GPS Navigation Using Three Satellites and a Precise Clock

GPS Navigation Using Three Satellites and a Precise Clock

MARK A. STURZA

Litton Aero Products, Canoga Park, Ca. Received March 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 radio- navigation 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 pro- viding 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) vectors from the satellites to the coordinate system.

Let x = [AX Ay AZ ATIT P = [Apl APZ APS Ap41T

CX, is the direction cosine between the LOS vector to the i th satellite and the j th coordinate.

The linearized equations can be solved for position and time corrections as a function of the pseudorange residuals as follows

x = H4-1 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,--- cov (x) = [ 1 -vy--- --vz- --- VT 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 uncor- related 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).


GDOP is defined by

GDOP = d/Trace [HdT H&l.

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:

GDOP2 = PDOP2 + TDOP2

where PDOP = d/v, + V, + Vz/up is position DOP

and TDOP = fir/a, is time DOP.

PDOP can further be partitioned into horizontal and vertical components

PDOP2 = HDOP2 + VDOP2

where HDOP = v’m/op is horizontal DOP

and VDOP = flZ/a, is vertical 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

a11 a1y a1z

Hs = [ 1 a2.x a2y a2z

w.z a3y a3r

1 = 11 1 1lT

and b is the clock bias. The position correction as a function of pseudorange residual is given by

x = H,-l [p - bll.

Thus the position correction error covariance matrix is

cov (x) = Ha-’ [a; I + cr$ 1 lTl HsmT

= a; [(H; H&l + 3 H,-’ 1 lT HzmT]

where ub2 in the clock bias error variance.


We see that the position error variances are a function of the diagonal elements of

[(H: Hs)-l + $ Hs-l 1 lT Hz-*].

So analogously to PDOP we define

PDOPs = J

Truce [(HT H&l + 5 H,-1 1 lT H,-7.

= 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, = d/Trace [(HTH,)-’ + k2H3-’ 1 lT H,-*I.

HDOPs and VDOPs are defined similarly 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) = < L+(t) - w-4 + w-w2 > Y 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 aver-

aging 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).


to-’ -

d -

OSCILLATOA

lo-=-

10-13 I I I I I I I I I I I J

100 IO' 103 5 lo4 P

z 105 g ,@ E x

10' 3 .- =

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, accel- eration, 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 sensitivities, the clock bias error variance is given by

ug (t) = a&J + u& (t - to)

Global Positioning System

7,ELAPSEO TIME ISECONDS)

Fig. e-Bias error variance of typical 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 by tracking one GPS satellite from an exactly known location, then

For clock updates accomplished by conventional time transfer methods, the clock bias error variances are:5


METHOD crm (SECONDS) TRANSIT 1 x 10-S TV LINE 10 1 x 10-G PORTABLE CESIUM 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 by four satellite GPS navigation, then

k2 (t) = TDOP2 (to) + & (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 18- satellite 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

Before and After Outage Quartz Crystal Clock Rubidium Clock

RELATIVE PERFORMANCE

HDOP VDOP 1.00 1.00 0.36 0.27 0.49 1.06


I , 1 I I 1 I i 5 10 15 20 25

TIME IMINUTESI t START OF EN0 OF OUTAGE OUTAGE

Fig. J-Plot of HDOP,, HDOP, (XTAL) and HDOPJ (Rb) during a 27-minute outage of the 18- satellite constellation.

1’ / /’ , / / / / / / / /

1’ /I

/’ / - VOOP

-- - - VOOPj IXTALI

/' . . . . . . . . . . . “DrJP3 ,Rb)

/ /’ / I /

a’

\

I-

I I I I I I

; 5 10 15 20 26

TIME IMINUTES) t

EEEoF ENOOF OUTAOE

Fig. I-Plot ofVDOP4, VDOP, (XZ’AL) and VDOP, (Rb) d wing a 27-minute outage of the 18- satellite constallation.


- HDOP

-- - - HDOP; (XTAL) . . . . . . . . . . . . “DlJP3 (Rb, 4-

3-

2-

I-

, I I I I I I 0 5 10 15 20 25 30

t TIME (MINUTES) END OF FOUR SATELLITE COVERAGE

Fig. 5-Plot of HDOP, (XTAL) and HDOPs (Rb) following end of Phase II constellation 4-satellite availability window.

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.


- VOOPq --- - VOOP3 (HAL) . . . . . . . . . . . voop3 ,Rb)

TIME (MINUTES)

Fig. 6-P& of VDOPS (XTAL) and VDOPS (Rb) following end ofPhase II constellation 4-satellite availability window.

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 Fre- quency Standard Unit.

REFERENCES

1.

2.

3.

4.

5.

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. Shively, Curtis A., “Reliability of NAVSTAR GPS for Civil Aviation,” ION Proceedings of the National Aerospace Meeting, Moffett Field, Ca., March 1982, pp 111-130. Jorgensen, P. S., “NAVSTARiGlobal Positioning System 18-Satellite Constellation” Navigation, Vol. 27, Number 2, pp 89-100. 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. Cashion, R. E., Klepczynski, W. J., and Putkovich, K., “The Use of Transit for Time Distribution” Navigation, Vol. 26, Number 1, pp 63-69.

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GPS Navigation Using Three Satellites and a Precise Clock