Christophe Macabiau Presented

COMPARISON OF THE LSAST AND MAPAS METHODS FOR AMBIGUITY RESOLUTION ON-THE-FLY

The use of the pseudorange information contained within the GPS carrier phase measurements enables to achieve a high level of positioning accuracy. But full access to this accuracy in real time for dynamic applications, like aircraft landing, requires the resolution 'onthe-fly' of the intrinsic ambiguity of the phase measurements. This can be done using one of the numerous ambiguity searching procedures, among which ones are LSAST and MAPAS. These two methods can be seen as sequential tests of the same multiple hypotheses. They use distinct decision criteria, and this results in different implementation constraints, run-time capabilities and performances. Both of them show a similar behaviour when applied to clean data, but MAPAS appears to be more robust when using data affected by multipath. The comparison starts with a parallel presentation of the mathematical developments involved in the design of the two methods, then their practical advantages are reviewed and some results are shown of their application to clean and corrupted data.

INTRODUCTION

A high level of positioning accuracy can be obtained through the use of the very precise pseudorange information contained within the GPS signal carrier phase measurements. However, this pseudorange observation is biased because of the ambiguous nature of the carrier phase measurements. Full access to the value of the pseudorange requires the resolution of that bias, called the phase measurement ambiguity. The procedures performing this resolution in real time in dynamic applications are said to perform the ambiguity resolution 'onthe-fly'.

Many techniques for ambiguity resolution on-thefly have been proposed so far, as summarized by [START_REF] Hatch | Comparison of several AROF kinematic techniques[END_REF]. These are for example the Ambiguity Function Method (AFM), described by Remondi [START_REF] Remondi | Pseudo-kinematic GPS Results Using the Ambiugity Function Method[END_REF] and Mader [START_REF] Mader | Kinematic GPS Phase Initialization using the Ambiguity Function[END_REF], the Least Squares Ambiguity Search (LSAST), presented by Hatch [START_REF] Hatch | Instantaneous Ambiguity Resolution[END_REF] and Lachapelle et al. [START_REF] Lachapelle | High-Precision GPS Navigation with Emphasis on Carrier-Phase Ambiguity Resolution[END_REF], the Fast Ambiguity Resolution Approach (FARA) described by Frei and Beutler (Frei and Beutler, 1990a), the Fast Ambiguity Search FIlter (FASF) described by Chen [START_REF] Chen | Fast Ambiguity Search Filter : A Noval Concept for GPS Ambiguity Resolution[END_REF], the optimized Cholesky decomposition method, described by Landau and Euler [START_REF] Landau | On-The-Fly Ambiguity Resolution for Precise Differential Positioning[END_REF], the integrated 'on-thefly' technique described by Abiddin (Abiddin, 1991), the ambiguity transform method presented by Teunissen [START_REF] Teunissen | A New Method for Fast Carrier Phase Ambiguity Estimation[END_REF] and the Direct Integer Ambiguity Search (DIAS), presented by Ming and Schwarz (Ming and Schwarz, 1995).

The Maximum A Posteriori Ambiguity Search (MA-PAS) method, jointly developped by SEXTANT AVION-IQUE, the Service Technique de la Navigation Aérienne (STNA) and the LTST, recently presented by the author (Macabiau, 1995), is another method for ambiguity resolution on-the-fly, which is based on the same principles as the LSAST.

These two procedures perform an active search of the value of the double differenced ambiguities of four particular satellites called the primary satellites. They both are multiple hypotheses sequential tests that process as many carrier phase measurements as necessary to isolate the best candidate in a predetermined set of three-integer combinations. However, the LSAST and MAPAS methods use different decision criteria to check the hypotheses: the LSAST is based on the weighted sum of the squared prediction errors obtained for the tested combination, while the MAPAS method uses the a posteriori probability of this candidate, conditionally on the prediction errors obtained for that candidate.

Thus these two algorithms have the same overall structure and only differ by the steps performing the the calculation of the decision criterion. This results in different implementation constraints and running capabilities. Both of them can raise ambiguities in seconds when the data conforms to the mathematical model employed. However, they show different performances when applied to data affected by unmodeled noise such as multipath.

This paper first recalls the measurement model employed, then compares the theoretical principles of the LSAST and MAPAS methods and reviews their practical advantages, presents the results of their application, and draws a conclusion on this comparison.

MEASUREMENT MODEL

The data used by the LSAST and MAPAS methods are the linearized double differences of the carrier phase measurements. 1

The observed carrier phase measurements are tracked to keep a constant ambiguity. As described by Rocken [START_REF] Rocken | The Global Positioning System : A New Tool for Tectonic Studies[END_REF], a first order model of these measurementsisasfollows:

i k=ft R k t Si k D i k 1 +f ion f trop N i +b i k+SA i k+" multi k where i k is the tracked carrier phase measurement of satellite i at epoch k f is the L1 carrier frequency and is the corresponding wavelength t R k and t Si k are respectively the receiver and satellite time equivalent phase offset with respect to GPS time D i k is the geometrical distance between the satellite i and the receiver ion and trop are the ionospheric and tropospheric propagation delays N i is the carrier phase measurement ambiguity SA i k is the perturbation introduced by the Selective Availability " multi k is the phase measurement error induced by the multipath propagation of the signal b i k is the phase measurement noise. In the following, we assume that b i k is a discrete white gaussian noise, having zero mean and variance 2 . The carrier phase tracking error " multi k induced by multipath can cause loss of lock and may confuse the ambiguity resolution procedure because of the corresponding distortion of the signal in space, as this error is usually left unmodeled by the procedure.

For example, when the disturbing signal is one specular reflected ray, this term can be expressed as in (2), as presented by Lippincott et al. (Lippincott et al., 1996): In that simple case, having modeled the Earth's surface as a flat plate, the lines of constant phase in space may be distorted as shown in figure 1.

" multi k= 1 2 2 tan 1 i kR ik iksinik R ik+
It can be shown that, in the case of one specular reflected ray, the error " multi k is bounded, as 0:25 " multi k 0:25 and will not cause a change in ambiguity, if left unmodeled. However, when several delayed signals interact, the error may be large enough to cause a fullcycle error and lead the ambiguity resolution procedure to raise wrong ambiguities.

In the next mathematical develomments, we assume " multi k = 0. We will examine the effects of this term on the performances of the two methods in the last two chapters.

In order to eliminate the disturbing terms such as the atmospheric delays and the clock errors, double differences are formed using the measurements obtained by another receiver and all the n k visible satellites.

These double differenced measurements are then linearized around a position estimate Xk, generally ob- tained by the use of LADGPS. A model of these quantities is k= Ck Xk N + Bk 3

where k is an n k 1 1 vector Xk is the 31 vector of the position estimation error: Xk= Xk Xk 2

Presented at DSNS 96, Saint-Petersburg, 1996

N is the n k 1 1 vector of the double differ- enced ambiguities C k is the n k 1 3 vector of the direction cosines Bk is the n k 1 1 vector of the phase mea- surement noise

The linear model given in ( 3) is the mathematical model used by both the LSAST and MAPAS methods.

COMMON PRINCIPLES OF LSAST AND MAPAS

The determination of the position is conditioned on the resolution of the double differenced ambiguity vector N . This resolution is done by testing thousands of possible values of N . These values are determined as the integer vectors associated with a position contained within a predefined search volume. This search volume is centered around the position estimate Xk, and its size depends on the uncertainty of that estimate.

The size of the trial set can be reduced if we note that only three of these ambiguities are independent in the noise-free model derived from (3). Thus the procedure looks for the best three-integer combination to be affected to the double differenced ambiguities of four particular satellites.

These satellites, called the primary satellites, are chosen according to their degree of visibility and their PDOP factor. They must stay visible as long as the resolution is not done, and their PDOP must not be too low nor too large, to guarantee a small number of candidates and a sufficient integrity. For example, the primary satellites can be chosen among the satellites of elevation greater than 7:5 o as the satellites yielding the closet PDOP to the arbitrary reasonable value of 7.5.

Once the primary satellites are identified, the model (3) can be split into 2 systems of equations:

P k= C P k Xk N P + B P k 4 S k= C S k Xk N S + B S k 5
where the first system is the system of the primary satellites, and the other system is the system of the non primary satellites, called the secondary satellites.

The initial search set, denoted N 0 , can be built as the set of the three-integer combinations N P abc = abc associated with a position contained within the search volume using (4) as:

XP abc k= S P k P k S P kN P abc where S P k is the pseudo-inverse of the primary sys- tem (4).

Thus, at each measurement epoch k, for each candidate N P abc = abc in the set N k , the value of the associated secondary ambiguities N S abc has to be deter- mined.

This can be done using only the phase measurements at epoch k, although a more robust estimate can be elaborated using all of the previous measurements.

The first method is based on the assumption that, for the true solution N P abc = N P , the vector ÑS abc k= S k C S k XP abc k 6 should be very close to N S under normal low noise con- ditions. Thus we can choose to set NS abc k=Round h ÑS abc k i But if the data is affected by multipath, NS abc k may be different from N S .

A more robust estimate can be obtained if all the NS abc k are averaged over time. We will then have, for each secondary satellite i:

NS abc i k=Round 2 4 k 1 X j=ki ÑS abc i j 3 5 7
where k i is the first epoch of lock on the signal transmit- ted by satellite i.

LSAST SPECIFIC STEPS

The estimate NS abc k is used by the LSAST to determine the phase measurements predictions for all the visible satellites, while the MAPAS method uses them to compute the phase measurements for the secondary satellites only.

Using the candidate ambiguity and its associated secondary ambiguity prediction, the LSAST computes the corresponding phase prediction error and its associated squared norm, then checks its validity against 2 values.

The complete ambiguity candidate is

Nabc k= N P abc NS abc k
This candidate is inserted in the complete model (3) to determine a prediction of the complete double differenced observations. We have

abc k= C k Xabc k Nabc k
with Xabc k = S kk S k Nabc k, where S k is the pseudo-inverse of the complete model. As presented by Leick for example [START_REF] Leick | A new Concept for GPS Phase Ambiguity Resolution On-The-Fly: The Maximum A Posteriori Ambiguity Search (MAPAS) Method[END_REF], if

N P abc = N P , then R L abc k=z abc k 1 c kz abc k 2 n k 4
where c k is the covraiance matrix of the complete vector of observations k.

Thus R L abc k can be checked against a likelihood threshold 1 L n k 4 . But this test, called the local test, uses only the current prediction errors and may not be very reliable.

This problem can be overcome if we note that where N T k = P k j=1 n j 4. Thus we can build a more reliable test, based on all the previous residuals. This test is called the global test.

Improper solutions are progressively rejected from the search set, and the best solution is rapidly identified as the candidate associated with the lowest global residual. Confidence in the fact that this candidate is the correct one can be gained by performing the following approxiamte verification, as presented by Frei and Beutler (Frei and Beutler, 1990b):

R G k R G abc k F F NT k NT k
where R G k is the second lowest global residual at epoch k.

MAPAS SPECIFIC STEPS

The MAPAS method uses the predicted value of the secondary ambiguities to compute the associated secondary phase observations S abc k= C S k XP abc k NS abc k and the corresponding phase prediction errors z S abc k.

Then, the value of the a priori probability density function is computed as PP k and SS k are the covariance matrices of the primary and the secondary observations.

The a posteriori probability is computed using Bayes' rule:

P N P abc = N P j z S abc k = fzS abc kjNP abc =NP X abc2N k f z S abc k j N P abc = N P Thus, if the a posteriori probability of a candidate is lower than a predefined threshold, then it is rejected from the set and will not be tested for at the next epoch. If this probability is larger than a preset decision threshold, then this candidate is elected as the correct solution.

ALGORITHMS IMPLEMENTATION AND RUN-TIME CAPABILITIES

As we can see from the previous description, the two algorithms only differ by the steps dedicated to the evaluation of the criterion. This implies a difference in the implementation constraints and in the execution times.

The LSAST checks the squared norm of the residuals against values of the cumulative probability density functions of the 2 and F laws. These values may be needed for a few thousands of degrees of freedom if the procedure is long to converge, and for several error probabilities if the thresholds are tightened up over time. This may be heavy to handle, especially during the tuning period of the procedure.

The MAPAS method evaluates straight probabilities of occurence, so the thresholds can be identified to the design error probabilities.

The execution times of two implementations of these procedures were compared on a statistical point of view (figures 2 and 3). The softwares were developped in ADA at the LTST and run on HP 712/80 workstations. This comparison is simply intended to give an order of magnitude of the difference between the execution times of the two procedures. Indeed, the computation are very specific to the implementation and the processing machines.

The ambiguity resolution trials are launched one after the after, using the satellite constellation visible from 0:00 a.m. till midnight. The phase data is generated every second for two receivers in a simulated dynamic configuration which is reset at the beginning of each trial. No multipath is assumed and the standard deviation of the measurement noise is set to = 3.8 mm. The performances of the two procedures are compared using clean data and data affected by multipath. For each trial we record the required number of samples and the success of the procedure. Simulated data are generated using a program providing theoretical single differences of phase for two receivers of given locations on Earth. One of the receivers is the reference station, of known location, and the other one is the moving receiver, following a certain simulated trajectory. The standard deviation of the phase measurement noise is set to = 3.8 mm. More than 10000 trials were performed in each case.

PERFORMANCES ON REAL AND SIMULATED DATA

The two procedures are first run on simulated data conforming to the multipath-free model (figures 4, 5, 6, 7). Then, specular multipath propagation on the Earth's surface is added, as it may be considered as the major contributing source of multipath during the final approach, as presented by Braasch [START_REF] Braasch | Characterization of GPS Multipath Errors in the Final Approach Environment[END_REF]. The moving receiver is assumed to follow a runway approach path. Any reflected ray of path lengh difference greater than 1.5 C/A code chip is not considered.

To get a first idea of the influence of multipath, we set the reflected to direct signal voltage ratio to = 0.1, without any consideration of the nature of the ground, of the antenna pattern and of the autocorrelation attenuation (figures 8, 9, 10, 11). Thus equation ( 2) is used in its simplest form.

In a second stage, to get a better knowledge of the influence of the Earth's surface, the ground is modeled as wet soil, and a simple model of the receiving an- 2) is used to model the measurement error induced by this ground reflection. The autocorrelation function employed has the ideal triangular shape, and an approximate value of the pseudorange measurement error is given by the weighted delay time of the reflected ray (figures 12, 13, 14, 15).

The success rate of the procedures can be improved by setting the a priori value of the standard deviation of the measurement noise to a larger value than the actual one, to take into account the effect of multipath. When this value is set to 7.6 mm, the global success rate for LSAST is 88.45%, and it is 95.36% for MAPAS. Finally, the two procedures are run on a 20 minute of real data severely affected by coloured noise such as multipath. These data were collected using an 8-channel SEXTANT AVIONIQUE NSS-100 GPS reat two points of known locations. The a priori value of the standard deviation of the measurement noise is set by using the variance of the phase tracking loop, giving an indication of the signal-to-noise ratio. Between 6 and 7 satellites were visible during this 20' segment.

CONCLUSION

The LSAST and MAPAS methods show similar performances when using clean data, but MAPAS seems to need more calculation power to raise the ambiguities.

Both of the procedures show significantly reduced performances when applied to data corrupted by coloured noise such as multipath, and this is a critical problem during the runway approach phase. However, MAPAS appears to be more efficient in counteracting this perturbing effect.

Ongoing investigations aim at the characterization of this robustness, which needs further validation in other real situations.

Figure 1 :

 1 Figure 1: Example of the deformation of the lines of constant phase in space in the case of one specular re ected signal from the Earth's surface. We chose here = 0.45, for a satellite with an elevation angle of 45 o . The curved continuous lines are the lines of constant phase resulting of the presence of the re ected signal, and are to bec ompared with the dashed lines in the ideal case. The straight line is the line of sight of the satellite.

Figure 2 :

 2 Figure 2: Average execution time of the LSAST simulation software as a function of the number of tracked satellites. The global average execution time is 11.2 s.

Figure 3 :

 3 Figure 3: Average execution time of the MAPAS simulation software as a function of the number of tracked satellites. The global average execution time is 14.2 s.

Figure 4 :Figure 5 :

 45 Figure 4: Average duration of the LSAST trials in seconds, as a function of the number of tracked satellites. The global average duration is 19.3 s.

Figure 6 :Figure 7 :

 67 Figure 6: Percentage of successful LSAST trials as a function of the number of tracked satellites. The global percentage is 99.73.

Figure 8 :Figure 9 :

 89 Figure 8: Average duration of the LSAST trials in seconds. The global average duration is 17.4 s.

Figure 11 :

 11 Figure 11: Percentage of successful MAPAS trials. The global percentage is 99.36.

Figure 12 :Figure 13 :

 1213 Figure 12: Average duration of the LSAST trials in seconds. The global average duration is 3.5 s.

Figure 14 :

 14 Figure 14: Percentage of successful LSAST trials. The global percentage is 20.97.

Figure 15 :

 15 Figure 15: Percentage of successful MAPAS trials. The global percentage is 58.34.

Presented at DSNS 96,Saint-Petersburg, 1996

ACKNOWLEDGEMENTS

The author wishes to thank the company SEXTANT AVIONIQUE for providing advices and technical assistance, as well as the technical services branch of the french civil aviation authority, the STNA, for supporting this research. He also wishes to thank Olivier Charleux for helping him in performing the LSAST study.


Thibaut Feydy

Nicolas Barnier
email: barnier@recherche.enac.frbrisset

Pascal Brisset

Nicolas Durand
email: durand@recherche.enac.fr

Mixed Conflict Model for Air Traffic Control

Airspace congestion is today the most critical issue European Air Traffic Management (ATM) has to face. Current real-time Air Traffic Control (ATC) is achieved by human controllers. One of their main tasks is to keep separation between aircraft, asking to the pilots to do basic avoidance manoeuvres. We propose here two mixed CSP models of this separation issue, combining discrete and continuous variables. An implementation of these models allows to produce optimal solutions for problems where numerous aircraft are conflicting.

Introduction

The capacities of European en-route Air Traffic Control (ATC) centres are far exceeded by a constant growth in air traffic demand, resulting in ever increasing flight delays.

To overcome this issue, novel ATM schemes are designed while keeping the hard constraint of a minimal 5 NM (9260 m) horizontal separation between every pair of aircraft to ensure safety. Nowadays, solutions to avoid conflicts are empirical, and human controllers rely on standard routes and traffic organization to devise them. However, the complexity of conflicts could tremendously grow within such future ATM systems, should the aircraft fly on direct routes, from taking-off airport to destination. Then human controllers would no longer be able to solve them efficiently by their own, thus requesting automated solvers.

Former approaches like [START_REF] Durand | Automatic aircraft conflict resolution using Genetic Algorithms[END_REF] use local search (namely genetic algorithms) to solve the conflict problem. These kind of meta-heuristics are well suited to solve large scale and difficult problems when no other relevant techniques are known, but stochastic search inherently lacks existence and optimality proofs.

We propose here to model the problem using CSP in order to get the optimal solution for a given criteria (e.g. the real cost of the manoeuvres, in time and/or fuel consumption). The difficulty to handle the required constraints is related to the fact that the separation must be kept at any time. One simple solution would be to discretize the time and set the constraints for each time step. However we choose here to express the separation between straight pieces of trajectory, using continuous constraints.

After presenting the models and some optimization strategies, we give some indication about their implementation, and eventually provide preliminary results validating the approach.

Mixed Models

An aircraft is characterized by its initial position, speed and heading, which are represented by floating point numbers. In our model we are only taking into account horizontal manoeuvres between aircraft at the same altitude.

Two aircraft are in conflict at a given time if the distance between them is smaller than the safety separation. The considered manoeuvres for maintaining separation involve deviations of the aircraft headings. The starting times of the manoeuvres and the deviation angles are discrete variables since it is more representative of the orders given to a pilot. More precise orders would indeed be irrelevant according to the pilots and aircraft equipment performances.

Horizontal TCAS Model

This horizontal TCAS1 model is the most simple for horizontal deviations: at the initial time, deviation angles are imposed to the aircraft headings in order to avoid conflicts, so there is only one discrete decision variable α i for the aircraft i. This simple model is for emergency situations and could be used for a real-time TCASlike system. This model is not suitable for common traffic control since it does not take into account returning back to the main trajectory.

Let -→

v ij be the relative speed and -→ p ij (t) the relative position between two aircraft i and j, and d the safety distance. We have -→

p ij (t) = -→ p ij (t 0 ) + -→ v ij (t -t 0 ).
There is no conflict between them at a given time t if -→ p ij (t) 2d 2 ≥ 0, meaning that the distance between them is greater than d. Two aircraft will not be in conflict if the discriminant of this polynomial is negative, which lead to an inequality constraint per couple of aircraft, stated as follow:

( -→ p ij (t 0 ) • -→ v ij ) 2 -( -→ p ij (t 0 ) 2 -d 2 ) -→ v ij 2 ≤ 0 (1) with -→ p ij (t 0 ) = x i (t 0 ) -x j (t 0 ) y i (t 0 ) -y j (t 0 ) -→ v ij = v i cos(θ i + α i ) -v j cos(θ j + α j ) v i sin(θ i + α i ) -v j sin(θ j + α j )
where θ i and θ j (initial headings), v i and v j (speeds) and x i (t 0 ), x j (t 0 ), y i (t 0 ), and y j (t 0 ) (initial positions) characterize the problem, whereas α i and α j are the decision variables.

Horizontal Human Controller Model

Now we consider that the aircraft is initially heading toward a waypoint. The autorised manoeuvres still are deviations from the original heading but they can be delayed: a decision variable t 1 for each manoeuvre starting time is added (see figure 1). Moreover each aircraft heads back to its original waypoint after some time δt, so another decision variable is added. The path of an aircraft p is then composed of three segments s p1 , s p2 , s p3 : during the first one, the aircraft is heading toward its waypoint (initial route); during the second one, the aircraft is deviated by an angle α; and during the last one, it is heading back toward its waypoint. Each segment of each aircraft trajectory is potentially in conflict with the segments of the other aircraft, resulting in 9 constraints per pair of aircraft.

Considering the polynomial similar to the one studied in (1), there is a conflict between two segments s ai and s bj if the three following conditions are verified:

there is a common time when the aircraft a is on the segment s ai and the aircraft b is on the segment s bj ; -the discriminant of the corresponding polynomial is positive;

at least one root of the polynomial is in the common time of both segments.

The negation of these statements leads us to the disjunction of reified constraints no common time ∨ is le(discriminant) ∨ root not in common time.

Optimization and Search Strategy

Different optimization strategies were studied, starting with the simple "minimize maximal deviation angle"(in absolute value). However, two solutions with the same maximal deviation angle can be very different from a human point of view, so we considered more precise optimization criteria. To improve the quality of solutions, an efficient strategy was to perform a multi-stage hierarchical optimization on the angles, corresponding to the following lexicographic minimization. Let

c(η) = card({|α i | = η})
be the number of (absolute values of) deviations equal to a given angle η, and {η 1 , . . . η n } their possible values sorted in decreasing order, the objective is then to lexicographically minimize the tuple: (c(η 1 ), . . . , c(η n )) = (c(30), c(20), c(10), c(0)) as in our application, deviations can range over {-30, -20, -10, 0, 10, 20, 30}.

Another optimization strategy is to maximize the number of aircraft not deviated. It is a relevant criterion for a human controller but the computation times are more costly. A third one, more focused at airlines concern, would be to minimize the total lengthening of the paths; however, this objective is yet too hard to achieve, at least for our complete CP system.

Implementation

To handle these problems the support for reals variables and constraints was added to FaCiLe [START_REF] Barnier | FaCiLe: a Functional Constraint Library[END_REF]. Programs may mix real and integer variables, as it is the case with our models.

Interval Arithmetic

Since it is difficult to make assumptions on the definition range of functions or the domains of variables, the arithmetic considered is an extended interval arithmetic. No exceptions are raised during operations over intervals but the domains of interval functions are consistently extended to infinite and empty values. The interval arithmetic library used to implement constraints is Filib++ [START_REF] Lerch | Filib++ -Interval Library, Specification and Reference Manual[END_REF][START_REF] Zilinskas | Comparison of packages for interval arithmetic[END_REF].

Propagation

The consistency algorithm implemented is based upon BC4 proposed by [START_REF] Benhamou | Revising hull and box consistency[END_REF] but enforces a weakening of the box ϕ -consistency [START_REF] Granvilliers | Box consistency through weak box consistency[END_REF]. The latter only considers intervals of a single width ϕ when looking for the leftmost and rightmost quasi-zeros, which can be costly when narrowing intervals of very different sizes.

While searching for the quasi-zeros, our FaCiLe implementation handles intervals of width αl + ϕ, l being the width of the narrowed interval and α ∈ [0, 1] an additional tuning parameter. The key idea behind this scheme is to work with a floating precision of the narrowing to efficiently handle domains of different sizes. Hence, costly computations of precise bounds are avoided for large intervals.

Integration within FaCiLe

FaCiLe is a Functionnal Constraint Library written in Ocaml. The module system of Ocaml allows to define functors, equivalent to functions at the module level. A functor builds a new module from a module taken as argument. The addition of variables over the reals to FaCiLe simply amounts to the application of the variables functor to a new domain module. This module provides a new type of domains and the necessary functions to handle them. Goals and constraints over these new variables can easily be written using the provided generic mechanisms.

Results

A common conflicting situation in ATC involves aircraft converging to the same waypoint. This has first been simulated with perfectly converging aircraft. Aircraft are disposed regularly on a circle, converging to the centre. The addition of noise to the positions, speeds and headings of aircraft breaks the symmetry and simulates a more realistic ATC event.

Problems up to 15 aircraft have been solved in a few seconds using the TCAS model and the hierachical optimization procedure (see figure 2), which is a reasonable computation time for emergency avoidance situations. For this problem, the deviation angles may range over {-30, -20, -10, 0, 10, 20, 30}, which are realistic ATC orders. Problems up to 30 aircraft have been solved as well in a few minutes. The horizontal human controller model have been used to optimally solve problems of two and three aircraft (see figure 3), which represent almost 90% of the total number of conflicts detected by the CATS 2 simulator [DA97] during a typical day of traffic. The same deviation angles were used for this model with a time step of 10 seconds over 10 minutes time periods.

Conclusion and Further Works

We have presented two mixed CSP models to optimally solve air traffic conflicts with horizontal manoeuvres: a simple TCAS-like model for emergency avoidance and a more complete human controller model to automate ATC. These models have been implemented with FaCiLe by adding real constraints and variables to the library.

Computational results for the first model compare well with former local search approaches [START_REF] Durand | Automatic aircraft conflict resolution using Genetic Algorithms[END_REF] like genetic algorithms, additionnally providing existence and optimality proofs. This model features a major improvement over the current TCAS, being able to solve conflicts involving more than 2 aircraft and issuing less drastic manoeuvres than changes of altitudes.

The second one is much more complex, and only conflicts involving 2 or 3 aircraft could be solved with the search strategies we have tried.

These results also demonstrates the versatility of FaCiLe and its ability to simply integrate new solvers. However, some work still has to be done to overcome the limitations of the "human" model (smarter strategies, more powerful propagations, relaxations of the model...), and the algorithms have to be tested on conflicts situations from real traffic data to be validated.

Fig. 1 .

 1 Fig. 1. Decision variables for the Horizontal Human Controller Model

Fig. 2 .

 2 Fig.2. Routes, at a given time, without and with conflicts solving for a 10 aircraft problem solved with the TCAS model.

Fig. 3 .

 3 Fig. 3. Conflict solving with return on main trajectory.

Traffic Control Avoidanc System: airborne device issuing emergency vertical manoeuvres to help short-term avoidance for a single pair of aircraft.


Sébastien Carcanague

/ Enac

France Olivier Julien

Olivier Julien

Willy Vigneau

A New Algorithm for GNSS Precise Positioning in Constrained Area

et de l'Espace" in 1999. He has been in charge of the Radionavigation unit at M3 Systems since 2002, and is especially managing projects related to signal processing studies applied to GPS, EGNOS and GALILEO. He is now technical director of M3 Systems and is following M3 Systems' development on GPS, EGNOS and Galileo algorithms and receivers. Recently, he has been particularly involved in several EGNOS and Galileo related projects as project manager and signal processing engineer.

Christophe MACABIAU graduated as an electronics engineer in 1992 from the ENAC in Toulouse, France. Since 1994, he has been working on the application of satellite navigation techniques

Phase Lock Loops, compared to the delay estimate made by the receiver Delay Lock Loops. PPP and RTK techniques are not common for mobile users travelling in difficult environment, like urban area, for several reasons: frequent carrier-phase cycle slips and loss of tracking, reduced number of visible satellites, strong multipath on carrier phase and code pseudorange measurements. Indeed, in a constrained environment, several GNSS signals can be blocked, and the received GNSS signals are usually a combination of several multipath components. The aim of the proposed paper is to present a new precise positioning algorithm, based on single-differenced (between stations) measurement that applies to constrained environments. This technique enables a decreased convergence time of RTK in these environments, and provides a better code-only position if the rover carrier phase measurements are not available. Indeed, the contribution of the reference receiver noise to the initial position estimation is drastically reduced by using a non ambiguous carrier phase combination instead of code measurements. The new model is based on a careful management of code-phase hardware biases that keeps the rover widelane ambiguity as an integer and makes use of the newly and freely available widelane satellite biases originally intended for PPP with ambiguity resolution. The advantages of using single-differenced observations instead of double-differenced measurements are also underlined in this paper.

INTRODUCTION 1.1 PRECISE POSITIONING IN URBAN ENVIRONMENT

Current GNSS positioning accuracy is sufficient for most civil users. However, a number of applications need a centimeter-level position: surveying, crustal motion monitoring, precise agriculture… These applications are classically using Real-Time Kinematic (RTK) and/or Precise Point Positioning (PPP) to provide this required accuracy. They both use the accuracy of the carrier phase measurements by estimating the unknown ambiguity, as a float or an integer. The 2 techniques are very efficient in open-sky environment, with good satellite geometry, few cycle slips and reduced multipath. New applications targeting users in urban areas would significantly benefit from a precise positioning capability in urban area such as Intelligent Transportation System (ITS), UAV navigation system, etc... However, it is difficult to directly transpose RTK or PPP to these applications since their environment is not adapted for these methods [START_REF] Kubo | Instantaneous RTK Positioning with Altitude-aiding for ITS Application[END_REF]:

 Multipath is very high on pseudorange and carrier phase  Cycle slips are frequent  Carrier phase availability is low  Satellite geometry is weak Figure 1 shows the cumulative density function of the duration between 2 consecutive tracking losses for GPS L1 C/A data collected at 1 Hz in Bordeaux's beltway during 1 hour and 20 minutes. The antenna was mounted on the roof of a car, which was driven along the beltway. The receiver was a Septentrio PolarX2. A satellite was considered visible from the first epoch that it was tracked until the last it was tracked. During this time interval, the occurrence of data gaps and their duration on each satellite were carefully checked. All satellites above 5 degrees were considered.

Figure 1 Cumulative density function of the duration between 2 consecutive tracking loss of GPS L1 C/A Figure 1 shows that about 70% of the carrier phase are continuously available for a duration smaller than 25 seconds. That is why in those conditions, nearinstantaneous ambiguity resolution is required [START_REF] Kubo | Instantaneous RTK Positioning with Altitude-aiding for ITS Application[END_REF][START_REF] De Jonge | Epoch-by-EpochTM Positioning and Navigation[END_REF].

As the ambiguity resolution convergence time, typically requires at least 30 minutes for PPP, it does not seem well suited for urban environment, even with recent efforts on cycle slip correction [START_REF] Banville | Improving Real-Time Kinematic PPP with Instantaneous Cycle-Slip Correction[END_REF]. The weak geometry and the high multipath would lengthen convergence time just as the low carrier phase availability. RTK thus seems to be the best a priori choice for GNSS precise positioning in urban environment.

RTK IN URBAN ENVIRONMENT

RTK is more robust to signal blockage and cycle slips than PPP because its convergence time can be significantly shorter and a precise position can be obtained more quickly. It is classically based on double differences of observables to eliminate common biases. An ambiguity resolution technique is then applied to solve for ambiguities as a set (geometry-based techniques) or independently (geometry-free techniques). However, previously stated effects of urban environment are very harmful:

 Multipath on pseudorange. Multipath reduces the accuracy of pseudoranges. It essentially impacts the accuracy of the float solution. It will result in a longer convergence time.  Multipath on carrier phase. They will mostly affect the estimation process of the ambiguities. The residuals will be larger and correct ambiguities can be rejected erroneously. Moreover, LAMBDA method [START_REF] Teunissen | Least-squares estimation of the integer GPS ambiguities[END_REF] classically used for ambiguity resolution is not optimal in the presence of biases [START_REF] Teunissen | Integer estimation in the presence of biases[END_REF] since the decorrelation phase of the method amplifies their effect [START_REF] Henkel | Modified LAMBDA for absolute carrier phase positioning in the presence of biases[END_REF].  Frequent cycle slips/complete loss of lock.

Carrier phase tracking loop is very sensitive to dynamics and the environment surrounding the receiver which can cause complete loss of lock. Some receivers can recover from this loss of lock in one or 2 seconds but other receivers can take up to 20 seconds [START_REF] De Jonge | Epoch-by-EpochTM Positioning and Navigation[END_REF]. During this time, the user won't be able to use one or several carrier phase measurements which will bring accuracy down. Cycle slips are also sometimes difficult to repair and force to estimate the ambiguity again. Moreover, in conventional RTK, a convergence time is usually required before reaching precise positioning level. Every time a cycle slip or a loss of lock occurs, the convergence time has to be restarted. Then instantaneous ambiguity resolution would be the best solution as in [START_REF] De Jonge | Epoch-by-EpochTM Positioning and Navigation[END_REF] and [START_REF] Wu | Fast Ambiguity resolution in RTK-GPS Positioning for Marine Navigation[END_REF] [START_REF] Odjik | What does "geometry-based" and "geometry-free"mean in the context of GNSS? GNSS Solution[END_REF]. Each of these effects makes single-epoch ambiguity resolution harder. Under these conditions, RTK algorithms have to be improved by first denoting that double difference is not optimal in the sense that:

 Although it removes important biases, code double differencing also adds the code noise and multipath measurement errors of the reference receiver in the double difference measurements.  Double difference further correlates the ambiguities [Enge,et al.,2nd Edition]. This is very important since the usual decorrelation step used by ambiguity resolution techniques amplifies biases when present.  The choice of the reference satellite(s) implies problems for the RTK program designer, to pass the change both in the ambiguities value and the accumulated covariance matrix in the positioning filter [START_REF] Teunissen | GPS for Geodesy[END_REF]]. To improve single-epoch ambiguity resolution in difficult environment, a new positioning model referred to as Optimized Between-stations Integer WidelANe (OBIWAN) will be introduced in this paper. It is optimized in the sense that spatially correlated errors affecting observations are cancelled by differentiation, while the noise and multipath added by the reference receiver in the code measurement of the rover are negligible. It is based on single difference of observations, which present a number of advantages compared to double difference model. Finally, code-phase biases are carefully handled, to allow the resolution of the widelane ambiguity as an integer. In the first part, the scenario will be presented. Necessary background on Melbourne-Wübbena combination and hardware biases will be presented in the second part. In the third part, the OBIWAN algorithm will be presented and detailed. Finally in the fourth part, the algorithm will be tested with real data showing that it improves singleepoch ambiguity resolution.

ASSUMPTIONS ON THE TARGET USER ENVIRONMENT

The technique described in this paper is based on the following assumptions:

 The system is composed of at least 2 stations: 1 reference receiver and 1 rover receiver  The reference receiver is of good quality in an open-sky environment, and is a dual-frequency receiver (L1/L2 or L1/L5) with only rare, if any, cycle slips. The rover receiver can be of a much lower quality and/or in a less friendly environment. However, it has to be dualfrequency and track the same signals as the reference receiver.  The 2 stations are very close (short baseline case: <10km). The tropospheric delay and the ionospheric delay affecting the observables of the 2 receivers are almost the same. Then, if we difference the observables from the 2 receivers, the residual ionospheric delay and the tropospheric delay remain at centimeter-level.

In practice, those requirements are not very restrictive. For instance, the city of Paris (intra-muros) is approximately contained in a circle with a diameter of 10km. 1 single good quality receiver could be used to improve positioning accuracy of thousands of potential users. Moreover, the price of dual frequency receivers is expected to drop in the upcoming years, with the full availability of L2C signal.

BACKGROUND FOR THE PROPOSED ALGORITHM

The code and phase measurement model associated to one satellite for a signal at the frequency L i can be written as:

where is the geometric range, is the receiver clock offset, is the satellite clock offset, is the ionospheric delay on frequency , T is the tropospheric delay, and are the receiver hardware delay of the code and carrier phase respectively, and are the satellite hardware delay of the code and the carrier phase, is the carrier phase integer ambiguity, is the wavelength of the signal, and are noise/multipath of code and carrier phase.

Because it is difficult to solve for the integer ambiguities of the raw phase measurements due to the very small wavelength of the carrier, many applications first go through the use of phase measurements combinations that create new measurements with extended wavelength. This is for instance the case for the widelane phase combination given by: with where is the carrier phase observable on the frequency .

The resulting widelane phase measurement is also ambiguous, but has kept an integer ambiguity with an longer associated wavelength.

The so-called Melbourne-Wübbena combination ( [START_REF] Melbourne | The Case for Ranging in GPS Based Geodetic System[END_REF] and [START_REF] Wübbena | Software Developments for Geodetic Positioning with GPS Using TI 4100 Code and Carrier Measurements[END_REF]) uses the widelane phase observable and the narrowlane code observable to isolate the widelane ambiguity. The narrowlane code combination is given by: with and the code observable on the frequency .

The Melbourne-Wübbena combination can then be obtained through [START_REF] Banville | Satellite and Receiver Phase Bias Calibration for Undifferenced Ambiguity Resolution[END_REF]:

(1) with and the Melbourne-Wubbena biases for receiver and satellite respectively, the noise of the code narrowlane combination and the noise of the carrier phase widelane combination.

The resolution of the widelane ambiguity using the Melbourne-Wübbena combination has been at the center of PPP algorithm developers for a few years, as it is the key of undifferenced ambiguity resolution. The reader might refer to [START_REF] Laurichesse | Integer Ambiguity Resolution on Undifferenced GPS Phase Measurements and its Application to PPP and Satellite Precise Orbit Determination[END_REF], [START_REF] Collins | Isolating and estimating undifferenced GPS integer ambiguities[END_REF] and [START_REF] Ge | Resolution of GPS Carrier-phase Ambiguities in Precise Point Positioning (PPP) with Daily Observations[END_REF] for more details on this technique. To do so, it is important to have access to the Melbourne-Wübbena satellite biases . These biases are now freely available and released with a week latency by the CNES-CLS IGS center (.wls products) [START_REF] Loyer | CNES-CLS IGS Analysis Center : Results and processing strategies[END_REF]. Once the satellite biases are removed, the user can determine both the Melbourne-Wübbena receiver bias and the ambiguities by using, for instance, a sliding averaging window.

In the next section, the new algorithm referred to as OBIWAN will be presented. It aims at fixing the widelane ambiguity more efficiently. Fixing the widelane ambiguity correctly leads to a decimeter-level position, which can be sufficient for many applications. It can also pave the way to L1 ambiguity resolution and centimeterlevel positioning [START_REF] Liu | Resolution Strategies in Single and Multiple Reference Station Scenarios[END_REF].

ALGORITHM PRESENTATION

OBIWAN ALGORITHM

As explained earlier, one of the disadvantages of the use of code double difference to initialize float solution is that it adds the noise/multipath error of the rover measurements with the noise/multipath error of the reference receiver measurements since these kinds of error are uncorrelated between receivers. Moreover, the double difference increases the correlation between the ambiguities which makes more complex the ambiguity resolution step. The algorithm proposed hereafter offers a solution to both problems.

First, let's form the widelane and the narrowlane combination on the reference receiver and the rover:

where the superscript 'ref' refers to the reference station and the superscript 'rov' refers to the rover.

Comparing equation (1) and equation ( 2), we can first denote that the Melbourne-Wübbena biases can be expressed as:

The OBIWAN algorithm is as such:

 Resolution of the widelane ambiguities on the reference receiver: The widelane ambiguities of the reference receiver are solved by removing satellite biases from Melbourne-Wübbena combination (equation ( 1)) and averaging over a sliding window in order to reduce measurement noise. The optimal window length depends on satellite elevation and goes from 5 minutes to 30 minutes with 1 Hz data [START_REF] Laurichesse | Integer Ambiguity Resolution on Undifferenced GPS Phase Measurements and its Application to PPP and Satellite Precise Orbit Determination[END_REF]. The float value obtained after the averaging can be decomposed into an integer ambiguity (specific for each satellite) and a hardware receiver bias (common to all satellites). This couple is defined modulo a widelane cycle. The output of this step is then a set of widelane ambiguities and a receiver bias. Once this estimation step is complete, the widelane carrier phase measurements of the reference receiver can be considered as a very precise and unambiguous observation. This step should not be a problem as the reference receiver is assumed to be in a friendly environment with a good satellite visibility.  Combinations using the unambiguous carrier phase measurements of the reference receiver and the reduced-noise code measurements of the rover.

Two observables are created in this step: o The first is a single difference of the widelane carrier phase observables. o The second is the Melbourne-Wübbena combination, with the phase of the receiver and the code of the rover This provides; i.e.:

where it was assumed that the stations are close enough so that the tropospheric delays and ionospheric delays are cancelling out and where is the difference of the 2 receiver clocks. It can be seen that equation (5b) contains the known Melbourne-Wübbena satellite bias of equation (4b). This is logical as from a satellite bias point of view, forming a combination of observables from the same receiver or from different receiver is similar.  Removing known reference receiver ambiguities and satellite biases: In this step, the user removes the known satellite biases and the estimated widelane ambiguity of the reference receiver (obtained from step 1) which gives:

where represents a new clock term that includes the hardware bias term of equation (5a) Denoting that , then the final measurement model is: with the correction.

To summarize, two new observables have been created that are:  An ambiguous precise observable that is the singledifferenced widelane carrier phase.  An associated unambiguous measurement that is the Melbourne-Wübbena combination, formed with the widelane phase of the reference receiver and the narrowlane code of the rover. This observable is corrected with the satellite widelane biases and the estimated widelane ambiguities of the reference receiver. These two new measurements can be used to solve for the widelane ambiguity of the rover based on a precise, single-differenced unambiguous measurement.

Simplifying notations, we have obtained a singledifferenced integer widelane model that is such that: (8) With a geometric term defining the vector from the reference receiver to the rover, a clock term common to every satellite, an integer ambiguity and being exactly the Melbourne-Wübbena receiver bias of the rover.

A scheme of the algorithm can be found on Figure 2. A first remark is that the unambiguous measurements (7b), I a very accurate positioning pseudorange as all the biases have been removed by the single difference and the noise introduced by the reference receiver carrier phase is negligible. This precise pseudorange should provide far better positioning accuracy than double differenced code measurements. Moreover, it is interesting to denote that an increase of the number of reference station would directly result in a higher accuracy as the noise in the solution most entirely comes from the rover.

Secondly, the integer ambiguity in equation ( 7a) is exactly the integer ambiguity obtained with Melbourne-Wübbena combination on the rover receiver. This property really simplifies combined geometry-free and geometry-based ambiguity resolution. For instance, widelane ambiguities of the rover can be estimated both with the previous algorithm, i.e. estimating the position/clock and all ambiguities altogether, and by averaging Melbourne-Wübbena combination of each satellite individually.

Thirdly, this technique is based on differences between stations to eliminate spatially correlated biases. In term of ionospheric delay, the residual bias is the same for equation ( 7a) and (7b). It is 1.28 times more important than the residual in meters obtained by differencing L1only measurements between the 2 stations. However, it is smaller in units of cycles, as the wavelength of the widelane is larger than the wavelength on L1. This should allow successful ambiguity resolution with baselines from a few kilometers to a few tenths of kilometers, depending on ionospheric activity.

LIMITATIONS OF OBIWAN FOR SINGLE-FREQUENCY USERS

The concept of using the unambiguous carrier phase of the reference receiver instead of the code seems new and no reference to it was found in the literature by the author. OBIWAN uses the specific property of the widelane carrier phase and the narrowlane code observations. It is difficult to extend this technique to single-frequency users for different reasons:

 The ambiguity is very hard to isolate on the reference receiver. In the code minus phase observable, an ionospheric term remains. For instance, with L1 data:

The advantage of the MW combination is that the ionospheric term is removed. Moreover, the code observable is noisier than the narrowlane observable. The code-phase receiver bias would also be very difficult to estimate for the same reason.  In a monofrequency observable, the phase wind-up would affect the single differenced carrier phase observable. This effect is both due to a rotation of the satellite antenna and the receiver antenna, since GPS signal is polarized.

In the case of between station observables, the receiver rotation contribution would remain. Any full clockwise rotation of the receiver antenna around the vertical axis would result in an apparent increase of 1 cycle in the carrier phase observable [START_REF] Banville | Antenna Rotation and Its Effects on Kinematic Precise Point Positioning[END_REF]. As the receiver rotation is very difficult to model without any external attitude information, a monofrequency version of the algorithm described in the first part couldn't be applied.

Since the widelane observable is not affected by the phase wind-up ( [START_REF] Banville | Improving Real-Time Kinematic PPP with Instantaneous Cycle-Slip Correction[END_REF]), no external information on receiver antenna rotation is required in our case.

TEST ON REAL DATA

POSITIONING ACCURACY PERFORMANCE

OBIWAN was tested on 3 IGS stations [START_REF] Dow | The International GNSS Service in a changing landscape of Global Navigation Satellite Systems[END_REF] situated in Washington, USN0, USN3 and NRL1 with data from May 1 st , 2010. USN0 was considered as the reference station. The baseline USN0-USN3 is approximately 0.3km long and the baseline USN0-NRL1 is approximately 11km long. Observation sampling rate was 1/30 Hz. Data from the 3 stations were synchronized and will be considered simultaneous in the following results. The signals tracked were GPS L1 P and GPS L2 P. Note that the technique could be applied to GPS L1/L5 receivers or GALILEO E1/E5a receivers, but available satellite Melbourne-Wübbena biases are currently only for GPS P1 and P2 code measurements. It is difficult to know if code measurements are carrier-smoothed, but it is highly probable that they are.

Cycle slips were first detected and repaired on the reference station and the widelane ambiguity was determined from the Melbourne-Wübbena combination.

Then, the narrowlane code is formed on the rover, combined with the widelane phase of the reference receiver. If the reference receiver and the rover have similar level of noise and multipath, this unambiguous measurement is expected to reduce noise by a factor of 2 compared to double difference with P1 only, from noise propagation laws. Error standard deviation is approximately divided by , as expected from noise propagation laws. Indeed, the pseudoranges from equation (7b) are 2 times more accurate but there are 2 times less observables compared to a filter including P1 and P2.

SINGLE-EPOCH AMBIGUITY RESOLUTION PERFORMANCE

The improvement in the initial position estimation brought by the use of the unambiguous widelane phase of the reference receiver instead of classic code measurements has been shown to be significant. The aim of this part is now to quantify the impact of the new model in term of single-epoch ambiguity success rate. As stated before, the advantage of the new model for ambiguity resolution is twofold. First the initial position and float ambiguities will be more precise than when using code double difference, as the noise/multipath contribution of the widelane carrier phase on the reference receiver is very small. Secondly, as we do not perform double difference, the ambiguities are less correlated than with the double difference model. As the decorrelation step of the LAMBDA method is responsible of biases amplification, reducing the correlation should also reduce bias amplification.

The LAMBDA method was implemented directly as provided in the Matlab version of Delft University. A least-square filter was first used to estimate the coordinates, the clock term and float ambiguities.

LAMBDA algorithm was then used, with float ambiguities and their associated covariance matrix as input. The comparison in terms of ambiguities resolution success rate was performed on 2 baselines. The first was between USN0 and USN3, which is approximately 0.3km and the second was between USN0 and NRL1 which is approximately 11km. Tropospheric delay and ionospheric delay were voluntarily not corrected on the last baseline, to see the impact of biases on the time to fix ambiguities.

To separate the impact of the decorrelation and the improved accuracy of the float solution on the success rate, we have computed the percentage of correct first fix with classic double difference, with the method exposed before and the method exposed before differenced between satellites. Differencing between satellites equation ( 7a) and (7b) avoids estimating the clock term and the receiver bias, but correlates ambiguities. The impact of ambiguities correlation can then be deduced. Indeed, if the bias is estimated correctly, the difference between single difference and double difference of the new algorithm observables should only come from the correlation of the ambiguities in the double difference, as they have the exact same accuracy in the initial float solution. Results can be found on Note that these values are below the values we can find in the literature (usually close to 100% fix rate). This might be due to the fact that we have not performed any validation of the ambiguities. Higher fix rate would have been obtained if we had rejected ambiguities that didn't pass the validation test. However, it seemed more relevant to compare raw fix rate before validation to see the impact of improved accuracy on ambiguity resolution. We can see that the biggest contribution to the success rate is the improved accuracy of the float solution in the ambiguity resolution process. However, the fact that the ambiguities ave not been correlated further with a double difference brings a little improvement of up to 0.5% in the case of longer baseline, which is more affected by biases. The reason why this improvement is only marginal might be the difficult step of accurately estimating the rover hardware bias. In any case, the new model improves single-epoch ambiguity resolution even if observables are between-satellite differenced, which avoids the bias estimation.

CONCLUSION AND FUTURE WORK

The OBIWAN algorithm presented in this paper reduces the contribution of the reference receiver to the initial float solution in terms of noise and multipath, as it uses only the unambiguous widelane carrier phase instead of code measurements. Moreover, integer ambiguity resolution becomes possible with a single-difference model using code and phase of the rover, which avoids further correlation of the ambiguities and greatly simplifies the work of the RTK program designer. It has been shown that the level of noise and multipath was approximately divided by in the position domain compared to a double difference of code on L1 and L2, if the noise and multipath level are similar on the 2 stations. The less correlated ambiguities also bring a modest improvement when carrier phase measurements are biased. However, the method has not been tested with a rover receiver that is in a difficult environment which will make the estimation of the Melbourne-Wübbena receiver hardware bias more difficult. In this paper, a single value was taken for a day, but average over shorter periods showed variations. However, it is difficult to determine from where those variations come from. It can be either:

 Multipath residuals that bias the average processing  Actual hardware bias variation If the estimation of this bias is too complicated due to the environment, the new method should be used with differences between satellites, to eliminate the issue of hardware bias estimation.

Figure 2

 2 Figure 2 Diagram of the OBIWAN algorithm. The code of the reference receiver is no longer used in the positioning filter while the widelane ambiguity is kept as an integer.

Table 1 Noise+multipath statistics of code double differences and the observable of equation (7b), for different double differences. Double-differenced carrier phase on L1 is subtracted to all observables to remove the geometric part. Standard deviation is then computed. Baseline length was 0.3km.

 1 Results from Table1confirm this theory. The accuracy on each pseudorange is approximately increased by a factor of 2. However, it is not totally fair to directly compare the level of noise/multipath. Indeed, the new method provides 2 times less observables to the position filter than double differences on both L1 and L2, as only the narrowlane combination on the rover can be used. Considering the observables P1 and P2, a baseline vector solution was then estimated with a least-square on a single-epoch basis. Results can be found on Table2.

			PRN	PRN	PRN	PRN
			double	double	double	double
			difference	difference	difference	difference
			1	2	3	4
	Std	of	0.56	0.64	0.56	0.49
	Double
	Difference
	P1 (m)
	Std	of	0.54	0.59	0.48	0.44
	Double
	Difference
	P2 (m)
	Std	of	0.27	0.31	0.25	0.21
	equation
	(7b)
	differenced
	between 2
	satellites
	(m)
	To compare similar quantities, equation (7b) is
	differenced between 2 satellites to eliminate the clock
	term.

Table 2 Estimated coordinates error statistics for a baseline of 0.3km

 2

Table 3 .

 3 The hardware bias of the rover was obtained by averaging over a whole day and by taking a unique value. However, more refined estimation on shorter time windows has to be performed for stations with less stable bias than USN0, USN3 and NRL1.

	Baseline length	Single-epoch	ambiguity
		resolution success rate
	0.1km	Double	92.30%
		difference
		OBIWAN	98.45%
		OBIWAN	98.30%
		differenced
		between
		satellites
	11km	Double	85.45%
		difference
		OBIWAN	93.65%
		OBIWAN	93.15%
		differenced
		between
		satellites

Table 3 Single-epoch ambiguity resolution success rate with different baseline, mask angle=20°, for an entire day.

 3

Presented at ION ITM 2011, San Diego

ACKNOWLEDGMENTS

The authors would like to thank Denis Laurichesse, Félix Pérosanz and Sylvain Loyer for their advices and remarks on this paper, as well as the Association Nationale de la Recherche et de la Technologie (ANRT) for the financial support.