Myriam Foucras
email: myriam.foucras@abbia.fr

Bertrand Ekambi

Fayaz Bacard

Olivier Julien

Christophe Macabiau

Optimal GNSS Acquisition Parameters when Considering Bit Transitions

Keywords: Acquisition, Bit sign transition, Probability of detection

This article focuses on the bit sign transition and its impact on the acquisition performance in terms of probability of detection. To do so, a theoretical study on the correlation process considering bit sign transition is done leading to the expression of the probability of detection, expressed in function of the bit transition location. Based on this, Monte-Carlo simulations were run to determine the acquisition performance degradations in terms of sensitivity losses and probability of detection for several GNSS civil signals. This allows determining the optimal acquisition parameters when bit sign transitions are considered during the acquisition process.

I. INTRODUCTION

In GNSS receivers, the acquisition process, the first stage of the signal processing module, consists in assessing the presence of GNSS signals and providing a rough estimation of the incoming signal parameters: the Doppler frequency and the code delay. To detect the presence of the signal, the received signal is correlated with a succession of locally generated replicas until the acquisition detector crosses a predefined threshold. One criteria of acquisition performance is the probability of detection which is the probability to detect the signal when the parameters of the local replica are (close to be) correct. This probability should be as high as possible but under unfavorable conditions, such as adverse environments, the detection becomes a challenge.

Initially, the GNSS signals were only defined on one component (such as GPS L1 C/A) but the new generation of signals has two components (such as GPS L1C, GPS L5, Galileo E1 OS, Galileo E5a/b…). One component referred to as data component contains the navigation message which can be seen as a random binary sequence from the point of view of the acquisition process. The other component is the pilot component, which does not carry any useful information but is modulated by a secondary code which is a known short binary sequence. For the acquisition process, with no aid and a priori information, the position of the bit transition on both components is not known and this implies considerable degradation on the acquisition performance as it was already presented by authors in [START_REF] Foucras | An Efficient Strategy for the Acquisition of Weak Galileo E1 OS Signals[END_REF].

The introduction of the pilot component and then the split of the total signal power between two components impact the way to process such a signal. Indeed, depending on the objective, these composite signals can be acquired by using correlator outputs based on one of the two components or based on both components. Considering the bit transition problem, one can wonder what the most suitable acquisition technique is, as when using both components, transitions can occur on both components.

The general context of this study is to acquire GNSS signals with a received C/N 0 (carrier to noise ratio) of 27 dB-Hz but with a high probability of detection (95%). As often neglected in the literature, a first work was done assuming that the data bits are constant during the correlation operation. The second step consists in studying the acquisition assuming no constancy because there is no reason that the integration interval is aligned with the data bit. Then, the motivations behind the presented investigation are to study the impact of the bit transition on the GNSS signals acquisition and provide optimal acquisition parameters accordingly. To do so, the mathematical model of the acquisition detector in presence of bit transitions is developed in order to express the probability of detection in presence of bit transitions. This is applied to a number of GNSS signals. Based on this, Monte-Carlo simulations were run to determine the optimal acquisition parameters which choice clearly depends on the received C/N 0 , the signal structure and the acquisition technique using one or both components for two-component signals. This study is led for different GNSS signals which present feature differences in signal structure.

The article is divided into the following sections:

• A first section serves as a review of the GNSS signals and the acquisition process when the bit transition is not considered.

• The second section is dedicated to the mathematical model of the correlator outputs and then the probabilities of detection in presence of bit sign transitions.

• The following section, based on Matlab simulations presents the acquisition performance losses in terms of probability of detection due to bit transitions and optimal acquisition parameters

• The last section concludes the paper remaining the main points of interest and results.

II. GNSS SIGNALS AND ACQUISITION MODELS

A. GNSS signals' model

In the case of this study, four GNSS signals are considered: GPS L1 C/A, GPS L1C, GPS L5 and Galileo E1 OS.

Where

• is the spreading code (PRN) period

• is the data bit duration

• is the secondary code bit duration

A bit transition is defined as the transition between 2 consecutive bits and during a bit transition, a sign transition can occur or not as illustrated in Fig. 1.

Bit (n-1) -1

Bit n -1

Bit (n+1) +1

Bit transition Bit sign transition Fig. 1: Bit sign transition scheme

1) GPS L1 C/A signal

The GPS L1 C/A signal is a unique data component [START_REF] Navstar | GPS Space Segment/User Segment L1C Interface (IS-GPS-800C)[END_REF]. The GPS L1 C/A signal acquisition performance depends on the correlation duration. Indeed, because the data bit duration is twenty times longer than the spreading code period, it is possible to use as correlation durations an integer divider of the data bit duration such as 1, 2, 4, 5, 10 or 20 ms [START_REF] Kaplan | Understanding GPS: Principles and Applications[END_REF].

The expression of the received GPS L1 C/A signal at the correlator input is:

/ , cos 2 (1)
Where

•

, designs the spreading code on the data component when stands for " " or on the pilot component for standing for " " • √2 is the amplitude of signal and is the total signal power at the correlator input

• is the data sequence • is the intermediate frequency • is the incoming Doppler frequency • is the incoming code delay •
is the incoming noise which is assumed to be a white noise with centered Gaussian distribution with a constant two-sided power spectral density equal to /2 dBW/Hz • is the initial phase

2) GPS L1C signal

The GPS L1C signal, defined in [START_REF] Navstar | GPS Space Segment/Navigation User Interfaces (IS-GPS-200G)[END_REF], provides a number of advanced features including a power difference in both components: 75% of power in the pilot component for enhanced signal tracking and 25% of power in the data component. At the end, the received signal is represented as follows (in-phase representation because the relationship between the two components is not yet specified [START_REF] Betz | Description of the L1C Signal[END_REF]): On the data component, the probability of a bit sign transition, noted as , is assumed to be 50%. On the pilot component, the probability of a bit sign transition depends on the secondary code structure. For the GPS L1C signal, there are 63 secondary codes of 1800 bits that were defined. For each one, the probability of a bit sign transition is given in Fig. 2. As it can be observed, the probability of a bit sign transition on the pilot component noted as is close to 0.5. In this work, it is assumed that for all of the GPS L1C secondary codes, = 0.5.

3) GPS L5 signal

The GPS L5 signal presents the particularity to also have a secondary code of length 10 bits (Neuman-Hofman) on the data component, . The two components are in quadrature [START_REF] Navstar | GPS Space Segment/User Segment L5 Interfaces (IS-GPS-705C)[END_REF] as it can be seen with its expression: , cos

Where √ and , .

On the GPS L5 pilot component, the secondary Neuman-Hofman code is unique for all signals and contains 20 bits (Fig. 3). As it can be observed, the probability of bit sign transition on the pilot component is 0.5. On the data component, the product of the data sequence and the NH will be seen as a data sequence with an increased rate (Fig. 4). This implies a non-uniform distribution of the bit sign transition. Knowing that there is a sign transition between the last and the first secondary code bits and that the data bit can change of sign during the NH 10 period change, the probability of a bit sign transition on the data component is:

1 10 1 1 2 9 5 9 0.55 1 2 (5)
The Galileo E5a and E5b signals have the same features as the GPS L5 signal but the probabilities and are not equal to ½ due to the presence of secondary codes on both components, as illustrated in Fig. 5 for the pilot component. In the following, the Galileo E5 a/b acquisition performance is not presented but seems to be relatively equivalent as GPS L5.

4) Galileo E1 OS signal

Galileo E1 OS differs from GPS signals by its spreading code period which is equal to 4 ms (for 4092 chips) instead of 1 ms (GPS L1 C/A and GPS L5) or 10 ms (GPS L1C). It is defined in [START_REF]European GNSS (Galileo) Open Service Signal In Space Interface Control Document (OS SIS ICD) Issue 1[END_REF] and the received signal expression is:

, , , , cos 2 (6)
Where

• √ •
, and , are the subcarriers on the data and pilot components defined by:

, √10 √11 1 √11 , √10 √11 1 √11 (7)
The secondary code on the pilot component is a unique 25bit code, represented on Fig. 6. Due to its number of bits (an odd number), the probability of a secondary code bit sign transition is 0.48. In the following, 0.5 is assumed. As illustrated in Fig. 7, the correlation process between the incoming signal and the local replica of one component provides in-phase and quadrature correlator outputs. It is considered that the correlation operation for one component of the GNSS signal and assuming that:

• There is no data bit sign transition during the correlation process (ideal case) • The correlation operation lasts for seconds • The parameters of the processed signal and the local replica are constant during the correlation operation such that the code delay error and the Doppler frequency error are constant and the carrier phase error at the beginning of the correlation process is

The in-phase and quadrature-phase correlator outputs can thus be modelled as:

2 cos sinc 2 sin sinc (8)
Where

• and are the noises at the correlator outputs which follow a centered Gaussian distribution and their variances are [START_REF] Bitar | Advanced GPS Signal Processing Techniques for LBS Services[END_REF] • is the autocorrelation function Assuming a local BOC(1,1) generation in this work, there is no distinction between the data autocorrelation and the pilot one for data/pilot signals. Moreover, the cross-correlation between the data and pilot spreading codes is assumed to be negligible due to the choice of the spreading codes as orthogonal as possible.

C. Acquisition process in absence of data 1) Acquisition based on one component

For signals with data and pilot components, an acquisition method consists in acquiring only one component or in considering both components to not suffer from a signal power loss. For the acquisition based on one component -and then the acquisition of the GPS L1 C/A signal-and for non-coherent summations, the acquisition detector is:

(9)
The acquisition process can be seen as a detection problem [START_REF] O'driscoll | Performance Analysis of the Parallel Acquisition of Weak GPS Signals[END_REF] because the purpose of the acquisition is to detect whether or not a signal from a given satellite is present at the receiver level. The hypothesis test based on Neyman-Pearson theorem, fully presented in [START_REF] Foucras | Probability of Detection for GNSS Signals with Sign Transitions[END_REF] and only briefly described here, can be modelled as: the null hypothesis assumes that the useful signal is absent whereas the alternative hypothesis assumes that the useful signal is present and the signal parameters are well estimated. Then, the acquisition detection problem compares the acquisition detector with a threshold denoted . The signal is declared present for a crossing of the threshold. The acquisition detector under the null and alternative hypothesis follows:

2 2 , , , (10)
Where , , is the non-centrality parameter ( designs the number of bit sign transitions occurring at )

Under the null hypothesis, the normalized acquisition detector follows a chi-square distribution with 2 degrees of freedom. The threshold can thus be determined knowing the desired probability of false alarm [START_REF] Foucras | Probability of Detection for GNSS Signals with Sign Transitions[END_REF], is set in this study to 10 as in [START_REF] Rtca | Assessment of Radio Frequency Interference Relevant to the GNSS L1 Frequency Band RTCA/DO-235B[END_REF].

1 1 , , , (11)
Where

, , , is the cumulative distribution function of a chi-square distribution with 2 degrees of freedom and a noncentrality parameter , , .

Under the alternative hypothesis

, the probability of detection can be determined because the normalized acquisition detector is a non-central chi-square distribution with 2 degrees of freedom and with a non-centrality parameter , , with the parameter , , :

, , sinc (12)

2) Acquisition based on both components

For the acquisition based on both components, the data and pilot components of the received signal are correlated separately with the local spreading codes of the data and pilot components [START_REF] Bastide | Analysis of L5/E5 Acquisition, Tracking and Data Demodulation Thresholds[END_REF]. The acquisition detector is then the sum of the two squared correlator outputs pairs given by the data and pilot components: [START_REF] Bastide | Analysis of L5/E5 Acquisition, Tracking and Data Demodulation Thresholds[END_REF] Under the null and alternative hypothesis, the normalized acquisition detectors are chi-square distributions with 4 degrees of freedom.

The non-centrality parameter under is the sum of the square expectations of the data and pilot correlator outputs

, , , , , .
III. BIT SIGN TRANSITION In reality, during the acquisition process, with no a priori information, there is no reason that the integration is aligned with the data bits since the receiver has not yet achieved bit synchronization. If this assumption is no longer considered, the present results become radically different. The following section intends to develop the associated mathematical model and to understand the impact of bit sign transitions on the acquisition of GNSS signals.

A. Correlator outputs in presence of bit transition

Without loss of generality, it is possible to consider that the correlation is performed between 0 and . Let us assume that a bit sign transition occurs at where 0 . As presented in [START_REF] Foucras | Probability of Detection for GNSS Signals with Sign Transitions[END_REF], the correlator outputs expressions can be shown to be equal:

B. Acquisiton process in presence of data

The acquisition detector, as the sum of the correlator outputs, will be affected by the presence of data sign transitions. Considering bit sign transition, the non-centrality parameter depending on [START_REF] Rtca | Assessment of Radio Frequency Interference Relevant to the GNSS L1 Frequency Band RTCA/DO-235B[END_REF] becomes dependent on the parameter (16) in function of the sign transition location expressed for one summation:

, , , , cos 1 2 cos cos 2 (16)
[11] provides a study on the non-centrality parameter and its dependence with the sign transition location , the Doppler frequency error ( [START_REF] O'driscoll | Performance Analysis of the Parallel Acquisition of Weak GPS Signals[END_REF] has presented initial results and [START_REF] Foucras | Probability of Detection for GNSS Signals with Sign Transitions[END_REF] brings a further investigation). Considering no Doppler and code delay errors, the first intuitive result is that the losses due to a bit sign transition can be important for a bit sign transition close to the center of the coherent integration interval as represented in Fig. 8. This induces potential strong degradation with respect to the probability of detection assessed without considering bit transitions. Let us go further by providing the theoretic probability of detection for each considered GNSS signal.

1) Acquisition based on one component a) Data/Pilot GNSS signals

In the case of the acquisition on one component and for the dwell time of ms where is the coherent integration time (assumed equal to the spreading code period) and the number of non-coherent summations, the maximum number of bit sign transitions is the number of non-coherent summations, which means .

At each spreading code period, a bit sign transition occurs with a probability of . The number of bit sign transitions can thus be modeled as a binomial distribution , . Then, the probability that bit sign transitions occurs is:

1 1 , , , (19)
The non-centrality parameter , , corresponds:

, , , , , , (20)

b) GPS L1 C/A

The case of GPS L1 C/A is particular because the spreading code period is shorter than the data bit duration. As explained in [START_REF] Foucras | Probability of Detection for GNSS Signals with Sign Transitions[END_REF] and illustrated by Fig. 9, over 20 successive 1-ms correlations, only one can be affected by a data bit transition which can involve a bit sign transition. The 19 other 1-ms correlations will be free of bit transition. The probability of detection can be generalized for any , the maximum number of bit transitions being . For a given , according to the bit sign transition location, there can be 1 or bit sign transitions. Indeed, for example, for 35 ms, in 25% of the cases, 1 bit sign transition occurs and in 75% of the cases, 2 bit sign transitions occur. This percentage of 1 bit transitions denoted is:

(21)
The probability of detection results in 2 sums:

, 2 1 1 2 2 1 1 2 1 2 (22)

2) Acquisition based on both components

In the case of an acquisition technique based on both data and pilot components and using dwell time of , the maximum number of bit sign transitions is 2 ( bit sign transitions per component). For instance, if 5, there can be 0, … , 10 bit sign transitions. In Fig. 10, the probabilities of

b) GPS L5 signal For the GPS L5 signal, as previously seen, the probability of bit sign transitions on the data component is not ½ ( 0.55 as for the Galileo E1 OS signal. The number of bit sign transitions on the data component should thus be considered, let us denote it . In the same way, the number on bit sign transitions on the pilot component can be defined but it is .

Over spreading code periods, is upper bounded by min , , there cannot be more bit sign transitions on the data component than the total number of bit sign transitions or the number of non-coherent summations. Furthermore, the number of data bit sign transitions on the data component should satisfy max , 0 because for example if 2 (maximum number of bit sign transitions), this implies that on each component, bit sign transitions occurs and then .

Then the probability of occurrence of bit sign transitions is:

, 1 1 , 1 , , (24) Where , , , 2 , ,
Then the probability of detection (18) becomes:

, , , , , (25)
c) GPS L1C signal The GPS L1C signal is characterized by a difference in power on both components. This implies a non-centrality parameter per component: , , , , and , , , , . The non-centrality parameter contains:

• terms from the data component ( considering bit sign transition and without bit transition) • terms from the pilot component (there are bit sign transitions and correlations without bit transition):

/ , , , , , , , , , (26)
Because the probability of bit sign transitions on each component is assumed to be one half, the probability of occurrence given in (24) can be simplified:

, 1 2 , 1 , / , (27)

IV. SIMULATIONS RESULTS

This work focuses only on the impact of bit sign transitions on the acquisition performance through the study of the probability of detection. In this context, for simulation, the incoming code delay and the Doppler frequency are assumed to be perfectly estimated. Moreover, the coherent integration time is chosen to be:

• An integer divider of the data bit duration and multiple of the spreading code period for GPS L1 C/A. As illustrated in Fig. 9, the dwell time is processed in blocks of 20 ms to respect the data bit duration. • The spreading code period which is the bit duration for data/pilot signals It seems to be clear that if a bit transition occurs at with in [0, ], the next bit transition occurs at ( for data/pilot signals). Furthermore, the bit sign transition location is supposed to follow a uniform distribution on [0, ].

A. GPS L1 C/A signal 1) General result on the probability of detection

The challenge for the GPS L1 C/A signal is to determine the optimal coherent integration time. Indeed, because the data bit duration is twenty times longer than the spreading code period, the coherent integration time could be whichever integer divider of the data bit duration. Without considering bit sign transition, it is preferable to choose long coherent integration time [START_REF] Bastide | Analysis of L5/E5 Acquisition, Tracking and Data Demodulation Thresholds[END_REF]; then for example for a dwell time of 20 ms, the best combination to have the high probability of detection is to set to 20 ms and to 1. As illustrated in Fig. 11 for a null frequency Doppler frequency error, the higher the coherent integration time, the higher the probability of detection is . Fig. 11: Probability of detection for 20 ms without bit sign transition (GPS L1 C/A) Fig. 12 and Fig. 13 give the probabilities of detection for a total integration time of 20 ms for the 6 values of integration time (integer divider of 20 ms) for two bit sign transition locations: at a quarter and in the middle of the coherent integration time. For a coherent integration time of 1 ms (black circle curve), there is a slight degradation on the probability of detection wherever is the bit sign transition. The most affected probability of detection is for a coherent integration time of 20 ms (red triangle curve). In the worst case (transition at 10 ms), even for high C/N 0 , the probability of detection stagnates at 0.5. This can be explained by the theoretical formula (22): there is a probability of one half that there is a bit sign transition and if it is the case, the probability is 0, if not, the probability is 1. III give the probability of detection for the optimal coherent integration time between 1, 2, 4, 5, 10, 20 ms depending on the bit sign transition location and the C/N 0 . For C/N 0 values higher than 31 dB-Hz, the probability of detection is 1. As presented in Fig. 12 and Fig. 13, for C/N 0 between 27 dB-Hz and 35 dB-Hz, the optimal choice for the coherent integration time is 4 or 5 ms. For higher C/N 0 , the signal is strong enough to be acquired with short coherent integration times because for a high coherent time, the probability of detection is highly affected. At the contrary, for lower C/N 0 , long coherent integration times are required to accumulate signal power. For low C/N 0 , the same study was done and presented in Table IV but for a total integration time of 100 ms. In this case, the probability of detection is close to 1 (the minimum is 0.92 for 25 dB-Hz and a bit sign transition at 5 ms). When the bit sign transition location gets close to the middle of the coherent integration time, the optimal choice for the coherent integration time is still 4 or 5 ms.

2) Average probability of detection

To conclude this section on the GPS L1 C/A signal, the average probability of detection on the bit sign transition location is given. For each coherent integration time, the probability of detection is computed for 51 different bit sign transition locations uniformly distributed between 0 and /2 and then the average value over the 51 obtained probabilities of detection is taken. Fig. 14 presents the result for a total integration time of 20 ms. As it can be observed, too short coherent integration times (1 or 2 ms) are not the best choices when considering bit sign transition. For very low C/N 0 , as previously said, the longer the coherent integration time is, the best is. For C/N 0 between 25 dB-Hz and 28 dB-Hz , the probabilities of detection for a coherent integration time between 4 ms and 20 ms are very close (at maximum difference of 1 dB). After 28 dB•Hz, the red curve ( 20 ms) falls compared with the other curves and the best choices for the coherent integration time is 4, 5 and 10 ms. For a probability of detection of 0.95, there is a loss of 2 dB due to bit transition for optimal coherent integration times. Indeed, without data modulation, the probability of detection is reached for a C/N 0 above 28.4 dB-Hz (20 ms) whereas a C/N 0 of 30.3 dB-Hz is required when the bit sign transition is considered and for a coherent integration time of 5 ms. Let us note that in the case of no data modulation (Fig. 11), there is only a loss of 0.4 dB for the same desired probability of detection and a coherent integration time (5 ms).

B. Galileo E1 OS signal 1) General result on the probability of detection

For a coherent integration of 4 ms, there can be 0 or 1 bit sign transition on each component. In each case (no bit transition, 1 on one of both components or 1 on both components), the associated probability of detection is represented in the left part of Fig. 15. For one bit sign transition on both components at , the probability of detection is null (blue curve). Indeed, the non-centrality parameter is null (17) but this case occurs only one time over 4 which explains the average probability (magenta curve) at 0.75 for high C/N 0 . 2) Probability of detection in the worst case location for data sign transition Fig. 16 presents the probabilities of detection for both acquisition techniques: on one component (data or pilot) and on both components on one spreading code period. Two curves (blue and red curves) correspond to the probabilities without considering bit sign transition. The acquisition on both components provides better probabilities of detection than acquisition on one component. The result stays true when a bit sign transition occurs in the middle of the integration interval that means at 2 ms (magenta and cyan curves). As it was the case for GPS L1 C/A and a coherent integration time equal to the data bit duration ( 20 ms), for high C/N 0 , the probability of detection is equal to 0.5 for the acquisition on one component (square cyan curve). As illustrated in Fig. 16 for 1 and Fig. 17, for 5 non-coherent summations, the probability of detection is still higher for acquisition on both components. Here, the probabilities of detection are given for the worst case that means a bit sign transition in the middle of the coherent integration time but the result is true whichever the bit sign transition location.

3) Average probability of detection

The average probability of detection on the bit sign transition location is computed for both acquisition techniques and presented in Fig. 18. Obviously, the higher probability of detection is for the acquisition on both components. We can remark that the losses between the two curves are around 2 dB which means that is less than in the case of no data modulation (Fig. 16). For a given probability of detection of 0.95 and a total integration time of 20 ms, if 31 dB-Hz is required to acquire Galileo E1 OS on both components, then:

• The received C/N 0 should be 3 dB higher in order to acquire the signal with bit sign transitions • If only one component is used, when not considering bit sign transition, 2 dB more are required. When considering bit sign transitions, 4 dB more are needed to reach the same probability of detection.

Fig. 18: Average probabilities of detection on for a total integration time of 20 ms (Galileo E1 OS)

C. GPS L5 signal 1) General result on the probability of detection

For the GPS L5 signal, the coherent integration time is 1 ms, that means the spreading code period. Compared to Galileo E1 OS, the probabilities of detection are lower because the integration time is shorter. The other difference with Galileo E1 OS is that the probability of a bit sign transition on the data component is not 0.5 ( 0.55), then this implies different values of probabilities of occurrence when only one component is acquired depending if it is the data or the pilot component (square cyan curve for the pilot component, green for the data component). 2) Probability of detection in the worst case location for data sign transition As it can be observed in Fig. 19, once again, for one spreading code period and bit sign transitions at 0.5 ms (middle of the coherent integration time), the probability of detection for one component is around 0.5 (for the data component, it is 1 0.45).

Fig. 19: Probabilities of detection on 1 ms (spreading code period) for acquisition in one or two components (GPS L5)

For a dwell time of 20 ms, when bit sign transitions occur at 0.5 for from 0 to 20 1, the probability of detection presents a loss higher than 3 dB (when not considering bit sign transitions). The losses seem to be lower than for Galileo E1 OS but the probabilities of detection are higher for Galileo E1 OS which can be explained by a longer coherent integration time (smaller impact due to bit transition but correlator outputs more noisy).

3) Average probability of detection

Due to the short coherent integration time, the average probability of detection (Fig. 21) for acquisition on both components is close to the probability of detection in the worst case (transition in the middle of the coherent integration time) in particular for high C/N 0 . Once again, it seems to be really preferable to acquire both components to limit the losses on the probabilities of detection due to bit sign transitions.

D. GPS L1C signal

For the GPS L1C signal, 75% of the signal power is concentrated on the pilot component, then the difference between the acquisition on both components or only on the pilot component can be anticipated as very small. As presented in Fig. 22 and Fig. 23, when modulation by a binary sequence is not considered both curves (acquisition of pilot or both components) are very close (red and blue circle curves). The difference between the probabilities of detection for an acquisition on the data component or on the pilot component is high due to the difference in signal power. Considering bit sign transitions, for low C/N 0 , the probabilities of detection for acquisition on both components or on pilot component match. For high C/N 0 , the difference tends to blur when the total integration time increases. As presented for Galileo E1 OS in Fig. 15, the probability of occurrence that the maximum number of bit sign transitions occurs becomes lower when the number of non-coherent summations increases and then the associated probability of detection (which has a null value) has a lower impact on the average probability of detection.

1) Probability of detection in the worst case location for data sign transition

2) Average probability of detection

This trend is confirmed by Fig. 24 where the difference between the two curves is very low. A loss of 5 dB on the probability of detection when considering or not bit sign transitions is observed.

E. Discussion

Table V and Table VI conclude this section by presenting the simulation results. In order to compare the performance of the 4 considered GNSS signals, the average probability of detection on the bit sign transition location is given for each signal and for a dwell time of 20 ms and for 3 values of C/N 0 . At 27 dB-Hz, the highest probability of detection is for the GPS L1 C/A signal. Because the signal is weak, for GPS L1 C/A, a long coherent integration time is better to accumulate the useful signal. For the same reason, the GPS L5 signal has the lowest performance in terms of probability of detection because the coherent integration time is 10 ms which is the longest for the other data/pilot signals. Clearly, for Galileo E1 OS and GPS L5, it is better to acquire both components to not suffer from the loss of 3 dB on the signal power. However, for the GPS L1C signal, it seems to be more interesting to acquire only the pilot component to reduce by one half the computation load and reach the same performance. Now let us interest in the received total C/N 0 for which the average probability of detection is equal to 0.95 and considering a dwell time of 20 ms. The GPS L1 C/A signal is the signal for which the C/N 0 is the lowest to detect with a probability of 95% when considering or not bit sign transitions. There is a loss of 1.9 dB (equivalent C/N 0 ) considering bit sign transitions which is low compared to 3 dB-Hz for Galileo E1 OS, 4.6 dB for GPS L5 and 5.1 dB for GPS L1C. When only one component is used, the degradations can be very high as 10 dB for GPS L1C.

CONCLUSIONS

This paper has shown interesting results related to the probability of detection when considering bit sign transitions. This work allows determining the optimal acquisition parameters to maximize the average probability of detection on the bit sign transition location. The first sections provide the expression of the probability of detection for several GNSS signals whichever the total integration time and considering a coherent integration time lesser or equal to the bit duration to avoid several bit sign transitions during the coherent integration interval. These first innovative results take into account the signal structure (probability of bit sign transition on each component, signal power distribution…) and on the acquisition technique on one or two components.

The simulation results highlight the optimal acquisition parameters in function of the considered GNSS signal. Concerning the GPS L1 C/A signal, this work focuses on the best coherent integration time considering bit sign transition. It was shown that a good tradeoff is a coherent integration time of 4, 5 or 10 ms (not an extremum value). Indeed, when the coherent integration time is too long (20 ms), the probability of detection is highly degraded by data bit sign transitions. If the coherent integration time is too short, particularly for low C/N 0 , even without considering bit sign transition, the signal is hardly acquired.

For data/pilot signals, if one component has more signal power, it is obvious that it should be the component to acquire. In the case of GPS L1C (75% on the pilot component), the acquisition on both components provides approximately the same probabilities of detection (considering or not bit sign transition) as the acquisition on the pilot component. In this configuration, it is thus a better choice to acquire only the pilot component to reach the same performance and in a reduced computation time and execution load. For data/pilot signals with the same signal power repartition (50% on each component), it is preferable to acquire both component to not suffer from signal power loss.

The comparison of GNSS signals acquisition performance highlights that the probability of detection when bit transitions occur is better for the signals with the longer spreading code period. The highest losses (around 5 dB) on the required sensitivity to reach a predefined average probability of detection due to bit sign transitions are for signals with the longest spreading code periods (GPS L1C). That means that for long spreading code period signals, the required C/N 0 is the highest which is the contrary of the case no bit transition. This work was presented for a dwell time of 20 ms but it can be extended at any dwell time. In particular, it can be interesting to study for a higher dwell time to acquire weak signals with a high probability of detection. For the GPS L1 C/A signal, the choice of the best coherent integration time would then be refined and not depending with the received sensitivity which is generally unknown. For the new signals (data/pilot structure), the bit sign transitions degrade considerably the acquisition performance. Thus, it appears necessary to consider bit sign transition insensitive acquisition method such as 1+1 ms and DBZPTI [START_REF] Foucras | A Novel Computationally Efficient Galileo E1 OS Acquisition Method for GNSS Software Receiver[END_REF]. This work can be extended following different axes. The first one consists in validating it on a set of real data. This permits to have a more realistic study on the acquisition performance taking into account other facts which can accentuate or reduce the losses due to bit sign transitions as the code Doppler effect [START_REF] Foucras | Detailed Analysis of the Impact of the Code Doppler on the Acquisition Performance of New GNSS Signals[END_REF]. From the point of view of the acquisition implementation and with an objective close to realtime processing, other aspects can be studied. Knowing that the probability of detection for the acquisition on both components on ms is the same as the probability of detection for the acquisition on one component but on 2 ms, it can be interesting to study the computational load for each acquisition technique in order to reduce the execution time but reaching the same acquisition performance.

  of the received signal on the data and pilot components respectively • is the secondary code on the pilot component • and are the subcarriers, which are defined by:

Fig. 2 :

 2 Fig. 2: Probability of bit transition on the GPS L1C pilot component in one secondary code period

Fig. 3 :

 3 Fig. 3: GPS L5 secondary code on the pilot component

Fig. 4 :

 4 Fig.4: The product of the data sequence and secondary code on the data component results in a binary sequence with a rate of 1kbit/s

Fig. 5 :

 5 Fig. 5: Probability of Galileo E5 secondary code bit sign transition in one spreading code period

Fig. 6 :

 6 Fig. 6: Galileo E1 OS secondary codeB. Correlator output in absence of dataUsually GNSS receivers generate a local replica of the incoming signal depending on the estimation of the code delay ̂ and Doppler frequency .

Fig. 7 :

 7 Fig. 7: Correlation process scheme

Fig. 8 :

 8 Fig. 8: Losses on the non-centrality parameter due to a bit sign transitionThe mathematical development of the non-centrality parameter permits to give it when the Doppler frequency error is null. It is:

  No bit transition 19 times over 20Bit transition 1 time over 20No bit sign transition Bit sign transition

Fig. 9 :

 9 Fig. 9: Scheme to determine the probability of detection (GPS L1 C/A)

Fig. 10 :

 10 Fig. 10: Probability of occurrence of bit sign transitions a) Galileo E1 OS signal For the Galileo E1 OS signal, the probability of a bit sign transition on each component is one half, then the number of bit sign transitions on spreading code periods is a binomial distribution 2 , . Because the total signal power is split in 50% on each component, , , and

  of bit sign transitions (on both components) Probability of occurence of n bit sign transitions On

Fig. 12 :

 12 Fig. 12: Probability of detection for ms and 1 bit sign transition at /4 (GPS L1 C/A)

Fig. 13 :

 13 Fig. 13: Probability of detection for 20 ms and 1 bit sign transition at /2 (GPS L1 C/A)

Fig. 14 :

 14 Fig. 14: Average probability of detection (on the bit sign transition location) for a 20 ms (GPS L1 C/A)

Fig. 15 :

 15 Fig.15: Probabilities of detection for 0, 1 or 2 transitions during the first spreading code period on both components in the middle of the integration interval (Galileo E1 OS)

Fig. 16 :

 16 Fig. 16: Probabilities of detection on 4 ms for acquisition on one or two components (Galileo E1 OS)

Fig. 20 :

 20 Fig.20: Probabilities of detection on 20 ms for acquisition in one or two components for a bit sign transition in the middle of the coherent integration time (GPS L5)

Fig. 21 :

 21 Fig. 21: Average probabilities of detection on for a total integration time of 20 ms (GPS L5)

Fig. 22 :

 22 Fig.22: Probabilities of detection on 20 ms for acquisition in one or two components for a bit sign transition in the middle of the coherent integration time (GPS L1C)

Fig. 23 :

 23 Fig.23: Probabilities of detection on 20 ms for acquisition in one or two components for a bit sign transition in the middle of the coherent integration time (GPS L1C)

Fig. 24 :

 24 Fig.24: Average probabilities of detection on for a total integration time of 20 ms (GPS L1C)

Table I

 I

	and [2] summarize the main technical features
	needed for this study.

TABLE I .

 I SIGNAL FEATURES

	Signal	Nb of comp Power	Modulation	PRN code length	Data	Secondary code
	GPS L1 C/A	1	BPSK	1023 chips 1 ms	20 ms	None
	GPS L1C	2 Data: 25% Pilot: 75%	TMBOC (6, 1, 1/11)	10 230 chips 10 ms	10 ms	Pilot 10 ms
	GPS L5	2 Data: 50% Pilot: 50%	BPSK	10 230 chips 1 ms	10 ms	Data: NH10 1 ms Pilot: NH20 1 ms
	Galileo E1 OS	2 Data: 50% Pilot: 50%	CBOC (6,1,1/11)	4092 chips 4 ms	4 ms	Pilot 4 ms

Table II and

 II Table

TABLE II .

 II PROBABILITY OF DETECTION FOR THE OPTIMAL INTEGRATION TIME

				Bit sign transition location
	C/N0 (dB•Hz)	25 27 29 31	0.49 0.81 0.98 1	0.41 0.34 0.29 0.27 0.25 0.25 0.25 0.25 0.72 0.61 0.56 0.53 0.50 0.48 0.47 0.47 0.94 0.90 0.86 0.82 0.80 0.79 0.78 0.78 1 0.99 0.99 0.98 0.98 0.98 0.97 0.97

TABLE III .

 III OPTIMAL COHERENT INTEGRATION TIME (IN MS) FOR

TABLE IV .

 IV OPTIMAL COHERENT INTEGRATION TIME (IN MS) FOR

TABLE V .

 V AVERAGE PROBABILITY OF DETECTION FOR 20 MS

					C/N0 (dB-Hz )
				27	30	35
	GPS L1 C/A	0.55 T I 10 ms	0.93 T I 5 ms	1 T I 20 ms
	Galileo	1 comp.	0.05	0.24	0.87
	E1 OS	2 comp.	0.13	0.51	0.98
		1	Data	0.01	0.05	0.61
	GPS L5	comp.	Pilot	0.01	0.06	0.66
		2 comp.	0.04	0.24	0.99
		1	Data	0.03	0.13	0.67
	GPS L1C	comp.	Pilot	0.27	0.64	0.90
		2 comp.	0.29	0.70	0.95
	TABLE VI. REQUIRED C/N0 FOR AN AVERAGE PROBABILITY OF DETECTION OF 95% 20 MS
				Without data	With data
	GPS L1 C/A	28.4 dB-Hz	30.3 dB-Hz
	Galileo		1 comp.		33 dB-Hz	36.8 dB-Hz
	E1 OS		2 comp.	30.9 dB-Hz	33.9 dB-Hz
			Data	34.9 dB-Hz	38 dB-Hz
		1 comp.
	GPS L5		Pilot	34.9 dB-Hz	37.5 dB-Hz
			2 comp.		33 dB-Hz	34.1 dB-Hz
			Data	35.1 dB-Hz	46 dB-Hz
		1 comp.
	GPS L1C		Pilot	30.3 dB-Hz	41.4 dB-Hz
			2 comp.


Jérémy Vezinet

Anne-Christine Escher

Christophe Macabiau

Alain Guillet

Video integration in a GNSS/INS hybridization architecture for approach and landing

Keywords: Hybridization, Video based navigation, EKF I Covariance prediction

Most of the current operations are supported by the Global Navigation Satellite System (GNSS) potentially augmented with Ground Based Augmentation System (GBAS), Satellite Based Augmentation Systems (SBAS) and Aircraft Based Augmentation System (ABAS). ABAS system integrates on-board information that can for example be provided by Inertial Navigation System (INS) so as to enhance the performance of the navigation system. GNSS/INS hybridization is already performed on current commercial aircraft. This solution allows reaching better performance in terms of accuracy, integrity, availability and continuity than the two separated solutions. Hybridization principle is based on the use of GNSS to estimate the inertial drift. It allows navigating in oceanic en-route operations down to non-precision approaches for long range aircraft. The most stringent requirements for precision approaches or automatic landings cannot be reached with the current GNSS/INS hybridization schemes. In the framework of my PhD stud, the idea is to extend the hybridization filter by including other data available onboard. The global objective of the work is to propose a system that will be able to fuse the multiple sources of information or measurement in a global architecture. The current paper focus on a solution that integrates a video system within an EKF GNSS/INS architecture during approach and landing of an aircraft.

INTRODUCTION

INS/GNSS hybridization is currently used on commercial aircraft in order to compensate drawbacks of the two systems independently. The hybridized solution constitutes a strong, robust, accurate solution for the estimation of most of the navigation parameters. Some solutions also include a integrity monitoring algorithm that can detect GNSS or INS failures. However some operations require new constraints and requirement that can't be achieved with the current navigation solution. In that way, one of the ideas was to use other sources of data and to propose other hybridization architectures.

The study of the possible sensors/systems and associated information/measurements available on board a commercial aircraft has been investigated in previous works, and a set of sensors and data to consider has been identified: a GNSS (potentially augmented with GBAS), an INS (different classes of inertial sensors will be considered) and a video system.

The aim of my work is to arrive to the definition of a global hybridization filter architecture based on the integration of multiple information/measurement sources or sensors (INS, GNSS receiver and a video system), depending on the operational environment: the aircraft operation considered, the sensors available during the considered operation, the performance requirement for supporting the considered operation. In a first part, the current paper, as a first step, will describe an algorithm for integrating optical angular measurements in a classical GNSS/INS Kalman Filter Hybridization solution. The optical angular measurements are supposed to be provided by a camera mounted on-board the aircraft, capable to detect and track features of known location on the runway.

Then, the paper will describe the model for the generation of the optical angular measurements, as well as the mechanization process of the integrated solution that relates the visual measurement and the state of the filter.

Finally, some simulations results of the proposed algorithm will be presented. The simulation results of the filter when integrating video will permit to validate the proposed integration and to constitute a basis for future simulations. Performance assessment is not done in the current paper. The goal is to describe the GNSS/INS EKF architecture and to present a solution for integrating video during approach and landing.

II. EKF ALGORITHM DESCRIPTION

A. EKF Model

The discrete state EKF is described by the following equations: [START_REF] Geas | Phase I of the GNSS Evolutionary Architecture Study[END_REF] Where and are respectively the state transition and observation functions, and are respectively the state vector and observation vector at instant , and are the process and observation noises which are both assumed to be zero mean Gaussian noises with covaria respectively.

B. State Transition Model

The EKF state vector is an error state ve the INS error estimation. The state transition implemented has been derived from classic I in a wander azimuth coordinate system. This done considering an INS composed of 3 gyrometers and a barometer. The INS mecha into account a 3 rd order baro-inertial loop.

The wander azimuth frame is locally de to the North-Eath-Down (NED) local nav follows: (

ector that describes model of the EKF INS mechanization s mechanization is accelerometers, 3 anization also takes efined with respect vigation frame as frame Where the angle is defin its computation is done as descr Frame definition:

In the entire paper, by conv notations for the frame involved , , is the inertial fra reference frame in which Newt origin of the inertial frame is g center of mass, the axis point axis points along the earth defined to complete the right-ha , , is the Earth-cen frame or e-frame). Its origin is The coordinate axis is fixed t aligned with the axis of the frame rotates relative to the in ⁄ 7.292115 10 / , , is the Geographic or n-frame). The geographic fr to the earth's geoid. The ax the ellipsoid along the ellipso toward the North and the a orthogonal, right-handed fram projection of the platform orig will use, the ECEF geodetic co the latitude, longitude and altitu , , is the body or frame). The body frame is rig interest, usually at a fixed poin The axis is defined in the fo defined pointing to the bottom defined pointing to the right of the right-handed orthogonal illustrated in Fig. 2. ed as the wander azimuth angle, ribed in [START_REF] Poor | A Geometric Description of Wander Azimuth Frames[END_REF]. vention we will use the following d: ame (i-frame). It is define as a ton's laws of motion apply. The given coincident with the earth's ts toward the vernal equinox, the h's spin axis and the axis is anded coordinate system. ntered earth-fixed frame (ECEFfixed to the center of the earth. to the earth and the axis is inertial frame. Then the ECEF nertial frame at a frequency of / c navigation frame (NED-frame frame is defined locally, relative xis points toward the interior of oid normal. The axis points axis points east to complete the me. The frame origin is the gin onto the earth's geoid. We oordinates system that introduces ude parameters , , . r mobile frame (b-frame or mgidly attached to the vehicle of nt such as the center of gravity. forward direction. The axis is m of the vehicle. The axis is f the vehicle in order to complete frame. All these frames are and reference frames [START_REF] Escher | Inertial Navigation and GPS/IRS Hybridization[END_REF] The last frame , , is the wander azimuth frame (w-frame). Wander azimuth reference frames solve the high latitude problem faced by the geographic frame. The definition is expressed in terms of the angular velocity of the frame with respect to the earth frame. In that way when the instantaneous rotation vector of the w-frame with respect to the e-frame expressed in the n-frame is as follows:

⁄ cos 0 (9)
Velocity error equation:

The INS error mechanization in wander azimuth frame is described by the Earth relative velocity error equation in the wander-azimuth frame [START_REF] Farrel | The global positioning system and inertial navigation[END_REF]:

ω ⁄ 2 ⁄ Ω ⁄ 2Ω ⁄ (10)
Where:

is the Earth-relative velocity error in the wander azimuth system coordinates is the specific force error in the wander azimuth system coordinates (i.e. the accelerometer biases) is the measured specific force is the local gravity vector error (in the current study, a Somigliana model is used for the local gravity [START_REF] Farrel | The global positioning system and inertial navigation[END_REF])

-
is the estimated Earth relative velocity expressed in the wander azimuth system coordinates -⁄ is the instantaneous rotation vector of the a-frame with respect to the b-frame, expressed in the a-frame coordinate system. Ω is the associated skew symmetric matrix.

Attitude error equation:

The second equation that describes the INS error mechanization is the attitude error equation [START_REF] Farrel | The global positioning system and inertial navigation[END_REF]:

Ω ⁄ ⁄ ⁄ (11)
Where:

-

Ω ⁄ ⁄ ⁄
is the attitude error vector -⁄ is the rotation rate error of the mobile frame with respect to the inertial frame (i.e. the gyrometer biases)

State vector:

The state vector of the implemented filter is as following:

-, are the horizontal angular position error in the wander azimuth frame.

is the baro-inertial altitude error.

is the output of the compensator of the feedback loop of a third order baro-inertial loop.

-, are the horizontal velocity error in the wander azimuth frame.

is the baro-inertial vertical speed (in along the down vertical axis).

-, , are the alignment angles errors in the wander azimuth frame.

-, , are the gyroscope bias errors in the aircraft frame.

-, , are the accelerometer bias errors in the aircraft frame.

is the GNSS receiver clock bias error.

is the GNSS receiver clock drift error.

-, … , are the GNSS pseudoranges correlated errors (for N satellites).

The linearized state transition matrix is then derived from the INS mechanization as it is done in [START_REF] Diesel | GPS/INS integration for civil aviation[END_REF].

The first two states are horizontal angular position error expressed in the wander azimuth frame. These parameters denoted and can be expressed in the NED frame as follows:

cos sin sin cos (12)
Then the invert transformation is:

cos sin sin cos (13)
These horizontal angular position errors (also called "small angles") are related to the errors in latitude and longitude by the following relations:

cos (14)
The transition matrix for the biases can be modeled as a first order Markov process with a driven noise and a correlation time that corresponds to the quality of the sensors.

The transition matrix for the GN correlated errors can be modeled as a first ord driven noise and a correlation time that c biggest error contribution in the pseudorange The transition matrix for the receiver cloc the two-state clock model as described in [START_REF] Farrel | The global positioning system and inertial navigation[END_REF]:

0 1 0 0 C. Observation Model
The observation model is described by t its linearized matrix form . In the g architecture presented in the current paper provided by two systems: a GNSS receiver an The video system observations and the observ video are presented in the next part.

For the GNSS receiver observation, the measurements, we will consider the follow model:

With: the geometrical range between the receiver a satellite antenna, is the speed of light, is bias, is the contribution of ps correlated in time, and is a white Gaussian By denoting, We obtain the following observation pseudorange measurement:

Then the linearized observation matrix on position and receiver clock bias. The thre the matrix are as follows:

1: 3
The other elements are:

4: 0 0 1 0 0 0 1 0
Where N is the size of the state vector.

SS pseudoranges der Markov with a corresponds to the measurements.

ck is obtained from :

The linearized observation m is detailed in the next part of the

III. VIDEO MEASU

A. Video Measurement Model

The video measurements ar video system composed of a algorithm.

The camera is supposed to order to provide measuremen detected in the scenery during airport environment, targets… navigation based on video ha study, a model for video mea This model is composed of a that describes the location of t the environment. In our study where the video system is able of the runway during the approa The current study is not f features by an image proces assumptions that the measurem output of the video system.

To summarize, the video runway corners (with know environment and to provide for illustrated in Fig. 3: These angular measureme measurements that will be use follows for one target: matrix for the video observation e paper.

-A measurement
UREMENT INTEGRATION re provided at each instant by a camera and a video processing o be mounted on the aircraft in nts that correspond to features approach and landing (runway, …). A previous study about as been done in [START_REF] Vezinet | State of the Art of Image-Aided Navigation Techniques for Approach and Landing[END_REF]. From this asurements has been proposed. set of two angular coordinates the image of a specific target in y [START_REF] Vezinet | State of the Art of Image-Aided Navigation Techniques for Approach and Landing[END_REF], we proposed a scenario e to detect and track the corners ach and landing.

focused on the detection of the ssing algorithm and made the ments are directly provided at the System is supposed to detect wn locations) in the runway r each feature the following data angle between the x-axis of the ne-of-Sight from the focal point et projected in the vertical plane and z-axis of the aircraft frame. ted .

angle between the x-axis of the ne-of-Sight from the focal point rget projected in the horizontal x-axis and y-axis of the aircraft is denoted . We obtain the following observation mo measurements: tan tan

B. Video Observation Model

The video observation model is a functi measurement to the state of the system. In o video observation model, we need to find depends the observation function. This ob will also be used in order to generate the vide The first and most simple dependency is as it is illustrated in Fig. 4. This figure observed by a camera during an approach runway. The coordinate of the point detec , can be related to its coordinate , frame (rotated of the roll angle with resp frame).

odel for the video (25)

on that relates the order to define the d on which states servation function eo measurements.

with the roll angle shows the image h when seeing the cted in the image in a horizontal pect to the image Where is the roll angle of

In the following, the de function with pitch, heading an detailed.

The entire description can These figures detail the relatio angles and the aircraft state. The angle denoted GCA ( related to the other paramet haversine function. This functi angle as following: is then:

Δ cos Δ 2 (34)
ngent of the measurements to the craft using a composition of the n equations ( 27) to (34).

consider the horizontal angular nt is illustrated in Fig. 6.

C. Linearized Observation Ma

The linearization of the ob computation of the Jacobian matrix has two lines for each matrix line is as follows:

The computation of the elem using the law of derivation for c if we denote:

, , cos sin

, , sin c , 1 , 1 , sin cos , sin sin , 2 asin 1 cos

acos cos co

These functions can be use observation model for the vide previous part. The computatio matrix needs the computation respect to each state of the state of these basic functions can matrix.

The current part details so algorithm for the integration o proposed in [START_REF] Vezinet | State of the Art of Image-Aided Navigation Techniques for Approach and Landing[END_REF] in a GNSS/bar integration architecture is done provide some results ob implementation.

IV. ESTIMA

The current part deals with of estimation of the proposed E GNSS/INS architecture is simu approach in the second sectio during this study, we used as r As a preliminary step of the global study, inertial measurements were provided by the dataset (more precisely, load factor data were used to generate accelerometer measurements), but we simulated the functioning of a baroinertial platform (a third order baro-inertial loop). In that way we simulated the baro-inertial estimations of the position, velocity and attitude. It is really important to understand that these first results are obtained from simulated values that could not be completely representative of the reality. The next step of the work will be to do the same simulations using real platform data. This preliminary step is useful in order to validate the complete scenario proposed for the study. In addition, the actual settings of the Kalman Filter (i.e. the values in the transition state covariance matrix and measurement noise covariance matrix) are related to the inertial measurements and to the inertial platform simulator. However they still can be improved. For the next step of the study, new settings will have to be determined in order to match with the real dynamic of the trajectory.

From the position, velocity and attitude parameters recorded during the reference flight and considered as true parameters, we simulate the inertial platform estimations. The aim of the filter was then to estimate the errors made by the simulated inertial platform. After estimation of the inertial errors by the filter, we could compensate the inertial parameters with these estimated errors and then have corrected estimations of the parameters simulated with the inertial platform software.

It can be seen in the following section, that the estimations obtained with the simulated platform does not completely correspond to realistic inertial platform outputs.

The GNSS measurements considered for all the evaluation results presented in this paper are iono-free GNSS/GALILEO measurements. For all these simulations we assumed the nominal 24 GPS constellation, a 27 GALILEO constellation, with 5° mask angle for GPS and 10° mask angle GALILEO, without code carrier smoothing, and unsmoothed measurement error models for tropo, noise, satellite clock and ephemeris as in [START_REF] Montloin | GNSS Multipath Failures Modes Analysis for Airport Surface Operations[END_REF]. Multipath error models are such that L1 and L5 multipath error are uncorrelated with standard deviation 0.13 0.53 m, assuming airborne multipath errors have a long correlation time.

A. Without Video

The first simulations present result without integrating video in the hybridization architecture. It only consisted in simulations of the GNSS/INS EKF architecture.

As the EKF estimates inertial platform errors, the estimation errors at the output of the platform simulator and at the output of the Kalman Filter are compared. Fig. 9 to Fig. 11 illustrates these differences. From the previous figures, we can see that simulated platform estimations do not illustrate the drifting behavior of a real platform.

In a second step, Fig. 12 to Fig. 14 present the GNSS/INS EKF estimation error of position, velocity and attitude. The estimation errors appear in blue and in red there is the 3-sigma covariance of the estimation error. . Converting it in meters we obtain a mean value around 6 .

Fig. 13. EKF velocity estimation error

The horizontal velocity error is around 0.2 / , but the covariance error is much higher. These results can be adjusted by further optimization of the settings of the filter. For attitude estimation, the EKF does not bring any improvement. The errors estimated by the EKF do not permit a good correction of the inertial attitude estimation. We could expect this result for attitude estimation because GNSS does not directly provide help for attitude estimation.

The results presented in this section were provided by the GNSS/INS EKF architecture, without video integration. These results show that the filter is able to improve estimation of the navigation parameters. However, the settings still need to be optimized, and the simulated parameters can be replaced by real data. In addition the multipath model for generation of the pseudorange measurements is conservative and not representative of real multipath delays. Next section will provide preliminary results using the video integration process detailed in the previous part of the paper without presenting a performance assessment.

B. With Video

The simulations done in this section only aimed at validating the process proposed for integrating video in a EKF architecture. The video measurements are generated with simulation software developed during this study. It uses the observation matrix described in the previous part. The input parameters of the software are the position and attitude profile of the aircraft on one hand and characteristics of the video system on the other hand. The characteristics are described by three parameters:

-The focal length is set at 5 mm -The resolution in pixel is set at 2160x4096 (4K) -The size of the sensor is set at 24x32 mm Using these parameters and the observation matrix described below, we generate video measurements for two points corresponding at the corners at the beginning of the runway. Their locations are illustrated in Fig. 15. These measurements are available for the final segment of the approach. Results before integration of video measurements are the same than without video. One of the main assumptions used in our study is that the video system is able to detect and lock precise interest points of known locations (corners of the runway). We actually started studies on video processing algorithm in order to try to develop a software able to do such a detection.

Finally, the video measurement simulation generate coordinate in pixels of the detected points in the image. Then these coordinates are turned into the optical angular measurements described in the previous part. Before the conversion a simple error model is added on the pixel coordinates (an error model following a White Gaussian distribution with a standard deviation of 1 pixel). For the following of the study, the error model will be developed.

In addition with the simulation of the video measurements, a "visibility" flag is generated, indicating that the point is actually seen by the camera. In our simulation, this flag is set at 0 when the point is not considered visible and at 1 when it is visible (i.e. when the aircraft is aligned with the runway). In addition we will only consider the approach and landing phases and not the take-off phase.

The results of the estimation of the navigation parameters using video are illustrated in Fig. 16 to Fig. 18. Analyzing the previous figures presenting the results of the simulations with video, we can first say that there is no significant improvement in terms of accuracy. Attitude seems to be slightly degraded However, as it is detailed before, the current assumptions of the simulations, which is the use of simulated inertial data, and the use of iono-free GPS/GALILEO measurements does not allow concluding on the improvements provided by the video in term of performance. This part only allows a preliminary validation of the integration of video measurements in the proposed architecture. One of the points that need to be improved is the use of simulated inertial data. In fact, as it is presented in the introduction of the current part, simulated inertial data are used in the current simulations. Also GNSS measurement errors and position errors are very small because we are in the case of iono-free GPS/GALILEO measurements. In that case, the results are not completely representative of the performance of the filter with video measurements. The main objective of the simulations was to validate the video integration architecture, using the video observation model.

In addition, the interest in adding video can be identified by optimizing the settings of the filter. This is a first step in detailing the improvement provided by video, but the settings of the filter and the parameter of the video will be optimized in the following and the advantage in integrating video when loss a GNSS will be studied. In addition with the optimization of the filter and the video parameters, the use of real data will allows assessing the true performance of integrating video in the current architecture.

As an additional simulation scenario, single frequency GPSL1C or degraded inertial classes can be intended in order to test the integration of video in degraded scenarios.

V. CONCLUSION

Interest in integrating an additional source of data in a navigation algorithm, other than aiding at the estimation of the navigation parameter, is to provide a source of measurement uncorrelated with the GNSS receiver. In fact, the first idea in adding a video system, able to provide visual measurements, was to dispose of another mean for providing information that can be used for navigation purpose. As it was presented in [START_REF] Vezinet | State of the Art of Image-Aided Navigation Techniques for Approach and Landing[END_REF], several methods can be used for navigation using video and in that study we wanted to use video as an autonomous means providing absolute geometrical information and in real time (i.e. at each frame of the video). The video process is not supposed to use an external database or reference. This method is supposed to be used during an approach and landing operation, because the runway needs to be visible, then we considered in this study that the visual measurements were available when the aircraft is aligned with the runway.

In the paper we first proposed a hybridization filter using an EKF and integrating video and GNSS code pseudorange measurements. The second part of the paper aimed at detailing the process for integrating the video measurements in the filter. A detailed description of the observation matrix is provided.

Finally some preliminary results, using simulated inertial data and iono-free GPS/GALILEO simulated data, showed that with the actual settings of the filter and characteristics of the video measurements, the video does not provide a significant improvement in terms of accuracy. One of the reasons of the lack of evidence of the improvement brought by video in the current results is that the use of the INS simulated data does not completely produce the real performance in terms of estimation of the navigation parameters. The use of simulated data allows matching the real dynamic of an aircraft and producing the same behavior than a Kalman filter. Also GNSS measurement are iono-free GPS/GALILEO measurements, reflecting a post 2020 situation but with a highly conservative multipath error model that can be refined. In that way, the integration of video in that scenario seems not to provide improvements. However, the interest of this study is that it gives an idea of the process for integrating an additional source of data in a global architecture. This constitutes a good approach for integrating various sensors in a future global hybridization architecture.

These results illustrated the ability of the proposed architecture to integrate other measurements provided by other sources. Further simulations have to be done in that way. This architecture can also be completed with other sensors and the filter model can be modified in order to test some particular Kalman filters: the Unscented Kalman Filter (UKF) or the Information Kalman Filter (IKF).

For the following of the study, the settings of the filter will be optimized and the parameters of the camera will be refined. The main change in the future simulations will be the use of a set of real data of the same trajectory, in order to assess the real performance of the current architecture.

The existing simulator will also allow us to try some combinations of sensors during approach and landing and during specific scenarios or failure cases. In that way degraded inertial classes an GNSS receivers will be tested. In particular, our study will also test the ability of the video measurements to compensate a GNSS outage and allows performing the current operation.

Fig. 1 .

 1 Fig. 1. Definition of the wander azimuth

Fig. 2 .

 2 Fig. 2. Coordinate a

  Fig. 3. Optical a

  angular measurements ents are converted in metrical ed in the observation model as tan tan tan tan Where and are white Gaussian no the total error on the coordinate of the det image. These measurements are related to pix image points as follows: tan tan Where is the focal length of the camera By denoting, tan tan , , , , , tan , , , , ,

Fig. 4 .

 4 Fig.4. Impact of roll a

First

  considering the meas taking into consideration the cu following scheme: angle on video measurements relation between the two angular n intermediate set of coordinates is described in (26). observation nd position of the aircraft will be be illustrated with the Fig. 5. ons between the measurements surement in the vertical plane, urvature of the Earth we have the

Fig. 5 .

 5 Fig. 5. Observation model for the vertical optical a

  We can finally relate the tan attitude and position of the airc different functions presented inIn a second way we will measurement. This measuremen

Fig. 6 .

 6 Fig. 6. Observation model for the horizontal optical

  way, we can relate the measurements to the attitude and position of t composition of the different functions prese (35) to (39).

  atrix bservation function is done by Matrix of the function. This h target and N columns. Each to describe the entire eo measurements detailed in the on of the linearized observation of a complicate derivation with e vector. Detailing the derivative help for the global linearized ome elements of the proposed of optical angular measurement ro-INS architecture. This global with an EKF. The next part will btained with the proposed ATION RESULTS the presentation of some results EKF. In a first section, only the ulated, video is integrated during on. In the simulations realized reference, a set of data recorded during a real flight. The trajectory performed during this flight is a three hour Toulouse-Toulouse flight. A 3D representation of the trajectory is illustrated in Fig. 7 and the horizontal and altitude profiles are illustrated in Fig. 8.

Fig. 7 .

 7 Fig. 7. 3D Trajectory

Fig. 9 .

 9 Fig. 9. IRS and GPIRS position estimation errors

Fig. 12 .

 12 Fig. 12. EKF position estimation error

Fig. 14 .

 14 Fig. 14. EKF attitude estimation error

Fig. 15 .

 15 Fig. 15. Corners location

Fig. 16 .

 16 Fig. 16. Position estimation with video