> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO
EDIT) < - What IDENTIFICATION NUMBER ??
1
EXCESS AMBIGUITY RESOLUTION ALGORITHMS ANALYSIS
IN INTERFEROMETRIC MEASURINGS OF SATELLITE RADIO
NAVIGATION SYSTEMS SIGNALS
K. Y. Kostyrev, A. M. Aleshechkin
Abstract— This paper describes the methods and
principles of objects angular data determination by
interferometric measuring of Satellite Radio
Navigation Systems signals. It considers the
ambiguity resolution problem solving by the excess
method. Use of frequency and phase shifts
differences is shown to decrease time expenses and
algorithm complexity for orientation definition. As a
result of the study we suggest some implementation
algorithms and describe their particular features.
Index
Terms—Satellites,
radio
interferometric measuring, ambiguity
navigation,
I. INTRODUCTION
Object angular data determination by interferometric
method with the signals of Satellite Radio Navigation
Systems (SRNS) involves the problem of ambiguity
resolution. It appears in the measured values of received
signals angular phase difference.
This problem can be solved by applying the excess
method [1], which uses integral property of phase
ambiguities. The main advantage allows us to reduce the
time required to determine the angular orientation of the
object Moreover algorithm may use satellite frequencies
both separately and in pairs (the difference between
these frequencies). This approach can increase speed and
reduce hardware costs for information processing.
II. EXCESS METHOD STEPS
Object angular data determination by SRNS with
excess method for ambiguity resolution by using
frequency and phase differences consists of the
following:
1. receiving signals from n SRNS satellites with
two or more spaced antennas. The antennas are located
along one or two axes of the measured object,
2. measuring the phase shift between received
signals from each satellite;
3. calculating the difference of phase shifts in
pairs for all frequencies of each satellite;
4. selecting differential integer ambiguities
values in measuring the phase shift for a minimum
constellation of 2 or 3 satellites. The step determines all
potential values of the angular orientation;
5. possible angular orientation values exclusion
that does not correspond with a priori data of the antenna
system orientation and the distance between spaced
antennas;
6. checking the remaining angular orientation
values with the differential ambiguities ΔNi calculation
for the measured phase shifts. Calculations are hold only
for satellites that are not included in the minimum
constellation;
7. angular orientation determination with the
differential phase shifts of all received satellite signals.
The step also includes the differential ambiguities
oversearching procedure. At the end the value
corresponding to the desired object angular orientation is
determined as the maximum of likelihood function.
III. GENERAL EQUATIONS
A. Main System
The values of the angular orientation for the
minimum and the full constellation of satellites are
determined by solving the system using (for example)
frequencies L1 and L2:

   k xi  X    k yi  Y    k zi  Z  12i  N12i ;
 2
2
2
2

X Y  Z  B ,
(1)
here i  1,..., n - the current number of satellites; n total number of satellites using for angular data
determination; k xi , k yi , k zi - directions-vectors cosines
from the object to i satellite in the current measuring
time measurement; 12i  1i  2i - measured and
adjusted for systematic error value of the difference
between signals phase shifts at L1 and L2 for i satellite
(in cycles); N12i  N1i  N 2i - value of the difference
between signals integer ambiguities at L1 and L2 for i
satellite (in cycles), satisfying the condition:
1
1
N12i   B 12i  0,5 ;  

, 2i , 1i - signals
2i
1i
wavelengths at L1 and L2 for i satellite; B - distance
between the antennas, when n  3 - known with high
accuracy, when n  3 - to be specified during equation
> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO
EDIT) < - What IDENTIFICATION NUMBER ??
system solving; X , Y , Z - unknown values of the second
antenna phase center relative coordinates to the first one
B. Base-length and elevation criteria
In common situations some values of parameters we
are interested in can’t be obtained. In that situation we
should remove such values with the use of specific
criteria. In fact some of the objects can be oriented not
accidentally, but have definite features that depend on
the object type. It allows us to short the number of
excesses and eliminate certainly unsuitable values. For
example, the base-length is known to be:
B  Ba priori  Bposs ,
(2)
value, Bposs - possible error between them.
Starting data input
Minimum satellite constellation determination
Possible angular values searching
Removal of redundant solutions with base-length
and elevation criteria
Extra satellites ambiguity calculating for
remaining possible angular values
Solving equation system for all satellites
Printing the solution with complies with likehood
function maximim
The same principle can be used for elevation or
azimuth values.
FINISH
C. Oversearching
The oversearching procedure takes place during
differential ambiguities calculation process for all
satellites. Here the number of excesses is increased by 2:
(3)
*
here N12i - calculated value, N12i
- alternatives.
Necessity of the oversearcing procedure is caused by
the presence of phase measurement error, which value
increases in the use of phase difference.
D. System Solving
We applied least-squares method for main system (1)
solving. In that way we obtain:
n
2
Q      kxi  x    k yi  y    kzi  z  Ô12i   min,
i1
START
Likehood function computing
here B - calculated value, Ba priori - a priori known
N12* i  N12i  1, N12i , N12i  1 ,
2
Fig.1. Block diagram of main algorithm points.
IV. RESULTS
During this study the excess method algorithms were
formed. We used such software as C + + Builder 2009,
Mathcad 14 and Orbitron 3.71.
Established program calculates satellites phase shifts,
determines the value of ambiguity and differences for all
satellites at minimum satellite constellation, computes
optimal object angular orientation values and probability
of correct ambiguity resolution phase measurements,
provides errors calculation in determining the azimuth,
elevation and the base-length.
(4)
here Ô12i  N12i  12i - differential full phase.
E. Block Diagram
The block diagram presents all main calculations
which are carried out at the main algorithm. The same
points may be used for separate frequencies and for their
pairs.
Fig.2. Functional dependence between root-mean-square
deviation value of azimuth and phase measuring error
with differential frequency L1-L2 when base-length is
variable
> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO
EDIT) < - What IDENTIFICATION NUMBER ??
REFERENCES
[1] Pat. 2379700 RU, C 1. Method of object angular
orientation by satellite radio navigation system signals./
A. M. Aleshechkin (RU),V. I. Kokorin (RU),
J. L. Fateev (RU). - 2008131246/09, 28.07.2008. Date
of publication: 20.01.2010 Bull. 2. – 17 p.
Fig.3. Functional dependence between correct ambiguity
resolution probability phase measuring error with
differential frequencies L1-L2, L1-L3, L2-L3. Baselength equals 0.5 m.
As a result, we obtained functional dependences
between correct ambiguity determination probability and
phase measuring error for GLONASS and GPS
frequencies L1, L2, L3 and their differences; between
determination errors of azimuth, elevation, base-length
and phase measuring error for GLONASS and GPS
frequencies L1, L2, L3 and their differences. Functional
dependences were found between correct ambiguity
determination probability, azimuth, elevation, baselength determination errors and the distance between the
antennas.
Also differential frequency algorithm was
implemented. As you can see at Fig.3 correct ambiguity
resolution probability values fall gradually, but such
meanings reveal themselves insufficiently. Differential
frequency of L2 and L3 demonstrates terrible results,
because the value of its wavelength is much larger than
used base-length.
V. CONCLUSION
Modeling results analysis showed that the developed
algorithm can be used to determine the angular
orientation of moving objects. Using frequencies and
phases differences reduces the time required to
determine the angular orientation. The oversearching
procedure increases correct ambiguity determination
probability.
We hope to achieve better results (high accuracy and
suitable correct ambiguity determination probability
value) in future by improving the algorithms, combining
separate satellite frequencies and differential satellite
frequencies. Moreover we are going to add multi-scale
measuring principles for our purposes.
3