Improving the GNSS Attitude Ambiguity
Success Rate with the Multivariate
Constrained LAMBDA Method
118
G. Giorgi, P.J.G. Teunissen, S. Verhagen, and P.J. Buist
Abstract
GNSS Attitude Determination is a valuable technique for the estimation of
platform orientation. To achieve high accuracies on the angular estimations, the
GNSS carrier phase data has to be used. These data are known to be affected by
integer ambiguities, whi ch must be correctly resolved in order to exploit the
higher precision of the phase observables with respect to the GNSS code data.
For a set of GNSS antennae rigidly mounted on a platform, a number of nonlinear
geometrical constraints can be exploited for the purpose of strengthening the
underlying observation model and subsequently improv ing the capacity of fixing
the correct set of integer ambiguities. A multivariate constrained version of the
LAMBDA method is presented and tested here.
118.1 Introduction
Attitude determination is an important issue in remote
sensing applications: the knowledge of the orientation
of the platform which carries the sensors (radars,
lasers, etc.) is required for the pointing procedures.
Although the accuracy of a stand-alone GNSS attitude
system might not be comparable with the one obtain-
able with other modern attitude sensors, a GNSS-
based system presents several advantages. The main
assets are that it is driftless and it requires less mainte-
nance. Many works investigated the feasibility and
performance of GNSS Attitude Determination, see
e.g. [1–6]. The key for a precise attitude estimation
is the ambiguit y resolution process, since only when
the integers inherent to the GNSS carrier phase
observations are correctly fixed one is able to
exploit the carrier phase data, which are of two orders
of magnitude more accurate than the GNSS code
observations. In this contribution we focus on the
problem of fixing the correct integer ambiguities for
data collected on a frame of antennae firmly mounted
on a rigid platform: the relative positions between the
antennae are assumed to be known and constant. In
such configurations, the baselines lengths and the
angles between them are known, resulting in a set of
nonlinear constraints posed on the baseline vectors
which can be exploited to strengthen the underlying
observation model. The set of GNSS phase and code
observations is cast into a linearized system solvable
G. Giorgi
S. Verhagen (*)
P.J. Buist
Delft Institute of Earth Observation and Space Systems (DEOS),
Delft University of Technology, PO Box 5058, 2600 Delft,
The Netherlands
e-mail: a.a.verhagen@tudelft.nl
P.J.G. Teunissen
Delft Institute of Earth Observation and Space Systems (DEOS),
Delft University of Technology, PO Box 5058, 2600 Delft,
The Netherlands
Department of Spatial Sciences, Curtin University of
Technology, GPO Box U1987, Perth, WA 6845, Australia
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136,
DOI 10.1007/978-3-642-20338-1_118,
#
Springer-Verlag Berlin Heidelberg 2012
941
in a least-squares sense, where both the integerness of
the ambiguities and the constraints on the baselines
must be fulfilled. The method is based on an extension
of the Integer Least Squares (ILS, [7]) principle,
employed to solve in a least-squares sense a linear
system of equations where some of the unknowns are
integer-valued. A well-known mechanized implemen-
tation of the ILS principle is the Least squares AMBi-
guity Decorrelation Adjustment (LAMBDA) method
[8], widely used for its high computational efficiency.
This method has been recently extended to accommo-
date those baseline applications where the distance
between the antennae is known and constant: the base-
line Constrained LAMBDA method was introduced in
[9, 10]. The inclusion of the baseline length constraint
results in a large improvement in the success rate, as
shown in [11–15].
We here present and test the performance of
the multivariate generalization of the Constrained
LAMBDA method [16], which solves the model
where the full set of nonlinear geometrical constraints
is taken into account, i.e. the different baseline lengths
and their relative positions.
118.2 Modeling of the Multi-antennae
GNSS Observations
We consider a set of m þ 1 antennae (m independent
baselines) simultaneously tracking the same n þ 1
GNSS satellites on a single frequency. The set of
linearized Double Difference (DD) GNSS phase and
code observations obtained on the m baselines can be
cast into a Gauss–Markov model as follows:
EðYÞ¼AZ þ GB Z 2 Z
nm
; B 2 R
3m
DðvecðYÞÞ ¼ Q
Y
(118.1)
where
E(·) is the expectation operator
Y is the 2n by m matrix whose columns are the
code and phase DD observations derived at each
baseline:
Y ¼ y
1
y
2
... y
m
½
Z is the n by m matrix whose columns are the integer-
valued ambiguities for each baseline:
Z ¼ z
1
z
2
... z
m
½
B is the 3 by m matrix whose columns are the real-
valued baseline coordinates:
B ¼ b
1
b
2
... b
m
½
A is the 2n by n design matrix which contains the
carrier wavelength
G is the 2n by 3 design matrix of line-o f-sight vectors
D(·) is the dispersion operator
Q
Y
is the 2nm by 2nm variance–covariance matrix of
the vector of observations vec (Y)
We make use of the vec operator, which stacks
the columns of a matrix below each other, to define
the variance–covariance matrix of the vector of
observations vec (Y). We assume that the antennae
are separated by short baselines, for which the
atmospheric effects can be neglected, and the only
real-valued unknowns to be estimated are the 3m
coordinates of the baseline vectors. Also, the short
baseline hypothesis allows us to make use of a unique
matrix of line-of-sight vectors G for al l the baselines.
A Gaussian-distributed error is assumed on the
observables Y.
Aiming to estimate a platform’s orientation solely
via GNSS measurements, two or more antennae are
assumed to be firmly mounted on the platform, which
is here considered as a rigid body. We introduce
a system of body axes (local) u
1
u
2
u
3
taken as to have
the first axis u
1
aligned with the first baseline, the
second axis u
2
perpendicular to u
1
and lying in the
plane formed by the first two baselines, and the third
axis u
3
oriented that u
1
u
2
u
3
form an orthogona l frame
(see Fig. 118.1, where f
i
is the ith baseline, and f
ij
indicates the jth coordinate of the baseline i). The
baseline coordinates expressed in the local frame are
collected in matrix F, and the relations hip between
these and the coordinates B expressed in the global
frame x
1
x
2
x
3
is:
B ¼ R F (118.2)
where R is the orthogonal matrix which describes
the relative orientation between the local and global
frames, i.e. the attitude of the platform. For notational
convenience, the rotation matrix and the local baseline
coordinates F are defined as [16]
942 G. Giorgi et al.
m 3
q ¼ 3
(
: RF ¼ r
1
; r
2
; r
3
½
f
11
f
21
f
31
f
m1
0 f
22
f
32
f
m2
00f
33
f
m3
2
6
4
3
7
5
m ¼ 2
q ¼ 2
(
: RF ¼ r
1
; r
2
½
f
11
f
21
0 f
22
m ¼ 1
q ¼ 1
(
: RF ¼ r
1
½f
11
½
(118.3)
where q is a parameter introduced to cope with the
case of m < 3 baselines. The relation (118.2)isa
linear transformation, which changes the unknowns
of the problem: the estimation of the baseline
coordinates B turns into the estimation of the
components of the matrix R, of which only three are
independent. Hence, in addition to the integer con-
straint on the matrix Z, also the orthogonality of the
matrix R has to be considered. This allows to rewrite
the set of baseline observations (118.1)as
EðYÞ¼AZ þ GRF Z 2 Z
nm
; R 2 O
3q
DðvecðYÞÞ ¼ Q
Y
(118.4)
This is the model that we aim to solve in a least-
squares sense. The standard LAMBDA method can be
employed to solve the system when the constraint on
the matrix R is disregarded, being the system solely
subject to the integer constraint on Z. Via a modifica-
tion of the LAMBDA method, the model (11 8.4)is
solvable in a rigorous least-squares sense taking into
account all the different constraints, which are the
integer nature of the entries of Z and the orthogonality
of the rotation matrix R. This is demonstrated in the
following section.
118.3 Integer Least Squares
We aim to solve for the model (118.4) in a rigorous
least-squares sense, minimizing the weighted squared
norm of the residuals. The solution of the model
(118.4) is derived with a three-steps procedure: first
obtain a float solution, then search for the integer
ambiguities and finally extract the orthogonal matrix
R. In this section we describe each of these steps.
118.3.1 The Float Solution
The float solution of (118.4) is the least-squares solu-
tion obtained disregarding both the integerness of the
matrix Z and the orthogonality of R:
N
vecð
^
ZÞ
vecð
^
RÞ
!
¼
I
m
A
T
F G
T
Q
1
Y
vecðYÞ
N ¼
I
m
A
T
F G
T
Q
1
Y
I
m
AF
T
G
(118.5)
where is the Kronecker product and we made use of
the property
vecðX
1
X
2
X
3
Þ¼ X
T
3
X
1
vecðX
2
Þ
vecð
^
ZÞ and vecð
^
RÞ are the float estimators of Z and R;
their v–c matrices are obtained via the inversion of the
normal matrix N:
Q
^
Z
Q
^
Z
^
R
Q
^
R
^
Z
Q
^
R
¼ N
1
(118.6)
If one assumes that the matrix Z is known, the
conditional solution of R (conditioned on the know-
ledge of the matrix Z) is obtained as
vecð
^
RðZÞÞ ¼ vecð
^
RÞQ
^
R
^
Z
Q
1
^
Z
vecð
^
Z ZÞ (118.7)
Fig. 118.1 The first baseline f
1
(Main antenna – Aux
1
antenna)
defines the first body axis u
1
, while the second body axis u
2
,
perpendicular to u
1
, lies in the plane formed by f
1
and f
2
(Main
antenna – Aux
2
antenna). u
3
is taken as to form an orthogonal
frame
118 Improving the GNSS Attitude Ambiguity Success Rate 943
The prec ision of the conditional solution
^
RðZÞ is
described by the v–c matrix
Q
^
RðZÞ
¼ Q
^
R
Q
^
R
^
Z
Q
1
^
Z
Q
^
Z
^
R
(118.8)
118.3.2 The Search for the Integer
Minimizer
Given the float estimator
^
Z, the conditional solution
^
RðZÞ and their v–c matrices, we can write the sum-of-
squares decomposition of the weighted squared norm
of the residuals of (118.4)as
vecðYÞðI
m
AÞvecðZÞðF
T
GÞvecðRÞ
2
Q
Y
¼
vecð
^
EÞ
2
Q
Y
þ
vec Z
^
Z
2
Q
^
Z
þ
vec
^
RðZÞR
2
Q
^
RðZÞ
(118.9)
where
^
E is the matrix of least-squares residuals. The
decomposition shows that the last term can always be
made zero for any value assumed by Z, by taking
R ¼
^
RðZÞ, if the orthogonality of R is disregarded.
The minimization of the least-squares residuals then
reduces to the well known case of finding the integer
minimizer of the second term, and the standard
LAMBDA method can be directly applied. When the
orthogonality constraint on the matrix R is taken, the
last term generally differs from zero, since
^
RðZÞ is
usually non orthogonal: this leads to a modification
of the search algorithm to be adopted, resulting in
a multivariate constrained version of the LAM BDA
method.
118.3.2.1 The LAMBDA Method
Disregarding the orthogonality of R, the integer-
valued minimizer of (118.9) equals
Z
U
¼ arg min
Z2Z
nm
vecðZ
^
ZÞ
2
Q
^
Z
(118.10)
The matrix
Z
U
has the minimum distance from the
float solution
^
Z in the metric defined by Q
^
Z
: since
no closed-form solution of (118.10) is known, the
estimation of the matrix
Z
U
involves a direct search
inside a set of suitable integer candidates:
O
U
w
2
¼ Z 2 Z
nm
j
vecðZ
^
ZÞ
2
Q
^
Z
w
2
no
(118.11)
The set O
U
, which geometrically draws an hyper-
ellipsoid centered in vecð
^
ZÞ and size/shape driven by
the entries of Q
^
Z
, is evaluated and the integer matrix Z
that minimizes the squared norm (118.10) is extracted.
The LAMBDA method is applied to perform the
search in an efficient and fast way; it works by
decorrelating the ambiguities performing an admissi-
ble (i.e. which preserves the integerness of the vari-
ables) transformation: the effect of the decorrelation
is to have a reduced set of integer candidates, among
which the matrix
Z
U
is quickly extracted.
118.3.2.2 The Multivariate Constrained
LAMBDA Method
The full-constrained least-squares minimization of
(118.9) is obtained by taking the minimization with
respect both the matrix of ambiguities Z and the
orthogonal matrix R:
Z
C
¼ arg min
Z2Z
nm
CðZÞ
CðZÞ¼
vecðZ
^
ZÞ
2
Q
^
Z
þ
vecð
^
RðZÞ
RðZÞÞ
2
Q
^
RðZÞ
(118.12)
with
vecð
RðZÞÞ ¼ arg min
R2O
3q
vecð
^
RðZÞRÞ
2
Q
^
RðZÞ
(118.13)
where O
3q
is the class of 3 q orthogonal matrices,
i.e. R
T
R ¼ I
q
. The integer minimizer
Z
C
weighs the
sum of tw o terms: the first is the distance with respect
to the float solution
^
Z weight ed by Q
^
Z
, and the second
is the distance between
^
RðZÞ and the solution of the
nonlinear constra ined least-squares problem (118.13).
The latter gives the orientation of the platform
RðZÞ by
minimizing in a least-squares sense the distance from
the matrix
^
RðZÞ, subject to the orthogonal constraint.
Note that for the single-baseline case (m ¼ 1) the
944 G. Giorgi et al.
problem reduces to the one addressed in [9, 10, 17]:
the method discussed here is a multivariate gene-
ralization. The minimizer of (118.12) is searched in
the set defined as
O w
2
¼ Z 2 Z
nm
jCðZÞw
2
(118.14)
Minimizing the cost function (118.12) in the set
O(w
2
) is a non-trivial task: the evaluation of C(Z)
involves the computation of a nonlinear constrained
least-squares problem, and if the set contains a
large number of candidates the search is very time-
consuming. Hence, the choice for the scalar w is
an important issue, since it strongly affects the time
dedicated to the minimization process.
To make the search more time-efficient, and to
cope with both the problems of setting the value of w
and computing (118.13) a large number of times, the
two algorithms coined as the Expansion and the Search
and Shrink approaches were developed [12–15]: by
using two functions that provide a lower and an
upper bound for the cost function C(Z) and that are
easier to evaluate (i.e. the computation of (118.13)is
avoided), the search for the integer minimizer
Z
C
is
computed efficiently and in a much faster way.
118.3.3 The Attitude Solution
The two above mentioned search methods provide the
ILS minimizer
Z of the expression (118.9), respectively
with (Constrained LAMBDA,
Z ¼
Z
C
) or without
(LAMBDA,
Z ¼
Z
U
) considering the orthogonality of
R.Notethatingeneral
Z
U
6¼
Z
C
. Given the integer
minimizer resolved, the conditional attitude solution
is obtained as in (118.7): the solution
^
RðZÞ is char-
acterized by a better accuracy (118.8), but it is in
general non-orthogonal. In order to obtain the sought
orthogonal attitude matrix, the following nonlinear
constrained least-squares problem has to be solved:
vecð
Rð
ZÞÞ ¼ arg min
R2O
3q
vecð
^
Rð
ZÞRÞ
2
Q
^
RðZÞ
(118.15)
where
Z ¼
Z
U
for the LAMBDA method and
Z ¼
Z
C
for the Constrained LAMBDA method. The nonline-
arity of the problem comes from the orthogonality of
R: via a suitable parameterization of the rotation
matrix, e.g. Euler angles or Quaternions [18], the
orthogonality is implicitly fulfilled, and the least-
squares solution of (118.15) can be solved for example
with the Newton method.
118.4 Testing the Method
The method presented was tested with simulated data
and with data collected during a kinematic experi-
ment. All the data sets were processed with both the
LAMBDA method and the Constrained LAMBDA
method, in order to compare the different performance
obtained in terms of single-epoch, single-frequency
success rate.
118.4.1 Simulation Results
Different sets of data were generated via a Monte Carlo
simulation, reproducing the set of baseline observations
according to the model (118.4). Table 118.1 sum-
marizes the set-up of the simulations: from the actual
GPS constella tion on 22 January 2008 (as seen from
Delft, The Netherlands), we selected five to eight
satellites, with corresponding PDOP values between
4.2 and 1.8. Two baselines were simulated, of 1 and
2 m length, forming an angle of 100
. For each of
the 24 scenarios, 10
5
samples were generated, aiming
to extract an accurate estimation of the success
rate, defin ed as the ratio of samples where the correct
integer ambiguity matrix has been fixed and the
total number of samples. The data sets were proces-
sed applying the LAMBDA and the Constrained
LAMBDA methods as described in Sect. 118.3.
Table 118.2 shows the single-frequency, single-epoch
Table 118.1 Simulation set up
Frequency L1
Number of satellite (PRNs)
5/6/7/8
Corresponding PDOP 4.19/2.14/
1.92/1.81
Undifferenced code noise
s
p
(cm)
30-15-5
Undifferenced phase noise
s
f
(mm)
3-1
Baselines f
i
(x
1
, x
2
, x
3
)
~
f
1
¼½1; 0; 0m
~
f
2
¼½0:35; 1:97; 0m
Samples simulated 10
5
118 Improving the GNSS Attitude Ambiguity Success Rate 945
