Piotr Bilinski
email: piotr.bilinski@inria.frwww.inria.fr

Francois Bremond
email: francois.bremond@inria.frwww.inria.fr

Statistics of Pairwise Co-occurring Local Spatio-Temporal Features for Human Action Recognition

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Introduction

In recent years, recognition of human actions has became one of the most popular topic in computer vision domain. It has many potential applications, such as video surveillance, video indexing, retrieving and browsing, sport event analysis, human-computer interface and virtual reality. Although various methods have been proposed and much progress has been made, action recognition still remains a challenging problem. The main issues are: variations in visual and motion appearance of both people and actions, occlusions, noise, enormous amount of video data and changes in viewpoint, scale, rotation and illumination.

Over the last decade, there have been many studies on the recognition of human actions in video. Most of the state-of-the-art approaches can be divided into four categories depending on the type of features used. The first group of methods uses silhouette information [START_REF] Davis | Hierarchical motion history images for recognizing human motion[END_REF][START_REF] Ahad | Motion history image: its variants and applications[END_REF][START_REF] Aggarwal | Human motion analysis: a review[END_REF][START_REF] Kim | In: Silhouette-based Human Activity Recognition Using Independent Component Analysis, Linear Discriminant Analysis and Hidden Markov Model[END_REF][START_REF] Lin | Recognizing actions by shape-motion prototype trees[END_REF]. The second category of techniques analyses object or motion trajectories [START_REF] Messing | Activity recognition using the velocity histories of tracked keypoints[END_REF][START_REF] Raptis | Tracklet descriptors for action modeling and video analysis[END_REF][START_REF] Kaaniche | Gesture recognition by learning local motion signatures[END_REF][START_REF] Wang | Action recognition by dense trajectories[END_REF]. However, both of these groups require precise algorithms, which is often very difficult to achieve due to such challenges as: low discriminative appearance, illumination changes, camera movement, occlusions, noise and drifting problems. The third group of methods uses local spatiotemporal features [START_REF] Laptev | On space-time interest points[END_REF][START_REF] Rapantzikos | Dense saliency-based spatiotemporal feature points for action recognition[END_REF][START_REF] Klaser | A spatio-temporal descriptor based on 3d-gradients[END_REF][START_REF] Gilbert | Fast realistic multi-action recognition using mined dense spatio-temporal features[END_REF][START_REF] Liu | Learning human actions via information maximization[END_REF]. Local spatio-temporal features have recently become a very popular video representation for action recognition. They have demonstrated promising recognition results for a number of action classes. They are able to capture both motion and visual appearance. Moreover, they are robust to scale variations, viewpoint changes and background clutter. Over the last decade, many algorithms have been proposed to detect local spatio-temporal interest points (e.g. Harris3D [START_REF] Laptev | On space-time interest points[END_REF], Cuboid [START_REF] Dollar | Behavior recognition via sparse spatio-temporal features[END_REF], Hessian [START_REF] Willems | An efficient dense and scale-invariant spatiotemporal interest point detector[END_REF] or Dense sampling [START_REF] Wang | Evaluation of local spatio-temporal features for action recognition[END_REF]) and represent them using spatio-temporal descriptors (e.g. HOG [START_REF] Laptev | Learning realistic human actions from movies[END_REF], HOG3D [START_REF] Klaser | A spatio-temporal descriptor based on 3d-gradients[END_REF], HOF [START_REF] Laptev | Learning realistic human actions from movies[END_REF], Cuboid [START_REF] Dollar | Behavior recognition via sparse spatio-temporal features[END_REF] or ESURF [START_REF] Willems | An efficient dense and scale-invariant spatiotemporal interest point detector[END_REF]). One of the most frequently used detectors in the literature is the Harris3D [START_REF] Laptev | On space-time interest points[END_REF], which is a space-time extension of the Harris operator. This algorithm is usually applied with Histogram of Oriented Gradients (HOG) and Histogram of Oriented Flow (HOF) descriptors [START_REF] Laptev | Learning realistic human actions from movies[END_REF]. The former describes local visual appearance and the latter characterizes local motion appearance of an interest point.

Local spatio-temporal features have been mostly used with a bag-of-words model. Together, these techniques have shown to achieve high recognition rate across various datasets. The bag-of-words model encodes global statistics of features, computing histogram of feature occurrences in a video sequence. This technique also has its own limitations. One of the main drawbacks of the bagof-words model is that it ignores local pairwise relations among the features. To overcome this limitation, contextual features from the fourth category could be used. Contextual features can capture human-object interactions [START_REF] Gupta | Objects in action: An approach for combining action understanding and object perception[END_REF], scene context information [START_REF] Li | What, where and who? classifying events by scene and object recognition[END_REF][START_REF] Marszalek | Actions in context[END_REF], figure-centric features [START_REF] Sun | Hierarchical spatiotemporal context modeling for action recognition[END_REF][START_REF] Wang | Action recognition with multiscale spatio-temporal contexts[END_REF][START_REF] Banerjee | Learning neighborhood co-occurrence statistics of sparse features for human activity recognition[END_REF][START_REF] Kovashka | Learning a hierarchy of discriminative space-time neighborhood features for human action recognition[END_REF], or pairwise relations between features [START_REF] Oikonomopoulos | An implicit spatiotemporal shape model for human activity localisation and recognition[END_REF][START_REF] Banerjee | Learning neighborhood co-occurrence statistics of sparse features for human activity recognition[END_REF][START_REF] Ta | Pairwise features for human action recognition[END_REF]. Oikonomopoulos et al . [START_REF] Oikonomopoulos | An implicit spatiotemporal shape model for human activity localisation and recognition[END_REF] have proposed to construct class-specific codebooks of local features and encode spatial co-occurrences of pairs of codewords. Then, the action model is classified based on the probabilistic voting framework. Liu et al . [START_REF] Liu | Learning human actions via information maximization[END_REF] have proposed to explore the correlation of the compact video-word clusters using a modified correlogram. Banerjee et al . [START_REF] Banerjee | Learning neighborhood co-occurrence statistics of sparse features for human activity recognition[END_REF] have proposed to learn local neighbourhood relationships between local features, and train a CRF based human activity classifier. The neighbourhood relationships are modelled in terms of pairwise co-occurrence statistics. However, these methods are restricted using discriminative power of individual local features and ignoring association between appearance and geometric information. Thus, the performance of these techniques mainly depends on a single type of applied features. Ta et al . [START_REF] Ta | Pairwise features for human action recognition[END_REF] have proposed to encode both appearance and spatio-temporal relations of local features. However, by calculating two independent codebooks (one codebook per feature type), this method ignores important association between appearance and geometric information.

To differ from those ideas, we propose a novel representation based on local spatio-temporal features and bag-of-words technique. Recent methods have typically focused on capturing global statistics of features. However, existing approaches ignore relations between the features, and thus may not be discriminative enough. Therefore, we propose a novel feature representation which captures statistics of pairwise co-occurring local spatio-temporal features. Our representation captures not only global distribution of features but also focuses on geometric and appearance (both visual and motion) relations among the features. Calculating a set of bag-of-words representations with different geometrical arrangement among the features, we keep an important association between appearance and geometric information. We evaluate our approach on two publicly available datasets for human action recognition (KTH and UCF-ARG datasets). We show that the proposed representation enhances the discriminative power of local features and improves action recognition performance.

The rest of the paper is organized as follows. In section 2, we present our novel action recognition approach. In section 3, we present obtained results from our extensive set of experiments. Finally, in section 4, we conclude with future directions of work.

Proposed Approach

We propose a novel feature representation which captures statistics of pairwise co-occurring local spatio-temporal features. Firstly, we detect local interest points and capture both motion and visual appearance around extracted points. Then, we create a set of bag-of-words representations with different geometrical arrangement among the features. Our representation captures not only global distribution of features but also focuses on geometric and appearance (both visual and motion) relations among the features. Moreover, calculating a set of bag-of-words representations, we keep an important association between geometric and appearance information. The technique presented in this section enhances the discriminative abilities of features and improves action recognition performance.

Feature Extraction

Local spatio-temporal features have demonstrated high recognition results for a number of action classes. Therefore, we use them as basic features for our approach.

For each video sequence, we extract local spatio-temporal points of interest and their local spatio-temporal descriptors. To detect interest points, we use the sparse Harris3D corner detector [START_REF] Laptev | On space-time interest points[END_REF]. To enhance the probability of capturing relevant information, we apply an algorithm searching over multiple spatial and temporal scales. Then, for each detected point, we compute HOG and HOF descriptors.

We highlight here that, all the mentioned algorithms in this section were selected based on their use in the literature and provide a good baseline for comparison with state-of-the-art techniques. However, our action representation method is independent of the type of detector and descriptor, and can be used together with any other algorithm.

Statistics of Pairwise Co-occurring Local Spatio-Temporal Features

In this section, we present our novel feature representation which captures statistics of pairwise co-occurring local spatio-temporal features. The following steps are applied for each video sequence independently. Firstly, we extract local spatio-temporal interest points P = {P 1 , ..., P n } (where P l = (x l , y l , t l )) in a video sequence (Section 2.1). Then, for every point P i we find its n-nearest neighbour points from the extracted set of points P:

F(P) {(P i , P j ) ∈ P 2 : i nn (P, i, j) ≤ n}, (1)
where i nn (P, i, j) = m means that point P j is the m-th nearest neighbour in order to point P i . To calculate the distance between two points, we use the Euclidean metric.

Then, to differentiate pairs of points between those that are close to each other from those that are far away from each other, we split the set F(P) to several smaller subsets based on the value of the function i nn :

S(P, a, b) = {(P i , P j ) ∈ F(P) : i nn (P, i, j) ∈ a, b)}, (2)
where 0 ≤ a ≤ b ≤ n.

As we mentioned in the previous section, each point P i is represented not only by its 3D position but also by the HOG-HOF descriptor. For simplicity, we indicate a HOG-HOF descriptor assigned to point P i as D(P i ). Therefore, to capture appearance (both visual and motion) relationship between two points, we represent each pair of points from the set S(P, a, b) as a concatenation of their descriptors:

D(P, a, b) = {D(P i )||D(P j ) : (P i , P j ) ∈ S(P, a, b)}, ( 3
)
where || is the concatenation operator. Finally, we represent each video sequence as a collection of sets of features D(P, a, b):

V(P, K) = (D(P, k 1 , k 2 ), D(P, k 2 , k 3 ), ..., D(P, k |K|-1 , k |K| )), (4)
where K = (k 1 , k 2 , ..., k |K| ). These sets vary in different geometrical arrangement among the features. Our novel representation of features captures geometric and appearance (both visual and motion) relations among the features. Moreover, calculating sets of features D(P, a, b), we keep an important association between geometric and appearance information. Thus, by using suitable designed features, we are able to overcome the limitation of the bag-of-words approach.

Action Representation

To represent videos, we apply the bag-of-words model for each feature class (i.e. HOG-HOF and D(P, ., .)) independently. We construct visual vocabularies from training videos clustering computed features. Then, we assign each feature to its closest visual world. The obtained histograms of visual world occurrences over video forms the final representation.

The amount of features D(P, ., .) extracted from all the training videos can be large. Therefore, to speed-up the approach, we propose to perform clustering in the following hierarchical manner. In the first step, we process each video sequence independently. To reduce the computational cost, we limit the number of features for each video sequence to F M AX using random sampling. Then, the obtained features are clustered. In the second step, we process all the training videos together and re-cluster all the obtained groups of features to create a final codebook representation.

Action Classification

To recognize an action, we use Multiple Kernel Learning (MKL) formulated for multi-class classification problem. We use MKL because it provides a natural method to combine different types of features. Given a list of base kernel functions, MKL searches for their linear combination which maximizes a performance measure. MKL considers a convex combination of n kernels:

K(H i , H j ) = n z=1 β z K z (H i , H j ), (5)
with β z ≥ 0 and n z=1 β z = 1. To compare two m-bins histograms H i = [H i (1), ..., H i (m)] T and H j = [H j (1), ..., H j (m)] T , we apply a χ 2 distance:

χ 2 (H i , H j ) = 1 2 m z=1 (H i (z) -H j (z)) 2 H i (z) + H j (z) ( 6
)
This distance is then converted into a χ 2 kernel using a multi-channel generalized Gaussian kernel:

K z (H i , H j ) = exp(- 1 A z χ 2 (H i , H j )) (7)
where A z is the normalization parameter set as in [START_REF] Laptev | Learning realistic human actions from movies[END_REF].

Experiments

Our experiments demonstrate the effectiveness of the proposed representation for a various of action categories. We evaluate our approach on two benchmark datasets for human action recognition -KTH and UCF-ARG datasets. Sample frames from video sequences of these datasets are presented in Figure 1.

The performed experiments demonstrate that our representation enhances the discriminative power of features and improves action recognition accuracy.

Implementation Details

In order to quantize local features, we use the k-means clustering technique. We use the L 2 norm metric to calculate the distance between features and visual words. We set the maximum amount of features extracted from a single video sequence to F M AX = 10 5 , which is a good compromise between the amount of data obtained from a video sequence and the time needed for clustering. We set the size of the codebook to 1000, which has shown empirically to give good results. In order to create statistics of pairwise co-occurring local spatio-temporal features, we set the parameter K to (

, which has shown empirically to give good results. In all our experiments, we apply the cross-validation technique to both gauge the generalizability of the proposed approach, and select the most discriminative statistics of pairwise co-occurring local spatio-temporal features. We use the Leave-One-Out Cross-Validation (LOOCV) technique, where videos of one person are used as the validation data, and the remaining videos as the training data. This is done repeatedly so that the videos of each person are used once as the validation data.

KTH Dataset

The KTH [START_REF] Schuldt | Recognizing human actions: A local svm approach[END_REF] 1 dataset contains six types of human actions: walking, jogging, running, boxing, hand waving and hand clapping. Each action is performed several times by 25 different subjects in four different scenarios: outdoors (s1), outdoors with scale variation (s2), outdoors with different clothes (s3) and indoors (s4). The dataset contains 599 video files. All sequences were recorded with 25 fps frame rate.

The dataset contains a set of challenges like: scale changes, illumination variations, shadows, different scenarios, cloth variations, inter and intra action class speed variations and low resolution (160 × 120 pixels spatial resolution).

We follow recent evaluations on the KTH dataset [START_REF] Liu | Recognizing realistic actions from videos "in the wild[END_REF][START_REF] Wu | Action recognition using context and appearance distribution features[END_REF][START_REF] Kim | Tensor canonical correlation analysis for action classification[END_REF][START_REF] Wu | Action recognition in videos acquired by a moving camera using motion decomposition of lagrangian particle trajectories[END_REF][START_REF] Jiang | Recognizing human actions by learning and matching shape-motion prototype trees[END_REF] using LOOCV scheme. In general, LOOCV assesses the performance of an approach with much more reliability than splitting-based evaluation schemes because it is much more comprehensive. Results from the experiments are presented in Table 1. Comparison of our approach with state-of-the-art methods in the literature using LOOCV technique is presented in Table 2. For scenarios s1, s2, s3 and s4, our approach obtains the recognition rate of 98.67%, 95.33%, 93.20% and 98.00% respectively. Overall, our approach obtains 96.30% recognition rate. The results clearly show that our representation enhances the discriminative power of features, improves action recognition performance and outperforms state-of-the-art techniques.

KTH

UCF-ARG Dataset

The UCF-ARG To the best of our knowledge there are not publicly available results for this dataset. Therefore, we compare our approach with popular baseline approach [START_REF] Laptev | Learning realistic human actions from movies[END_REF]. We use Harris3D to detect local spatio-temporal interest points and HOG-HOF descriptors to represent 3D video patches in the neighbourhood of detected points. Then, we apply bag-of-words model to represent video sequences and SVM for classification. The results from the experiments are presented in Table 3. The baseline approach obtains 80.98% recognition rate. Our proposed statistics of pairwise co-occurring local spatio-temporal features improve action recognition rate achieving 82.05% accuracy. We observe that also on this dataset, our representation enhances the discriminative power of local features and improves action recognition performance.

Method

Year Recognition Rate Laptev et al . [START_REF] Laptev | Learning realistic human actions from movies[END_REF] 2008 80.98% Our method 82.05% Table 3. UCF-ARG dataset: Comparison of our approach with baseline state-of-theart method.

Conclusions and Future Work

We have proposed a novel feature representation which captures statistics of pairwise co-occurring local spatio-temporal features. Our representation captures not only global distribution of features but also focuses on geometric and appearance (both visual and motion) relations among the features. Calculating a set of bag-of-words representations with different geometrical arrangement among the features, we keep an important association between appearance and geometric information. The proposed approach has been evaluated on two public benchmark datasets for human action recognition. Obtained results have demonstrated that our technique enhances the discriminative power of features and improves action recognition performance. In the future work, we intend to evaluate our technique using different interest point detectors and descriptors. We also intend to examine different machine learning techniques to combine various types of features.

Fig. 1 .

 1 Fig. 1. Sample frames from video sequences of the KTH (first row) and UCF-ARG (second row) datasets.

Table 1 .

 1 KTH dataset: Evaluation of our approach. The table shows the recognition rate overall and for each scenario independently.

		Recognition Rate
	s1		98.67%
	s2		95.33%
	s3		93.20%
	s4		98.00%
	s1-s4		96.30%
	Method	Year	Recognition Rate
	Ta et al . [27]	2010	93.0%
	Liu et al . [29]	2009	93.8%
	Wu et al . [30]	2011	94.5%
	Kim et al . [31]	2007	95.33%
	Wu et al . [32]	2011	95.7%
	Lin et al . [33]	2011	95.77%
	Our method		96.30%

Table 2 .

 2 KTH dataset: Comparison of our approach with state-of-the-art methods in the literature.

  2 (University of Central Florida -Aerial camera, Rooftop camera and Ground camera) dataset is a multiview human action dataset. It contains 12 actors performing ten types of human activities: boxing, carrying, clapping, digging, jogging, open-close trunk, running, throwing, walking and waving. Except for open-close trunk, all the other actions are performed 4 times by each actor in different directions. The open-close trunk action is performed 3 times.In total, we use 468 video sequences from the ground camera. The dataset is recorded using a high-definition camcorder (Sanyo Xacti FH1A camera) with 60 fps frame rate and spatial resolution of 1920 × 1080 pixels.The dataset contains a set of challenges like: different shapes, sizes and ethnicities of people, scale changes, shadows, cloth variations, inter and intra action class speed variations, and different scenarios.

http://www.nada.kth.se/cvap/actions/

http://vision.eecs.ucf.edu/data/UCF-ARG.html

Acknowledgements. This work was supported by the Région Provence-Alpes-Côte d'Azur. However, the views and opinions expressed herein do not necessarily reflect those of the financing institution.


P Achatz

W Gajewski

E Bustarret

C Marcenat

R Piquerel

C Chapelier

T Dubouchet

O A Williams

K Haenen

J A Garrido

M Stutzmann

Low temperature transport in highly boron-doped nanocrystalline diamond

Keywords: numbers: 73.23.HK, 71.30.+h, 73.22.-f, 73.61.Cw

We studied the transport properties of highly boron-doped nanocrystalline diamond thin films at temperatures down to 50 mK. The system undergoes a doping-induced metal-insulator transition with an interplay of intergranular conductance g and intragranular conductance g0, as expected for a granular system. The conduction mechanism in the case of the low-conductivity films close to the metal-insulator transition has a temperature dependence similar to Efros-Shklovskii type of hopping. On the metallic side of the transition, in the normal state, a logarithmic temperature dependence of the conductivity is observed, as expected for a metallic granular system. Metallic samples far away from the transition show similarities to heavily boron-doped single crystal diamond. Close to the transition, the behaviour is richer. Global phase coherence leads in both cases to superconductivity (also checked by a.c. susceptibility), but a peak in the low-temperature magnetoresistance measurements occurs for samples close to the transition. Corrections to the conductance according to superconducting fluctuations account for this negative magnetoresistance.

I. INTRODUCTION

Depending on the microscopic parameters, a granular material can vary from a relatively good metal to an insulator, by changing the strength of electron tunneling coupling between conducting grains. Only recently, new insights have been gained on the theoretical understanding of such granular systems 1 , which have been applied to the experimental low temperature transport behaviour in highly boron-doped nanocrystalline diamond. Indeed, renewed interest in doping-induced electronic phase transitions came up as superconductivity was discovered in heavily boron-doped diamond 2 , and was confirmed by several other groups (see Ref. 3

and references therein).

The metal-insulator transition in highly boron-doped diamond has been analyzed in ion-implanted films 4 and more recently in single crystalline diamond 5 . In this paper, we report on the metal-insulator transition and superconductivity in highly boron-doped nanocrystalline diamond (B:ncd) thin films, demonstrating the influence of the granular structure on the transport properties at low temperatures.

II. EXPERIMENTAL

Nanocrystalline diamond films were grown in a microwave plasma-enhanced chemical vapor deposition (MPCVD) as described elsewhere 6 . Doping with boron was achieved by adding trimethylboron to the gas phase with molar concentrations B/C reaching 10000 ppm. These films were grown on quartz substrates (suitable for transport measurements), which were either prepared by mechanical polishing with diamond nanoparticles (Bmec prefix) or by ultrasonic seeding in an aqueous colloid of 10 nm sized diamond powder (Bus prefix) 7 . Electrical resistivity measurements were performed from room temperature down to 360 mK using a Quantum Design Physical Properties Measurements System. A four terminal configuration has been used for the electric contact to the films. In the same setup, magnetoresistance measurements were performed with perpendicular fields up to 8 T. A dilution fridge was used at lower temperatures with perpendicular fields up to 14 T. The change in the mutual inductance of a small coil was used for the a.c. magnetic susceptibility measurements. The boron concentration n B has been obtained from Secondary Ion Mass Spectroscopy (SIMS) depth profiles of 11 B -, 12 C -, and 11 B 12 C -ions.

III. RESULTS AND DISCUSSION

The temperature dependence of the normalized resistance ρ/ρ(300K) in the range 300 K down to 350 mK is shown in Fig. 1 (a) for several samples. The highly doped samples with n B ≥ 1.3 × 10 21 cm -3 also show superconductivity with critical temperatures up to T c = 2.1 K. A.c. susceptibility measurements confirmed that this superconductivity has not a filamentary nature. As illustrated by Fig. 1 (b), the onset of the diamagnetic screening corresponds to the transition to zero resistance. The critical boron concentration n c for the metal-insulator transition is about 2.6 × 10 20 cm -3 (Ref. 6). Similar values have been found in the case of single crystal and polycrystalline diamond 3 . However, as will be shown be- low, the metal-insulator transition is not only due to the increasing doping within the grains, but also to the increase in intergranular coupling between metallic grains. Fig. 2 (a) shows the temperature dependence of the normalized conductivity for the weakly-doped samples Bus100 and Bmec2500, which are close to the metalinsulator transition. A Efros-Shklovskii (ES) type of hopping dependence following ln(σ) ∝ (T 0 /T ) 0.5 is observed over a remarkably wide range of temperatures, from 300 K down to 20 K. Multiple cotunneling processes can lead to Efros-Shklovskii-like variable range hopping in granular metal arrays with weak intergrain coupling 8,9 . An estimate of the localization length ξ from the experimentally determined T 0 values gives ξ ≈ 1 nm. Taking into account an acceptable variation of the dielectric function for this estimate, the localization length would be still much smaller than the grain size in our samples. The situation corresponds to well-coupled insulating grains (in agreement with the SIMS values) rather than weakly coupled metallic grains, with the localization length comparable to the interatomic distance between the impurities 10 . However, Efros-Shklovskii type of tem- perature dependence at such high temperatures has not been observed in single crystal diamond samples within the same doping range, where Mott variable range hopping has been seen without a crossover to ES hopping at low temperatures 5 . This remains an open question which one can probably assign to the granularity of the samples.

The physical quantities characterizing the transport in a granular material are the dimensionless intragranular conductance g 0 , and the intergranular (tunneling) conductance g. Both correspond to one spin component and are measured in the units of the quantum conductance e 2 / : g = G/(2e 2 / ) with G the conductance ([G] = Ω -1 ). Other important parameters are the grain charging energy E c = e 2 /4πk B ǫ 0 ǫ r d (d is the grain size), and the mean level spacing δ = (νV ) -1 (ν is the density of states and V the volume of the grain) 1 . If the constituent grains are made of a superconductor mate-rial, the behaviour of such a system can be quantified by adding one more energy parameter, the superconducting gap ∆ of the material of a single grain. The interplay between Josephson coupling J and grain charging energy E c determines whether global macroscopic superconductivity is built up in the sample or not 1 . A "granular system" is obtained in the limit g ≪ g 0 , whereas the case g ∼ g 0 can be viewed as a homogeneously disordered system.

Samples with g > g c exhibit metallic transport whereas samples with g < g c show insulating behaviour, corresponding to metallic grains with strong and weak intergrain coupling, respectively. Here, g c is the critical conductance which is related to E c and δ as

g c = (1/2πD) ln(E c /δ) ( 1
)
where D is the dimensionality 1 . For a grain size d ≈ 150 nm and a typical density of states ν ≈ 10 20 eV -1 cm -3 one gets E c ≈ 20K and δ ≈ 1mK. One could expect two-dimensional effects, as the grain size is of the same order of magnitude as the thickness and, thus, g c ≈ 0.52 or g c ≈ 0.77 for a three-dimensional or two-dimensional system, respectively. These values lie between the estimated conductances for samples Bus100 and Bus1000.

For samples with g > g c , one should find a logarithmic temperature dependence (for two-as well as three-dimensional systems) of the conductivity at low temperatures 1 . This behaviour has been reported for several granular metallic systems [START_REF] Radhakrishnan | [END_REF]12 . As can be seen in Fig. 2 (b), some of the nanocrystalline samples show a logarithmic temperature dependence at low temperatures, clearly indicating a metallic behaviour in the granular limit g ≪ g 0 . A crossover to superconducting behaviour occurs for the highly conducting samples (Bus4000 to Bus10000). The temperature at which the deviation from the logarithmic temperature dependence takes place increases with doping. Electron-electron interaction is responsible for the logarithmic temperature dependence; at higher temperatures interactions with phonons lead to a power law dependence (whole temperature range for Bus1000).

The superconducting properties of a granular system are in many ways determined by the properties of the grains. The superconducting gap ∆ exists in each grain and its value is close to the gap magnitude in the bulk, provided ∆ ≫ δ 1 . Anderson showed that the superconducting gap is suppressed or can even be fully destroyed if the latter inequality is not satisfied 13 . In our case, δ = (νV ) -1 ≈ 1mK and ∆ ≈ 300µV = 3.6K 14 and, thus, these effects can be neglected. Sufficiently strongly coupled grains (g ≫ 1) can maintain (global) superconducting coherence because the coupling reduces the phase fluctuations.

In FIG. 3: Magnetoresistance of sample Bus4000 at several constant temperatures, revealing regions with negative and positive magnetoresistance, respectively. by free charges usually seen in metals. In the case of strong coupling (g ≫ 1) and strong Coulomb interaction (E c ≫ ∆) the charging energy is renormalized to ≈ ∆/g. Therefore, for g ≫ 1 the effective Coulomb energy is always smaller than the Josephson coupling J ≈ g∆, and superconductivity occurs at sufficiently low temperatures 16,17 . With E c ≈ 20K this is the case for the most highly-doped samples. These samples follow a transport behaviour typically seen for disordered metals. Measurements of the magnetoresistance as well as the critical magnetic field phase diagram reveal similar results compared to highly boron-doped single crystal diamond 5 . Fig. 3 shows the magnetoresistance of sample Bus4000 at several constant low temperatures in the range from 100 up to 500 mK and with perpendicular fields up to 14 T. Bulk superconductivity (superconducting grains) is destroyed by application of extremely strong fields, the positive magnetoresistance in the high-field limit resembles the behaviour seen in metallic single crystal and nanocrystalline diamond films 18,19 . By decreasing the magnetic field a region with negative magnetoresistance is found. Only at sufficiently weak magnetic fields the resistivity starts to decrease and the sample displayed superconducting behaviour. Similar behaviours have been reported in other granular systems [20][21][22] .

One possibility to explain the negative magnetoresistance of a granular metal is due the suppression of the density of states (DOS) due to superconducting fluctuations 23 . One fluctuation contribution comes from nonequilibrium Cooper pairs forming a new transport channel (δσ AL ) 24 , and another from coherent scattering of electrons forming a Cooper pair on impurities (δσ MT ) 25,26 . Both corrections enhance conductivity and lead to a positive magnetoresistance. The formation of nonequilibrium Cooper pairs results in a fluctuational gap in the one-electron spectrum 27 . This suppression of the density of states has been shown to lead to a correction of the conductivity δσ DOS 23 . While in conventional superconductors δσ AL and δσ MT are more important than δσ DOS , in granular systems the DOS correction plays a very important role 23,28 . The conductivity δσ DOS is negative and its absolute value decreases with increasing magnetic field, thus leading to the negative magnetoresistance. This is what is seen for sample Bus4000 in Fig. 3.

Another possibility is based upon the fact that, as is the case for sample Bus4000, g is close to one, approaching the quantum resistance R 0 = h/(2e) 2 ≈ 6.4kΩ for which a superconductor-insulator transition is expected. In the case of a superconductor-insulator transition, similar behaviours of the magnetoresistance have been reported [29][30][31] . The origin of such a transition is still under debate. Some theories claim that it may be understood in terms of Cooper-pair scattering out of the superconducting condensate into a Bose-glass 32,33 , others claim that a percolation description of the superconductor-insulator transition is more adequate 34 .

IV. CONCLUSION

In summary, we report on the metal-insulator transition in highly boron-doped nanocrystalline diamond films. The metal-insulator transition is induced by the increasing doping, thereby also controlling the intragranular conductance. Low temperature transport measurements suggest that metallic nanocrystalline diamond follows dependencies for the conductivity typical for a granular system with metallic/superconducting grains and strong or low intergranular coupling. On the insulating side of the transition, Efros-Shklovskii-like variable range hopping was observed, while on the metallic side a logarithmic temperature dependence of the conductivity was found. The granular structure influences the superconducting properties and the magnetoresistance in highly boron-doped metallic nanocrystalline diamond films. Tuning of the microscopic parameters leads to rich behaviour and similarities to the superconductorinsulator transition are pointed out, explained on the basis of superconducting fluctuations and the importance of granularity.

FIG. 1 :

 1 FIG. 1: (a) Temperature dependence of the normalized resistance ρ/ρ(300K) from 300 K down to 350 mK. The weakly doped insulating sample Bmec2500 shows thermally activated transport. The higher doped metallic samples show also superconductivity; (b) a.c. magnetic susceptibility measurements clearly indicate the non-filamentary nature of superconductivity. The onset of the diamagnetic response corresponds to the transition to zero resistance.

FIG. 2 :

 2 FIG. 2: (a) Temperature dependence of the normalized conductivity for the weakly doped samples Bus100 and Bmec2500, following a Efros-Shklovskii hopping over a wide range of temperatures. The T0 values of both samples are indicated; (b) Logarithmic temperature dependence of the normalized conductivity for several highly doped samples as expected for a granular metal (T in units of Kelvin).

  the approach of Chakravarty et al. (see Ref. 15) the originally strong Coulomb interaction is reduced by the electron tunneling to other grains. This renormalization corresponds to the screening of the Coulomb interaction

V. ACKNOWLEDGEMENTS

This work was partly financially supported by the Research Programs G.0068.07 and G.0430.07 of the Research Foundation Flanders (FWO), the IAP-P6/42 project 'Quantum Effects in Clusters and Nanowires', and by Nanosystems Initiative Munich, NIM.


Kossi Tiassou

Karama Kanoun

Mohamed Kaâniche

Christel Seguin
email: christel.seguin@onera.fr

Chris Papadopoulos
email: chris.papadopoulos@airbus.com

Online model adaptation for aircraft operational reliability assessment

Keywords: Reliability, online evaluation, mission planning, aircraft, operability

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

All along its mission achievement, an aircraft has to fulfill operational requirements involving the current online situation. In particular, before each flight, some conditions called dispatch requirements, must be met otherwise the aircraft is not allowed to fly or its mission has to be adapted to fit the current aircraft configurations capability. The dispatch requirements concern principally the current operational state of the aircraft's equipment, the intended mission and the ability to conveniently achieve maintenance activities. It is worth mentioning that aircraft mission profiles are not identical and maintenance facilities are not the same at all airports. Usually, there are more facilities at the aircraft operator's base than at the other airports. Additionally, while a given equipment failure may not allow dispatch for some mission profile, it may require only some simple adjustments like deactivating the equipment solicitation in other cases.

An aircraft mission consists of a sequence of flights and stops. Not meeting the operational requirements before or during a flight can cause significant disruptions that may lead to heavy economic loss [START_REF] Ahmadi | Assessment of Operational Consequences of Aircraft Failures: Using Event Tree Analysis[END_REF] due, for instance, to inoperability and compensation given to passengers. Therefore, special attention must be continuously paid to the aircraft operational capability. The missions and maintenance activities must be adjusted regularly. One can base the assignment of a suitable mission to a given situation and maintenance activities planning on operational reliability assessment using models. The model should be able to capture and to adapt to the various situations that may be encountered during missions' achievements.

Our work aims at developing an assessment approach, based on dependability modeling, that makes it possible to continuously assess the ability of an aircraft to keep operating up to a given time or location. Our approach is based on developing a generic stochastic dependability model that can be dynamically updated to represent the current state of the aircraft, with regards to the planned/ intended mission. The model is structured in such a way that different kinds of measure can be done, considering a mission profile or not, maintenance activities or not. The global objective is to assess the operational reliability of the aircraft with regards to operational interruptions. In this paper, an interruption corresponds to a delay, a flight cancellation, an in-flight turn back or a diversion from the initial plan.

The work does not address safety, which is dealt with in previous published work (see e.g., [START_REF] Prescott | Aircraft safety modeling for time-limited dispatch[END_REF]). It is mainly dedicated to aircraft operability and more specifically to model adaptation online, during the aircraft mission. To the best of our knowledge, there are almost no publications related to aircraft operability modeling for online assessment. The studies related to aircraft operability are about dispatch reliability and are carried out for design enhancement [START_REF] Saintis | Computing in-service aircraft reliability[END_REF][START_REF] Bineid | Development of an aircraft systems dispatch reliability design methodology[END_REF]. In [START_REF] Ahmadi | Assessment of Operational Consequences of Aircraft Failures: Using Event Tree Analysis[END_REF], the operational risks of aircraft systems failures were studied using event tree analysis, but the work did not deal with the operability assessment during the aircraft mission. This paper presents the modeling approach, together with its online adaptation management. Section 2 provides a description of a typical aircraft's mission. Section 3 outlines the role that the model-based reliability assessment can play in an aircraft's operability improvement. Section 4 presents the modeling and update approach. Section 5 presents a formal description of the model and identifies the changes that may affect it during missions' achievement. Section 6 is devoted to the conclusion and future work.

MISSION DESCRIPTION

An aircraft mission consists in performing a predefined set of flights under some maintenance conditions. The achievement of the mission is such that each flight is followed by a stop where the aircraft is prepared for the next flight. At each stop, the aircraft is inspected and the discrepancies that are reported during the previous flight are checked. If a component is found inoperative, a dispatch decision is taken based on the requirements of the next flight. The flight captain refers to an approved document called Minimum Equipment List (MEL) where the components are listed with the status Go, Goif or Nogo.

The Go status is the case where the aircraft can fly with the component inoperative. It is worth noting that even in this case, attention must be paid to the future behavior of the aircraft since a subsequent failure may prevent dispatch.

For the Goif status, the flight can be achieved provided that other equipment are operative and/or some technical (operational or maintenance) procedures are feasible:

• Go-if-o: Some operational procedures must be carried out or feasible to allow the dispatch. It concerns essentially a limitation in the functionalities that will be available during the flight.

• Go-if-m: Some maintenance procedures must be carried out. These include time limited dispatch requirements and have an impact on the planned maintenance activities, e.g., the failed component must be repaired within a period of ten days.

The Nogo status prevents the aircraft from flying. In this case, the component must be repaired before any flight.

Figure 1 summarizes the possible outcomes of the dispatch status.

Dispatch is allowed if i) there is no "No Go" and ii) all "Go If" conditions are acceptable.

When dispatch is allowed, the flight can be aborted or diverted if the aircraft capability is degraded. Procedures stated in the Flight Crew Operating Manual (FCOM) are used to determine whether the flight must be diverted (or aborted depending on the location) or not.

Adverse situations while operating an aircraft lead to operational interruptions such as, flight delays, cancellations, in flight turn-back and diversions [START_REF] Saintis | Computing in-service aircraft reliability[END_REF][START_REF] Ahmadi | Assessment of Operational Consequences of Aircraft Failures: Using Event Tree Analysis[END_REF]. Our work mainly deals with operational interruptions caused by systems failures and the inability to maintain within an acceptable time.

ROLE OF MODEL-BASED RELIABILITY ASSESSMENT

The problem is to assess, during the aircraft's operations, its ability to satisfy the operational requirements, in the presence of unforeseen events, and initiate an adapted corrective action to prevent adverse situations. There is a need to have good control on the aircraft's behavior while in service, and to be able to predict, with a good level of precision, the events that may be encountered. This includes an ability to cope with random events and a capacity to ensure that the solutions adopted are suitable. Model-based dependability assessment is well suited to support this process.

In our approach the model is essentially used to evaluate the probability to continue operating until a given period of time, known as reliability measure, and the estimated probability will determine whether a corrective action must be initiated or not.

The model can be used while planning the missions and during their achievement. To plan the mission, the model can be used to estimate the period of time during which the aircraft system can be operated without reaching an adverse state. This allows us to determine the mission profile to be assigned to the aircraft. Once a mission is assigned to the aircraft, the model can be used during its achievement, both on ground and while in flight, to assess the ability to succeed in continuing on the remaining part of the mission, and readapt it if necessary. The following describes how the model can be used during the mission.

On ground

The dispatch decision process takes into consideration the ability to achieve the upcoming flight. Using the model, this dispatch decision may take into account the ability to continue until the next airport where there are enough facilities to fix the problems that may be encountered. To do so, the model is updated with the current state of the aircraft, the failure distributions of the system components, the considered mission profile and maintenance activities.

During the flight

During the flight, the model can be updated, after the occurrence of major events that may affect the operability, to provide an indication on the reliability of the remaining part of the mission. The result may help in selecting the most appropriate maintenance or operations planning actions in order to improve the ability to achieve the whole mission. When a major event happens, the model is updated and run to assess the aircraft ability to continue the mission. The outcome may be used to support the decision, by the operations control centre, to continue the flight or revise the planned mission. In case of a decision to divert the flight, the model can also be used to determine a convenient diversion airport. In case of emergency (due for example, to problems that may affect safety), the model can be used once a diversion airport is selected, to re-assess the operational reliability.

Maintenance planning

Assessing the success of a mission is a means for evaluating the concordance of a maintenance program with the mission. Different maintenance strategies can be compared considering various alternatives and priorities for performing component repairs. The best strategy can be determined based on the estimated probability of mission accomplishment without operational interruption.

THE MODELING APPROACH

As stated in Section 2, the aircraft has to fulfill a set of operational requirements from the MEL (dispatch requirements) before flying and operational requirements from the FCOM (in-flight requirements) during the flight. We distinguish:

The minimal requirements (Min_Sys_Req): from the MEL, independent of the mission profile and which must be fulfilled to operate the aircraft whatever the mission profile.

The mission profile requirements (M_Prof_Req), which are specific to the mission profile.

The evaluations will be based on the fulfillment of both of these requirements.

Quantitative measure

The objective is to evaluate the probability of occurrence of an adverse event that may lead to an operational interruption. We distinguish two cases of evaluation:

• While planning a mission: the aircraft system reliability (SR) is evaluated with regard to Min_Sys_Req in order to determine the maximum number of flight hours that can be achieved, without maintenance. This is used to determine the length of the mission or to plan maintenance activities.

• Once a mission is assigned to the aircraft and during its achievement: the probability to achieve the mission without an operational interruption, referred to as Mission Reliability (MR). MR is evaluated with regard to Min_Sys_Req and M_Prof_Req in order to determine whether a preventive action (change in the mission profile or maintenance actions) must be initiated or not.

Structure of the model

Assessing the aircraft operational reliability based on dependability models is clearly convenient to suitably manage its mission. However, the assessment will be useful only with an adequate modeling approach. The model must not only represent efficiently the aircraft and its missions but also be easily updatable to be adapted efficiently to i) the current real state of the aircraft, ii) to changes in the initial mission, and iii) to changes in the maintenance facilities at the next airports. The model must be thus designed in such a way so as to facilitate the updates.

Figure 2 shows the overall structure of the model. In this paper, we focus on the main philosophy for structuring and describing the model rather than on a specific modeling formalism. For the sake of separation of concerns and in order to make the model flexible, the relevant elements of the global problem that need to be captured appear explicitly. Four levels are distinguished.

Operational level: It describes the succession of periods during which the aircraft is either flying or on ground (i. e., the mission profile). A distribution is specified to define the duration of each period.

Requirements level: It corresponds to the aggregation of the operational requirements Min_Sys_Req and M_Prof_Req, expressed as a function of the states of the aircraft components and the mission profile. These requirements are formulated as Boolean expressions, representing the different combinations that do not lead to an operational interruption.

System: It describes the aircraft system behavior. The system is decomposed into subsystems and atomic components according to its design logic or its functions. This level describes the components failure scenarios.

Maintenance level: It describes the maintenance possibilities at the various airports involved in the mission. It is intended to represent the predefined maintenance facilities available at the airports. This has an impact on the repair time of the system components at a given stop. The maintenance activity itself is modeled at the system level.

Figure 2 Overall structure of the dependability model

The above structure of the model is of great help for model updating. The model can be easily adapted when changes occur in a given level. For instance, when a component, which must be maintained at a main maintenance base airport, fails, the system level sub model should be updated with this new state and the new failure distributions of the other components. The operational level sub model should also be updated with the remaining parts of the mission. Therefore, the global model construction process must be designed in such a way so as to cope with these issues and facilitate the update. The model construction and update are presented in the following.

Model construction and update

The model construction is based on a dynamic model building process, taking into account the different categories of updates that may generally happen. The global model building consists in an initial basic modeling work and the establishment of instructions that can be used for the updates online. The different kinds of updates that may generally affect the model are identified in the following.

Overview of the dependability model's kinds of update

Three kinds of updates may take place.

Update of the initial state of the model: this corresponds to the case where for example some system components have failed and/or the aircraft is at a new step of the achievement of the mission. The reliability is evaluated to assess the impact of the new state on the mission's achievement. In this case, only the states of the components are updated; the events distributions and the mission profile may remain the same.

Update of parameters or event distributions: this corresponds to the case where i) new failure distributions have been prognosticated for the system components, or ii) the mission's flights durations have changed or iii) the maintenance facilities at the next airports have changed, affecting the mean time to repair the components. The update can concern the parameters of the distribution laws or the laws themselves. It can have an impact also on the model processing techniques. Analytical methods always require some restrictions on the events distributions laws used in the model, e.g. some formalisms do not allow non-exponential laws to be managed. In this case, simulation techniques are used to process the models.

The update affecting the structure of the model: this corresponds to the case where the mission profile has changed. For example while considering various alternatives of mission profile. Changing the mission profile may imply also changes in maintenance scenarios (the maintenance activities take place at the airports included in the mission profile) and M_Prof_Req (each mission profile has its specific requirements). Therefore, different operational, requirements and maintenance levels sub models corresponding to each alternative, have to be considered.

Due to the need of reconstruction implied by these possible changes, the method to obtain the global model must be clearly established.

Construction of the model

We consider the following dynamic model building methods:

Dynamic Composition of pre-developed sub-models: In order to build the global model online, one may use pre-developed sub-models corresponding to pre-identified and analyzed situations. The sub-models are built with coherent linking interface that can be used to compose them. The sub models can be gathered in a library and the construction of the global model consists just in the composition of some selected ones. The drawback of this technique for an online model construction is that all the possible scenarios must be identified and modeled once and for all.

Completion of a model: One may have an initial model, which can be completed in order to create a global model corresponding to a more general situation. To do so, the initial model provides a "public" interface that is used to develop and integrate the additional sub-models. The "public" interface is a set of identifiable variables provided by the initial model. These variables are used by the additional sub-model to depict its behavior.

For our case, there will be an initial construction that can be completed by sub-models whether specified online or retrieved from a library. The construction of the global model consists then of two steps: the construction of an initial model, done once and for all, and its completion by the mission profile information online. The initial model describes the static part of the model and provides an interface for the initialization of its variables. The interface also provides the information regarding the composition with the additional models.

Model update management

The approach to manage the model update consists in using some configuration files that can be updated with the current states of the components and the mission profile. The parameters that are set in the configuration files will be used to update and complete the model before the (re-)evaluations. Moreover, the update of the configuration files will be notified to an evaluation manager, which will determine its relevance in order to initiate the processing of the model.

The evaluation manager can correspond to a human operator or to an automatic algorithm that determines when the identified changes in the mission profile, or the system state should lead to an update of the model, or most probably a mix of both.

Figure 3 shows the update scenario.

It is expected that the system update will rely on diagnosis and prognosis modules (that are run and updated at runtime either automatically by diagnosis and prognosis online mechanisms embedded in the control systems or by the crew) to parameterize the system level sub model. Data from the flight plans will be used to configure the operational level sub model.

Figure 3 Update of the model

As stated in section 3, the evaluation can be done either at the operator's request or automatically in order to notify the operator of changes that may affect the ability to continue the mission.

The aim is to make the data exchange, between the evaluation manager and the model, independent of the modeling formalism used for building and processing the dependability model. The update is possible by someone who is aware that there are changes requiring a model update, and who can easily enter the changes. Some kinds of update are automatically done by other runtime processes.

Additionally, updating and processing the model can be performed either online or on ground while the aircraft is still in flight, without waiting for the aircraft landing, in order to obtain as early as possible the new assessment results (i. e., the updated system reliability and/or the updated mission reliability).

A formal description of the underlying dependability model is given in the following, together with the update specification and integration.

FORMAL DESCRIPTION, POSSIBLE CHANGES AND BUILDING THE MODEL

The formal description of the model helps us in identifying more clearly the major variables affected by changes in the model. The description of the scenarios to specify the changes deals with the data exchange while updating the model.

Formal description

We formally present in the following the different components of the model by detailing the information involved in each level. To facilitate the comprehension, an UML-like graphical representation of the different elements is given in figure 4.

Figure 4 General description of the model

Operational level

A mission is composed of a sequence of flights, which are actually achieved considering a preflight period on ground (named "ground period" in this paper) to get ready, and the flight period itself. Let Gp denote a preflight period on ground and Fp a flight period. The couple CF =(Fp, Gp) or CF= Fp!Gp represents the complete flight achievement process, from the flight preparation activities to the flight end. In the remainder of this paper, the operator "!" symbolizes a succession of activities or periods.

A mission M composed of n flights is then formulated as follows:

! M = • i=1..n CF i = • i=1..n (Gp i ,Fp i ).
Each flight period can be decomposed into phases that are distinguished by the system functionalities required for their success. Assuming that a given number p of phases can be identified for each flight, we have Fp= Ph 1 ! Ph 2 ! … Ph p . The order of the phases in this notation is important in the sense that the phases are achieved successively. Each phase Ph j has a duration determined by a distribution law DPh j that may be deterministic.

Let I denote the occurrence of an interruption during the mission. I might occur during a flight period or a ground period.

The interruption of a flight is defined as the interruption of one of its phases. A phase interruption is defined as the loss of its requirements fulfillment when the phase is ongoing. At this level, the requirements fulfillment of a phase Ph i is represented by a Boolean variable RPh i , which indicates whether the requirements are fulfilled or not. These requirements are defined at the requirements level.

At each ground period, it should be ensured that the aircraft meets the requirements to achieve the next flight and some activities should complete before departure time; otherwise an interruption occurs. A ground period can consist of: i) scheduled maintenance (SM) activities and other ground activities (OGA) or ii) scheduled maintenance activities extended by unscheduled maintenance (UM) activities, followed by the other ground activities.

Accordingly, each ground period can be represented by the succession of the corresponding activities: Gp=SM ! UM ! OGA. SM and OGA have an associated duration determined by a distribution law, which depends on the considered location. The extension of an SM activity with an UM activity depends on the operational state of the system and the maintenance facilities. An UM activity generally takes place when the dispatch requirements (denoted DR) are not met. It is generally dedicated to the repair of the critical system components that are needed to perform the flight. These are explicitly identified in DR.

A ground period has a deterministic planned duration pd(Gp), which indicates the maximum time beyond which an interruption of the flight occurs.

From the above representation, it appears that the definition of the operational level requires the knowledge of the number n of flights composing the mission, the phases composing each flight and their durations, as well as the ground period activities durations, and the sequencing of the flights and ground periods. The planned duration of the ground period activities, beyond which the flight is interrupted, should also be defined.

Requirements level

This level describes the requirements to be satisfied for the successful achievement of the mission defined at the operational level, taking into account the decomposition of the mission into successive flight and ground periods.

The successful accomplishment of the phases Ph j of a flight is conditioned by the availability of a group of functions f 1 f 2 … f nj delivered by the aircraft system. Thus, the availability of these functions corresponds to the requirements to be satisfied during each phase to ensure the successful evolution of the flight. These requirements, denoted as RPh i , can be defined by a Boolean expression identifying the combination of functions that need to be available for achieving the corresponding phase. Alternatively, these requirements can be defined through the identification of the combination of function losses that would lead to the interruption of the flight phase.

The dispatch requirements DR i to be fulfilled by the system during a ground phase Gp i can be defined in a similar way. These requirements, defined by a Boolean expression, are determined by: i) the availability of some required functionalities f 1 , f 2 , … f nf and ii) the possibility of performing some maintenance activities (Ma) within the planned duration pd(Gp i ) :

DR i =f (f 1 , f 2 , … f nf , Ma).
It is noteworthy that the duration of an unscheduled maintenance period UM correspond to the time needed to perform the unscheduled maintenance actions.

To summarize, a requirement is a combination of functions needed at the operational level, to allow dispatch or the successful accomplishment of a flight phase. The requirements associated to a given mission composed of n flight cycles result from the aggregation of the requirements associated to each flight cycle of the mission. For a given flight CF i , DR i and RPh j=1..p are the requirements related to ground and flight phases. For dispatch requirement DR i , the required functions are almost the same whatever the flight to be achieved. Therefore, we gathered the required combinations of functions needed by every flight in a requirement called Min_Sys_Req. We denote by M_Prof_DR i the additional requirements that are specific to the mission profile under investigation. These mission profile specific requirements can be related to the availability of some functions or to the achievement of the maintenance activities required to dispatch the flight, if any.

Accordingly, DR i = Min_Sys_R ! M_Prof _DR i . M_Prof _DR i is by default true.

The requirements are represented using Boolean expressions, based on the availability of system functions. The availability of each function is derived from the analysis of the availability of the system components contributing to its implementation. The mapping between system function states and system components states is achieved at the system level of the modeling approach. Concretely a function is characterized by its state, which is defined by a conditional function based on the system components state. In the following, we use the notation:

f k=1,2,.. = g (C 1 S, C 2 S, … C nk S); C 1 S, C 2 S, … C nk S
are the variables representing the state information of the components involved in the accomplishment of f k , g is a function formulating the relation between the components states and the function f k .

System level

The system can be seen as a set of components C l with possible dependencies among them. It is represented using stochastic state space techniques. Each system component is subject to failures events and maintenance activities. More generally, the state of a component C l , represented by C l S, may take different values identified as its domain C l SD. C l SD can be decomposed into two domains C l SD = Operational (C l SO) ! Failed (C l SF). Usually, at least two distinct states are associated to each component: C l SD = {ok, ko}. Failure events and maintenance activities are then defined as change of the state variable value respectively from C l SO to C l, SF, and from C l SF to C l SO. The specification should also include the definition of the probability distributions describing the occurrence of failures events (denoted as Fdistr i ), and maintenance activities durations (denoted as Mdistr i ). Generally, exponential distributions are assumed for these events, characterized respectively by instantaneous failure rates "(t), and repair rate µ(t). Usually, these rates are assumed to be time independent. A maintenance strategy should also be defined with priority level is associated to each component in order to determine the order of the maintenance activities when several components are failed.

It appears from this description that the relevant characteristics that define a system component are its state, its failure distribution law, and its maintenance duration distribution law.

As maintenance activities depend on the facilities available, the repair rate may depend on the considered location.

Maintenance level

Maintenance activities at each ground period are characterized by the resources available such as spares and technicians. Considering a nominal maintenance distribution for each system component C l , we consider an impact function MI Gp which is used to determine the additional delay that the ability to have the facilities necessary for the maintenance tasks at the considered ground period Gp, may add on the nominal maintenance duration. A similar approach is proposed in [START_REF] Kumar | Reliability, Maintenance and Logistic Support: A Life Cycle Approach[END_REF].

It is at this level that the deferred maintenance activities are managed together with the global maintenance program.

When several components are failed, there may be a prioritization in the maintenance activities, inducing an order in the components maintenance. The management of the priorities is done at this level. It is done by allowing the maintenance activities by priority level. The maintenance of two or more failed components of the same priority level is non deterministic.

The parameters specifying maintenance duration in the different ground periods included in the mission profile are the parameters that can change at this level.

Interface between the levels

The interface between the operational level and the requirements level corresponds to the list of requirements ((RPh l , …RPh p ) i , DR i ) i=1..n corresponding to the flight phases and ground periods.

The interface between the requirements level and the system level is the list of functions (f 1 , f 2 , … f nf ) provided by the system, on which is based the specification of the requirements.

The interface between the system level and the maintenance level is the maintenance facilities impact on the system components maintenance (MI Gp ).

The interface between maintenance level and the operational level is the information about the current ground period and whether it is ongoing or not.

Changes in the model and specification update

The changes concern essentially the mission profile, the system components and the maintenance policies. The changes that may occur are described together with how they can be reported to the model in the following.

Changes in the different levels

Operational level: The changes concern the definition of the number n of flights, the parameters of each flight, and the ground period parameters. The parameters of a flight are the duration distribution laws of the phases Ph 1 , Ph 2 , … Ph p . For a ground period, the parameters concern the total duration allocated to the ground activities, the estimation of scheduled maintenance activities (SM) and the other activities (OGA) durations. These durations are specified using estimated values given whether by an operator, or based on historical data.

Requirements level: When the mission profile changes, the functions required for the achievement of the flights in the new mission profile may change. That is for each flight Fp i in the new profile, the required functions allowing it and those required for each of its phases have to be indicated. The specification of the requirements consists in defining combinations of predefined functionalities. The requirements can be specified by selection of predefined ones or combination established by the operator, based on known functionalities. However, it will be hardly necessary to change the requirements expressed in the initial model.

System level:

The changes concerning the system components are their initial state and their failure and maintenance distribution law. For the state of the components, it consists in changing its current value to another value of its domain. The impact on the global model is just the change of its initial configuration. For failure and maintenance distributions, it consists in considering new distribution function or new values for the distribution function parameters, in order to have a better fitting of the event occurrence distribution.

Maintenance level:

The parameters that must be updated at the maintenance level are the maintenance influence function MI Gp related to the ground periods involved in the mission profile. These are defined whether by maintenance crew via the specification of facilities available, or based on historical maintenance data related to the ground periods.

Model specification update

From an external viewpoint, the changes will be considered as mission profile change, component state change, failure prognosis, and maintenance duration estimation. The mission profile change will most likely involve changes in several levels of the model. It may concern the operational level, the requirements level and the maintenance level. The state change and failure prognosis concern the system level. The integration of the changes to the model will be done considering specification from external actors/processes. The scenarios to specify the changes that must be integrated are described in the following.

Mission profile:

The specification of a new mission profile is based on the flights to achieve. To simplify the task, it may be possible to use predefined objects. All the flight and ground periods that need to be taken into account must be specified following the order of their achievement. The specification of a mission profile will be done based on the following operations. • Specification of the additional dispatch requirements (M_Prof_DR).

• Specification of the phases Ph 1 , Ph 2 , … Ph p . For each phase:

-Its duration DPh i : the external actor/process provides an estimated deterministic duration and, in case it is not deterministic, specifies the probability distribution function characterizing DPh i .

-Its requirements stated as Boolean combination of functions For the specification of requirements, the user may select requirements related to a predefined flight or define them by combining listed functionalities using "and", "or" and "not" operators. This task is adapted to our targeted users, who are aircraft operators, as they are familiar with the functionalities and only basic logical operators are used.

" Specification of a ground period:

o Indication of its planned duration.

o Specification of scheduled maintenance (SM) duration: similar to a flight phase duration specification.

o Definition of the other activities duration.

" Specification of maintenance policies: Indication of the list (MI Gp1 , MI Gp2 … MI Gpn ) corresponding to the ground periods included in the mission profile.

The maintenance impact function cannot be given directly by maintenance crew. They should be determined based on basic information that can be provided by maintenance crew. The estimation can be based on information like the technician availability and the time to repair some typical components.

Component state

The indication of a new value for a component state. This can be done either by a diagnosis process, which assesses the component state and provides the information concerning the current state, or via a human machine interface where an operator indicates the new state value.

Prediction of component failure and maintenance activity duration

They concern the specification of an event distribution. New failure distribution and maintenance duration distributions can be specified either by a human actor or a process. In case of a human actor, the distribution is specified by giving values like the estimation of the time to the event occurrence, which can be used as parameters for a predetermined distribution function.

The distribution function can also be given directly by an external process (a prognosis module). The possible distribution functions must be identified and standardized for all the interacting modules.

The model building

As stated in the modeling approach, the global model building consists in an initial model that must be completed online based on the update scenario.

Initial construction of the model

The initial model consists essentially of the system level model. Only the components whose structure will not change are represented. The system level model is constructed and all the functions that may be used at the requirements level are defined. As Min_Sys_Req are common to all the missions and independent of missions' profile, they are also represented at the requirements level in the initial model. Figure 5 shows the initial content of the model. The system level sub model is constructed leaving the state of the basic system components and events distributions as parameters to be instantiated in the global model.

The system level sub model provides as output the state of the functions that are needed by the requirements expression. Min_Sys_Req is to be combined with the additional requirements of the mission profile, which will be specified, based on the functions f k=1,… , when considering the operational level.

This initial model, parameterized with the initial states and failure distribution information, corresponds to the model that will be used to evaluate the system reliability measure, which is the probability to meet Min_Sys_Req for a given duration.

Construction of the global model

The completion of the previous model, taking into account the mission profile, forms the global model. Figure 6 shows the elements that must be added to the initial model in order to obtain the global model in the case of a single flight period Fp composed of three phases, preceded by a ground period Gp. One has to specify the expected duration of each phase (Ph1, Ph2, Ph3), and the requirements corresponding to each of them. The expected duration of ground period activities must also be specified, together with M_Prof_DR, which completes Min_Sys_Req in the condition to allow the upcoming flight.

Figure 6 The global model realization

The dynamics of a flight period is that, when a phase is ongoing, its requirements must remain fulfilled up to the end of its duration. The phases take place successively indicating the requirements that must be fulfilled at each time point of the flight profile. A flight period interruption occurs when the requirements of the phase that is ongoing are not met.

CONCLUSION AND FUTURE WORK

The objective of the work is to develop a model that helps improve the operability of an aircraft all along its mission. A modeling approach has been developed distinguishing the main components involved in the global problem. The model is adaptable, based on an update method, in order to represent correctly the situations for which it will be used.

In this paper, we concentrated on the approach to develop the dependability model, especially its formal description, and on how it can be updated online in order to adapt to changes in the current situation of the aircraft. The aim is to re-assess the mission operability as soon as some changes take place. The current description has been made independent of the formalism that must be used to build the model. For a concrete implementation, one has to choose an appropriate formalism. We have considered AltaRica and SANs as formalisms for the implementation. AltaRica was used for the purpose of safety study (see e.g., [START_REF] Bieber | Safety Assessment with AltaRica -Lessons learnt based on two aircraft system studies[END_REF]), to model a number of aircraft subsystems on which the work is based. SAN is an extension of Petri nets. SAN and the associated Möbius tool [START_REF] Daly | Möbius: An extensible tool for performance and dependability modeling[END_REF] provide compositional operators that are convenient to master the complexity of the model.

Figure 1

 1 Figure 1 Dispatch status outcomes

"o

  Specification of a flight: Indication of a predefined flight if the flight profile has been already defined, or o Definition of a new flight profile:

Figure 5

 5 Figure 5 The initial model's content

The corresponding AltaRica and SAN models [6] [8] were developed using a subsystem of an aircraft. The model updates have been achieved manually. Future work will consider concrete implementations of the model dynamic construction, taking as examples the major changes that may happen.