(Horizontal DOP) and VDOP (Vertical DOP), as a function of three-satellite geom- etry and clock stability. ... still wishes to utilize GPS. This can be accomplished ...
GPS Navigation Using Three Satellites and a Precise Clock MARK A. STURZA Litton Received
Aero March
Products,
Canoga
Park, Ca.
1983
ABSTRACT Navigation using GPS generally requires that the user track four satellites to resolve his 3-D spatial position and time bias. There are several reasons why it is desirable to navigate while tracking only three satellites: -The proposed 18 satellite GPS constellation will exhibit substantial periods of poor four satellite geometry over large geographic areas several times a day. -Failure of a GPS satellite in orbit. will result in periods of only three-satellite availability over large geographic areas. -During establishment of the operational GPS constellation, there will be long periods of three-satellite GPS coverage. Three-satellite GPS navigation can be accomplished by equipping the user with a precise clock. The required stability of the clock is a function of the maximum allowable PDOP (Position Dilution of Precision) and the time interval between updates of the clock. Clock updates can be accomplished by tracking four GPS satellites, tracking one GPS satellite from a known location, or conventional time transfer methods. This paper presents a formula for computing PDOP and its components, HDOP (Horizontal DOP) and VDOP (Vertical DOP), as a function of three-satellite geometry and clock stability. This formula is used to plot HDOP and VDOP versus time for representative high quality quartz crystal and low-cost rubidium clocks for two scenarios of satellite geometry. The stability and environmental sensitivity of high quality quartz crystal and low-cost rubidium clocks are discussed.
INTRODUCTION The NAVSTAR Global Positioning System (GPS) is a satellite-based radionavigation system intended to provide highly accurate three dimensional position and precise time on a continuous global basis. When the system becomes fully operational in late 1988, it will consist of 18 satellites in six orbital planes inclined at 55”. Each plane will contain three satellites spaced 120” apart in 12-hour orbits. The relative phasing of the satellites from one orbital plane to the next is 40”. Each satellite will continually transmit navigation signals at Ll = 1575.42MHz and L2 = 1227.6MHz consisting of the P-code ranging signal (10.23MBPS), the C/A code ranging signal (l.O23MBPS), and 50BPS data providing satellite ephemeris and clock bias information. Navigation using GPS is accomplished by passive triangulation. The GPS user equipment measures the pseudorange to four satellites, computes the position of the four satellites using the received ephemeris data; and processes 122
Sturza: Navigation
with Precise Clock
123
the pseudorange measurements and satellite positions to estimate three-dimensional user position and precise time. There are situations where the user only has three satellites available and still wishes to utilize GPS. This can be accomplished by equipping the user with a precise clock. Three situations where three satellite and a precise clock GPS navigation would be desirable are discussed in the following section. THREE SATELLITE
SITUATIONS
Poor Geometry The lgsatellite GPS constellation described above has been shown’ to exhibit areas of poor four satellite geometry at certain times. A user relying on four satellite GPS navigation in these outage areas will experience extremely poor navigation performance. Four outage areas occur at the same time and repeat at the same location twice a day. The outages disappear and a new set of four appear 40 minutes later at other locations. Thus, there are a total of 72 outage areas on the earth at various times. The duration of the outages ranges from 5 to 30 minutes with an average of 10 minutes. Satellite
Failure
Failure of a GPS satellite in orbit would result in periods of only three satellite coverage of approximately 30-minute duration over large geographic areas2. It may require months for a replacement satellite to be launched. Constellation
Build-up
Five GPS satellites will be maintained in orbit during the 1983 to 1986 time period. This will result in 4 to 12 hours per day of three satellite coverage depending on location. From 1986 to 1988, the number of satellites in orbit will be increased from 5 to 18 satellites. This will increase the area and time of three satellite coverage. By the end of 1987, three satellite coverage will be essentially continuously global. FOUR SATELLITE
PERFORMANCES
Before deriving performance equations for three satellite and a precise clock navigation, four satellite performance equations are reviewed. For four satellite GPS navigation, the linearized navigation equations are given by3 Hqx = p where x is the four element position and time correction vector p is the four element pseudorange residual vector
124
Global Positioning
System
and H4 is the direction cosine matrix mapping the LOS (line-of-sight) from the satellites to the coordinate system.
vectors
Let x = [AX Ay AZ ATIT P =
CX,is the direction j th coordinate.
cosine between
[Apl
APZ
APS
Ap41T
the LOS vector to the i th satellite
and the
The linearized equations can be solved for position and time corrections function of the pseudorange residuals as follows x = H4-1
as a
Q.
Since this relationship is linear, it holds for the errors in the position and time correction and the pseudorange residual errors. Thus, G = H4-kp where E, is the position and time correction error vector and E, is the pseudorange residual error vector. To obtain the position and time error variances, the covariance matrix of the position and time correction errors is computed. This matrix has the form
v,---vy--cov (x) =[--vz---VT 1
where V,, V,, V, and VT are the position and time variances and the off-diagonal elements are functions of the correlations between the errors. The covariance matrix is given by cov (x) = E[E, Exrl = H4-l cov (p) HT = [H,T cov (p)-’ H,l-’
Now making the assumption that the pseudorange residual errors are uncorrelated from satellite to satellite and equal for each satellite, the pseudorange residual covariance matrix reduces to cov (p) = a,?. Thus
cov (x) = up2 [HhT H41 -l. We see that the position and time error variances are functions of the diagonal elements of [H4T H,]-’ this leads to the concept of GDOP (geometric dilution of precision).
Sturza: Navigation
with Precise Clock
125
GDOP is defined by
[HdTH&l.
GDOP = d/Trace Thus GDOP = VV,
+ V, + V, + V,/ u p.
GDOP is the amplification factor of pseudorange error variance to the combined position and time error variance. GDOP can be partitioned into separate position and time variances as follows:
where
GDOP2
= PDOP2 + TDOP2
PDOP
= d/v,
and TDOP
= fir/a,
PDOP can further
where
+ V, + Vz/up is position
be partitioned
DOP
is time DOP. into horizontal
PDOP2
= HDOP2 + VDOP2
HDOP
= v’m/op
and VDOP
= flZ/a,
and vertical
is horizontal is vertical
components
DOP
DOP.
GDOP, PDOP and TDOP are invariant with respect to coordinate system while HDOP and VDOP only make sense in a LTP (local tangent plane) system. THREE SATELLITE
AND A PRECISE CLOCK PERFORMANCE
A performance measure analogous to PDOP for four satellite GPS navigation can be developed for three satellite and a precise clock navigation. The linearized navigation equations for this case are given by p = Hzx+bl where
x = [A.xAyAzlT p = [API bz 4dT Hs = 1 =
[ 1 a11
a1y
a1z
a2.x
a2y
a2z
w.z
a3y
11
1
a3r
1lT
and b is the clock bias. The position correction
as a function x = H,-l
Thus the position
correction
of pseudorange [p - bll.
error covariance
cov (x) = Ha-’
residual
matrix
is
[a; I + cr$ 1 lTl HsmT
= a; [(H; H&l where ub2 in the clock bias error variance.
+ 3
H,-’
1 lT HzmT]
is given by
126
Global Positioning
We see that elements of
the position
error
[(H: Hs)-l So analogously
System
variances
+ $
are a function
of the diagonal
Hs-l 1 lT Hz-*].
to PDOP we define
PDOPs
=
Truce [(HT H&l
+ 5
H,-1 1 lT H,-7.
J
= vvz + vy+ V*lu, We define k2 = ab2/ap2 to be the ratio of clock bias error variance to pseudorange residual error variance then PDOP,
[(HTH,)-’
= d/Trace
HDOPs and VDOPs are defined similarly
+ k2H3-’ 1 lT H,-*I. to HDOP and VDOP such that
PDOPs2 = HDOPs2 + VDOPs2 HDOP, = v’~/cJ, VDOP, = flz/a, As in the four satellite case, PDOPs is invariant to coordinate system, and HDOPs and VDOPs only make sense in a LTP system. We see that PDOP, is a function of three satellite geometry and clock bias error variance. CLOCK STABILITY
AND SENSITIVITY
Clock bias error variance is a function of clock stability, environmental sensitivity, and time since the last clock update. A clock is mechanized by counting cycles from an oscillator. The four most common types of oscillators are quartz crystal, rubidium, cesium beam, and hydrogen maser. Due to size, weight, power and cost considerations only quartz crystal and rubidium oscillators are practical for airborne GPS navigation equipment. The stability and environmental sensitivity of high quality quartz crystal and low cost rubidium oscillators are discussed. Oscillator stability is measured by means of Allan variance u2
(T)
=
2TW13
where 4(t) is the oscillator phase at time t T is averaging time and < > denotes infinite time averaging. Figure 1 shows representative plots of Allan variance as a function of averaging time for high quality quartz crystal and low cost rubidium oscillators. The up-turn of the plots is a result of quartz aging for the quartz crystal oscillator and cell component aging for the rubidium oscillator.4 The clock bias error variance due to oscillator stability is given by (Tba (7)
=
Tu,
(7).
Sturza: Navigation
with Precise Clock
127
to-’ -
d
-
OSCILLATOA
lo-=-
I
10-13
I
I
IO'
100
I
I
I
I
I
I
I
I
J
103
5 P
lo4
z
105
g 3 .-
,@
E x =
10'
7,AVEAAGlNGTlME(SECONOS)
Fig,
I-Allan
variance
of typical
high
quality
quartz
and
low cost rubidium
oscillators.
Plots of obS(T) as a function of elapsed time since last clock update for clocks utilizing the high quality quartz crystal and low cost rubidium oscillators described above are shown in Figure 2. It can be seen that for elapsed times, less than one minute, the quartz crystal clock has slightly better performance; but that for long elapsed times, greater than one hour, the rubidium clock has significantly superior performance. Both quartz crystal and rubidium oscillators are sensitive to environmental conditions. Quartz crystal oscillators are more sensitive to temperature, acceleration, vibration, shock, and radiation; while rubidium oscillators are more sensitive to pressure and magnetic field. The bias errors resulting from these sensitivities will dominate the bias errors due to stability shown in Figure 2 for long elapsed times. In a typical air transport environment, Figure 2 is a good model of clock performance for elapsed times less than 1 hour. CLOCK UPDATES Neglecting the effect of environmental variance is given by
sensitivities,
ug (t) = a&J + u& (t - to)
the clock bias error
Global Positioning
7,ELAPSEO
Fig.
e-Bias
error
variance
TIME
of typical
System
ISECONDS)
XTAL
and Rb clocks.
o$ (t) is the clock bias error variance at time t c& is the clock bias error variance at the time, to, of the last clock update &, (t - to) is the clock bias error variance due to clock stability. The clock bias error variance due to stability, &, was discussed in the preceeding section and is shown in Figure 2. The clock bias error variance at the time of the last clock update, &, depends on how the update was accomplished. If the update was accomplished by four satellite navigation, then C& = TDOP2 . cr?. If the update was accomplished known location, then
by tracking
one GPS satellite
For clock updates accomplished clock bias error variances are:5
by conventional
time transfer
from an exactly
methods,
the
Sturza:
Navigation
129
with Precise Clock
METHOD TRANSIT TV LINE 10 PORTABLE CESIUM
crm (SECONDS) 1 x 10-S 1 x 10-G 1 x 10-G
The factor k used in the calculation of PDOPs and defined by lzZ = u&E is given by k2 (t) = [&o + a& (t - to)l/u; Thus if the last clock update was accomplished then k2 (t) = TDOP2 (to) +
by four satellite
GPS navigation,
& (t - to) , us
Nominal values for the pseudorange residual error variance, a;, are 5 meters (1.7 x 10m8 seconds) for dual frequency P-code operation and 20 meters (6.7 x 10ds seconds) for single frequency C/A-code operation. PLOTS OF HDOP AND VDOP Three satellite and a precise clock navigation is evaluated for two scenarios by plotting HDOPs and VDOPB. The first scenario is navigation during the 27-minute outage of the 18satellite constellation occurring at N 5” latitude, E 35” longitude, and 0 meters altitude from 0326 GMT to 0353 GMT on July 1, 1985. The second scenario is navigation at N 34” 11.5’ latitude, W 118” 35.4’ longitude and 200 meters altitude on February 20, 1983 following the four satellite visibility window provided by NAVSTAR’s 3, 4, 5, and 6 currently in orbit. The four satellite visibility window ends at 1105 GMT. Figure 3 is a plot of HDOP4, HDOP9 (quartz crystal clock, XTAL), and HDOP, (rubidium clock, Rb) during the l&satellite constellation outage. HDOP, has an average value of 1.75 for the five-minute intervals before and after the outage. HDOP, (XTAL) has an average value of 4.9 during the outage with minimum and maximum values of 4.7 and 5.7 respectively. HDOP, (Rb) has an average value of 3.6 during the outage with minimum and maximum values of 2.5 and 5.7 respectively. Figure 4 is a plot of VDOP4, VDOP, (XTAL), and VDOP, (Rb) during the 18-satellite constellation outage. VDOP, averages 1.8 for the five minutes before and after the outage. VDOP, (XTAL) has an average value of 6.7 during the outage with minimum and maximum values of 2.0 and 12.6 respectively. VDOPs (Rb) has an average value of 1.7 during the outage with minimum and maximum values of 1.6 and 2.0 respectively. Using HDOP, and VDOP, before and after the outage as references the average relative performance of three satellite and a precise clock navigation during the outage is RELATIVE Before and After Outage Quartz Crystal Clock Rubidium Clock
HDOP 1.00 0.36 0.49
PERFORMANCE VDOP 1.00 0.27 1.06
Global Positioning
130
I
,
1 5
i
System
I 10
I 15 TIME
1 20
I 25
IMINUTESI
t EN0 OF OUTAGE
START OF OUTAGE
Fig. J-Plot of HDOP,, satellite constellation.
HDOP,
(XTAL)
and HDOPJ
/ /’
/I
(Rb)
1’
/
/ //
during
/
/ //
a 27-minute
, /’
outage
of the 18-
/ 1’
-
VOOP
----
VOOPj
IXTALI
. . .. . . . . . . . “DrJP3 ,Rb) /
a’
/
I /
/'
/’
\ II
I 5
I 10
;
I 15 TIME
I 20
I 26
IMINUTES)
t ENOOF OUTAOE
EEEoF
Fig. I-Plot ofVDOP4, satellite constallation.
VDOP,
(XZ’AL)
and
VDOP,
(Rb)
d wing
a 27-minute
outage
of the 18-
Sturza:
Navigation
131
with Precise Clock
-
HDOP
----
HDOP;
(XTAL)
.... ... ..... “DlJP3 (Rb,
4-
3-
2-
I-
I 5
, 0
I 10
t
I 15 TIME
I 20
I 25
I 30
(MINUTES)
END OF FOUR SATELLITE COVERAGE
Fig. 5-Plot availability
of HDOP, window.
(XTAL)
and HDOPs
(Rb)
following
end of Phase
II constellation
4-satellite
Figure 5 is a plot of HDOP, during the four satellite visibility window; and HDOP, (XTAL) and HDOPB (Rb) following the four satellite visibility window. HDOP, has an average value of 2.5 during the last ten minutes of the four satellite visibility window. HDOP, (XTAL) starts at 5.7 and increases to 9.7 after 9 minutes. HDOP, (Rb) starts at 5.4 and increases to 9.7 after 30 minutes. Figure 6 is a plot of VDOP, during the four satellite visibility window; and VDOP, (XTAL) and VDOP, (Rb) following the four satellite visibility window. VDOP, has an average value of 3.4 during the last ten minutes of the four satellites visibility window. VDOP, (XTAL) starts at 6.6 and increases to 9.0 after 4 minutes. VDOP, (Rb) starts at 6.3 and increases to 9.7 after 21 minutes. The plots of HDOP, (XTAL), HDOPs (Rb), VDOPs (XTAL), and VDOP, (Rb) continue to increase until only two satellites are visible, then they become infinite. CONCLUSION In conclusion it has been shown that three-satellite GPS navigation can be accomplished by equipping the user with a precise clock. The navigation performance provided, as measured by HDOP, and VDOPB, has been shown to be a function of three satellite geometry and clock stability. Three satellite and a precise clock GPS navigation during a typical outage of the 1%satellite constellation has been shown to have l/z to l/3 the position accuracy of four satellite GPS navigation before and after the outage depending on clock quality. Three satellite and a precise clock GPS navigation following the end of the four satellite visibility window provided by NAVSTAR’s 3, 4, 5, and 6 was shown to provide up to 30 minutes of additional GPS navigation depending on performance requirements and clock stability.
132
Global Positioning
System
---
-
VOOPq VOOP3(HAL)
........... voop3 ,Rb)
TIME(MINUTES)
Fig. 6-P& availability
of VDOPS window.
(XTAL)
and
VDOPS
(Rb)
following
end ofPhase
II constellation
4-satellite
ACKNOWLEDGEMENTS The author wishes to thank Dr. Daniel Chen of the Litton Aero Products Division for generating the LOS vectors for the 18 satellite constellation used in the calculations of HDOP, and VDOP,. And Dr. Tae Kwon of the Litton Guidance and Control Systems Division for discussing Litton’s Rubidium Frequency Standard Unit. REFERENCES 1. Kruh, Pierre, “The NAVSTAR Global Positioning System Six-Plane l&Satellite Constellation,” NTC Record-1981, New Orleans, Louisiana, November 1981, pp E9.3.1-8. of NAVSTAR GPS for Civil Aviation,” ION 2. Shively, Curtis A., “Reliability Proceedings of the National Aerospace Meeting, Moffett Field, Ca., March 1982, pp 111-130. 3. Jorgensen, P. S., “NAVSTARiGlobal Positioning System 18-Satellite Constellation” Navigation, Vol. 27, Number 2, pp 89-100. 4. Fruehouf, Hugo, “Precision Oscillators and their Role and Performance in Navigation Systems,” PLANS 82 Record, IEEE AES Society, Atlantic City, N.J., December 1982, pp 289-301. 5. Cashion, R. E., Klepczynski, W. J., and Putkovich, K., “The Use of Transit for Time Distribution” Navigation, Vol. 26, Number 1, pp 63-69.
