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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.

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 [

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 [

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 [

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 [

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 [

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 Section

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.

This is the process of using the
available observations to estimate a real-valued (

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.

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.”

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.

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 [

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.

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.

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).

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.

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,

With the addition of a
third frequency on GPS and the deployment of Galileo, new widelane phase
combinations are possible. From (

Characteristics of modernized GPS and GALILEO open service frequencies and signals.

Signal | Modulation | Frequency (MHz) | Wavelength (m) | Chipping rage (Mc/s) | |
---|---|---|---|---|---|

Modernized GPS | L1 C/A | BPSK | 1575.42 | 0.190 | 1.023 |

L2 C | BPSK | 1227.60 | 0.244 | 1.023 | |

L5 | BPSK | 1176.45 | 0.254 | 10.23 | |

Galileo | E1-E2 | MBOC | 1575.42 | 0.190 | 1.023 |

E5b | AltBOC | 1207.14 | 0.248 | 10.23 | |

E5a | AltBOC | 1176.45 | 0.254 | 10.23 |

GPS and Galileo widelane wavelengths in metres.

GPS | ||
---|---|---|

L1 | ||

0.8619 | L2 | |

0.7514 | 5.8610 | L5 |

Galileo | ||
---|---|---|

E1-E2 | ||

0.8140 | E5b | |

0.7514 | 9.7684 | E5a |

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 [

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 [

The Least squares AMBiguity
Decorrelation Adjustment (LAMBDA) method is a generic method for ambiguity
resolution [

The general mathematical model for geometry-based carrier-phase GNSS positioning can be described as follows:

(1) Estimate the ambiguities as real values,

(2) Determine the integer value

(3) Compute the fixed estimates of parameter

However, it is difficult to
quantify (

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 (

Generally speaking, if the
carrier phase ambiguities are resolved to their correct integer values, the
accuracy of the baseline estimate obtained from (

The baseline solution
obtained from (

Assuming two noncolinear
(nonparallel) baseline vectors are available, (

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.

The signal characteristics
of modernized GPS and Galileo have been investigated in great detail by many
researchers [

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 [

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., [

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.

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

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 to

Finally, for the triple-frequency
case, the number of ambiguities increases again to

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.

According to the Galileo
Mission High Level Definition document [

Combined GPS and GALILEO constellation on September 30, 2007 (GPS PRN: 1~32 (noninclusive), GALILEO PRN: 36~65).

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.).

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 Table

Keplerian elements and other orbital information used to simulate LEO users.

ENVISAT | ISS | |
---|---|---|

Semimajor Axis (a) | 7155 km | 6712.5 km |

Eccentricity (e) | 0.0001030 | 0.0001202 |

Inclination (e) | ||

Right-ascension of ascending node (Ω) | ||

Argument of Perigee ( | ||

Mean anomaly | ||

Mean orbital altitude | 790 km | 338.7 km |

Period | 101-minutes | 91.31-minutes |

Typical ground track for
ENVISAT shown in magenta with the Galileo constellation. The circle containing the black cross indicates the initial ENVISAT location (

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 (

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.

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.

Figure

Availability, HDOP, and VDOP of Galileo as viewed from a polar orbiting LEO (ENVISAT) over a 24 hour period.

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 Figure

Latitude of ENVISAT as a function of time.

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 (1

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 0.01

The Kalman covariance update equation is given by

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 (

Figure

Sample Probability of Incorrect Fix upper bounds for single-frequency, dual and triple-frequency cascading for ENVISAT.

Figure

Sample Probability of Incorrect Fix upper bounds for the EWL, WL, and E1/E2 ambiguities in triple-frequency cascading for ENVISAT.

Figure

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.

Figure

Sample estimated fixed
solution accuracy for single, dual, and triple-frequency cascading.

Figure

Probability of Incorrect fix upper bounds for single-frequency, dual and triple-frequency cascading for ENVISAT.

Probability of Incorrect Fix upper bounds for the EWL, WL, and E1/E2 ambiguities in triple-frequency cascading for ENVISAT.

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.

Figure

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.

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

With two systems, the
number of satellites observed roughly doubles, as shown in Figure

Availability, HDOP, and VDOP of GPS/Galileo as viewed from a polar orbiting LEO (ENVISAT) over a 24-hour period.

Figure

Probability of Incorrect Fix upper bounds for single-frequency, dual and triple-frequency cascading for ENVISAT using two GNSS.

Probability of Incorrect Fix upper bounds for the EWL, WL, and E1/E2 ambiguities in triple-frequency cascading for ENVISAT using two GNSS.

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 Figure

Probability of Incorrect Fix upper bounds for single-frequency, dual and triple-frequency LAMBDA for ENVISAT using two GNSS.

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.

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.