# Comparing Multicarrier Ambiguity Resolution  Methods for Geometry-Based GPS and Galileo Relative Positioning and Their Application to Low Earth  Orbiting Satellite Attitude Determination

Authors: Kyle O'Keefe; Mark Petovello; Wei Cao; Gérard Lachapelle; Eric Guyader
Journal: International Journal of Navigation and Observation
 (2009)

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## Abstract

This paper presents an evaluation of several GNSS multicarrier ambiguity (MCAR) resolution techniques for the purpose of attitude determination of low earth orbiting satellites (LEOs). It is based on the outcomes of the study performed by the University of Calgary and financed by the European 6th Framework Programme for Research and Development as part of the research project PROGENY. The existing MCAR literature is reviewed and eight possible variations of the general MCAR processing scheme are identified based on two possible options for the mathematical model of the float solution, two options for the estimation technique used for the float solution, and finally two possible options for the ambiguity resolution process. The two most promising methods, geometry-based filtered cascading and geometry-based filtered LAMBDA, are analysed in detail for two simulated users modelled after polar orbiting LEOs through an extensive covariance simulation. Both the proposed Galileo constellation and Galileo used in conjunction with the GPS constellation are tested and results are presented in terms of probabilities of correct ambiguity resolution and float and fixed solution baseline accuracies. The LAMBDA algorithm is shown to outperform the cascading method, particularly in the single-frequency dual-GNSS system case. Secondly, more frequencies and multiple GNSS always offer improvement, but the single-frequency dual-system case is found to have similar performance to the dual-frequency single-system case.


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## Body

## 1. Introduction

PROGENY (PROvision of Galileo Expertise, Networking and support for International Initiatives) is a research and technological development project launched by the European GNSS
Supervisory Authority (GSA), in the frame of the 6th Framework Programme. PROGENY
consists of a series of activities supporting the innovation and international
initiatives in relation to the Galileo programme. In particular, the project
has established a platform for scientific and technological cooperation with
different regions worldwide, and has run a set of targeted studies in
cooperation with international partners.This paper presents the
results of the study performed by the University of Calgary Department of
Geomatics Engineering related to the definition of a method for LEO satellite
attitude determination, using Multiple Carrier Ambiguity Resolution (MCAR).GNSS-based attitude determination
is accomplished by kinematic carrier-phase GNSS techniques. Namely, a number of
short baselines are established on the vehicle with known coordinates in the
vehicle body frame. Carrier-phase GNSS is used to determine local level frame
(east, north, vertical) components of the same baselines and the knowledge of
the baselines in both frames is then used to establish the rotation angles between
the two frames.In order for attitude to be
determined precisely, the GNSS baseline components must be estimated using
fixed carrier-phase ambiguities. There are presently two main approaches to
dual-frequency kinematic double-difference ambiguity resolution, and these two
methods can be generalized to the case where modernized GPS and Galileo provide
additional observations on additional frequencies.The first approach is to
estimate a position based on pseudorange measurements and also form the wide lane
observable by differencing the L1 and L2 phase measurements. Either the
pseudorange or the pseudorange-derived position can be used to provide an
initial estimate of the widelane ambiguity. The widelane can generally be
resolved quickly over short baselines typically associated with attitude
determination (i.e., order of a few metres). Once resolved, the next step is
either to use the fixed widelane phase range, or alternately the fixed widelane
position estimate as a starting point to estimate a float solution for the L1
ambiguity. Because a fixed widelane phase range is more precise than a
pseudorange measurement, it becomes possible to estimate the L1 float ambiguity
with sufficient confidence to allow it to be resolved quickly. The sequence of
steps between pseudorange, widelane, and L1 has led to this method being called
“cascading.” Triple-frequency variations of this algorithm have been proposed
for both modernized GPS and Galileo where an additional step is added to make
use of the third frequency to form an even longer wavelength widelane
observable before cascading down to the widelane observable. These methods are
generally referred to as either three carrier ambiguity resolution (TCAR), multiple carrier ambiguity resolution (MCAR), or simply cascading integer
resolution (CIR) methods [1].The second approach to
multiple frequency kinematic ambiguity resolution involves using Teunissen’s
Least squares AMBiguity Decorrelation Adjustment (LAMBDA) method to determine
an optimal linear combination of a set of float ambiguities for the purposes of
ambiguity resolution [2, 3]. In
this method, any set of float ambiguities may be estimated and then a linear
combination is found that minimizes the correlation between the set of
ambiguity states.In both these methods,
either geometry-free or geometry-based processing may be used. In geometry-free
processing, double-difference observations from each satellite are treated
independently until ambiguities are resolved at which point the fixed phase
range is used to compute a position solution. In Geometry-based processing, the
float ambiguities and the baseline vector are estimated together and the
baseline vector is then improved once the ambiguities are resolved [4, 5].In the geometry-based case,
the inclusion of observations from more satellites, or satellites from more
than one system (e.g., GPS and Galileo) has been previously shown to
result in improved ambiguity resolution performance. Likewise, the addition of
a third frequency has been shown to improve geometry-free ambiguity resolution [6].The major contribution of
this paper is that it presents, to our knowledge, the first large-scale
comparison through simulation of geometry-based LAMBDA to geometry-based
cascading for any application (terrestrial or orbiting, positioning or attitude
determination). Most previous work comparing the two methods was theoretical
only and addressed only the geometry-free case [7]
though the geometry-based case is addressed in [8].This paper aims to
demonstrate the effectiveness of various multicarrier ambiguity resolution
methods for the purpose of attitude determination of a Low Earth Orbiting (LEO)
satellite. The remainder of the paper is divided into three sections. Following
a brief review of carrier phase GNSS in Section2, a set of simulation
scenarios are defined in Section 3 in order to assess several ambiguity
resolution methods for two typical LEO orbiting users tracking GNSS satellites.
Performance is assessed by simulating the use of Galileo alone, and Galileo and
GPS together with one, two and three frequencies. The results of this series of
simulations are presented in Section 4 with an emphasis on the performance of
each method in each scenario in terms of probably of correct ambiguity
resolution.


## 2. Background

GNSS data processing for
attitude determination can be divided into four steps listed in what follows. The first
three steps are identical to the process of carrier-phase GNSS positioning
without attitude determination.  The
difference is that for attitude determination the baseline(s) being determined
connects two or more points on a vehicle with known coordinates in the body
frame of the vehicle. Attitude determination is then implemented in the fourth
step.(1) Float Ambiguity Solution
This is the process of using the
available observations to estimate a real-valued (float) estimate of the
carrier phase ambiguities.  These
ambiguity estimates may, if necessary, be filtered over time in order to
reduce their uncertainty.(2) Ambiguity Resolution
This is the process of resolving the
float ambiguities to integer values.
The output from the ambiguity resolution process is a set of
integer carrier phase ambiguities.
It is noted, however, that the ambiguities are not necessarily
guaranteed to be correct.(3) Fixed Ambiguity Baseline Solution
The integer ambiguities are used, along
with the corresponding carrier phase measurements, to generate an estimate
of the relative position vector between the two receivers involved in the
double-difference.  This relative
position vector is usually called the baseline vector, or simply
“baseline.”(4) Convert Baseline Solution to Attitude Solution
This is the process of using the known
baseline vectors in the body frame of the vehicle (spacecraft) and
baseline vectors obtained from the GNSS solution in the local-level frame
to estimate the attitude of the vehicle.Each of these steps is
discussed in more detail in the following sections.


### 2.1. Float Solution

The primary objective of
the float solution is to obtain an initial, real-valued, estimate of the
carrier phase ambiguities.  The actual
implementation of the float solution will depend on the data processing
strategy adopted, but these strategies can be broken down into two categories:
geometry-free and geometry-based [9, 10].


#### 2.1.1. Geometry-Free Approach

The geometry-free approach
is not concerned with the position of the receiver and instead aims to estimate
the double-difference range and ambiguity to each satellite, along with any
significant systematic errors.  The
pseudorange and carrier phase measurements made on one or more frequencies are
the inputs into the system.  The state
vector usually consists of the range to the satellite, the ambiguities to be
estimated and optionally an ionospheric error term.  The latter is usually only included when the
residual double-difference ionospheric error is nonnegligible.  For the case at hand however, because the
receivers are all located within a few metres of each other, the ionosphere
term can be safely neglected.The fact that each
satellite is treated separately is both an advantage and a disadvantage.  It is an advantage because it provides a
relatively simple implementation and it does not depend on the number of
satellites in view, nor their distribution in the sky.  It is a disadvantage because it does not
exploit the fact that the measurements to all of the different satellites are
related via the position of the receiver.
In other words, no information is shared between filters estimating each
double-difference ambiguity, which generally degrades performance.  A further disadvantage is that the
pseudorange errors—particularly multipath—can significantly degrade reliability.


#### 2.1.2. Geometry-Based Approach

In contrast to the
geometry-free approach, the geometry-based approach explicitly estimates the
baseline vector between the two receivers along with the ambiguities and any
other systematic errors.  Again, for the
short baselines involved in this application, these systematic errors need not
be considered.  The state vector is
usually divided into two components, a vector of ambiguities to the various
satellites and the remaining states such as position, velocity, and so forth.  In this way, all of the observations are
linked together via the position information which provides geometric strength
to the solution.  This also implies that
the ambiguities for all the satellites are estimated together, instead of on a
satellite-by-satellite basis, as with the geometry-free approach.The main disadvantage of the geometry-based approach is that it is dependent on the number and
distribution of the satellites being tracked.
As such, if the number of satellites tracked decreases below four, or if
the distribution of satellites in the sky is unsatisfactory, performance will
suffer.  That said, for the application
at hand, and for the planned number of GNSS satellites in orbit (Galileo
with/without the addition of GPS), this is not expected to play a major role.


#### 2.2. Carrier Phase Ambiguity Resolution

In practice, there are
several strategies of ambiguity resolution, and in the present section, the
implementation details of each will be described. As discussed earlier, ambiguity
estimation techniques can be broadly classified as either geometry-based or geometry-free,
depending on whether or not baseline vector components are estimated
simultaneously with the ambiguity states. Ambiguity resolution methods can be
further classified as instantaneous or filtered depending on whether or not
more than a single epoch of observations is used in the estimation process.
Finally, multifrequency methods can be further divided between those that use
specific linear combinations of the various frequencies (referred to as
cascading methods in this paper) and those that attempt to estimate the optimal
combination on the fly (LAMBDA).


#### 2.2.1. Widelaning, Cascading Methods and Lambda

In the simplest sense,
ambiguity resolution can be accomplished by comparing an absolute measurement
(a code pseudorange) with a relative measurement (a phase measurement) to
determine the bias between the two (the ambiguity). Conceptually, this requires
that the code pseudorange be accurate enough that one can confidently determine
the phase ambiguity. In practice, this means that the uncertainty on the code
measurement must be significantly less than the carrier wavelength so that the
code measurement can place you definitively in a particular carrier phase
cycle. In low frequency radio-navigation systems this is relatively easy but in
GNSS, the wavelengths are short and the code measurements are relatively noisy.
For example, the GPS L1 wavelength is approximately 19 cm but a typical double-differenced code measurement may have errors on the order of 50 cm or more.Widelaning
If measurements on more than
one frequency are available, it is possible to form linear combinations of
these measurements that have larger effective wavelengths. Specifically, a
widelane (WL) observation is formed when two phase observations are subtracted
from each other. The resulting linear combination has a frequency equal to the
difference between the frequencies of the two original observations,(1)fWL=fa−fb,(2)λWL=cfWL=cfa−fb.
With current dual-frequency
GPS, a widelane observation(3)φWL=φL1−φL2 can be formed. The
resulting combination has an effective wavelength of approximately 86 cm,
making it more reasonable to estimate the widelane ambiguity from a between
receiver single-differenced code pseudorange measurement. Once the widelane
ambiguity has been determined, a fixed widelane phase range can be formed that
can then be used to estimate the L1 ambiguity.
With the addition of a
third frequency on GPS and the deployment of Galileo, new widelane phase
combinations are possible. From (2) it should be noted that a widelane wavelength is inversely
proportional to the difference in the frequencies of the two signals involved.
For this reason L2 and L5 in GPS and E5a and E5b in Galileo can be used to form
very long wavelength widelanes that are often referred to as the extra-widelane
(EWL) combination.  For GPS L2-L5, the EWL wavelength is 5.861 m. For Galileo E5a-E5b it is 9.768 m. The existence of
these very long wavelengths forms the basis for a number of integer cascading ambiguity
resolution algorithms. Table 1 lists
the modernized GPS and Galileo frequencies and wavelengths while possible
widelane combinations are listed in Table 2.Table 1
Characteristics of modernized GPS and GALILEO open service frequencies and signals.
SignalModulationFrequency (MHz)Wavelength (m)Chipping rage (Mc/s)Modernized GPSL1 C/ABPSK1575.420.1901.023L2 CBPSK1227.600.2441.023L5BPSK1176.450.25410.23GalileoE1-E2MBOC1575.420.1901.023E5bAltBOC1207.140.24810.23E5aAltBOC1176.450.25410.23Table 2
GPS
and Galileo widelane wavelengths in metres.
(a)GPSL10.8619L20.75145.8610L5(b)GalileoE1-E20.8140E5b0.75149.7684E5aCascading Algorithms
The use of the L2-L5 widelane
as the first step in a cascaded ambiguity resolution for GPS has been proposed
by many researchers, particularly for the geometry-free case [11, 12], and
similar algorithms have been proposed for Galileo [13, 14]. When
the EWL ambiguity is resolved, the resulting phase-range is then used to
estimate a shorter wavelength widelane ambiguity and this process is continued
until the L1 ambiguity is resolved at which point a carrier-phase position
solution is computed from the fixed L1 phase measurements. GPS methods have
traditionally been called CIR (cascaded integer resolution) methods while
Galileo methods have been referred to as TCAR or MCAR (triple- or multicarrier
ambiguity resolution) methods.
Most additional previous
work focuses on finding better linear combinations of the three phase
measurements (other than EWL, WL, L1) to increase the likelihood of successful
ambiguity resolution in the presence of large differential atmospheric errors
for long baseline surveying applications [15–19]. There
have been some attempts to use triple-frequency GPS and triple-frequency
Galileo simultaneously in geometry-based ambiguity resolution techniques [4, 5]
including the use of the two common frequencies (L1 and L5) to obtain an
additional double-differenced measurement between the two systems [6].The Lambda Approach to Multifrequency Ambiguity Resolution
The Least squares AMBiguity
Decorrelation Adjustment (LAMBDA) method is a generic method for ambiguity
resolution [20]. The
method can be applied to any set of float ambiguities that have been jointly
estimated (meaning that the float ambiguities share a covariance matrix). The
method can either be applied to the multifrequency geometry-free case or to
single- or multifrequency geometry-based cases. Its excellent performance on
short baselines is well known and is described in [21]. In
the geometry-free case two or three ambiguities are being estimated for a
single double-difference satellite pair. In the geometry-based case, several
ambiguities and baseline vectors are being estimated. In both cases, the key to
the use of the LAMBDA method is in the fact that double-differenced ambiguities
that are estimated together are usually highly correlated.  Earlier search-based techniques would
estimate float ambiguities for either of the two above-mentioned  cases and then search a
large volume of possible ambiguity sets around the float solution for the best
fitting fixed ambiguity solution. Unfortunately, due to the high correlation of
the ambiguity states, the search volume is typically very large making these
methods very time consuming [22]. The
LAMBDA method also employs a search, but prior to the search, it attempts to
find a linear transformation of the ambiguity set being estimated that
decorrelates the ambiguities from each other while maintaining their integer
nature.  Unlike the cascading methods
mentioned earlier, which rely on specific linear combinations of the phase
measurements to facilitate ambiguity resolution, the LAMBDA method uses the
covariance matrix of the float ambiguity solution to find the optimal linear
combination to facilitate ambiguity resolution. It has been previously shown
that the various cascading schemes (where the linear combinations allowed are
restricted the EWL, WL, and L1) are theoretically suboptimal compared to the
LAMBDA- derived linear combinations in the geometry free case [7]. The LAMBDA
method’s application to the multifrequency ambiguity resolution problem has been
studied extensively in [6, 23–28].


#### 2.2.2. Geometry-Based AR and Assessing Probability of Correct Fix

The general mathematical model for geometry-based carrier-phase GNSS
positioning can be described as follows:(4)z=Hx+v=[H1H2]×[x1x2]+v,where z is the observation vector, x
is the state vector to be estimated in the
filter, comprised of the ambiguities sub-vector x1 and the other parameters x2 to be estimated, for example, position, ionospheric
and/or tropospheric estimates, H
is the design matrix for the full state
vector x,
and H1 and H2 are the corresponding design matrices for states x1 and x2,
v
is the measurement noise, generally assumed
to be white and to follow a Gaussian normal distribution.
There are three steps to resolve the ambiguities to their integer
values, which can be described as follows.(1) Estimate the ambiguities as real values,x^1,
with the other state parameters, x^2.
The integer nature of the ambiguities is ignored and the estimates are referred
to as float estimates. The corresponding covariance matrix of the errors of the
state vector can be the partitioned into the covariance of the ambiguity errors,
the covariance matrix of the other states, and a cross-covariance term(5)P=[Px^1Px^1x^2Px^2x^1Px^2].(2) Determine the integer valuex⌣1 of ambiguities based on the float estimates x^1.
Various methods have been proposed in the process of integer fixing, as
discussed earlier.(3) Compute the fixed estimates of parameterx⌣2 based on the earlier fixed integer ambiguities.
The fixed estimates and their variance can be formulated from the solution in the aforementioned
Step 1 as(6)x⌣2=x^2−Px^2x^1Px^1−1(x^1−x⌣1),(7)Px⌣2=Px^2−Px^2x^1Px^1−1Px^1x^2, where the covariance of the
fixed estimates Px⌣2 is valid only under the assumption that the
ambiguities have been fixed to their correct values. In the aforementioned Step 2, the
process of obtaining the integer values of the ambiguities can be defined as a
mapping of the real space to the integer space. Then the probability that a
given integer vector x⌣1 is equal to a particular integer vector z,
and can be assessed as(8)P(x⌣1=z)=P(x^1∈Sz)=∫Szpx^1(s)ds, where px^1 is the probability density function (PDF) of
the float ambiguities [29]. This defines the probability of correctly
resolving the ambiguities, or the probability of correct fix (PCF).However, it is difficult to
quantify (8) numerically because of the complexity of the so-called pull-in
region and the computational load of the integration process. Thus a
simplification or approximation is required. There have been various bounds
proposed to approximate the PCF [29]. A
method of ambiguity bootstrapping is widely used and adopted to determining a
lower bound of the PCF [29–31]. The
evaluation is based on the following expressions:(9)P(x⌣B=x)=∏i=1n(2Φ(12σx^i∣I)−1),(10)Φ(x)=12π∫−∞xe−(1/2)n2dn. In (9), x⌣B is the bootstrapped integer ambiguity vector, σx^i|I is the conditional standard deviation of
ambiguity i conditioned on the previous I=(1,2,…,i−1) ambiguities, and Φ(x) describes the area under the normal distribution.There are many methods to
fix the float ambiguities to integer values in Step 2. In the case of ambiguity
rounding, the conditional nature of the conditional standard deviations are
ignored and an estimate of the PCF can be obtained from the variances of the
float ambiguities. It has been previously observed that (9) is not invariant to ambiguity parameterizations [29–32]. In
the remainder of this paper the PCF is quantified in terms of Probability of Incorrect Fix (PIF), which is equal to one minus PCF. Likewise, the lower bound of
probability of correct fix becomes an upper bound on the probability of
incorrect fix. In this case “good” performance is represented as a small upper bound
(significantly less than 1) while “poor” performance is represented by an upper
bound on PIF that is close to one (suggesting that incorrect ambiguity
resolution is likely).


#### 2.3. Fixed Ambiguity Baseline Solution and Transformation to Attitude Solution

Generally speaking, if the
carrier phase ambiguities are resolved to their correct integer values, the
accuracy of the baseline estimate obtained from (6) is on the order 1–3 cm in each coordinate direction.  However,
if one or more ambiguities are resolved to an incorrect integer value, then the
baseline solution will contain errors on the order of the carrier phase
wavelength (e.g., approximately 19 cm at L1).  This is why the ambiguity
resolution process is so critical. Fixed position accuracy can be obtained from
the covariance matrix of the fixed solution obtained with (7).The baseline solution
obtained from (6) can then be used to compute the attitude of the vehicle.  We denote this solution in the local level
frame as δxℓ.  Assuming the baseline vector between any two
antennas is also known in the body frame, δxb,
then the following relationship will hold:(11)δxℓ=Rbℓδxb, where Rbℓ is the rotation matrix that transforms a
vector in the body frame into the local level frame.  The assumed convention for the rotation
matrix is(12)Rbℓ=R3(α)R1(−β)R2(−γ), where γ, β, and α are respectively the roll, pitch and azimuth
of the vehicle, and R1, R2, and R3 are rotation matrices about the primary (x),
secondary (y), and tertiary (z) axes respectively.  It is noted that other orders of rotations
can also be used without loss of generality.Assuming two noncolinear
(nonparallel) baseline vectors are available, (11), (12) allow for the estimation of the attitude parameters using the known
baseline vectors in the body frame and the measured baseline vectors in the
local level frame.  Mathematically,(13)δx^ℓ=δxℓ+εδxℓ=Rbℓδxb+εδxℓ, where εδxℓ are the errors in the baseline vector
estimated from the GNSS data. Estimation of the attitude parameters can be
performed using least-squares or Kalman filtering. The covariance matrix of the
attitude solution can be obtained from the covariance matrices of baselines
though covariance propagation [33].


## 2.1. Float Solution

The primary objective of
the float solution is to obtain an initial, real-valued, estimate of the
carrier phase ambiguities.  The actual
implementation of the float solution will depend on the data processing
strategy adopted, but these strategies can be broken down into two categories:
geometry-free and geometry-based [9, 10].


### 2.1.1. Geometry-Free Approach

The geometry-free approach
is not concerned with the position of the receiver and instead aims to estimate
the double-difference range and ambiguity to each satellite, along with any
significant systematic errors.  The
pseudorange and carrier phase measurements made on one or more frequencies are
the inputs into the system.  The state
vector usually consists of the range to the satellite, the ambiguities to be
estimated and optionally an ionospheric error term.  The latter is usually only included when the
residual double-difference ionospheric error is nonnegligible.  For the case at hand however, because the
receivers are all located within a few metres of each other, the ionosphere
term can be safely neglected.The fact that each
satellite is treated separately is both an advantage and a disadvantage.  It is an advantage because it provides a
relatively simple implementation and it does not depend on the number of
satellites in view, nor their distribution in the sky.  It is a disadvantage because it does not
exploit the fact that the measurements to all of the different satellites are
related via the position of the receiver.
In other words, no information is shared between filters estimating each
double-difference ambiguity, which generally degrades performance.  A further disadvantage is that the
pseudorange errors—particularly multipath—can significantly degrade reliability.


### 2.1.2. Geometry-Based Approach

In contrast to the
geometry-free approach, the geometry-based approach explicitly estimates the
baseline vector between the two receivers along with the ambiguities and any
other systematic errors.  Again, for the
short baselines involved in this application, these systematic errors need not
be considered.  The state vector is
usually divided into two components, a vector of ambiguities to the various
satellites and the remaining states such as position, velocity, and so forth.  In this way, all of the observations are
linked together via the position information which provides geometric strength
to the solution.  This also implies that
the ambiguities for all the satellites are estimated together, instead of on a
satellite-by-satellite basis, as with the geometry-free approach.The main disadvantage of the geometry-based approach is that it is dependent on the number and
distribution of the satellites being tracked.
As such, if the number of satellites tracked decreases below four, or if
the distribution of satellites in the sky is unsatisfactory, performance will
suffer.  That said, for the application
at hand, and for the planned number of GNSS satellites in orbit (Galileo
with/without the addition of GPS), this is not expected to play a major role.


### 2.2. Carrier Phase Ambiguity Resolution

In practice, there are
several strategies of ambiguity resolution, and in the present section, the
implementation details of each will be described. As discussed earlier, ambiguity
estimation techniques can be broadly classified as either geometry-based or geometry-free,
depending on whether or not baseline vector components are estimated
simultaneously with the ambiguity states. Ambiguity resolution methods can be
further classified as instantaneous or filtered depending on whether or not
more than a single epoch of observations is used in the estimation process.
Finally, multifrequency methods can be further divided between those that use
specific linear combinations of the various frequencies (referred to as
cascading methods in this paper) and those that attempt to estimate the optimal
combination on the fly (LAMBDA).


### 2.2.1. Widelaning, Cascading Methods and Lambda

In the simplest sense,
ambiguity resolution can be accomplished by comparing an absolute measurement
(a code pseudorange) with a relative measurement (a phase measurement) to
determine the bias between the two (the ambiguity). Conceptually, this requires
that the code pseudorange be accurate enough that one can confidently determine
the phase ambiguity. In practice, this means that the uncertainty on the code
measurement must be significantly less than the carrier wavelength so that the
code measurement can place you definitively in a particular carrier phase
cycle. In low frequency radio-navigation systems this is relatively easy but in
GNSS, the wavelengths are short and the code measurements are relatively noisy.
For example, the GPS L1 wavelength is approximately 19 cm but a typical double-differenced code measurement may have errors on the order of 50 cm or more.Widelaning
If measurements on more than
one frequency are available, it is possible to form linear combinations of
these measurements that have larger effective wavelengths. Specifically, a
widelane (WL) observation is formed when two phase observations are subtracted
from each other. The resulting linear combination has a frequency equal to the
difference between the frequencies of the two original observations,(1)fWL=fa−fb,(2)λWL=cfWL=cfa−fb.
With current dual-frequency
GPS, a widelane observation(3)φWL=φL1−φL2 can be formed. The
resulting combination has an effective wavelength of approximately 86 cm,
making it more reasonable to estimate the widelane ambiguity from a between
receiver single-differenced code pseudorange measurement. Once the widelane
ambiguity has been determined, a fixed widelane phase range can be formed that
can then be used to estimate the L1 ambiguity.
With the addition of a
third frequency on GPS and the deployment of Galileo, new widelane phase
combinations are possible. From (2) it should be noted that a widelane wavelength is inversely
proportional to the difference in the frequencies of the two signals involved.
For this reason L2 and L5 in GPS and E5a and E5b in Galileo can be used to form
very long wavelength widelanes that are often referred to as the extra-widelane
(EWL) combination.  For GPS L2-L5, the EWL wavelength is 5.861 m. For Galileo E5a-E5b it is 9.768 m. The existence of
these very long wavelengths forms the basis for a number of integer cascading ambiguity
resolution algorithms. Table 1 lists
the modernized GPS and Galileo frequencies and wavelengths while possible
widelane combinations are listed in Table 2.Table 1
Characteristics of modernized GPS and GALILEO open service frequencies and signals.
SignalModulationFrequency (MHz)Wavelength (m)Chipping rage (Mc/s)Modernized GPSL1 C/ABPSK1575.420.1901.023L2 CBPSK1227.600.2441.023L5BPSK1176.450.25410.23GalileoE1-E2MBOC1575.420.1901.023E5bAltBOC1207.140.24810.23E5aAltBOC1176.450.25410.23Table 2
GPS
and Galileo widelane wavelengths in metres.
(a)GPSL10.8619L20.75145.8610L5(b)GalileoE1-E20.8140E5b0.75149.7684E5aCascading Algorithms
The use of the L2-L5 widelane
as the first step in a cascaded ambiguity resolution for GPS has been proposed
by many researchers, particularly for the geometry-free case [11, 12], and
similar algorithms have been proposed for Galileo [13, 14]. When
the EWL ambiguity is resolved, the resulting phase-range is then used to
estimate a shorter wavelength widelane ambiguity and this process is continued
until the L1 ambiguity is resolved at which point a carrier-phase position
solution is computed from the fixed L1 phase measurements. GPS methods have
traditionally been called CIR (cascaded integer resolution) methods while
Galileo methods have been referred to as TCAR or MCAR (triple- or multicarrier
ambiguity resolution) methods.
Most additional previous
work focuses on finding better linear combinations of the three phase
measurements (other than EWL, WL, L1) to increase the likelihood of successful
ambiguity resolution in the presence of large differential atmospheric errors
for long baseline surveying applications [15–19]. There
have been some attempts to use triple-frequency GPS and triple-frequency
Galileo simultaneously in geometry-based ambiguity resolution techniques [4, 5]
including the use of the two common frequencies (L1 and L5) to obtain an
additional double-differenced measurement between the two systems [6].The Lambda Approach to Multifrequency Ambiguity Resolution
The Least squares AMBiguity
Decorrelation Adjustment (LAMBDA) method is a generic method for ambiguity
resolution [20]. The
method can be applied to any set of float ambiguities that have been jointly
estimated (meaning that the float ambiguities share a covariance matrix). The
method can either be applied to the multifrequency geometry-free case or to
single- or multifrequency geometry-based cases. Its excellent performance on
short baselines is well known and is described in [21]. In
the geometry-free case two or three ambiguities are being estimated for a
single double-difference satellite pair. In the geometry-based case, several
ambiguities and baseline vectors are being estimated. In both cases, the key to
the use of the LAMBDA method is in the fact that double-differenced ambiguities
that are estimated together are usually highly correlated.  Earlier search-based techniques would
estimate float ambiguities for either of the two above-mentioned  cases and then search a
large volume of possible ambiguity sets around the float solution for the best
fitting fixed ambiguity solution. Unfortunately, due to the high correlation of
the ambiguity states, the search volume is typically very large making these
methods very time consuming [22]. The
LAMBDA method also employs a search, but prior to the search, it attempts to
find a linear transformation of the ambiguity set being estimated that
decorrelates the ambiguities from each other while maintaining their integer
nature.  Unlike the cascading methods
mentioned earlier, which rely on specific linear combinations of the phase
measurements to facilitate ambiguity resolution, the LAMBDA method uses the
covariance matrix of the float ambiguity solution to find the optimal linear
combination to facilitate ambiguity resolution. It has been previously shown
that the various cascading schemes (where the linear combinations allowed are
restricted the EWL, WL, and L1) are theoretically suboptimal compared to the
LAMBDA- derived linear combinations in the geometry free case [7]. The LAMBDA
method’s application to the multifrequency ambiguity resolution problem has been
studied extensively in [6, 23–28].


### 2.2.2. Geometry-Based AR and Assessing Probability of Correct Fix

The general mathematical model for geometry-based carrier-phase GNSS
positioning can be described as follows:(4)z=Hx+v=[H1H2]×[x1x2]+v,where z is the observation vector, x
is the state vector to be estimated in the
filter, comprised of the ambiguities sub-vector x1 and the other parameters x2 to be estimated, for example, position, ionospheric
and/or tropospheric estimates, H
is the design matrix for the full state
vector x,
and H1 and H2 are the corresponding design matrices for states x1 and x2,
v
is the measurement noise, generally assumed
to be white and to follow a Gaussian normal distribution.
There are three steps to resolve the ambiguities to their integer
values, which can be described as follows.(1) Estimate the ambiguities as real values,x^1,
with the other state parameters, x^2.
The integer nature of the ambiguities is ignored and the estimates are referred
to as float estimates. The corresponding covariance matrix of the errors of the
state vector can be the partitioned into the covariance of the ambiguity errors,
the covariance matrix of the other states, and a cross-covariance term(5)P=[Px^1Px^1x^2Px^2x^1Px^2].(2) Determine the integer valuex⌣1 of ambiguities based on the float estimates x^1.
Various methods have been proposed in the process of integer fixing, as
discussed earlier.(3) Compute the fixed estimates of parameterx⌣2 based on the earlier fixed integer ambiguities.
The fixed estimates and their variance can be formulated from the solution in the aforementioned
Step 1 as(6)x⌣2=x^2−Px^2x^1Px^1−1(x^1−x⌣1),(7)Px⌣2=Px^2−Px^2x^1Px^1−1Px^1x^2, where the covariance of the
fixed estimates Px⌣2 is valid only under the assumption that the
ambiguities have been fixed to their correct values. In the aforementioned Step 2, the
process of obtaining the integer values of the ambiguities can be defined as a
mapping of the real space to the integer space. Then the probability that a
given integer vector x⌣1 is equal to a particular integer vector z,
and can be assessed as(8)P(x⌣1=z)=P(x^1∈Sz)=∫Szpx^1(s)ds, where px^1 is the probability density function (PDF) of
the float ambiguities [29]. This defines the probability of correctly
resolving the ambiguities, or the probability of correct fix (PCF).However, it is difficult to
quantify (8) numerically because of the complexity of the so-called pull-in
region and the computational load of the integration process. Thus a
simplification or approximation is required. There have been various bounds
proposed to approximate the PCF [29]. A
method of ambiguity bootstrapping is widely used and adopted to determining a
lower bound of the PCF [29–31]. The
evaluation is based on the following expressions:(9)P(x⌣B=x)=∏i=1n(2Φ(12σx^i∣I)−1),(10)Φ(x)=12π∫−∞xe−(1/2)n2dn. In (9), x⌣B is the bootstrapped integer ambiguity vector, σx^i|I is the conditional standard deviation of
ambiguity i conditioned on the previous I=(1,2,…,i−1) ambiguities, and Φ(x) describes the area under the normal distribution.There are many methods to
fix the float ambiguities to integer values in Step 2. In the case of ambiguity
rounding, the conditional nature of the conditional standard deviations are
ignored and an estimate of the PCF can be obtained from the variances of the
float ambiguities. It has been previously observed that (9) is not invariant to ambiguity parameterizations [29–32]. In
the remainder of this paper the PCF is quantified in terms of Probability of Incorrect Fix (PIF), which is equal to one minus PCF. Likewise, the lower bound of
probability of correct fix becomes an upper bound on the probability of
incorrect fix. In this case “good” performance is represented as a small upper bound
(significantly less than 1) while “poor” performance is represented by an upper
bound on PIF that is close to one (suggesting that incorrect ambiguity
resolution is likely).


### 2.3. Fixed Ambiguity Baseline Solution and Transformation to Attitude Solution

Generally speaking, if the
carrier phase ambiguities are resolved to their correct integer values, the
accuracy of the baseline estimate obtained from (6) is on the order 1–3 cm in each coordinate direction.  However,
if one or more ambiguities are resolved to an incorrect integer value, then the
baseline solution will contain errors on the order of the carrier phase
wavelength (e.g., approximately 19 cm at L1).  This is why the ambiguity
resolution process is so critical. Fixed position accuracy can be obtained from
the covariance matrix of the fixed solution obtained with (7).The baseline solution
obtained from (6) can then be used to compute the attitude of the vehicle.  We denote this solution in the local level
frame as δxℓ.  Assuming the baseline vector between any two
antennas is also known in the body frame, δxb,
then the following relationship will hold:(11)δxℓ=Rbℓδxb, where Rbℓ is the rotation matrix that transforms a
vector in the body frame into the local level frame.  The assumed convention for the rotation
matrix is(12)Rbℓ=R3(α)R1(−β)R2(−γ), where γ, β, and α are respectively the roll, pitch and azimuth
of the vehicle, and R1, R2, and R3 are rotation matrices about the primary (x),
secondary (y), and tertiary (z) axes respectively.  It is noted that other orders of rotations
can also be used without loss of generality.Assuming two noncolinear
(nonparallel) baseline vectors are available, (11), (12) allow for the estimation of the attitude parameters using the known
baseline vectors in the body frame and the measured baseline vectors in the
local level frame.  Mathematically,(13)δx^ℓ=δxℓ+εδxℓ=Rbℓδxb+εδxℓ, where εδxℓ are the errors in the baseline vector
estimated from the GNSS data. Estimation of the attitude parameters can be
performed using least-squares or Kalman filtering. The covariance matrix of the
attitude solution can be obtained from the covariance matrices of baselines
though covariance propagation [33].


## 2.1.1. Geometry-Free Approach

The geometry-free approach
is not concerned with the position of the receiver and instead aims to estimate
the double-difference range and ambiguity to each satellite, along with any
significant systematic errors.  The
pseudorange and carrier phase measurements made on one or more frequencies are
the inputs into the system.  The state
vector usually consists of the range to the satellite, the ambiguities to be
estimated and optionally an ionospheric error term.  The latter is usually only included when the
residual double-difference ionospheric error is nonnegligible.  For the case at hand however, because the
receivers are all located within a few metres of each other, the ionosphere
term can be safely neglected.The fact that each
satellite is treated separately is both an advantage and a disadvantage.  It is an advantage because it provides a
relatively simple implementation and it does not depend on the number of
satellites in view, nor their distribution in the sky.  It is a disadvantage because it does not
exploit the fact that the measurements to all of the different satellites are
related via the position of the receiver.
In other words, no information is shared between filters estimating each
double-difference ambiguity, which generally degrades performance.  A further disadvantage is that the
pseudorange errors—particularly multipath—can significantly degrade reliability.


## 2.1.2. Geometry-Based Approach

In contrast to the
geometry-free approach, the geometry-based approach explicitly estimates the
baseline vector between the two receivers along with the ambiguities and any
other systematic errors.  Again, for the
short baselines involved in this application, these systematic errors need not
be considered.  The state vector is
usually divided into two components, a vector of ambiguities to the various
satellites and the remaining states such as position, velocity, and so forth.  In this way, all of the observations are
linked together via the position information which provides geometric strength
to the solution.  This also implies that
the ambiguities for all the satellites are estimated together, instead of on a
satellite-by-satellite basis, as with the geometry-free approach.The main disadvantage of the geometry-based approach is that it is dependent on the number and
distribution of the satellites being tracked.
As such, if the number of satellites tracked decreases below four, or if
the distribution of satellites in the sky is unsatisfactory, performance will
suffer.  That said, for the application
at hand, and for the planned number of GNSS satellites in orbit (Galileo
with/without the addition of GPS), this is not expected to play a major role.


## 2.2. Carrier Phase Ambiguity Resolution

In practice, there are
several strategies of ambiguity resolution, and in the present section, the
implementation details of each will be described. As discussed earlier, ambiguity
estimation techniques can be broadly classified as either geometry-based or geometry-free,
depending on whether or not baseline vector components are estimated
simultaneously with the ambiguity states. Ambiguity resolution methods can be
further classified as instantaneous or filtered depending on whether or not
more than a single epoch of observations is used in the estimation process.
Finally, multifrequency methods can be further divided between those that use
specific linear combinations of the various frequencies (referred to as
cascading methods in this paper) and those that attempt to estimate the optimal
combination on the fly (LAMBDA).


## 2.2.1. Widelaning, Cascading Methods and Lambda

In the simplest sense,
ambiguity resolution can be accomplished by comparing an absolute measurement
(a code pseudorange) with a relative measurement (a phase measurement) to
determine the bias between the two (the ambiguity). Conceptually, this requires
that the code pseudorange be accurate enough that one can confidently determine
the phase ambiguity. In practice, this means that the uncertainty on the code
measurement must be significantly less than the carrier wavelength so that the
code measurement can place you definitively in a particular carrier phase
cycle. In low frequency radio-navigation systems this is relatively easy but in
GNSS, the wavelengths are short and the code measurements are relatively noisy.
For example, the GPS L1 wavelength is approximately 19 cm but a typical double-differenced code measurement may have errors on the order of 50 cm or more.Widelaning
If measurements on more than
one frequency are available, it is possible to form linear combinations of
these measurements that have larger effective wavelengths. Specifically, a
widelane (WL) observation is formed when two phase observations are subtracted
from each other. The resulting linear combination has a frequency equal to the
difference between the frequencies of the two original observations,(1)fWL=fa−fb,(2)λWL=cfWL=cfa−fb.
With current dual-frequency
GPS, a widelane observation(3)φWL=φL1−φL2 can be formed. The
resulting combination has an effective wavelength of approximately 86 cm,
making it more reasonable to estimate the widelane ambiguity from a between
receiver single-differenced code pseudorange measurement. Once the widelane
ambiguity has been determined, a fixed widelane phase range can be formed that
can then be used to estimate the L1 ambiguity.
With the addition of a
third frequency on GPS and the deployment of Galileo, new widelane phase
combinations are possible. From (2) it should be noted that a widelane wavelength is inversely
proportional to the difference in the frequencies of the two signals involved.
For this reason L2 and L5 in GPS and E5a and E5b in Galileo can be used to form
very long wavelength widelanes that are often referred to as the extra-widelane
(EWL) combination.  For GPS L2-L5, the EWL wavelength is 5.861 m. For Galileo E5a-E5b it is 9.768 m. The existence of
these very long wavelengths forms the basis for a number of integer cascading ambiguity
resolution algorithms. Table 1 lists
the modernized GPS and Galileo frequencies and wavelengths while possible
widelane combinations are listed in Table 2.Table 1
Characteristics of modernized GPS and GALILEO open service frequencies and signals.
SignalModulationFrequency (MHz)Wavelength (m)Chipping rage (Mc/s)Modernized GPSL1 C/ABPSK1575.420.1901.023L2 CBPSK1227.600.2441.023L5BPSK1176.450.25410.23GalileoE1-E2MBOC1575.420.1901.023E5bAltBOC1207.140.24810.23E5aAltBOC1176.450.25410.23Table 2
GPS
and Galileo widelane wavelengths in metres.
(a)GPSL10.8619L20.75145.8610L5(b)GalileoE1-E20.8140E5b0.75149.7684E5aCascading Algorithms
The use of the L2-L5 widelane
as the first step in a cascaded ambiguity resolution for GPS has been proposed
by many researchers, particularly for the geometry-free case [11, 12], and
similar algorithms have been proposed for Galileo [13, 14]. When
the EWL ambiguity is resolved, the resulting phase-range is then used to
estimate a shorter wavelength widelane ambiguity and this process is continued
until the L1 ambiguity is resolved at which point a carrier-phase position
solution is computed from the fixed L1 phase measurements. GPS methods have
traditionally been called CIR (cascaded integer resolution) methods while
Galileo methods have been referred to as TCAR or MCAR (triple- or multicarrier
ambiguity resolution) methods.
Most additional previous
work focuses on finding better linear combinations of the three phase
measurements (other than EWL, WL, L1) to increase the likelihood of successful
ambiguity resolution in the presence of large differential atmospheric errors
for long baseline surveying applications [15–19]. There
have been some attempts to use triple-frequency GPS and triple-frequency
Galileo simultaneously in geometry-based ambiguity resolution techniques [4, 5]
including the use of the two common frequencies (L1 and L5) to obtain an
additional double-differenced measurement between the two systems [6].The Lambda Approach to Multifrequency Ambiguity Resolution
The Least squares AMBiguity
Decorrelation Adjustment (LAMBDA) method is a generic method for ambiguity
resolution [20]. The
method can be applied to any set of float ambiguities that have been jointly
estimated (meaning that the float ambiguities share a covariance matrix). The
method can either be applied to the multifrequency geometry-free case or to
single- or multifrequency geometry-based cases. Its excellent performance on
short baselines is well known and is described in [21]. In
the geometry-free case two or three ambiguities are being estimated for a
single double-difference satellite pair. In the geometry-based case, several
ambiguities and baseline vectors are being estimated. In both cases, the key to
the use of the LAMBDA method is in the fact that double-differenced ambiguities
that are estimated together are usually highly correlated.  Earlier search-based techniques would
estimate float ambiguities for either of the two above-mentioned  cases and then search a
large volume of possible ambiguity sets around the float solution for the best
fitting fixed ambiguity solution. Unfortunately, due to the high correlation of
the ambiguity states, the search volume is typically very large making these
methods very time consuming [22]. The
LAMBDA method also employs a search, but prior to the search, it attempts to
find a linear transformation of the ambiguity set being estimated that
decorrelates the ambiguities from each other while maintaining their integer
nature.  Unlike the cascading methods
mentioned earlier, which rely on specific linear combinations of the phase
measurements to facilitate ambiguity resolution, the LAMBDA method uses the
covariance matrix of the float ambiguity solution to find the optimal linear
combination to facilitate ambiguity resolution. It has been previously shown
that the various cascading schemes (where the linear combinations allowed are
restricted the EWL, WL, and L1) are theoretically suboptimal compared to the
LAMBDA- derived linear combinations in the geometry free case [7]. The LAMBDA
method’s application to the multifrequency ambiguity resolution problem has been
studied extensively in [6, 23–28].


## 2.2.2. Geometry-Based AR and Assessing Probability of Correct Fix

The general mathematical model for geometry-based carrier-phase GNSS
positioning can be described as follows:(4)z=Hx+v=[H1H2]×[x1x2]+v,where z is the observation vector, x
is the state vector to be estimated in the
filter, comprised of the ambiguities sub-vector x1 and the other parameters x2 to be estimated, for example, position, ionospheric
and/or tropospheric estimates, H
is the design matrix for the full state
vector x,
and H1 and H2 are the corresponding design matrices for states x1 and x2,
v
is the measurement noise, generally assumed
to be white and to follow a Gaussian normal distribution.
There are three steps to resolve the ambiguities to their integer
values, which can be described as follows.(1) Estimate the ambiguities as real values,x^1,
with the other state parameters, x^2.
The integer nature of the ambiguities is ignored and the estimates are referred
to as float estimates. The corresponding covariance matrix of the errors of the
state vector can be the partitioned into the covariance of the ambiguity errors,
the covariance matrix of the other states, and a cross-covariance term(5)P=[Px^1Px^1x^2Px^2x^1Px^2].(2) Determine the integer valuex⌣1 of ambiguities based on the float estimates x^1.
Various methods have been proposed in the process of integer fixing, as
discussed earlier.(3) Compute the fixed estimates of parameterx⌣2 based on the earlier fixed integer ambiguities.
The fixed estimates and their variance can be formulated from the solution in the aforementioned
Step 1 as(6)x⌣2=x^2−Px^2x^1Px^1−1(x^1−x⌣1),(7)Px⌣2=Px^2−Px^2x^1Px^1−1Px^1x^2, where the covariance of the
fixed estimates Px⌣2 is valid only under the assumption that the
ambiguities have been fixed to their correct values. In the aforementioned Step 2, the
process of obtaining the integer values of the ambiguities can be defined as a
mapping of the real space to the integer space. Then the probability that a
given integer vector x⌣1 is equal to a particular integer vector z,
and can be assessed as(8)P(x⌣1=z)=P(x^1∈Sz)=∫Szpx^1(s)ds, where px^1 is the probability density function (PDF) of
the float ambiguities [29]. This defines the probability of correctly
resolving the ambiguities, or the probability of correct fix (PCF).However, it is difficult to
quantify (8) numerically because of the complexity of the so-called pull-in
region and the computational load of the integration process. Thus a
simplification or approximation is required. There have been various bounds
proposed to approximate the PCF [29]. A
method of ambiguity bootstrapping is widely used and adopted to determining a
lower bound of the PCF [29–31]. The
evaluation is based on the following expressions:(9)P(x⌣B=x)=∏i=1n(2Φ(12σx^i∣I)−1),(10)Φ(x)=12π∫−∞xe−(1/2)n2dn. In (9), x⌣B is the bootstrapped integer ambiguity vector, σx^i|I is the conditional standard deviation of
ambiguity i conditioned on the previous I=(1,2,…,i−1) ambiguities, and Φ(x) describes the area under the normal distribution.There are many methods to
fix the float ambiguities to integer values in Step 2. In the case of ambiguity
rounding, the conditional nature of the conditional standard deviations are
ignored and an estimate of the PCF can be obtained from the variances of the
float ambiguities. It has been previously observed that (9) is not invariant to ambiguity parameterizations [29–32]. In
the remainder of this paper the PCF is quantified in terms of Probability of Incorrect Fix (PIF), which is equal to one minus PCF. Likewise, the lower bound of
probability of correct fix becomes an upper bound on the probability of
incorrect fix. In this case “good” performance is represented as a small upper bound
(significantly less than 1) while “poor” performance is represented by an upper
bound on PIF that is close to one (suggesting that incorrect ambiguity
resolution is likely).


## 2.3. Fixed Ambiguity Baseline Solution and Transformation to Attitude Solution

Generally speaking, if the
carrier phase ambiguities are resolved to their correct integer values, the
accuracy of the baseline estimate obtained from (6) is on the order 1–3 cm in each coordinate direction.  However,
if one or more ambiguities are resolved to an incorrect integer value, then the
baseline solution will contain errors on the order of the carrier phase
wavelength (e.g., approximately 19 cm at L1).  This is why the ambiguity
resolution process is so critical. Fixed position accuracy can be obtained from
the covariance matrix of the fixed solution obtained with (7).The baseline solution
obtained from (6) can then be used to compute the attitude of the vehicle.  We denote this solution in the local level
frame as δxℓ.  Assuming the baseline vector between any two
antennas is also known in the body frame, δxb,
then the following relationship will hold:(11)δxℓ=Rbℓδxb, where Rbℓ is the rotation matrix that transforms a
vector in the body frame into the local level frame.  The assumed convention for the rotation
matrix is(12)Rbℓ=R3(α)R1(−β)R2(−γ), where γ, β, and α are respectively the roll, pitch and azimuth
of the vehicle, and R1, R2, and R3 are rotation matrices about the primary (x),
secondary (y), and tertiary (z) axes respectively.  It is noted that other orders of rotations
can also be used without loss of generality.Assuming two noncolinear
(nonparallel) baseline vectors are available, (11), (12) allow for the estimation of the attitude parameters using the known
baseline vectors in the body frame and the measured baseline vectors in the
local level frame.  Mathematically,(13)δx^ℓ=δxℓ+εδxℓ=Rbℓδxb+εδxℓ, where εδxℓ are the errors in the baseline vector
estimated from the GNSS data. Estimation of the attitude parameters can be
performed using least-squares or Kalman filtering. The covariance matrix of the
attitude solution can be obtained from the covariance matrices of baselines
though covariance propagation [33].


## 3. Simulation Test

In this section, the
accuracy of a short baseline is simulated for two different LEO satellites. The
accuracy is highly dependent on the ability to correctly resolve integer
ambiguities, and as such the first step is to assess the ambiguity resolution
performance.


### 3.1. Simulation Parameters

The signal characteristics
of modernized GPS and Galileo have been investigated in great detail by many
researchers [34]. Only
the GPS civilian signals and Galileo open service (OS) signals will be
considered in this study, their characteristics are listed in Table 1.Note that the open services
of modernized GPS and Galileo have two frequencies in common, but the
third frequency on each is unique. Because of this, it could be possible to form
double-difference observations between the two systems using L1 and E1/E2 phase
measurements and also L5 and E5a [6].
However, this is not possible for L2 and E5b. Double-differencing between
systems was not conducted in this study and for all three frequencies a base
satellite was assigned for each system.No official documents on the range accuracy of future GPS and Galileo
signals have been released but several studies quantify the signal performances
of future GNSS systems (e.g., [35]). The code and carrier observations are
affected by systematic errors and random noise. However, for attitude
determination applications, where the baseline is very short, the only error
sources that will not be completely cancelled by differencing are the code and
phase multipath and the receiver noise. Code multipath and noise depend on the
structure of the code and the design or the receiver. Phase noise can be
assumed to be a few percent of a carrier cycle for all of the signals and
similarly phase multipath can be shown to have a maximum amplitude of one
quarter of the wavelength and typical values of less than a few centimetres [36].


#### 3.1.1. Frequency Combinations

In all of the cascading methods
described earlier only a single code measurement is used for each satellite. In
principle, this should be the code measurement with the smallest measurement
noise and the smallest multipath. The same can be said for the LAMBDA-based
methods. In principle, it is possible to use multiple code observations in a
geometry-based least-squares or filtered LAMBDA solution, but in practice,
little is gained from this, particularly in short baselines where there are no
residual ionospheric errors.In the geometry-based LAMBDA and
cascading simulations presented in what follows, a single code observation from each
satellite is used in conjunction with one, two or three phase measurements from
each satellite. The following three frequency combinations are assessed.Single-Frequency
In this case, only L1 and E1/E2
code and phase measurements are used. LAMBDA is applied to decorrelate the
estimated ambiguities and obtain PCF estimates. This is contrasted with what
could be called “single-frequency cascading” or simply single-frequency
ambiguity resolution without the application of LAMBDA. In both cases, the
total number of double-difference ambiguities estimated is(nGPS−1)+(nGal−1) where nGPS and nGal are the number of GPS and Galileo satellites
tracked, respectively.Dual-Frequency
In the dual-frequency case, L2
and E5b phase observations are added to the single-frequency case described
earlier. In both the LAMBDA and cascading scenarios this doubles the number of
ambiguities to be estimated to2×[(nGPS−1)+(nGal−1)]. With LAMBDA, the ambiguities
consist of some linear combination of all the ambiguities on each frequency (as
determined by the algorithm) while with cascading the ambiguities are
specifically the Widelane ambiguities (which are resolved first) followed by
the L1 and E1/E2 ambiguities.Triple-Frequency
Finally, for the triple-frequency
case, the number of ambiguities increases again to3×[(nGPS−1)+(nGal−1)].
Again, for LAMBDA, the ambiguities being estimated are some linear combination
of the ambiguities of the three frequencies while for triple-frequency
cascading the extra-widelane (EWL) ambiguities are first resolved, followed by
the widelane (WL) ambiguities, and finally the L1 and E1/E2 ambiguities.
In both the cascaded and LAMBDA cases, the same input covariance
matrix of the ambiguities is used, however, in the LAMBDA cases the LAMBDA
algorithm is allowed to determine the optimal decorrelating transformation while
in the cascading cases a fixed set of linear combinations (EWL, WL, and L1) are
used.


##### 3.1.2. Constellations

According to the Galileo
Mission High Level Definition document [37], the
space segment comprises a constellation of a total of 30 MEO satellites in 3
orbital planes inclined at 56 degrees at 23616 km altitude. For the
simulation, the satellites were assumed to be equally spaced in each plane. The
satellites in the second and third planes where advanced by 12 and 24 degrees
in mean anomaly with respect to the first plane. The relative orientation of the
GALILEO constellation with respect to the GPS constellation has not yet been
determined and due to the different inclination and orbital radius of the two
systems, will not be constant over time either [38]. As a
result, the ascending nodes of the three orbital planes of the simulated
GALILEO constellation were arbitrarily assigned right ascensions of 0, 120, and
240 degrees respectively. For fair comparison, the real GPS constellation
consisting of 30 satellites as it existed at the start of GPS week 1430 was
used. GPS is officially a 24 satellite constellation but has had on the order
of 30 satellites for the past several years. At the time of writing, there were
32 GPS satellites but a 30 GPS satellite constellation was chosen because
previous studies have shown that small differences in the number of satellites
(e.g., between 24 and 32) does not greatly affect the results in terms of
positioning accuracy or ambiguity resolution. However, differing numbers of
satellites can lead to conclusions about one constellation providing better
performance than another and in a vast majority of cases the constellation with
more satellites provides better performance [23, 39] . To
distinguish the two constellations, the Galileo satellites have been
arbitrarily assigned the numbers 36 through 65. A combined constellation
is shown in Figure 1.Figure 1
Combined GPS and GALILEO constellation on
September 30, 2007 (GPS PRN: 1~32 (noninclusive), GALILEO PRN: 36~65).


##### 3.1.3. Masking Environment

It is assumed that the satellite-borne
antennas are facing in the radial direction, have hemispheric gain patterns and
an unobstructed view of the sky. An isotropic mask angle of 5 degrees is
assumed to limit use of low elevation signals that are likely to be corrupted
by large multipath (from other surfaces on the satellite, e.g.).


##### 3.1.4. User Motion and Orbit Descriptions

The simulated users are two
low earth orbiting (LEO) satellites in either highly inclined (near polar) or moderately
inclined orbits. Two typical LEO users were modelled after two existing LEO
satellites. ENVISAT, a European space agency earth observation satellite was
selected as a model for a polar orbiting LEO. The satellite is in a near-polar
sun-synchronous orbit with an orbital period of 101-minutes and an altitude of
approximately 790 km.
The International Space Station (ISS) was selected as an example of a LEO in a
moderately inclined orbit.  The Keplerian
elements used for each satellite and some other information are given in Table3. It
should be noted that these orbital elements are approximate values taken from
public sources and are neither synchronized with each other nor with the
simulated GNSS constellations. They are meant to be representative of each type
of user on an arbitrary day and time. Typical ground tracks for the two
satellites as well as the ground tracks for the GPS and Galileo satellites are
shown in Figures 2 and 3. Note
that the GNSS satellites are in medium earth orbit, far above the user LEOs. In
the two figures, the blue portions of the GNSS satellite ground tracks indicate
periods when they are in the field of view of the LEO and are being used in the
solution.Table 3
Keplerian elements and
other orbital information used to simulate LEO users.
ENVISATISSSemimajor
Axis (a)7155 km6712.5 kmEccentricity
(e)0.00010300.0001202Inclination
(e)98.5°51.6°Right-ascension of ascending node (Ω)161.5°51.6°Argument of Perigee (ω)0°331.9°Mean anomaly263.19°28.22°Mean orbital altitude790 km338.7 kmPeriod101-minutes91.31-minutesFigure 2
Typical ground track for
ENVISAT shown in magenta with the Galileo constellation. The circle containing the black cross indicates the initial ENVISAT location (~80°S, ~60°W). The GNSS satellites are labelled by PRN number. The
blue portion of their ground tracks indicates the GNSS satellite is being used
in the solution.Figure 3
Typical ground track for
the ISS shown in magenta with the combined GPS + Galileo constellation. The circle containing the black cross indicates the initial ISS
location (~0°N, ~50°E). The GNSS satellites are labelled by PRN number. The
blue portion of their ground tracks indicates the GNSS satellite is being used
in the solution.


##### 3.1.5. Simulation Time and Data Rates

The trajectories of the two
user LEOs and the two GNSS constellations were modelled for an arbitrary 24-hour period. In this 24-hour period, filtered geometry-based LAMBDA and
cascading ambiguity resolution schemes were implemented with a reset interval
of one-minute such that 1440 intervals were evaluated. This was done to create
many samples of the initial convergence phase of the filter, which is of
interest in terms of ambiguity resolution performance. A data rate of one
observation per ten seconds (0.1 Hz) was assumed for the purposes of evaluating
ambiguity resolution methods.At the beginning of each
one-minute interval, the Kalman filters used in each method were reset and
re-dimensioned to handle the number of satellites that were above the elevation
mask at the start of the interval and were above the elevation mask at the end
of the interval. This was done to avoid having to introduce or remove ambiguity
states part way though the one-minute filtering interval.


### 4. Data Analysis and Results

Results from using Galileo alone
are now presented and discussed followed by a section where the same
simulations are repeated but with GPS and Galileo being used together. Each
section begins with satellite availability and dilution of precision results,
followed by 1, 2, and 3 frequency geometry-based cascading probability of
correct fix results, followed by 1, 2, and 3 frequency geometry-based LAMBDA
probability of correct fix results. The first section concludes with a
discussion of fixed ambiguity baseline solution estimates and corresponding
attitude angle error estimates and these results can be generalized to the dual
GNSS case.


#### 4.1. Galileo Results

##### 4.1.1. Satellite Availability and Dilution of Precision

Figure4 shows
a plot of the number of available satellites and the corresponding HDOP and
VDOP values as a function of time for ENVISAT. Results for the ISS were very
similar, and have been omitted to save space. The main difference between the
two is that the VDOP of the polar orbiting ENVISAT tests tends to be worse as
the satellite travels over the poles due to the lack of overhead GNSS
satellites in the polar regions.Availability, HDOP, and
VDOP of Galileo as viewed from a polar orbiting LEO (ENVISAT) over a 24 hour
period.
(a)(b)(c)As expected, VDOP values
are larger than HDOP. This is due the geometrical distribution of satellites in
GNSS and the fact that observability of the vertical direction is limited by
the correlation between the vertical position state and the receiver clock
offset. This effect remains despite the elimination of the clock offset term in
the double-differencing process. A major result of this is that generally the
vertical baseline component will be poorer than the horizontal components,
making pitch and roll more difficult to estimate than azimuth when receiver
antennas are configured in the horizontal plane.The DOP values are also
periodic with the orbital period of the LEO as can be seen by comparing the DOP
time series to the latitude of the LEO shown in Figure5. This
is especially true for polar satellites since when they are over the poles,
they will only see GNSS satellites on the horizon around them (since GNSS
satellites are generally in 54 to 56 degree inclined orbits). As a result,
baseline solutions and corresponding attitude estimation quality will vary
periodically with the LEO orbit period.Figure 5
Latitude of ENVISAT as a
function of time.


##### 4.1.2. Cascading with Galileo

Results of the
geometry-based cascading schemes are now presented. A single filtered 1-minute
interval (of the 1440 1-minute intervals simulated) consisting of 6 simulated
observations occurring at time = 0, 10, 20, 30, 40, and 50 seconds is presented
in detail first. The results for all 1440 trials are then shown together. An
arbitrary 1-minute segment for the ENVISAT was selected. For this, and every 1-minute segment, the following procedure is used. First, the assumption is made
that the system is warm started meaning that the receiver has approximate
coordinates for itself and has acquired and is tracking all visible satellite
before the start of the 1-minute segment. At the beginning of the 1-minute
segment, the “rover” receiver in the two receiver pair estimates its position
with respect to the “base” receiver to an accuracy of 1 m (1σ) in
each dimension in the satellite body frame. This initial value is based on
typical differential code positioning accuracy. A Kalman filter is then
initialized with this initial estimate for the antenna position, and an initial
ambiguity estimate variance that corresponds also to 1 m (but is expressed in
terms of cycles).The combined position and
float ambiguity filter is then updated once every 10 seconds with a single code
observation from each satellite, and a phase observation for each frequency being
used from each satellite for the particular scenario. E1/E2 phase observations
are used to estimate both an E1/E2 ambiguity and WL, E5b is used to estimate
the WL and the EWL, and E5a contributes only the EWL.  Due to the short baseline length and fixed
nature of the antennas, the position (baseline component) states are modelled
as a random walk processes with a process noise of 0.01m2/s (about
a mean value determined from an existing orbital and attitude model for the
satellite) and ambiguity states are modelled as random constant processes.
After each Kalman update, the covariance matrix of the ambiguities is analysed
to determine the lower bound on the probability of correct fix of the full set
of ambiguities using the technique described in Section 2.2.The Kalman covariance
update equation is given by(14)Pk+=Pk−−Pk−HkT[HkPk−HkT+Rk]−1HkPk−, where Hk is the design matrix, Rk is the measurement error covariance matrix, and Pk− and Pk+ are the covariance matrices of the errors of
the states before and after the update respectively. Note that for simplicity
the Kalman gain matrix is included in, but not shown as a separate quantity in (14). The prediction step assumes that the baseline components, and thus also
the direction cosines in H are expressed in terms of the local level
frame of the satellite, thus the transition matrix of the system is an identity
matrix and is not shown. The Kalman covariance prediction equation is simply(15)Pk−=Pk−1++Q, where Q contains process noise for the baseline
component states and is zeros in all rows and columns corresponding the
ambiguity states.For the cascading technique,
the probability of partially fixing each step (the EWL alone, the EWL and WL,
and all three) is also evaluated. As a final step, the estimated covariance of
the fixed position states is computed using (7). The results of this can then be used to compute the estimated
accuracy of the derived attitude angles.Figure6 shows
the overall probability of incorrect fix upper bounds for single-, dual-, and
triple-frequency cascading (over one 1-minute interval). To interpret this
figure, consider the red line near the top. This is the probability of
incorrectly resolving the E1/E2 ambiguity as a function of time in the case
that only a single-frequency receiver is used. The line decreases as a function
of time indicating increasing odds of correct ambiguity resolution.  However, the dual- and triple-frequency
cascading options, shown by the green and blue lines, respectively, show that
dual and triple-frequency cascading methods offer several orders of magnitude
improvement. The y-axis is logarithmically scaled and for example 10−4 indicates a probability of correct fix of 99.99%. Note that these are the
probabilities of fixing all of the ambiguities, and in dual- and triple-frequency cases there are two and three times as many ambiguities to fix. But
even though there are more ambiguities to fix, the ability to cascade substantially
increases the chances of correctly resolving the ambiguities.Figure 6
Sample Probability of Incorrect Fix upper bounds for single-frequency, dual and triple-frequency cascading for ENVISAT.Figure7 shows
the probability of incorrect fix for each stage of the cascading process for
the triple-frequency case. The blue line represents PIF for the EWL ambiguities
as a function of time. Note that resolving the EWL ambiguities is more or less
assured. Though it is not visible, there is a green line plotted almost
directly under the red line. This green line represents the probability of
correctly resolving the WL after resolving the EWL. The reason it is not visible
is that resolving the WL more or less ensures resolving the E1/E2 ambiguity (in
the short baseline for attitude determination case) meaning that the
probability of resolving the E1/E2 ambiguity after resolving with WL is close
to unity. The result is that the probability of resolving all three is only
very slightly less than the probability of resolving the EWL and the WL alone.
Note that the red line, representing the probability of resolving the E1/E2
ambiguities after resolving the EWL and WL is identical to the probability of
resolving all of the ambiguities in triple-frequency cascading which is plotted
as the blue line in Figure 6.Figure 7
Sample Probability of
Incorrect Fix upper bounds for the EWL, WL, and E1/E2 ambiguities in triple-frequency cascading for ENVISAT.Figure8 shows
the corresponding float solution baseline component accuracies as a function of
time for the three scenarios shown in Figure 6. The
three baseline component labels, E, N, and h, correspond to the longitude,
latitude, and vertical direction in the local level frame of the spacecraft. Note that the scale of the vertical subplot is double
those of the two horizontal components corresponding the decreased accuracy
provided by GNSS in the vertical direction. Note how float solution is
relatively poor, on the order of 10 to 20 cm. In this regime, attitude determination on
a short baseline would not be feasible.Sample estimated float solution accuracy for single, dual and
triple-frequency cascading. E, N, h indicate
longitude, latitude and vertical components in the local level frame of the
spacecraft.
(a)(b)(c)Figure9 shows the corresponding baseline accuracy components for the fixed
solution. Note that the accuracy is more or less constant as a function of
time. These estimated accuracies are the accuracies assuming the ambiguities
have been correctly resolved and this figure does not take into account the
fact that at the beginning of the filtering interval the likelihood, for
example, of correctly resolving the E1/E2 ambiguity with a single-frequency
receiver is quite low. However, this accuracy level does represent the steady
state assuming the ambiguities have been correctly resolved and is useful as
the attitude accuracy can be derived directly from this baseline accuracy. For
small errors, such as those shown in Figure 9, and a simple antenna configuration with two or three antennas in
the horizontal plane, the corresponding attitude errors expressed in radians
are well approximated by the ratio of the horizontal positioning accuracy to
baseline length for azimuth, and by the ratio of the vertical error to baseline
length for the roll and pitch errors.Sample estimated fixed
solution accuracy for single, dual, and triple-frequency cascading.E, N, h indicate longitude, latitude, and vertical components in the
local level frame of the spacecraft.
(a)(b)(c)Figure10 shows
the PIF as a function of time for single, dual and triple-frequency cascading
for ENVISAT. All 1440 simulations are plotted on one figure to demonstrate the
range of PIF for varying LEO to Galileo satellite geometry. As can be seen, the
sample result shown in Figure 6 was
in fact one of the best cases and even with three frequencies, the PIF can still
be relatively high (e.g., only 10%) after a minute of filtering. Also, the
larger performance gain is from one frequency to two, with all of the PIF
samples for the single-frequency case being worse than any of the dual or
triple-frequency samples whereas there is some overlap between good dual-frequency results and relatively poor triple-frequency results. Figure 11 shows
all of the samples of EWL, WL and E1/E2 PIF for the triple-frequency case as
discussed earlier in the context of Figure 7.
Again, the sample result shown earlier is one of the better samples. From Figure 11 it can be seen that the probability of resolving the EWL in a
triple-frequency cascading scheme is very high and very consistent between
samples while there is more variability between the probabilities of resolving
the WL in the second step in the cascading scheme. It can also be seen that
there is little difference in the probability of resolving the WL and the
probability of resolving all three, indicating again that once the widelanes
are resolved in triple-frequency cascading, resolving the E1/E2 ambiguities is
relatively easy.Figure 10
Probability of Incorrect fix upper bounds for single-frequency, dual and triple-frequency cascading for ENVISAT.Figure 11
Probability of Incorrect
Fix upper bounds for the EWL, WL, and E1/E2 ambiguities in triple-frequency
cascading for ENVISAT.


##### 4.1.3. Lambda with Galileo

With the LAMBDA method, as
opposed to cascading, all of the ambiguity states are estimated together and
ambiguity resolution is facilitated by the LAMBDA algorithm deciding what
linear combination of the original ambiguities most decorrelates the ambiguities
thus making them easiest to fix. In terms of the covariance simulation, the
same Kalman filter equations are used, only in this case the design matrix
reflects the fact that the original ambiguities (and not widelanes) are being
estimated. The result of the float solution is then sent to the LAMBDA
algorithm for decorrelation.Figure12 shows
the PIF results for the LAMBDA method and can be compared directly the
corresponding cascading results shown in Figure 10.  Note that there is improvement in all three
cases (single-, dual-, and triple-frequency) but most notably the triple-frequency ambiguity resolution PIF (the blue lines) is significantly reduced
compared to cascading demonstrating the potential of the LAMBDA method to find
optimal linear combinations of the ambiguities that provide more ambiguity
resolution potential compared the EWL, WL, and E1/E2 used in the cascading
method.Figure 12
Probability of Incorrect Fix upper bounds for single-,
dual- and triple-frequency LAMBDA for ENVISAT.One important result to
note is that less is gained by adding a third frequency in terms of ambiguity
resolution for very short baselines. In other words, there is a larger change in
PIF from the single-frequency case to the dual-frequency case than there is
from the dual- to triple-frequency case. The addition of the second frequency
allows the LAMBDA algorithm to automatically form a widelane combination, which
for a short baseline such as this, is more than sufficient for ambiguity
resolution. The addition of the third frequency provides only a marginal
improvement as the code error in this case is smaller that the extra-wide lane
wavelength and there is no differential ionosphere error affecting either the code
or phase measurements in this application. This is similar to the results of
using cascading (found in the TCAR/CIR literature) where the largest gains from
using three frequencies are found in longer baseline applications. A final note
about the LAMBDA results is that  close inspection shows a small number of dual and
triple-frequency trials where the PIF appears to increase counter-intuitively
from one epoch to the next. This is not an error. An unfortunate feature using
the bootstrapped lower bound for the PCF as an estimate of the integer least
squares PCF is that the lower bound is not invariant to the decorrelating
linear combination found using LAMBDA. Occasionally, the LAMBDA algorithm
changes its linear combination from one filter epoch to the next and this can
result in discontinuities, both decreasing and increasing, in the PCF lower
bound. It should be noted that the actual PCF does not change, only value of
the bound.


##### 4.2. GPS/Galileo Results

In this section, the
results presented for Galileo are now repeated for the case that both GPS and
Galileo are being used.


##### 4.2.1. Availability and Dilution of Precision

With two systems, the
number of satellites observed roughly doubles, as shown in Figure13. The
corresponding DOP values also decrease. As in the Galileo only case, the
largest (poorest) DOP values occur when the fewest satellites are tracked and
the results are periodic with the orbital period of the LEO satellite.Availability, HDOP, and
VDOP of GPS/Galileo as viewed from a polar orbiting LEO (ENVISAT) over a 24-hour period.
(a)(b)(c)


##### 4.2.2. Cascading with GPS/Galileo

Figure14 shows
the dual GNSS probabilities of incorrect fix bounds for cascading for ENVISAT
analogous to those presented in Figure 10 for
the Galileo only case. The same patterns are observed, though with roughly
double the number of observations and double the number of ambiguities to
estimate, the performance improvement is only marginal.  Likewise, the probability of successfully
resolving each stage in the three-frequency case, shown in Figure 15, is
only marginally better than the results for Galileo only shown in Figure 11.Figure 14
Probability of Incorrect
Fix upper bounds for single-frequency, dual and triple-frequency cascading for
ENVISAT using two GNSS.Figure 15
Probability of Incorrect
Fix upper bounds for the EWL, WL, and E1/E2 ambiguities in triple-frequency
cascading for ENVISAT using two GNSS.


##### 4.2.3. Lambda with GPS/Galileo

Use of the LAMBDA method
with two systems results in a major improvement in the probability of correct
fix than can be obtained with a single-frequency receiver. Though the dual- and
triple-frequency cases are also improved with LAMBDA, the significant
improvement can be seen in the red lines in Figure16 when
compared to the Galileo only case shown in Figure 12. Most
striking is the absence of cases where the probability of incorrect fix hovers
near the top of the figures (in the 0.1 to 1.0 range). This can be explained by
the fact that the LAMBDA method is able to exploit the geometric diversity of
the single-frequency signals from the two systems by forming linear
combinations of all of the L1 and E1/E2 ambiguities while in the Galileo only
case, approximately half as many ambiguities are available. There is less
improvement with the dual and triple-frequency cases because these cases
already have a diversity of ambiguities on different frequencies for the LAMBDA
algorithm work with even in the Galileo only case.Figure 16
Probability of Incorrect
Fix upper bounds for single-frequency, dual and triple-frequency LAMBDA for
ENVISAT using two GNSS.


#### 4.3. Galileo versus GPS/Galileo: Recommendations

Based on these results it
can be concluded that the LAMBDA approach is superior to the cascading approach
in all cases. However the LAMBDA method is particularly useful in the case of a
single-frequency dual-GNSS application. However to obtain the most reliable
ambiguity resolution, multiple frequencies are required when using both one or
two GNSS constellations. If power and weight of the GNSS receiver payload is no
object, a dual-system triple-frequency receiver using LAMBDA will provide the
best performance. However, if power and weight are an issue, a dual-frequency
single-system receiver offers similar performance to a single-frequency
dual-system one.  Similar results were
obtained with the ISS as the simulated user and are therefore not shown.While these results
consider the case of attitude determination for an orbiting user, they can be
generalized to ground based users. Of course while the absolute performance of
each algorithm will depend on measurement accuracies, satellite geometry, and
user dynamics, it is reasonable to expect the relative performance of the
algorithms to remain unchanged.


## 3.1. Simulation Parameters

The signal characteristics
of modernized GPS and Galileo have been investigated in great detail by many
researchers [34]. Only
the GPS civilian signals and Galileo open service (OS) signals will be
considered in this study, their characteristics are listed in Table 1.Note that the open services
of modernized GPS and Galileo have two frequencies in common, but the
third frequency on each is unique. Because of this, it could be possible to form
double-difference observations between the two systems using L1 and E1/E2 phase
measurements and also L5 and E5a [6].
However, this is not possible for L2 and E5b. Double-differencing between
systems was not conducted in this study and for all three frequencies a base
satellite was assigned for each system.No official documents on the range accuracy of future GPS and Galileo
signals have been released but several studies quantify the signal performances
of future GNSS systems (e.g., [35]). The code and carrier observations are
affected by systematic errors and random noise. However, for attitude
determination applications, where the baseline is very short, the only error
sources that will not be completely cancelled by differencing are the code and
phase multipath and the receiver noise. Code multipath and noise depend on the
structure of the code and the design or the receiver. Phase noise can be
assumed to be a few percent of a carrier cycle for all of the signals and
similarly phase multipath can be shown to have a maximum amplitude of one
quarter of the wavelength and typical values of less than a few centimetres [36].


### 3.1.1. Frequency Combinations

In all of the cascading methods
described earlier only a single code measurement is used for each satellite. In
principle, this should be the code measurement with the smallest measurement
noise and the smallest multipath. The same can be said for the LAMBDA-based
methods. In principle, it is possible to use multiple code observations in a
geometry-based least-squares or filtered LAMBDA solution, but in practice,
little is gained from this, particularly in short baselines where there are no
residual ionospheric errors.In the geometry-based LAMBDA and
cascading simulations presented in what follows, a single code observation from each
satellite is used in conjunction with one, two or three phase measurements from
each satellite. The following three frequency combinations are assessed.Single-Frequency
In this case, only L1 and E1/E2
code and phase measurements are used. LAMBDA is applied to decorrelate the
estimated ambiguities and obtain PCF estimates. This is contrasted with what
could be called “single-frequency cascading” or simply single-frequency
ambiguity resolution without the application of LAMBDA. In both cases, the
total number of double-difference ambiguities estimated is(nGPS−1)+(nGal−1) where nGPS and nGal are the number of GPS and Galileo satellites
tracked, respectively.Dual-Frequency
In the dual-frequency case, L2
and E5b phase observations are added to the single-frequency case described
earlier. In both the LAMBDA and cascading scenarios this doubles the number of
ambiguities to be estimated to2×[(nGPS−1)+(nGal−1)]. With LAMBDA, the ambiguities
consist of some linear combination of all the ambiguities on each frequency (as
determined by the algorithm) while with cascading the ambiguities are
specifically the Widelane ambiguities (which are resolved first) followed by
the L1 and E1/E2 ambiguities.Triple-Frequency
Finally, for the triple-frequency
case, the number of ambiguities increases again to3×[(nGPS−1)+(nGal−1)].
Again, for LAMBDA, the ambiguities being estimated are some linear combination
of the ambiguities of the three frequencies while for triple-frequency
cascading the extra-widelane (EWL) ambiguities are first resolved, followed by
the widelane (WL) ambiguities, and finally the L1 and E1/E2 ambiguities.
In both the cascaded and LAMBDA cases, the same input covariance
matrix of the ambiguities is used, however, in the LAMBDA cases the LAMBDA
algorithm is allowed to determine the optimal decorrelating transformation while
in the cascading cases a fixed set of linear combinations (EWL, WL, and L1) are
used.


#### 3.1.2. Constellations

According to the Galileo
Mission High Level Definition document [37], the
space segment comprises a constellation of a total of 30 MEO satellites in 3
orbital planes inclined at 56 degrees at 23616 km altitude. For the
simulation, the satellites were assumed to be equally spaced in each plane. The
satellites in the second and third planes where advanced by 12 and 24 degrees
in mean anomaly with respect to the first plane. The relative orientation of the
GALILEO constellation with respect to the GPS constellation has not yet been
determined and due to the different inclination and orbital radius of the two
systems, will not be constant over time either [38]. As a
result, the ascending nodes of the three orbital planes of the simulated
GALILEO constellation were arbitrarily assigned right ascensions of 0, 120, and
240 degrees respectively. For fair comparison, the real GPS constellation
consisting of 30 satellites as it existed at the start of GPS week 1430 was
used. GPS is officially a 24 satellite constellation but has had on the order
of 30 satellites for the past several years. At the time of writing, there were
32 GPS satellites but a 30 GPS satellite constellation was chosen because
previous studies have shown that small differences in the number of satellites
(e.g., between 24 and 32) does not greatly affect the results in terms of
positioning accuracy or ambiguity resolution. However, differing numbers of
satellites can lead to conclusions about one constellation providing better
performance than another and in a vast majority of cases the constellation with
more satellites provides better performance [23, 39] . To
distinguish the two constellations, the Galileo satellites have been
arbitrarily assigned the numbers 36 through 65. A combined constellation
is shown in Figure 1.Figure 1
Combined GPS and GALILEO constellation on
September 30, 2007 (GPS PRN: 1~32 (noninclusive), GALILEO PRN: 36~65).


#### 3.1.3. Masking Environment

It is assumed that the satellite-borne
antennas are facing in the radial direction, have hemispheric gain patterns and
an unobstructed view of the sky. An isotropic mask angle of 5 degrees is
assumed to limit use of low elevation signals that are likely to be corrupted
by large multipath (from other surfaces on the satellite, e.g.).


#### 3.1.4. User Motion and Orbit Descriptions

The simulated users are two
low earth orbiting (LEO) satellites in either highly inclined (near polar) or moderately
inclined orbits. Two typical LEO users were modelled after two existing LEO
satellites. ENVISAT, a European space agency earth observation satellite was
selected as a model for a polar orbiting LEO. The satellite is in a near-polar
sun-synchronous orbit with an orbital period of 101-minutes and an altitude of
approximately 790 km.
The International Space Station (ISS) was selected as an example of a LEO in a
moderately inclined orbit.  The Keplerian
elements used for each satellite and some other information are given in Table3. It
should be noted that these orbital elements are approximate values taken from
public sources and are neither synchronized with each other nor with the
simulated GNSS constellations. They are meant to be representative of each type
of user on an arbitrary day and time. Typical ground tracks for the two
satellites as well as the ground tracks for the GPS and Galileo satellites are
shown in Figures 2 and 3. Note
that the GNSS satellites are in medium earth orbit, far above the user LEOs. In
the two figures, the blue portions of the GNSS satellite ground tracks indicate
periods when they are in the field of view of the LEO and are being used in the
solution.Table 3
Keplerian elements and
other orbital information used to simulate LEO users.
ENVISATISSSemimajor
Axis (a)7155 km6712.5 kmEccentricity
(e)0.00010300.0001202Inclination
(e)98.5°51.6°Right-ascension of ascending node (Ω)161.5°51.6°Argument of Perigee (ω)0°331.9°Mean anomaly263.19°28.22°Mean orbital altitude790 km338.7 kmPeriod101-minutes91.31-minutesFigure 2
Typical ground track for
ENVISAT shown in magenta with the Galileo constellation. The circle containing the black cross indicates the initial ENVISAT location (~80°S, ~60°W). The GNSS satellites are labelled by PRN number. The
blue portion of their ground tracks indicates the GNSS satellite is being used
in the solution.Figure 3
Typical ground track for
the ISS shown in magenta with the combined GPS + Galileo constellation. The circle containing the black cross indicates the initial ISS
location (~0°N, ~50°E). The GNSS satellites are labelled by PRN number. The
blue portion of their ground tracks indicates the GNSS satellite is being used
in the solution.


#### 3.1.5. Simulation Time and Data Rates

The trajectories of the two
user LEOs and the two GNSS constellations were modelled for an arbitrary 24-hour period. In this 24-hour period, filtered geometry-based LAMBDA and
cascading ambiguity resolution schemes were implemented with a reset interval
of one-minute such that 1440 intervals were evaluated. This was done to create
many samples of the initial convergence phase of the filter, which is of
interest in terms of ambiguity resolution performance. A data rate of one
observation per ten seconds (0.1 Hz) was assumed for the purposes of evaluating
ambiguity resolution methods.At the beginning of each
one-minute interval, the Kalman filters used in each method were reset and
re-dimensioned to handle the number of satellites that were above the elevation
mask at the start of the interval and were above the elevation mask at the end
of the interval. This was done to avoid having to introduce or remove ambiguity
states part way though the one-minute filtering interval.


## 3.1.1. Frequency Combinations

In all of the cascading methods
described earlier only a single code measurement is used for each satellite. In
principle, this should be the code measurement with the smallest measurement
noise and the smallest multipath. The same can be said for the LAMBDA-based
methods. In principle, it is possible to use multiple code observations in a
geometry-based least-squares or filtered LAMBDA solution, but in practice,
little is gained from this, particularly in short baselines where there are no
residual ionospheric errors.In the geometry-based LAMBDA and
cascading simulations presented in what follows, a single code observation from each
satellite is used in conjunction with one, two or three phase measurements from
each satellite. The following three frequency combinations are assessed.Single-Frequency
In this case, only L1 and E1/E2
code and phase measurements are used. LAMBDA is applied to decorrelate the
estimated ambiguities and obtain PCF estimates. This is contrasted with what
could be called “single-frequency cascading” or simply single-frequency
ambiguity resolution without the application of LAMBDA. In both cases, the
total number of double-difference ambiguities estimated is(nGPS−1)+(nGal−1) where nGPS and nGal are the number of GPS and Galileo satellites
tracked, respectively.Dual-Frequency
In the dual-frequency case, L2
and E5b phase observations are added to the single-frequency case described
earlier. In both the LAMBDA and cascading scenarios this doubles the number of
ambiguities to be estimated to2×[(nGPS−1)+(nGal−1)]. With LAMBDA, the ambiguities
consist of some linear combination of all the ambiguities on each frequency (as
determined by the algorithm) while with cascading the ambiguities are
specifically the Widelane ambiguities (which are resolved first) followed by
the L1 and E1/E2 ambiguities.Triple-Frequency
Finally, for the triple-frequency
case, the number of ambiguities increases again to3×[(nGPS−1)+(nGal−1)].
Again, for LAMBDA, the ambiguities being estimated are some linear combination
of the ambiguities of the three frequencies while for triple-frequency
cascading the extra-widelane (EWL) ambiguities are first resolved, followed by
the widelane (WL) ambiguities, and finally the L1 and E1/E2 ambiguities.
In both the cascaded and LAMBDA cases, the same input covariance
matrix of the ambiguities is used, however, in the LAMBDA cases the LAMBDA
algorithm is allowed to determine the optimal decorrelating transformation while
in the cascading cases a fixed set of linear combinations (EWL, WL, and L1) are
used.


### 3.1.2. Constellations

According to the Galileo
Mission High Level Definition document [37], the
space segment comprises a constellation of a total of 30 MEO satellites in 3
orbital planes inclined at 56 degrees at 23616 km altitude. For the
simulation, the satellites were assumed to be equally spaced in each plane. The
satellites in the second and third planes where advanced by 12 and 24 degrees
in mean anomaly with respect to the first plane. The relative orientation of the
GALILEO constellation with respect to the GPS constellation has not yet been
determined and due to the different inclination and orbital radius of the two
systems, will not be constant over time either [38]. As a
result, the ascending nodes of the three orbital planes of the simulated
GALILEO constellation were arbitrarily assigned right ascensions of 0, 120, and
240 degrees respectively. For fair comparison, the real GPS constellation
consisting of 30 satellites as it existed at the start of GPS week 1430 was
used. GPS is officially a 24 satellite constellation but has had on the order
of 30 satellites for the past several years. At the time of writing, there were
32 GPS satellites but a 30 GPS satellite constellation was chosen because
previous studies have shown that small differences in the number of satellites
(e.g., between 24 and 32) does not greatly affect the results in terms of
positioning accuracy or ambiguity resolution. However, differing numbers of
satellites can lead to conclusions about one constellation providing better
performance than another and in a vast majority of cases the constellation with
more satellites provides better performance [23, 39] . To
distinguish the two constellations, the Galileo satellites have been
arbitrarily assigned the numbers 36 through 65. A combined constellation
is shown in Figure 1.Figure 1
Combined GPS and GALILEO constellation on
September 30, 2007 (GPS PRN: 1~32 (noninclusive), GALILEO PRN: 36~65).


### 3.1.3. Masking Environment

It is assumed that the satellite-borne
antennas are facing in the radial direction, have hemispheric gain patterns and
an unobstructed view of the sky. An isotropic mask angle of 5 degrees is
assumed to limit use of low elevation signals that are likely to be corrupted
by large multipath (from other surfaces on the satellite, e.g.).


### 3.1.4. User Motion and Orbit Descriptions

The simulated users are two
low earth orbiting (LEO) satellites in either highly inclined (near polar) or moderately
inclined orbits. Two typical LEO users were modelled after two existing LEO
satellites. ENVISAT, a European space agency earth observation satellite was
selected as a model for a polar orbiting LEO. The satellite is in a near-polar
sun-synchronous orbit with an orbital period of 101-minutes and an altitude of
approximately 790 km.
The International Space Station (ISS) was selected as an example of a LEO in a
moderately inclined orbit.  The Keplerian
elements used for each satellite and some other information are given in Table3. It
should be noted that these orbital elements are approximate values taken from
public sources and are neither synchronized with each other nor with the
simulated GNSS constellations. They are meant to be representative of each type
of user on an arbitrary day and time. Typical ground tracks for the two
satellites as well as the ground tracks for the GPS and Galileo satellites are
shown in Figures 2 and 3. Note
that the GNSS satellites are in medium earth orbit, far above the user LEOs. In
the two figures, the blue portions of the GNSS satellite ground tracks indicate
periods when they are in the field of view of the LEO and are being used in the
solution.Table 3
Keplerian elements and
other orbital information used to simulate LEO users.
ENVISATISSSemimajor
Axis (a)7155 km6712.5 kmEccentricity
(e)0.00010300.0001202Inclination
(e)98.5°51.6°Right-ascension of ascending node (Ω)161.5°51.6°Argument of Perigee (ω)0°331.9°Mean anomaly263.19°28.22°Mean orbital altitude790 km338.7 kmPeriod101-minutes91.31-minutesFigure 2
Typical ground track for
ENVISAT shown in magenta with the Galileo constellation. The circle containing the black cross indicates the initial ENVISAT location (~80°S, ~60°W). The GNSS satellites are labelled by PRN number. The
blue portion of their ground tracks indicates the GNSS satellite is being used
in the solution.Figure 3
Typical ground track for
the ISS shown in magenta with the combined GPS + Galileo constellation. The circle containing the black cross indicates the initial ISS
location (~0°N, ~50°E). The GNSS satellites are labelled by PRN number. The
blue portion of their ground tracks indicates the GNSS satellite is being used
in the solution.


### 3.1.5. Simulation Time and Data Rates

The trajectories of the two
user LEOs and the two GNSS constellations were modelled for an arbitrary 24-hour period. In this 24-hour period, filtered geometry-based LAMBDA and
cascading ambiguity resolution schemes were implemented with a reset interval
of one-minute such that 1440 intervals were evaluated. This was done to create
many samples of the initial convergence phase of the filter, which is of
interest in terms of ambiguity resolution performance. A data rate of one
observation per ten seconds (0.1 Hz) was assumed for the purposes of evaluating
ambiguity resolution methods.At the beginning of each
one-minute interval, the Kalman filters used in each method were reset and
re-dimensioned to handle the number of satellites that were above the elevation
mask at the start of the interval and were above the elevation mask at the end
of the interval. This was done to avoid having to introduce or remove ambiguity
states part way though the one-minute filtering interval.


## 3.1.2. Constellations

According to the Galileo
Mission High Level Definition document [37], the
space segment comprises a constellation of a total of 30 MEO satellites in 3
orbital planes inclined at 56 degrees at 23616 km altitude. For the
simulation, the satellites were assumed to be equally spaced in each plane. The
satellites in the second and third planes where advanced by 12 and 24 degrees
in mean anomaly with respect to the first plane. The relative orientation of the
GALILEO constellation with respect to the GPS constellation has not yet been
determined and due to the different inclination and orbital radius of the two
systems, will not be constant over time either [38]. As a
result, the ascending nodes of the three orbital planes of the simulated
GALILEO constellation were arbitrarily assigned right ascensions of 0, 120, and
240 degrees respectively. For fair comparison, the real GPS constellation
consisting of 30 satellites as it existed at the start of GPS week 1430 was
used. GPS is officially a 24 satellite constellation but has had on the order
of 30 satellites for the past several years. At the time of writing, there were
32 GPS satellites but a 30 GPS satellite constellation was chosen because
previous studies have shown that small differences in the number of satellites
(e.g., between 24 and 32) does not greatly affect the results in terms of
positioning accuracy or ambiguity resolution. However, differing numbers of
satellites can lead to conclusions about one constellation providing better
performance than another and in a vast majority of cases the constellation with
more satellites provides better performance [23, 39] . To
distinguish the two constellations, the Galileo satellites have been
arbitrarily assigned the numbers 36 through 65. A combined constellation
is shown in Figure 1.Figure 1
Combined GPS and GALILEO constellation on
September 30, 2007 (GPS PRN: 1~32 (noninclusive), GALILEO PRN: 36~65).


## 3.1.3. Masking Environment

It is assumed that the satellite-borne
antennas are facing in the radial direction, have hemispheric gain patterns and
an unobstructed view of the sky. An isotropic mask angle of 5 degrees is
assumed to limit use of low elevation signals that are likely to be corrupted
by large multipath (from other surfaces on the satellite, e.g.).


## 3.1.4. User Motion and Orbit Descriptions

The simulated users are two
low earth orbiting (LEO) satellites in either highly inclined (near polar) or moderately
inclined orbits. Two typical LEO users were modelled after two existing LEO
satellites. ENVISAT, a European space agency earth observation satellite was
selected as a model for a polar orbiting LEO. The satellite is in a near-polar
sun-synchronous orbit with an orbital period of 101-minutes and an altitude of
approximately 790 km.
The International Space Station (ISS) was selected as an example of a LEO in a
moderately inclined orbit.  The Keplerian
elements used for each satellite and some other information are given in Table3. It
should be noted that these orbital elements are approximate values taken from
public sources and are neither synchronized with each other nor with the
simulated GNSS constellations. They are meant to be representative of each type
of user on an arbitrary day and time. Typical ground tracks for the two
satellites as well as the ground tracks for the GPS and Galileo satellites are
shown in Figures 2 and 3. Note
that the GNSS satellites are in medium earth orbit, far above the user LEOs. In
the two figures, the blue portions of the GNSS satellite ground tracks indicate
periods when they are in the field of view of the LEO and are being used in the
solution.Table 3
Keplerian elements and
other orbital information used to simulate LEO users.
ENVISATISSSemimajor
Axis (a)7155 km6712.5 kmEccentricity
(e)0.00010300.0001202Inclination
(e)98.5°51.6°Right-ascension of ascending node (Ω)161.5°51.6°Argument of Perigee (ω)0°331.9°Mean anomaly263.19°28.22°Mean orbital altitude790 km338.7 kmPeriod101-minutes91.31-minutesFigure 2
Typical ground track for
ENVISAT shown in magenta with the Galileo constellation. The circle containing the black cross indicates the initial ENVISAT location (~80°S, ~60°W). The GNSS satellites are labelled by PRN number. The
blue portion of their ground tracks indicates the GNSS satellite is being used
in the solution.Figure 3
Typical ground track for
the ISS shown in magenta with the combined GPS + Galileo constellation. The circle containing the black cross indicates the initial ISS
location (~0°N, ~50°E). The GNSS satellites are labelled by PRN number. The
blue portion of their ground tracks indicates the GNSS satellite is being used
in the solution.


## 3.1.5. Simulation Time and Data Rates

The trajectories of the two
user LEOs and the two GNSS constellations were modelled for an arbitrary 24-hour period. In this 24-hour period, filtered geometry-based LAMBDA and
cascading ambiguity resolution schemes were implemented with a reset interval
of one-minute such that 1440 intervals were evaluated. This was done to create
many samples of the initial convergence phase of the filter, which is of
interest in terms of ambiguity resolution performance. A data rate of one
observation per ten seconds (0.1 Hz) was assumed for the purposes of evaluating
ambiguity resolution methods.At the beginning of each
one-minute interval, the Kalman filters used in each method were reset and
re-dimensioned to handle the number of satellites that were above the elevation
mask at the start of the interval and were above the elevation mask at the end
of the interval. This was done to avoid having to introduce or remove ambiguity
states part way though the one-minute filtering interval.


## 4. Data Analysis and Results

Results from using Galileo alone
are now presented and discussed followed by a section where the same
simulations are repeated but with GPS and Galileo being used together. Each
section begins with satellite availability and dilution of precision results,
followed by 1, 2, and 3 frequency geometry-based cascading probability of
correct fix results, followed by 1, 2, and 3 frequency geometry-based LAMBDA
probability of correct fix results. The first section concludes with a
discussion of fixed ambiguity baseline solution estimates and corresponding
attitude angle error estimates and these results can be generalized to the dual
GNSS case.


### 4.1. Galileo Results

#### 4.1.1. Satellite Availability and Dilution of Precision

Figure4 shows
a plot of the number of available satellites and the corresponding HDOP and
VDOP values as a function of time for ENVISAT. Results for the ISS were very
similar, and have been omitted to save space. The main difference between the
two is that the VDOP of the polar orbiting ENVISAT tests tends to be worse as
the satellite travels over the poles due to the lack of overhead GNSS
satellites in the polar regions.Availability, HDOP, and
VDOP of Galileo as viewed from a polar orbiting LEO (ENVISAT) over a 24 hour
period.
(a)(b)(c)As expected, VDOP values
are larger than HDOP. This is due the geometrical distribution of satellites in
GNSS and the fact that observability of the vertical direction is limited by
the correlation between the vertical position state and the receiver clock
offset. This effect remains despite the elimination of the clock offset term in
the double-differencing process. A major result of this is that generally the
vertical baseline component will be poorer than the horizontal components,
making pitch and roll more difficult to estimate than azimuth when receiver
antennas are configured in the horizontal plane.The DOP values are also
periodic with the orbital period of the LEO as can be seen by comparing the DOP
time series to the latitude of the LEO shown in Figure5. This
is especially true for polar satellites since when they are over the poles,
they will only see GNSS satellites on the horizon around them (since GNSS
satellites are generally in 54 to 56 degree inclined orbits). As a result,
baseline solutions and corresponding attitude estimation quality will vary
periodically with the LEO orbit period.Figure 5
Latitude of ENVISAT as a
function of time.


#### 4.1.2. Cascading with Galileo

Results of the
geometry-based cascading schemes are now presented. A single filtered 1-minute
interval (of the 1440 1-minute intervals simulated) consisting of 6 simulated
observations occurring at time = 0, 10, 20, 30, 40, and 50 seconds is presented
in detail first. The results for all 1440 trials are then shown together. An
arbitrary 1-minute segment for the ENVISAT was selected. For this, and every 1-minute segment, the following procedure is used. First, the assumption is made
that the system is warm started meaning that the receiver has approximate
coordinates for itself and has acquired and is tracking all visible satellite
before the start of the 1-minute segment. At the beginning of the 1-minute
segment, the “rover” receiver in the two receiver pair estimates its position
with respect to the “base” receiver to an accuracy of 1 m (1σ) in
each dimension in the satellite body frame. This initial value is based on
typical differential code positioning accuracy. A Kalman filter is then
initialized with this initial estimate for the antenna position, and an initial
ambiguity estimate variance that corresponds also to 1 m (but is expressed in
terms of cycles).The combined position and
float ambiguity filter is then updated once every 10 seconds with a single code
observation from each satellite, and a phase observation for each frequency being
used from each satellite for the particular scenario. E1/E2 phase observations
are used to estimate both an E1/E2 ambiguity and WL, E5b is used to estimate
the WL and the EWL, and E5a contributes only the EWL.  Due to the short baseline length and fixed
nature of the antennas, the position (baseline component) states are modelled
as a random walk processes with a process noise of 0.01m2/s (about
a mean value determined from an existing orbital and attitude model for the
satellite) and ambiguity states are modelled as random constant processes.
After each Kalman update, the covariance matrix of the ambiguities is analysed
to determine the lower bound on the probability of correct fix of the full set
of ambiguities using the technique described in Section 2.2.The Kalman covariance
update equation is given by(14)Pk+=Pk−−Pk−HkT[HkPk−HkT+Rk]−1HkPk−, where Hk is the design matrix, Rk is the measurement error covariance matrix, and Pk− and Pk+ are the covariance matrices of the errors of
the states before and after the update respectively. Note that for simplicity
the Kalman gain matrix is included in, but not shown as a separate quantity in (14). The prediction step assumes that the baseline components, and thus also
the direction cosines in H are expressed in terms of the local level
frame of the satellite, thus the transition matrix of the system is an identity
matrix and is not shown. The Kalman covariance prediction equation is simply(15)Pk−=Pk−1++Q, where Q contains process noise for the baseline
component states and is zeros in all rows and columns corresponding the
ambiguity states.For the cascading technique,
the probability of partially fixing each step (the EWL alone, the EWL and WL,
and all three) is also evaluated. As a final step, the estimated covariance of
the fixed position states is computed using (7). The results of this can then be used to compute the estimated
accuracy of the derived attitude angles.Figure6 shows
the overall probability of incorrect fix upper bounds for single-, dual-, and
triple-frequency cascading (over one 1-minute interval). To interpret this
figure, consider the red line near the top. This is the probability of
incorrectly resolving the E1/E2 ambiguity as a function of time in the case
that only a single-frequency receiver is used. The line decreases as a function
of time indicating increasing odds of correct ambiguity resolution.  However, the dual- and triple-frequency
cascading options, shown by the green and blue lines, respectively, show that
dual and triple-frequency cascading methods offer several orders of magnitude
improvement. The y-axis is logarithmically scaled and for example 10−4 indicates a probability of correct fix of 99.99%. Note that these are the
probabilities of fixing all of the ambiguities, and in dual- and triple-frequency cases there are two and three times as many ambiguities to fix. But
even though there are more ambiguities to fix, the ability to cascade substantially
increases the chances of correctly resolving the ambiguities.Figure 6
Sample Probability of Incorrect Fix upper bounds for single-frequency, dual and triple-frequency cascading for ENVISAT.Figure7 shows
the probability of incorrect fix for each stage of the cascading process for
the triple-frequency case. The blue line represents PIF for the EWL ambiguities
as a function of time. Note that resolving the EWL ambiguities is more or less
assured. Though it is not visible, there is a green line plotted almost
directly under the red line. This green line represents the probability of
correctly resolving the WL after resolving the EWL. The reason it is not visible
is that resolving the WL more or less ensures resolving the E1/E2 ambiguity (in
the short baseline for attitude determination case) meaning that the
probability of resolving the E1/E2 ambiguity after resolving with WL is close
to unity. The result is that the probability of resolving all three is only
very slightly less than the probability of resolving the EWL and the WL alone.
Note that the red line, representing the probability of resolving the E1/E2
ambiguities after resolving the EWL and WL is identical to the probability of
resolving all of the ambiguities in triple-frequency cascading which is plotted
as the blue line in Figure 6.Figure 7
Sample Probability of
Incorrect Fix upper bounds for the EWL, WL, and E1/E2 ambiguities in triple-frequency cascading for ENVISAT.Figure8 shows
the corresponding float solution baseline component accuracies as a function of
time for the three scenarios shown in Figure 6. The
three baseline component labels, E, N, and h, correspond to the longitude,
latitude, and vertical direction in the local level frame of the spacecraft. Note that the scale of the vertical subplot is double
those of the two horizontal components corresponding the decreased accuracy
provided by GNSS in the vertical direction. Note how float solution is
relatively poor, on the order of 10 to 20 cm. In this regime, attitude determination on
a short baseline would not be feasible.Sample estimated float solution accuracy for single, dual and
triple-frequency cascading. E, N, h indicate
longitude, latitude and vertical components in the local level frame of the
spacecraft.
(a)(b)(c)Figure9 shows the corresponding baseline accuracy components for the fixed
solution. Note that the accuracy is more or less constant as a function of
time. These estimated accuracies are the accuracies assuming the ambiguities
have been correctly resolved and this figure does not take into account the
fact that at the beginning of the filtering interval the likelihood, for
example, of correctly resolving the E1/E2 ambiguity with a single-frequency
receiver is quite low. However, this accuracy level does represent the steady
state assuming the ambiguities have been correctly resolved and is useful as
the attitude accuracy can be derived directly from this baseline accuracy. For
small errors, such as those shown in Figure 9, and a simple antenna configuration with two or three antennas in
the horizontal plane, the corresponding attitude errors expressed in radians
are well approximated by the ratio of the horizontal positioning accuracy to
baseline length for azimuth, and by the ratio of the vertical error to baseline
length for the roll and pitch errors.Sample estimated fixed
solution accuracy for single, dual, and triple-frequency cascading.E, N, h indicate longitude, latitude, and vertical components in the
local level frame of the spacecraft.
(a)(b)(c)Figure10 shows
the PIF as a function of time for single, dual and triple-frequency cascading
for ENVISAT. All 1440 simulations are plotted on one figure to demonstrate the
range of PIF for varying LEO to Galileo satellite geometry. As can be seen, the
sample result shown in Figure 6 was
in fact one of the best cases and even with three frequencies, the PIF can still
be relatively high (e.g., only 10%) after a minute of filtering. Also, the
larger performance gain is from one frequency to two, with all of the PIF
samples for the single-frequency case being worse than any of the dual or
triple-frequency samples whereas there is some overlap between good dual-frequency results and relatively poor triple-frequency results. Figure 11 shows
all of the samples of EWL, WL and E1/E2 PIF for the triple-frequency case as
discussed earlier in the context of Figure 7.
Again, the sample result shown earlier is one of the better samples. From Figure 11 it can be seen that the probability of resolving the EWL in a
triple-frequency cascading scheme is very high and very consistent between
samples while there is more variability between the probabilities of resolving
the WL in the second step in the cascading scheme. It can also be seen that
there is little difference in the probability of resolving the WL and the
probability of resolving all three, indicating again that once the widelanes
are resolved in triple-frequency cascading, resolving the E1/E2 ambiguities is
relatively easy.Figure 10
Probability of Incorrect fix upper bounds for single-frequency, dual and triple-frequency cascading for ENVISAT.Figure 11
Probability of Incorrect
Fix upper bounds for the EWL, WL, and E1/E2 ambiguities in triple-frequency
cascading for ENVISAT.


#### 4.1.3. Lambda with Galileo

With the LAMBDA method, as
opposed to cascading, all of the ambiguity states are estimated together and
ambiguity resolution is facilitated by the LAMBDA algorithm deciding what
linear combination of the original ambiguities most decorrelates the ambiguities
thus making them easiest to fix. In terms of the covariance simulation, the
same Kalman filter equations are used, only in this case the design matrix
reflects the fact that the original ambiguities (and not widelanes) are being
estimated. The result of the float solution is then sent to the LAMBDA
algorithm for decorrelation.Figure12 shows
the PIF results for the LAMBDA method and can be compared directly the
corresponding cascading results shown in Figure 10.  Note that there is improvement in all three
cases (single-, dual-, and triple-frequency) but most notably the triple-frequency ambiguity resolution PIF (the blue lines) is significantly reduced
compared to cascading demonstrating the potential of the LAMBDA method to find
optimal linear combinations of the ambiguities that provide more ambiguity
resolution potential compared the EWL, WL, and E1/E2 used in the cascading
method.Figure 12
Probability of Incorrect Fix upper bounds for single-,
dual- and triple-frequency LAMBDA for ENVISAT.One important result to
note is that less is gained by adding a third frequency in terms of ambiguity
resolution for very short baselines. In other words, there is a larger change in
PIF from the single-frequency case to the dual-frequency case than there is
from the dual- to triple-frequency case. The addition of the second frequency
allows the LAMBDA algorithm to automatically form a widelane combination, which
for a short baseline such as this, is more than sufficient for ambiguity
resolution. The addition of the third frequency provides only a marginal
improvement as the code error in this case is smaller that the extra-wide lane
wavelength and there is no differential ionosphere error affecting either the code
or phase measurements in this application. This is similar to the results of
using cascading (found in the TCAR/CIR literature) where the largest gains from
using three frequencies are found in longer baseline applications. A final note
about the LAMBDA results is that  close inspection shows a small number of dual and
triple-frequency trials where the PIF appears to increase counter-intuitively
from one epoch to the next. This is not an error. An unfortunate feature using
the bootstrapped lower bound for the PCF as an estimate of the integer least
squares PCF is that the lower bound is not invariant to the decorrelating
linear combination found using LAMBDA. Occasionally, the LAMBDA algorithm
changes its linear combination from one filter epoch to the next and this can
result in discontinuities, both decreasing and increasing, in the PCF lower
bound. It should be noted that the actual PCF does not change, only value of
the bound.


#### 4.2. GPS/Galileo Results

In this section, the
results presented for Galileo are now repeated for the case that both GPS and
Galileo are being used.


#### 4.2.1. Availability and Dilution of Precision

With two systems, the
number of satellites observed roughly doubles, as shown in Figure13. The
corresponding DOP values also decrease. As in the Galileo only case, the
largest (poorest) DOP values occur when the fewest satellites are tracked and
the results are periodic with the orbital period of the LEO satellite.Availability, HDOP, and
VDOP of GPS/Galileo as viewed from a polar orbiting LEO (ENVISAT) over a 24-hour period.
(a)(b)(c)


#### 4.2.2. Cascading with GPS/Galileo

Figure14 shows
the dual GNSS probabilities of incorrect fix bounds for cascading for ENVISAT
analogous to those presented in Figure 10 for
the Galileo only case. The same patterns are observed, though with roughly
double the number of observations and double the number of ambiguities to
estimate, the performance improvement is only marginal.  Likewise, the probability of successfully
resolving each stage in the three-frequency case, shown in Figure 15, is
only marginally better than the results for Galileo only shown in Figure 11.Figure 14
Probability of Incorrect
Fix upper bounds for single-frequency, dual and triple-frequency cascading for
ENVISAT using two GNSS.Figure 15
Probability of Incorrect
Fix upper bounds for the EWL, WL, and E1/E2 ambiguities in triple-frequency
cascading for ENVISAT using two GNSS.


#### 4.2.3. Lambda with GPS/Galileo

Use of the LAMBDA method
with two systems results in a major improvement in the probability of correct
fix than can be obtained with a single-frequency receiver. Though the dual- and
triple-frequency cases are also improved with LAMBDA, the significant
improvement can be seen in the red lines in Figure16 when
compared to the Galileo only case shown in Figure 12. Most
striking is the absence of cases where the probability of incorrect fix hovers
near the top of the figures (in the 0.1 to 1.0 range). This can be explained by
the fact that the LAMBDA method is able to exploit the geometric diversity of
the single-frequency signals from the two systems by forming linear
combinations of all of the L1 and E1/E2 ambiguities while in the Galileo only
case, approximately half as many ambiguities are available. There is less
improvement with the dual and triple-frequency cases because these cases
already have a diversity of ambiguities on different frequencies for the LAMBDA
algorithm work with even in the Galileo only case.Figure 16
Probability of Incorrect
Fix upper bounds for single-frequency, dual and triple-frequency LAMBDA for
ENVISAT using two GNSS.


### 4.3. Galileo versus GPS/Galileo: Recommendations

Based on these results it
can be concluded that the LAMBDA approach is superior to the cascading approach
in all cases. However the LAMBDA method is particularly useful in the case of a
single-frequency dual-GNSS application. However to obtain the most reliable
ambiguity resolution, multiple frequencies are required when using both one or
two GNSS constellations. If power and weight of the GNSS receiver payload is no
object, a dual-system triple-frequency receiver using LAMBDA will provide the
best performance. However, if power and weight are an issue, a dual-frequency
single-system receiver offers similar performance to a single-frequency
dual-system one.  Similar results were
obtained with the ISS as the simulated user and are therefore not shown.While these results
consider the case of attitude determination for an orbiting user, they can be
generalized to ground based users. Of course while the absolute performance of
each algorithm will depend on measurement accuracies, satellite geometry, and
user dynamics, it is reasonable to expect the relative performance of the
algorithms to remain unchanged.


## 4.1. Galileo Results

### 4.1.1. Satellite Availability and Dilution of Precision

Figure4 shows
a plot of the number of available satellites and the corresponding HDOP and
VDOP values as a function of time for ENVISAT. Results for the ISS were very
similar, and have been omitted to save space. The main difference between the
two is that the VDOP of the polar orbiting ENVISAT tests tends to be worse as
the satellite travels over the poles due to the lack of overhead GNSS
satellites in the polar regions.Availability, HDOP, and
VDOP of Galileo as viewed from a polar orbiting LEO (ENVISAT) over a 24 hour
period.
(a)(b)(c)As expected, VDOP values
are larger than HDOP. This is due the geometrical distribution of satellites in
GNSS and the fact that observability of the vertical direction is limited by
the correlation between the vertical position state and the receiver clock
offset. This effect remains despite the elimination of the clock offset term in
the double-differencing process. A major result of this is that generally the
vertical baseline component will be poorer than the horizontal components,
making pitch and roll more difficult to estimate than azimuth when receiver
antennas are configured in the horizontal plane.The DOP values are also
periodic with the orbital period of the LEO as can be seen by comparing the DOP
time series to the latitude of the LEO shown in Figure5. This
is especially true for polar satellites since when they are over the poles,
they will only see GNSS satellites on the horizon around them (since GNSS
satellites are generally in 54 to 56 degree inclined orbits). As a result,
baseline solutions and corresponding attitude estimation quality will vary
periodically with the LEO orbit period.Figure 5
Latitude of ENVISAT as a
function of time.


### 4.1.2. Cascading with Galileo

Results of the
geometry-based cascading schemes are now presented. A single filtered 1-minute
interval (of the 1440 1-minute intervals simulated) consisting of 6 simulated
observations occurring at time = 0, 10, 20, 30, 40, and 50 seconds is presented
in detail first. The results for all 1440 trials are then shown together. An
arbitrary 1-minute segment for the ENVISAT was selected. For this, and every 1-minute segment, the following procedure is used. First, the assumption is made
that the system is warm started meaning that the receiver has approximate
coordinates for itself and has acquired and is tracking all visible satellite
before the start of the 1-minute segment. At the beginning of the 1-minute
segment, the “rover” receiver in the two receiver pair estimates its position
with respect to the “base” receiver to an accuracy of 1 m (1σ) in
each dimension in the satellite body frame. This initial value is based on
typical differential code positioning accuracy. A Kalman filter is then
initialized with this initial estimate for the antenna position, and an initial
ambiguity estimate variance that corresponds also to 1 m (but is expressed in
terms of cycles).The combined position and
float ambiguity filter is then updated once every 10 seconds with a single code
observation from each satellite, and a phase observation for each frequency being
used from each satellite for the particular scenario. E1/E2 phase observations
are used to estimate both an E1/E2 ambiguity and WL, E5b is used to estimate
the WL and the EWL, and E5a contributes only the EWL.  Due to the short baseline length and fixed
nature of the antennas, the position (baseline component) states are modelled
as a random walk processes with a process noise of 0.01m2/s (about
a mean value determined from an existing orbital and attitude model for the
satellite) and ambiguity states are modelled as random constant processes.
After each Kalman update, the covariance matrix of the ambiguities is analysed
to determine the lower bound on the probability of correct fix of the full set
of ambiguities using the technique described in Section 2.2.The Kalman covariance
update equation is given by(14)Pk+=Pk−−Pk−HkT[HkPk−HkT+Rk]−1HkPk−, where Hk is the design matrix, Rk is the measurement error covariance matrix, and Pk− and Pk+ are the covariance matrices of the errors of
the states before and after the update respectively. Note that for simplicity
the Kalman gain matrix is included in, but not shown as a separate quantity in (14). The prediction step assumes that the baseline components, and thus also
the direction cosines in H are expressed in terms of the local level
frame of the satellite, thus the transition matrix of the system is an identity
matrix and is not shown. The Kalman covariance prediction equation is simply(15)Pk−=Pk−1++Q, where Q contains process noise for the baseline
component states and is zeros in all rows and columns corresponding the
ambiguity states.For the cascading technique,
the probability of partially fixing each step (the EWL alone, the EWL and WL,
and all three) is also evaluated. As a final step, the estimated covariance of
the fixed position states is computed using (7). The results of this can then be used to compute the estimated
accuracy of the derived attitude angles.Figure6 shows
the overall probability of incorrect fix upper bounds for single-, dual-, and
triple-frequency cascading (over one 1-minute interval). To interpret this
figure, consider the red line near the top. This is the probability of
incorrectly resolving the E1/E2 ambiguity as a function of time in the case
that only a single-frequency receiver is used. The line decreases as a function
of time indicating increasing odds of correct ambiguity resolution.  However, the dual- and triple-frequency
cascading options, shown by the green and blue lines, respectively, show that
dual and triple-frequency cascading methods offer several orders of magnitude
improvement. The y-axis is logarithmically scaled and for example 10−4 indicates a probability of correct fix of 99.99%. Note that these are the
probabilities of fixing all of the ambiguities, and in dual- and triple-frequency cases there are two and three times as many ambiguities to fix. But
even though there are more ambiguities to fix, the ability to cascade substantially
increases the chances of correctly resolving the ambiguities.Figure 6
Sample Probability of Incorrect Fix upper bounds for single-frequency, dual and triple-frequency cascading for ENVISAT.Figure7 shows
the probability of incorrect fix for each stage of the cascading process for
the triple-frequency case. The blue line represents PIF for the EWL ambiguities
as a function of time. Note that resolving the EWL ambiguities is more or less
assured. Though it is not visible, there is a green line plotted almost
directly under the red line. This green line represents the probability of
correctly resolving the WL after resolving the EWL. The reason it is not visible
is that resolving the WL more or less ensures resolving the E1/E2 ambiguity (in
the short baseline for attitude determination case) meaning that the
probability of resolving the E1/E2 ambiguity after resolving with WL is close
to unity. The result is that the probability of resolving all three is only
very slightly less than the probability of resolving the EWL and the WL alone.
Note that the red line, representing the probability of resolving the E1/E2
ambiguities after resolving the EWL and WL is identical to the probability of
resolving all of the ambiguities in triple-frequency cascading which is plotted
as the blue line in Figure 6.Figure 7
Sample Probability of
Incorrect Fix upper bounds for the EWL, WL, and E1/E2 ambiguities in triple-frequency cascading for ENVISAT.Figure8 shows
the corresponding float solution baseline component accuracies as a function of
time for the three scenarios shown in Figure 6. The
three baseline component labels, E, N, and h, correspond to the longitude,
latitude, and vertical direction in the local level frame of the spacecraft. Note that the scale of the vertical subplot is double
those of the two horizontal components corresponding the decreased accuracy
provided by GNSS in the vertical direction. Note how float solution is
relatively poor, on the order of 10 to 20 cm. In this regime, attitude determination on
a short baseline would not be feasible.Sample estimated float solution accuracy for single, dual and
triple-frequency cascading. E, N, h indicate
longitude, latitude and vertical components in the local level frame of the
spacecraft.
(a)(b)(c)Figure9 shows the corresponding baseline accuracy components for the fixed
solution. Note that the accuracy is more or less constant as a function of
time. These estimated accuracies are the accuracies assuming the ambiguities
have been correctly resolved and this figure does not take into account the
fact that at the beginning of the filtering interval the likelihood, for
example, of correctly resolving the E1/E2 ambiguity with a single-frequency
receiver is quite low. However, this accuracy level does represent the steady
state assuming the ambiguities have been correctly resolved and is useful as
the attitude accuracy can be derived directly from this baseline accuracy. For
small errors, such as those shown in Figure 9, and a simple antenna configuration with two or three antennas in
the horizontal plane, the corresponding attitude errors expressed in radians
are well approximated by the ratio of the horizontal positioning accuracy to
baseline length for azimuth, and by the ratio of the vertical error to baseline
length for the roll and pitch errors.Sample estimated fixed
solution accuracy for single, dual, and triple-frequency cascading.E, N, h indicate longitude, latitude, and vertical components in the
local level frame of the spacecraft.
(a)(b)(c)Figure10 shows
the PIF as a function of time for single, dual and triple-frequency cascading
for ENVISAT. All 1440 simulations are plotted on one figure to demonstrate the
range of PIF for varying LEO to Galileo satellite geometry. As can be seen, the
sample result shown in Figure 6 was
in fact one of the best cases and even with three frequencies, the PIF can still
be relatively high (e.g., only 10%) after a minute of filtering. Also, the
larger performance gain is from one frequency to two, with all of the PIF
samples for the single-frequency case being worse than any of the dual or
triple-frequency samples whereas there is some overlap between good dual-frequency results and relatively poor triple-frequency results. Figure 11 shows
all of the samples of EWL, WL and E1/E2 PIF for the triple-frequency case as
discussed earlier in the context of Figure 7.
Again, the sample result shown earlier is one of the better samples. From Figure 11 it can be seen that the probability of resolving the EWL in a
triple-frequency cascading scheme is very high and very consistent between
samples while there is more variability between the probabilities of resolving
the WL in the second step in the cascading scheme. It can also be seen that
there is little difference in the probability of resolving the WL and the
probability of resolving all three, indicating again that once the widelanes
are resolved in triple-frequency cascading, resolving the E1/E2 ambiguities is
relatively easy.Figure 10
Probability of Incorrect fix upper bounds for single-frequency, dual and triple-frequency cascading for ENVISAT.Figure 11
Probability of Incorrect
Fix upper bounds for the EWL, WL, and E1/E2 ambiguities in triple-frequency
cascading for ENVISAT.


### 4.1.3. Lambda with Galileo

With the LAMBDA method, as
opposed to cascading, all of the ambiguity states are estimated together and
ambiguity resolution is facilitated by the LAMBDA algorithm deciding what
linear combination of the original ambiguities most decorrelates the ambiguities
thus making them easiest to fix. In terms of the covariance simulation, the
same Kalman filter equations are used, only in this case the design matrix
reflects the fact that the original ambiguities (and not widelanes) are being
estimated. The result of the float solution is then sent to the LAMBDA
algorithm for decorrelation.Figure12 shows
the PIF results for the LAMBDA method and can be compared directly the
corresponding cascading results shown in Figure 10.  Note that there is improvement in all three
cases (single-, dual-, and triple-frequency) but most notably the triple-frequency ambiguity resolution PIF (the blue lines) is significantly reduced
compared to cascading demonstrating the potential of the LAMBDA method to find
optimal linear combinations of the ambiguities that provide more ambiguity
resolution potential compared the EWL, WL, and E1/E2 used in the cascading
method.Figure 12
Probability of Incorrect Fix upper bounds for single-,
dual- and triple-frequency LAMBDA for ENVISAT.One important result to
note is that less is gained by adding a third frequency in terms of ambiguity
resolution for very short baselines. In other words, there is a larger change in
PIF from the single-frequency case to the dual-frequency case than there is
from the dual- to triple-frequency case. The addition of the second frequency
allows the LAMBDA algorithm to automatically form a widelane combination, which
for a short baseline such as this, is more than sufficient for ambiguity
resolution. The addition of the third frequency provides only a marginal
improvement as the code error in this case is smaller that the extra-wide lane
wavelength and there is no differential ionosphere error affecting either the code
or phase measurements in this application. This is similar to the results of
using cascading (found in the TCAR/CIR literature) where the largest gains from
using three frequencies are found in longer baseline applications. A final note
about the LAMBDA results is that  close inspection shows a small number of dual and
triple-frequency trials where the PIF appears to increase counter-intuitively
from one epoch to the next. This is not an error. An unfortunate feature using
the bootstrapped lower bound for the PCF as an estimate of the integer least
squares PCF is that the lower bound is not invariant to the decorrelating
linear combination found using LAMBDA. Occasionally, the LAMBDA algorithm
changes its linear combination from one filter epoch to the next and this can
result in discontinuities, both decreasing and increasing, in the PCF lower
bound. It should be noted that the actual PCF does not change, only value of
the bound.


### 4.2. GPS/Galileo Results

In this section, the
results presented for Galileo are now repeated for the case that both GPS and
Galileo are being used.


### 4.2.1. Availability and Dilution of Precision

With two systems, the
number of satellites observed roughly doubles, as shown in Figure13. The
corresponding DOP values also decrease. As in the Galileo only case, the
largest (poorest) DOP values occur when the fewest satellites are tracked and
the results are periodic with the orbital period of the LEO satellite.Availability, HDOP, and
VDOP of GPS/Galileo as viewed from a polar orbiting LEO (ENVISAT) over a 24-hour period.
(a)(b)(c)


### 4.2.2. Cascading with GPS/Galileo

Figure14 shows
the dual GNSS probabilities of incorrect fix bounds for cascading for ENVISAT
analogous to those presented in Figure 10 for
the Galileo only case. The same patterns are observed, though with roughly
double the number of observations and double the number of ambiguities to
estimate, the performance improvement is only marginal.  Likewise, the probability of successfully
resolving each stage in the three-frequency case, shown in Figure 15, is
only marginally better than the results for Galileo only shown in Figure 11.Figure 14
Probability of Incorrect
Fix upper bounds for single-frequency, dual and triple-frequency cascading for
ENVISAT using two GNSS.Figure 15
Probability of Incorrect
Fix upper bounds for the EWL, WL, and E1/E2 ambiguities in triple-frequency
cascading for ENVISAT using two GNSS.


### 4.2.3. Lambda with GPS/Galileo

Use of the LAMBDA method
with two systems results in a major improvement in the probability of correct
fix than can be obtained with a single-frequency receiver. Though the dual- and
triple-frequency cases are also improved with LAMBDA, the significant
improvement can be seen in the red lines in Figure16 when
compared to the Galileo only case shown in Figure 12. Most
striking is the absence of cases where the probability of incorrect fix hovers
near the top of the figures (in the 0.1 to 1.0 range). This can be explained by
the fact that the LAMBDA method is able to exploit the geometric diversity of
the single-frequency signals from the two systems by forming linear
combinations of all of the L1 and E1/E2 ambiguities while in the Galileo only
case, approximately half as many ambiguities are available. There is less
improvement with the dual and triple-frequency cases because these cases
already have a diversity of ambiguities on different frequencies for the LAMBDA
algorithm work with even in the Galileo only case.Figure 16
Probability of Incorrect
Fix upper bounds for single-frequency, dual and triple-frequency LAMBDA for
ENVISAT using two GNSS.


## 4.1.1. Satellite Availability and Dilution of Precision

Figure4 shows
a plot of the number of available satellites and the corresponding HDOP and
VDOP values as a function of time for ENVISAT. Results for the ISS were very
similar, and have been omitted to save space. The main difference between the
two is that the VDOP of the polar orbiting ENVISAT tests tends to be worse as
the satellite travels over the poles due to the lack of overhead GNSS
satellites in the polar regions.Availability, HDOP, and
VDOP of Galileo as viewed from a polar orbiting LEO (ENVISAT) over a 24 hour
period.
(a)(b)(c)As expected, VDOP values
are larger than HDOP. This is due the geometrical distribution of satellites in
GNSS and the fact that observability of the vertical direction is limited by
the correlation between the vertical position state and the receiver clock
offset. This effect remains despite the elimination of the clock offset term in
the double-differencing process. A major result of this is that generally the
vertical baseline component will be poorer than the horizontal components,
making pitch and roll more difficult to estimate than azimuth when receiver
antennas are configured in the horizontal plane.The DOP values are also
periodic with the orbital period of the LEO as can be seen by comparing the DOP
time series to the latitude of the LEO shown in Figure5. This
is especially true for polar satellites since when they are over the poles,
they will only see GNSS satellites on the horizon around them (since GNSS
satellites are generally in 54 to 56 degree inclined orbits). As a result,
baseline solutions and corresponding attitude estimation quality will vary
periodically with the LEO orbit period.Figure 5
Latitude of ENVISAT as a
function of time.


## 4.1.2. Cascading with Galileo

Results of the
geometry-based cascading schemes are now presented. A single filtered 1-minute
interval (of the 1440 1-minute intervals simulated) consisting of 6 simulated
observations occurring at time = 0, 10, 20, 30, 40, and 50 seconds is presented
in detail first. The results for all 1440 trials are then shown together. An
arbitrary 1-minute segment for the ENVISAT was selected. For this, and every 1-minute segment, the following procedure is used. First, the assumption is made
that the system is warm started meaning that the receiver has approximate
coordinates for itself and has acquired and is tracking all visible satellite
before the start of the 1-minute segment. At the beginning of the 1-minute
segment, the “rover” receiver in the two receiver pair estimates its position
with respect to the “base” receiver to an accuracy of 1 m (1σ) in
each dimension in the satellite body frame. This initial value is based on
typical differential code positioning accuracy. A Kalman filter is then
initialized with this initial estimate for the antenna position, and an initial
ambiguity estimate variance that corresponds also to 1 m (but is expressed in
terms of cycles).The combined position and
float ambiguity filter is then updated once every 10 seconds with a single code
observation from each satellite, and a phase observation for each frequency being
used from each satellite for the particular scenario. E1/E2 phase observations
are used to estimate both an E1/E2 ambiguity and WL, E5b is used to estimate
the WL and the EWL, and E5a contributes only the EWL.  Due to the short baseline length and fixed
nature of the antennas, the position (baseline component) states are modelled
as a random walk processes with a process noise of 0.01m2/s (about
a mean value determined from an existing orbital and attitude model for the
satellite) and ambiguity states are modelled as random constant processes.
After each Kalman update, the covariance matrix of the ambiguities is analysed
to determine the lower bound on the probability of correct fix of the full set
of ambiguities using the technique described in Section 2.2.The Kalman covariance
update equation is given by(14)Pk+=Pk−−Pk−HkT[HkPk−HkT+Rk]−1HkPk−, where Hk is the design matrix, Rk is the measurement error covariance matrix, and Pk− and Pk+ are the covariance matrices of the errors of
the states before and after the update respectively. Note that for simplicity
the Kalman gain matrix is included in, but not shown as a separate quantity in (14). The prediction step assumes that the baseline components, and thus also
the direction cosines in H are expressed in terms of the local level
frame of the satellite, thus the transition matrix of the system is an identity
matrix and is not shown. The Kalman covariance prediction equation is simply(15)Pk−=Pk−1++Q, where Q contains process noise for the baseline
component states and is zeros in all rows and columns corresponding the
ambiguity states.For the cascading technique,
the probability of partially fixing each step (the EWL alone, the EWL and WL,
and all three) is also evaluated. As a final step, the estimated covariance of
the fixed position states is computed using (7). The results of this can then be used to compute the estimated
accuracy of the derived attitude angles.Figure6 shows
the overall probability of incorrect fix upper bounds for single-, dual-, and
triple-frequency cascading (over one 1-minute interval). To interpret this
figure, consider the red line near the top. This is the probability of
incorrectly resolving the E1/E2 ambiguity as a function of time in the case
that only a single-frequency receiver is used. The line decreases as a function
of time indicating increasing odds of correct ambiguity resolution.  However, the dual- and triple-frequency
cascading options, shown by the green and blue lines, respectively, show that
dual and triple-frequency cascading methods offer several orders of magnitude
improvement. The y-axis is logarithmically scaled and for example 10−4 indicates a probability of correct fix of 99.99%. Note that these are the
probabilities of fixing all of the ambiguities, and in dual- and triple-frequency cases there are two and three times as many ambiguities to fix. But
even though there are more ambiguities to fix, the ability to cascade substantially
increases the chances of correctly resolving the ambiguities.Figure 6
Sample Probability of Incorrect Fix upper bounds for single-frequency, dual and triple-frequency cascading for ENVISAT.Figure7 shows
the probability of incorrect fix for each stage of the cascading process for
the triple-frequency case. The blue line represents PIF for the EWL ambiguities
as a function of time. Note that resolving the EWL ambiguities is more or less
assured. Though it is not visible, there is a green line plotted almost
directly under the red line. This green line represents the probability of
correctly resolving the WL after resolving the EWL. The reason it is not visible
is that resolving the WL more or less ensures resolving the E1/E2 ambiguity (in
the short baseline for attitude determination case) meaning that the
probability of resolving the E1/E2 ambiguity after resolving with WL is close
to unity. The result is that the probability of resolving all three is only
very slightly less than the probability of resolving the EWL and the WL alone.
Note that the red line, representing the probability of resolving the E1/E2
ambiguities after resolving the EWL and WL is identical to the probability of
resolving all of the ambiguities in triple-frequency cascading which is plotted
as the blue line in Figure 6.Figure 7
Sample Probability of
Incorrect Fix upper bounds for the EWL, WL, and E1/E2 ambiguities in triple-frequency cascading for ENVISAT.Figure8 shows
the corresponding float solution baseline component accuracies as a function of
time for the three scenarios shown in Figure 6. The
three baseline component labels, E, N, and h, correspond to the longitude,
latitude, and vertical direction in the local level frame of the spacecraft. Note that the scale of the vertical subplot is double
those of the two horizontal components corresponding the decreased accuracy
provided by GNSS in the vertical direction. Note how float solution is
relatively poor, on the order of 10 to 20 cm. In this regime, attitude determination on
a short baseline would not be feasible.Sample estimated float solution accuracy for single, dual and
triple-frequency cascading. E, N, h indicate
longitude, latitude and vertical components in the local level frame of the
spacecraft.
(a)(b)(c)Figure9 shows the corresponding baseline accuracy components for the fixed
solution. Note that the accuracy is more or less constant as a function of
time. These estimated accuracies are the accuracies assuming the ambiguities
have been correctly resolved and this figure does not take into account the
fact that at the beginning of the filtering interval the likelihood, for
example, of correctly resolving the E1/E2 ambiguity with a single-frequency
receiver is quite low. However, this accuracy level does represent the steady
state assuming the ambiguities have been correctly resolved and is useful as
the attitude accuracy can be derived directly from this baseline accuracy. For
small errors, such as those shown in Figure 9, and a simple antenna configuration with two or three antennas in
the horizontal plane, the corresponding attitude errors expressed in radians
are well approximated by the ratio of the horizontal positioning accuracy to
baseline length for azimuth, and by the ratio of the vertical error to baseline
length for the roll and pitch errors.Sample estimated fixed
solution accuracy for single, dual, and triple-frequency cascading.E, N, h indicate longitude, latitude, and vertical components in the
local level frame of the spacecraft.
(a)(b)(c)Figure10 shows
the PIF as a function of time for single, dual and triple-frequency cascading
for ENVISAT. All 1440 simulations are plotted on one figure to demonstrate the
range of PIF for varying LEO to Galileo satellite geometry. As can be seen, the
sample result shown in Figure 6 was
in fact one of the best cases and even with three frequencies, the PIF can still
be relatively high (e.g., only 10%) after a minute of filtering. Also, the
larger performance gain is from one frequency to two, with all of the PIF
samples for the single-frequency case being worse than any of the dual or
triple-frequency samples whereas there is some overlap between good dual-frequency results and relatively poor triple-frequency results. Figure 11 shows
all of the samples of EWL, WL and E1/E2 PIF for the triple-frequency case as
discussed earlier in the context of Figure 7.
Again, the sample result shown earlier is one of the better samples. From Figure 11 it can be seen that the probability of resolving the EWL in a
triple-frequency cascading scheme is very high and very consistent between
samples while there is more variability between the probabilities of resolving
the WL in the second step in the cascading scheme. It can also be seen that
there is little difference in the probability of resolving the WL and the
probability of resolving all three, indicating again that once the widelanes
are resolved in triple-frequency cascading, resolving the E1/E2 ambiguities is
relatively easy.Figure 10
Probability of Incorrect fix upper bounds for single-frequency, dual and triple-frequency cascading for ENVISAT.Figure 11
Probability of Incorrect
Fix upper bounds for the EWL, WL, and E1/E2 ambiguities in triple-frequency
cascading for ENVISAT.


## 4.1.3. Lambda with Galileo

With the LAMBDA method, as
opposed to cascading, all of the ambiguity states are estimated together and
ambiguity resolution is facilitated by the LAMBDA algorithm deciding what
linear combination of the original ambiguities most decorrelates the ambiguities
thus making them easiest to fix. In terms of the covariance simulation, the
same Kalman filter equations are used, only in this case the design matrix
reflects the fact that the original ambiguities (and not widelanes) are being
estimated. The result of the float solution is then sent to the LAMBDA
algorithm for decorrelation.Figure12 shows
the PIF results for the LAMBDA method and can be compared directly the
corresponding cascading results shown in Figure 10.  Note that there is improvement in all three
cases (single-, dual-, and triple-frequency) but most notably the triple-frequency ambiguity resolution PIF (the blue lines) is significantly reduced
compared to cascading demonstrating the potential of the LAMBDA method to find
optimal linear combinations of the ambiguities that provide more ambiguity
resolution potential compared the EWL, WL, and E1/E2 used in the cascading
method.Figure 12
Probability of Incorrect Fix upper bounds for single-,
dual- and triple-frequency LAMBDA for ENVISAT.One important result to
note is that less is gained by adding a third frequency in terms of ambiguity
resolution for very short baselines. In other words, there is a larger change in
PIF from the single-frequency case to the dual-frequency case than there is
from the dual- to triple-frequency case. The addition of the second frequency
allows the LAMBDA algorithm to automatically form a widelane combination, which
for a short baseline such as this, is more than sufficient for ambiguity
resolution. The addition of the third frequency provides only a marginal
improvement as the code error in this case is smaller that the extra-wide lane
wavelength and there is no differential ionosphere error affecting either the code
or phase measurements in this application. This is similar to the results of
using cascading (found in the TCAR/CIR literature) where the largest gains from
using three frequencies are found in longer baseline applications. A final note
about the LAMBDA results is that  close inspection shows a small number of dual and
triple-frequency trials where the PIF appears to increase counter-intuitively
from one epoch to the next. This is not an error. An unfortunate feature using
the bootstrapped lower bound for the PCF as an estimate of the integer least
squares PCF is that the lower bound is not invariant to the decorrelating
linear combination found using LAMBDA. Occasionally, the LAMBDA algorithm
changes its linear combination from one filter epoch to the next and this can
result in discontinuities, both decreasing and increasing, in the PCF lower
bound. It should be noted that the actual PCF does not change, only value of
the bound.


## 4.2. GPS/Galileo Results

In this section, the
results presented for Galileo are now repeated for the case that both GPS and
Galileo are being used.


## 4.2.1. Availability and Dilution of Precision

With two systems, the
number of satellites observed roughly doubles, as shown in Figure13. The
corresponding DOP values also decrease. As in the Galileo only case, the
largest (poorest) DOP values occur when the fewest satellites are tracked and
the results are periodic with the orbital period of the LEO satellite.Availability, HDOP, and
VDOP of GPS/Galileo as viewed from a polar orbiting LEO (ENVISAT) over a 24-hour period.
(a)(b)(c)


## 4.2.2. Cascading with GPS/Galileo

Figure14 shows
the dual GNSS probabilities of incorrect fix bounds for cascading for ENVISAT
analogous to those presented in Figure 10 for
the Galileo only case. The same patterns are observed, though with roughly
double the number of observations and double the number of ambiguities to
estimate, the performance improvement is only marginal.  Likewise, the probability of successfully
resolving each stage in the three-frequency case, shown in Figure 15, is
only marginally better than the results for Galileo only shown in Figure 11.Figure 14
Probability of Incorrect
Fix upper bounds for single-frequency, dual and triple-frequency cascading for
ENVISAT using two GNSS.Figure 15
Probability of Incorrect
Fix upper bounds for the EWL, WL, and E1/E2 ambiguities in triple-frequency
cascading for ENVISAT using two GNSS.


## 4.2.3. Lambda with GPS/Galileo

Use of the LAMBDA method
with two systems results in a major improvement in the probability of correct
fix than can be obtained with a single-frequency receiver. Though the dual- and
triple-frequency cases are also improved with LAMBDA, the significant
improvement can be seen in the red lines in Figure16 when
compared to the Galileo only case shown in Figure 12. Most
striking is the absence of cases where the probability of incorrect fix hovers
near the top of the figures (in the 0.1 to 1.0 range). This can be explained by
the fact that the LAMBDA method is able to exploit the geometric diversity of
the single-frequency signals from the two systems by forming linear
combinations of all of the L1 and E1/E2 ambiguities while in the Galileo only
case, approximately half as many ambiguities are available. There is less
improvement with the dual and triple-frequency cases because these cases
already have a diversity of ambiguities on different frequencies for the LAMBDA
algorithm work with even in the Galileo only case.Figure 16
Probability of Incorrect
Fix upper bounds for single-frequency, dual and triple-frequency LAMBDA for
ENVISAT using two GNSS.


## 4.3. Galileo versus GPS/Galileo: Recommendations

Based on these results it
can be concluded that the LAMBDA approach is superior to the cascading approach
in all cases. However the LAMBDA method is particularly useful in the case of a
single-frequency dual-GNSS application. However to obtain the most reliable
ambiguity resolution, multiple frequencies are required when using both one or
two GNSS constellations. If power and weight of the GNSS receiver payload is no
object, a dual-system triple-frequency receiver using LAMBDA will provide the
best performance. However, if power and weight are an issue, a dual-frequency
single-system receiver offers similar performance to a single-frequency
dual-system one.  Similar results were
obtained with the ISS as the simulated user and are therefore not shown.While these results
consider the case of attitude determination for an orbiting user, they can be
generalized to ground based users. Of course while the absolute performance of
each algorithm will depend on measurement accuracies, satellite geometry, and
user dynamics, it is reasonable to expect the relative performance of the
algorithms to remain unchanged.


## 5. Conclusions

In this paper, the
effectiveness of two approaches to multiple-frequency carrier phase ambiguity
resolution was evaluated for case of attitude determination onboard a low-earth
orbiting satellite.  The evaluation is based
on simulations of a polar orbiting LEO and an inclined orbiting LEO. The
overall conclusion is that the geometry-based LAMBDA method is superior to the
geometry-based cascading method in terms of probability of correct ambiguity
resolution, and time required to achieve a particular probability of correct
fix. Further conclusions may also be made about the various frequencies and
systems available. The use of two GNSS provides a significant increase in
ambiguity resolution performance while the addition of the third frequency on
each provides only marginal improvement compared to the dual-frequency case.



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*Source: 592073-2009-05-27.xml*