Precise kinematic GPS positioning

Herbert LANDAU Institute of Astronomical and Physical Geodesy

University FAF Munich Werner-Heisenberg-Weg 39

D-8014 Neubiberg, F.R. Germany

PRECISE K I N E M A T I C GPS P O S I T I O N I N G

EXPERIENCES ON A LAND VEHICLE

USING TI 4100 RECEIVERS AND SOFTWARE

Abstract

Tests in relative kinematic positioning using TI 4100 GPS receivers on a land

vehicle were carried out in September 1986 in the testnet "Werdenfelser Land" south

o f Munich. Over the northern east-west traverse several stations with distances o f 400

to I000 m were kinematically coordinated by use o f dual-frequency code and carrier

phase measurements. All cycle slips were removed by a specific algorithm based on

Kalman filtering. Position agreements o f centimeter-level were reached in a comparison

o f forward and backward run at sly stations. The paper presents a description o f the

applied model and an analysis o f the results.

1. Introduction

As soon as the NAVSTAR/GPS system wil l be ful ly established the precise relative positioning of a moving GPS receiver on land, sea, and space will play an important role in different fields like airborne gravimetry, airborne photogrammetry, laser profiling, and offshore seismic observations. Various applications of the GPS carrier phase in static positioning techniques have shown the possibility to reach cm- accuracy. The major problem in applying the carrier phase in kinematic positioning is the occurrence of cycle slips. It is "relatively" easy to overcome this problem in static positioning techniques, but it is more diff icult in kinematic applications since the position of the moving receiver is changing and in the beginning unknown Solving the problem of cycle slip detection and fixing is therefore the key to precise kinematic positioning by use of the carrier phase.

Results of tests in an aircraft were presented by Mader (1986), showing decimeter accuracy. First tests concerning kinematic positioning on a land vehicle were carried out and described by Remondi (1985a, b. 1986). The subject of his work was to demonstrate the use of L-band carrier phase measurements in kinematic surveys under the assumption that no loss of lock occurs. Remondi describes the use of changing the tracking bandwidth to 16 Hz while moving and to 0.7 Hz while standing on the marks in order to avoid loss of lock during travelling between the stations. In the test described in the following this change was not possible since the TI 4100s were operating

Bull. Gdod. 63 (1989)pp. 85-96.

85

Herbert LANDAU

with the commercial TI software. Under practical considerations the assumption to maintain phase lock with a large bandwidth during the whole observation session seems more or less theoretical. In practice a lot of cycle slips occur in kinematic GPS applications due to signal blockage between satellite and moving receiver.

Further results of land vehicle positioning were presented by Cannon et al. (1986), Kleusberg et al. (1986), Lachape|le et al. (1986) and Seeber etal. (1986). All these publications point out the problem of cycle slip fixing and end up with accuracies of about 0 . 5 m - 2 m .

Methods of cycle slip fixing for a moving receiver are described by Goad (1986), Mader (1986) and Bastos and Landau (1988).

The present paper will describe a relative kinematic positioning test, discuss the models for cycle slip fixing and positioning and present results obtained from the processing of dual-frequency code and carrier phase data.

2. Descr ip t ion o f the Test Measurements

The tests were carried out in September 1986 on an east-west traverse of the inertial test network "Werdenfelser Land" located about 100 km south of Munich (Rein et al., 1986). The practical work was prepared and done by members of the Institute of Astronomical and Physical Geodesy using T I 4 ] 0 0 receivers and the commercial operating software. The traverse consists of seven control points with separations of about 400m up to 1000m and a length of 3 k.m (F igure I ) . The seven control stations were all equipped with survey tripods prepared to hold the GPS antenna. A private car was used to transport the GPS receiver with the antenna mounted on its top. During the whole observation session a static receiver was observing on a reference station 6 km away from the traverse.

ILl O

I--I

J

7 6 5 0 250 500 750 t000

i t" 2t ' f,l" 22 ' i~, *~ 23 ' i t " 24' LONGITUDE

Fig. I - K i n e m a t i c test traverse

The static receiver was working in user dynamic class 0 and the moving receiver in class ] corresponding to tracking bandwidths of 0.'7 Hz and 5 l-!z,

respectively. The maximum acceleration allowed in class ] is 6 m/s 2 . No larger band width was chosen since this would increase the noise level in the measurements. Since the commercial TI operating software does not allow changing of the bandwidth in

86

PRECISE KINEMATIC GPS POSITIONING

the operating mode this would result in worse positioning results. In addition it becomes more dif f icult to f ix cycle slips for a moving receiver if the noise level increases. The measurement sampling rate for both receivers was three seconds.

The whole observation session of ] h ]2 m can be subdivided into two parts �9

- 15 minute static positioning part with the "moving" receiver on starting point ] in order to get an initial positioning result,

- a b o u t one hour kinematic positioning part with the receiver travelling from station ! to 7 and back, stopping at each control station for 30 seconds.

Therefore stations ] to 6 are positioned twice by the moving receiver. Since at the momen.t no reference positions from land surveys or static GPS applications are available for the stations these double occupancies wilt be used later on to discuss the accuracy of the results.

3. Data Description

Unfortunately the satellite geometry was not very good during the t e s t - the PDOP values were ranging from 5.3 to 5.9.

In addition, low elevation angles of ]5 ~ were used in order to have a sufficient time span with one satellite configuration. This led to some difficulties during processing caused by multipath and atmospheric effects, f~b le I shows the quality of the data for the moving receiver. Several cycle slips occurred in each tracker with bad data sequences averaging 20 to 30 seconds in length.

Velocities of up to 20 m/s and accelerations of up to 1 m/s 2 were reached during observation.

Table 1

Observational data statistics for the moving receiver

Eleva t ion range

Average

signal / noise

ratio

Bad data

Cycle slips

L 1

L 2

t 1

L 2

23 ~ _ 18 ~

42 dB

41 dB

6 .9%

6.9 %

10

S A T E L L I T E S

66 ~ _ 67 ~

41 dB

40 dB

6.3 %

6.4 %

11

15 ~ _ 35 ~

42 dB

41 dB

5.4 %

5.6 %

12

15 ~ _ 37 ~

43 dB

41 dB

] 8.8 %

8 . 8 %

10

As shown in Table 1 the average signal to noise ratio, the number of cycle slips, and the percentage of bad data are similar for all channels.

In order to analyze the dependency of the occurrence of cycle slips on sudden

87

Herbert LANDAU

changes in the vehicle velocity the latter was plotted versus the t ime in combination with cycle slips. Figure2 shows the velocity of the car during the forward and backward runs. The stops at the control stations are indicated by periods with zero velocity. Nearly all cycle slips occurred during movements and in most cases we have a loss of phase lock in more than one tracker. It seems that there is a high correlation between velocity changes and cycle slips. Similar effects are shown by Cannon et al. (1986). From the practical work in the field the author remembers, however, that signal blockage by trees was a major reason for the cycle slips.

C_ a.~ r

(J r r s

O

E

o ~

O

>

>-

I

I 4b 6'4

Time reference in minutes

Fig. 2 - Cycle slip occurrence and vehicle velocity (triangle indicates cycle sl ip)

4. Cycle Slip Correction Algorithm

Loss of phase lock is a serious problem in kinematic positioning when using carrier phase measurements. It is necessary to avoid cycle slips or f ix them in order to achieve accuracies less than the carrier wavelength of about ]9 cm for L ] frequency.

The cycle slips can be detected and fixed by an algorithm described in the following. }t is based on dual-frequency code and carrier phase measurements. An alternative to that method is to increase redundancy by more than 4 simultaneously tracked satellites using different hardware.

In principle the algorithm is using two observation combinations:

The phase-range combination to find approximate values for the cycle slip values

A p ( t ) = X (',I) ( t ) - r ( t l ) ) - ( p ( t ) - p ( t l ) ) (4.1)

with

88


X .... L l or L 2 wavelength

@ .... carrier phase for the actual epoch t and the initial epoch t ]

p .... observed range derived from code measurements

and the differenced phase data combination applied already by Goad (1986) for static data

6 r (t) 4,, (t) f ' = - ~ 2 (t) (4.2) f2

where

fl and f2 are thefrequenciesin L~ and L 2

~t and dl) 2 are the carrier phases in each frequency.

Madcr (1986) already proposed to use these quantities in combination with linear regression to f ix the cycle slips. Instead of linear regression, however, a Kalman f i l ter was applied to correct the cycle slips in the kinematic test presented here.

The filtering technique is based on the fol lowing first-order differential equation, which is representing the dynamics of the ~inear system

_~(t)= F_(t) x(t)+w(t) (4.3)

with

( t) system state vector

_w (t) random forcing function

F ( t ) dynamics matrix

The discrete Kalman fi l ter we are using in our approach is described by the relations (Gelb 1974).

--Xk = -~k- I - X k - l + Wk-I ' w k ~ N (O_ ,Qk ) (4.4)

Zk = ~k X-k + Vk ' v k ~ N ( O , ~k ) (4.5)

(4.6)

(4.7)

(4.8)

(4.9)

(4.10)

The state and covariance extrapolation equations are

_Xk (-) = 2k - t --Xk-t (+) 1"

Pk(-) = -Ok-, -Pk-1 (+) 2k-I + O--k-t

The measurement updating relations are given by

X k(+) = X_k(--) + K_ k [z k-H kx k (-)]

P--k (+) = [[--KR -Hk] Pk ( - )

~k = ~ ( - ) -Hk r [--~k -Pk (- ) ~J~+ Bk]-'

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Herbert LANDAU

where ( - ) indicates extrapolated quantities and (+ ) indicates updated quantities.

The applied dynamic system is a simple third-order process corresponding to a quadratic polynomial (Goad 1986) with

0 1 01 F = 0 0 1 (4.11)

0 0 0

The transition matrix ~ is computed from the relation

1 At A t : / 2 1

_@ = e l 'a t with ~ = 0 l At (4.12)

0 0 1

where A t is the observation sampling rate (here At = 3 sec).

-I he general form of the state vector is given by

4 1 3

built by the considered observable and its derivatives. Depending on the sampling rate of the receiver the profit by the introduction of the second derivative might be questionable. However a sampling interval of less than on second is desirable for kinematic applications.

A Kalman filter is applied to phase-range combinations in L 1 and L 2 to find

approximate values for the jumps in carrier phase and afterwards another filter is applied to the differenced phase data to minimize

A6&b = ~dP(t2) -- 8~ ( t l ) (4.14)

resulting in the correct combination of cycle slips in L 1 and L 2 .

It is shown by Bastos and Landau (1988) that the algorithm is very well behaving in applications to static and kinematic data.

Figures 3 and 4 describe the application of the algorithm to T I 4 t 0 0 data. In a first step the observed phase-range values are filtered in order to find approximate cycle slip values. The observed data in Figure3 show a periodic behaviour of 3-5 minutes with an amplitude of ]0 cycles, which can be explained by effects of multipath on the measurements (Bishop and Klobuchar, 1985). The filtering technique allows the separation of this effect from the data and results in more or less pure noise. The results of filtering the differenced phase data quantity (eq. 4.2) are given in Figure 4. The noise of these data is much smaller in comparison to the variation. The fi lter is correcting the data for a trend probably caused by variations in the ionosphere and multipath effects, and leaves noisy residuals which can be used to f ix the cycle slips by relation (4.14).

90


~o I

4J

b - I

o

,

ib 25 a5 45 55 65 75 Time re fe rence in minutes

Fig. 3 - Phase-range combination for L 1 frequency. (All functional values are given

in wavelengths o f L ! ) .

,.-I o

"I. .,-t 0

I Q~

10

C. O2,

r-1

o

o

(n t~ 0

ib 2b '

Time reference in minutes

Fig. 4 - Differenced phase data (static receiver). (Al l functional values are given in wavelengths o f ]. t ) .

The data for the static receiver, given in Figure 4, seem easy to handle. This becomes more difficult in case of a moving receiver. Figure 5 shows some data for the same satellite for the moving receiver. The differenced phase data are much more affected by small variations due to multipath. In the data presented here a cycle slip occurred at time reference t = ]5.2. The figure shows the data after the correction proving that the filtering technique was able to model the variations due to multipath and resulting in residuals less than 0.] cycles.

91

Herbert LANDAU

t~

r

.i-I g l a~

r r

' O

t ~ r

r

03 r~

Q

N

o

tvl

o

o

t ~ &

r

xb ab Time reference in minutes

Fig. 5 - Differenced phase data (moving receiver). (Al l functional values are given in wavelengths of L 1 ) .

When processing cycle slips in the way described above the required precision for the determination of approximate values by use of the phase-range combination is + 3 cycles, whereas the precision for the filtered estimates of the differenced phase data before and after the irregularity is +O.]4cycles (Bastos and Landau, 1988). Under these circumstances cycle slip periods longer than ] - 2 minutes are dif f icult or impossible to process if the data are affected by multipath in the way shown in the figures due to changes in the dynamic environment for a moving receiver.

All 36 cycle slips were successfully fixed using the algorithm described.

5. Relative Positioning Model

The relative positioning technique described in the following is using differences of observations between the moving receiver and the static receiver. A detailed description of the method can be found in Landau (1988).

The pseudo-range observation equation is given by the relation

Ap i ( t ) = A p i 0 ( t ) -- C- At (t) (5.1)

where Ap i ( t ) . . . observed pseudo-range difference between the receivers to satellite i ,

Ap i o ( t ) . . . geometric range difference to satellite i

At (t) . . . receiver clock difference, and

c . . . speed of light.

The carrier phase observation is defined by the retation

Aq bi (t) = APoi ( t ) . f / c -- f - At ( t ) + N i (5.2)

92

PRECISE K INEMATIC GPS POSITIONING

where

A @i (t) . . . observed carrier phase difference to satellite i ,

f . . . carrier frequency for L 1 or L2 ,and

N i . . . phase ambiguity for satellite i .

Please note, that in the equations above the observed quantities are assumed to be already corrected for atmospheric effects.

In the following it will be assumed, that the phase ambiguity was determined beforehand, so that the carrier phases can be modelled in analogy to the pseudo-ranges This determination can be done in different ways :

| . By a derivation from known initial positions of both receivers,

2. By simply exchanging the GPS antennas between static and moving receiver if possible ( R e m o n d i , 1986),

3. Or by observing a sufficient time span before starting and computing the initial conditions from these observations.

In our case the last method was chosen to determine the phase ambiguities.

A filtering technique was used to optimally combine the two observation types

in one model, The applied Kalman filter is of 12 th order and is in principle based at the equations already described in the previous chapter.

The general form of the state vector is given by

x_ = (5.3)

-Y

where 5 t is a vector describing the behaviour of the receiver clock difference

Sit = At (5.4)

At"

with At . . . time difference

A[ . . . frequency difference

A t * . . . frequency drift difference.

The vectors L, ~_, and "[" are corresponding to the position, velocity, and

acceleration of the vehicle in each coordinate direction. By use of Kalman filtering we are able to derive for each time epoch position, velocity, and acceleration estimates, even for those which are not identical with measurement epochs. This circumstance among other advantages makes the Kalman filter a very powerful tool for kinematic positioning.

93

Herbert LANDAU

Depending on the dynamics and the sampling rate it might become useless to include

second derivatives.

6. Analysis of Results

The algorithm described above was applied to the L z and L 2 frequency of

code and carrier phase observations. Due to the absence of comparison values the author is comparing the different coordinates of the forward and backward run in order to get some idea about the precision of the positions. The results are summarized in Table 2.

Table 2

Differences between forward and backward run in centimeters

S t a t i o n

1

2

3

4

5

6

Carrier phases

L 1

Ax Ay Az

- I - I l

- 4 - 2 0

- 1 - 1 - 4

0 0 - 2

5 1 2

2 1 0

Pseudo ranges

t 2

Ax Ay Az

- 4 - 2 0

- 3 0 - 3

- 7 - 1 . - 4

2 0 - 2

6 - 2 - 2

- 2 0 3

L 1

Ax Ay &z i

- 10 41 14

93 153 45

- 5 2 7 --2 - 2 7 6

471 84 217

483 117 451

269 24 177

Ax

t 2

Ay Az

35 - 41 -- 45

- 4 2 0 - 1 8 2 194

244 22 167

289 56 224

- 1 8 2 - 1 2 4 - - 11

- 1 6 5 - 2 6 - - 2 1 1

The table shows discrepancies of up to 7 cm between the forward and the backward run when using carrier phase measurements. The differences may be caused by the bad geometry and by ionospheric and multipath influences. A better agreement might be possible for a superior satellite geometry and higher elevation angles. The differences in the code measurement derived positions (Table2)seem to confirm this assumptions since differences of up to 5 m look a bit too large.

A combined filtering of code and carrier phase measurements was performed, but yielded in results identical with the pure application of carrier phases, since both observation types have the same geometrical strength when the ambiguities are fixed.

Analyzing the differences in positioning results of L 2 and L l carrier phase

we find a good agreement (Table 3). The differences are again up to 7 cm and the RMS values of coordinate differences are about 1 - 2 cm.

7. Conclusions

It was shown that a relative positioning for a separation of about 6 km between reference and mocing receivers with an accuracy of about | - 2 cm is possible by use of dual-frequency carrier phase measurements. The key to achieve such accuracies is the correction of all cycle slips. It should be mentioned that this cycle slip correction algorithm is suited for the processing of dual-frequency carrier phase data in connection with P-code measurements only.

94


Table 3

Comparison of positions derived from L 2 frequency with.positions derived

from L 1 (Discrepancies are given in centimeters).

Station L I / L 2

1

2

3

4

5

6

7

6

5

4

3

2

1

Ax Ay Az

m

Average + 0.4 + 0.2 - 0.7

RMS _+ 1.9 _+ 0.5 _.+ 1.1

- 0 . 1 - 0 . 1 - 0 . 1

0.6 - 0 . 4 0.2

6.8 0.8 0.6

- 1 . 2 0.0 - 1 . 4

- 2 . 7 - 0 . 7 2.3

4.7 1.0 - 1 . 7

- 2 . 5 - 0 . 6 - 1 . 3

1.1 0.7 1.1

- 1 . 2 0.2 - 1 . 7

0.6 0.0 - 2 . 4

0.5 0.5 0.1

1.6 1.5 - 3 . 4

- 3 . 4 0.9 - 0 . 9

Contributions to relative positioning with GPS concerning land vehicle, marine and aircraft positioning by use of GESAR operating software and different tracking bandwidths will follow. By comparing GPS results with photogrammetry (aircraft) and precise infrared sensors (hoovercraft) we will be able to analyze the GPS positioning accuracy during movement.

Acknowledgement

The suppor t of the German Research Foundat ion is gratefu l ly acknowledged.

0

0 0

95

Herbert LANDAU

REFERENCES

L. BASTOS, H. LANDAU (1988) : Fixing Cycle Slips in Dual-Frequency Kinematic GPS-Applications Using Kalman Filtering. In : Manuscripta Geodaetica, Vol. 13, No. 4, pp. 249--256.

G.J. BISHOP, J.A. KLOBUCHAR (1985) : Multipath Effects on the Determination o f Absolute Ionospheric Time Delay from GPS Signals. Radio Science, Vol. 20, No. 3, pp. 388--396.

M.E. CANNON, K.P. SCHWARZ, R.V.C. WONG (1986) : Kinematic Positioning with GPS. An Analysis o f Road Tests. In : Proceedings of the Fourth International Geodetic Symposium on Satellite Positioning. April 2 8 - May 2, 1986, Austin, Texas, Vol. 2, pp. 1251--1267.

A. GELB (1974) : Applied Optimal Estimation. The M.I.T. Press, Massachusetts Institute of Technology, Cambridge, USA.

C. GOAD (1986) : Precise Positioning with the Global Positioning System. Proceedings of the Third International Symposium on Inertial Technology for Surveying and Geodesy, September 16-20 , 1985, Banff, Canada, pp. 745--756.

G.W. HEIN, B. EISSFELLER, H. LANDAU (1986) : Comparative Study. o f the Ferranti, Honeywell, and Litton Inertia/Surveying System - Testnetz Werdenfelser Land. Proceedings of the Third International Symposium on Inertial Technology for Surveying and Geodesy, September 16--20, 1985, Banff, Canada, pp. 525-546.

A. KLEUSBERG, S.H. QUEK, D.E. WELLS, G. LACHAPELLE, J. HAGGLUND (1986) : Relative Positioning Techniques for Moving Platforms. In : Proceedings of the Fourth International Geodetic Symposium on Satellite Positioning, April 28--May 2, 1986, Austin, Texas, Vol. 2, pp. 1299--1310.

G. LACHAPELLE, J. HAGGLUND, W. FALKENBERG, P. BELLAMEARE, M. CASEY, M. EATON (1986) : GPS Land Kinematic Positioning Experiments. In : Proceedings of the Fourth International

Geodetic Symposium on Satellite Positioning, April 28--May 2, 1986, Austin, Texas, Vol. 2, pp. 1327--1344.

H. LANDAU (1988) ; Zur Nutzung des Global Positioning Systems in Geodaesie und Geodvnamik : Modellbildung, Softwareentwicklung und Analyse. Ph.D. Thesis (in preparation). Institute of Astronomical and Physical Geodesy, University FAF Munich.

G. MADER (1986) : Dynamic Positioning Using GPS Carrier Phase Measurements. Manuscripta Geodaetica, Vol. 11/4, pp. 272--277.

B.W. REMONDI (1985a) . Performing Centimeter Accuracy Relative Surveys in Seconds Using GPS Carrier Phase. In : Proceedings of the First International Symposium on Precise Positioning with the Global Positioning System, Rockville, MD, pp. 789--798.

B.W. REMONDI (1985b) : Performing Centimeter-Level Surveys in Seconds with GPS Carrier Phase : Initial Results. NOAA Technical Memorandum NOS NGS--43.

B.W. REMONDI (1986) : Performing Centimeter-Level Surveys in Seconds with GPS Carrier Phase : Init ial Results. In : Proceedings of the Fourth International Geodetic Symposium on Satellite Positioning, April 2 8 - May 2, 1986, Austin, Texas, Vol. 2, pp. 1229--1250.

G. SEEBER, A. SCHUCHARDT, G. WUBBENA (1986) : Precise Positioning Results with TI 4100 GPS Receivers on Moving Platforms. In : Proceedings of the Fourth International Geodetic Symposium on Satellite Positioning, April 28--May 2, 1986, Austin, Texas, Vol. 2, pp. 1269-- 1286.

R e c e i v e d : 2 2 . 0 4 . 1 9 8 8

A c c e p t e d : 2 3 . 0 9 . 1 9 8 8

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Precise kinematic GPS positioning