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CPS
Modeling for
Designing
Aerospace
Vehicle Navigation
Systems
JOHN
J.
DOUGHERTY
HOSSNY EL-SHERIEF
DANIEL J. SIMON
GARY
A.
WHITMER
1RW
Systems Integration Group
The complexity
of
the design of a Global Positioning System
(GPS)
user
segment,
as
well as the performance demanded
of
the components, depends
on
user
requirements
such
as
total
navigation accuracy.
Other
factors,
Cor
instance
the
expected
satellite/vehicle geometry
or
the accuracy
of
an
accompanying
inertial navigation system,
can
also affect the
user
segment
design
Models
of
GPS
measurements
are
used to predict
user
segment
performance
at
various levels. Design curves
are
developed which
illustrate
the
relationship
between user requirements,
the
user
segment design,
and
component performance.
I.
INTRODUCTION
Because of the versatility provided
by
its global
availability and the passive nature of the user
segment, the Global Positioning
System (GPS)
is
being used
in
a wide range of aerospace applications.
Among these are on-board navigators and trajectory
references for range safety and for testing inertial
navigation systems.
GPS
is
a satellite navigation
system developed and maintained
by
the United States
Department of Defense. It includes 24 satellites in
semigeosynchronous orbit providing continuous global
coverage and excellent navigation accuracy
[1].
A GPS user segment comprises the hardware and
software employed
by
the user to obtain navigation
information from GPS. The user segment must
be
designed so that some user performance requirement
is
met. For a GPS user segment employed as a
component of an on-board navigation system, the
requirement
is
usually on the total navigation accuracy.
For a GPS user segment
as
an
autonomous navigator,
the requirement
is
usually on the GPS navigation
accuracy itself. For a GPS user segment as a trajectory
reference for testing inertial navigation systems, the
requirement
is
usually on the ability to achieve test
objectives such as estimating the inertial navigation
system accuracy
[2,
3].
GPS user segment designs can
be
broadly classified
into
two
categories: receiver- and translator-based
designs. A GPS receiver processes GPS signals to
estimate its own position and velocity. This information
can be used directly,
or
can be combined with other
navigation estimates (from an inertial navigation
system, for instance)
to
get a best-estimate of the
vehicle position and velocity
[4-6].
A GPS receiver
must compensate for known measurement errors in
real-time
[7].
A GPS translator,
on
the other hand,
is
a relatively simple device whose function
is
to
frequency shift ("translate") the GPS signals from one
frequency band to another, such as a telemetry band.
The translated signal
is
then retransmitted to a ground
receiving station, where it
is
time-tagged and processed
or recorded for later processing.
Oftentimes an application will require the use
of
a receiver-based user segment. For instance, using
GPS for on-board navigation usually demands a
receiver. On the other hand, when using a GPS user
segment as a navigation reference for testing inertial
navigation systems, a translator-based segment offers
several advantages, including low cost, weight, and
power consumption and high reliability. Furthermore,
ground postprocessing of the signals allows for the
use of highly accurate satellite orbital information not
available in real-time and the use of highly detailed
corrections.
It
also allows analysts to iteratively edit the
data and respond to anomalous conditions. The result
is
accuracy better than that achievable
by
a receiver
doing real-time navigation.
The performance factors that affect the design
of
a GPS user segment are considered here. Models
presented previously
in
the literature (and referenced
throughout this work) are used
to
predict GPS
performance as quantified
by
several specific
measures. Although the results apply to various other
applications, GPS used as a navigation reference for
testing inertial navigation systems is considered as a
specific example.
The inertial navigation system
is
a
key
component
of
aircraft, missiles, sounding rockets, launch vehicles,
and other aerospace systems.
It
generally comprises
three
or
more accelerometers, three
or
more gyros,
and associated hardware and electronics. The inertial
instruments (the accelerometers and gyros, known
collectively as the inertial measurement unit or
IMU)
provide the navigation computer with the acceleration
and attitude data necessary to generate velocity and
position information
[3J.
The velocity and position data
are in turn used
by
the guidance and control computer
to achieve mission objectives, such as intercepting a
target
or
inserting a payload into orbit. Errors in the
IMU
data result in errors in the navigated state and
hinder the achievement
of
these objectives.
Flight testing
is
an important tool in evaluating the
contribution
of
the
IMU
to errors in the navigated
state. Estimating the source and magnitude
of
the
IMU
errors requires a separate trajectory reference.
In the past, ground-based radars
or
a second on-board
IMU
have been used to provide the reference. The
second option is usually prohibitive in terms
of
both
cost and payload restrictions, while radars suffer
from limitations in both geometry and accuracy.
Recent flight-testing has demonstrated that a GPS
user segment
can
provide a small, light, affordable,
and accurate trajectory reference system for evaluating
IMU
errors
[8J.
Section II describes how the GPS user segment
design affects the ability to meet user requirements.
Important features of a user segment design, as well
as their effect
on
GPS data quality, are described. The
GPS error model used in the study
is
presented. Also
included
is
an
outline of the methods used to flight-test
IMUs and the manner in which the data are processed.
Several different measures can be used to
assess how well the user segment
is
performing.
The measures may quantify one-dimensional
or
three-dimensional accuracy, or they may reflect the
ability
to
meet overall user requirements. Section
III defines these measures, including those used in
flight-testing IMUs.
Parametric studies were performed to assess the
sensitivity
of
the instrumentation system performance
to the GPS user segment design. Section IV presents
results which
can
be used to determine the complexity
of
a user segment required to achieve given GPS
navigation accuracies as well as the broader flight-test
objectives.
Finally, conclusions are presented in Section
V.
Included
is
a discussion
of
how the design curves
Fig.
1.
User segment design.
developed in the previous sections can
be
used for
other applications of GPS.
II.
USER
REQUIREMENTS
AND
THE
GPS
USER
SEGMENT
A.
Contributors to System Performance
The GPS user segment functions as part
of
a
system designed to achieve some application-specific
objective. The ability to meet performance
requirements which quantify that objective depends
on the design of the user segment as well as other
factors relating to the system performance. This
relationship
is
illustrated in Fig.
1.
Each box represents
either a measure of performance
or
a factor affecting
performance; boxes higher in the figure depend
on
boxes connected to them from below.
At the top
of
the figure
is
the user's requirement
on the performance of the whole system. For
GPS/inertial navigation system hybrids, the user's
requirement would typically be on the total navigation
accuracy. For an autonomous GPS navigation system,
the user requirement would be
on
the GPS navigation
accuracy. For the case
of
GPS used as a trajectory
reference for flight-testing inertial navigation systems,
the user requirement would
be
on measures, such as
estimation uncertainties,
of
the ability to estimate the
errors
of
the system.
The three-dimensional measurement accuracy
of
the GPS user segment can
be
determined
independently
of
other components in the user's
system, as illustrated in the second level of Fig.
1.
It
depends on the satellite geometry, the vehicle flight
path, and the one-dimensional GPS measurement
accuracy
[9].
In
general for a GPS receiver, the
measurements are in the form
of
satellite-to-receiver
TABLE I
Ionospheric Refraction Correction
I-a
Accuracy (feet) (Gauss-Markov)
Range
(XRAI
~
Delta Range (X
OR
/)
Single Frequency
25.
1.6
Dual Frequency
8.
0.02
Note:
~AI
10
12
s;
rhRI = 200
s.
Roman symbols here correspond to italic symbols
in
text.
TABLE
II
Measurement Correction Parameters
Type Symbol
I-a
Accuracy
Coarse
Fmc
Antenna Phase Center Location (feet) rc
AlA
0.10
0.10
0.10
0.05
0.05
0.05
Ephemeride Accuracy (HLC frame)
Position (feet)
rc
Xi
.up
II.
40.
24.
7.
25.
15.
Velocity (rt/sec) rc
.005
.003
.003
.003
.002
.002
Tropospheric Refraction Scale Factor (nd)
GM
XrsF'
0.3 0.02
Note: rc =random constant,
GM
= Gauss-Markov process.
r~SF
=
2000
s.
Underlined symbols here correspond to boldface symbols
in
text.
Roman symbols here correspond to italic symbols
in
text.
range, determined from the codes modulated onto
the GPS signal, and the change in that range (i.e.,
delta range) as derived from the phase of the GPS
signaL For a GPS translator, the measurements are
in the form
of
satellite-to-receiver-to-ground recorder
range and delta range. The measurement aceuracy
depends
on
the
receiver or translator design
[10,
11],
the antenna design, the accuracy
of
the satellite
ephemeris data, relativity and atmospheric effects,
and
fixed
characteristics of GPS
[12].
Contributors to
GPS accuracy are summarized in Tables I and II. The
data are based on the literature
[13,
14]
and flight-test
experience.
Receivers and translators can be designed to
process the
Ll
(1575.42 MHz)
or
L2
(1227.60 MHz)
signals or both. Processing two frequencies allows for
better ionospheric refraction corrections, as shown
in
Thble I
[1].
In addition, receivers and translators ean
be designed to process one
or
both of the GPS codes.
The GPS
LI
signal
is
quadrature modulated
by
two
pseudorandom codes, a 1.023 Mbit/s coarse/acquisition
(C/A) code
and
a 10.23 Mbit/s precision (P) code
[7].
The type
of
code used determines the range precision
which can be achieved. Note that the required
bandwidth
is
2 MHz for the CIA code and 20 MHz for
the P code.
The design and calibration
of
the antenna affects
the accuracy of the phase-derived delta range
measurement.
The
antenna phase induces error
through three different mechanisms, as discussed
in Section lIB. The accuracy of the phase center
calibration also affects the calculation
of
vehicle
reference point to phase center lever arm, effectively
introducing measurement errors. Table II presents the
phase center location uncertainty for both a fine and
coarse calibration.
The
GPS satellite ephemerides are obtained either
in real-time from the GPS navigation message
[15]
or from satellite tracking data spanning a period
of
several days both before and after the time
of
interest.
The accuracy
of
the ephemerides can
be
expressed as
position and velocity standard deviations in height,
long-track, cross-track (HLC) coordinates; Thble
II presents the values used for this study
[13].
The
HLC coordinate frame is a right-handed, noninertial
coordinate system rotating with the satellite orbital
motion. The first axis
is
parallel
to
a line segment
connecting the Earth's center and the satellite; the
TABLE III
User Segment Configurations
Configuration
I
Code
I
Frequency
I
Meas. Calc.
I
A
P
dual
fine
B P
dual
coarse
C
CIA
single
fine
D
CIA
single
coarse
third
is
parallel to the satellite orbital angular velocity
vector.
The
second axis completes the orthogonal set.
Two
different data correction schemes are
considered. The coarser correction scheme adjusts
the GPS measurements for satellite clock phase and
frequency, drift in the translator carrier frequency, and
changes in the signal path length due to ionospheric
and tropospheric refraction. A coarse correction
for relativistic effects
is
also built into the GPS
clock frequency. A finer approach does the coarse
corrections plus precise corrections for general and
special relativistic effects due to the vehicle motion and
higher accuracy tropospheric refraction corrections
based
on
weather data. These corrections are
summarized in Tables
II and III. Fixed characteristics
of GPS include the satellite clock phase and frequency
accuracy after correction. Contributing to the GPS
delta range resolution
are
the carrier wavelength,
errors in the phase tracking loop, and atmospheric
effects.
B.
GPS
Error Model
The various contributors to GPS measurement
errors were modeled and then simulated to assess
their impact on the user segment performance. A
description of the model used in the simulation follows.
Although this model applies to a translator-based user
segment, it can
be
used for receivers
by
taking the
receive time and location to be coincident with the
translation time and location.
The GPS range measurement
is
modeled as
Ri(tk) = ri(tk) +
Si(tk)TCkH(tr)X~
+ (t;;' - to)Si(t;; l
CkH(tj;*)X~
+
(tt
-
to)X~F
+
c/10
9
X~p
+ Bi(tk)XfsF(tk)
+ XkAI(tk) + Si(tk)T CRB(tk)XLA + vk(tk) (1)
where
Ri
is
the measured range from the ith satellite to
the vehicle to the ground;
ri
is
the true range;
tk
is
the ground receive time;
t;;
is
the vehicle translation time;
tj;*
is
the satellite transmission time;
to
is
the reference time;
Si
is
the unit vector from the vehicle to the
ith
satellite;
CkH
is
the direction cosine matrix from the HLC
frame for the ith satellite to the reference frame;
CRB
is
the direction cosine matrix from the vehicle
body frame
to
the referene frame;
c
is
the speed
of
light;
Bi
is
the tropospheric refraction correction for the
ith
satellite;
X~,
X~,
X~F'
X~p,
XfSF' and XkAI are
per-satellite GPS errors (see Tables
I-II);
XLA
are
global GPS errors (see Table II);
vk
is
the range measurement noise for the
ith
satellite.
Values for
Bi vary from 100
ft
on the ground to
zero above approximately 50
mi
altitude.
The
1
(J
value
for satellite clock phase error
X~
p
is
10 ns; the 1
(J
value for satellite clock frequency error
X~F
is
one
part in 10
12
. The 1
(J
values for
vk
are
25
ft for
CIA
code and 5 ft for P code.
The GPS delta range measurement
is
modeled as
follows.
Di(tk) =
i(tk)
+ [Si(t;;)T
CkH(tr)
.
- Si(tk-l)T
CkH(t;;~I)]X~
+
[(tr
- to)Si(tk)TckH(t;;*)
-
(t;;~1
- to)Si(tk_l)T
CkH(t;;:'l)]X~
+ (t;;' -
t;;~l)X~F
+ Bi(tk)XfsF(tk)
- Bi
(tk-l)XfsF(tk-l)
+ XbRI(tk) -
XbRI(tk-I)
+ XbRA(tk) - XbRA(tk-J)
+
[si(t;;l
CRB(tk) -
Si(t;;_ll
CRB(t;;_l)]XLA
+ XbR(tk) + XSR(tk) +
V~C(tk)
-
V~c(tk-l)
(2)
where
Di(tk)
is
the measured delta range from the ith
satellite to the vehicle to the ground over the interval
(tk-l,td;
d
i
(tk)
is
the true delta range;