Overview of Pulsar Navigation: Past, Present and Future Trends
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Overview of Pulsar Navigation:
Past, Present and Future Trends
PETER J. BUIST, STEVEN ENGELEN, ARASH NOROOZI, PREM SUNDARAMOORTHY,
SANDRA VERHAGEN, and CHRIS VERHOEVEN
Delft University of Technology
Received October 2010; Revised April 2011
ABSTRACT: In this contribution we will provide an overview of the work that has been done on pulsar naviga-
tion and show a new direction in pulsar-based navigation research. Up until now the focus has been on X-ray
pulsars, whereas our focus will be on the possibility of using radio pulsars. The radio frequency range has been
neglected because the radio-frequency pulses were assumed to be too weak to detect with antennas of a reasona-
ble size. We will demonstrate that with a relatively small antenna radio pulses can be detected even on Earth.
In our discussion we will make a comparison of pulsar navigation with GNSS and the differences are analyzed
in a detailed discussion on both navigation methods.
INTRODUCTION pulsars in our solar system [3–9]. These studies
show that navigation using pulsars is both feasible
Pulsar-based navigation is not a new idea – ever
and reasonably accurate for space applications.
since Jocelyn Bell [1] discovered her ‘‘little green
As pulsar navigation is based on natural objects,
men’’ were actually rapidly rotating neutron stars,
some irregularities are expected, yet the achievable
their existence and properties have been studied.
accuracy [9, 10] would at least approach that of
Her discovery dates back to 1968, and four years
common deep space navigation principles.
later, NASA installed a plaque on their Pioneer 10
Moreover, pulsars, unlike man-made navigation
and 11 spacecraft, as well as a more extensive
satellites, are immune to solar flares or hostile
golden record on both Voyager spacecraft which con-
attempts at disabling them; and due to their broad-
tained a map (Figure 1), designed by F. Drake [2],
band nature, jamming their signals is rather diffi-
showing the location of the Sun with respect to the
cult. These aspects, combined with the fact that there
center of the galaxy using the direction of 14 known
are already over 1800 known pulsars in the celestial
pulsars, along with their pulse periods. The lengths
sphere, gives them a very firm base for governmental
of the lines in the figure show the relative distances
and civilian use, as well as military applications not
of the pulsars to the Sun. The pulse periods are indi-
only for space but also on Earth if we would be able to
cated by long binary numbers corresponding to the
detect them.
pulsars. Since these periods change over the time,
Radio pulsar navigation - utilizing signals which
not only the location of the Sun but also the epoch of
can indeed be detected on Earth - has not received
launch of the probe can be calculated.
much attention for a long time, even for interplan-
The idea was that by grossly over-defining the
etary applications, since the signal strength of ra-
map (three pulsar sources would suffice), another
dio pulsars was deemed too weak to be useful [6, 8,
civilization would be able to determine the ratios of
9]. Advances in signal processing, however, would
the pulse arrival times, and link them to the original
enable faster and more accurate detections [11],
pulsar source, allowing them to find the location of
using a smaller antenna size, and hence recently
Earth.
interest has spiked once more for using these as a
Since then, however, for navigation on or near to
navigation source.
Earth, most navigation methods have focused on
In this contribution we will discuss pulsar-based
satellite-based systems like the United States’
navigation. In the section ‘‘Previous Work on Pulsar
Global Positioning System (GPS), although studies
Navigation,’’ we will provide an overview of what
have been performed on navigation using X-ray
has been done in this field, which is, as we will see,
mostly on X-ray pulsars. In the section ‘‘Principles
NAVIGATION: Journal of The Institute of Navigation
Vol. 58, No. 2, Summer 2011
of Pulsar Navigation in Comparison with GNSS,’’
Printed in the U.S.A. we will compare the principles of pulsar-based navi-
153allow access to very high precision timing for satel-
lites or remote observatories at planetary or terres-
trial surfaces.
PREVIOUS WORK ON PULSAR NAVIGATION
A number of studies have been performed on
navigating using pulsar signals. Chester and But-
man [4] published a paper in 1981 on using X-ray
pulsars for deep space navigation. Back then, only
17 X-ray pulsars had been identified, yet they con-
cluded that for any orbits beyond Saturn, such a
system (with a 0.1 m2 X-ray detector) would sur-
pass the achievable accuracy using the Deep Space
Network.
Sheikh performed his doctoral research [7] on
navigation using pulsars, and he likewise concluded
that navigation would best be suited using X-ray
sources. His work suggests that initial position
determination is possible with several tens of kilo-
Fig. 1–The pulsar map, engraved in the plaques launched with meters in accuracy. While using a delta-correction
pioneer 10, 11 and voyager 1 and 2 (adapted from: [2]). The line method, which relies on a position estimate from an
representing the relative distance of the Sun to the center of the
galaxy is not shown independent source (e.g., a deep-space network
ranging measurement), the position solution can be
determined within 100 m and a velocity resolution
within 10 mm/s Root Mean Square (RMS), using a
gation with satellite-based navigation. In the sec-
1 m2 detector, and 500 s of integration time. He also
tion ‘‘Principles of Pulsar Radio Signal Detection,’’
concluded that pulsar based navigation would be
the detection of radio pulsars is discussed and the
the only system known to date for deep space appli-
section ‘‘Pulsar Experiment’’ will show preliminary
cations which would be capable of both relative nav-
results of such an experiment. However, we will first
igation, as well as absolute navigation. This is a
discuss some examples of pulsar-based navigation
unique feature, as it would enable deep space mis-
for (inter)planetary and terrestrial applications.
sions to navigate much more precisely in faraway
orbits, as the system is independent of range or
angular effects commonly associated with radio
Applications
ranging using a deep space network. Moreover, the
Global Navigation Satellite Systems or GNSS system would allow for autonomous navigation,
(like the aforementioned GPS, the European Gali- reducing the load on ground-station operators and
leo, etcetera) have proven that the applications for removing the lag due to large distances, when per-
navigation systems are quite difficult to predict. forming orbit-correction manoeuvres.
This is even more true with navigation systems Emadzadeh [14] subsequently studied using
which use very robust transmitters of a natural X-ray pulsars for a relative navigation system. This
source, such as pulsars. One immediately apparent would allow two (or more) spacecraft in formation to
application, aside from the traditional satellite determine their relative location, as well as their
navigation device, is a feed-back of pulsar informa- absolute location in deep space, based on matching
tion to the radio astronomy community, since a the arrival times of specific pulses in both space-
multitude of receivers could actively monitor the craft. He showed in simulations that depending on
currently known catalogue of pulsars. the number of pulsars, their characteristics, and
As LORAN-C is phased out [12], more commer- geometric distribution for an hour’s worth of obser-
cially viable applications can be envisaged. A pul- vation, accuracies on the order of 1 km would be
sar-based navigation device could provide a (naval) achievable.
back-up for the current GPS navigation systems, The European Space Agency has performed a
just like LORAN-C, should these devices prove feasibility study [6] on navigation using pulsar
manageable for naval use. timing information. They conclude that navigation
With their remarkable stability, some pulsars using radio pulsar signals is technologically chal-
are expected to compete with atomic clocks as a lenging due to the required antenna size, but that
timing reference [13]. Such a pulsar receiver would accuracies of less than 1000 km would be achieva-
154 Navigation Summer 2011ble within a limited time span. They also state
that in order to achieve as high an accuracy as
possible, a large bandwidth is required. Moreover,
they state that X-ray pulsars would require even
longer integration times, and hence smoother or-
bital paths, compared to radio pulsars, as they
found that for the most useful pulsar candidates,
only around 90 photons per hour would be
detected. Moreover, they proposed to establish a Fig. 3–Pulse model
universal pulsar time.
Recently, interest in X-ray pulsar-based naviga-
once. However, not all neutron stars are necessarily
tion has sparked in China [15], and advanced fil-
detectable as pulsars. The beams from some neu-
tering techniques have been shown in simulations
tron stars may never pass Earth as they are not
to increase the performance of such systems to
pointing in our direction and will therefore not be
achieve accuracies of less than 100 m in the posi-
detected.
tion domain after filtering of a few hours of obser-
As discussed, pulsars can be extremely accurate
vations.
pulse sources. This property could make them ideal
All studies conclude that prior information gained
for Time-of-Arrival-based navigation methods. How-
using, e.g., an Inertial Measurement Unit (IMU), is
ever, the transmit time and position of the source is
highly beneficial as it decreases the search space
not precisely defined [6]. In this section, we will dis-
considerably.
cuss the pulsar signals in more detail in comparison
with GNSS.
PRINCIPLES OF PULSAR NAVIGATION IN
COMPARISION WITH GNSS Position Description in a Reference Frame
In this section we will discuss pulsar navigation One important difference between pulsar-based
in comparison with GNSS. Pulses in a wide fre- and GNSS-based navigation is the applied refer-
quency range can be received occurring at regular ence frame. GNSS systems make use of reference
intervals which corresponds to a beam (or beams) frames which are Earth Centered and rotating
being emitted from a rotating neutron star (Figure with the Earth. This means that they are so-called
2). According to the lighthouse model, the pulsar Earth Centered Earth Fixed frames. One well-
emits radiowaves and particles along its magnetic known example is WGS84 for GPS.
axis. As the neutron star is rotating, observers For pulsar navigation, a Barycentric (Solar cen-
detect pulses with a distinct period. Figure 3 shows tered) reference frame is applied which is quasi-
pulses from a pulsar and the period of the signal, or inertial. It is quasi-inertial as the Sun has relative
the time between two pulses from the same beam is motion towards other celestial bodies like the
the time that it takes for the neutron star to rotate pulsars.
For GNSS systems, the GNSS satellite positions
are known in their reference frame at the meter
level in real-time from navigation messages and
post-processed positions become even more precise.
For pulsars, this information is not available, but
as we will see in the next section, this information
is not necessary to estimate a user’s position.
Observation Model
In this section a general model for the range
between a receiver r and transmitter s is devel-
oped, where the transmitter can be either a GNSS
satellite or a pulsar.
GNSS Observations
First we will discuss the general equations for
the code and carrier observations for a GNSS sig-
Fig. 2–Pulsar model nal [16–19]. All GNSSs now available and under
Vol. 58, No. 2 Buist et al.: Overview of Pulsar Navigation 155development are transmitting at a number of com- tween receiver r and transmitter s at time t, ssr; f ¼
mon frequencies in the L-band. At present, only tr(t) 2 ts(t 2 ssr; f ) is the signal traveling time
the Russian Glonass is using frequency division between reception at receiver r at tr(t) and trans-
multiple access but will change to code division mission from transmitter s at ts(t 2 ssr; f ). The other
multiple access which is standard in the other terms of the equations will be explained in detail in
GNSSs [17]. the remainder of this section.
The code observation, Psr; f (t), is equal to the dif-
ference between the receiver time at signal recep-
tion, tr(t), and satellite time at signal transmission,
tsf (t,t 2 ssr;f ), multiplied by the speed of light, c. The Pulsar Observations
frequency f indicates the corresponding carrier fre-
For pulsars we will use the code phase notation
quency. For GNSS, this ‘‘true’’ range between user
as the equivalent of the pulse phase observation
and GNSS satellite can be calculated as the position
and consider it a code signal consisting of predomi-
of the second is known and the first can be esti-
nately zeros and only a 1 during a code period
mated using the code observations. This measure-
which would be the pulse period. We will see later
ment is usually called pseudorange as it is biased
on that like the code observation the pulsar obser-
due to the fact that satellite and receiver clocks are
vation is also ambiguous.
not synchronized, and it is based on the transit
As indicated in Figure 4, each pulsar transmits
time from satellite to user. The GNSS signal is gen-
a wideband signal, ranging from low frequency ra-
erated onboard the satellite and time stamped with
dio waves to gamma ray radiation or from 100
the transmission time. At the receiver side, a rep-
MHz to 85 GHz [20] depending on the pulsar’s
lica of the signal is generated and the messages are
characteristics. For pulsar navigation, the pulse
decoded [17, 18]. Without bit synchronization, the
frequency fn of the pulsar n is utilized. The pulse
receiver will measure only the code phase offset,
wave fronts and pulse phase observation for an ob-
wsr;c . The integer number Nsr;c , of code cycles, kc, that
server at a certain position and at time t are
have occurred is initially unknown [17–19]. Each
shown in Figure 5. In this figure the spiral repre-
GNSS code has its own length or period, e.g., the
sents the pulsar. It emits beams of radio waves
coarse/acquisition (C/A)-code of GPS has a 1-msec
which sweep through space as the star rotates,
period, and is constantly repeated in the transmit-
like lighthouse beams, thus from afar pulsars seem
ted signal. wsr;c is provided by the delay lock loop
to flicker or pulsate at their rotation periods.
that keeps the replica signal aligned with the
Therefore in our solar system, shown by an image
received GNSS signal [17–19]. With enough
of the Sun and Earth, the wavefront can be consid-
decoded data and an estimation of the receiver clock
ered as pulse waves. For simplicity the beam radi-
offset, fr(t), Nsr;c (sometimes referred to as the code
ation axis is perpendicular to the rotation axis. In
ambiguity) can be determined.
general this will not be the case (see Figure 2). If
The GNSS receiver also tracks the Doppler
the pulse phase of the pulsar at time T0 is known,
shifted carrier typically with a phase lock loop and
the phase evolution can be written as [6]:
this accumulated observable is referred to as the
carrier phase observation, Fsr; f (t). wsr;fn ðtÞ ¼ wsr;fn ðT0 Þ þ fn ðt T0 Þ
The general equations for the introduced code
and carrier observations for a GNSS signal are: X
m
f m ðt T0 Þm
n
þ ½cycle (3)
m!
Psr;f ðtÞ ¼ cðtr ðtÞ tsf ðt; t ssr; f ÞÞ þ esr; f n¼2
¼ kc Nr;cs
þ cwsr;c þ esr; f where fmn ’s are the known multiple derivatives of
the pulse frequency, which for the pulsar-based
¼ qsr; f ðt; t ssr; f Þ
navigation application could be ignored for the
þ Ir;s f þ Tr;s f þ c½fr ðtÞ fs ðt; t ssr; f Þ order 3 or 4 and higher as these higher order terms
þ c½dr; f ðtÞ dsf ðt; t ssr; f Þ þ esr; f ð1Þ are known to be stable over periods longer than
months [6].
Usr; f ðtÞ ¼ qsr; f ðt; t ssr; f Þ Ir;s f þ Tr;s f þ c½fr ðtÞ
fs ðt; t ssr; f Þ þ c½dr; f ðtÞ dsf ðt; t ssr; f Þþ
þ kf ½/r; f ðt0 Þ þ /sf ðt0 Þ þ kf Nr;s f þ esr; f ð2Þ
where Psr;f (t) is the code observation on frequency
f at time t and Fsr;f (t) is the carrier phase observa-
tion between receiver r and transmitter s on fre-
quency f. qsr; f (t,t 2 ssr; f ) is the geometric distance be- Fig. 4–Electromagnetic spectrum
156 Navigation Summer 2011In the model, kf depends for GNSS on the carrier
frequency f, kc depends on the code period, and kfn
for pulsar navigation depends on the pulsar fre-
quency fn.
Atmospheric Delays on GNSS and Pulsar
Observations
In Eqs. (1) and (2), Tsr;f is the tropospheric and Isr;f
is the ionospheric error between receiver r and
transmitter s at frequency f. Tropospheric delays
cannot be neglected up to an altitude of 16 km above
Earth’s surface at the equator and 9 kilometers
above the poles [17], whereas the ionosphere will
influence signals up to altitudes of more than hun-
dreds of kilometers. The ionospheric effect is fre-
quency-dependent and therefore if observations are
obtained from the same transmitter at two or more
frequencies, a so-called ionosphere-free observation
can be applied [17].
Signals in the L-band from pulsars are affected
the same way as GNSS signals by the atmosphere,
but the X-ray emissions are completely absorbed by
the Earth’s atmosphere, and therefore the use of
that wavelength is limited to space or planetary
bodies without an atmosphere. For pulsars, depend-
ing on the frequency band, the signals are also
affected by the interstellar medium. This effect will
be discussed further in the section ‘‘Principles of
Fig. 5–Principle of pulsar navigation (not on scale)
Pulsar Radio Signal Detection’’ where an experi-
ment in which we observed a pulsar is described.
For the received pulse phase, wsr;fn (t), Doppler
shift also has to be taken into account, but will not
be discussed in detail in this contribution. For a Other Error Terms for GNSS and
more detailed discussion on the observation model Pulsar Observations
for pulsars we refer to [6–9]. In Figure 5 the pulsar
spatial period or wavelength is indicated as fr(t) and fs are clock errors of the receiver r and
kfn ¼ fcn , thus ignoring the variation in pulse fre- transmitter s, respectively. For a pulsar, this last
quency with c as the speed of light. error is caused by irregularities in the pulsar fre-
quency. For example, pulsar ‘‘hiccups’’ have been
observed [20]. For a GNSS signal, the information
on the satellite clock error is transmitted in the
Wavelengths of GNSS and Pulsar Observations
data message with moderate accuracy and a poste-
In Eq. (2), /r,f(t0) is the initial phase on fre- riori this information is available with a higher ac-
quency f in the receiver r, and /sf (t0) is the initial curacy. For pulsars this information is only avail-
phase of the carrier phase at the frequency f in the able a posteriori.
transmitter s. kf is the wavelength and as only a dr,f(t) and dsf (t,t 2 ssr;f ), dr,f(t) and dsf (t,t 2 ssr;f ) are
fractional carrier phase can be measured when a the code and carrier phase delays respectively at
signal is acquired, Nsr;f is the number of complete the receiver r and, for GNSS, in the transmitter s.
carrier phase cycles that is unknown at the signal For pulsars, dsf (t) are the irregularities in the char-
lock-on by the receiver. The phase equation pro- acteristics of the pulsar other than the pulsar fre-
vides an alternative way, if one is able to resolve quency which is modeled in fs(t,t 2 ssr;f ) [20].
the integer ambiguity problem, to utilize GNSS For GNSS, information on the orbit of the satel-
observations as will be discussed later on. lite is transmitted in the data messages in what is
For the pulsar observations in (3), wsr;fn (t) is the called an ephemeris. For pulsars, this information
pulse cycles (both fractional and complete) that is of course not available in the pulsar signal and,
have occurred since T0 and wsr;fn (T0) is the difference as we will see next, also not required in the com-
in initial pulse phase between receiver and pulsar. monly proposed navigation approaches.
Vol. 58, No. 2 Buist et al.: Overview of Pulsar Navigation 157esr;f and esr;f are residual unmodeled error terms definitions as applied by the GNSS community.
on code and carrier observations, respectively. For According to this definition, in absolute positioning
GNSS, esr;f has a typical value of dm-level and esr;f the coordinates of a receiver at an ‘‘unknown’’
is of mm-level. For pulsars, esr;f depends on the pul- point are determined with respect to a reference
sar’s characteristics and the integration time of frame (normally centered in the Earth) by using
the observation. For X-ray pulsars, [9, 10] give a the ‘‘known’’ positions of the GNSS satellites being
typical order of accuracy of hundreds of kilometers tracked. In relative positioning, the coordinates of
for a short integration time (this follows directly a receiver at an ‘‘unknown’’ point are determined
from the pulsar spatial period with a typical pulse with respect to another receiver at a ‘‘known’’
frequency on the order of 1000 Hz), and better point.
than 100 m for an integration time of more than For pulsar-based navigation, absolute positioning
one week. is generally defined as relative to the Barycentric
For radio observations from the pulsar, the accu- center of our solar system, which as it is relative
racy of an observation for a 1 s integration time is to a known point, in GNSS terms is relative posi-
the same as for X-rays as it is determined by the tioning. For pulsar-based navigation, relative posi-
pulse frequency. However, as for X-ray observations tioning is often defined as relative to a previous
a limited number of photons are observed (typically coordinate. We will apply the relative positioning
90 a day according to [6]), but, as radio observations definition of GNSS for pulsar navigation in this
can be made for every pulse period, the expected contribution, but with respect to a reference point
accuracy after integration is much higher. The im- rather than a receiver. Another important differ-
provement in accuracy must be investigated by ence between pulsar- and GNSS-based navigation
experiments as described in a later section. is that the positions of the transmitters are not
known in the former.
As code observations are available for GNSS, we
Linearized Observation Model can calculate the absolute position from the meas-
If we neglect the atmospheric effect and omit the ured range between receiver and satellite. If we
frequency dependence for simplification, the obser- would like to exploit the higher accuracy of the
vation equations can be linearized to [16]: carrier phase observation, we would have to esti-
mate the biases and resolve the ambiguities as
given in Eq. (2). This is more recently attempted
DPsr ðtÞ ¼ usr Dr þ c½Dfr ðtÞ Dfs ðt; t ssr Þ
in an application called Precise Point Positioning
þ c½dr ðtÞ ds ðt; t ssr Þ þ esr ð3Þ [21, 22]. For pulsar-based navigation, the observa-
DUsr ðtÞ ¼ usr Dr þ c½Dfr ðtÞ Dfs ðt; t ssr Þ tions are always relative to some assumed refer-
þ c½dr ðtÞ ds ðt; t ssr Þ þ k½/r ðt0 Þ þ /s ðt0 Þ ence point and therefore a position can only be cal-
culated using relative positioning in GNSS terms
þ k Nrs þ esr ð4Þ
which will be discussed next.
Relative navigation can also be based on deter-
where usr is the line-of-sight vector. For pulsar-based mining the Doppler shift in the received signals,
navigation, the precise positions of the pulsars in which will result in the user’s velocity. After inte-
an absolute reference frame are not available. How- gration, the receiver displacement can be deter-
ever, the positions of celestial bodies are known in mined. Doppler-based navigation has been used in
celestial coordinates and therefore we can calculate the early stages of the GPS system [17]. In this
their directional locations. Because of the scale of contribution we will not discuss these methods in
user position and the GNSS satellite, a user has to further detail.
calculate the line-of-sight using a reasonable esti-
mate of his position, whereas for pulsar navigation,
the line-of-sight to the transmitter can be assumed Differencing the Observation Equations for GNSS
to be the same anywhere in the solar system.
Differencing of observations has its origin in
For pulsar-based navigation some other terms in
interferometry [23]. For GNSS, differencing the ob-
the linearized observation model are omitted. For
servation equations is often applied to explicitly
a more detailed discussion on the observation
eliminate common error terms. If we take the differ-
model for pulsars we refer again to [6–9].
ence between observations at two receivers, referred
to as Receivers 1 and 2, collected at the same obser-
Positioning vation epoch we can write the single difference
equation as:
A number of different definitions of the terms
absolute and relative positioning can be found in
literature. In this contribution we will apply the DPsr12 ðtÞ ¼ usr2 Dr12 þ cDfr12 þ cdr12 þ esr12 (5)
158 Navigation Summer 2011DUsr12 ðtÞ ¼ usr2 Dr12 þ cDfr12 þ cdr12 þ k Nrs12 þ esr12 E½y ¼ Az þ Gb (9)
(6)
D½y ¼ Qyy (10)
where ()12 ¼ ()2 2 ()1.
It can be observed that the transmitter’s clock
error and instrumental delays are eliminated from where E[y] is the expected value and D[y] is the
the equations. dispersion of y. y is the vector of observed minus
For GNSS, we often take the difference of two computed double differenced code and carrier phase
single differences (the so called double difference) to observations of the order 2n, so y ¼ [DPsr12 12
(t) . . .
s1n T
explicitly eliminate the receiver clock errors from DPsr1n
12
(t) DF s12
r12 (t) . . . DF r12 ] . A is a design matrix of
the equations: dimension 2n 3 n containing the wavelengths kf
that link the data vector to the unknown vector of
DPsr12
12
ðtÞ ¼ usr12
2
Dr12 þ esr12
12
(7) ambiguities, z. The G matrix of dimension 2n con-
tains directional information in the form of the
DUsr12
12
ðtÞ ¼ usr12
2
Dr12 þ k Nrs12 þ esr12
12
(8) line-of-sight vectors, [usr12 . . . usr1n ]T. The variance
12 12
matrix of y is given by the positive definite matrix
A triple difference observation is obtained by dif- Qyy which is assumed to be known. For GNSS,
ferencing two double difference observations from mostly the same noise values for carrier observa-
a different observation time t, which is mathemati- tions on the same frequency are assumed. The
cally equivalent to solving a system with equations unknowns in this model are the three values of the
for two epochs. position vector b and an integer ambiguity vector,
[Nsr12
12
. . . Nsr1n
12
]T, containing an ambiguity for each
double differenced carrier observation. Observa-
Differencing the Observation Equations for
tions from at least four satellites are required to
Pulsars
solve the system of equations using single epoch
For pulsar-based navigation, a different approach data.
is usually applied. The most common one, as detailed
in [6–9] uses a reference time-of-arrival almanac, in
Pulsar-Based Relative Positioning
which the time-of-arrival of the pulse trains of each
pulsar is listed. Comparing the relative times of ar- For pulsar-based navigation, we can make use of
rival of the pulses at a location in space different the model from the section ‘‘GNSS-Based Relative
from the reference location will render a set of possi- Positioning,’’ but as discussed in the section ‘‘Ob-
ble solutions for the spacecraft’s location. Multiple servation Model,’’ only the equivalent of the code
pulsar signals are added to resolve the ambiguity of observations is available. Moreover, as each pulsar
the method, as well as possible clock errors. As dis- observation has its own pulse frequency, for every
cussed, because of the large distance between a re- observed pulsar with its unique pulse frequency,
ceiver and the pulsars, the celestial coordinates of fn, the design matrix, A, contains a different kfn for
the pulsars can be considered as constant during each observation and the variance matrix will con-
long periods and the pulse frequency and its deriva- tain different noise values for each observation
tives are well-known. Therefore, the phase of the depending on the observed pulsars’ characteristics.
pulse frequency at time t can be calculated. Conse- As each observation will add an unknown inte-
quently, it is common to take a single difference ger ambiguity, the system of equations cannot be
between the phase predictions at a known location solved using data from a single epoch. For a static
and the observations from a receiver at its current application, if five pulsars are observed, the system
location. By also differencing the uncertainties, could be solved using data from two epochs as we
delays at the pulsar’s side are removed, and by dif- would have seven unknowns and eight observa-
ferencing two single difference observations, the re- tions.
ceiver clock error and instrumental delays are also For completeness we will introduce two more
removed. The relative position vector as obtained methods here for pulsar-based navigation based on
from solving the equations is defined as the position time differencing. The time-of-arrival drift can be
of the receiver relative to the known reference posi- used to determine the Doppler shift of the signals,
tion (usually the Barycentric center). as detailed by Kestilä [11]. This would allow for a
crude relative positioning after integration. A more
exotic navigation method is also presented by Kes-
GNSS-Based Relative Positioning
tilä [11]. As is clear from Eq. (3), each pulsar has
The single frequency model for linearized double a characteristic spin-down rate, which is highly
differenced observation equations as given by Eqs. linear for any conceivable time-frame. If the space-
(7) and (8) in [16], can be written as: craft was to determine the characteristic age of the
Vol. 58, No. 2 Buist et al.: Overview of Pulsar Navigation 159sffiffiffiffiffiffiffiffiffi
pulsar, it would be able to compare this age to the
e 2 ne
age at a given reference location, which would allow fp ¼ (11)
it to determine the distance traveled with respect to pme
that location, as due to the limited speed-of-light,
the pulse train will show an age difference at differ- where ne is the electron number density, and e and
ent locations in space. This method, however, me are the charge and mass of an electron, respec-
requires extremely accurate clocks, as well as very tively.
long integration times. On the other hand, the The effect can be observed as a change in propa-
method does not involve any ambiguity, and will gation velocity of each frequency component. This
render the exact distance with respect to the point effect is called dispersion [20]. Depending on the
of reference. traveling time of the pulse in the ISM (or the dis-
Other pulsar signal properties proposed in litera- tance of the pulsar to the receiver), the effect on
ture for use in navigation, such as pulse intensity, the velocity change will vary. This effect causes the
are more challenging due to the irregularities pulse shape to change from a narrow pulse to a
involved in the emitted signals [20]. wider shape. In order to be able to compensate for
this shape deformation, each pulsar is associated
with a value called dispersion measure (DM) and
is determined by [20]:
PRINCIPLES OF PULSAR RADIO
SIGNAL DETECTION Zd
DM ¼ ne dl (12)
As discussed, most research on pulsar-based
navigation has been on X-ray pulsars. In the next 0
two sections, we will discuss the possibility of
where ne is the electron number density and d is
using radio pulsars for navigation.
the distance measured in parsec (a unit of length
In order to detect a pulsar in the radio frequency
used in astronomy: 1 pc 3.1 3 10þ16 m). DM is
range, an RF front-end is required. This front-end
usually expressed in pc/cm31. The reference for DM
consists of two main blocks. The first block is the
is Earth. DM basically shows how much the pulse
analog processing part and the second block is the
has been affected by ISM. Considering that the
digital pulse detection part (Figure 6). The analog
antenna and receiver are tuned to receive at center
block receives the pulsar signal at frequency f0,
frequency, f0, with a bandwidth of Df, the ISM
selects the required bandwidth Df, down converts
effect can be removed by using a filter with the fol-
the signal to an intermediate frequency (IF), and
lowing transfer function (H) [20]:
amplifies the signal to the level which can be digi-
8 9
tized. After the digitization, the second block applies :f0 þf fp2 ;
i2p
cd
de-dispersion and folding algorithms to shape the Hðf0 þ f Þ ¼ e 2ðf0 þf Þ
(13)
pulse, remove noise, and find the pulsar. The output
will be fed to the back-end which calculates the nav- where f0 is the center frequency, c is the speed of
igation solution. light, d is again the distance to the pulsar, and fp
The concept of the analog block can be found in is the average plasma frequency. Eq. (13) uses f0 þ
[24]. Here we will focus on de-dispersion and fold- f as a variable to emphasize that the received sig-
ing algorithms. The pulse while traveling from the nal frequency is centered at f0 and frequency f is
pulsar to the user passes through the interstellar limited by the receiver bandwidth, i.e., |f| Df/2.
medium (ISM). This medium which is a cold ion- The process of removing the effect of ISM is called
ized plasma, affects different frequency compo- de-dispersion.
nents in the pulse in a similar way as the iono- For GNSS, even for the new signals with wider
sphere around the Earth. bandwidth than the original GPS signals, the dis-
The index of refraction differs for each frequency persive character of the ionosphere is still too
and depends on the plasma frequency, fp, which is small to require the introduced de-dispersion tech-
defined by the following equation [20]: niques [25].
Another task of the front-end is finding the pulse
trails. Since each pulsar has a unique period,
P ¼ f1n , a technique called epoch folding can be
applied in order to detect different pulsars [20].
The process is similar to acquisition in GNSS, how-
ever, without generating a replica signal by the
pulsar receiver.
Folding is similar to integration except that in
Fig. 6–Pulsar receiver architecture folding, the data is broken into a sequence of dis-
160 Navigation Summer 2011Fig. 8–Pulse shape of pulsar B0329þ54, the right side provides
zoom-in [26]
PULSAR EXPERIMENT
In order to validate the approaches for de-disper-
sion and folding described in the previous section,
we established an experiment. Conventionally large
dish antennae with diameter of more than 20 m are
used to detect pulsars in radio frequencies. In this
experiment, we used one phased array antenna
Fig. 7–Epoch folding process
from the Low Frequency Array (LOFAR) radio tele-
scope. This antenna consists of a 5 m by 5 m array
with 16 antenna elements. The distance between
each adjacent element is 1.25 m. By using beam
crete intervals corresponding to the period of the forming techniques, the antenna is able to track the
expected pulsar and then added (or folded) ensur- pulsar. The pulsar we tracked was B0329þ54 which
ing that the pulsar signal is reinforced with each is one of the strongest pulsars visible in the north-
fold, while the noise approaches a mean zero. To ern hemisphere [6]. Figure 8 shows the shape of
illustrate the folding process (see Figure 7), data is this pulsar at 102.755 MHz based on the European
generated by adding uniformly distributed noise to Pulsar Network (EPN) database [26]. Table 1 shows
a periodic pulse. To simulate the noise, pseudoran- the parameters of this pulsar and measurement
dom values are drawn from the standard uniform setup.
distribution on the interval [21, 1]. The pulse train Using the de-dispersion method and epoch folding
is simulated as a square wave with a normalized algorithm as explained previously, we will investi-
amplitude of 1/10000, period P, pulse width W, and gate after what observation time, Tobs, the pulse
W/P ratio 0.01. First a sequence of data for one pe- shape is distinguishable. Figures 9 through 12
riod (t ¼ 0 to t ¼ P) is stored in an array (array A). show the results of the folding process after 20,000
The length of array A depends on the sampling s, 40,000 s, 60,000 s, and 80,000 s of folding. As can
rate (fs) and the period (P), i.e., La ¼ P 3 fs. The be seen, the pulse shape starts to increase as the
next sequence of data (t ¼ P to t ¼ 2P) with length folding process continues and is distinguishable af-
n is stored in another array (array B). These two ter 60,000 s of folding and remains higher than the
arrays are added element by element and stored in threshold afterwards.
array A again. The next sequence of data (t ¼ 2P The pulse detection criteria are pulse shape and
to t ¼ 3P) with length La is read and stored in pulse peak. The amplitude of the pulse should be
array B and added to array A. The process is
repeated until a clear pulse shape is distinguished Table 1—Experiment Data
after a total number of foldings, nf. Since all (1 Jy (Jansky) 5 1026 W m2 Hz1 [20])
points, except the pulse position, are random val-
ues with zero mean, by adding them for infinite Pulsar flux density 347.4 mJy
time, the average value should become zero while DM 26.776 pc/cm3
the values at the pulse position will increase. Pulse period (P) 714.57 ms
Center frequency (f0) 139.0625 MHz
The detection time varies depending on the
Band width (Df) 8.984375 MHz
strength of the pulsar signal, the background noise
Sampling frequency (fs) 762.9394 Hz
level, and the pulse period.
Vol. 58, No. 2 Buist et al.: Overview of Pulsar Navigation 161Fig. 9–Signal shape after 20,000 s of folding. At t ¼ 0.31 s the Fig. 11–Signal shape after 60,000 s of folding. At t ¼ 0.31 s
pulse amplitude is lower than noise the pulse amplitude is higher than noise and the shape is
distinguishable
Fig. 10–Signal shape after 40,000 s of folding. At t ¼ 0.31 s the Fig. 12–Signal shape after 80,000 s of folding. At t ¼ 0.31 s the
pulse amplitude is lower than noise but increasing pulse amplitude grows higher than the noise and the shape
remains the same as before
higher than a certain threshold with respect to the
maximum noise peaks. The detection condition of a the pulse period, W is the pulse width, Tsys is sys-
pulsar signal is selected similarly to the detection tem noise temperature, and np is 1 for single polar-
criteria applied in GPS for weak signals [27–29]: ization observation or 2 if two orthogonal polarized
the peak value should be at least twice the second signals are summed, respectively.
largest peak. At Tobs ¼ 60,000 s the amplitude of Using an antenna with a larger aperture, select-
the folded signal at pulse position is 40% higher ing a frequency band with less background noise,
than the rest of the peaks and is increasing. At avoiding multipath, and applying advanced signal
80,000 s the detection criteria is met. processing techniques will improve the detection
There are a number of factors which can affect time.
the pulse detection time. The relation between As discussed previously, the SNR values for pul-
SNR and observation time can be determined by sar observations for terrestrial and space applica-
the following equation [20]: tions will differ as pulsars signals, like GNSS
9 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi signals, are affected by Earth’s atmosphere. The
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8T WðP WÞ
>
> peak >
> typical atmospheric loss for L-band signals is,
SNR ¼ np tobs Df : ; (14)
Tsys P according to [17], around 0.5 dB. Other than that,
a radio range pulsar receiver will not be too differ-
where tobs is the observation time, Df is the signal ent for terrestrial and space applications as, differ-
bandwidth, Tpeak is the pulse peak amplitude, P is ent from GNSS, a Barycentric reference frame is
162 Navigation Summer 2011applied in which both Earth and spacecraft move 4. Chester, T. J. and Butman, S. A., ‘‘Navigation Using
relative to the Sun. However, for non-static users X-Ray Pulsars,’’ NASA Technical Reports, N81-
on Earth, movements in general are much less pre- 27129, 1981, pp. 22–25.
dictable than the orbits of spacecraft. 5. Hanson, J. E., ‘‘Principles of X-ray Navigation,’’
Ph.D. Thesis, Stanford University, 1996.
In the experiment presented in this section we
6. Sala, J., Urruela, A., Villares, X., Estalella, R., and
showed that with a relatively small phased array
Paredes, J. M., ‘‘Feasibility Study for a Spacecraft
antenna we were able to detect a radio pulsar sig- Navigation System relying on Pulsar Timing Infor-
nal after 60,000 s of folding. This proves that the mation,’’ ARIADNA Study, 03/4202, European Space
above mentioned approach is valid and it can be Agency, June 2004.
used for further processing. More sophisticated 7. Sheikh, S. I., ‘‘The Use of Variable Celestial X-ray
detection criteria could be applied to decrease the Sources for Spacecraft Navigation,’’ Ph.D. Thesis,
required detection time. University of Maryland, 2005.
8. Sheikh, S. I., Pines, D. J., Ray, S. R., Wood, K. S.,
Lovellette, M. N., and Wolff, M. T., ‘‘Spacecraft Navi-
CONCLUSIONS gation Using X-Ray Pulsars,’’ Journal of Guidance,
Control, and Dynamics, Vol. 29, No. 1, 2006, pp. 49–
In this contribution we explained the principles
63.
of pulsar-based navigation by comparing it with 9. Sheikh, S. I., Golshan, A. R., and Pines, D. J.,
well-known satellite-based navigation systems such ‘‘Absolute and Relative Position Determination Using
as GNSS. We derived the observation equations for Variable Celestial X-Ray Sources,’’ Proceedings of the
both GNSS and pulsar-based navigation and dis- 30th Annual AAS Guidance and Control conference,
cussed in detail the differences between the two 2007, pp. 855–874.
navigation methods. Up until now most work in 10. Hanson, J., Sheikh, S. Graven, P., and Collins, J.,
this area has been focused on X-ray observations. ‘‘Noise Analysis for X-Ray Navigation Systems,’’ Pro-
The radio frequencies emitted by pulsars were con- ceedings of the Position, Location and Navigation
sidered too weak to be useable for navigation appli- Symposium (PLANS), 2008 IEEE/ION, 5–8 May
2008, pp. 704–713.
cations; however, we think that with modern proc-
11. Kestilä, A. A., Engelen, S., Gill, E. K. A., Verhoeven,
essing and antenna techniques it could be possible
C. J. M., Bentum, M. J., and Irahhauten, Z., ‘‘An
to develop a system with an antenna of reasonable Extensive and Autonomous Deep Space Navigation
size to track radio pulsars in space and perhaps System Using Radio Pulsars,’’ Proceedings of the 61st
even on Earth. The experiment described in this International Astronautical Congress, Prague, 2010,
contribution showed that with a relatively small IAC-10.B2.4.8.
phased array antenna, it is possible to detect a ra- 12. US Coast Guard notice, Retrieved October 13, 2010,
dio pulsar after applying folding. http://www.navcen.uscg.gov/?pageName¼loranMain
13. Allan, D. W. ‘‘Millisecond Pulsar Rivals Best Atomic
Clock Stability,’’ 41st Annual Frequency Control
Symposium, 1987, pp. 2–11.
ACKNOWLEDGMENTS
14. Emadzadeh, A. A., ‘‘Relative Navigation between
We would like to acknowledge the people at Two Spacecraft Using X-ray Pulsars,’’ Ph.D. Thesis,
ASTRON, who generously provided the LOFAR University of California, Los Angeles, 2009.
data; in particular Joeri van Leeuwen, as he 15. Liu, J., Ma, J., Tian, J.-W., Kang, Z.-W., and White,
helped with importing the raw data, and Stefan P., ‘‘X-Ray Pulsar Navigation Method for Spacecraft
with Pulsar Direction Error,’’ Journal of Advanced
Wijnholds. Moreover Mark Bentum from ASTRON
Space Research, 2010, doi:10.1016/j.asr.2010.08.019.
is acknowledged. The anonymous reviewers and
16. Teunissen, P. J. G. and Kleusberg, A., GPS for Geod-
Christian Tiberius are thanked for their valuable esy, Springer, Berlin, Heidelberg, New York, 1998.
comments during the review process. 17. Misra, P. and Enge, P., Global Positioning System,
Signals, Measurements, and Performance, Ganga-
Jamuna Press, 2001.
REFERENCES 18. Kaplan, E. and Hegarty, C., Understanding GPS:
Principles and Applications, 2nd Ed., Artech House,
1. Hewish, A., Bell, S. J., Pilkington, J. D. H., Scott, P. 2006.
F., and Collins, R. A., ‘‘Observation of a Rapidly Pul- 19. Braasch, M. S. and Van Dierendonck, A. J., ‘‘GPS Re-
sating Radio Source,’’ Nature, Vol. 217, February 24, ceiver Architectures and Measurements,’’ Proceed-
1968, pp. 709–713. ings of the IEEE, Vol. 87, No. 1, 2002, pp. 48–64.
2. Sagan, C., Salzman, L., and Drake, F., ‘‘A Message 20. Lorimer, D. and Kramer, M., Handbook of Pulsar As-
from Earth,’’ Report, Science, Vol. 175, 25 February tronomy, Cambridge University Press, 2005, ISBN:
1972, pp. 881–884. 978–0521828239.
3. Downs, G. S., ‘‘Interplanetary Navigation Using Pul- 21. Zumberge, J. F., Heftin, M. B., Jefferson, D. C. Wat-
sating Radio Sources,’’ NASA Technical Reports, kins, M. M., and Webb, F. H., ‘‘Precise Point Posi-
N74-34150, 1974, pp. 1–12. tioning for the Efficient and Robust Analysis of GPS
Vol. 58, No. 2 Buist et al.: Overview of Pulsar Navigation 163Data from Large Networks,’’ Journal of Geophysical 26. European Pulsar Network database, Retrieved
Research, Vol. 102, No.B3, March 10, 1997, pp. October 13, 2010, http://www.mpifr-bonn.mpg.de/div/
5005–5017. pulsar/data/browser.html
22. Le, A. Q., ‘‘Achieving Decimetre Accuracy with Sin- 27. Jung, J., ‘‘Implementation of Correlation Power Peak
gle Frequency Standalone GPS Positioning,’’ Proceed- Ratio Based Signal Detection Method,’’ Proceedings
ings of the 17th International Technical Meeting of of the 17th International Technical Meeting of the
the Satellite Division of The Institute of Navigation Satellite Division of The Institute of Navigation (ION
(ION GNSS 2004), Long Beach, CA, September GNSS 2004), Long Beach, CA, September 2004, pp.
2004, pp. 1881–1892. 486–490.
23. Counselman, C. C., Hinteregger, H. F., and Shapiro, 28. Borre, K., Akos, D. M., Bertelsen, N., Rinder, P., and
I. I., ‘‘Astronomical Applications of Differential Inter- Jensen, S. H., A Software-Defined GPS and Galileo
ferometry,’’ Science, Vol. 178, 1972, pp. 607–608. Receiver: A Single-Frequency Approach, ISBN
24. Behzad, R., RF Microelectronics, 1998, Prentice-Hall 0817643907, Birkhauser, 2007.
Inc. 29. Geiger, B. C., Soudan, M., and Vogel, C., ‘‘On the
25. Pany, T., Eissfeller, B., and Winkel, ‘‘Analysis of the Detection Probability of Parallel Code Phase Search
Ionospheric Influence on Signal Propagation and Algorithms in GPS Receivers,’’ Proceedings of the
Tracking of Binary Offset Carrier (BOC) Signals For 2010 IEEE 21st International Symposium on Perso-
Galileo and GPS,’’ Proceedings of the 27th General nal Indoor and Mobile Radio Communications
Assembly of the International Union of Radio Sci- (PIMRC), doi: 10.1109/PIMRC.2010.5672040, 2010,
ence, Maastricht, 2002. pp. 865–870.
164 Navigation Summer 2011You can also read
