !!!!
!! !!!
Superior codes have been discovered using quadriphase
sequences. These new sequences maintain a large family
while approaching the Welch bound—resulting in a 3 dB
improvement over Gold codes10. Quadriphase sequences,
or !! sequences, differ from binary sequences in that they
have a four symbol alphabet: !! !!! !!! ! !! .
These sequences are easily generated using shift registers,
and they are straightforwardly implemented in
transmitters and receivers. There is a rich class of
alternative, near-optimal codes with non-binary
alphabets9; however, they require a significant increase in
bit-depth, e.g. 3-phase codes with alphabet:
!! ! !! !!!! ! ! !!! !!!! . In contrast to other non-binary
codes, quadriphase sequences can be implemented with
single-bit symbol generators in each the in-phase and
quadrature signal components. Additionally, as with the
Gold codes, !! sequences maintain a constant power
envelope for the transmitter.
Several classes of
quadriphase sequences are readily available in the
literature,
including
families
!! !! !! !! !!!and!!
[Boztas10, Tang11, Jiang12]. The properties of these codes
are summarized in Table 1.
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
The discussion above considered the cross-correlation for
codes without Doppler, as is typical of the literature.
Equation (2) can be modified into the following form
!
(6)
!! !!! !! ! ! ! ! !!!"#
!!!!!! !
!!!
where ! is the Doppler separation, normalized as
described below, between the codes.
This, more
generalized, formulation is relevant for the processing
involved herein, since the received signals will often have
differing frequencies due to the various motion of the
transmitters. Figure 2 depicts the worst case crosscorrelation for several Z4 codes and the C/A code. At
zero-Doppler, the performance closely follows the
predicted bounds in Table 1.
512
Figure 3: UAS CONOP
Figure 2: Worst-case code correlation in the presence of
Doppler
In this scenario, UAS-borne RDs range on one or more
UDs, one of which is notionally represented as a convoy
member. The two-way ranging links (orange) also carry
low bandwidth data: geolocation to the UDs, and
(possibly) barometer information to the UASs. The UAS
constellation knows its position via GPS (if equipped with
anti-jam capabilities), a higher layer of UASs, or PTAN.
Because the UD clock is eliminated, the ranging data on
the two-way links is actual geometric range, not
pseudorange. The UASs share the ranging data via robust
communication links (blue), compute the position of each
RD, and infill the solution to the UDs. The solution is
height constrained, either by barometers on the UDs, or
by DTED maps on board the UASs. There are only two
positional variables being solved for (latitude and
longitude) so with favorable geometry, only two UASs
are required. Additionally, with the UASs arranged at the
vertices of symmetric polygons the Horizontal Dilution
Of Precision (HDOP) degrades less rapidly from its best
value. Since no Doppler data is being used in the
solution, the solution is not corrupted by UD motion.
Redundant solutions are always preferred for robustness.
A similar CONOP can be used with aerostats.
As expected, at zero-Doppler the longer codes have better
cross-correlation suppression, and the Z4 codes
outperform the C/A codes by about 3 dB. For Dopplers
away from zero, the performance degrades. This is
unsurprising since the Doppler effectively randomizes the
codes: Random codes tend to perform worse than Z4 or
Gold codes.
Doppler in Equation 6 is normalized such that “1” refers
to an ! that would result in one full-cycle in a single code
epoch. Across frequencies the Z4 code outperforms the
Gold code of the same length, although by a small margin
at some Doppler shifts. The C/A code has an advantage
in this figure since its code set only consists of 32
members. The Z4 code of the same length contains 1025
members.
Note that determining the composite interference from
multiple simultaneous transmitters is not well-represented
by a simple worst-case analysis that multiplies the !!"#
by the number of simultaneous users. The worst-case
results are not representative of the average crosscorrelation: each point on the figure is the single-worst
cross-correlation across all code pairs and all lags. It is
unlikely that multiple codes would each contribute worstcase noise since that would require a pathological
alignment of the lag and Doppler between the sequences.
Additionally, even if the lag and Doppler were to align in
a worst-case mode, the interferer signals would not be
coherent since their relative phase is random.
Link Analysis and Effects of Jamming
A UAS-borne RD is vastly closer to the UD than the GPS
constellation: a slant range of 22 km has a space loss that
is 60 dB less than that of GPS, all else being equal
(receiver antenna gain, frequency, signal bandwidth).
This gives a UAS-borne RD a tremendous potential for
overpowering jamming by brute force. The Effective
Isotropic Radiated Power (EIRP) of GPS is approximately
648 W (acknowledging that the GPS constellation is
running about 3 dB hot with respect to guaranteed power
levels)5. Because of its isoflux antenna all Earth-fixed
GPS users see this EIRP. Under the aforementioned “all
else being equal” conditions, an RD with an EIRP of just
0.6 mW has the same jam resistance as GPS. If the RD
EIRP is instead 6 W, the jammer will have to increase its
power by 40 dB with respect to what was effective for
GPS. This alone may force the enemy to use much higher
power jammers, or deploy many of them much closer to
IV. PHYSICAL ARCHITECTURE 1: UAS OR
BALLOON-BORNE RD’S
General Concept of Operation
The basic concept of operation (CONOP) for a UAS twoway local positioning system is shown in Figure 3:
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
513
the UDs than would be the case for a successful L1 GPS
jamming system. In either case, the number of jammers
with flux capable of jamming a locked-on receiver would
be far less. This would possibly enable CRPA antennas,
even on handheld equipment, to suppress them, e.g. the
Toyon CRPA13.
jammer power. In the above we have assumed that
J 0 >> N 0 . Thus jam resistance is enhanced linearly by
the ratio of integration time to chip time and quadratically
by the ratio of jammer distance to UD-RD distance. For
ground propagation, the factor ! usually results in losses
that grow with increasing d.
Noiselike Jammers
For tone jammers, a frequency excision process can
remove most of the jammer energy by transforming the
digitized time domain data to the frequency domain,
searching for peaks, zeroizing these frequency bins, and
transforming back to the time domain. We will therefore
focus our attention on noiselike jammers.
If the J-RD link is actually ground propagation and the
RD-UD link is freespace, assuming that ! =1 is
conservative. If we define a required post-correlation
SNR we can calculate the ratio of ranging link distance to
jammer distance at which the jammer “burns through” is:
(11)
For simplicity, we assume that the jammer power is
distributed uniformly over the main lobe of the signal so
that an effective single-sided jammer noise spectral
density can be defined:
This is plotted as a function of integration time for various
values of the parameter ! !!! !!!! , assuming a required
SNR of 13 dB, and a chipping rate of 10 MHz. The
results are shown in Figure 4 This shows that even when
the effective power ratio advantage of the jammer with
respect to the RD is 20 dB, if the integration time is 20 ms
or greater the RD can be 4 times as far away from the UD
as the jammer. If the effective powers from the jammer
and RD are essentially equal, the RD can be 63 times
further away than the jammer.
J 0 = J! ,
(7)
where ! is the chipping period and J is the total jammer
power presented to the receiver processing. The jammer
power on the ranging link is given by:
! 1 "$
J = PJ GJRGRJ #
# ,
" 4! d &%
2
(8)
'
!s$
2 ! T $ PT 1
=
#" d &%
SNR* #" ! &% PJ' "
*
where PJ is the jammer transmit cable power, GJR is the
jammer transmit gain in the direction of the ranging link
receiver (UD or RD), GRJ is the ranging link receiver
gain in the jammer direction, ! is a factor representing
losses over and above free space for a ground-based
jammer, and d is the distance from the jammer.
Similarly, the ranging-link carrier power presented to the
receiver (UD or RD) will be
(9)
! 1 "$
C = PT GTRGRT #
" 4! s &%
2
,
where PT is the ranging link transmit power, GTR ( GRT ) is
the ranging link transmitter (receiver) gain in the direction
of the receiver (transmitter), and s is the slant range from
the RD to the UD. The effective post-correlation SNR is
SNR =
(10)
Figure 4: Jammer Burn Through Distance Ratio (Jammer
Freespace Propagation)
The situation becomes even more favorable if the RD-UD
link is free space while the J-UD link is ground
propagation so that ! > 1 .
2CT
J0
!T$ P G G
= 2 # & T TR RT
" ! % PJ GJRGRJ
2
! d$ 1
#" &%
s "
IF SNR and Bit Depth
The Intermediate Frequency (IF) Jammer-to-Signal ratio
is of great interest when greater than unity as it
determines the number of bits required in the analog to
digital converter (A/D) so that the signal does not fall
below the least significant bit (LSB). In the above
analysis the IF (pre-correlation) SNR is:
2
'
!T $ P ! d$ 1
= 2 # & T' # &
" ! % PJ " s % "
where
PT' = PT GTRGRT is
an
effective
,
ranging-link
transmitter power and P = PJ GJRGRJ is an effective
'
J
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
514
(12)
C PT'
=
J PJ'
Since this value is dependent on d, the previous nondimensional charts can no longer be used. Instead, we
will focus on an individual use-case. Solving for the postcorrelation SNR with the Egli model in place:
2
"d # 1
.
$ %
&s' !
If we look at the inverse of this ratio at the burn-through
point, we obtain:
(13)
SNR =
2
! J$
!T$
#" &% =
# & .
C * ! ' SNR* " " %
2CT
J0
2
!T $ P G G ! d$ 1
= 2 # & R RU UR # &
" ! % PJ GJU GUJ " s % "
This ratio IF SNR ratio is plotted in Figure 5 for a
required post-correlation SNR of 13 dB and freespace
propagation (!=1). We see that when 20 ms integration is
used to achieve the post-correlation SNR, the IF SNR is
about 33 dB. At 6 dB per bit, the A/D could thus be as
coarse as six bits.
(17)
'
4
2
!T$ P !d $ #
= 2 # & R' # 2 &
" ! % PJ " s % ( hJ hU )2
,
where
c
4" ' 40 '10 6
= 0.5929 (non-dimensional) .
!=
If we again impose a required SNR, we can solve for the
jammer burn through as
(18)
Egli Ground Propagation Model Applied to the RD to UD
link
We now show that the jam resistance of the RD to UD
link in the case of a ground-based jammer is superior to
the generic analysis above, which assumed free space
propagation for both links. If we use the Egli ground
propagation model14, the jammer power presented to the
UD is:
(14)
2
( hJ hU )2
2
2
.
UAS Use Case
First we will consider a UAS flying at 6000 ft. (1.8 km) at
maximum slant range corresponding to a 30° mask angle.
In this case the UD-RD slant range is 3.8 km. The
minimum jammer distance is plotted in Figure 6 with
hJ = hU = 3m . Note that even when the jammer has a 20
dB effective power advantage, it can be no further away
than 15 m for a 20 ms integration time.
Figure 5: IF C/J at Burn Through
! 40 $
J = PJ GJU GUJ # &
" f %
1 ! PJ' ( hJ hU )
d* = SNR* ! s
2 T PR' ! 2
4
.
d4
where f is in MHz. In this equation, we have replaced the
generic subscript “R” used in Equation (8) with “U” since
we are specifically considering ground-based UDs.
Rearranging:
(15)
40 $
!
J = PJ GJU GUJ #
" c / ! /10 6 &%
! 1 "$
= PJ GJU GUJ #
#
" 4! s &%
2
( hJ hU )2
d4
Figure 6: Minimum Jammer Standoff for Burn Through
(UAS case)
2
.
Static Geometries and HDOP Sensitivity
The effectiveness of the local PNT system is a function of
the quantity of RDs and their geometry relative to the
UDs. A practical and quantitative way measure the
effectiveness is Horizontal Dilution Of Position (HDOP).
The fractional loss with respect to free space is thus:
(16)
" 4" ! 40 !10 6 %
! =$
'&
c
#
2
( hJ hU )2
d2
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
515
The perspective in the figure is of a 2D slice in the x-y
plane. In each case the horizontal distance (in the x-y
plane) between each of the RDs and the UD is L. The RD
height, h, in the z-dimension is not discernible in this
view. Thus, the elevation angle of the RDs as viewed
from the UD is given as
For simplicity, the analysis below will not consider UD
velocity and clock rate terms. The position coordinates
are defined in a local geodetic plane, with x and y
dimensions corresponding to easting and northing, and z
corresponding to altitude. The sensitivity matrix, !, for
the one-way range is given by the standard form,
(19)
!!!! !! !
!
!!
!!!! !! !
!!
For two-directional ranging, the sensitivity matrix is
(APPENDIX A):
!!
!!!!! !! !
!
:
!!!!! !! !
!!
! ! ! ! ! !! ;
!"#$ ! !!!! ! !!!! :
As discussed above, at minimum the one-way ranging
system requires at least 3 RDs and the two-way ranging
system requires at least 2 RDs.
:
The best case HDOP values, presented in Table 2, emerge
at the (0, 0) location in the figures. It is evident that the
two-way system outperforms the one-way system for this
figure-of-merit.
Figure 7 shows three simple formations of 2, 3, and 4
RDs:
Table 2: Best HDOP for Static Formations
# RDs
2
3
4
Figure 7: RD Formations
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
!
The one-way ranging system performs worse than the
two-way system. This is most notable in regions outside
the polygon formed by the RDs—in the area outside the
polygon, the line-of-sight vectors in H become
increasingly parallel. In the one-way ranging system,
which includes a clock state, this parallelization creates
correlation between clock and the ranging states and
reduces positioning performance. The two-way ranging
system excels because it does not have a clock state.
This can be thought of as a direct measurement of twoway range, uncorrupted by clock.
HDOP is then
computed as the sum of the first two diagonal elements of
Q:
(21)
(22)
!
Unconstrained RD Visibility with respect to Elevation
Figure 8 (a) and (b) provide HDOP contour plots for the
case of one-way ranging in the 3 and 4 RD formations,
respectively. Similarly Figure 8 (c) and (d) contain
HDOP contours for 3, and 4 RD formations for two-way
ranging. The images are of the x-y plane at z equal to 0.
Each point on the figures corresponds to the HDOP that
would be observed by a UD at that location. The RD
positions are fixed—the dark rings note their locations.
The units on the axes are in multiples of L, which as
defined above is the horizontal distance between the (0, 0)
position and the RDs. The height of the RDs corresponds
to a 30° elevation angle when viewed from (0, 0). Note
that the scales on the contours differ from image to image.
where !!!! !! ! is the unit-vector pointing from the UD
position, !, to the RD position, !! . !! is given by the
vector !!!!! , and represents an altitude measurement,
e.g. from a barometer. This approach implicitly assumes
that all measurements are of identical quality, including
the altitude measurement.
(20)
!!" ! !"#!!
(23)
!
! ;
!
!
516
One-way
Ranging
!
1.3
1.2
Two-way
Ranging
1.1
.9
.8
(a) HDOP Contours for One-way Ranging System with a 3 RD Static
Formation
(b) HDOP Contours for One-way Ranging System with a 4 RD Static
Formation
(c) HDOP Contours for Two-way Ranging System with a 3 RD Static
Formation
(d) HDOP Contours for Two-way Ranging System with a 4 RD Static
Formation
Figure 8: HDOP Contour Plots
One great advantage of the two-way ranging architecture
is that it is possible to obtain a solution with only two
platforms, although there is a two-point ambiguity. This
ambiguity would be rapidly resolved by constellation
motion. The HDOP contours for this case are shown in
Figure 9. Note that unlike the three and four RD cases
above, the best HDOP is not found at the (0,0) location.
Figure 10 provides another metric of performance for
each of the formations. This figure presents the worstcase HDOP value for a UD located within a circle
centered at (0, 0). This plot is a measure of the spatial
extent of the HDOP’s stability. From the figure it is
apparent that as the UD travels away from the (0, 0)
position, the one-way system degrades more rapidly than
the two-way one. As discussed above, this is due to the
clock state.
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
Figure 9: HDOP Contours for Two-way Ranging System
with a 2 RD Static Formation
517
Constrained RD Visibility with respect to Elevation
Because the height is explicitly constrained through an
altitude measurement, e.g. by way of a barometer or
terrain map, the optimal HDOP is achieved at the lowest
possible elevation, i.e. h equal to zero. However, line-ofsight obstructions limit the visibility of low elevation
RDs. The following analysis incorporates this obstruction
by masking RDs below a 30° elevation threshold.
The 3 and 4 RD configurations from Figure 7 serve as the
basis of the formations. Figure 11 (a) and (b) are the oneway ranging contours for 3 and 4 RDs, respectively.
Figure 11 (c) and (d) are the similar plots for two-way
ranging. (Note that these plots are on the same scale.)
Figure 10: Worst HDOP within a radius of centrally located
UD
(a) HDOP for various RD elevations, One-way Ranging, 3 RDs
(b) HDOP for various RD elevations, One-way Ranging, 4 RDs
(c) HDOP for various RD elevations, Two-way Ranging, 3 RDs
(d) HDOP for various RD elevations, Two-way Ranging, 4 RDs
Figure 11: Elevation Constrained HDOP
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
518
As the UASs in the tracks pass each other, the HDOP will
be approximated by the 3 and 4 RD formations that were
presented in the figures above. Figure 13 (a) and (b) are
the HDOP under tessellation for the one-way ranging
system; Figure 13 (c) and (d) are for two-way ranging.
Large Area Tesselations
In order to attain a larger coverage area, a simple
racetrack flight pattern is proposed in Figure 12, in which
the UASs fly in repeatable racetrack formations:
Generally, the result is the expected extension of untessellated results above. One distinction, however, is
that for tessellated patterns the advantage of two-way
ranging over one-way ranging is slightly diminished.
E.g., for the un-tessellated patterns, compare Figure 11 (b)
and (d). In this case, there is a marked advantage of twoway ranging—there is a larger region with satisfactory
(blue) HDOPs. In contrast, for the tessellated patterns
compare Figure 13 (b) and (d). In these figures, the
extent of the good, blue HDOP region is about the same
in one- and two-way ranging.
Figure 12: Simple Racetrack UAS Formation
(a) HDOP for various RD elevations, One-way Ranging, 3 RDs,
Tessellated
(b) HDOP for various RD elevations, One-way Ranging, 4 RDs,
Tessellated
(c) HDOP for various RD elevations, Two-way Ranging, 3 RDs,
Tessellated
(d) HDOP for various RD elevations, Two-way Ranging, 4 RDs,
Tessellated
Figure 13: Tessellated HDOP
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San Diego, CA, January 24-26, 2011
519
In the tessellated cases, the notable distinction is that in
the regions where the one-way system becomes
completely degenerate (white, infinite HDOP), the twoway system is poor, but typically not singular.
For non-degenerate cases (e.g., a UD in a pit or RD
operating at tree top level) L max is significantly greater
than the turning radius of a UAS allowing us to model
UAS movement as a nonholonomic point mass.
Let us initially assume that there exists a known set of
UASs that are tasked to function as RDs for a known set
of UDs. Further, let us assume that communications are
available between the UASs. The strategy autonomous
UASs should use to minimize HDOP varies by the
operating conditions. Specifically, UAS strategy is
dictated by (1) the density of UASs within the operating
area, the ratio of UASs to UDs, (2) the stability of the
UAS team, and (3) the ability of the UASs to reliably
communicate with each other.
The following figures present the effect on HDOP if a
single UAS becomes disabled. Figure 14 and Figure 15
are analogous to Figure 13 (b) and (d), respectively, with
the distinction that that UAS at (L, 0) has been removed.
In this scenario, naturally, the HDOP performance is
degraded. The degradation is less with the two-way
system, which is particularly discernable for RD
elevations over 45°.
Static Grid
We define the population of UASs as saturating the
environment when the number of UASs, n, in the
constellation can completely cover the engagement area.
If the density of the UAS saturates the operating area, the
UAS fleet can provide effective HDOP by forming a
static grid as shown Figure 13 (a) and (b). If edge effects
are ignored, RD saturation occurs when:
(25)
Figure 15: HDOP for various RD elevations, Two-way
Ranging, 4 RDs, Tessellated, One UAS Removed
For fixed altitude, h, optimization of HDOP results in
“pressure” for L to grow to a maximum value defined by
a minimum mask angle!!!"!!"# :
!!"# !
Similarly, we define the population of UDs as saturating
when the number of UDs, m, satisfies the equation:
!
!"#!!!!"!!"# !
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
! ! !"#$#!%!"&!!"#!
&
! ! !!!"#
UASs can form a static grid autonomously by selfassigning themselves to the intersection of a twodimensional lattice overlaying the operating area. As
long as the maximum range between adjacent UAS is less
than !!!"# ! !, coverage from at least four UASs at
oblique angle is assured. When high Quality of Service
(QoS) vehicle-vehicle communication is available, a
central controller can use a resource allocation algorithm
to assign vehicle locations.
When high QoS
communications is not available and membership in the
UAS set is ad hoc, static or quasi-static assignments will
produce grids with gaps as shown earlier in Figure 14 and
Figure 15. The UAS formation can autonomously
remove gaps as they form by using physicomimetic
control algorithms15.
Spears showed that robots
responding to virtual physics forces “emitted” by peer
robots can be used to drive a multi-robot system to a
desired configuration including hexagonal and square
lattice structures.
Spears also showed that
physicomimetic control is effective at coordinating ad hoc
robot communities, a property that supports self-healing
reorganization that will be required by UAS performing
as RDs.
Figure 14: HDOP for various RD elevations, One-way
Ranging, 4 RDs, Tessellated, One UAS Removed
(24)
!!
520
(26)
!
!! &
!
We base our saturation ratio on Figure 13, which showed
that four RDs are capable of providing good HDOP for
small L. In UD saturation, each UD may be individually
served by a UAS group. Note that in the case of two-way
ranging only 2 UAVs are required to provide PVT
information.
Mission Level Autonomy
The most challenging operating conditions are those in
which neither the UAS nor the UDs are saturated and both
UAS and UD team composition are subject to change.
JHU/APL has developed a control strategy called Mission
Level Autonomy (MLA) that is shown to be effective at
controlling the behavior of a system of autonomous
vehicles in these conditions. MLA was adapted from a
potential fields approach called dynamic co-fields. Based
on insect models of cooperation and coordination,
problems are solved heterarchically, rather than
hierarchically. Instead of centralized control, decisionmaking is decentralized, occurring on each individual
system node. These nodes, in this case air vehicles,
coordinate indirectly by altering the environment and
reacting to the environment as they pass through it. MLA
is highly scalable, from one vehicle to as many as the
communications network can handle. The conditions
under which MLA is useful are summarized in Table 3:
Figure 16: Fields that are (a) attractive, (b) repulsive, and (c)
complex
Consider a vehicle i to be a point particle with fixed
position Pi in Euclidean space at a fixed time. The
locations of the influencing entities that produce the fields
are likewise fixed in Euclidean space at a fixed time. The
force !!" !associated with entity j located at !! upon vehicle
i located at !! !is directed along the line-of-sight, and is a
function of the distance between i and j:
!!" ! ! !! ! !! !!" .
(27)
Entity j could be another UAS, a ground vehicle, a
Special Operations Forces (SOF) team, or even a point on
the ground. When the force is greater than zero, the field
is attractive and i moves toward j. When the force is less
than zero, the field is repulsive and i moves away from j.
Typically, there are multiple entities, each producing an
attractive, repulsive, or complex field, and the total force
on the vehicle is a summation of these fields:
Table 3: MLA Applicability Scenarios
(28)
!! ! !
! !! ! ! !! !
!!" ! !&
!
Each vehicle constructs the potential field locally. This
approach is similar to Zambonelli’s co-fields16 and
Arkin’s Behavioral robotics17. Dynamic Co-fields (DCF)
improved robot performance in dynamic situations by
incorporating dynamics into the motivating fields. The
total force affects the vehicle trajectory, which follows the
gradient of the potential field. In practice it is easier to
construct the gradients directly rather than explicitly
construct the potential field; since the gradient is a linear
operator, the gradients resulting from the influence of
each vehicle on the system of vehicles as a whole can be
found individually and summed.
The foundation of the MLA concept is the creation of
virtual potential fields. These fields are associated with
all mission-critical entities in each vehicle’s model of the
world. Attractive fields are produced by entities of
special interest to the vehicle, such as a ground object it
wants to follow or a radio signal it wants to track.
Repulsive fields are produced by entities that the vehicle
wants to avoid. This might include other air vehicles,
obstacles such as buildings, or enemy fire. Complex
fields can be produced when under certain conditions the
entity is attractive, and under other conditions the entity is
repulsive. See Figure 16:
The potential field is dynamic, based on changing selfknowledge of the environment, new information from
other vehicles, and mission redefinition. The initial
mission definition establishes goals and contingencies,
and is loaded onto the air vehicle before flight. Mission
parameters may or may not be rescoped during flight.
Vehicle self-knowledge and knowledge from other
vehicles are represented as beliefs.
These beliefs
comprise the situational awareness (e.g., sensor data) or
International Technical Meeting of The Institute of Navigation,
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521
operational environment (e.g., location in space) of the
vehicle. Beliefs are obtained through direct observation,
observations communicated from other vehicles, or
instructions from by human operators. Beliefs are
semantically described by the tuple (x, y, [z], n[], t, [!])
which includes a location within R3 space, a list n
enumerating the types of which the belief is a member,
the time t for which the belief is valid, and an optional
probability value ! used to express uncertainty. Sharing
of beliefs is a cornerstone of MLA. Beliefs are shared
over a reliable, robust communications framework that
was developed for belief transfer and employs a modular
multilayered architecture.
This robustness has been proven through seven years of
tests and demonstrations conducted by JHU/APL.
Examples of some of these include:
• Hop of video surveillance data over a 7 km link using
five small UASs that autonomously reconfigured
whenever a UAS returned to base to refuel or a new
UAS was introduced (August 2007).
• Heterogeneous team of UASs and unmanned ground
vehicles (UGVs) autonomously locating and tracking a
ground vehicle (September 2004).
• A 40-foot unmanned sea surface vehicle (USSV)
autonomously tracking and following a manned sea
surface vehicle (April 2007).
• Demonstration of persistent road surveillance using a
heterogeneous team of UASs and UGVs (August 2006).
• Multiple demonstrations of UAS teams autonomously
sensing and characterizing a chemical plume (2005
through 2009).
• Two UASs autonomously detecting and geolocating a
ground radio beacon (August 2007).
For systems consisting of a large number of vehicles, the
communications bandwidth can be quickly consumed.
The communications architecture maximizes efficient
utilization of bandwidth by limiting the number and size
of transmissions. This includes scaling belief decay and
the range of belief broadcasts depending on the number of
vehicles, physical operating area, mission complexity, and
network topology. The architecture also allows operation
without
assuming
continuous
bidirectional
communication between all swarm members. Though
this communications architecture optimizes the
communications of belief, one of the advantages of MLA
is the ability to operate with a delay tolerant network.
Even in the absence of successful sharing of beliefs, MLA
enables a vehicle to accomplish a mission (albeit in a
degraded fashion) using only the mission definition and
its beliefs based solely on self-knowledge.
For application to the Local Navigation Service (LNS)
problem, the UASs may be called upon to:
1. Provide area coverage over a complete battle space. If
the space is large it is likely that high altitude, very
capable UASs, such as RQ-4B Global Hawk would be
used, and essentially fly an optimal pattern. The
assumption is that these high flyers may be able to
access GPS even in a highly jammed environment
because of their standoff, the employment of CRPA and
beam-forming antennas, and ultra-tight IMU coupling.
Alternatively, they could position themselves with
PTAN. If precise timing is required as part of the LNS
they will require high precision clocks. As previously
described, when the dimensions of the battle space are
small relative to the height divided by the cosine of the
minimum elevation angle, the optimal solution reduces
to all the high flying UASs flying in a circle around the
center of the battle space at as large a standoff as
possible subject to the minimum elevation constraint. In
this case, MLA adds value only to heal the UAS
constellation when a member is removed (shot down,
runs out of fuel, etc.) or added. This, MLA can
certainly accommodate with ease.
2. Provide a “middle layer” composed of RQ-1 Predator
class UASs that navigate with RF links to the “high
flyers” and serve as references either for ground forces
directly or for even smaller “low flyers” that ultimately
provide LNS services to the ground forces. The
assumption is that the transmit power is lower for the
Predators than the Global Hawks, but sufficient to
overcome area jamming because of the proximity. The
MLA for this middle layer may be somewhat more
complex than the very high flyers because the
positioning of the assets may have to restrict the RF
MLA belongs to a class of bio-inspired autonomy
concepts that has several advantages over other strategies
such as market-based approaches or consensus variables
techniques, including high scalability (from 1 to n),
tolerance of communications network delay or disruption,
system self-healing capability, and timely convergence to
a solution. This latter attribute is perhaps the most
important for this application. The primary disadvantage
of MLA over other concepts is that MLA does not
guarantee convergence to the most optimal solution.
However, in the highly dynamic UAS space, optimal
solutions are ineffective (and potentially disastrous) if not
arrived at very quickly.
The other attributes are important as well: the number of
vehicles can change significantly from mission-tomission, or even within a single mission; communications
cannot always be assured; and vehicles may fail or require
refueling, requiring reorganization of the system to
accommodate fewer vehicles or replacement vehicles.
MLA supports these exigencies. Thus, perhaps its
greatest attribute is robustness.
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
522
link distance to the near-ground elements to overcome
jamming. Thus they are somewhat reactive to their
beliefs about the “low flyers”.
3. Provide concentrated Local Navigation service to
moving ground forces with “low flyers”, such as RQ-11
Ravens, RQ-7 Shadows, or ScanEagles. These UASs
will be able to transmit at much lower power levels than
the upper layer(s), but are much closer to the ground
elements. In this case the first requirement of MLA is
to allocate the low flyer fleet so that each ground
element can see the minimum number of UASs
required for positioning (two for two-way ranging, and
three for one-way ranging). This allocation problem
will have to take account of the limited number of
UASs, the priority ranking of the ground elements, and
any other missions that the UASs are performing, such
as optical surveillance. The allocation will be based on
the current beliefs of each UAS about itself and other
elements of the constellation, and the deployment and
priority of ground forces. This allocation will work
best if the beliefs are close to true, at least for entities
close by. The allocations are determined independently
by the individual UASs, without requirement for
consensus. Because the UASs are tasked with identical
mission rules, UASs in a close geographic area sharing
beliefs will organically converge to similar solutions. If
there are too few UASs to provide LNS service to all
the ground elements, triage will have to be performed,
and reallocation follows the same methodology as
allocation. After the basic resource allocation objective
is met, MLA can be used to deploy the UASs in a near
optimal, or at least reasonable way. Physical line-ofsight between the UAS and UD will be a baseline
metric for spatial positioning. Sum-of-squared HDOP
weighted by user priority is another metric that could be
employed, with communications network quality-ofservice and received signal strength being two others.
solution while all three UASs are providing two-way
navigation.
The third mission goal benefits the most from MLA. We
have therefore chosen to study it with a simulation of a
low-flyer UAS fleet operating under MLA to provide
LNS to a small number of moving ground elements.
Since there are three teams operating in the target space,
and three UASs support each team, nine Shadow UASs
are required to support the mission. For a flat, featureless
topography, no obstructions, ideal communications links,
and static UD, the optimal geometry for providing twoway navigation to the team consists of the three UASs
equidistant from the UD and separated by 120° (as shown
in Figure 7). In the absence of these ideal conditions, the
MLA tries to optimize the UAS positioning based on a set
of beliefs and autonomy rules.
For this scenario, the UASs are RQ-7 Shadow 200s (see
Figure 17) launched and maintained at a FOB near the
target area. These vehicles have a 3.4 m length, a 4.3 m
wingspan, and empty weight of 90 kg. They are launched
from a mobile hydraulic launcher. Shadows have an
endurance of 6 hr. and cruising speed of 148 km/hr. The
flight ceiling of a Shadow is 4572 m above Mean Sea
Level (MSL). The typical payload includes electrooptical (EO) and infrared (IR) cameras. In addition to the
EO/IR sensors, these Shadows are each equipped with an
RD and GPS. In this scenario, the operating altitudes will
be based on topography, optimal geometries, and
positioning rules (e.g., maintaining a 30° elevation to the
target UD). For the purposes of this scenario, it is
assumed that GPS jamming is ineffective against the
UASs at any if these operational altitudes. In a multilayered system, the Shadows may be getting their position
information from a high-flyer such as a Predator or
Global Hawk.
Figure 17: RQ-7 Shadow
The notional scenario consists of three small teams of
soldiers patrolling an R2 space of 400 km2, roughly a box
20 km on a side. This entire target area is assumed to be
GPS-denied due to jamming. Each team uses a two-way
UD to determine position from dedicated RDs assigned to
the team. A constellation of three UASs, each with an
RD, service each team. Two UASs provide the minimum
RDs required for two-way nav. The third UAS acts as a
replacement vehicle for refueling events or in case of a
crash, or as a communication node, climbing to high
altitudes to provide communications (voice, sensor data,
PNT data) coverage to the other teams or back to the
Forward Operating Base (FOB). Furthermore, the RD
aboard this third UAS can provide an overdetermined
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
The beliefs in this case include self-knowledge (e.g., UAS
position, UAS dynamics such as velocity and turning
radius), the surrounding environment (e.g., geographic
features, stored DTED data, other vehicles, enemy fire,
jamming signals), beliefs from nearby UASs (e.g., their
self-knowledge and sensing of the environment), and a set
of navigation-based metrics (e.g., line-of-sight to the UD,
HDOP, service priority, signal quality).
523
The autonomy rules generate the attractive and repulsive
fields based on these beliefs and the mission objectives.
For instance, an attractive field might be created around a
point in space that minimizes HDOP based on the current
solution and the positions of the other UASs. A repulsive
field might be created around the other UASs to prevent
them from getting too close, not just for flight safety but
also to maintain angular diversity for better navigation
solutions. Because the UASs and soldier teams move,
and the environment is in flux, the fields change
constantly, and thus the gradient created by these fields
changes constantly.
reliable communications between the command center at
the FOB and the UASs in the field. Even if this could be
realized, it severely limits the range of the UASs and,
likely, the ability to optimize their positions.
Control Law for Local UAS Formations
Effective HDOP may be autonomously provided by
assigning sub-teams of UAS to cover each UD (or cluster
of UDs). The assignment of UASs to subteams can be
easily provided by a centralized search-based resource
allocation algorithm if high QoS communications are
available and UAS team membership is static or at least
quasi-static. Subteams can provide minimal HDOP for
individual UDs by interacting as Kuramoto oscillators18.
Kuramoto showed that orbiting particles can become
phase and frequency locked by following the control law:
The mission may consist of more than simply providing
PNT navigation solutions to the UDs. One example
would be providing a communications link back to the
FOB. In this case, an attractive field is periodically
created at an altitude sufficient to provide the
communications relay. While one UAS provides the
communications link at altitude, the two others
reconfigure to optimize the PNT navigation solution for a
two-UAS constellation. This same reconfiguration occurs
when one UAS has to return to base for refueling, or if a
UAS crashes or gets shot down.
(29)
!
!"#!" !!! ! !! !&
!!!
where N is the number of orbiters (UASs), ! is the orbital
velocity, K is a constant. Pacifico19 demonstrated that
particles can become anti-phase and frequency locked by
following the control law:
Communication links between the UASs are certainly
desired, but not necessary. Because of the distances
between the teams, and shading from geographic
obstacles such as mountains, it is possible that all UASs
will not be able to communicate with all other UASs. It is
further possible that one UAS, or a group of UASs, will
be completely isolated from the others. As described
above, MLA can operate even in the presence of
unreliable communications, albeit with a degraded
solution.
(30)
!
! ! !! !
!
!
!"#!" !!! ! !! !
!!!
By selecting a radii of L and adhering to Pacifico’s
control law with four UAS
(31)
! ! !! !
!
!
!
!"#!" !!! ! !! !&
!!!
A team of four UAS can achieve an antiphase orbit
around a UD.
Some communications between the three UAS
constellations can be advantageous. Just as the individual
three-UAS constellation can reconfigure and self-heal
upon the loss of a UAS, the three constellations can
reallocate UASs between the constellations. For instance,
if in one constellation two of the three UASs require
refueling, a single UAS from another constellation can fly
over to fill in and join the remaining UAS. Note that
these reallocations are derived independently aboard each
UAS based on the beliefs, mission definition, and
autonomy rules; there is no central controller or direct
cooperation between UASs.
MLA Results
Experiments were conducted to assess the utility of MLAenabled UAS providing localization support for multiple
users.
Fields were constructed that included two
elements: (1) an attraction to users, and (2) a repulsion
from other UASs. Previous experiments had indicated
that fields based upon Pacifico’s anti-phase oscillators19
would generate self-organizing teams of orbiting UASs
evenly distributed around a moving object. Anti-phase
oscillators are inefficient at minimizing HDOP as evennumbered UAS will align themselves on opposite sides of
a UD, unproductively aligning their signals. To eliminate
this undesired behavior the UAS field included a
repulsive well co-located with the each UAS and a second
antipodal repulsive well of lower magnitude. This
generated the effective orbiting behavior shown in Figure
18, in which red dots are UDs, the blue dots are UAS and
grey shadings are DCF field intensities.
Humans are incapable of commanding the UAS behaviors
produced by the MLA. A single operator cannot control
that many vehicles, and multiple operators trying to
coordinate with each other would produce only chaos.
There are too many beliefs to process, the mission rules
are too complex, and the UASs travel too fast for humans
to operate with the same speed and efficiency as the
MLA. Furthermore, human control necessarily requires
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
!
! ! !! !
!
524
control link and varying UAS characteristics (e.g., time of
flight and mean time to failure).
V. PHYSICAL ARCHITECTURE 2:
OPERATIONALLY RESPONSIVE SATELLITEBORNE RD’S
Responsive Constellation Reconstitution
The ability to rapidly reconstitute a critically impaired or
otherwise lost PNT capability is intimately predicated
upon the capabilities of both the launch and space
segments. The United States is presciently aware of this
dependency and has, as a consequence, established the
Operationally Responsive Space (ORS) Office in 2007 to
cultivate the enabling architectures, technologies,
procedures, logistics tails, and capabilities to deploy
needed warfighter solutions on a timescale of days to a
week20. As part of their developmental efforts, the ORS
is establishing a qualified stable of Launch Vehicle (LV)
options that will afford Low Earth and Highly Elliptical
Orbit (LEO/HEO) access, in a manner consistent with
objective timescales. Detailed in Table 4 is a listing of
the reference orbits that ORS has established as
operationally relevant to envisioned mission needs, and
equally determined to be serviceable by their LV options,
including the Minotaur I and IV, Space-X Falcon-1e and 9, and Raptor21,22.
Figure 18: MLA simulation of UASs
!"#$%&'
To measure the effects we used the average HDOP-1 for
all UDs as a metric. During the simulations a set of UD
performed a random walk over a planar world with
dimension x, y " 10Lmax. As a baseline we generated
results for a satisficing grid of station-keeping UAS that
produced 0.84 HDOP-1.
Approximately twenty
simulations were run in which the number of UAS were
varied. The results of these experiments are shown in
Figure 19.
(#$"
("
!#'"
!#&"
!#%"
!#$"
!"
Table 4: ORS reference HEO for communications missions
(" $" )" %" *" &" +" '" ," (!"
()*'"+,-./0'1()*2("3'
Figure 19: MLA simulated HDOP-1
The results show that once the number of UAS vehicles
reaches four times the number of UDs, the UD HDOP
produced by MLA is slightly better than the HDOP
produced by the grid (between 0.88 and 1.05 HDOP-1 for
MLA as compared to 0.84 HDOP-1 for the grid). Note
that in the results for cases with low UAS ratios the
performance is less than what might be expected as,
intuitively, two UAS should be sufficient to provide
effective geolocation for a single UD. The lower than
expected performance is because rapid acquisition of new
UDs requires constant exploration of the space.
The methodology and results of Kantsiper et al<= were
extended to the noted HEO reference orbits with a focus
on the PNT mission preferring extended dwell durations
over specific theaters. For a representative Southwest
Asia area of regard, a nominal latitude of 40° is
emphasized. >(7!)$& <? shows that only 5-10 satellites,
deployed individually across three-five critically inclined
orbital planes separated by right ascension of ascending
node, are needed to provide the requisite continuous
coverage.
Future MLA Work
The results presented here describe an initial exploration
into MLA controlled geolocation providing UAS swarms.
Further understanding will require more expressive
simulations that explore MLA-UAS as a variety of
operational features vary. Features we would like to
investigate include: terrain models, particularly complex
line-of-sight models; varying the number of UAS; varying
the number of UD; varying the size and shape of the
operating area; varying communications service for the
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
525
obstructions. If circular LEO constellations (e.g. Walker),
similar to that employed by I77N, are instead utilized,
much of the theater-optimized coverage is proportionally
reduced in favor of global access.
Figure 20: # of Satellites Required for Continuous Coverage
While most of the orbital inclinations are served by
CONUS launch facilities, truly responsive, global access
is only afforded by a limited number of national means.
Among several options investigated by JHU/APL, was the
potential for leveraging the operationally deployed lift
capability of Submarine Launched Ballistic Missiles
(SLBM). Among the candidate platforms considered, the
most capable is that associated with the 80” full-caliber
Ohio class missile tube24. A conceptual satellite design
based on this promising opportunity is shown in >(7!)$&
<@. >(7!)$& <@ (a) shows the stowed and deployed
configurations of the microsat. >(7!)$& <@ (b) shows
details of the microsat’s subsystems, including the multiband communications payload, which is readily
accommodated within the volumetric constraints of the
spacecraft structure, along with all other subsystems
needed to provide required propulsion, power, pointing
control, and data link necessary to execute the mission.
&
>(7!)$& <?& shows that 5-10 satellites will afford
continuous coverage to theaters of interest (38-46 latitude
highlighted in purple). This estimate reflects a 10°
elevation angle constraint typical for communications
applications desiring unobstructed link paths above
typical terrestrial surface features and (man-made)
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
526
(a) PNT microsatellite concept showing both stowed and deployed configurations.
(b) PNT microsatellite internal packaging design.
Figure 21: Notional Microsatellite Design
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
527
A Two-Way LEO System
We will focus our analysis on a two-way LEO system
because:
1. It is easier to rapidly replenish a LEO
constellation that has suffered battle damage.
2. Doppler measurements from a LEO system are
much more sensitive to geolocation than MEO or
HEO constellations.
3. A two-way system removes UD clock effects
from the solution and enables single epoch
geolocation for static RD’s.
Use Case: SOF Operating in Denied Airspace
We consider here the case of Special Operations Forces
(SOF) operating in the early stages of a conflict in which
airborne jammers have not been eliminated. We will
consider the 800 km polar ORS RO-A orbit. The twoway link analyses corresponding to each direction of a
1575.42 MHz link (the existing L1 frequency band may
not be optimal for this application and is used as a
reference point only) are shown in&A,6#$&B&,"'&A,6#$&C.
Table 5: RD to UD (Space to Ground) Link Analysis
RD=>UD
The challenges presented by operation from low Earth
orbit relative to a MEO (GPS) constellation are much
larger rail-to-rail Doppler (+/-21 kHz vice a 4 kHz), and
high chirp (~250 Hz/sec at L-band for a 40° Elevation
pass).
LINK CHARACTERISTICS
Frequency
Height
Elevation
Slant Range
Space Loss
UD Antenna Gain
RD Antenna Gain
RD Passive Gain (cable, T/R switch, cavity filter)
Other Fade
Total Link Losses
SIGNAL CHARACTERISTICS
Chipping Rate
Code Length
Code Epoch Time
Equivalent Code Epoch Distance
JAMMER CHARACTERISTICS
Effective Jammer Power (=J*G_JU*G_UJ)
Jammer Standoff
Jammer Space Loss (Free Space)
J0
N0
J0 + N0
LINK CLOSURE
RD Transmit Cable Power
UD Receive Power
N-Packets Integrated UD
Total UD Integration Time
UD Receiver Noise Figure
Maximum Doppler Scalloping Loss
Postcorrelation SNR (UD)
As shown in Figure 22, global single-ping geolocation
(with two-point ambiguity) is possible by intersecting a
ranging sphere and Doppler cone with the surface of the
Earth. That a single epoch solution exists for static UDs
is of immense importance to a sparse constellation that
may have to be continually replenished to replace battle
damage. Spaceborne operation of the system also offers
the possibility of interacting with large numbers of
individual tags simultaneously with Code Division
Multiple Access (CDMA). If Z4 family ! codes are used
to address the UDs, the ratio of peak correlation energy to
code cross correlation energy for large N is about N. As
long as the difference in link loss is not greater than this,
the probability of false activation of a UD is low. Even if
this happens, the UD will not be falsely identified because
it would fail verification in subsequent transactions.
Since the space loss from all visible tags varies only a few
dB from a LEO platform, the restriction on dynamic range
of the link-loss falls primarily on the fade environment.
1575.42
800.00
30.00
1395
-159
-3
4
-3
-3
-164
Units
MHz
km
deg
km
dB
dB
dB
dB
dB
dB
10 MHz
8191
819.1 mu-sec
244 km
1000
17
-121
-134
-174
-134
W
km
dB
dBm/Hz
dBm/Hz
dBm/Hz
200
-111
175
143.3425
4
0.579
13
W
dBm
number
msec
dB
dB
dB
Table 6: UD to RD (Ground to Space) Link Analysis
UD=>RD
LINK CHARACTERISTICS
Frequency
Height
Elevation
Slant Range
Space Loss
UD Antenna Gain
RD Antenna Gain
RD Passive Gain (cable, T/R switch, cavity filter)
Other Fade
Total Link Losses
SIGNAL CHARACTERISTICS
Chipping Rate
Code Length
Code Epoch Time
Equivalent Code Epoch Distance
JAMMER CHARACTERISTICS
Effective Jammer Power (=J*G_JR*G_RJ)
Jammer Standoff
Jammer Space Loss (Free Space)
J0
N0
J0 + N0
LINK CLOSURE
UD Transmit Cable Power
RD Receive Power
N-Packets Integrated RD
Total UD Integration Time
UD Receiver Noise Figure
Maximum Doppler Scalloping Loss
Postcorrelation SNR (UD)
Figure 22: Single Ping Geolocation from a LEO Satellite
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
Param
Param
1575.42
800.00
30.00
1395
-159
-3
4
-3
-3
-164
Units
MHz
km
deg
km
dB
dB
dB
dB
dB
dB
10 MHz
8191
819.1 mu-sec
244 km
1000
1395
-159
-172
-174
-170
W
km
dB
dBm/Hz
dBm/Hz
dBm/Hz
0.10
-144
80
65.528
4
0.579
13
W
dBm
number
msec
dB
dB
dB
The scenario chosen for this analysis corresponds to the
UD viewing the satellite at a mask angle of 30°, 1121 km
away from the sub-satellite point as shown in Figure 23.
We assume 1kW effective power jammer for the
downlink (RD=>UD) and uplink (UD=>RD).
The
528
jammer is elevated on an AeroStat, aircraft, or UAS, so
that its power decays via freespace propagation. The
jammer standoff from the UD is 17 km. The downlink
and uplink are essentially balanced with a post-correlation
SNR of 13 dB. Fade of 3 dB can be tolerated with a
spacecraft transmit power of 200W through a 4 dB gain
isoflux antenna. The UD antenna is assumed to have a -3
dBi gain. With these assumptions, a balanced link (in
terms of post-correlation SNR) is achieved with a UD
transmit power of only 0.1W. We have assumed a 10
MHz chipping rate and a code length of 8191 so that each
code epoch is 0.82 millisecond. This results in the same
jammer rejection as the P(Y) code, a pseudorange
ambiguity of 244 km (almost as long as the C/A code)
and a code cross correlation rejection of about 39 dB, if
the Z4 family ! codes are used. For a balanced link, the
UD coherently combines 75 code repeats, while the RD
combines 80. (If the system were operating against only
thermal noise, a balanced link would require that the RD
integrate more packets than the UD, according to the ratio
of their transmit power. The proximity of the jammer to
the UD changes the situation so that the number of
packets combined on RD and UD is nearly equal in this
case.)&&
elevation of 40° are shown as the top three plots in Figure
24, as a function of the time since closest approach
The bottom plot is the “single-ping” geolocation error
assuming a two-way Doppler velocity measurement noise
standard deviation (SD), ! Dop-2-Way , of 0.5 Hz, and a range
noise SD, ! R-2-Way , of 1 m.
The East-North-Up
coordinates were equivalent to cross/track-along/track-up
in this case. The Doppler velocity noise assumes that
after a successful initial transaction, both RD and UD
coherently process a full one-second of data. The
Doppler measurement noise is consistent with the
Cramer-Rao lower bound for one second coherent
processing. The vertical height constraint SD, ! Height Const ,
was 10 m. The single-ping covariance was given by:
(
!1
C = H t R1!Way
H
the
)
equivalent
!1
2
2
2
#$ ,
, ! NORTH
, ! UP
where diag (C) = !"! EAST
1-way
measurement
2
2
2
R1!Way = diag(! R-2-Way
, ! Dop-2-Way
, ! Height
Const ) / 2 ,
sensitivity matrix H is:
! H
# R2
H = # H D2
#
#" ê z
noise
is
and the
$
&
&.
&
&%
The first two rows are as given in Appendix B and
ê z = [ 0, 0,1] is the upward pointing unit vector in the
East-North-Up (at the UD) system. Note that for this
scenario, the pass was visible for about 200 seconds, the
rail-to-rail Doppler was +/- 20 kHz, and the maximum
chirp was about 225 Hz/sec. The position was always
well estimated with the one-sigma of each component less
than 10 m. The along track (north for polar orbit)
geolocation error is dominated by the Doppler
measurements, while the cross track (east) is dominated
by the range measurements.
Figure 23: 30°(green) and 10°(yellow) Mask Angle
Footprint for 800 km Height
Geolocation Covariance Analysis
The elevation angle, Doppler, and chirp for the 800 km
ORS RO-A polar orbit ranging on a UD with maximum
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
529
Figure 24: Orbital Parameters and Covariance Analysis
architecture we studied resembles a cellular
communication network with the UAS-borne RDs
corresponding to the Base Station Terminals (BSTs) of a
cell system. The UDs, of course, correspond to the
cellular handsets. We have shown that such a system is
very robust to jamming. We examined the optimal
laydown of the UASs and how HDOP is affected by the
various geometries. It was shown that the HDOP of UDs
served by small clusters of RDs as well as regular patterns
of RDs benefited greatly from two-way ranging. We also
described how the use of Mission Level Autonomy
(MLA) algorithms enables a limited number UAS
constellation to serve ground operations in a robust way.
VI. SUMMARY
We have analyzed the tradeoffs involved in designing a
local or theater GPS system. Our operating assumption
was that such a system would be unconstrained by legacy
GPS architecture, equipment, or waveforms. It could
therefore employ more efficient cyclostationary Z4
ranging codes than the legacy BPSK codes. It could also
reap the benefits of two-way measurement and
communication including (1) the elimination of UD clock
error, (2) the great reduction of the UD infill bandwidth
by computing the UD position on the RD, (3) code
ambiguity elimination, and (4) (perhaps) the use of
cognitive radio techniques to resist jamming. The
existence of robust cellular telephony employing lowpower compact user handsets in a very crowded spectrum
shared by many users testifies to the viability of this
approach. Because of this, the UAS-based physical
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
For operations over very large areas of denied airspace,
we considered geolocation of static UDs from two-way
links from satellites consistent with specifications
developed in the Operationally Responsive Satellite
(ORS) program. Two-way satellite links that can measure
530
Doppler as well as range enable single satellite, single
epoch geolocation. This reduces the number of satellites
required in the constellation by about a factor of two. A
notional satellite design consistent with launch from an
Ohio-class submarine was presented.
are the measurements noise. c is the speed of light. The
odd rows are the measurements from the RDs to the UD,
and the even rows are the measurements from the UD to
the RDs. ! is a zero-mean multi-vairate Gaussian with
covariance: The measurement noises are independent,
identically distributed Gaussians with covariance R:
For future investigations, the full impact of Cognitive
Radio (CR) on jam resistance for two-way systems should
be evaluated. It would seem that the ability to change
frequency and waveforms would assist in defeating
jamming. However, as stated in Burbank4:
(3)
(4)
!
!
!
!
!
!
!!!! !! !
!
.
!!!! !! !
The DOP matrix for the weight-least squares solution is
given by:
!!
!!
! ! !! ! ! !! !!
!!
(7)
!
! !! ! ! !!
!!
This can be placed into the following form:
APPENDIX A: HDOP CALCULATIONS
The measurements for the two-way ranging system are
modeled as:
(8)
!!
!!
!
!!
!
!! ! !!
!!
!!
where M is a diagonal matrix with entries ! ! , and !! is
the sensitivity matrix. Note that M is identically the
covariance for the measurement for the one-way ranging
system. With equal covariance matrices, HDOP between
one- and two-way ranging are comparable using the
unweighted HDOP formulation: ! ! ! !! .
!!!!!
!!!
where ! is the UD position, !!" is the UD clock error, !!
is the ith RD position, !!"! is the ith RD clock error, and !!
International Technical Meeting of The Institute of Navigation,
San Diego, CA, January 24-26, 2011
!! ! !!
!
!!!!! ! !!!
!! ! !
(6)
The authors express their gratitude to Lawrence Karr of
RoundTrip LLC for his assistance in understanding the
more subtle concepts involved in two-way range and
Doppler measurement using spread spectrum signals.
!
!
!!!! !! !
!
.
!!!! !! !
Let define the following matrix:
The authors express their gratitude to Steve Jones and
Jack Burbank of APL for their assistance in understanding
cognitive radio operation in the presence of jamming.
! ! !! ! ! !!" ! !!"!
!!!!
!
! ! !! ! !
!
(5)
This study was convened and funded by the U.S. Army.
!!
! ! ! !!
!
! ! ! !!
! ! ! ! !!
!! !
The measurement sensitivity matrix for !! is given by:
ACKNOWLEDGMENTS
(1)
!
!
!
!! is zero-mean Gaussian vector with covariance C:
It thus appears that the benefits of CR against jamming
are highly scenario dependent: One of the most important
questions is what spectrum allocation constraints apply to
the ranging system, and whether it can adopt a strategy of
competing with a jammer at a section of spectrum that is
favorable to it.
! ! !! ! ! !!"! ! !!"
!
!
!
The RD-to-UD and the UD-to-RD measurement noises
are of the same variance because the SNR on the links are
balanced. Adjacent measurements in ! are added to
remove the clock dependence giving:
“A traditional frequency-follower jammer will perform no
worse against a CR network as compared to the case of a
traditional frequency hopping network, if the jammer is
properly designed and configured. In the case of
wideband barrage jamming, (i.e. all possible frequency
channels of operation are attacked), the jammer will
perform at least as well against a CR network compared
to the traditional wireless network case (hopping or fixed
in frequency).”
! ! !! ! ! !!" ! !!"!
!
! ! !! ! ! !!"! ! !!"
!
! ! !! !
!
(2)
531
APPENDIX B: SINGLE PING COVARIANCE
11
Tang, X., Udaya, P. “A Note on the Optimal
Quadriphase Sequences Families.” IEEE Transactions on
Information Theory, Vol. 53, No. 1, January 2007
12
Jiang, W., et al. “New Optimal Quadriphase Sequences
with Larger Linear Span.” Information Theory for
Wireless Networks, 2007 IEEE Information Theory
Workshop , July 2007
13
http://www.toyon.com/aj_gps.asp
14
Egli, John J. (Oct. 1957). "Radio Propagation above 40
MC over Irregular Terrain". Proceedings of the IRE
(IEEE) 45 (10): 1383–1391.
15
D:&E9$,)%;&F:&E9$,)%:&G%("7&H)+(/(*(,#&10I%(*%&+.&
J."+).#&H7$"+%:&!"#!$$$#!"%&'"(%)*"(+#,*"-&'&".&#*"#
!"-*'/(%)*"0#!"%&++)1&".&#("2#345%&/5K&<L@5<LL;&@MMM:#
16
M. Mamei, F. Zambonelli, L. Leonardi "Co-Fields: A
Unifying Approach to Swarm Intelligence", 3rd
International Workshop on Engineering Societies in the
Agents' World, Madrid (E), Sept. 2002, LNAI.
17
R. Arkin, Behavior-Based Robotics, MIT Press, 1998.
18
Y. Kuramoto, Chemical Oscillations, Waves, and
Turbulence, Springer, Berlin, 1984.
19
D. Pacifico, D. Scheidt, Decentralized Anti-Phase
Synchronization of Mobile Ad hoc networks (MANETS),
NTSD Tech-Memo, Johns Hopkins University Applied
Physics laboratory, 2003.
20
BAA-ORS-08-01 – Launch, Range, and Modeling and
Architecture solicitation, www.fbo.gov.
21
Sandhoo, G.P., Rogers, A.Q., Stadter, P.A., et al.,
Standards for Responsive Small Satellites, ESA Small
Satellites Systems and Services Symposium, Rhodes,
Greece, May 26-30th, 2008.
22
Welsh, J.S., Operationally Responsive Space (ORS)
Technology Needs, Digest of the 7th International
Symposium of the International Academy of Astronautics
on Small Satellites for Earth Observation, Berlin,
Germany, May 4-8, 2009.
23
Kantsiper, B.L., Stadter, P.A., Benson, J.H., Stewart,
P.L., ORS HEO Constellations for Continuous
Availability, 2007 Responsive Space Conference, Los
Angeles, CA, April 23-26th, 2007.
24
Rogers, Aaron, Anderson, Charles, Lotito, Napolillo,
David, Scherock, Jeff, Sharer, Peter, Wadsley, Brian,
Submarine Launched Space Payloads :History, Utility,
and Challenges, Submarine Technology Symposium,
Laurel, MD, May 11-13, 2010.
The single ping covariance was based on having three
measurements at each epoch: two-way range, two-way
Doppler, and a height pseudo measurement. The two-way
(1x3) range sensitivity, modified as in APPENDIX A to
work with an equivalent 1-way measurement noise, is:
(9)
H R2 =
!r2"Way
= 2 ê t
!rU
,
where r2!Way is the two-way range measurement, rU is the
UD position in the ENU system, and ê = ê ( rR , rU ) is the
UD-RD line of sight unit vector. The Doppler sensitivity
matrix (1x3) is
t
t
!D2"Way
2 v I " êê
(10)
H D2 =
=
!rU
!
r
(
)
where D is the first-order Doppler frequency shift due to
relative motion, v is the RD-UD relative velocity, ê is
the line-of-sight vector, and r is the UD-RD distance.
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