Electronics and Communications in Japan, Part 1, Vol. 76, No. 1, 1993 Tnnrlstcd from Denshi Joho Tushin Gakkai Ronbunlhi, Vol. 75-B-1I. No. 2, February 1992, pp. 138-144

Distribution of Positioning Error in a Global Positioning System

Kuniharu Okuda, Member

Yuge National College of Maritime Technology, Ehime, Japan 794-25

Akio Yasuda, Member

Tokyo University of Mercantile Marine, Tokyo, Japan 135

SUMMARY

The distribution of the positioning error in a global positioning system (GPS) depends on the allocation of satellites. Thus, the agreement between the covariance ellipse derived from the allocation of satellites and that derived from the positioning error distribution of the actually measured data is examined for two reception systems, i.e., the sequential and the parallel processing systems. The result shows that the two covariance ellipses agree well, although the situation depends a little on the shape of the ellipse as well as the azimuth and elevation of the satellite.

The size of the ellipse is approximately one-half in the parallel processing system compared to that of the sequential one. It is proved by the results of the analysis that the distribution of the error is anticipated before the measurement or the positioning during navigation and is represented by an ellipse. The method is applied to a navigating ship. It is shown that the display of the error distribution together with the positioned point is useful to the navigator in determining the steering of the ship in the sense that the reliability for the position of the ship is inCteased.

Key words: Global positioning system; error dis- tribution; covariance ellipse.

1. Introduction

It is well known that error distribution in the posi- tioning by GPS depends on the allocation of satelIites [ 1- 31. Considering that it will be useful information to the user if the error distribution is anticipated before the measurement, we attempted a quantitative evaluation of the error distribution by the following procedures [l]:

In the determination of GDOP, which is used as the evaluation coefficient for the positioning accuracy, the elements of the covariance matrix determined by the allocation of satellites are proportional to the variance and covariance of the variables of the bivariate normal distribution. Thus, we find the covariance ellipse wm- sponding to the equi-probability curve. Then we deter- mine the other covariance ellipse from the positioning error distribution actually measured at the fixed point and compared these two covariance ellipses.

In the previous study, only the timedivision switching receiver was employed in the actual measure- ment. By contrast, two types of receivers, i.e., time-divi- sion switching and the parallel processing, are used si- multaneously in the present study to analyze the posi- tioning data in detail. Another point is that the positioned point is displayed during navigation with the covariance ellipse, based on the satellite allocations. Since the area

95 ISSN8756-6621/93/00014o!l5 1993 Scripta Technics, Inc.

-

2. Determination of Covariance 2 W L 320 i b , , "\ 86; 13[ 1 7 *

-0 2240: w :S.V.No. .c -- I2 12 -

Ellipse

J .S .7 (hour)

This ellipse is called the covariance ellipse [4]. The square-root of C is proportional to the length of the axis of the ellipse. The probability P that the stochastic vari- ables x' and y' are observed to be inside the ellipse is a function of C, being given by P = 1 - exp(-C/2).

The covariance ellipse for the actual measurement is determined as follows. Let x' and y' be the errors in the positioning in the longitude and latitude, respectively. Their standard deviations and the correlation coefficient are calculated. By substituting the results into Eq. (1). the ellipse is determined.

The covariance ellipse based on the satellite alloca- tions is determined as follows. Using the magnification factors am and an, the elements of the covariance matrix which is deduced from the allocation of satellites and used to deduce the DOP (HDOP) in two-dimensional and PDOP threedimensional positioning), of the standad deviation a. of the pseudo-ranging error, ax and uy in Eq. (1) are given by

The correlation coefficient p, is given by

(3)

where aq is also an element of the covariance matrix.

In Eq. (2), a, is determined by

s is-\: w 1 - 1 :

2

\ 6 -

/ 9

r 13

3

16'

6

12 '

17

10 14 18 22 0 4 8

J . S . T ( h o u r ) 31 16 31 17

Fig. 1. Azimuth and elevation of GPS satellites.

where d , is the square root of standard deviation of the directional positioning errors ox and uy and given by

If uo is determined beforehand as a common parameter for dl satellites, it is pssible to anticipate the error distribution before the measuremeut or for each measure- m t during navigation. Since the two-dimmsional posi- tion is sufficient for the ship navigation, the ellipse, which gives the distributed area of positioning results, is a projection on the x-y plane of the ellipsoid in the case of threedimensional positioning.

3. Method of Data Acquisition

3.1. Fiied-point measurement

The measurement at the fixed point is executed using receiver JLR-4000 of JRC CO. and G'IT-3000 of Sony Co. JLR-4000 is a timedivision switching receiver which has one processing channel and measures the pseudo-range by sequentially switching the signals from the satellites. The positioning is made for each second. GTT-3000 is a parallel processing receiver which has four processing channels. The positioning is made for each 0.1 s.

The antenna is placed on the terrace of Yuge Na- tional College of Maritime Technology at the height of 27 m above sea level. The position of the antenna is 34'15.153" in latitudeand 133'12.406'E in longitude, as read from the chart of the Tokyo geodetic system, converting the data into WGS-84 geodetic system. The result of measurements in the two receivers is read into the PC-9801 automatically for each 5 s and stored on the floppy disk. The recorded data are the date, time, posi- tion, height, satellite vehicle number used in the position, and PDOP (or HDOP). For G'IT-3000, the azimuth and elevation of the satellite used in the positioning also are read. The data were acquired from March 15 to 17, 1990 using two receivers. Figure 1 shows the temporal varia- tion of the azimuth and elevation of the satellites used in the analysis from March 16 to 17 [6].

3.2. Measurement during navigation

The measurement during navigation is made by installing JLR-4000 on the training ship Yuge-maru (336 gross ton) of Yuge National College of Maritime Tech. As in the case of the fixed-point measurement, the result of positioning is recorded automatically on floppy disk using PC-9801. The velocity and the course of the ship obtained by the receiver are recorded in addition to the data in the fixed-point measurement. The azimuth and elevation of the satellite used in the positioning are read from the display of the receiver and are recorded.

The PPI image of the radar also is recorded on the video tape to determine the position of the ship as refer- ence data for the GPS positioning.

The ship followed a route, starting from the Taka- matsu Harbor of Kagawa Prefecture, directed to Yuge, through the Bisan Set0 North Traffic Route. While the

1990.7.22 16:05 - 16:55 N

16:05 S

Fig. 2. Allocation of GPS satellites on running.

ship was anchored at Takamatsu Harbor, the fixed-point measurement is made, and the difference between the average of the measured values and the position on the chart is determined. The deviation is added to the posi- tion determined by GPS during navigation. The deviation is -0.231' in the latitude direction and +0.167' in the longitude direction.

The combination of satellites used in this measure- ment is 11-17-19 and 02-11-17, and the average HDOP is 1.4. When the data are converted from the geodetic system of WGS-84 used in GPS to the Japanese geodetic system, -1 1 " to -12" must be added to the latitude and +9" to +lo" must be added to the longitude for the measurement around this latitude. However, the result of conversion does not agree with the position on the chart. Consequently, the forementioned method of correction is applied.

Figure 2 shows the azimuth and elevation of the satellite used in the navigation. In this figure, the zenith is placed at the center and the horizon is placed at the circle periphery. The time shown in the figure is at the time when the ship passed the Bisan Set0 North Traffic Route.

97

Table 1. Result of positioning by timedivision and comparison of covariance ellipses (March 15-17, 1990).

(0.18)

0.13 (0.02)

I 1 1

( 1.00111) (0.004’) (0.008’) (0.005m) (1.47)

6.39111 15.126’ 12.117’ 0.006nm 3.60 ( 0.92d (0.001’) (0.002’) (0.001nm) (0.05)

combination

ratio

03-1 1-13-14

03-13-18-19

06- 12- 16- I7

03- 18- 19 0.97

7 0.89 (0.13)

5 1.05 (0.07)

10 1.17 (0.14)

09-12-16-17

AXlS

iifference

-0.82’ (3.47‘)

8.48’ ( 13.49’)

13.61 (6.61’)

-5.57’ (4.86’)

1.89‘

angular

-

(lI.lO*)

-5.84’ (25.13’)

Average of total samples

Eccen- tricity

0 0 Latitude Longitude d , I I DOP

0.69 (0.02)

7.02111 ( 1.281)

15.107’ I 12.398’ 1 0.0llnm (0.005’) (0.005’) (0.002nm)

3.01 (0.17)

0.45 I 6 . 3 h 115.121’ 112.417’ I 0.010nm I 3.06

0.66 1 5.80111 1 15.123’ 1 12.418’ 1 0.013nm 1 5.71 (0.04) ( 0.561) (0.002’) (0.005’) (0.002nm) (0.69) -~

0.48 (0.10)

0.29 (0.06)

~~

6.301~ ( 0 . 5 2 ~ )

6.68111 ( 0.87m)

15.130’ (0.ool’)

15.129’ (0.002’

12.423’ (0.003’ )

12.416’ (0.002’ )

0.006nm (0.00 I nm)

0.005nm (0.001nm)

3.97 (0.13)

~~

2.96 (0.13)

4. Result of Analysis and Discussions

4.1. Fixed-point measurement

Tables 1 and 2 compare the covariance ellipse based on the actual measurement and that based on the satellite allocations. Table 1 is the result of timedivision switching system, and Table 2 is the result of the parallel processing system. The positioning data are from March 15 to 17, 1990. A sample is composed of 60 positioning data, where the signals are received continuously for 5 min for the same satellite combination in the time-divi- sion switching system and the parallel processing system. For each satellite combination, the short axis ratio, the axis angular difference, eccentricity, standard deviation a, of the measurement error of the pseudo-range, lati- tude and longitude (34”N and 133% is used as the refer- ence), d, and the average of DOP are deduced. The figure in the parentheses on the lower side is the standard deviation for the average.

The short axis ratio is defined here, being 4- culated using the short axis of the ellipse by actual

measurement as the numerator and the value by the satellite allocation as the denominator. The long axis does not differ much in the two cases; and, consequently, the distribution spread is compared by the short axis. The axis angular difference is the angular difference between the principal axis of the two ellipses. The clockwise direction is taken as positive, and the principal axis of the ellipse based on the satellite allocation is used as the reference. The eccentricity is the value of the ellipse based on the satellite allocation. DOP is HDOP for the combination of three satellites and is PDOP for the com- bination of four satellites.

DOP is almost less than 7, indicating that the posi- tioning accuracy is satisfactory. The short axis ratio is around 1 for both reception systems, and the standard deviation is not very large. The axis angular difference is around 0 for the average of the timedivision switch- ing system. It seems that the direction of the principal axis is nearly the same, but the standard deviation may reach 12 deg, depending on the combination of satellites. In the parallel processing system, there are satellite combinations where the average of the axis angular

98

Table 2. Result of positioning by parallel processing and comparison of covariance ellipses (March 15-17, 1990)

11.44M ~ ( 0.90m)

I I, I

02- 06-09

03- 18- 19

03- 11 -13-14

03-13-18-19

06- 12- 1 6- 17

09-12-16-17

combination tricity

8 0.88 -3.00’ 0.69 (0.19) (1.W) (0.02)

29 0.94 1.48’ 0.45 (0.11) (12.21’) (0.18)

7 1.02 1.12’ 0.43 (0.14) (4.75’) (0.02)

5 1.02 1.32’ 0.66 (0.05) (4.27’1 (0.04)

10 1.10 -0.33’ 0.48 (0.18) (10.70’) (0.10)

~ ~~

11 0.96 -3.71’ 0.20 (0.07) (12.46’) (0.06)

15.136’ (0.005’ )

15.134’ (0.002’)

N 34’15.127’N

12.455’ 0.02411~ (0.005’ ) (0.003nm)

12.462’ 0.012nm (0.004’) (0.001nm)

by ‘the a I I Mat i on of satel I i tes

# : S . V . N o . S +: Position by CPS

Fig. 3. C~mparison of two covariance ellipses.

Average of

13.98111 ( 1.37111)

12.49m ( 1.33~1)

12.21111 ( 0.93111)

10.64m ( I . l l n l )

11.57111 ( 0.78111)

otal samples

Latitude Longitude d, I I 15.095’ 12.4 18’ 0 . 0 2 3 ~ (0.005’ ) (0.006’ (0.003nn)

15.174’ 12.430’ 0.021nm (0.045’) H- (0.008’) (0.01 Inn)

15.127’ 1 12.462’ 1 0.012nm (0.003’) (0.001’) (0.002nm)

DOP

3.01 (0.17)

3.07 (1.47)

3.60 (0.05)

5.71 0.69)

3.97 (0.13)

2.96 (0.13)

difference is large. The standard deviation also exceeds 25 deg.

There are two reasons for this property. One is the shape of the ellipse. When the eccentricity is small and the shape of the ellipse is close to a circle, the axis angu- lar difference exhibits a large standard deviation. This implies that the principal axis may change greatly by a d l variation of the actual data when the shape is close to a circle. In the parallel processing system, 4, and variation of actually measured data are smaller than those in the timedivision switching system. In addition, the angle of the principal axis varies greatly when the shape is close to the circle.

Figure 3 is a comparison example of the covariance ellipse based on the measured data and that based on the satellite allocations. The measured data remain in the ellipse based on the satellite allocation, although the direction of the principal axis differs greatly (axis angu- lar difference A0 is A 0 - -24.5’). Those ellipses are

99

0,a : Satellite Combinations o f 02-06-09 0 , . : Satellite Combinations o f 03-18-19

n Difference of ~ 10,. Axial Ansle U $ 8 v

Y- b 2 Y- E 1 " .- I I

0 10 20 30 40

Time (min)

Fig. 4. Difference of axial angles and HDOP (in 5 min).

drawn by setting C - 6 in Eq. (1) and the average value as the center. It should contain 95 percent of the mea- sured points. The thick-line ellipse is based on the mea- sured data. The thin-line ellipse is based on the satellite allocation. The dotted-line circle is the circle with radius Zf- (which contains 95 percent of the measured points in the radial deviation). The plus mark indicates the lneasud point.

The other reason is the azimuth and elevation values of the satellite. The ellipse based on the satellite allocations is drawn based on these two parameters for the satellite. The mepsured data in this study are record- ed at the rate of one sample in 5 min. The satellite is moving during this period. The azimuth and the elevation values to determine the ellipse based on the satellite allocations are those at the middle of 5 min. However, depending on the satellite allocations, it may happen that the principal axis may change greatly in 5 min. This may be one of the reasons to produce a large difference.

Figure 4 is an example of the temporal variation of the principal axis direction. For the satellite combination of 03-18-19, there is a great change (black square); but for the combination 02-06-09, the change is small (white

. .. . , . .

16:45

lo00 I

500

133' 0

Fig. 5. Positions and covariance ellipses by GPS on running in Bisan Set0 N o d Traffic Route.

100

Table 3. Parameters of covariance ellipse during navigation

Axis angular

difference

Short axis

90. 7.22 Satellite combination PDOP

16110 106-09-11-14 I 109.0111

16:25

16:30

16:35

16:15 106-09-11-14 I 107.7111

02-09-11-14 155.11

02-09-11-14 149.6111

02-09-11-14 147.9111

16:20 I 02-09-11-14 1 162.4111

53.5m

55. l m

56.3111

7 1 . 0 ~

60.9111

64.2n

6.1' 3 . 8

4.3' 4 . 1

2.8' 4 . 4

- 0.2' 5 . 0

- 3.9' 6 .1

-10.2' 9.0

~~

16:40 102-06-11-14 I 102.9111

16145

16:50

16155

02-09-1!-14 162.2111

02-09-11-14 208.5111

02-09-11-14 348.9111

square). The black and white circles indicate, respective- ly, the HDOP. It is proved that the standard deviation of the axis angular difference is larger for the satellite combinations with a large change of the principal axis direction.

In the timedivision switching system, d , is approximately 0.009 to 0.023 nautical mile. However, that of the parallel processing system is approximately 0.005 to 0.013 nautical mile, which is approximately one-half of the timedivision switching system. The reason for the reduction is that the difference of the receiving system is reflected on the difference of the distribution of the measured data, which produces a difference in the standard deviation uo of the pseudo- range measuremt error. The average value of uo in the timedivision switching system is approximately 10.6 to 14.0 m, while that of the parallel procesSing system is

I I

I I 53.4111 I 17.1' 1 3.2

53.8111 I 18.3' I 3 . 5

54.3111 1 18.8' 1 3.8

52.2111 1 7.6' I 3.7

67.1111 I -16.0' 1 16.7

approximately 5.8 m to 7.0 m [7], which is approximate- ly one-half of the timedivision switching system.

4.2. Measurement during navigation

The data were acquired on the ship which sailed on the Bisnn Set0 North Traffic Route. This route is so narrow that the comparison of the GPS measurement point and the actual track is easy.

Figure 5 shows the GPS measurement points in the navigation on the Bisan Seto North Traffic Route, to- gether with the covariance ellipse with 95 percent proba- bility with the measurement point as the center for each minute. The plus sign indicates the GPS measurement point, and the dotted line is the route on the chart. The cross mark indicates the positioning by the radar for each

101

2 min. The positioning by the radar is made independent- ly on video tape, which may contain the human individu- al error. Consequently, no quantitative comparison is made between the GPS positioning and the radar posi- tioning. Since the time is recorded on the tape at 1-min intervals, it is difficult to align the radar positioning time to the GPS time. The time difference is several to several tens of seconds.

To draw the ellipse during navigation, a,, is set as 14.53 m, which includes 95 percent of a, in Table 1. Table 3 shows the parameters of the covariance ellipse and PDOP for each 5 min from 16:05 to 1655. PDOP changes greatly from 3.2 to 16.7, changing from the satisfactory positioning accuracy to deteriorated values. Even if DOP is the same, the shape of the ellipse depe- nds on the satellite allocations. In this study, the long axis of the ellipse is 349 m and the short axis is 67 m for PDOP of 16.7. Whether or not this value implies suffi- cient accuracy for a ship sailing through a narrow chan- nel, the reliability for the positioning result is improved by indicating the position not by a point but by an area.

5. Conclusions

The accuracy of the GPS system should be evaluat- ed by the comparison of the track by GPS and that deter- mined by the radar. However, this is not executed in this study by the following various reasons. The purpose of this study is to display the position by an area The indi- vidual human error is contained in the analysis of the track by the radar. The measurement time cannot exactly be aligned. The conversion of the data between the geo- detic systems seems to contain a problem. The error distribution of GPS positioning depends on the satellite combination. It is difficult to discuss the accuracy at the same level, since the two data have different contents to permit a quantitative analysis.

To display the result of positioning by an ellipse, as is considered in this study, the standard deviation of the pseudo-range measurement error should be deter- mined beforehand for each receivinglprocessing system, and it is necessary also to combine the result into the software. A system is already on the market which

combines the GPS receiver with a plotter so that the track of the ship is displayed. The forementioned software should desirably be combined with such systems.

In the case considered in this navigation, the ship should not leave the route and the navigator should pay attention to the measurement error in the sidewise direc- tion. The data thus are useful as the reference data in the positioning by cross bearing and radar when the ellipse is long in the sidewise direction of the navigation. It should be noted that not all of the scheduled satellites were launched in February 1991 and DOP values are still changing.

Acknowledgement. The authors are grateful for the assistance in the data acquisition during navigation by crews of the training ship of Yuge National College Maritime Tech.

1.

2.

3.

4.

5.

6.

7.

REFERENCES

A. Yasuda, M. Hirata, K. Okuda, and H. Imatsu. The error distribution and GDOP in GPS. ROC. Jap. Inst. Navig., 79, pp. 25-31 (Sept. 1988). K. Okuda. Accuracies of position fixes obtained by the shipping GPS receiver. Jour. Jap. Inst. Navig., 94, pp. 39-43 (Dec. 1987). Y. Matsuno, Y. Yamanaka, N. Shimada, M. Uchia- yama, and M. Hidaka On result of positioning by GPS. Jour. Jap. Inst. Navig., 102. pp. 17-26 (Dec. 1989). S. Brandt (Trs.: H. Yoshiro, H. Takahashi. and Y. Koyanagi). Statistical and Computational Method in Data Analysis. North-Holland Publishing compa- ny (1976). Jap. Geod. Soc. GPS-Accurate positioning by satellite, pp. 121-140. Jap. Geod. SOC. (Nov. 1989). T. Ishibashi, K. Shimada, and M. Higashiguchi. Experimental results for GPS by a vehicle on the expressways. Trans. (B-II) I.E.I.C.E., Japan, J72-B-

S. Kimura. Perspectives of NAVSTAIUGPS. Radio Navig., 26, pp. 23-36 (1980).

II, 10, pp. 557-564 (&t. 1989).

102

AUTHORS (from left to right)

Kuniharu Okuda graduated in 1973 from the Dept. of Fishery, Nihon University, and obtained a Master's degree in 1976 from Tokyo University Maritime and became an Assistant. He was an Assistant and later a Lecturer. Since 1983, he has been an Assoc. Prof., Yuge Nat'l College Maritime Tech. He is engaged in research on NNSS, pseudo-radar image, GPS, and remote sensing. He is a member of the Jap. Inst. Navig.; Inf. Roc. Soc., Jap.; and Jap. Soc. Remote Sensing.

Akio Yasuda graduated in 1966 from the Dept. of Electrical Eng., Nagoya Institute of Technology, and obtained a Dr. of Eng. degree from Nagoya University. Served as Assistant at Nagoya University and Assoc. Prof. at Tokyo University of Mercantile Marine. Since 1987, he has been Prof. at Tokyo University of Mercantile Marine. He is engaged in research on plasma measurement by laser, development of marine wave meter, BS and GMS reception on board, satellite communica- tion and positioning. He is a member of IEEJ, Laser Soc. Jap.; Jap. Inst. Navig.; Jap. Soc. Plasma Science and Nuclear Fusion Research; and IEEE.

103