OCTOPUS: Multi antennae GPS/GLONASS
RTK System
Lev Rapoport, Ivan Barabanov, Alexander Khvalkov, Alexey Kutuzov, Javad Ashjaee,
Javad Positioning Systems
BIOGRAPHY
Lev Rapoport, MSEE’76, PhD’82, DrSc’95. Since 1994
he worked for Ashtech. Since 1998 he has been with
Javad Positioning Systems.
Ivan Barabanov, MS’93, PhD’95. Since 1997 he has been
with Javad Positioning Systems.
Alexander Khvalkov, MSEE’91. Since 1997 he has been
with Javad Positioning Systems.
Alexey Kutuzov, MS’94, PhD’97. Since 1998 he has been
with Javad Positioning Systems.
Javad Ashaee, PhD, pioneered high precision GPS at
Trimble Navigation. He founded Ashtech in 1987 and
founded Javad Positioning Systems where he is involved
in technical development.
ABSTRACT The real time kinematic system for precise
carrier phase differential processing of the measurements
from several dual band dual systems JPS receivers is
considered in the paper.
The system is composed of four JPS Eurocard receivers
exchanging with measurements and results of processing.
The system is based on the general purpose kinematic
engine. It includes carrier phase differential ambiguity
filter, on-the-fly ambiguity resolution, coordinates and
velocity estimators.
The system has two options: (i) free processing or
monitoring, (ii) attitude determination. First option
supposes
several
kinematic
RTK’s
working
simultaneously, while under second option the whole
antennae array is supposed to move as a rigid body. The
second option applies additional geometry constraints due
to the constant between antennae distances.
The paper is organized the following way. The
architecture is discussed at first. Then the methods are
described. The performance results illustrating special
features of the system are included into the text where
description of the features comes. Mathematical details are
placed in the Appendixes.
1. ARCHITECTURE
The system is composed of four JPS Eurocard [1]
receivers. First three of them are rovers with RTK engines
working inside and will be referred to as R1, R2, R3.
Fourth receiver, the master one, will be referred to as M.
All receivers are supposed to be moving. We use a term
‘rover’ only in the context of relative positioning. Every
pair Ri - M works as rover-base pair. The master receiver
sends to Ri the differential corrections. The rover
receivers send back to master the base line coordinates.
When the attitude determination option is switched on the
master receiver calculates the rotation matrix from the
body frame coordinate system to WGS84 or to the local
coordinate system. Also the master receiver accepts
commands from user and translates them to the commands
for rovers.
Rovers communicate only with master and do not
communicate with outside world. All receivers are
connected to the common board which provides data
exchange through RS-232 ports.
The architecture of the system is shown schematically on
the Figure 1 below.
R
1
R
2
R
3
M
BOARD
USER
Figure 1.
2. RTK ENGINE
The RTK engine is composed of two separate parts:
1.
The single base line RTK engine responsible for
estimation of the base line between Ri and M for all
i=1,2,3. It is implemented into rover receivers
software. In the general case all antennae are
supposed to be moving. That is the case for attitude
determination option. The latency between time tag
and the output time includes differential corrections
transmission (M-Ri) delay, processing delay,
backward data transmission (Ri-M) delay, and a nonlinear adjustment delay if attitude (orientation) is
calculated. When an antenna connected to the master
receiver is known to be fixed, the differential
corrections from M are extrapolated in Ri’s from the
master receiver time tag to the rover receiver instant
time to decrease the latency and increase the output
rate.
2. The attitude determination engine (ADE) responsible
for orientation. ADE is implemented into the master
receiver software and works only under attitude
determination option. This engine calculates a
rotation matrix from the body frame system to
specified coordinate system. It processes the base
lines received from Ri’s, tagged with the same time.
The rotation matrix is a solution to certain non-linear
least squares problem. Under free processing
(monitoring) option the receiver M only registers
processing results of three RTK tagged with the same
time and outputs them to user.
The engine operates at 10 Hz rate in the time tagged mode
for single base line; at 20 Hz rate in the in-time mode for
single base line; at 10 Hz rate in the attitude determination
mode.
2.1. NOTATIONS
Let:
- lower index s be for a satellite, lower index
for a receiver, r=R1,R2,R3,M, lower index
-
k
r
be
be for
an epoch;
n k be the number of satellites involved into
-
upper
R
b
d
be the band index, b=1,2 (L1,L2);
be the d-dimensional real-valued Euclidean
vector space,
-
d ×m
R
⋅ be the norm;
be the space of real-valued
d × m
matrices;
-
I d ∈ R d × d be the identity matrix;
-
Zd
-
symbol
ρ
be the integer-valued vector space;
T
r ,s ,k
be for matrix transpose;
be the distance between r-th receiver and s-
th satellite at the k-th epoch;
-
ρ r , k = ( ρ r ,1,k , L , ρ r ,nk , k ) T ;
-
x r , y r , z r are the Cartesian coordinates of the
receiver r;
dt r , M be the between-receiver time scales shift,
r=R1,R2,R3;
-
X r = ( x r , y r , z r , dt r , M ) T ;
-
N
(b )
r,M
∈ R
n
k
be the vector of first difference
float-valued ambiguities (just float for the sake of
brevity) of the pair r-M;
-
Nˆ
(b )
r,M
be the vector of first difference ambiguities
with integer between-components differences;
T
T
-
N r , M = ( N r( 1, M) , N r( ,2M) ) T ;
-
Λ ( b ) = diag
(λ1
(b )
,..., λ n k
(b )
) be a
diagonal matrix of wavelengths;
-
Yr = ( x r ,− x M , y r − y M , z r − z M )T b
e the base line vector, r=R1,R2,R3;
-
J r , k ∈ R n k × 3 be the Jacobian matrix of the
vector-function
ρ
r ,k
with respect to variables
x r , y r , z r , composed of directional cosines; is
supposed to be calculated at the initial point of rough
approximation;
-
c stands for the light speed, C = (c,..., c) T ∈ R nk ;
-
Ar , k = [J r , k
-
µ r , k ∈ R n k be the vector of code measurements;
-
ϕ r ,k ∈ R nk
C ]∈ R n k × 4
be the vector of carrier phase
measurements;
-
processing at the k-th epoch;
-
-
σ µ2 and σ ϕ2 are the variances of the code and phase
m2 ;
be the distance between antennae . r , r ′
measurements noise, both are measured in
-
l r ,r′
Other technical notations will be introduced where that is
necessary.
2.2. SINGLE BASE LINE RTK ENGINE
The engine is fed with dual band (L1/L2) and dual system
(GPS/GLONASS) first difference code and carrier phase
input data flow. The quantities which are usually neglected
when forming double differences, are considered as
variables to be estimated. Special form of double
differences is used only when resolving integer
ambiguities.
Another special feature of the engine is that it uses
constant between-antennae distance when estimating
ambiguities under attitude determination and heading
options.
Further considerations are based on the following
linearized measurement model. Note that we use
linearized equations since second partial derivatives are
negligible and the Newton method to solve the least
squares problem turns to the Gauss-Newton method.
Given rough approximation
~
X
r ,k
Note that all GPS and GLONASS measurements come in
the same systems (1) and (2). The only difference between
to the position of the
receiver r at the k-th epoch, the linearized measurement
equations with respect to the correction ∆ X r , k take
Λ(bk ) .
the form
Ak ,r ∆X k ,r =
Ak ,r ∆X k ,r + Λ(kb ) N r( b ) =
Λ(kb ) (ϕ k(b,r) − ϕ k( b, M) ) − ( ρ k( b,r) − ρ k(b, M) ) + Γk(,br) + ξ k
(b = 1,2), where
When attitude determination option is switched on, all
between – antennae distances are supposed to be constant.
In this case all single base line RTK’s work in the heading
option when the base line length is constant. So, an
essentially non – linear constraint
(1)
µ k(b,r) − µ k(b, M) − ( ρ k(b, r) − ρ k(b, M) ) + γ k( b,r) + ε k
(2)
Yr ,k = l r , M
ε k , ξ k is the measurement noise which
is supposed to be white with the variance
σ
2
µ
,σ
2
ϕ
(3)
is added to the system (1), (2) to be considered as
additional measurement available with no noise at every
epoch.
The main part of the single base line RTK engine is the
ambiguities estimator. Its flow chart is shown on the
Figure 2 below.
.
Tropospheric and ionospheric delays [2] are modeled and
included into corrections
λ(bk ) in the matrix
GPS and GLONASS is in the values of
γ k(b,r) , Γk(b,r) .
ϕk
µk
+
AkT = LQ
Code
outliers
detector
τ k ,τ k′ ,τ k′′
AkT = L Q
Cycle
slips
catcher
ϕk
Float
ambiguities
EKF filter
Nk
N̂ k
Ambiguities
resolution
µk
Σ
Figure 2
The lower index r is omitted for the sake of brevity.
Consider the blocks in details.
2.2.1. CODE OUTLIERS DETECTOR.
It checks the code measurements (1) for consistency with
the hypothesis for the structure of the noise ε k . It solves
GLONASSL1, GPSL1-GLONASSL2 delays ( τ k ,τ k′ ,τ k′′
respectively) are estimated by a first order astatic feedback
loop. Let us rewrite (1) as (lower index
Ak ∆X k = h
(b )
k ,
(b )
where hk
r
is omitted)
b=1,2,
(1')
are right hand sides from (1). Let also the set
the code – differential least squares problem for (1) with
respect to the correction ∆X k , r at every epoch and
of the satellites indexes
crosses out some measurements if that is necessary to
indexes are for GPS and another n k − n k
χ - test. To solve the least squares problem the
T
A = LQ factorization is used. The rank - one
satisfy
2
modification of the LQ – factorization [3] is used to
recalculate the correction after removing every one
measurement.
Along with solution of the code-differential least squares
problem, the interchannel GPSL1-GPSL2, GPSL1-
is divided into two parts
{1,...nk } = {1,...n } ∪ {n + 1,...n k } , where first n kG
G
k
G
k
G
GLONASS. Let e, e′ be the vectors
e = (1, L 1G , G0 , L , 0 ) T ,
1
nk
nk +1
nk
e = (0, L 0G , G1 ,L , 1 ) T
1
nk
nk +1
nk
are for
Let the system (1') be solved by least squared method with
respect to
∆X k and η
(b )
k
be the residuals, (b=1,2). The
following recursive formulae show the operation of the
interchannel delays estimator:
gradient-like descent algorithm. The Figure 3 shows
typical behaviour of the estimator. The vertical axis is for
delays in meters, horizontal axis is for time in seconds.
That is seen from the graph that delays converge to their
stationary values.
2.2.2. CYCLE SLIPS CATCHER.
It solves least squares problem for carrier phase first
difference equations (2) with respect to the correction
∆X k ,r at every epoch and the correction N k −1,r or
Nˆ k −1,r taken from previous epoch. It crosses out the
measurements, which violates χ criterion, the same way
as code outlier detector does. For all crossed out
measurements the float ambiguity filter is reset.
2
2.2.3. FLOAT AMBIGUITIES FILTER.
The float ambiguity filter estimates the float values of the
first difference (Ri-M) ambiguities N k , r on the base of
e T η k( 2)
, τ0 = 0,
nkG
e′T η k(1)
τ k′ = τ k′ −1 + α
, τ 0′ = 0 ,
n k − nkG
e′T η k( 2)
τ k′′ = τ k′′−1 + α
, τ 0′′ = 0 ,
n k − nkG
τ k = τ k −1 + α
(1)
k +1
h
=µ
(1)
k +1, r
−µ
(1)
k +1, M
−
( ρ k(1+)1, r − ρ k(1+)1, M ) + γ k(1+)1,r − τ k′ e′,
hk( 2+1) = µ k( 2+)1, r − µ k( 2+)1, M −
( ρ k( 2+)1, r − ρ k( 2+)1, M ) + γ k( 2+)1,r − τ k e − τ k′′e′,
continuous carrier phase and code data flows passed
through the input check (see Figure 2). Its numerical
scheme is a recursive version of the least squares method
for equations (1), (2), (3) and can be considered as
extended Kalman filter (EKF) [4].
Under assumption that ambiguities N k , r do not vary in
time while coordinates
x k , r , y k ,r , z k ,r , dt k ,r , M
vary independently from epoch to epoch, we need to
estimate N k , r having at hand its previous estimation
N k −1,r , current measurements (1), (2) and constraint (3).
Staying within EKF framework we write the following
least squares problem (lower index
does not lead to misunderstanding):
r
is omitted where it
where α is a small steepest descent parameter. This
method can also be considered as discrete analogue of the
Fk (∆X , N ) =
1
∑ 2σ
b =1, 2
+
1
∑ 2σ
b =1, 2
2
ϕ
2
µ
( Ak ∆X − hk(b ) ) T Wk(b ) ( Ak ∆X − hk(b ) )
( Ak ∆X + Λ(kb ) N ( b ) − g k(b ) ) T Wk(b ) ( Ak ∆X + Λ(kb ) N ( b ) − g k(b ) )
(4)
1
+ ( N − N k −1 ) T S k −1 ( N − N k −1 )
2
1
q
+ ((( ~
x k + ∆x − x k , M ) 2 + ( ~
y k + ∆y − y k , M ) 2 + (~
z k + ∆z − z k , M ) 2 ) 2 − l r , M ) 2 → min
∆x , N
2
The minimizers of (4) are
∆X k ,r and N k ,r . First term is
responsible for measurement equations (1), second one is
responsible for measurements (2), both at the k-th epoch.
Third term in (4) combines new measurements with
previous estimation of the float ambiguity vector N k −1, r .
Fk −1 is represented as second order
Taylor-series expansion at the minimum point ∆X k −1, r
The cost function
and
N k −1,r . Its first partial derivatives are zero at the
minimum point. So, third term in (4) is a cost for deviation
of the current estimation of the ambiguities N k , r from
their previous value
N k −1,r measured in terms of the
quadratic part of the cost function
Fk −1 . S k −1 is the
matrix of the second partial derivatives of the function
Fk −1 , restricted to the variables N and calculated at the
minimum point ∆X k −1, r and N k −1, r :
S k −1
∂ 2 Fk −1 ∂ 2 Fk −1
=
−
∂N∂∆X
∂N 2
In the expression (4) above
−1
 ∂ 2 Fk −1  ∂ 2 Fk −1


2 
 ∂∆X  ∂∆X∂N
Wk(b ) is a diagonal weighting
matrix,
hk(b ) = µ k(b ) − µ k( b, M) − ( ρ k( b ) − ρ k(b, M) ) + γ k( b ) ,
are right hand sides of the equations (1),
g k(b ) = Λ(kb ) (ϕ k(b ) − ϕ k( b, M) ) − ( ρ k(b ) − ρ k(b, M) ) + Γk(b )
are right hand sides of the equations (2).
Last term in (4) is the quadratic penalty function for the
constraint (3), q is the penalty parameter [5].
The expressions for second partial derivatives in the
recursive expression
option when the constraint (3) is supposed to be applied at
every epoch.
(i) In this case the last term in (4) is omitted. The
function Fk is quadratic and minimization is
performed using only one iteration (6) or
n k Quasi-
Newton iterations.
(ii) Under second case the minimization in (4) is
performed numerically by Quasi-Newton iterations.
To compare behaviour of RTK under heading option with
constraint (3) taken into account and without it, we had
performed the following experiment. Given the base line
with constant length l = 55.8 m, we ran RTK for 24
hours under free processing mode and 24 hours under
heading mode with constraint (3). Since a dual band
processing forces ambiguity to be resolved at first several
epoch, we have chosen a single band L1 only and
compared the ambiguity resolution time for two cases said
above. The results are on the figure 4 where the dashed
line represent the integral distribution of the ambiguity
resolution time for the case (i) while the solid line
represent the integral distribution for the case (ii). That it
easily seen that applying the additional constraint (3) in
the case when we are certainly sure of the constant length
makes it possible to achieve fixed solution faster. Note
oncemore that the heading option for single base line RTK
is a part of the attitude option of the whole system. The
attitude determination method is described below. It takes
into account all between-antennae distances.
−1
 ∂ 2 Fk  ∂ 2 Fk


, (5)
2 
 ∂∆X  ∂∆X∂N
are presented in the Appendix 1, S 0 = 0 .
To minimize cost function (4) with respect to ∆X and
N the Newton method can be used. It takes several
∂ 2 Fk
∂ 2 Fk
−
Sk =
∂N 2 ∂N∂∆X
iterations
 ∂ 2 Fk

 ∆X k   ∂∆X 2

 =
2
 N k   ∂ Fk

 ∂N∂∆X
∂ 2 Fk 

∂∆X∂N 
∂ 2 Fk 

∂N 2 
−1
 ∂Fk

 ∂∆X
 ∂Fk

 ∂N


 (6)



to achieve convergence. Typically it takes 2-4 iterations
with non-linear constraint (3). Under free processing with
no constraint (3), it takes only one Newton iteration which
turns to Gauss-Newton iteration for this case. Every
iteration costs
(4 + nk ) 3 arithmetic operations.
We use BFGS Quasi-Newton update formula [6] and lowrank modifications of the Choletsky factors [3] to reduce
the cost of the iteration to
(4 + nk ) 2 , keeping super-
linear convergence rate.
In all calculations the matrix A = LQ factorization is
used.
Consider two cases: (i) free processing when between
antennae distance is not taken into account, (ii) heading
T
Figure 4.
2.2.4. AMBIGUITY RESOLUTION.
To resolve ambiguity we minimize the quadratic form
Qk ( N ) = ( N k − N ) T S k ( N k − N ) →m min , (7)
N ∈D
k
N k is the float estimation of the ambiguities
vector, S k is defined in the expression (5), m k is the
where
m
number of ambiguities. To define the set D k over which
the minimization in (7) is taken, let us suppose that
components in N are ordered the following way:
(1)
 N GPS

(1)
( 2)
 ( 2)  N GPS
∈ R p1 , N GPS
∈ R p2 ,
N 
(1)
(2)
N =  GPS
, N GLN
∈ R p3 , N GLN
∈ R p4 ,
(1) 
 N GLN  p1 + p 2 + p 3 + p 4 = mk .
 N ( 2) 
 GLN 
Let also
p1 p1 +1
mk
p1 p1 +1
p1 + p 2 p1 + p 2 +1
mk
e3 =
(0,L 0 , 1 , L ,
p1 + p 2
1
e 4 = (0, L
1
p1 + p 2 +1
0
,
1
,
0
,L 0 )T ,
p1 + p2 + p3 p1 + p 2 + p3 +1
1
mk
,L, 1 ) T .
p1 + p 2 + p3 p1 + p 2 + p3 +1
mk
Z
mk
Y3 ) ∈ R 3×3 be the base line matrix
in the WGS-84 coordinate system. It is tagged with
the time of the k-th epoch;
-
(
Y0 = Y1
0
Y2
0
0
)
Y3 ∈ R 3×3 be the base line
⊕ {α 1e1 : α 1 ∈ R }⊕ {α 2 e2 : α 2 ∈ R
1
⊕ {α 3 e3 : α 3 ∈ R 1 }⊕ {α 4 e4 : α 4 ∈ R 1 }
−1
Q = YY0 is not good since does not
guarantee that the matrix Q is really orthogonal. That is
The simplest way
due to the noise affecting the base lines estimated by
single base line RTK’s. We use the following approach.
Then
D mk =
Y = (Y1 Y2
antennae do not belong to a plane. The problem is to
evaluate the rotation matrix from the body frame to WGS84. The rotation matrix to another Cartesian system is
calculated the same way.
e 2 = (0, L 0 , 1 ,L , 1 , 0 , L 0 ) T ,
1
-
matrix in the body frame system.
The matrices Y and Y0 both are non – singular since the
e1 = (1, L 1 , 0 , L , 0 ) T ,
1
Let:
1
}
Let N̂ k be the minimizer for (7). It follows from above,
that there are four refference satellites; the ambiguities for
them are float – valued. All between – components
differences of N̂ k are integer among the components of
the same system and the same band. The problem (7) is
easily reduced to the partially integer quadratic
minimization with 4 float and m k − 4 integer variables.
At first we adjust the matrix Y such that it is closest to Y
among the matrices which have the same mutual between
columns scalar products as the matrix Y0 . This property
guarantees prescribed between antennae distances. Then
calculate
Q = Y Y0
−1
(8)
Now the matrix Q is guaranteed to be orthogonal. To
formulate the adjustment problem one needs to choose the
matrix norm to measure the closeness of two matrices. We
use the Frobenius norm:
F
2
F
= tr ( F T F ) = tr ( FF T ) ,
Another special features we use when resolve ambiguities:
(i) Partial resolution. Every epoch the set of all
ambiguities is partitioned into three subsets: the set
of ambiguities already resolved, the set of
ambiguities to be resolved, the set of ambiguities to
be float-valued.
(ii) Two criteria: absolute and relative contrasts to check
if the ambiguities are resolved correctly.
(iii) Using of LLL algorithm [7] as sequential low-rank
modifications of the Choletsky factorization of the
matrix S k .
where tr (⋅) denotes the trace of the matrix.
The adjustment problem is formulated as following:
(iv) Using of the constant distance constraint (3) when
searching for the integer minimum.
matrix Y has the same lengths of the columns and the
same scalar products between columns as the matrix Y0
2.3. ATTITUDE DETERMINATION ENGINE
Under attitude determination mode the master receiver
sends differential corrections to rovers R1,R2,R3. The
rovers work under heading option with constant base line
length (constraint (3)). The master receivers gets back
from rovers the base line coordinates Yk , R1 , Yk , R 2 , Yk , R 3 .
has. The solution to the problem (9), (10) is obtained the
following way:
Further we use the notation
Y1 , Y2 , Y3 for brevity. All
base line vectors are of the correct length, but the angles
between them are not. To take into account the angles, or
all six between antennae distances, the attitude
determination engine performs certain non – linear
adjustment.
Y −Y
2
→ min
F Y ∈R 3× 3
(9)
under constraint
Y T Y = Y0T Y0
(10)
So, we are looking for the matrix Y which is closest to
Y in the Frobenius norm among those which have the
same Grammian as the matrix Y0 has. In other words, the
Y = YM
(11)
where M is the symmetric and positive definite solution
of the following quadratic matrix equation:
MY T YM = Y0Y0T
The proof is in the Appendix 2
(12)
tr ((Y − Y ) T (Y − Y )) =
3. APPENDIX 1
Let:
tr (Y T Y ) − 2 trY T Y + tr (Y T Y ) =
Y + ∆Y = (( ~
x + ∆x − x M ) 2
1
+ (~
y + ∆y − y M ) 2 + ( ~
z + ∆z − z M ) 2 ) 2
tr (Y0 Y0 ) − 2 trY T Y + tr (Y T Y )
T
1
p( X ) = 12 ( Y + ∆Y − l r , M ) 2 ,
Y − lr ,M
α=
Y
And so, the problem (9), (10) is equivalent to the problem
tr (Y T Y ) → max
l r ,M
, β =
under the condition (10).
Applying the Lagrangian multiplier method to the
optimization problem (9’), (10) we obtain
Y
Then
∂P
∂∆Y
∆Y = 0
∂2P
∂∆Y 2
 x − xM

= αY = α  y − y M
z−z
M

= αI + β
∆Y = 0
1
2
Y


,


L(Y ) = − tr (Y T Y )
+
 S
S k −1 = 
S
S
(1, 2 )
k −1
(2)
k −1
S

,


∂ 2 Fk
∂N∂∆X
=
∆X = 0
1
σ ϕ2
∂ Fk
=
∂N 2
(1)
(1) (1)
(2)
1  Λ k Wk Λ k + S k −1
T
σ ϕ2 
σ ϕ2 S k(1−,12)
Y
the matrix of the Lagrangian multipliers. That is simple
algebra to obtain that the stationary point condition for
(13) takes the form:
− Y + YΠ = 0
Substitution to (10) gives
Π −1Y T YΠ −1 = Y0Y0T
−1
Denoting Π by
two conditions.
M we obtain (11) and (12) from last
Note now that there are 2 = 8 solutions of the quadratic
matrix equation (12). All eight provides necessary
conditions for the optimum in (9), (10) due to the
stationary conditions of the Lagrangian. Only one of them
is positive definite. And only one of them provides true
optimum. To find it we need to try all eight solutions and
choose one which provides the least value of the criterion
(9). But it is clear, that under moderate noise, the matrix
M in (11), (12) will be close to identity matrix and
solution to the problem (9), (10) will be positive definite.
The Figure 5 below shows the results of processing which
illustrate this conclusion. We installed the antennae to the
ground at the known points. The between antennae
distances were 5 - 15 m. The system worked at 1Hz rate.
The solid line on the figure presents a deviation of the
rotation matrix (constant in our case) from its true value.
The dashed presents a deviation of the matrix M from
3
= wAkT WAk + q
∂2P
∂∆Y 2
1
Y
2
(13)
Y = YΠ −1 .
Then
= wAkT WAk + q(αI + β
((Yi , Y j ) − (Yi 0 , Y j0 )) → min
or
 N (1) 
N =  ( 2 )  .
N 
∆X = 0
ij
λij = λ ji are Lagrangian multipliers for the
condition (10), (⋅,⋅) denotes the inner product. Let Π be
where the partition of the matrix into the blocks
corresponds to the partition of the vector
∂ 2 Fk
∂∆X 2
∑λ
where
YY T ,
1
1
+ 2 ), W = W (1) + W ( 2 ) ,
2
σ µ σϕ
(1)
k −1
(1, 2 ) T
k −1
3
i , j =1
Let also
w=(
(9’)
Y
YY T ),
 Wk(1) Λ(k1) Ak 
 (2) (2)  ,
W Λ A 
k
k 
 k
2


(2) 
+ S k −1 
σ ϕ2 S k(1−,12 )
Λ(k2)Wk( 2 ) Λ(k2 )
So, the recursive equation (5) can be rewritten in terms of
notations above.
4. APPENDIX 2
First of all, taking into account condition (10) we rewrite
(9) as
identity matrix:
M −I .
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