master: lecture 7 - upc universitat politècnica de...

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Master of Science in GNSS @ J. Sanz & J.M. Juan

1

Lecture 7

Carrier-based Differential Positioning.

Ambiguity Resolution

Techniques

Contact: [email protected]

Web site: http://www.gage.upc.edu

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2

Authorship statement

The authorship of this material and the Intellectual Property Rights are owned by J. Sanz Subirana and J.M. Juan Zornoza.

These slides can be obtained either from the server http://www.gage.upc.edu, or [email protected]. Any partial reproduction should be previously

authorized by the authors, clearly referring to the slides used.

This authorship statement must be kept intact and unchanged at all times.

5 March 2017

http://www.gage.upc.edu/

mailto:[email protected]

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Contents

3

1. Linear model for DGNSS: Double Differences

1.1. Differential Code and carrier based positioning

1.2. Precise relative Positioning

1.3. The Role of Geometric Diversity

2. Ambiguity resolution Techniques

2.1. Resolving ambiguities one at a time

- Single-frequency measurements

- Dual-frequency measurements

- Three-frequency measurements

2.2 . Resolving ambiguities as a set: Search techniques

- Least-Squares Ambiguity Search Technique

- LAMBDA Method

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Linear model for Differential Positioning

Single difference ( ) ( ) ( ) ( ) j j j j

ru ru u r

P

j j j j j

ru ru ru ru ru ru ruP c t T I K

L

j j j j j j j

ru ru ru ru ru ru ru ru ruL c t T I N b

Double difference

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) jk jk k j k k j j

ru ru ru ru u r u r

P

jk jk jk jk jk

ru ru ru ru ruP T I

L

jk jk jk jk jk jk jk

ru ru ru ru ru ru ruL T I N

Reference Station

User

P

k k k k k


P

j j j j j


The same for carrier:

Now are cancelled:• Receiver clock • Receiver code instrumental delays • Receiver carrier instrumental

delays

Carrier ambiguities N are integer

Sat jSat k

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1 1( ) sat sat

ru ruL L

PRN06

PRN30

PRN06

PRN30

PLAN-GARR: 15km

1( )L

j j j j j j j

ru ru ru ru ru ru ru ru ruL L c t T I N b

1 1( ) sat sat

ru ruP P

1( )P

j j j j j

ru ru ru ru ru ru ruP P c t T I K

Dif. Tropo. and Iono. :Small variations

Dif. Instrumental delays and carrier ambiguities:constant

Dif. Wind-up: Very small

Dif. Receiver clock:Main variations Common for all satellites

Single-Difference of measurements

(corrected by geometric range!!)

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1 1

sat sat

ru ruL L

PRN06-PRN30

PLAN-GARR: 15km

PRN06-PRN30

PLAN-GARR: 15km

1( )L


ru ru ru ru ru ru ruL L T I N

1( )P

jk jk jk jk jk

ru ru ru ru ruP P T I

Dif. Tropo. and Iono. :Small variations

Carrier ambiguities: constant

Dif. wind-up: negligible

1 1

sat sat

ru ruP P

Double-Difference of measurements

(corrected by geometric range!!)

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Single difference ( ) ( ) ( ) ( ) j j j j

ru ru u r

P

j j j j j


where:0 0 0 0

ˆ ˆ ˆ j j j j j j

ru ru u ru ru site ru ephρ r ρ ε ρ ε

L

j j j j j j j

ru ru ru ru ru ru ru ru ruL c t T I N b

Double difference

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) jk jk k j k k j j

ru ru ru ru u r u r

P

jk jk jk jk jk

ru ru ru ru ruP T I

where:0 0 0 0 0

ˆ ˆ ˆ ˆjk jk jk jk k k j j

ru ru u ru ru site ru eph ru eph ρ r ρ ε ρ ε ρ ε

L


ru ru ru ru ru ru ruL T I N Reference Station

User

0 0 0 ;jk k j

ru ru ru ru u r r r rbeing:

Sat jSat k

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Exercise:

Consider the Single Differences of geometric range:

where

Show that the Double Differences are given by:

j j j

ru u r

being:

0 0 0 0ˆ ˆ ˆ j j j j j j

ru ru u ru ru site ru ephρ r ρ ε ρ ε

0 0 0 0 0ˆ ˆ ˆ ˆjk jk jk jk k k j j


0 0 0 ;jk k j

ru ru ru ru u r r r r

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( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) jk jk k j k k j j

ru ru ru ru u r u r

P

jk jk jk jk jk

ru ru ru ru ruP T I

where:0 0 0 0 0

ˆ ˆ ˆ ˆjk jk jk jk k k j j


L



For short baselines (e.g. up to 10 km) and if the reference

station coordinates are accurately known, we can assume:

Reference Station

User

0

0 ; 0 ; 0

ˆ 0

0

jk jk jk

ru ru ru

j j

ru eph

site ru u

T I

ρ ε

ε r r

With these simplifications, we have:

Note for baselines up to 10 km

the range error of broadcast orbits is less than 1cm

(assuming ).10 m j

eph

9

0 0ˆjk jk jk jk

ru ru u u P ruP ρ r

0 0ˆ

L

jk jk jk jk jk

ru ru u u ru ruL N ρ r

ru u r r r r

Remark: 0 0 0

jk jk jk jk jk jk

ru ru u u r rP P P

Sat j Sat k

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Differential code and carrier positioning

For larger distances, the atmospheric propagation effects (troposphere, ionosphere) must be removed with accurate modelling. Wide area users will require also orbit corrections.

10

As with the SD, the left hand side of previous equations can be spitted in two terms: one associated to the reference station and the other to the user. Then, the differential corrections can be computed for code and carrier as:

0 0ˆ

P

jk jk jk jk jk

u u P u u ruP PRC ρ r

• The user applies this differential correction to remove/mitigate common errors:

Where the carrier ambiguities N are integer numbers and must be estimated together with the user solution.

0 0;jk jk jk jk jk jk

P r r L r rPRC P PRC L

0 0ˆ

L

jk jk jk jk jk jk

u u L u u ru ruL PRC N ρ r

Remark: 0 0 0

jk jk jk jk jk jk

ru ru u u r rP P P

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0 0

0 0

0 0

0 0

1

,1

1,1

,1

, 11

, 1

, 1

1

ˆ ˆ 0 0Pref

ˆ ˆ 1 0Pref

Pref ˆ ˆ 0 0

Prefˆ ˆ 0 1

TR

u uR

P TR u

Ru u

RLru

TR nn R

P R nu uruR n

TLn R

u u

N

N

ρ ρ

rρ ρ

ρ ρ

ρ ρ

Reference Station

User

11

0

0

, , , ,

, , , ,

Pref

Pref

R k R k R k R k

P u u P

R k R k R k R k

L u u L

P PRC

L PRC

The previous system for navigation equations is written in matrix notation as:

where

Differential code and carrier positioning

The user applies this differential correction to remove/mitigate common errors

0 0ˆ

P

jk jk jk jk jk

u u P u u ruP PRC ρ r

0 0ˆ

L

jk jk jk jk jk jk

u u L u u ru ruL PRC N ρ r0 0 0ˆ ˆ ˆjk k j

u u u ρ ρ ρwhere

Carrier ambiguities

ur

Sat jSat k

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– The reference station with known coordinates, computes differential corrections:

– The user receiver applies these corrections to its own measurements to remove SIS errors and improve the positioning accuracy.

Reference station(known Location)

Actual SV Position

Broadcast SV Position

Differential Message Broadcast

PRC

Measured Pseudoranges

Calculated Range ref

Pref

Puser

User

Differential positioning

0 0;jk jk jk jk jk jk

P r r L r rPRC P PRC L

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Correlations among the DD Measurements

We assume uncorrelated measurements (both code and carrier). Then, the covariance matrix is diagonal:

2

P PP I2

L LP I where, we can assume: 50 ; 5P Lcm mm

1 1 0 0;

0 0 1 1

k

u

k kru r

jjuru

j

r

X

X X

XX

X

2

2

2

2

2

1 00 0 0

1 1 0 0 1 00 0 02

0 0 1 1 0 10 0 0

0 10 0 0

SD

X

X

X X

X

X

P I

Let X represent the code P or the Carrier L measurement.

• The single difference (SD) and its covariance matrix can be computed as:

Thence, if the measurements are uncorrelated, so are they in single differences, but the noise is twice!

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Correlations among the DD Measurements

• Now, the double difference (DD) and its covariance matrix can be computed as:

1 1 0;

0 1 1

k

rujk

ru j

rujllruru

XX

XX

X

2

2 2

2

2 0 0 1 01 1 0 2 1

0 2 0 1 1 20 1 1 1 2

0 0 2 0 1

DD

X

X X X

X

P

Thence, even if the original measurements are uncorrelated, the double differences are correlated.

Note: The removal of the relative “User”—“Reference-station” common bias (e.g. relative receiver clock) in DD is done at the expense of one observation and the introduction of a correlation between measurements.

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Single and double differences of receivers/satellites

rov ref

k R

k R

rov ref

Receiver errors affecting both satellites are removed (e.g. Receiver clock)

SIS errors affecting both receivers are removed (e.g. Satellite clocks,...)

Receiver errors common for all satellites do not affect positioning (as they are assimilated in the receiver clock estimate). Thence:

- Only residual errors in single differences between sat. affect absolute posit.- Only residual errors in double differences between sat. and receivers affect

relative positioning.

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Single and double differences of receivers/satellites

rov ref

k R

k R

rov ref

Receiver errors affecting both satellites are removed (e.g. Receiver clock)

SIS errors affecting both receivers are removed (e.g. Satellite clocks,...)

When comparing SD and DD one might suggest that in the DD formulation there is even further error reduction, positively influencing the results in positioning. This is however not true, since in the SD case the mean value of unmodelled effects is absorbed by the receiver clock. If the DD correlations are taken into account, the positioning results in both cases are the same. However the DD formulation has the advantage that it allows the direct estimation of the ambiguities.

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Contents

17












- LAMBDA Method

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ˆj j j j

ru u r u ru ρ r

Relative Positioning

The following relationship can be obtained from the figure, where we assume that the baseline is shorter than the distance to the satellite by orders of magnitude:

Applying the same scheme to a second satellite “k”

ˆ k k k k

ru u r u ruρ r

Thence, the double differences of ranges are:

ˆ ˆ ˆ jk k j k j jk

ru ru ru u u ru u ruρ ρ r ρ r

ReferenceStation

ru u rr r r

ur rr

To Earth Centre

j

u

j

r

j j j

ru u r

Satellite-j

ˆ j

uρ

j j j

ru u r

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Exercise:

Demonstrate the following relationship between the baseline and the differential range [*]:

2

jj j j u ru

ru u r ruj j

u u ru

ρ rr

ρ ρ r

[*] This result is from [RD-7]

• Taking in and leads to the approximate expression previously found.

• and depend on the baseline , which is the vector to estimate.Nevertheless, it is not very sensitive to changes in such baseline and can be computed iteratively, computing the navigation solution starting from .

0rur

ˆ j j j j

u r rue r2

j

u ruj

j j

u u ru

ρ r

ρ ρ rwith

Comments:

The previous expression can be written as:

j ˆ je

j ˆ je

rur

0rur

/ 2ˆ

/ 2

jj u ru

j

u ru

ρ re

ρ r

Reference StationUser

j

uρ

/ 2rur

ru u rr r r

ˆ je


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Solution

Consider the following relations:

j j

ru r ur ρ ρ

/ 2ˆ

/ 2 2

j j jj u ru r u

j j

u ru u ru

ρ r ρ ρe

ρ r ρ r

2 2

ˆ2

j j j j

u ru ru r u

j j j j

r u r u

j j j j

r u u ru u

ρ r e r ρ ρ

ρ ρ ρ ρ

ρ r ρ

Then:

ˆ j j j j

u r rue r

2

j

u ruj

j j

u u ru

ρ r

ρ ρ r

2ˆ

jj j u ru

j j

u u ru

ρ re

ρ ρ rwith:

Reference StationUser

j

uρ

/ 2rur

ru u rr r r

ˆ je


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As commented before, for short baselines (e.g. less than 10km), we can assume that ephemeris and propagation errors cancel, thence:

ˆ P

jk jk jk

ru u ru ruP ρ r

ˆ L

jk jk jk jk

ru u ru ru ruL Nρ r

Thence, the double differences of ranges are:

ˆ ˆ ˆ jk k j k j jk

ru ru ru u u ru u ruρ ρ r ρ r

Note that these equations allows a direct estimation of the baseline, without needing an accurate knowledge of the reference station coordinates.


P

jk jk jk jk jk

ru ru ru ru ruP T I

L



P

jk jk jk

ru ru ruP

L

jk jk jk jk

ru ru ru ruL N

0 ; 0

0

jk jk

ru ru

jk

ru

T I

ReferenceStation

ru u rr r r

ur rr

To Earth Centre

j

u

j

r

j j j

ru u r

Satellite-j

ˆ j

uρ

Note: but, at least the approximate coordinates of reference station or user are needed to compute

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1

,1

,

1,1

,1,

, 1 1, 1,

, 1

, 1

ˆ ˆ 0 0

ˆ ˆ 1 0

ˆ ˆ 0 0

ˆ ˆ 0 1

TR

R u u

r uT

R ruR

u uRr u

ru

TR n n RR nr u u uruR n

Tr u n R

u u

P

LN

PN

L

ρ ρ

rρ ρ

ρ ρ

ρ ρ

22

The previous system for navigation equations is written in matrix notation as:

Carrier ambiguities


ˆP

jk jk jk

ru u ru ruP ρ r

ˆ L

jk jk jk jk

ru u ru ru ruL Nρ rˆ ˆ ˆjk k j

u u u ρ ρ ρwhere

ReferenceStation

ru u rr r r

ur rr

To Earth Centre

j

u

j

r

j j j

ru u r

Satellite-j

ˆ j

uρ

Baseline vector

DD Code and Carrier measurements

ru u r r r r

urrr

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1 L


ru ru ru ru ru ru ruL L T I N

1 P

jk jk jk jk jk

ru ru ru ru ruP P T I


1 1

sat

ruL L 1 1

sat

ruP P

PRN06-PRN30

PLAN-GARR: 15km

PRN06-PRN30

PLAN-GARR: 15km

Variation of the Baseline projection over the unit Line-Of-Sight vector

ˆjk jk

ru u ru ρ r

ReferenceStation

ru u rr r r

ur rr

To Earth Centre

j

u

j

r

j j j

ru u r

Satellite-j

ˆ j

uρ

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1 1

sat

ruL L 1 1

sat

ruP P

PRN06-PRN30

PLAN-GARR: 15km

PRN06-PRN30

PLAN-GARR: 15km

PRN06-PRN30

IND1-IND2: 7 m

PRN06-PRN30

IND1-IND2: 7m

1 1

sat

ruL L 1 1

sat

ruP P

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In this approach, the reference station broadcast its time-tagged code and carrier measurements, instead of the computed differential corrections.

Thence, the user can form the double differences of its own measurements with those of the reference receiver, satellite by satellite, and estimate its position relative to the reference receiver.

Notice that, the baseline can be estimated without needing an accurate knowledge of reference the station coordinates. Of course, the knowledge of the reference station coordinates would allow the user to compute its absolute coordinates.


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Time synchronization issues:

There is an important and subtitle difference between the previous approach of relative positioning (which does not need to know the reference station coordinates) and the differential positioning approach based on the knowledge of the reference station coordinates.

• The differential corrections wary slowly, and its useful life can be up to several minutes with S/A=off.

• But, the measurements change much faster. The range rate can be up to 800m/s and, thence, a synchronizer error of 1millisecond can lead up to more than 1/2 meter of error.

• As commented before, real-time implementation entails also latencies, that can be up to 2 seconds, thence, a extrapolation technique must be applied to the measurements to reduce error due to latency and epoch mismatch (to <1cm if ambiguities are intended to be fixed).

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PRC L1

dL1

Receiver: JAVAD TRE_G3TH DELTA3.3.12

RRC

~ 660 m/s

up to ~800 m/s

recdt

d ~ 2cm/sRRC

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L1

dL1

L1

dL1

1 recdL d dt

300 km/450s 667m/srec

dt

1 recdL d dt

~ 660 m/s

up to ~800 m/s

recdt

d

1 ms jump of receiver clock adjust

300Km=1ms

Receiver: JAVAD TRE_G3TH DELTA3.3.12

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300Km=1ms

COMMENTS

Real-Time implementation entails delays in data transmission, which can reach up to 1 or 2 s.

• Differential corrections vary slowly and its useful life is of several minutes (S/A=off)

• But, the measurements change much faster:

•The range rate d/dt can be up to 800m/s

and, therefore, the range can change by more than half a meter in 1 millisecond.

Moreover the receiver clock offset can be up to 1 millisecond (depending on the

receiver configuration).

•Thence, the reference station measurements must be :•Synchronized to reduce station clock mismatch: station clock can be estimated to within 1ms

•Extrapolated to reduce error due to latency: carrier can be extrapolated with error < 1cm.

1mmdtsta

dL1

1 recdL d dt

~ 660 m/s

up to ~800 m/s

recdt

d

RRC= PRC/t

RRC

RRC ~ 1 cm/s

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Contents

30












- LAMBDA Method

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Let us consider again the problem of relative positioning for short baselines. We have previously found the following equation for DD carrier measurements, assuming short baselines (e.g. < 10km)

ˆ ˆ L

jk k j jk jk

ru u u ru ru ruL Nρ ρ r

The Role of Geometric Diversity: Triple differences

This is from [RD-3]

As the ambiguities are constant along continuous carrier phase arcs, an option could be to take differences on time. Thence, if the user and reference receiver are stationary we can write the “triple differences” as:

ˆ ˆ L

jk k j jk

ru u u ru ruL ρ ρ r

2 112

13 3 1

1

1

ˆ ˆ

ˆ ˆ

ˆ ˆ

T

u uru

T

ru u uru

KT

Kru

u u

L

L

L

ρ ρ

ρ ρr ν

ρ ρ

For simplicity, we assign (j=1) to the

reference satellite

2 1( ) ( )jk jk jk

ru ru ruL L t L t

where:

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This is from [RD-3]

ˆ ˆ L

jk k j jk

ru u u ru ruL ρ ρ r

2 112

13 3 1

1

1

ˆ ˆ

ˆ ˆ

ˆ ˆ

T

u uru

T

ru u uru

KT

Kru

u u

L

L

L

ρ ρ

ρ ρr ν

ρ ρ

Now, we have a “clean” equations system involving only the baseline vector to estimate. But the geometry is very weak (the associated DOP will be large number) and the position estimates will be in general worse than those from double differences.

The Role of Geometric Diversity: Triple differences

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Let us now consider a simple model for the estimation of the relative position vector from SD carrier measurements, assuming short baselines (e.g. <10km):

ˆ j j j j

ru u r u ruρ r

ReferenceStation

ru u rr r r

ur rr

To Earth Centre

j

u

j

r

j j j

ru u r

Satellite-j

ˆ j

uρ

L

j j j j

ru ru ru ru ru ruL c t N b ˆ L

j j j j

ru u ru ru ru ruL d Nρ r

ru ru rud c t b where

Estimation of position and change in position:

the role of Geometric Diversity This is from [RD-3]

11 1

2 22

ˆ 1

ˆ 1

ˆ 1

T

uru ru

T

ruru ruu

ru

K KT

Kru ru

u

L N

L N

d

L N

ρ

rρν

ρ

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Previous system can be arranged as:

1 0

1 0 1 0

1 0

( ) ( )( ) ( ) ( ) ( )

( ) ( )

ru ru

ru ru

ru ru

t tt t t t

d t d t

r rL L G G ν

1 0

1 0 1 1 0

1 0

( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( )

ru ru

ru ru

ru ru

t tt t t t t

d t d t

r rL L G G G ν

Considering now differences between two epochs t0 and t1 , and assuming no cycle-slips:

ruL G

1 1

1 0

1 1

1 01 0

1 1

1 0

ˆ ˆ( ) ( ) 0

ˆ ˆ( ) ( ) 0( ) ( )

ˆ ˆ( ) ( ) 0

T

u u

T

u u

T

u u

t t

t tt t

t t

ρ ρ

ρ ρG G

ρ ρ

Estimation of changes in baseline vector and clock bias is tied to the geometry matrix at time t1 . This can be well determined.

Estimation of absolute valueof baseline vector is tied to the change in geometry matrix at time t1 . This would be poor determined if such change is not significant.

cannot be estimated at all!

0( )rud t

Estimation of position and change in position:

the role of Geometric Diversity This is from [RD-3]

11 1

2 22

ˆ 1

ˆ 1

ˆ 1

T

uru ru

T

ruru ruu

ru

K KT

Kru ru

u

L N

L N

d

L N

ρ

rρν

ρ

1 1( ) ( )t t G G1 1 0( ) ( ) ( )ru ru rut t t r r r

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- Least-Squares Ambiguity Search Technique.

- LAMBDA Method.

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Ambiguity resolution Techniques

To simplify notation, when different frequencies are considered, we will remove the subscript “ru”.

q

jk jk jk jk

q q q LL N

As commented before, the ambiguity terms are integer numbers, and we can take benefit of this property to fix such ambiguities applying integer ambiguity resolution techniques.

; 1,2 q

jk jk jk

q PP qK sat. in view K ‘SD’

K(K-1) ‘DD’, but only K-1 DDs are

linearly independent

Take the highest elevation as the reference satellite to minimize

measurement error.

As a driven problem to study the ambiguity fixing, we will consider the problem of differential positioning in DD for short baselines (e.g. < 10 km).

In general we will consider that we have Code and Carrier measurements in different frequencies (q=1,2…), i.e. P1, P2, L1, L2…

, , Lq

jk jk jk jk

q ru ru q q ru ruL N

, ; 1,2 Pq

jk jk jk

q ru ru ruP q

We assume the following measurement errors:

0.5m

0.5cm

q

q

P

L

1m

1cm

jkq

jkq

P

L

, , , ; 1,2q

jk jk jk jk jk

q ru ru ru q ru P ruP T I q

, , , ,q

jk jk j jk jk jk jk

q ru ru ru q ru q ru q ru L ruL T I N ,

0

0

0

j

ru

jk

q ru

jk

ru

T

I

Short baseline

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- LAMBDA Method.

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11ˆ

1

15jk jk

N P

38

Resolving ambiguities one at a Time

A simple trial would be (for instance using L1 and P1):

Too much error (5 wavelengths)!

11 1 1 jk jk jk jk

LL N

11 jk jk jk

PP1 1

1

1

ˆ

jk jkjk

roundoff

L PN

11 1 1 1 jk jk jk jk

PL P N

1

1

1 20cm

1m

1cm

jk

jk

P

L

1N

1N

1/ 2N

1/ 2N

Note that, assuming a Gaussian distribution of errors, guarantee only the 68% of success

1ˆ 1/ 2jk

N

N

As the ambiguity is constant (between cycle-slips), we would try to reduce uncertainty by averaging the estimate on time, but we will need 100 epochs to reduce noise up to ½ (but measurement errors are highly correlated on time!)

Good

Fail

N̂

Similar results with L2, P2 measurements

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- LAMBDA Method.

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Resolving ambiguities one at a Time Dual frequency measurements: wide-laning with the Melbourne-Wübbenacombination

1

2

1 1 1

2 2 2

jk jk jk jk

L

jk jk jk jk

L

L N

L N

1

2

1

2

jk jk jk

P

jk jk jk

P

P

P

ˆ

jk jkjk W N

W

W roundoff

L PN

1 1 2 2

1 2

1 1 2 2

1 2

N

W

jk jkjk jk jk

N P

jk jkjk jk jk jk

W W W L

f P f PP

f f

f L f LL N

f f

1 2

1 2

86.2 m

W

W

N N N

cc

f f

Now, with uncorrelated measurements from 10 epochs will reduce noise up to about ¼.

ˆ

1 710.8

86.2jk jk

NWN P

W

cm

cm

N

jk jk jk jk

W N W W PL P N

1 2

1 2

1 2 1 1 2 2

1 2 1 2

jk jk jk jk jk

L L

jk jk jk

W L L

L L N N

N N

1

1

/ 2 71 m

6 6cm

jk jkN

jk jkW

P P

L L

c

1 2 21

1 2

ˆˆ

jk jk jkjk W

roundoff

L L NN

Fixing N1 (after fixing NW)

1 1ˆ

1 2

1 1.42 1/ 4

5.4jk jk

LN

cm

cm

2 1ˆ ˆ ˆ jk jk jk

WN N N

1

1

2

2 1

19.0cm

24.4cm

5.4cm

1cmjkL

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44

Once the integer ambiguities are known, the carrier phase measurements become unambiguous pseudoranges, accurate at the centimetre level (in DD), or better.

Thence, the estimation of the relative position vector is straightforward following the same approach as with pseudoranges.

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Exercises:

1) Consider the wide-lane combination of carrier phase measurements

, where LW is given in length units (i.e. Li= i fi ).

Show that the corresponding wavelength is:

Hint:LW= W fW ; fW = f1 – f2

2) Assuming L1, L2 uncorrelated measurements with equal noise L,

show that:

1 2

W

c

f f

1 1 2 2

1 2

W

f L f LL

f f

2

12 112

212

1;

1WL L

f

f

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- LAMBDA Method.

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Three Frequency measurements:

1 2 2 5

; ;

i W EW

i

c c c

f f f f f

1 2

2 5

W

EW

N N N

N N N1

1

12

12

25

25

10,712m

1

10,707 m

1

N

EN

P P

P P

1 1 2 2

1 2

2 2 5 5

2 5

N

EN

N P

EN P

f P f PP

f f

f P f PP

f f

1 1 2 2

1 2

2 2 5 5

2 5

W

EW

W W W L

EW EW EW L

f L f LL N

f f

f L f LL N

f f

1

1

12

12

25

25

15,7 m

1

133,3 m

1

W

EW

L L

L L

c

c

With three frequency systems, having two close frequencies it is possible to generate an extra-wide-lane signal to enable the single epoch ambiguity fixing.

; 1,2,5 ii i i LL N i

We still consider the above problem of relative positioning in DD for short baselines (e.g. < 10 km) Ionosphere, troposphere and wind-up differential errors cancel.

We drop here the superscript (jk) for simplicity

2 2

12 1 2

2 2

25 2 5

/ (77 / 60)

/ (24 / 23)

f f

f f

GPS Frequency Wavelengths Combinations

L1 154 x 10.23 MHz 1 =0.190 m 2 1 =0.054 m

L2 120 x 10.23 MHz 2 =0.244 m W = 0.862 m

L5 115 x 10.23 MHz 5 =0.255 m EW =5.861 m

Exercise:Justify the previous

expressions for .

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GPS Frequency Wavelengths Combinations

L1 154 x 10.23 MHz 1 =0.190 m 2 1 =0.054 m

L2 120 x 10.23 MHz 2 =0.244 m W = 0.862 m

L5 115 x 10.23 MHz 5 =0.255 m EW =5.861 m

ˆ

EW ENEW

EW roundoff

L PN

ˆˆ

EW EW EW W

W

Wroundoff

N L LN

1 2 21

1 2

ˆˆ

W

roundoff

L L NN

1 2 2 5; W EWN N N N N N

1

1

12

12

25

25

10,71m

1

10,71m

1

N

EN

P P

P P

N

EN

N P

EN P

P

P

W

EW

W W W L

EW EW EW L

L N

L N

1

1

12

12

25

25

15,7 m

1

133,3 m

1

W

EW

L L

L L

c

c

ˆ

1 0.71m0.12

5.861mENEWN P

EW

ˆ

1 33.3cm0.39

86.2cmEWWN L

W

1 1ˆ

1 2

1 1.42 1/ 4

5.4LN

cm

cm

1

2

1 1 1

2 2 2

L

L

L N

L N

1 2

1 2

1cm

1m

L L

P P

2 2

12 1 2

2 2

25 2 5

/ (77 / 60)

/ (24 / 23)

f f

f f

We still consider the above problem of relative positioning in DD for short baselines (e.g. < 10 km) Ionosphere, troposphere and wind-up differential errors cancel .

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ˆ

EW ENEW

EW roundoff

L PN

ˆˆ

EW EW EW W

W

Wroundoff

N L LN

1 2 21

1 2

ˆˆ

W

roundoff

L L NN

1

1

12

12

25

25

1

1

1

1

N

EN

P P

P P

1

1

12

12

25

25

1

1

1

1

W

EW

L L

L L

ˆ

1

ENEWN P

EW

ˆ

1

EWWN L

W

1 1ˆ

1 2

12

LN

Galileo Frequency Wavelengths Combinations

E1 154 x 10.23 MHz 1 =0.190 m 2 1 =0.058 m

E5b 118 x 10.23 MHz 2 =0.248 m W = 0.814 m

E5a 115 x 10.23 MHz 3 =0.255 m EW =9.768 m

2 2

12 1 2

2 2

23 2 3

/ (77 / 59)

/ (118 /115)

f f

f f

N

EN

N P

EN P

P

P

W

EW

W W W L

EW EW EW L

L N

L N

1

2

1 1 1

2 2 2

L

L

L N

L N

1 2

1 2

1cm

1m

L L

P P

Exercise:

Repeat the previous study for the Galileo signals E1, E5b and E5a

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ˆ

EW ENEW

EW roundoff

L PN

ˆˆ

EW EW EW W

W

Wroundoff

N L LN

1 2 21

1 2

ˆˆ

W

roundoff

L L NN

1

1

12

12

25

25

10,71m

1

10,71m

1

N

EN

P P

P P

1

1

12

12

25

25

15.4cm

1

154.9cm

1

W

EW

L L

L L

ˆ

1 0.71m0.07

9.768mENEWN P

EW

ˆ

1 54.9cm0.67

81.4cmEWWN L

W

1 1ˆ

1 2

1 1.4cm2 1/ 4

5.8cmLN

2 2

12 1 2

2 2

23 2 3

/ (77 / 59)

/ (118 /115)

f f

f f

N

EN

N P

EN P

P

P

W

EW

W W W L

EW EW EW L

L N

L N

1

2

1 1 1

2 2 2

L

L

L N

L N

Exercise:

Repeat the previous study for the Galileo signals E1, E5b and E5a

Galileo Frequency Wavelengths Combinations

E1 154 x 10.23 MHz 1 =0.190 m 2 1 =0.058 m

E5b 118 x 10.23 MHz 2 =0.248 m W = 0.814 m

E5a 115 x 10.23 MHz 3 =0.255 m EW =9.768 m

1 2

1 2

1cm

1m

L L

P P

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- LAMBDA Method.

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G(t)

Linear Model:

52

Each epoch brings a set of (K-1) equations for each frequency.Note: nt is the number of epochs

As a driven problem to study the ambiguity fixing, we will consider problem of differential positioning in DD for short baselines (e.g. < 10 km). To simplify, we will consider only carrier measurements at a single or dual frequency.

Resolving Ambiguities as a set

12 12 12 12

13 13 13 13

1 1 1 1 1 1 1 1

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

q

q

q

q i i q L i

q i i q L i

K K K K

q i i q L i

L t t N t

L t t N t

L t t N t

Static position Equations Unknowns

Single frequency (K-1)* nt 3+(K-1)

Dual frequency 2(K-1)*nt 3+2(K-1)

In principle, the estimation of ambiguities in this system is not a big problem if we can wait enough time and the unmodelled errors are not so large.

Kin. position Equations Unknowns

Single frequency (K-1)* nt 3* nt+(K-1)

Dual frequency 2(K-1)*nt 3* nt+2(K-1)

0 0ˆ( ) ( ) ( ) ( ) jk jk jk

i i i it t t tρ r

( ) ( ) ( ) q

jk jk jk jk

q i i q L iL t t N t

( ) ( ) ( ) i i it t ty G r N ν

Prefit-residualy(t)

We can estimate all parameters (position and ambiguities) as a set by considering the over-dimensioned system of linear equations and solving it by the LS criterion.

4, 2tK n 5, 4tK n

1,2q

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For static positioning, considering two epochs (for instance):

( ) ( ) ( ) ( ) i i i it t t ty G r N ν Single freq.: K=K-1

Dual freq.: K=2(K-1)K 3

matrix

3

vector

vector

K

vector

K

1 1 1

( ) ( ) ( )

( ) ( ) ( )

i i i

i i i

t t t

t t t

y G νIr N

y G I ν

y G r AN νIn general, mixing several epochs, we will write:

Using the least-squares criterion, we can look for a real valued 3-vector and a -vector of integers that minimizes the cost function (sum of squared residuals):

( , ) c r N y G r A N Weighted norm can be taken as well

rNK

The problem can be easily reformulated for the kinematic case. Kalman filtering can be applied as well.

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Different strategies can be applied:

• To Float the ambiguities (i.e. treating the ambiguities as real numbers).• To Search ambiguities over a limited set of integers to ‘find the best

solution’.• To solve as an Integer Least-Squares problem.

For an observation span relatively long, e.g. one hour, the floated ambiguities would typically be very close to integers, and the change in the position solution from the float to the fixed solution should not be large.

As the observation span becomes smaller, ambiguity resolution play a more important role. But very short observation spans implies the risk of wrong ambiguity fixing, which can degrade the position solution significantly.

The performance, is thence measured by:

1. Initialization time2. Reliability (or, correctness) of the integer estimates


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- LAMBDA Method.

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Search techniques

Position domain

Ambiguity Function Method (AFM)

ARCE

…………

Ambiguity domain

LSAS (Hatch, 1990)

LAMBDA (Teunissen, 1993)

MLAMBDA (Chang et al. 2005)

OMEGA (Kim and Langley, 2000)

FASF (Chen and Lachappelle, 1995)

IP (Xu et al., 1995)

……………

Strategy:

• Define a volume to be searched

• Set up a grid within this volume

• Define a cost function (e.g. the sum of squared residuals)

• Evaluate the cost function at each grid point

Solution corresponds to the grid point with the lowest value of the cost function

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A conceptually simpler approach would consist on:

• Estimate the floated solution and its uncertainty

• Define as a volume to be searched and

evaluate the cost function (the RMS residuals) over the 6 ambig.:

The previous search must be done for each satellite in view.

- If there are 5 satellites tracked 4 DD ambiguities 64 = 1 296 combinations

- If there are 8 satellites tracked 7 DD ambiguities 67 =279 376 combinations

ˆˆ(e.g. =2502347.74 cycles, 0.6cycles)

NN

ˆe.g. 3 2cyclesN

2502345, ,2502350

The integer ambiguity solution corresponding to the smallest RMS residuals is used to select the candidate. However if two or more candidates give roughly similar values of RMS, the test can not be resolute.A ratio test (of 2 or 3, depending of the algorithm) between the two smallest RMS

is often used to validate the test. If the ratio is under these values, no integer solution can be determined and is better to use the floated solution.

Search techniques( )k ˆ rˆ r

ˆr

N̂

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- LAMBDA Method

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LAMBDA Method

59

Consider again the previous problem of estimating , a 3-vector of real numbers, and a (K-1)-vector of integers, which are solution of

y G r AN ν min yW

y G r AN

r

N

To better exploit the internal correlations [*], we consider now the covariance 1y yW P

[*] Remember that DD measurements are correlated, as already seen.

It can be shown the following orthogonal decomposition:

ˆ ( ) ˆ

2 22 2 2ˆ ˆˆ ˆ( )

r N

NWy WWy W

y G r A N y G r A N r r N N N

Let be the float solution and covariance matrix:

ˆ;

ˆ

r

N

ˆ ˆˆ ,

ˆ ˆˆ ,

ˆCov

ˆ

r r N

r N N

P Pr

P PN

Residual of real-valued floated solution ˆˆ,r N

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Thence, we have to find a 3-vector of real numbers, and a (K-1)-vector of integers minimizing:

ˆ ( ) ˆ

2 22 2 2ˆ ˆˆ ˆ( )

r N

NWy WWy W

y G r A N y G r A N r r N N N

and r N

r N

This term is irrelevant for minimization since it does not depend on

This term can be made zero for any N

ˆ

2

1

ˆ ˆˆ ,

ˆmin

ˆˆ ˆ( )

NW

r N N

N N N

r r N r W W N N

Float solution and covariance matrix:

ˆ;

ˆ

r

N

ˆ ˆˆ ,

ˆ ˆˆ ,

ˆCov

ˆ

r r N

r N N

P Pr

P PN

1

ˆ ˆˆ ˆ, ,

r N r NW P1

ˆ ˆ

N N

W P

This term must be minimized over the integers

The vectors and are often referred to as the fixed user solution and fixed ambiguity.

r N

LAMBDA Method

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ˆ

2

ˆˆ ˆ ˆ( )

T

cN

NWN N N N N W N N

The integer search: Finding the integer vector N that minimizes the cost function

• A diagonal WN matrix would mean that the integer ambiguity estimates are

uncorrelated.

• If the weight WN matrix is diagonal, the minimizing of the cost function is trivial. The

best estimate is the float ambiguity rounded to the nearest integer.

1

ˆ ˆ

N N

W P

ˆ ˆ1 1

ˆ ˆ2 2

2

ˆ 2

1/ 0

0 1/

N N

N N

NW

ˆ ˆ ˆ ˆ1 1 2 2

2 2

1 1 2 2

2 2

ˆ ˆ

( )

N N N N

N N N Nc N

In practice, the estimated (float) ambiguities are highly corre-lated and the ellipsoidal region stretches over a wide range of cycles. This is specially the case when the measurements are limited to a single epoch or only a few epochs.

Thence, points that appears much further away from the floated solution may have lower values of cost function than those which appear nearby. In this context, the search for integer vectors can by extremely inefficient.

1N̂

2N̂

61

Ellipse parallel to coordinate axes

LAMBDA Method

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Thence, we are looking for an “integer-valued” transformation matrix Z that makes the matrix W as close as possible to a diagonal matrix and with quite similar axes (spherical).

ˆ ˆ

N Z N

N Z N Note that(i.e. it is a volume-preserving transformation)

1, integers det( ) 1 Z Z Zˆ ˆ T

N NP Z P Z

Z

•To improve the computational efficiency of the search, the float ambiguities should be transformed so that the elongated ellipsoid turns into a sphere-like. Thus, the search can be limited to the neighbours of the floated ambiguity.

• Moreover, this transformation should make the matrix W as close as possible

to diagonal (i.e. decorrelating the ambiguities). Note: W is a positive definite matrix and thence, can be always diagonalized (as a real-valued matrix) with orthogonal eigenvectors. But the problem here is that the integer ambiguities N must be

transformed preserving its integer nature!

Moreover, the inverse of transformation matrix Z-1 must be also integer, to transform

back the results after finding the ambiguities

Pictures from

[RD-6]

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Exercise:

Show that:

1, Integers det( ) 1 Z Z Z

That is, Z is a volume-preserving transformation

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Decorrelation: Computing the Z-transform

The following conditions must be fulfilled:1. Z must have integer entries2. Z must be invertible and have integer entries3. The transformation Z must reduce the product of all

ambiguity variances.Note that (i.e. it is a volume-preserving transformation)

1, integers det( ) 1 Z Z Z

Gauss manipulation over matrix P=W-1 can be applied to find-out the matrix Z.

1 1 1 2

1 2 2 2

ˆ ˆ ˆ ˆ

ˆ

ˆ ˆ ˆ ˆ

N N N N

N N N N

p p

p pNP

1

1

1 0

1

Z

2

2

1

0 1

Z

1 2ˆ ˆ ˆ ˆint /

i ii N N N N

p p

Transforms N2 (N1 remains unchanged)


Start transforming first the element with largest variance.1

1

1

1 0

1

Z

21

2

1

0 1

Z

Note: Inverse matrices have also integer entries

From [RD-4]

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Gauss manipulation over matrix P=W-1 can be applied to find-out the matrix Z

1 1 1 2

1 2 2 2

ˆ ˆ ˆ ˆ

ˆ

ˆ ˆ ˆ ˆ

N N N N

N N N N

p p

p pNP

1

1

1 0

1

Z

2

2

1

0 1

Z

1 2ˆ ˆ ˆ ˆint /

i ii N N N N

p p



ˆ

53.4 38.4

38.4 28.0

N

P1.05

ˆ1.30

N

1

1 0

2 1

Z 1 int 10.4 / 4.6 2

ˆ ˆ1 1

1 0 4.6 10.4 1 2 4.6 1.2

2 1 10.4 28.0 0 1 1.2 4.8

T

N N

P Z P Z

2

1 1

0 1

Z 2 int 38.4 / 28.0 1

ˆ ˆ2 2

1 1 53.4 38.4 1 0 4.6 10.4

0 1 38.4 28.0 1 1 10.4 28.0

T

N NP Z P Z

Step 1:

Step 2:

Example:

The half, at most!

We transform first the element with largest variance (in this case N1)

In general,to increase the number of small off-diagonal elements, we have to transform first the elements with largest variance

Example from [RD-4]

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2

1 1

0 1

Z 1

1 0

2 1

Z

ˆ

53.4 38.4

38.4 28.0

N

Pˆ

4.6 10.4

10.4 28.0

N

P ˆ

4.6 1.2

1.2 4.8

N

P

1 2

1 0 1 1 1 1

2 1 0 1 2 3

Z Z Z

ˆ

1 1 53.4 38.4 1 2 4.6 1.2

2 3 38.4 28.0 1 3 1.2 4.8

T

NN

P Z P Z

Example and pictures from

[RD-4]

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1 1

2 3

Z

1 1 1.05 0.25ˆ ˆ

2 3 1.30 1.80

N Z N

0.25 0int

1.80 2

N

10 3 1 0 2

2 2 1 2 2

N Z

ˆ

53.4 38.4

38.4 28.0

N

P1.05

ˆ1.30

N

Example:

ˆ

4.6 1.2

1.2 4.8

N

P

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ˆ

53.4 38.4

38.4 28.0

N

P1.05

ˆ1.30

N

Example:

11 12

12 22

p p

p p

P

1

2

0

0

P

1 11 22

2 11 22

1

2

1

2

p p w

p p w

2 2

11 22 12( ) 4w p p p

12

11 22

2tan 2

p

p pf

ˆ

81.14 0

0 0.25

N

P

1

2

81.14 9.0

0.25 0.5

tan 2 3.02 35 85

f f

1N

f2

2N 2N

1N

1

Let P be a symmetric and positive-definite matrix:

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Consider again the previous problem of estimating , a 3-vector of real numbers, and a (K-1)-vector of integers, which are solution of

r

N

y G r AN ν

The solution comprises the following steps:

1. Obtain the float solution and its covariance matrix:

2. Find the integer vector N which minimizes the cost function

a) Decorrelation: Using the Z transform, the ambiguity search space is

re-parametrized to decorrelate the float ambiguities. b) Integer ambiguities estimation (e.g. using sequential conditional least-

squares adjustment, together with a discrete search strategy).c) Using the Z-1 transform, the ambiguities are transformed to the

original ambiguity space.

3. Obtain the ‘fixed’ solution r, from the fixed ambiguities N.

ˆ;

ˆ

r

N

ˆ ˆˆ ,

ˆ ˆˆ ,

r r N

r N N

P P

P P

1

ˆ ˆ

N N

W P ˆ

2

ˆˆ ˆ ˆ( )

T

cN

NWN N N N N W N N

y AN G r ν

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b1) Integer rounding

This is the simplest way.

Just to round-up the ambiguity vector entries to its nearest integer

Several approach can be applied:

• Integer rounding

• Integer bootstrapping

• Integer Least-Squares

• …..

b) Integer ambiguities estimation

0.25 0int

1.80 2

N

For instance, in the previous example:

1ˆ ˆint( ), , int( )KN NN

Comment:In principle, the previous transformation Z

is not required by the estimation concept; it is only to achieve considerable gain in speed in the computation process [RD-5].

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b2) Integer bootstrapping (from [RD-6])

It makes use of integer rounding, but it takes some of the correlations between the ambiguities into account.

1. We start with the most precise floated ambiguity (here we will assume )

2. Then, the remaining float ambiguities are corrected taking into account

their correlation with the last ambiguity.

1

1 | |

2

ˆ ˆ ˆ1 1| 1 ,

2

ˆ ˆ ˆ1 | 1 |,2

înt

ˆ ˆ înt int

ˆ ˆ înt int

n n n

i I i I

n n

n n n n n nN N N

n

n I i I iN N Ni

N N

N N N N N

N N N N N

|ˆ

i IN Stands for the i-th ambiguity obtained through a conditioning of the

previous sequentially rounded ambiguities. 1, , I i n

ˆ T

NP L DL

| |

2

ˆ ˆ ˆ,

j i I i Iij N N N

l

Using the triangular decomposition

ˆnN

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Float

2N

1N

2 1 2

2

ˆ ˆ ˆ1 1 2 2,ˆ ˆ

N N NN N N N

2 2ˆnint 0

N N

2 1 2

2

ˆ ˆ ˆ1 1|2 1 2 2,ˆ ˆ ˆnint nint 1

N N NN N N N N

Example and pictures from [RD-6]

Conditional estimate

1|2N̂

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73

b3) Integer Least Squares (ISL) (from [RD-6])

1. The target to find the integer vector N which minimizes the cost function

2. The integer minimiser is obtained through a search over the integer grid points on the n-dimensional hyper-ellipsoid:

• Where c2 determines the size of search region.• The solution is the integer grid point N, inside the ellipsoid, giving the

minimum value of cost function c(N).

1ˆ

21

ˆˆ ˆ ˆ( )

T

cN

NPN N N N N P N N

Using the triangular decomposition: ˆ T

NP L DL

1 1 2ˆ ˆ c T

TN N L D L N N

1 2

ˆˆ ˆ c

T

NN N P N N

Defining:

ˆ ˆ( ) ( ) T TN N L N N L N N N N

|

| |

2

ˆ

2

ˆ ˆ ˆ,

where:

i I

j i I i I

i N

i j N N N

d

l

1 2c T

N N D N N

2 2 2

1 1 2 2 2

1 2

( ) c

n n

n

N N N N N Nc N

d d d

1

ˆ ˆ

N N

W P

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74

But depends on .

ˆ( ) TL N N N N

1

ˆ

ˆ ˆ ; 1, 2, ,1

n n

n

i i j i ji

j i

N N

N N N N l i n n

iN 1ˆ ˆ, ,i nN N

1/2 1/2

1/2 1/22 21/2 2 1/2 2

1 1 1 1 1

1/2 1/22 2

1/2 2 1/2 2

1 1 1 1

2 2

ˆ ˆ

ˆ ˆ

c c

c c

c c

n n n n n

n n n n n n n n n n n

n n

j j j n j j j

j j

N d N N d

N d N N d N N d N N d

N d N N d N N d N N d

1 2c T

N N D N N 2 2 2

1 1 2 2 2

1 2

( ) c

n n

n

N N N N N Nc N

d d d

Search region bounds:

Acceptance test: The integer ambiguity solution corresponding to the smallest RMS residuals is used to select the candidate. However if two or more candidates give roughly similar values of RMS, the test can not be resolute.A ratio test (of 2 or 3, depending of the algorithm) between the two smallest

RMS is often used to validate the test.

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75

Ellipsoid size: selecting the candidates for the acceptance test

The size of the ellipsoidal search region is controlled by c2

Therefore, the performance of the search process is highly dependent on c2:• A small c2 may result in a ellipsoidal region that fails to contain the solution.• A too large value for c2 may result in high time-consuming for the search process.

Search with enumeration: When the number of required candidates is at most n+1 (with n=dim(N)), the following procedure can be applied to set the value c2 :

• The best determined ambiguity is rounded to its nearest integer. The remaining ambiguities are then rounded using their correlations with the first ambiguity:

• In each step of the conditional rounding procedure, two candidates are taken: The nearest and second-nearest, and conditional rounding is proceeded in both cases.

• If p candidates are requested, the values of cost function c(N) are ordered in ascending order and c2 is chosen equal to the p-th value.

1

1 | |

2

ˆ ˆ ˆ1 1| 1 ,

2

ˆ ˆ ˆ1 | 1 |,2

ˆnint

ˆ ˆ ˆnint nint

ˆ ˆ ˆnint nint

n n n

i I i I

n n

n n n n n nN N N

n

n I i I iN N Ni

N N

N N N N N

N N N N N

If more than n+1 candidates are requested, the volume of the search ellipsoid can be used

([RD-6]).

based on the bootstrapping

estimator

1 2c T

N N D N N

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76

Search with shrinking technique: practical example


This is an alternative to the previous strategy, based on shrinking the search ellipsoid during the process of finding the candidates.

In the next example, we have to choose 6 candidates:

Fist candidate is the bootstrapped solution

The other 5 candidates are found by choosing the conditional “Ambiguity-1” to the 2nd, 3rd, 4th and 5th nearest integers

Round “Ambiguity-2” to the second near integer and round the new conditional estimate for “Ambiguity-1”

The candidate with the largest c2 is removed.

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77


Thence, the best 6 candidates are found (in the ISL sense). The one with the smallest cost function c(N) value is the actual ISL solution.

2nd nearest 3rd nearest

4th nearest is out of the

region.

No more candidates at

this step

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The integer ambiguity solution corresponding to the smallest RMS residuals is used to select the candidate.

However if two or more candidates give roughly similar values of RMS, the test can not be resolutive.

A ratio test (of 2 or 3, depending on the algorithm) between the two

smallest RMS is often used to validate the test.

If the ratio is under these values, no integer solution can be determined

and is better to use the floated solution.

Acceptance Test

1ˆ

1

ˆˆ ˆ ˆ

T

RMS

N

NPN N N N P N N

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Examples with MATLAB (octave)

79

Note: This document uses the transposed matrix ZT, but the

principle is the same.

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80

load large Q, a

[Qz,Zt,Lz,Dz,az,iZ] = decorrel (Q,a);

imagesc(Q)colorbar

imagesc(Qz)colorbar

Z

Qz=Lz’*diag(Dz)*Lz

1

ˆ ˆQ N N

P W

Examples with MATLAB (octave)

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[L,D] = ldldecom(Q)

Q=L’*diag(D)*LQz=Lz’*diag(Dz)*Lz

Qz= Z*Q*Z’Q= inv(Z)*Qz* inv(Z’)

az= Z*aa= inv(Z)*az

Z =3 0 -4 -3 -5 -4 -4 2 -2 1 -3 1-0 -1 1 -1 -2 4 4 -3 4 1 0 13 5 -2 -2 1 -1 -2 1 -1 -4 -1 -1-5 -2 3 2 4 3 -3 -2 -2 1 -3 -14 5 1 4 2 6 5 2 -4 1 2 -4-8 -4 1 0 0 -3 2 3 2 -1 -0 44 -7 -0 1 0 -4 -1 -7 3 -5 -1 22 -1 -8 -1 2 -4 1 2 -4 2 2 -2-3 2 3 10 -8 -2 -5 0 -4 1 -4 0-1 6 8 -1 2 1 2 7 3 -2 6 1-8 7 -8 3 -6 -1 1 0 0 3 -1 -18 1 6 -3 5 4 -5 -3 0 -0 1 -3

[Qz,Zt,Lz,Dz,az,iZ] = decorrel (Q,a); Z=Zt’

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load large Q, a

Integer rounding

round(a)

[ - 28491 65753 38830 5004 -29196 -298 -22201 51236 30258 3899 -22749 -159]

Decorrelation + Integer rounding

[Qz,Zt,Lz,Dz,az,iZ] = decorrel (Q,a)azfixed=round(az);afixed=iZ*azfixed

[ -28537 65473 38692 4939 -29228 -504 -22237 51018 30150 3849 -22774 -320]

Decorrelation + bootstrapping

[Qz,Zt,Lz,Dz,az,iZ] = decorrel (Q,a)azfixed=bootstrap(az,Lz);afixed=iZ*azfixed

[ -28451 65749 38814 5025 -29165 -278 -22170 51233 30245 3916 -22725 -144]

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[ -28451 65749 38814 5025 -29165 -278 -22170 51233 30245 3916 -22725 -144]

Decorrelation + ILS with enumeration search


[azfixed,sqnorm] = lsearch (az,Lz,Dz,6);afixed=iZ*azfixed

c(N) -28451 65749 38814 5025 -29165 -278 -22170 51233 30245 3916 -22725 -144 15.0 -28279 65862 38805 5170 -29061 -192 -22036 51321 30238 4029 -22644 -77 31.6 -28727 65935 39032 4844 -29337 -178 -22385 51378 30415 3775 -22859 -66 33.9 -28546 66062 39027 4998 -29228 -83 -22244 51477 30411 3895 -22774 8 34.5-28229 65518 38583 5197 -29056 -500 -21997 51053 30065 4050 -22640 -317 34.7 -28365 65586 38683 5084 -29124 -418 -22103 51106 30143 3962 -22693 -253 35.5

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[ -28451 65749 38814 5025 -29165 -278 -22170 51233 30245 3916 -22725 -144]

Decorrelation + ILS with search-and-shrink


[azfixed,sqnorm] = ssearch (az,Lz,Dz,6);afixed=iZ*azfixed

c(N) -28451 65749 38814 5025 -29165 -278 -22170 51233 30245 3916 -22725 -144 15.0 -28279 65862 38805 5170 -29061 -192 -22036 51321 30238 4029 -22644 -77 31.6 -28727 65935 39032 4844 -29337 -178 -22385 51378 30415 3775 -22859 -66 33.9 -28546 66062 39027 4998 -29228 -83 -22244 51477 30411 3895 -22774 8 34.5-28229 65518 38583 5197 -29056 -500 -21997 51053 30065 4050 -22640 -317 34.7 -28365 65586 38683 5084 -29124 -418 -22103 51106 30143 3962 -22693 -253 35.5

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References

[RD-1] J. Sanz Subirana, J.M. Juan Zornoza, M. Hernández-Pajares, GNSS Data processing. Volume 1: Fundamentals and Algorithms. ESA TM-23/1. ESA Communications, 2013.

[RD-2] J. Sanz Subirana, J.M. Juan Zornoza, M. Hernández-Pajares, GNSS Data processing. Volume 2: Laboratory Exercises. ESA TM-23/2. ESA Communications, 2013.

[RD-3] Pratap Misra, Per Enge. Global Positioning System. Signals, Measurements, and Performance. Ganga-Jamuna Press, 2004.

[RD-4] B. Hofmann-Wellenhof et al. GPS, Theory and Practice. Springer-Verlag. Wien, New York, 1994.

[RD-5] P. Joosten and Tiberius, C. LAMBDA: FAQs. GPS Solutions (2002), 6:109-114.

[RD-6] Sandra Verhagen and Bofeng L., LAMBDA software package. MATLAB implementation, Version 3.0. Mathematical Geodesy and Positioning, Delft University of Technology.

[RD-7] XW. Chang, CC. Paige, L. Yin, 2005, Code and carrier phase based short baseline GPS positioning: computational aspects. GPS Solutions, 2005 (2005) 9:72-83, DOI 10.1007/s10291-004-0112-8 Springer.

85

http://link.springer.com/article/10.1007/s10291-004-0112-8

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86

Thank you

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87

Backup slides

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Least-Squares Ambiguity Search technique

88

This technique requires an approximate solution, which can be obtained from code range measurements. The search area can be defined by surrounding the approximate position by a 3 region (i.e. ).

y G r N ν 1ˆ ˆ

N y G r

Thence, K-4 equations are redundant

K-1 equations with 3+(K-1) unknowns.

Then , given the 3-D position, the (K-1) integerambiguities can be resolved automatically.

That is, the (K-1) integer ambiguities are constrained to three degrees of freedom.

ˆ r

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89

The technique is based on exploiting the constrains on the integer ambiguities:

• The tracked satellites into two groups: 4 sat. (with good DOP) + N-4 sat.

• The primary group of 4 sat., is used determine the possible ambiguity sets (given an initial position estimate and its associated uncertainty )

Then, the corresponding position estimates are computed.

• The remaining N-4 secondary satellites are used to eliminate candidates of the possible ambiguity sets: Each position estimate is checked against measurements from the

secondary group of N-4 satellites. With the correct position estimate, the difference between the

measured and computed carrier phase should be “close to an integer” for each satellite pair.

( ) ( )

1ˆ ?k k

y G r N

( )kN

( )ˆ kr

ˆ r

( ) ( ) ( )

1ˆ ˆ k k k

r N y G r N

1

( ) ( ) ( )ˆ

T

k k kN r G G G y N

( )k ˆ rˆ r

ˆr

master: lecture 7 - upc universitat politècnica de...

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