Design of kinematic DGNSS filters with consistenterror covariance information

H.K. Lee, C. Rizos and G.-I. Jee

Abstract: Consistent and realistic error covariance information is important for positionestimation, error analysis, fault detection, and integer ambiguity resolution for the applicationsof global navigation satellite systems. In designing a position domain carrier-smoothed-code filterwhere incremental carrier-phases are used for time-propagation, formulation of consistent errorcovariance information is not easy due to correlation of successive propagation noises. To provideconsistent and correct error covariance information, two recursive filter algorithms are proposedbased on carrier-smoothed-code techniques: the stepwise optimal position projection filter, andthe stepwise unbiased position projection filter. A Monte Carlo simulation result shows that theproposed filter algorithms actually generate consistent error covariance information and theneglecting of carrier-phase noise induces optimistic error covariance information. It is also shownthat the stepwise unbiased position projection filter is attractive since its performance is good and itscomputational burden is moderate.

1 Introduction

To achieve good navigation performance under high-dynamics a filter usually requires two types of sensors: asensor with fast output rate for time-propagation and asensor with slow output rate for measurement update. Forthis reason a Global Navigation Satellite System (GNSS)receiver is usually paired with inertial or dead-reckoningsensors [1–3]. If a vehicle’s movement is fully known inadvance a mathematical model describing the vehicle’sdynamics can completely replace the role of fast-ratesensors. In reality, however, the vehicle’s movement cannotbe fully known in advance. If the vehicle’s trajectory ispartially known and the fast-rate aiding sensors are notavailable, a stochastic model can marginally replace thefunction of such sensors. For this reason the well-knownGlobal Positioning System (GPS) Kalman filter [1–3]implemented within receivers typically use simplifieddynamic models representing stationary, constant-speed,or constant-acceleration conditions. The performance ofsuch a GPS Kalman filter is strongly dependent on theconsistency between the assumed dynamic model and thereceiver’s actual motion. If this consistency condition is notsatisfied the performance of the GPS Kalman filter issignificantly degraded.

Unlike other positioning sensors, a GNSS receiverprovides diverse measurements. This diversity of measure-ments results in substantial improvement in positioningaccuracy and fault detection ability. The improvement inpositioning accuracy obtained by eliminating an assumeddynamic model in the range domain was first introducedin [4]. Subsequently, several filters have been proposed toeliminate the need for explicit receiver dynamic models inthe position domain. In general these algorithms use thecode measurements for measurement update and thecarrier measurements for time-propagation. The threemost representative algorithms are the complementaryfilter in [5], the simple and modified time-differentialfilters of [6], and the phase-connected filter proposedin [7].

In addition to the accuracy a good navigation filtershould also provide a reliable measure of how accuratethe position estimates are. Owing to this requirement, anavigation filter usually computes an error covariancematrix as a statistical measure. An analysis of previousresearch [4–8] confirms that a detailed treatment of thistopic is hard to find. Since carrier-phase noise is muchless than pseudorange noise, it is usually neglected orapproximated. This neglect of carrier-phase noise gen-erates inconsistencies in the error covariance information(as seen later). Consistent (and realistic) error covarianceinformation plays an important role in position estimation,fault detection, integer ambiguity resolution, and erroranalysis of differential GNSS.

Extending the conventional carrier-smoothed-code tech-niques, this paper proposes two position domain filteralgorithms: the stepwise optimal position projection filter(SOPF) and the stepwise unbiased position projectionfilter (SUPF). Compared with the conventional carrier-smoothed-code filters, the proposed filters fully consider thecomplex cross-correlation generated by carrier noises andare computationally efficient since they require minimalfour states. Since most of the presented derivation procedureis gain-independent, an extension to other variationalalgorithm is easy.

q IEE, 2004

IEE Proceedings online no. 20041026

doi: 10.1049/ip-rsn:20041026

H.K. Lee was with the Satellite Navigation & Positioning Group,University of New South Wales, as a visiting postdoctoral fellow and isnow with the School of Avionics and Telecommunication, HankukAviation University, Kyunggi-do, Korea

C. Rizos is with the Satellite Navigation & Positioning Group, School ofSurveying and Spatial Information Systems, University of New SouthWales, Sydney, Australia

G.-I. Jee is with the Department of Electronics Engineering, KonkukUniversity, Seoul, Korea

Paper first received 2nd June 2003 and in revised form 21st July 2004.Originally published online 5th November 2004

IEE Proc.-Radar Sonar Navig., Vol. 151, No. 6, December 2004382

2 Design of kinematic filter with consistent errorcovariance

The pseudoranges and carrier-phases measured by a singlereceiver contain a variety of error sources, includingreceiver clock bias, thermal noise, satellite clock bias,ionospheric delay, tropospheric delay, and multipathdisturbance If a user’s receiver and a reference receiverare located close by, the common-mode error sources suchas the satellite clock bias, ionospheric delay, and tropo-spheric delay can be effectively eliminated [1–3]. This typeof data combination is referred to as ‘single differencing’.The multipath error can be effectively detected andmitigated by various other methods [9–16]. The correctedpseudorange ~rrj;k and carrier-phase ~ffj;k (assuming thecommon-mode errors and multipath error have beeneliminated) can be modelled [3] as

~rrj;k ¼ eTj;kðxj;k � xu;kÞ þ bu;k þ vj;k; vj;k � ð0; rrÞ

~ffj;k ¼ eTj;kðxj;k � xu;kÞ þ bu;k þ nj;k þ lNj nj;k � ð0; rfÞ

ð1Þ

where

ej;k : line of sight (LOS) vector from the receiver tothe jth satellite

xj;k : Earth-centred Earth-fixed (ECEF) position ofthe jth satellite

xu;k : ECEF receiver positionbu;k : receiver clock biasNj : unresolved integer ambiguityvj;k; nj;k : white noise terms in code and carrier

measurementsrr; rf : uniform noise strength values of code and

carrier measurements

The true state vector Xk considered in this paper iscomposed of the three-dimensional position sub-vectorxu;k and the scalar clock bias bu;k

Xk :¼xu;k

bu;k

� �ð2Þ

The symbols �XXk; d �XXk; �PPk; XXk; dXXk; and PPk are used torepresent the a priori state estimate, a priori estimationerror, a priori error covariance matrix, a posteriori stateestimate, a posteriori estimation error, and a posteriori errorcovariance matrix at the kth step, respectively,

d �XXk � ðO; �PPkÞ; dXXk � ðO; PPkÞd �XXk :¼ �XXk �Xk; dXXk :¼ XXk �Xk

d �XXk ¼d�xxu;k

d�bbu;k

� �; dXXk ¼

dxxu;k

dbbu;k

" #�XXk ¼

�xxu;k

�bbu;k

� �XXk ¼

xxu;k

bbu;k

" #

ð3Þ

The a priori estimate �XXkþ1 of either the SOPF or SUPF atthe ðk þ 1Þth step is generated after the time-propagation

that combines the a posteriori estimate XXk and theincremental carrier-phase measurements f ~ffj;kþ1 � ~ffj;kg atthe kth step. The a posteriori estimate XXk at the kth step isgenerated by the measurement update at the kth step,combining both the a priori estimate �XXk and the new codemeasurements f ~rrj;kg at the kth step. As seen later, betweenthe proposed SOPF and SUPF the differences arise only inthe gain matrix formulation for time-propagation. Thus twofilters are derived at the same time. If it is necessary todiscriminate the variables of the two different filters the

superscripts o and s are used to denote the SOPF and SUPF,respectively.

2.1 Measurement update by pseudoranges

The indirect measurement zj;k with respect to d �XXk formeasurement update is formed by the following equation:

zj;k ¼ ~rrj;k � eTj;kðxj;k � �xxu;kÞ � �bbu;k ð4Þ

According to (1)–(4), the indirect measurement zjk satisfies

the condition

zj;k ¼ hj;kd �XXk þ vk;j

hj;k :¼ eTj;k �1

� �ð5Þ

Stacking the indirect measurements fzj;kgj¼1;2;...;J into asingle vector Zk; one obtains

Zk ¼ Hkd �XXk þ vk; vk � ðOJ1; rrIJJÞ ð6Þ

where

Zk :¼

z1;k

z2;k

..

.

zJ;k

26664

37775; Hk :¼

h1;k

h2;k

..

.

hJ;k

26664

37775; vk :¼

v1;k

v2;k

..

.

vJ;k

26664

37775 ð7Þ

Assuming a gain matrix Kk; the a priori estimate �XXk is

updated to the a posteriori estimate XXk as follows:

XXk ¼ �XXk � KkZk ð8ÞAccordingly, the estimation error is updated by (1), (3), and(8)

dXXk ¼ ðI44 � KkHkÞd �XXk � Kkvk ð9Þ

2.2 Time-propagation by incremental carrier-phases

To derive the exact measurement equation for time-propagation it is necessary to define the variables Dxu;k;Dbu;k; Dxj;k; and Dej;k as the increments of the user position,user receiver clock bias, the jth satellite’s position, and theLOS vector from the kth step to the ðk þ 1Þth step,respectively,

Dxu;k :¼ xu;kþ1 � xu;k; Dbu;k :¼ bu;kþ1 � bu;k

Dxj;k :¼ xj;kþ1 � xj;k; Dej;k :¼ ej;kþ1 � ej;k ð10ÞThen, according to (1) and (10), the incremental carrier-phase ð ~ffj;kþ1 � ~ffj;kÞ satisfies the condition

~ffj;kþ1 � ~ffj;k ¼ ðej;k þ Dej;kÞTðxj;kþ1 � xu;kþ1Þ þ nj;kþ1

þ bu;kþ1 � eTj;kðxj;k � xu;kÞ � nj;k � bu;k

¼ � ðej;k þ Dej;kÞTDxu;k þ nj;kþ1 � nj;k

þ Dbu;k þ eTj;kDxj;k þ DeT

j;kðxj;kþ1 � xu;kÞð11Þ

If one forms the indirect measurement oj;kþ1 for time-propagation as

oj;kþ1 :¼ eTj;kDxj;k þ DeT

j;kðxj;kþ1 � xxu;kÞ � ð ~ffj;kþ1 � ~ffj;kÞð12Þ

oj;kþ1 satisfies the following relationship according to (7)and (10)–(12):

IEE Proc.-Radar Sonar Navig., Vol. 151, No. 6, December 2004 383

oj;kþ1 ¼ hj;kþ1DXk þ wj;kþ1; DXk ¼ ðD xu;kÞT Dbu;k

� �T

ð13Þwhere

wj;kþ1 :¼ �D eTj;kdxxu;k � nj;kþ1 þ nj;k ð14Þ

Stacking oj;kþ1 for all the satellites into a vector Vkþ1 yieldsthe following linear equation:

Vkþ1 ¼ Hkþ1DXk þ Wkþ1 ð15Þwhere

Vkþ1 :¼

o1;kþ1

o2;kþ1

..

.

oJ;kþ1

26664

37775; Wkþ1 :¼

w1;kþ1

w2;kþ1

..

.

wJ;kþ1

26664

37775 ð16Þ

According to (14) and (16), the equivalent measurementnoise vector Wkþ1 can further be decomposed as

Wkþ1 ¼ �DHkdXXk � nkþ1 þ nk ð17Þwhere

DHk :¼ Hkþ1 � Hk

nk :¼ ½ n1;k n2;k . . . nJ;k �T � ðOJ1; rfIJJÞ ð18Þ

The decomposition of the measurement vector Wkþ1 in (17)is used later in deriving the gain matrix for time-propagation. For temporal use, the gain matrix is simplydenoted by Ukþ1: The gain matrix Ukþ1 should satisfy thecondition

UkHk ¼ I44 ð19Þfor all k in order to prevent biased estimation. By

pre-multiplying Ukþ1 to both sides of (15), an estimate DXk

ð ~ffkþ1; ~ffk; XXkÞ of the incremental state DXk can be obtained

DXk ð ~ffkþ1; ~ffk; XXkÞ ¼ Ukþ1Vkþ1 ¼ DXk þ Ukþ1Wkþ1

ð20Þwhere

~ffk :¼ ½ ~ff1;k~ff2;k . . . ~ffJ;k �T ð21Þ

Using the estimated incremental state DXkð ~ffkþ1; ~ffk; XXkÞthe a posteriori estimate XXk at the kth step is propagated intime to the a priori estimate �XXkþ1 at the ðk þ 1Þth step

�XXkþ1 ¼ XXk þ DXk ð ~ffkþ1; ~ffk; XXkÞ ð22ÞBy utilising (3), (17), (20) and (23), the estimation error ispropagated in time

d �XXkþ1 ¼ ðI44 � Ukþ1DHkÞdXXk � Ukþ1ðnkþ1 � nkÞ ð23Þ

2.3 Covariance recursion and gain formula

The recursion equations of the a priori and a posterioriestimation error d �XXk and dXXk are derived by combining (9)and (23) in two different orders as follows:

dXXk ¼ ðI44 � KkHkÞðI44 � UkDHk�1ÞdXXk�1

� ðI44 � KkHkÞUkðnk � nk�1Þ � Kkvk

d �XXk ¼ ðI44 � UkDHk�1ÞðI44 � Kk�1Hk�1Þd �XXk�1

� ðI44 � UkDHk�1ÞKk�1vk�1 � Ukðnk � nk�1Þð24Þ

As shown in (24) the a priori estimation error d �XXk isindependent of the pseudorange noise vk Thus the errorcovariance matrix is updated according to (3) and (9)

PPk ¼ ðI44 � KkHkÞ �PPkðI44 � KkHkÞT þ rrKkKTk

Kk ¼ �PPkHTk Hk

�PPkHTk þ rrIJJ

��1¼ 1

rrPPkHT

k ð25Þ

Since dXXk�1 is independent of nk; as shown in (24), thefollowing relationship holds:

E dXXk nTk

� �¼ �ðI44 � KkHkÞUk; E dXXk nT

kþ1

� �¼ O4J

ð26Þ

Utilising (3), (23) and (26) the error covariance matrix ispropagated in time as follows:

�PPkþ1 ¼ðI44�Ukþ1DHkÞPPkðI44�Ukþ1DHkÞT þ2rFUkþ1

UTkþ1�rFðI44�Ukþ1DHkÞðI44�KkHkÞUkUT

kþ1

�rFUkþ1UTk ðI44�KkHkÞTðI44�Ukþ1DHkÞT

ð27Þ

In (27), effects of small carrier-phase noise sequencefnkþ1 � nkgk¼0;1;2;... are fully considered. The correlationbetween ðnkþ1 � nkÞ of Vkþ1 and ðnk � nk�1Þ of Vk isconsidered as shown by the third and fourth terms of (27).As approximations of (27), the following simple equationscan be considered:

�PPkþ1 ¼ ðI44 � Ukþ1DHkÞPPkðI44 � Ukþ1DHkÞT ð28Þ

�PPkþ1 ¼ ðI44 � Ukþ1DHkÞPPkðI44 � Ukþ1DHkÞT

þ rtuneUkþ1UTkþ1 ð29Þ

where rtune denotes the tuning value to consider the time-propagation by incremental carrier-phase. In a strict sense(28) means that carrier-phase measurements are noise-free.Equation (29) means that the sequence fnkþ1 � nkgk¼0;1;2;...

is erroneously considered as white. Effects of theseapproximations can be seen later in the simulation part.

The covariance recursions of (25) and (27) are indepen-dent of gain formulation. It is now explained how theproposed SOPF and SUPF formulate gains. For the gain Kk

for measurement update the SOPF and SUPF commonlyutilise the well known Kalman gain. In formulating the gainUk for time-propagation, however, the SOPF and SUPF arebased on different derivations. The SOPF gain Uo

kþ1 isformulated based on the following background. Since noconstraint is desired on receiver movement, the unbiasedestimation of the displacement vector DXk at each time stepis the most concern for stepwise position projection in time.According to the classical linear gaussian estimation theory[2, 3], the optimal unbiased estimate DXo

k of DXk in theleast-squares sense can be obtained by applying theweighted pseudo-inverse to the measurement vector Vkþ1

DXok ¼ Uo

kþ1Vkþ1 ð30Þ

where

Uokþ1 :¼ HT

kþ1 Qokþ1ð Þ�1Hkþ1

� ��1HT

kþ1 Qokþ1ð Þ�1

Qokþ1 :¼ E Wkþ1WT

kþ1

� �ð31Þ

According to (17) and (31) the error covariance matrixQo

kþ1 is

IEE Proc.-Radar Sonar Navig., Vol. 151, No. 6, December 2004384

Qokþ1 :¼E �DHkdXXo

k�nkþ1þnk

��DHkdXXo

k�nkþ1þnk

�Th i

¼DHkPPokðDHkÞTþ2rFIJJþrFDHk I44�Ko

kHkð ÞUok

þrF Uokð ÞT I44�Ko

kHkð ÞTðDHkÞT

ð32Þ

Equation (32) shows all the sources that affect the SOPFgain Uo

kþ1 for time-propagation. It is observed that thesatellite geometry change DHk combined with the previous

step’s estimation error PPok and the gains Kk and Uo

k affect thegain Uo

kþ1 through the weighting matrix Qokþ1: Although the

gain Uokþ1 for SOPF is based on good theoretical back-

ground, its appearance is complex as shown in (31) and (32).If the satellite geometry change DHk is small, which holdsfor GNSS, the weighting matrix Qo

kþ1 approaches a scaledidentity matrix. For this reason the SUPF utilise an equally-weighted pseudo-inverse matrix Us

kþ1 as follows:

DXsk ¼ Us

kþ1Vkþ1 ð33Þ

where

Uskþ1 :¼ HT

kþ1 Qskþ1ð Þ�1Hkþ1

� ��1HT

kþ1 Qskþ1ð Þ�1

Qskþ1 ¼ IJJ ð34Þ

As easily verified, the SUPF gain Uskþ1 also does not generate

bias error in time-propagation and its computational burdenis much less than that of Uo

kþ1: However, the SUPF wouldgenerate a more deteriorated performance than the SOPF ifDHk is not negligible. Pseudolite [17] applications are atypical example for this since LOS directions to adjacentpseudolites are sensitive to receiver movements.

Denoting Ukþ1 for both Uokþ1 and Us

kþ1 and Qkþ1 for bothQo

kþ1 and Qskþ1; it can be verified that

I44�Ukþ1DHk ¼ HTkþ1ðQkþ1Þ�1Hkþ1

� ��1

HTkþ1ðQkþ1Þ�1Hkþ1 � HT

kþ1ðQkþ1Þ�1DHk

� �¼ Ukþ1Hk ð35Þ

Substituting (35) into (23) and (27), the time-propagation ofthe estimation error and error covariance matrix of SUPF issimplified as follows:

d �XXskþ1 ¼ Us

kþ1½HkdXXk � ðnkþ1 � nkÞ��PPs

kþ1 ¼ Uskþ1

hð1 þ 2rF=rrÞHkPPs

kHTk

� 2rFHk HTk Hk

��1HT

k þ 2rFIi

Uskþ1ð ÞT ð36Þ

The derived SOPF and SUPF are summarised in Tables 1and 2, where a selection matrix Gk is used to considerabrupt satellite inclusions and outages. The selectionmatrix Gk consists of 0 s and 1 s where 1 denotes themeasurement that is valid at both the kth and ðk � 1Þthstep, simultaneously. Thus the selection matrix Gk mapsfrom the J-dimensional measurement vector Yk thatconsiders all the channels to the reduced pðkÞ-dimensionalmeasurement vector Y�

k

Y�k ¼ GkYk ð37Þ

Table 1: Stepwise optimal position projection filter

Initialisation

Ck0 ¼ Gk0Hk0

XXk0 ¼ E½X k0jGk0 ~��k0�

PPk0 ¼ r� CTk0Ck0

� ��1

F k0 ¼ CTk0Ck0

� ��1CT

k0

Time-propagation

Ckþ1 ¼ Gkþ1Hkþ1; Lk ¼ Gkþ1Hk ;Dkþ1 ¼ Ckþ1 � Lk ; Mk ¼ Gkþ1GTk

Jkþ1 ¼ Dkþ1PPk DTkþ1 þ 2rFI�ðkþ1Þ�ðkþ1Þ

þ rFDkþ1ðI44 � K k Ck ÞF k MTk þ rFMk F T

k ðI44 � K k Ck ÞT DT

kþ1

F kþ1 ¼ CTkþ1J

�1kþ1Ckþ1

� ��1CT

kþ1J�1kþ1

�XXkþ1 ¼ XXk þ F kþ1Gkþ1Vkþ1

�PPkþ1 ¼ F kþ1 Lk PPk LTk þ rF

2I�ðkþ1Þ�ðkþ1Þ � Lk ðI44 � K k Ck ÞF k MTk

�Mk F Tk ðI44 � K k Ck Þ

T LTk

" #( )F T

kþ1

Measurement update

K k ¼ �PPk CTk Ck

�PPk CTk þ r�I�ðkÞ�ðkÞ

h i�1

XXk ¼ �XXk � K kGk Z k

PPk ¼ ðI44 � K k Ck Þ�PPk I44 � K k Ck

Tþr�K k K T

k

Table 2: Stepwise unbiased position projection filter

Initialisation

Ck0 ¼ Gk0Hk0

XXk0 ¼ E½X k0jGk0 ~��k0�

PPk0 ¼ r� CTk0Ck0

� ��1

Time-propagation

Ckþ1 ¼ Gkþ1Hkþ1; Lk ¼ Gkþ1Hk ; F kþ1 ¼ CTkþ1Ckþ1

� ��1CT

kþ1

�XXkþ1 ¼ XXk þ F kþ1Gkþ1Vkþ1

�PPkþ1 ¼ F kþ1ð1 þ 2rF=r�ÞLk PPk LT

k � 2rFLk LTk Lk

� ��1LT

k

þ2rFI�ðkþ1Þ�ðkþ1Þ

( )F T

kþ1

Measurement update

K k ¼ �PPk CTk Ck

�PPk CTk þ r�I �ðkÞ�ðkÞ

h i�1

XXk ¼ �XXk � K kGk Z k

PPk ¼ ðI44 � K k Ck Þ�PPk ðI44 � K k Ck Þ

T þ r�K k K Tk

IEE Proc.-Radar Sonar Navig., Vol. 151, No. 6, December 2004 385

3 Simulation

To investigate the characteristics of the proposed filters, aMonte Carlo simulation has been performed. In thesimulation the GPS Kalman filter where the dynamicmodel is approximated, the position domain filter wherethe carrier noises are approximated by (28) and (29), and theproposed two filters are compared. For the Monte Carlosimulation 100 ensembles of error sequences were gener-ated. During each trial of 3600 seconds, satellite outageswere assumed to have occurred at both 2261 and 3517seconds. The simulation configuration is shown in Fig. 1(the true positions of satellites and receivers are generated toprovide true ranges). To consider the amplification of noiseby single differencing, the variances of the code and carrierthermal noise are set as 1.5 m and 0.015 m, respectively,which are larger than those for a single receiver. Thereceiver clock bias is generated by a second-order Markovmodel appropriate for a crystal oscillator [2, 3]. Thetrajectory of the moving receiver is shown in Fig. 2(the trajectory repeats a straight run of 195 s followed by acircular turn of 5 s). During each straight run the velocityis maintained at 10m=s; and each circular turn generates acentripetal acceleration of 3m=s2:

To compare the performance of the four filters, meanvalues f ~mmðkÞgk¼0;1;2;...;3600 and 1s values f ~ssðkÞgk¼0;1;2;...;3600

of the error distances are computed based on the MonteCarlo error statistics

~mmðkÞ ¼ 1

N � 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXN

i¼1

½xxði; kÞ � xðkÞ�( )2

þXN

i¼1

½yyði; kÞ � yðkÞ�( )2

þXN

i¼1

½zzði; kÞ � zðkÞ�( )2

vuut

~ssðkÞ ¼ 1

N � 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXN

i¼1

½xxði; kÞ � xðkÞ�2 þXN

i¼1

½yyði; kÞ � yðkÞ�2 þXN

i¼1

½zzði; kÞ � zðkÞ�2vuut

ð38Þ

where N indicates the number of ensembles, k indicates thetime index, ½xðkÞ; yðkÞ; zðkÞ� indicates the true position, and½xxði; kÞ; yyði; kÞ; zzði; kÞ� indicates the estimated position by theith ensemble of data used in the Monte Carlo simulation. Inaddition to f ~mmðkÞgk¼0;1;2;...;3600 and f ~ssðkÞgk¼0;1;2;...;3600; 1svalues fssðkÞgk¼0;1;2;...;3600 of error distances based on eachfilter’s error covariance matrix are also computed from

ssðkÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipxxðkÞ þ pyyðkÞ þ pzzðkÞ

qð39Þ

where ½pxxðkÞ; pyyðkÞ; pzzðkÞ� indicates the diagonal elementsof the error covariance matrix that correspond to theposition errors.

The simulation results are shown in Figs. 3, 4 and 5.In all the graphs the dashed line indicates the meanvalues f ~mmðkÞgk¼0;1;2;...;3600 based on the Monte Carloerror statistics, the solid line indicates the 1s valuesf ~ssðkÞgk¼0;1;2;...;3600 based on the Monte Carlo errorstatistics, and the dotted line indicates the 1s valuesf ~ssðkÞgk¼0;1;2;...;3600 based on the filter’s error covariance

Fig. 1 Simulation configuration

Fig. 2 Trajectory of moving receiver Fig. 3 Performance of GPS Kalman filter

IEE Proc.-Radar Sonar Navig., Vol. 151, No. 6, December 2004386

matrix. Figure 3 shows the performance of the conventionalGPS Kalman filter based on the constant velocity model. Asindicated by the dashed and solid lines, the GPS Kalmanfilter accumulates large error periodically when the receiverundergoes circular turns during the Monte Carlo simulation.

The Fig. 4a shows the performance of the positiondomain code-carrier filter where carrier noise is neglected as

shown in (28). As shown by the solid line, the periodicaccumulation of large errors does not occur during thecircular turns since, unlike the GPS Kalman filter, noassumption was used regarding the receiver dynamics. Theabrupt error jumps that appear in the solid line at both 2261seconds and 3517 seconds are due to satellite outages. Asshown by the solid line, the position domain filter thatneglects carrier noise quickly reduces actual estimationerror to less than 0.2 m. However, as shown by the dashedline, it is optimistic of its position estimates compared to theactual errors.

Figure 4b shows the performance of the position domaincode-carrier filter that approximates, as shown in (29), theincremental carrier noise sequence fnkþ1 � nkgk¼0;1;2;... aswhite. If the assumed variance of white noise rtune is set asð0:015mÞ2; its actual estimation error is deteriorated around0.3 m as shown by the solid line. In addition, it is pessimisticof its position estimates compared to the actual errors, asshown by the dashed line. The mismatch between the actualerror statistics and the filter’s error covariance matrix shownin upper and lower plots of Fig. 4 would get worse if the rawL1 carrier-phase measurements are replaced by the iono-free or wide-lane carrier-phase measurements, due toamplification of carrier-phase noise.

All the plots in Fig. 5 show the simulation results of thetwo proposed filters. In each plot of Fig. 5, actual estimationerrors are quickly reduced less than 0.2 m as shown by thesolid line. It is hard to discriminate between the solid anddotted lines, since they are close to each other. This meansthat the error covariance matrix generated by each of theproposed two filters is consistent with the Monte Carlo errorstatistics. By comparing Figs. 5a and 5b, it can be seen thatthe performance of the SOPF and SUPF are practicallyidentical, though the SOPF is theoretically superior to theSUPF. Thus it can be concluded that the SUPF is the mostattractive among the compared filters since its performanceis good, its computational burden is not great, and itprovides consistent error covariance information.

4 Conclusions

To provide consistent error covariance matrices for single-differenced GNSS data processing, we have proposed twofilter algorithms based on the carrier-smoothed-codetechnique: the SOPF and the SUPF. Through simulation itwas verified that the GPS Kalman filter is easily biased dueto any inconsistency between the assumed dynamic modeland the actual receiver movements. It was also shown thatthe inconsistency of the error covariance matrix cannot beeliminated if the carrier-smoothed-code technique, withoutconsidering the effects of carrier noises, is implemented.It was shown that all the proposed filters provide unbiasedestimates with consistent error covariance matrices. Amongthe proposed two filters the SUPF was more attractive sinceit is computationally not too complex and its performance isgood.

5 Acknowledgment

This work was supported by the Post-doctoral FellowshipProgram of the Korea Science & Engineering Foundation(KOSEF).

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Fig. 4 Performance of carrier-smoothed-code filter

a Neglecting carrier noiseb Approximating incremental carrier-phase noise as white sequence

Fig. 5 Performance of proposed stepwise unbiased positionprojection filter and optimal position projection filter

a Unbiasedb Optimal

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