Yves Chaumette
email: chaumette.yves@free.fr

Francis Rousseaux
email: francis.rousseaux@univ-reims.fr

Tests and perception, a vaster scope

Keywords: perception, gesture, pulsation, model, whorl, arrow, negation I

Tests are based on a principle: to get results conform to expectations according to some rules. However the comparison of both processes is delimited by perception. Usually these perceptions of input data and output data are overlooked, yet they do exist. These four operations are modelled by a square reproducing a well-know diagram in the mathematical theory of categories. The subjective side induced by perception is reduced to a minimum in program testing, and objectivity is assumed with an analysis of criteria, functions and partition testing, hence of results of tests. However it can't be ignored in Non Functional Requirements, test expectations -which can themselves be tested -or about testing the understanding of business needs. Phenomenology can be modelled and these models can help to broaden the scope of tests, that's what this paper is aiming at. By dividing a math graph in semi-edges, one gets whorls -i.e. arrows without defined ends -which have a law of composition. Whorls are thus modelling perceptions and gestures, taking into account the gestation and the undefined. Plato already mentioned that a definition is based on undefined elements. Thus whorls broaden the scope of interactions in integrating the movement. In a larger frame, loop arrows describe a return to One self in a rotation -through the Other -a pulsation is describing a return to oneself -though the Same -and spirals or whorls -in a mixed way -are describing relation to oneself in progression.

INTRODUCTION

Tests are the most analytical, detailed, rigorous part of a project, as the goal is to check the realization of what has been planned. Rene Guitart in La Pulsation mathématique [START_REF] René | La pulsation mathématique[END_REF] develops the idea that mathematics are not merely formal demonstration, with precise definitions and strict logics, it is also a building and exploratory activity in broadening the scope and imagining conceptual relationships. Rigour in the foundations balances with opening to possibilities, concentration on reasoning balances with synthetic view. This paper is aiming at applying such a pulsation to tests, showing the usefulness of perception. It will proceeds through a mathematical structure lying behind the principle of test, and will broaden the categorical structure. A new structurewhorls -can be introduced for modelling the idea of phenomenology thinkers describing perception. Negation and identity produce different behaviours for results and perceptions, this observation validates the new structure of whorls and opens the way to a third world: the world of poles, be they questions, instruments, purposes or values. These three worlds are then applied to tests, taking into account the results, the campaigns of tests and their objectives.

II. EXAMPLE: A QUERY

A. Context of the query

We'll take as an example of test and perception a quest for references. Our intent is to find references and we formulate in our mind, we build a query with keywords and we observe the items displayed by a research engine. So we question the matching of our expectations and the logics of the engine. Let us assume we are looking for a reference of the double path principle of testing.

B. Observations

At the words "Tests principles software" Google Scholar gives about 1,3 millions of answers. As the first page displays links with cells, lungs, one chooses to avoid the word biology: there remains only 913 000 answers; if chemistry and medicine are removed, 846 000 answers remains. However, the first page displays already 7 books on testing software. The book of Myers [START_REF] Myers | The art of software testing[END_REF], quoted 2525 times, can be downloaded, the SUDOC mentions it has been first published in 1979, a long way ago. The second edition is dated 2008 on Google Scholar yet the copyright is dated 2004. Some interesting features are found:

• "when you test a program, you want to add value to it" (page 10) • "the intent of finding errors" (page 11). These 2 sentences reminds that testing is a purposeful activity. • Ten principles are listed (page 16), they are clearly empirical. • The first principle (page 16) states that it is necessary to define the expected output, and it goes on in requiring the description the input data and of the expected output data. Though the double path is not mentioned, this paragraph means it.

• The 6 th principle broadens the view of checking: "seeing whether the program does what is it is not supposed to do". • Each principle is then developed and the second principle states "A programmer should avoid attempting to test his or her own program" this alludes to the double path principle (neatly formulated below), and the psychological reason lies in the programmer misunderstanding (page 17) • "Errors tends to come in clusters" (page 19) the reason is not known but it may be a symptom of the quality of this block of program. A second book by Gilb and Finzy appears on this page, The PDF file enunciates principles (30 in number) without development, it can't be used in a theoretical paper for logics is missing. A third book by Perry in 2006 appears on this page, however the bibliometrics gives 0 to the rate of download and 3 for quotation count. And appears to be downloaded the book of Pohl and al [START_REF]RDF Resources Description Frameworks[END_REF], it contains 473 pages and page VI one finds the "need for 2 distinct processes : domain and engineering" we call them expected and operational. The page VI adds variability which is close to laterality, thus coming in the line of this paper. The second web page displays a presentation by Beck (a famous designer) and a copy of 9 pages from Boehm (another famous author) dated from 1991. The following page mentions a book by Mc Graw with 7 pages and a document of 2 pages about design and Testability. And time elapses after 40 minutes.

What did happen? The purpose was at the start, perception all the way, actions were performed and results arrived, some useful some useless. This paper will make a trial approach for modelling this behaviour.

III. THE CATEGORICAL STRUCTURE LYING BEHIND THE

TESTS

Test consists in checking that one gets what has been expected [3 page VI]. This double path is sometimes forgottten when one is using the design of realization for implementing the tests, in this way this design is not checked. The same phenomenon happpens when one is testing the specifications and not going back to the business needs, their capture is assumed to be fine. This double path becomes a closed circuit when it is perceived that the initial state is the same in both paths and the final state too. This double perception at the beginning and at the end is usually overlooked for it vanishes through the analysis of objectives, functionalities, criteria and tests scenarios. So perception is most often a mere observation of equality of records which can be dealt with a comparison program and hence produces only a return code.

When perception is noticed the double path draws a square, which is a well-known diagram in the mathematical theory of category. The mathematical theory of categories invented by Eilenberg and Mac Lane in 1945 [START_REF] Lane | Categories for the working mathematician[END_REF] is the language in which all mathematical findings -algebraic structures and logics of propositions -are nowadays described. A category is a set -or a class -of arrows -with a source and a target -equipped with a partial law of composition. This law of composition is associative. The sources and targets of these arrows are identified with loops which are neutral elements for the compositions of arrows [4 p 289]. Bailly & Longo [START_REF] Bailly | Mathématiques et sciences de la nature[END_REF] noticed that all transformations -in sciences -are seen as a triad: initial state, transition, final state. Thus an arrow with its source and target is modelling any interaction, the basis of systemic paradigm. In the semantic web, for example RDF (Resources Description Frameworks) [START_REF]RDF Resources Description Frameworks[END_REF] uses triples.

IV. THE TREND OF INTENTIONALITY, OPENING, LATERALITY

Perception has long been thought, including Kant [START_REF] Emmanuel | Critique of pure Reason[END_REF] as a relationship between an object and a subject and this is based on the same triad noticed above. However the statute of the object was questioned by Hegel [8 §1, §214] and Schopenhauer [START_REF] Schopenhauer | The world as will and representation, by Payne, Dover Publications 1969 Le monde comme volonté et comme représentation[END_REF] went on. Husserl [START_REF] Husserl | The crisis of european sciences and Transcendental phenomenology, The book Eddy[END_REF] noticed that consciousness is an opening, an expectation; in hanging up the attention, perception becomes a dyad: someone in relation to … ; the object is not yet grasped. This conception has been followed by Barbaras, Patocka [commented in Barbaras 11], Badiou [START_REF] Badiou | La logique des mondes[END_REF]. This last author even used the word laterality, a very geometric word. This remark concerns perception in the field of knowledge, and Gilles Chatelet [resumed in 13] transposed it in the field of action, it is a gesture, a part of gestation. A gesture takes place in a world "in the making", a world where things are happening and are not yet being defined. A gesture is thus a process without defined beginning nor end. One may notice the use of the present participativeand not the past participative -in the language describing a gesture. This pleads for a dyad basis and not a triad. How can it be modelled ? Indeed, a vector is a dyad with a point and an opening but it goes straight in its direction and it lacks laterality: exploration of the surrounding space. The concept of vector relates to an age of certainty, which fits well with Descartes and Newton.

V. WHORL, A DYADIC STRUCTURE

In order to find a dyad, one may come back to the graph underlying the category of arrows. Instead of a triad based on two vertices and an edge, one is looking for a 2-based structure. A vertex with half an edge -like people stretching one's hand -is a poor structure, it would soon fall on the usual graph. Half an edge around a vertex and another half edge won't give much, it will also lead to a graph. To remove both ends of an edge is a worthwhile hypothesis as this gesture of removing dots is used in some areas of mathematics [START_REF] Guitart | La pulsation du ternaire au binaire[END_REF]. A whorl is thus an oriented edge without ends, and whorls obey to a partial law of composition, as do the arrows. This law is also associative, if s 1 ° s 2 and s 2 ° s 3 do exist then (s 1 ° s 2 )° s 3 = s 1 ° (s 2 ° s 3 )

Whorls have interiority and a relational ability. Whorls can project on a category giving a loop. Thus two composed whorls project and give an arrow. Several whorls projecting on the same loop can have a (partial) order. Whorls can have universal solutions, a set of whorls can be factorized through one whorl. This may be used to describe a cone: the perception of an object is common to many perceptions, and it is almost in that way that Kant described the notion of an object: as the limit of a cone. The concept of cone was introduced in categories of arrows, yet it may be applied with whorls, an universal solution is then the invariant or filtering of the many elements composing the cone On a dyadic basis too, an object results of an invariant impression common to many impressions. For example, a tree appears as such as we can have many looks at it and a part of the impression remains similar in these different points of view. Such an impression has no subject and no object, that what means "removing the ends"; the laying down and definition of the object as well as the subject are hanged up.

In mathematics, a topos [4 p 289] is a special category with a final object and a classifier object: this enables to say that some propositions are true or not, and thus to assert judgments. Toposes are useful as most of the operations that can be made with sets can be made in them: this name topos (meaning place in Greek) reminds that these operations are located. Whorls are weakened arrows, sets of whorls are weakened categories, they can't have a final object and thus they can't be classified as true; the language of whorls escapes judgement. Judgement and perception are complementary, noticed Myers and Briggs [START_REF] Briggs-Myers | Manual: a guide to the development and use of the Myers-Briggs Type indicator[END_REF], following Jung [START_REF]Psychological Types[END_REF].

VI. NEGATION, IDENTITY AND 3 WORLDS

As noticed Hegel with the concept of nothingness [8 §40] and Badiou [START_REF] Badiou | La logique des mondes[END_REF], negation operates in one way with actions -or logical propositions-and in another way with perceptions. Perceptions can't be negated, they have a reverse side [12 page 146]: one can perceive there is not any more an object there, however this perception can't be erased. Similarly, the play of a test scenario might show defaults, but this play can't be ignored. A perception is an enquiry within a space of possible results, however this space has to be founded, usually it is a question that arouses the enquiry and brings about answers. It is a purpose or a value that arouses gestures, moves and which brings about activities that compose a project. The foundation of the space of possible results might be called a pole, as a pole of attraction; it has no negation, no inverse, and no reverse. The question 'why' can't be negated, it matters more or less, the same is true for an instrument, it is a tool for measuring a grandeur or a dimension, but it is not a measurement neither a result. A pole can be viewed as a blackhole as the geometry of actions collapse. The three worlds of actions, gestures, and poles can be seen in most of our deeds; for example a paper deals with a topicsits pole -, it follows a logical thread while keeping an overview of the goal -a whorl -and transmits ideas, procedures or results -arrows modelling interactions. Each of this world has a special movement related to identity, the return to Self. One may return to the Self classically in two ways through the Same or through the Other [START_REF] Plato | Plato's complete works[END_REF]. 1) The return to Self through the Other depicts a movement to the outward, a rotation which goes out of the centre, explores the outside and comes back from the periphery. This movement is a loop, an arrow returning to itself, such is the definition of objects within a category.

2) The return to self through the Same depicts a movement inward, never giving up its centre and it depicts a pulsation.

3) A gesture -or inquiry -being the interface between a pole and an action has a composed movement: action gives a rotation while the pulsation produced by a pole expresses through radii i.e. movements forward. A whorl is not stable and does not return to itself, yet several whorls may produce a ring which is stable. Thus a gesture goes back to the Same through a spiralling movement, exploring the surroundings and coming back in a ring composed of many whorls. Spirals are almost ignored in the western theory (save in the Waltzes of Vienna) while they are promoted in the eastern conception as movement of Vishnu [START_REF] Daniélou | Mythes et dieux de l'Inde[END_REF], in Chinese martial arts and dervish.

VII. THE 3 WORLDS IN THE TESTS

Three worlds appear in front of negation, what do they give to the testing process? Results of a test can be good, bad or fuzzy. These results come through campaigns that are composed of plays of scenarios, these plays can't be negated, they are enquiries in the software, finally an objective of tests is creating the space for these campaigns. An objective is thus a type with several instances of play which can take value on the set of possible results. The various objectives are themselves ordered according to a tree, as tests are aroused by the intent of checking the developed code, the fundamental intent being to "assess the quality of computer software" [Wikipedia quoted in 19] and hence detecting the failures. But this order of the various objectives of testing escapes the scope of this paper.

The pulsation described by Guitart is obvious in testing. In the most detailed part, tests concern a program and the results may be good or bad, when tests extend to a package the results may be fuzzy. When integration grows, the results concern the attainement of SLA and perception matters more. When the functioning of a whole system is considered, what matters is the objective and the attention turn to the poles of testing campaigns. Thus the pulsation applies both to results and to the scale of the three worlds.

VIII. THE DOUBLE PATH PRINCIPLE REVISITED

The diagram of the double path may be reviewed with the three worlds, the movement forward in time between the as-is state and the to-be state is expressing the pulsation of the purpose. This purpose arouses preparation of the test campaigns and observation of its results. A whorl projects on an arrow; preparation of tests induces planning, study, creation of test games, and observation produced the action of comparison of expected and obtained results. The pulsation in the three worlds applies also to the testing life-cycle. The pulse expands about the expectation, starting with the overview of the system purpose, going to testing objectives, campaigns, functions, scenarios, and test cases; this marks the most detailed part. All this study of preparing tests is an enquiry going from a pole to many planned interactions. Then the pulse contracts back, starting from results, integrating packages, non-regression tests, and ending in comparing the system results with objectives, this comparison refers to the purpose of the system. This can be modelled in a life cycle shown in figure 8. In the diagram all actions are in italics, the objects resulting from an action are embedded in rectangles. Most of gestures and perceptions are producing actions, such as "building of the query", however they are shown as gestures. Ideas that do not correspond to the initial quest are all gathered under the whorl as 'Other intent', they include some sentences by Myers [2 page 16 , 6 th Principle], design and testatibility, the documents from Beck and Boehm.

The initial intent has produced results: references were found, thus the expectation of finding references and the approach by the selected key words have been found worthwhile; the understanding of the algorithm of a research engine has been confirmed, in spite of the 134 000 answers. However, the notion of principle has shown to be a guide for action thus weakening its scope. With this refinement and the new perspectives the intent has slightly moved. This move evidenced in this example can be observed in our usual quests.

X. THREATS TO VALIDITY

A human trend is to set limits in order to define. Any budget requires a beginning and an end. To consider something without ends is strange and it might be a reason to reject whorls and gestures. Obviously it is possible to consider only results, yet as noticed Hegel "the naked result is the corpse which has let the trend behind itself" [20 page 45]. Gestation enriches the process and can describe creativeness. However, whorls can be like enzymes who catalyze a reaction yet do not appear in the result. That's the way Guitart [START_REF] René | Pour une théorie mathématique du geste structurant en mathématique après Bourbaki[END_REF] describes the use of some mathematical structures.

The natural trend in front of novelty is to come back to a well-known domain. If any perception gives way to an action, i.e. any whorl projects onto an arrow, the conception of whorls would be useless, though negation operates differently; in that case usual ontology would be convenient. However if the rate of conversion is one for 10 or even of 1 for one million -one object standing for one million perceptions -indeed the world of perceptions would have some uses and not only a formal status.

XI. PERSPECTIVES

This 3-world frame can be applied to describe priorities of an application in the business line, it can be used for describing movement in real-time computing when results arrived well after the process and the movement can't be cut in slices. Parallel computing might benefit of this model as the process matters more than the transitory results. Finally, modelling the intent and the gestures might be useful for developing an application, if one can see the project in an intentional view.

Figure 1 .

 1 Figure 1. The square diagram of test

Figure 2 .

 2 Figure 2. Example of category with 3 elements

Figure 3 .

 3 Figure 3. Projection of whorls onto arrows

  [4 p 55, 71].

Figure 4 .

 4 Figure 4. Cones of arrows and whorls

Figure 5 .

 5 Figure 5. 3 results for a test

Figure 6 .

 6 Figure 6. 3 levels in tests

Figure 7 .

 7 Figure 7. The square diagram of test revisited This diagram might be extended to the relationship of the business line and the development: business line is expecting some uses of the application which are fulfilled or not by the deployed software.

Figure 8 .

 8 Figure 8. General life cycle of testing Let us come back to our example and apply this modelling to it.

Figure 9 .

 9 Figure 9. Model of the example query


Charles Deledalle

Samuel Vaiter

Gabriel Peyré

Jalal M Fadili

Charles H Dossal

Gabriel Peyré Ceremade

Unbiased Risk Estimation for Sparse Analysis Regularization

Keywords: Sparsity, analysis regularization, inverse problems, risk estimator, GSURE

published or not. The documents may come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Problem statement. Many problems in image processing can be cast as recovering an image f 0 ∈ R N from the forward observation model

y = Φ 0 f 0 + w, (1)
where it is assumed throughout that w ∼ N (0, σ 2 Id P ), and Φ 0 : R N → R P is a bounded linear operator which is typically ill-behaved since it models an acquisition process that entails loss of information so that P N . Typical cases covered by the above degradation model are entry-wise masking (inpainting), convolution (acquisition blur), Radon transform (tomography) or a random sensing matrix (compressed sensing). Linear inverse problems are among the most active fields in signal and image processing. In order to regularize them and reduce the space of candidate solutions, one has to incorporate some prior knowledge on the typical structure of the This work has been supported by the European Research Council (ERC project SIGMA-Vision) and the French National Research Agency (project NatImages).

original signal or image f 0 . This prior information accounts for the smoothness of the solution and can range from the uniform smoothness assumption to more complex geometrical priors. To capture the complexity of image structures, we have witnessed a great deal of investigation where researchers spanning a wide range of viewpoints have advocated the use of sparsity in inverse problems.

Sparse regularization.

Embarking from the degradation model ( 1), a popular approach to find sparse solutions to linear inverse problems, is to solve the following convex but non-smooth problem

x ⋆ ∈ argmin x∈R Q 1 2 ||y -Φx|| 2 + λ||D * x|| 1 , (P(y))
where Φ = Φ 0 Ψ, Ψ : R Q → R N is a redundant synthesis dictionary, that maps coefficients x to images f = Ψx, D * is an analysis linear operator and λ > 0 is the regularization parameter. This formulation encompasses several well-known problems in sparse recovery:

• Synthesis-type regularization, e.g. [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF][START_REF] Tibshirani | Regression shrinkage and selection via the Lasso[END_REF]: in which case D * = Id Q , and we are seeking a sparse set of coefficients x ⋆ and the solution signal or image is synthesized from these representation coefficients f ⋆ = Ψx ⋆ .

• Analysis-type regularization, e.g. [START_REF] Elad | Analysis versus synthesis in signal priors[END_REF][START_REF] Vaiter | Robust sparse analysis regularization[END_REF]: which seeks a signal or image f whose coefficients D * f are sparse, with Ψ = Id N . Some problems are inherently of analysis-type such as the total variation regularization where D * = ∇ : R N → R 2N computes a discretized gradient of the image (a 2-D vector field) [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF]. Another example of analysis regularization is the Fused Lasso [START_REF] Tibshirani | Sparsity and smoothness via the fused Lasso[END_REF], where D is the concatenation of a 1-D discrete derivative and the (weighted) identity.

(P(y)) is versatile enough to allow hybridization of analysis and synthesis regularizations, e.g. to structure the synthesis coefficients in Ψ in a way that their gradient is sparse.

Solving (P(y)). Although it is convex, solving (P(y)) is rather challenging given its non-smoothness. Moreover, the ℓ 1 -norm is composed with D * and therefore the proximity operator is not computable in closed-form for general D. This precludes the use of popular iterative soft-thresholding (actually the forward-backward proximal splitting) without sub-iterating. We therefore appeal to a more elaborate primaldual splitting algorithm originating from the non-smooth optimization realm. For instance, we here use the relaxed Arrow-Hurwicz algorithm as revitalized in [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF]. This algorithm achieves full splitting where all operators are applied separately: proximity operators of 1 2 || • || 2 and λ|| • || 1 , Φ, D and their adjoints. See [START_REF] Vaiter | Robust sparse analysis regularization[END_REF] for more details.

Previous work

Analysis vs synthesis regularization. It is known that in general, analysis and synthesis prior formulations can differ significantly, unless Ψ = Id N and D is orthogonal or vice versa. Some insights can be found in [START_REF] Elad | Analysis versus synthesis in signal priors[END_REF] on the relation and distinction between analysis and synthesis-based ℓ 1 -sparsity regularizations. Moreover, there has been a recent wave of interest in the theoretical guarantees of analysis sparse recovery, see e.g. [START_REF] Vaiter | Robust sparse analysis regularization[END_REF] and references therein.

Degrees of freedom (DOF). Degrees of freedom (DOF) is a familiar phrase in statistics. DOF is often used to quantify the complexity of a statistical estimation procedure. However, there is no exact correspondence between the DOF and the number of parameters in the model. The SURE theory [START_REF] Stein | Estimation of the mean of a multivariate normal distribution[END_REF] gives a rigorous definition of DOF for any estimation procedure including non-linear ones.

For synthesis ℓ 1 regularization, an unbiased estimate of the DOF was established in the overdetermined case in [START_REF] Zou | On the "degrees of freedom" of the lasso[END_REF], and extended to the general setting in [START_REF] Kachour | The degrees of freedom of penalized l1 minimization[END_REF]. An unbiased estimate of the DOF for analysis ℓ 1 regularization is given in [START_REF] Vaiter | Robust sparse analysis regularization[END_REF], see also [START_REF] Tibshirani | The solution path of the generalized Lasso[END_REF].

Risk estimation. The DOF plays an important role in model validation and selection, and its unbiased estimates provide unbiased estimates of the risk on the prediction Φ 0 f ⋆ through the SURE [START_REF] Stein | Estimation of the mean of a multivariate normal distribution[END_REF]. These unbiased estimates of the risk can serve as a basis for automatic ways to choose the parameters of the reconstruction algorithm, e.g. λ in (P(y)).

For instance, the SURE has been extensively used in sparse denoising and deconvolution, e.g. [START_REF] Luisier | The SURE-LET approach to image denoising[END_REF][START_REF] Dupé | A proximal iteration for deconvolving Poisson noisy images using sparse representations[END_REF]. Recently, a generalization of the SURE (GSURE) [START_REF] Eldar | Generalized SURE for exponential families: Applications to regularization[END_REF] has been developed for noise models within the multivariate canonical exponential family. This allows to estimate the risk on a projected version of f ⋆ . Indeed, in the scenario where Φ 0 is rank-deficient or redundant, the GSURE can at best estimate the risk in ker(Φ 0 ) ⊥ = Im(Φ * 0 ). The projected GSURE has been applied to (non-iterative) wavelet-vaguelet deconvolution [START_REF] Pesquet | A SURE approach for digital signal/image deconvolution problems[END_REF], and to sparse synthesis regularization [START_REF] Vonesch | Recursive risk estimation for non-linear image deconvolution with a wavelet-domain sparsity constraint[END_REF][START_REF] Giryes | The projected GSURE for automatic parameter tuning in iterative shrinkage methods[END_REF] with the iterative soft-thresholding algorithm.

Contributions

The main contribution of this paper is a rigorous derivation of the exact expression of the projected GSURE associated to (P(y)). To this end, we capitalize on our recent work on local affine parameterization of solutions to (P(y)) [START_REF] Vaiter | Robust sparse analysis regularization[END_REF]. Unlike previously proposed projected GSURE computations in [START_REF] Vonesch | Recursive risk estimation for non-linear image deconvolution with a wavelet-domain sparsity constraint[END_REF][START_REF] Giryes | The projected GSURE for automatic parameter tuning in iterative shrinkage methods[END_REF], we have a closed-form expression that can be implemented efficiently once the solution of the inverse problem is computed. Moreover, our framework elegantly covers in a unified way synthesis and analysis sparse regularizations.

SPARSE ANALYSIS SENSITIVITY

Preliminaries. Analysis regularization enforces that sparse solutions of (P(y)) belong to some co-space G J = ker(D * J ) where J is the co-support of a solution x ⋆ J = I c where I = supp(D * x ⋆ ).

To be able to study the local sensitivity of the solution w.r.t. y, it is important that the forward operator Φ = Φ 0 Ψ is wellbehaved over the sparsity enforcing space G J . This is formalized through the assumption

Ker Φ ∩ G J = {0}. (H J )
When (H J ) holds we define the following mapping

A [J] = U (U * Φ * ΦU ) -1 U * .
where U is a matrix whose columns form a basis of G J .

Local sensitivity. For λ > 0, there exists a finite union of hyperplanes H λ ⊂ R Q , such that the following theorem holds.

Theorem 1. Let y ∈ H λ and let x ⋆ be a solution of (P(y)).

Let I = Supp(D * x ⋆ ) and s = sign(D * x ⋆ ). Suppose that (H J ) holds. Define

∀ȳ ∈ R Q , x(ȳ) = A [J] Φ * ȳ -λA [J] D I s I .
There exists an open neighborhood B of y such that for every ȳ ∈ B, x(ȳ) is a solution of (P(ȳ)).

The exact definition of H λ is quite technical, and for obvious space limitation, we refer to [START_REF] Vaiter | Robust sparse analysis regularization[END_REF] for details and a proof.

SURE RISK ESTIMATOR

Projected GSURE. In the following, we denote Π the orthogonal projector on ker(Φ 0 ) ⊥ = Im(Φ * 0 ), where it is assumed that ker(Φ 0 ) = {0}. This projector is Π = Φ * 0 (Φ 0 Φ * 0 ) + Φ 0 , where + stands for the Moore-Penrose pseudo-inverse.

We first notice that by strict convexity of the quadratic part, even if (P(y)) admits several solutions x ⋆ (that are obviously function of y), all f ⋆ = Ψx ⋆ share the same image under Π. We can thus define unambiguously µ(y) = Πf ⋆ as a (single-valued) mapping of the observation y.

Denoting µ 0 (y) = Φ * 0 (Φ 0 Φ * 0 ) + y the maximum likelihood estimator, the projected GSURE is defined following [START_REF] Pesquet | A SURE approach for digital signal/image deconvolution problems[END_REF] as:

GSURE(y) =||µ 0 (y) -µ(y)|| 2 -σ 2 tr((Φ * 0 Φ 0 ) -1 ) + 2σ 2 div Φ * 0 y (µ(y)),
where div Φ * 0 y (µ(y)) is the divergence of µ(•) w.r.t. Φ * 0 y. Note that this estimator is introduced in [START_REF] Pesquet | A SURE approach for digital signal/image deconvolution problems[END_REF] in the context of wavelet-vaguelet deconvolution, but it is easily seen to hold for arbitrary inverse problems and regularizations as long as the mapping µ is weakly differentiable in the sense of [START_REF] Stein | Estimation of the mean of a multivariate normal distribution[END_REF]. Note also that up to a constant, which does not depends on λ, our definition of GSURE is equivalent to the one introduced in [START_REF] Eldar | Generalized SURE for exponential families: Applications to regularization[END_REF][START_REF] Vonesch | Recursive risk estimation for non-linear image deconvolution with a wavelet-domain sparsity constraint[END_REF].

Theorem 1 immediately implies our main result, that allows to compute the GSURE.

Theorem 2. Except on a subset of R N of measure zero, the mapping y → µ(y) is C ∞ , and

div Φ * 0 y (µ(y)) = tr ΠΨA [J] Ψ * .
A direct consequence of this result is that the above expression of the GSURE is an unbiased estimate of the risk on Im(Φ * ), i.e.

E w (||Πf 0 -µ(y)|| 2 ) = E w (GSURE(y)) .
Numerical computation of the GSURE. The following proposition gives a way to compute efficiently the divergence term which boils down to solving a linear system.

Proposition 1. One has tr ΠΨA [J] Ψ * = E Z ( Ψν(Z), µ 0 (Z) ) (2)
where Z ∼ N (0, Id P ), and where for any z ∈ R P , ν = ν(z) solves the following linear system

Φ * Φ D J D * J 0 ν ν = Φ * z 0 . (3)
Proof. We have

tr ΠΨA [J] Ψ * = tr ΦA [J] Ψ * Φ * 0 (Φ 0 Φ * 0 ) + .
Hence denoting ν(z) = A [J] Φ * z, and using the fact that for any matrix U , tr(U ) = E Z ( Z, U Z ), we arrive at (2).

We then use the fact that A [J] Φ * , the inverse of Φ on G J , is the mapping that solves the following linearly constrained least-squares problem

A [J] Φ * z = argmin h∈G J ||Φh -z|| 2 .
The closed-form solution to this problem is given by (3).

In practice, the empirical mean estimator is replaced for the expectation in (2), hence giving

1 k k i=1 Ψν(z i ), µ 0 (z i ) WLLN -→ tr ΠΨA [J] Ψ * ,
for k realizations z i of Z. The numerical computation of ν(z i ) is achieved by solving the symmetric linear system (3) with a fast conjugate gradient solver.

NUMERICAL ILLUSTRATIONS

In this section, we exemplify the usefulness of our GSURE estimator which can serve as a basis for automatically tuning the value of λ. This is achieved by computing, from a single realization of the noise w, the parameter that minimizes the value of GSURE(y) for y = Φf 0 + w.

Sparse Synthesis Regularization In this example, Φ 0 is the (circulant) convolution operator associated to a Gaussian kernel of width 2 pixels, and the input PSNR is set to 27.78 dB. The regularization is of synthesis-type (D * = Id Q ) in an orthogonal wavelet dictionary Ψ. Fig. 1(a) depicts the projected risk and its GSURE estimate with k = 5 as a function of λ. The curves are indeed unimodal and coincide even with k = 5 and a single noise realization. Consequently, GSURE provides a high-quality estimate of λ minimizing the risk. A close in on the central area of the degraded and deconvolved (using the optimal λ) images is shown in Fig. 1(b)-(c) for visual inspection of the restoration quality.

Total Variation Regularization

We consider a compressed sensing setting where Φ 0 is a random partial DCT measurement matrix with an under-sampling ratio P/N = 0.5, and the PSNR is set to 27.50 dB. The regularization is of analysistype, anisotropic total variation regularization (D * = ∇, Ψ = Id N ). The GSURE is estimated with k = 1. The results observed on the deconvolution example are confirmed in this compressed sensing experiment both visually and qualitatively, see Fig. 2.

CONCLUSION

We proposed a grounded and computationally efficient framework to unbiasedly estimate the projected risk in ℓ 1regularized inverse problems handling both synthesis and analysis sparsity priors. Its usefulness has been illustrated on

Fig. 1 .Fig. 2 .

 12 Fig. 1. Deconvolution problem (with a Gaussian PSF) with wavelet synthesis regularization, and σ = 10. (a) Projected risk and its GSURE estimate 1 using k = 5 random realizations. (b) y. (c) µ(y) at the optimal λ.automatic choice of the regularization parameter for several inverse problem instances. The extension of this approach to other sparsity-promoting penalties, e.g. mixed norms for block structured sparsity, is currently under investigation.1

Without impacting the optimal choice of λ, the two curves have been vertically shifted to the same range of values for visualization purposes.


E Bringuier
email: eric.bringuier@upmc.fr

Connection between thermophoresis and thermodiffusion in a liquid binary mixture

Keywords: statistical physics, liquids, Soret effect user supplied): thermophoresis, binary mixture, suspension thermophoresis, suspension, thermodiffusion, binary mixture, Galilean invariance PACS: 66.10.C-, 47.57.E-, 05.70.Ln 1

Connection between thermophoresis and thermodiffusion in a liquid binary mixture

2 also called thermal diffusion [START_REF] Chapman | A note on thermal diffusion[END_REF][START_REF] Mason | Thermal diffusion in gases[END_REF][START_REF] Wiegand | Thermal diffusion in liquid mixtures and polymer solutions[END_REF], of the two components; it is a partial demixtion induced by the temperature gradient, which is opposed by the remixing effect of ordinary diffusion. In the viewpoint of thermophoresis, the particles move through a quiescent medium unperturbed by the guest particles. In the viewpoint of thermodiffusion, the molecules of the suspending medium are treated on the same footing as the suspended particles.

It appears that the connection between thermophoresis and thermodiffusion has been much less investigated than the one between the isothermal phenomena, and is subject to confusions due to the disparate concepts involved in the two areas [START_REF] Bringuier | On the notion of thermophoretic velocity[END_REF]. The purpose of the present paper is to dispel those confusions. We shall stay at the phenomenological level throughout, with no attempt at investigating the underlying transport mechanism(s) or making any assumption thereabout. The paper is structured as follows. Section 2 expounds the statistical tools needed to describe the transport of particles in a medium which is not necessarily homogeneous. Section 3 reviews two experiments measuring the so-called thermophoretic velocity acquired by a suspended particle under the applied temperature gradient. Special attention is paid to the conditions under which that velocity is measured as they are an integral part of the notion of thermophoretic velocity. Next, section 4 applies the statistical tools of section 2 to the measurements reviewed in section 3. The observables used in the thermophoresis and thermodiffusion viewpoints are then related to each other. It appears that the often unspoken specification of a Galilean reference frame is crucial in establishing the link between both viewpoints. Section 5 gathers the conclusions.

Constructing a Galilean-invariant equation of matter transport

This section describes the statistical framework needed to handle particle transport, which is often not deterministic in contrast to particle motion in mechanics. This comes about because the particle interacts with the infinitely many thermally fluctuating degrees of freedom of the medium. The latter will be allowed to be inhomogeneous; that feature is mandatory if temperature T varies with position, since many relevant physico-chemical parameters (such as the mass density or the viscosity of the medium) are functions of T.

The laboratory frame where transport is observed is taken to be Galilean. Considering only one coordinate x for simplicity and denoting time by t, the simplest description of the non-deterministic motion x(t) of a particle suspended in the medium involves the average position 〈x〉 and the variance 〈(x -〈x〉) 2 〉. The notation 〈 〉 means the averaging over a statistical ensemble of particles in which each particle has a distinct trajectory x(t).

Consider first a homogeneous medium. Starting from an ensemble of N particles sharply localized at position x 0 at zero time, i.e. a particle number density n(x, t=0) = N δ(x - x 0 ) where δ is the Dirac distribution, it is expected that, at time t > 0, n(x, t) be a Gaussian distribution,

n(x, t) = N [2π〈(x -〈x〉) 2 〉] 3/2 exp ( -(x -〈x〉) 2 2〈(x -〈x〉) 2 〉 ) , (1)
where the mean and variance are linear in time:

〈x〉 = x 0 + v d t, (2)
〈(x -〈x〉) 2 〉 = 2Dt. ( 3
)
The reasons for (1)-( 3) are threefold [START_REF] Beckmann | Elements of Applied Probability Theory[END_REF][START_REF] Mathews | Mathematical Methods of Physics[END_REF]: (i) because of homogeneity, the rates (d〈x〉/dt) and (d〈(x -〈x〉) 2 〉/dt) are independent of 2) is called the drift (or migration) velocity and D in ( 3) is called the diffusivity;

x 0 ; v d in (
(ii) it is also assumed that the process is stationary, i.e. the rates are independent of the origin of time;

(iii) because the random increments of the particle's position are independent but obey the same probability law owing to (i) and (ii), the central limit theorem of probabilities entails a Gaussian distribution of x after a sufficiently long time t.

We notice that (1)-( 3) are equivalent to ∂n ∂t

+ ∂j ∂x = 0, (4) j
(x, t) = v d n - ∂(Dn) ∂x . (5)
Equation ( 4) expresses the local conservation of particles and j is the conserving particlecurrent density.

In actuality, statement (iii) and expression [START_REF] Bringuier | From mechanical motion to Brownian motion, thermodynamics and particle transport theory[END_REF] do not hold over a short time t and n(x, t) is not Gaussian. This is well known in neutron transport when the scattering events are infrequent, i.e. when the neutron's mean free path approaches the container's size [START_REF] Weinberg | The Physical Theory of Neutron Chain Reactors[END_REF]. In electron transport in a solid medium, scattering events are so frequent that the deviation from the Gaussian shape can only be 'observed' in a numerical simulation of transport performed over a very short duration or very small length. It is then found that expression (5) of the current density has to be supplemented with paradiffusion terms such as ∂ 2 (Pn)/∂x 2 and higher-order spatial derivatives [START_REF] Bringuier | The current equation in strong electric fields[END_REF], skewing n(x, t) with respect to the Gaussian shape. This provides the most complete description of the non-deterministic motion, which includes moments or cumulants of x(t) of all orders [START_REF] Ryskin | Simple procedure for correcting equations of evolution: Application to Markov processes[END_REF]. However, if only one paradiffusion term is used in j, namely ∂ 2 (Pn)/∂x 2 , then negative values of n(x, t) are found at some positions [START_REF] Bringuier | The current equation in strong electric fields[END_REF].

Marcinkiewicz [START_REF] Marcinkiewicz | Sur une propriété de la loi de Gauß[END_REF] has shown that it is necessary to include all high-order derivatives in order that positivity of n(x, t) be preserved. That is the reason why, although the inclusion of one paradiffusion term improves the description of transport over short durations [START_REF] Bringuier | The current equation in strong electric fields[END_REF], in practice the expansion of j is truncated after the first two terms, corresponding to the central limit theorem. We follow that practice in the following.

D(x 0 ) = 1 2 ( d〈(x -〈x〉) 2 〉 dt ), (6)
depend on the initial position x 0 of the sharp group of particles. Local particle conservation (4) is ensured with the current density j given again by expression [START_REF] Bringuier | From mechanical motion to Brownian motion, thermodynamics and particle transport theory[END_REF] provided that v d and D be considered as functions of position. Those functions should be smooth, i.e. vary little over the mean free path, defined as the length needed for decorrelation of the instantaneous velocity due to the random changes of that velocity in scattering events. We stress that, for the coefficient of n in (5) to mean the local drift velocity of a sharply localized ensemble of particles (the velocity of their centroid according to ( 6)), the other contributions to j should be total spatial derivatives.

If j = v d (x) n + (∂f/∂x) with f
vanishing on the container's borders, then an integration by parts of ( 4)-( 5) (the Ostrogradsky-Gauss theorem in three dimensions) proves that (d〈x〉/dt) = v d (x 0 ) for n(x, t=0) = N δ(xx 0 ) [START_REF] Bringuier | On the notion of thermophoretic velocity[END_REF][START_REF] Van Kampen | Stochastic Processes in Physics and Chemistry[END_REF][START_REF] Bringuier | Kinetic theory of inhomogeneous diffusion[END_REF]. This has been checked experimentally on Brownian particles in an inhomogeneous (albeit isothermal) medium [START_REF] Lançon | Drift without flux: Brownian walker with a space-dependent diffusion coefficient[END_REF][START_REF] Blawzdziewicz | Comment on 'Drift without flux: Brownian walker with a space-dependent diffusion coefficient[END_REF].

If in addition f = 0, it is easily verified that n(x, t=0) = N δ(x -x 0 ) evolves into n(x, t) = N δ(x -X(t))
where X(t) satisfies the following differential equation:

dX dt = v d (X(t)), X(t=0) = x 0 . (8)
Then, position evolves as a deterministic function X(t) of time. The contributions to j other than v d (x) n, namely (∂f/∂x) with f = -Dn + ∂(Pn)/∂x + ..., are the diffusion and paradiffusion current densities [START_REF] Bringuier | The current equation in strong electric fields[END_REF]. Once paradiffusion is dropped, we are left with a non-homogeneous drift-diffusion equation, often labelled 'Fokker-Planck equation' [START_REF] Van Kampen | Stochastic Processes in Physics and Chemistry[END_REF]. It is a continuous-time self-repeating application of the central limit theorem of probabilities making allowance for a smooth change of v d and D with x and possibly with t [16]1 .

Whether in the homogeneous or inhomogeneous case, the particle transport equation ( 4)-( 5) exhibits manifest Galilean invariance. We can see it by observing transport in a different Galilean frame, such that x → x -Vt, with V denoting the translation velocity of the new frame with respect to the first one. Then, from ( 6) and ( 7), v d → v d -V transforms as a vector while D → D is a Galilean scalar2 . If the current density (5) be written as

F o r P e e r R e v i e w O n l y j(x, t) = (v d -∂D ∂x ) n -D( ∂n ∂x ), (9)
then the coefficient of n in ( 9) cannot be interpreted as the local rate of change of 〈x〉 in any frame.

Observation of a thermophoretic velocity

The first observation of the thermophoresis of particles in a liquid was reported by McNab and Meisen [START_REF] Mcnab | Thermophoresis in liquids[END_REF]. They were studying spherical polystyrene latex particles, of diameter ranging between 0.8 and 1.0 µm, dilute in an aqueous medium. The liquid suspension was subjected to a vertical gradient of temperature. The vertical drift, or migration, velocity was computed from the transit time, i.e. the time needed by a particle to move over a given distance in the medium. The actions of gravity and of the temperature gradient were assumed to be additive.

As the particle Reynolds number never exceeded 0.1, the terminal gravitational velocity (corrected for buoyancy) was obtained from the Stokes law and it was subtracted from the observed velocity in order to isolate the effect of the temperature gradient alone.

In the more recent experiment of Regazzetti and co-workers [START_REF] Regazzetti | Experimental evidence of thermophoresis of non-Brownian particles in pure liquids and estimation of their thermophoretic velocity[END_REF], the temperature gradient is horizontal, so that no gravitational/buoyant correction to the horizontal velocity is needed. The time t required for the drift of the particle across the horizontal thickness w of the channel is measured in three experiments, and 〈t〉 denotes the mean value. The thermophoretic velocity is operationally defined3 by equation ( 1) of [START_REF] Regazzetti | Experimental evidence of thermophoresis of non-Brownian particles in pure liquids and estimation of their thermophoretic velocity[END_REF] as w/〈t〉, i.e.

v d = ( dx d〈t〉 ), (10)
instead of v d = (d〈x〉/dt) of section 2. Both definitions of v d are consistent with each other insofar as the uncertainty [〈(t -〈t〉) 2 〉] 1/2 is much less than the mean 〈t〉. Since xx 0 ≈ v d t to zero order in the uncertainties, the condition [〈(t -〈t〉

) 2 〉] 1/2 << 〈t〉 amounts to [〈(x -〈x〉) 2 〉] 1/2 << v d 〈t〉, i.e. 2D << v d 2 〈t〉 ≈ v d (x -x 0 ). ( 11
)
The left-hand side is computed by means of the Stokes-Einstein formula to be about 10 -12 m 2 /s while the right-hand side is about 10 -9 m 2 /s [START_REF] Mcnab | Thermophoresis in liquids[END_REF].

In [START_REF] Mcnab | Thermophoresis in liquids[END_REF], the liquid is enclosed in a rigid container. Since the volume is constant, the pressure is not controlled by the experimentalist. It is determined by the system itself through the relation of state of the suspension, as discussed in the appendix. Although the value of pressure is not known, we show in section 4.1 that its horizontal gradient vanishes very quickly while its vertical gradient offsets gravity. In [START_REF] Regazzetti | Experimental evidence of thermophoresis of non-Brownian particles in pure liquids and estimation of their thermophoretic velocity[END_REF], the control of the flow rate demands a vertical pressure gradient associated with the viscous resistance of the channel; that gradient adds up to the gravitational pressure gradient. As for the horizontal pressure gradient, it is shown in section 4.1 to vanish very quickly. Therefore the thermophoretic velocity is either measured in the absence of a pressure gradient [START_REF] Regazzetti | Experimental evidence of thermophoresis of non-Brownian particles in pure liquids and estimation of their thermophoretic velocity[END_REF] or corrected for the effect of that gradient [START_REF] Mcnab | Thermophoresis in liquids[END_REF].

In both experiments, the temperature gradient over the particle's path xx 0 is established after a typical time τ = (xx 0 ) 2 /∆ where ∆ is the thermal diffusivity of the suspension. As the time needed to cross (x -

x 0 ) is 〈t〉 = (x -x 0 )/v d , we have τ/〈t〉 = v d (x -x 0 )/∆. ( 12
)
In water under standard conditions, ∆ ≈ 10 -7 m 2 /s. The typical measured thermophoretic velocities [START_REF] Mcnab | Thermophoresis in liquids[END_REF] were about 5x10 -6 m/s and (xx 0 ) ≈ 2x10 -4 m, so that τ/〈t〉 ≈ 10 -2 << 1. Therefore, the temperature gradient, just as the pressure gradient, may be regarded as static during thermophoresis.

Connection with thermodiffusion

Unary versus binary transport

From the previous section, the phenomenology of thermophoresis calls for the transport description of section 2 in which diffusion is neglected. The evolution of the liquid medium where particles are suspended is not considered. Basically, thermophoresis is regarded as a phenomenon of unary transport, i.e. there is one mobile species B in a quiescent medium of A molecules. The transport of B obeys a local equation of conservation (in three dimensions)

∂n B ∂t + div j B = 0, (13)
where n B is the number density of B and the vector field j B (r, t) is the conserving particlecurrent density of B at position r. It is n B u B (r, t), where the transport velocity u B is the average velocity of the ensemble of particles contained in a macroscopically small, but microscopically large, volume element located about r. From section 2, a first approximation to j B is v d n B and a second approximation, including the spreading about the centroid, is

v d n B -∇ ∇ ∇ ∇(Dn B )
, where ∇ ∇ ∇ ∇ denotes the gradient operator.

In the viewpoint of thermodiffusion [START_REF] Chapman | A note on thermal diffusion[END_REF][START_REF] Mason | Thermal diffusion in gases[END_REF][START_REF] Wiegand | Thermal diffusion in liquid mixtures and polymer solutions[END_REF], one is considering a binary mixture of the species A and B where they play symmetrical roles. Just as B, A is locally conserved, so that (13) is supplemented with where n A is the number density of A, j A = n A u A is the conserving current density of A and u A is the transport velocity of A. The evolution of the mixture composition, defined as the particle fraction x B = n B /n where n = n A + n B is the total particle-number density, is obtained [START_REF] Bringuier | Anatomy of particle diffusion[END_REF] by combining the equations of continuity ( 13) and ( 14):

∂n A ∂t + div j A = 0, (14)
n( Dx B Dt ) + div J B = 0, (15)
where 16) is the particle interdiffusion current density and x A = n A /n. In [START_REF] Bringuier | The current equation in strong electric fields[END_REF], the advective derivative

J B = x A j B -x B j A (
D/Dt is ∂/∂t + u . ∇ ∇ ∇ ∇ where u = x A u A + x B u B is
J B = nx A x B (u B -u A ), (17)
so that mixing stops when A and B are transported at the same velocity.

It is equally possible to describe the evolution by considering the mass fraction and current densities instead of the particle fraction x B and current densities j B and J B , and we shall do so in the appendix. The mass and particle fractions are homographic functions of each other, involving the ratio of masses m A of A and m B of B, and the mass and particle densities are related likewise.

The main concern of this paper is the unary limit of guest particles B being very dilute in a host medium of molecules A. Then, one intuitively expects the medium to be hardly perturbed by the transport of a few guest particles B. In [START_REF] Weinberg | The Physical Theory of Neutron Chain Reactors[END_REF], n A is expected to be independent of time and u A to vanish in the laboratory frame. But is that actually possible? In equilibrium n A is a function of the local thermodynamic state (p, T, x B ) of the mixture. The equilibrium relation n A = n A (p, T, x B ) retains its validity when the change in x B is slow; that is one usual assumption, or rather approximation, of non-equilibrium thermodynamics, which is borne out in experiments and in numerical studies of the molecular dynamics [START_REF] Kjelstrup | Criteria for validity of thermodynamic equations from non-equilibrium molecular dynamics simulations[END_REF]. Since p and T do not vary with time, we have

∂n A ∂t = ( ∂n A ∂x B ) p, T ( ∂x B ∂t ). ( 18
)
The upshot is that interdiffusion, i.e. (∂x B /∂t) ≠ 0, prevents us from taking div j A = 0 in [START_REF] Weinberg | The Physical Theory of Neutron Chain Reactors[END_REF] and thereby from considering that u A = 0. Therefore the medium of molecules A may not be regarded as being at rest. More precisely, relation [START_REF] Van Kampen | Stochastic Processes in Physics and Chemistry[END_REF] tells us that the medium of molecules A is locally expanded or contracted because of its change in composition x B . The medium may not be taken as a reference frame (in the sense of mechanics) because it is a non-rigid fluid instead of a solid. The condition of rigidity is div u A = 0 while the actual velocity field is such 4)-( 5) pertinent in thermophoresis cannot be straightforwardly recovered from the equations of binary transport ( 15)-( 16) used in thermodiffusion.

that div u A = -(∂/∂t + u A . ∇ ∇ ∇ ∇)
That conclusion is further supported by the fact that J B is a Galilean-invariant vector field, whereas j B → j Bn B V in a Galilean frame moving at velocity V with respect to the laboratory frame. Because of those distinct behaviours with respect to Galilean invariance, J B cannot be carelessly identified with j B , as one would like in letting x A → 1 and u A = 0 in [START_REF] Marcinkiewicz | Sur une propriété de la loi de Gauß[END_REF].

Consider now the mass transport velocity

v = (m A n A u A + m B n B u B )/ρ (19) where ρ = m A n A + m B n B
is the mass density. That velocity appears in the local conservation of mass in fluid dynamics [START_REF] Batchelor | An Introduction to Fluid Dynamics[END_REF],

∂ρ ∂t + div(ρv) = 0, ( 20
)
and it is the solution of the Navier-Stokes equation [START_REF] Batchelor | An Introduction to Fluid Dynamics[END_REF],

ρ( ∂ ∂t + v . ∇ ∇ ∇ ∇)v = -∇ ∇ ∇ ∇p + ρg + η∆v, ( 21
)
where g is the Earth's gravity field and η is the shear viscosity (we ignore the volume viscosity and the position dependence of η for simplicity). In mechanical equilibrium, (i) there is no horizontal pressure gradient and a vertical pressure gradient offsets gravity;

(ii) the solution v(r, t) of equation ( 21) is constant and uniform. Its value v in the laboratory frame is determined by the breaking of Galilean invariance due to the rigid container. The latter exerts a viscous braking force on the liquid and makes v vanish with respect to the container, which may be taken as a Galilean reference frame and usually is at rest in the laboratory. From the dimensional analysis of [START_REF] Blawzdziewicz | Comment on 'Drift without flux: Brownian walker with a space-dependent diffusion coefficient[END_REF], it is seen that mechanical equilibration (v = 0) is achieved over a typical transient time L 2 /ν, where L is a typical length of the container and ν = η/ρ is the momentum diffusivity of the liquid [START_REF] Batchelor | An Introduction to Fluid Dynamics[END_REF] (ν ≈ 10 -6 m 2 /s in water under standard conditions). That time is much shorter than the typical time for interdiffusion, namely L 2 /D BA , where D BA is the mutual diffusivity appearing in the phenomenological equation written in section 4.2: D BA ≈ 10 -12 m 2 /s for suspended spheres [START_REF] Mcnab | Thermophoresis in liquids[END_REF][START_REF] Regazzetti | Experimental evidence of thermophoresis of non-Brownian particles in pure liquids and estimation of their thermophoretic velocity[END_REF] and ≈ 10 -9 m 2 /s for ethanol dilute in water [START_REF] Tyn | Temperature and concentration dependence of some binary liquid systems[END_REF]. Because the momentum density ρv and the composition x B evolve over disparate time scales, it is possible to consider that v approximately vanishes everywhere in the container's frame while u Au B does not. In other words, because of its faster time scale, mechanical equilibration adjusts to the slower 'chemical equilibration', i.e. interdiffusion [START_REF] De Groot | Non-Equilibrium Thermodynamics[END_REF][START_REF] Wannier | Statistical Physics[END_REF]. The relative error incurred in approximating v to zero is of the order of D BA /ν ≈ 10 -3 or less in the examples considered. From v ≈ 0 in the container's frame and from definitions ( 16) and ( 19), we are now in a position to connect j B in that frame with J B , as

F o r P e e r R e v i e w O n l y 9 J B ≈ [x A + ( m B m A )x B ] j B . (22)
It is emphasized that the connection ( 22) is specific to liquid media where D BA /ν << 1. In gaseous media where D BA /ν ≈ 1 [START_REF] Deen | Analysis of Transport Phenomena[END_REF][START_REF] Bird | Transport phenomena[END_REF], it would not hold. Indeed it is not found in the literature on thermodiffusion in gases [START_REF] Chapman | A note on thermal diffusion[END_REF][START_REF] Mason | Thermal diffusion in gases[END_REF].

The mathematical relation between the rates of change of n

B = n(p, T, x B ) x B and of x B is ∂n B ∂t = [n + x B ( ∂n ∂x B ) p, T ]( ∂x B ∂t ). (23)
Furthermore, the vanishing of the mass transport velocity yields the advective derivative

Dx B Dt = ∂x B ∂t + (1 -m B m A )x B u B . ∇ ∇ ∇ ∇x B . ( 24
)
It is found after some algebra that the evolutions [START_REF] Mathews | Mathematical Methods of Physics[END_REF] of n B and ( 15) of x B are compatible

provided that ∂ ∂t (m A n A + m B n B ) = 0. ( 25
)
This is just the local conservation of mass (20) as we have considered that v = 0 after the typical transient time L 2 /ν of mechanical equilibration.

Relation between thermophoretic velocity and thermodiffusion coefficient

The first step in the phenomenology of thermodiffusion was to write an equation of evolution [START_REF] Bringuier | The current equation in strong electric fields[END_REF] of the mixture composition involving an interdiffusion current. The second step is to assume that the latter current responds to the local departures from thermodynamic equilibrium in a linear fashion. Since the independent intensive thermodynamic state variables are p, T and x B , and the experiment is performed under uniform pressure (in case of horizontal transport), those departures are ∇ ∇ ∇ ∇x B and ∇ ∇ ∇ ∇T. The linear response to those departures is written as [START_REF] De Groot | Non-Equilibrium Thermodynamics[END_REF][START_REF] Landau | Fluid Mechanics[END_REF]]

J B = -n(D BA ∇ ∇ ∇ ∇x B + D T,BA x A x B ∇ ∇ ∇ ∇T), (26)
where D BA is called the mutual diffusivity and D T,BA the thermodiffusion coefficient. Other, equivalent definitions of linear-response coefficients exist but they are not of interest to our concern.

Section 4.1 has related J B to the unary current density j B in the laboratory frame where the container is at rest. Writing j B as v d n B -∇ ∇ ∇ ∇(Dn B ) and expressing it as a linear combination of ∇ ∇ ∇ ∇x B and ∇ ∇ ∇ ∇T, we have to equate the coefficients of the latter two gradients on both sides of relation [START_REF] Bringuier | Anatomy of particle diffusion[END_REF]. The calculation is elementary but somewhat lengthy. It involves

∇ ∇ ∇ ∇D = ∂D ∂T ∇ ∇ ∇ ∇T + ( ∂D ∂x B ) ∇ ∇ ∇ ∇x B , (27)
∇ ∇ ∇ ∇n = nβ∇ ∇ ∇ ∇T + ( ∂n ∂x B ) ∇ ∇ ∇ ∇x B , (28)
D BA = D[1 + x B ( ∂ ln(nD) ∂x B )][x A + ( m B m A )x B ], (29)
-x A D T,BA ∇ ∇ ∇ ∇T = [v d -( ∂D ∂T + βD)∇ ∇ ∇ ∇T][x A + ( m B m A )x B ]. (30)
In most cases, one is interested in guest particles B dilute in a host medium of molecules A, so that x B is close to zero. To the lowest order in x B , the previous relations simplify to

D ≈ D BA , (31)
v d ≈ (-D T,BA + ∂D ∂T + βD)∇ ∇ ∇ ∇T. (32)
In [START_REF] Kirkwood | Flow equations and frames of reference for isothermal diffusion in liquids[END_REF] we have regained a known result of the literature on isothermal diffusion [START_REF] Einstein | Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen[END_REF][START_REF] Smoluchowski | Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen[END_REF][START_REF] Perrin | La Structure discontinue de la matière[END_REF][START_REF] Bringuier | Anatomy of particle diffusion[END_REF]: the unary diffusivity D is approximately equal to the binary (mutual) diffusivity D BA . In contrast, the thermophoretic velocity (32) is not proportional to the thermodiffusion coefficient D T,BA .

As was shown earlier [START_REF] Bringuier | On the notion of thermophoretic velocity[END_REF], the main departure from proportionality stems from the dependence of the diffusivity D ≈ D BA on temperature.

Conclusions

The purpose of this paper was to establish a connection between thermophoresis (the motion of a suspended particle under a temperature gradient) and thermodiffusion (the partial demixtion of a binary mixture under a temperature gradient). The phenomenology of thermodiffusion involves an interdiffusion current which is Galilean invariant whereas the phenomenology of thermophoresis involves a velocity which depends on the choice of a reference frame. Connecting the two phenomenologies demands the specification of a frame.

We have shown that the apparently natural frame, namely the suspending liquid, is not at rest with respect to the laboratory frame as soon as suspended particles move and, more importantly, the suspending liquid may not be considered as a Galilean reference frame because it is not rigid.

We have examined the time scales underlying the measurement of the thermophoretic velocity. The temperature and pressure gradients may be regarded as static during thermophoresis; the latter gradient should vanish, otherwise its contribution has to be subtracted from the measured velocity. In the thermodiffusion viewpoint, the composition gradient evolves over a 'chemical equilibration' time scale much longer than the thermal and mechanical equilibration time scales. Because the latter equilibration entails a (horizontally) uniform pressure, the mass transport velocity of the mixture approximately vanishes everywhere in the container's frame with a relative error D BA /ν. That equilibration provides a link between the transport velocities of A and B in that frame, which in turn connects the unary quantities (v d , D) used in thermophoresis with the binary quantities (D T,BA , D BA ) used in thermodiffusion. The ladder of time scales for mechanical, thermal and 'chemical' equilibrations is specific to the liquid state; it does not arise in the gaseous state where ν, ∆ and D BA have similar orders of magnitude. Thereby, besides providing a more rigorous derivation than our previous work [START_REF] Bringuier | On the notion of thermophoretic velocity[END_REF], the present paper has generalised the connection between v d and D T,BA to arbitrary particle fractions. In the framework of linear response [START_REF] De Groot | Non-Equilibrium Thermodynamics[END_REF][START_REF] Landau | Fluid Mechanics[END_REF], ω A j B -ω B j A is a linear combination of ∇ ∇ ∇ ∇ω B and ∇ ∇ ∇ ∇T on the same pattern as equation [START_REF] De Groot | Non-Equilibrium Thermodynamics[END_REF]. Consequently div(ω A j B -ω B j A ) is of second order in the gradients (it contains second-order spatial derivatives and products of first-order derivatives). The time variation of pressure accompanying the transport of particles is thus negligible in the linear-response framework. The latter may be used only for weak composition gradients, i.e. a nearly homogeneous liquid. As a counter-example, the mixing of pure water with pure ethanol gives rise to a noticeable contraction of volume under constant pressure (up to 3.5 % under standard conditions) or equivalently a change in pressure under constant volume. In case of very large ∇ ∇ ∇ ∇ω A and ∇ ∇ ∇ ∇ω B such as occurs at the border between pure A and pure B, a linear-response ('small-signal') analysis is not appropriate [START_REF] Kirkwood | Flow equations and frames of reference for isothermal diffusion in liquids[END_REF].

  If the medium is not homogeneous, v d and D are functions of position. Accordingly, the formulae (2) and (3) are not valid any longer. The definitions of v d and D should now be local, i.e. v d (x 0 ) = ( d〈x〉 dt ),

  (∂ ln n/∂T) p, x B denotes the coefficient of thermal expansion of the liquid mixture. The outcome of the calculation is

  evolves according to an equation similar to[START_REF] Bringuier | The current equation in strong electric fields[END_REF], namely B + div(ω A j B -ω B j A ) = 0, (A.[START_REF] Bringuier | From mechanical motion to Brownian motion, thermodynamics and particle transport theory[END_REF] where the argument of the divergence operator is the mass interdiffusion current density. Since v = 0 owing to local mechanical equilibration, from (A.4) and (A.5) we get dp dt = 1 ρ (∂ρ/∂ω B ) (∂ρ/∂p) div(ω A j B -ω B j A ). (A.6)

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Ryskin [16] remarks that the label 'Fokker-Planck equation' has often been given to an equation having the same pattern but with a wrong definition of D(x 0 ), not consistent with Galilean invariance.

If the medium were anisotropic, D would actually be a second-order tensor D ij instead of D δ ij in the isotropic case (δ ij is the Kronecker symbol).

Actually in their equation[START_REF] Einstein | Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen[END_REF] w is replaced by wd, where d is the diameter of a particle as it can be a significant fraction of w.

Appendix

This appendix examines the behaviour of pressure in a constant-volume experiment [START_REF] Mcnab | Thermophoresis in liquids[END_REF]. First, we acknowledge the existence of a relation of state of the mixture (or suspension) between the volume per particle or its inverse, pressure p, temperature T and composition. Taking quantities per unit mass, instead of per particle, that relation takes the form ρ = ρ(p, T, ω B ), (A.1) where ρ is the mass density and ω B = m B x B /(m A x A + m B x B ) is the mass fraction of B. Relation (A.1) states that the specific volume 1/ρ is a dependent state variable if p, T and ω B are taken as the independent intensive state variables. That equilibrium relation holds to a good approximation when variations occur slowly so that no dynamical equation is needed.

Secondly, the conservation of mass in a rigid box of horizontal length L and lateral area A is expressed by

For ease of reasoning, we have considered that transport occurs horizontally, along ξ varying between 0 and L, instead of vertically, in the presence of a time-independent temperature field T(ξ). The composition ω B varies with time t over the time scale L 2 /D BA . As argued in section 4.1, the horizontal pressure gradient ∂p/∂ξ vanishes very quickly over the time scale L 2 /ν as the momentum diffusivity ν is much larger than the particle mutual diffusivity D BA . That is why the position dependence of p is ignored. In (A.2) the time dependence of p compensates for that of ω B so that the total mass keeps constant.

Once the mass transport velocity v has vanished after the fast transient of mechanical equilibration, (A.2) is replaced by [START_REF] Lançon | Drift without flux: Brownian walker with a space-dependent diffusion coefficient[END_REF]


Impact of Wireless Medium Access Protocol on the Quadrotor Formation Control

Jose Alfredo Guerrero and Yacine Challal and Pedro Castillo

Abstract-In this paper, the impact of the medium access protocols on the average consensus problem over wireless networks for a group of quadrotors is established. We study the case of a group of quadrotors communicating over a wireless network considering both directed and undirected graphs of information flow. It turns out that the media access control (MAC) protocols have a direct impact in both convergence time and average consensus solution, i.e. the solution of the average consensus is no longer the average of the initial conditions. It will be shown that the solution for the average consensus problem over a wireless network depends directly on the MAC algorithm. Simulations are provided to demonstrate the theoretical results.

I. INTRODUCTION

In recent years, much attention has been directed toward multiple robot systems with global or local communications, such as swarming [1], [START_REF] Gazi | Stability of an Asynchronous Swarm With Time-Dependent Communication Links[END_REF], flocking [START_REF] Olfati-Saber | Flocking for Multi-Agent Dynamic Systems: Algorithms and Theory[END_REF], [START_REF] Tanner | Stable Flocking of Mobile Agents, Part I: Fixed Topology[END_REF], [START_REF] Tanner | Stable Flocking of Mobile Agents, Part II: Dynamic Topology[END_REF], [START_REF] Lee | Stable Flocking of Multiple Inertial Agents on Balanced Graphs[END_REF], consensus [START_REF] Ren | Consensus Seeking in Multi-vehicule Systems with a Time Varying Reference State[END_REF] [8], [START_REF] Guerrero | Robust Consensus Tracking of Leader-Based Multi-Agent Systems[END_REF], formations [START_REF] Fax | Information Flow and Cooperative Control on Vehicle Formations[END_REF], [START_REF] Lafferriere | Decentralized control of vehicle formations[END_REF], [START_REF] Dunbar | Distributed receding horizon control for multi-vehicle formation[END_REF], etc. Different approaches have been proposed in the literature for coordination of multiple autonomous robot systems such as leader/follower [START_REF] Chen | ISS-Based Robust Leader/Follower Trailing Control[END_REF], [START_REF] Kristiansen | Output Feedback Control of Relative Translation in a Leader-Follower Spacecraft Formation[END_REF], virtual structure [START_REF] Leonard | Virtual Leaders, Artificial Potentials and Coordinated Control of Groups[END_REF], [START_REF] Beard | A Coordination Architecture for Spacecraft Formation Control[END_REF] and behavioral control [START_REF] Arrichello | Formation Control of Marine Vessels using the Null-Space-Based Behavioral Control[END_REF] and [START_REF] Balch | Behavior-based Formation Control for multirobot teams[END_REF]. In the particular case of multi-agent (multiple mobile robot) consensus problem, most of the literature is focused on modeling the information flow network using either fixed or switching topologies. This means that the communication network does not suffer of time delay and packet drop problems. The main difference between fixed and switching topologies is that the switching topology case takes into account that every mobile robot has a limited range of interaction with its neighbors, i.e., the mobility of the robots affects the information exchange topology.

The distributed nature of multiple robot control system over wireless networks represent an interesting research problem. Data loss, data corruption and time delay over lossy network are key factors that may lead to performance degradation and even cause instability. Recent work on networked control over noisy communication channels includes [START_REF] Sahai | The necessity and sufficiency of anytime capacity for control over noisy cummincation link[END_REF], [START_REF] Takikonda | Control over Noisy Channels[END_REF]. In most of the cases the packet drop phenomena is modeled as a random process without any specification of its probabilistic distribution [START_REF] Lin | Robust Consensus of Multi-Agent Systems with Noise[END_REF], [START_REF] Zhang | Robust Consensus for Uncertain Multi-agent Systems on Directed Communication Topologies[END_REF]. In [START_REF] Hu | Stability Robustness of Networked Control Systems with Respect to Loss[END_REF], the authors consider the packet drop process as a Bernoulli process and develop stability conditions under these conditions. Another way to model the packet loss phenomena has been described J. A. Guerrero, Y. Challal and P. Castillo are with the HEUDIASYC laboratory UMR 6599 CNRS-UTC, BP. 20529, 60200 Compiègne, France (jguerrer, ychallal, castillo)@hds.utc.fr. This work was supported by the SIRENE project.

in [START_REF] Ling | Soft Real-Time Scheduling of Networked Control Systems with Dropouts Governed by a Markov Chain[END_REF], where a Markov chain has been used to model the packet dropout process.

This paper addresses the issue of packet loss as well as packet delay in a multi-agent aerial system considering a wireless network. For this end, we use the widely adopted network simulator (NS2) which provides a rich simulation environment modeling the different network communication layers: the physical layer (modulation, frequencies, signal and radio propagation models, wired and wireless channels, ...), link layer (different medium access control algorithms: TDMA, CSMA/CD, CSMA/CA, ALOHA, ...), routing layer (routing protocols over wired and wireless networks as well as mobile ad hoc networks), transport layer (TCP, UDP, RTP, etc.), and the application layer with a rich sample of applications for typical traffic generation scenarios. It will be assumed, for simplicity, that every agent in the multivehicle system broadcasts its information to its neighbors considering a fixed topology of information exchange, i.e. the mobility of the vehicles does not affect the information flow network. It is assumed that the neighbors of the i thquadrotor are always in the i th -quadrotor broadcasting range. Different network media access control protocols have been designed for wireless networks. Likewise, in this paper, we analyze the impact of the following algorithms on the multiple quadrotor average consensus problem: Carrier sense multiple access with collision avoidance (CSMA/CA) and time division multiple access (TDMA). This paper is organized as follows: A brief description of the multi-quadrotor dynamics and control over perfect communication links is introduced in section II. In Section III, the multi-quadrotor average consensus over wireless network analysis is presented. A stability analysis to improve the convergence of the consensus is proposed in section IV. Simulation results are illustrated in section V. And finally, conclusion and future work are discussed in section VI

II. MULTI-QUADROTOR CONSENSUS

A. Quadrotor dynamic model and control

Let us consider a group of N -quadrotor helicopters with the following dynamical model [START_REF] Lozano | Unmanned Aerial Vehicles Embedded Control[END_REF], [START_REF] Guerrero | Flight Formation of Multiple Rotorcrafts[END_REF]:

ẍi = -F i sin(θ i ) (1) ÿi = F i cos(θ i ) sin(ϕ i ) (2) zi = F i cos(θ i ) cos(ϕ i ) -1 (3) φi = τ ϕ,i (4) θi = τ θ,i (5) ψi = τ ψ,i (6)
where F i is the thrust force vector, (x i , y i , z i ) and (ϕ i , θ i , ψ i ) represent the position and orientation of the i th -quadrotor.

It was proved in [START_REF] Lozano | Unmanned Aerial Vehicles Embedded Control[END_REF] that the following control strategy stabilize the previous system

F i = -k 1,i ż -k 2,i (z i -z d i ) + 1 cos(ϕ i ) cos(θ i ) (7) τ ψ,i = -k 3,i ψi -k 4,i (ψ i -ψ d i ) (8)
τ θ,i = -σ 4 ( θi + σ 3 ( θi + θ i + σ 2 ( θi + 2θ i (9) -ẋi + σ 1 ( θi + 3θ i -3 ẋi -x i )))) τ ϕ,i = -σ 4 ( φi + σ 3 ( φi + ϕ i + σ 2 ( φi + 2ϕ i + ẏi (10) + σ 1 ( φi + 3ϕ i + 3 ẏi + y))))
where k 1,i , k 2,i , k 3,i and k 4,i are positive constant; z d and ψ d are the desired altitude and heading for the i th -quadrotor, respectively. Observe that the nonlinear control laws ( 7)-( 10) guarantee the stabilization of the i th -quadrotor in closed loop system such that

lim t→∞ z i = z d lim t→∞ ψ i = ψ d lim t→∞ x i = 0 lim t→∞ y i = 0

B. From individual to collective behavior

In order to model the interactions among helicopters, a graph-based theoretical approach has been considered. [START_REF] Lee | Stable Flocking of Multiple Inertial Agents on Balanced Graphs[END_REF] has established that the kinematic model of the x-position for a group of quadrotors can also be written as

ẋi = ūi ∀i = 1, .., n;
with leader-based multi-quadrotor consensus achieved using the following algorithm

ūi = - ∑ j∈Ni (x i -x j ) (11)
where N i is the set of vehicles transmitting their information to the quadrotor i. Noticed that [START_REF] Lafferriere | Decentralized control of vehicle formations[END_REF] ensures the consensus agreement in the sense of lim t→∞ |x i -x j | = 0. Hence, the position consensus among quadrotors yields

ẋ = -Lx ( 12
)
where L is the Laplacian matrix of the information exchange graph, more details see [START_REF] Guerrero | Robust Consensus Tracking of Leader-Based Multi-Agent Systems[END_REF], [START_REF] Guerrero | Flight Formation of Multiple Rotorcrafts[END_REF]. Therefore, the controllers ( 7)-( 10) can be improved to the case of multi-quadrotor consensus with the form

F i = -a 1,i ż -a 2,i ( ∑ j∈Ni (z i -z j ) -z d ) + 1 cos(ϕ i ) cos(θ i ) (13) τ ψ,i = -a 3,i ψi -a 4,i ( ∑ j∈Ni (ψ i -ψ j ) -ψ d ) ( 14
)
τ θ,i = -σ 4 ( θi + σ 3 ( θi + θ i + σ 2 ( θi + 2θ i -ẋi (15)
+ σ 1 ( θi + 3θ i -3 ẋi -( ∑ j∈Ni (x i -x j )))))) τ ϕ,i = -σ 4 ( φi + σ 3 ( φi + ϕ i + σ 2 ( φi + 2ϕ i + ẏi (16) + σ 1 ( φi + 3ϕ i + 3 ẏi -( ∑ j∈Ni (y i -y j ))))))
which implies that

lim t→∞ |(z j -z i )| = z d lim t→∞ |(ψ j -ψ i )| = ψ d i lim t→∞ |(x j -x i )| = 0 lim t→∞ |(y j -y i )| = 0
In order to illustrate the performance of the previous control strategy, simulations were carried, over the x-axis, out considering a 10-quadrotor platoon over a perfect communication network (no delay, no packet loss) with cyclic topology. The initial conditions are x i (0) = i, ∀i = 0, 1, ..., [START_REF] Guerrero | Robust Consensus Tracking of Leader-Based Multi-Agent Systems[END_REF].

Figure 1 shows the consensus response. Observe in this figure that the convergence time is small and the solution of the average consensus over the x-axis is the average of the initial conditions, i.e., the platoon achieves consensus to 4.5. Thus, the previous nonlinear control laws guarantees the position synchronization of the quadrotor platoon, see [START_REF] Guerrero | Flight Formation of Multiple Rotorcrafts[END_REF].

In literature, several simulations using the previous (similar or different) controllers have been carried out in order to prove the stability of the consensus or of the flight formation trajectory, see [START_REF] Fax | Information Flow and Cooperative Control on Vehicle Formations[END_REF], [START_REF] Lafferriere | Decentralized control of vehicle formations[END_REF], [START_REF] Dunbar | Distributed receding horizon control for multi-vehicle formation[END_REF], [START_REF] Chen | ISS-Based Robust Leader/Follower Trailing Control[END_REF]. Most of the assumptions in these works are considering wireless and perfect communication between the vehicles. In addition, the main results in flight formation are in general illustrated only in simulations.

As it is known, in a real multi-vehicle flight formation system, each aircraft collects information from its sensors and then exchange its information, employing wireless communication, with other autonomous vehicles in the network. Packet delay and packet loss in wireless becomes a major issue of study that must be taken into account when stabilizing multi-aerial vehicles. The goal of this work is to prove, in simulations, the impact of the wireless network communication in a multi-vehicle consensus.

III. MULTI-AGENT CONSENSUS OVER WIRELESS NETWORKS

Let us consider the case of a N-quadrotor formation flying over a wireless communication channel. From the automatic control point of view, the key factors when using wireless channels are: end-2-end time delay, packet dropout rate, network connectivity and noise. Let us assume that the mobility of the agents does not affect the network connectivity and neglect the noise from sensors. Then, we focus our attention on the packet dropout rate and the end-2-end time delay problems. It has been shown in the communications literature [START_REF] Lian | Performance Evaluation of Control Networks: Ethernet, ControlNet and DeviceNet[END_REF], [START_REF] Liu | Wireless medium access control in networked control systems[END_REF] that both packet dropout rate and the end-2-end time delay are determined by the Medium Access Control protocols. Our study considers two common technologies for wireless communications: CSMA/CA used in IEEE802.11 and TDMA used in GSM.

A. CSMA/CA

This is a distributed random access algorithm used in many standards such as Wifi IEEE802.11. This scheme uses a carrier sense multiple access with collision avoidance (CSMA/CA) mechanism for resolving the problem of access to the communication medium. This implies that when a node detects a collision, it stops transmitting and waits for a random time before retransmitting. More precisely, the protocol CSMA/CA works as follows [29]:

1) a carrier sensing scheme is used 2) a data station that intends to transmit sends a jam signal, 3) after waiting a sufficient time for all stations to receive the jam signal, the data station transmits a frame 4) while transmitting, if the data station detects a jam signal from another station, it stops transmitting for a random time and then tries again. We distinguish two variants: with and without channel reservation. In CSMA/CA with channel reservation, the transmitter first transmits a RTS (Request to Send) for channel reservation, and waits for receiving a CTS (Clear To Send) from the destination before starting to send the data frame.

CSMA/CA is not suitable for real time communications, since it does not guarantee a upper bound for the delay before sending a data frame.

B. TDMA

This is a time slotted scheme used in many standards such as GSM. Accordingly with [29], TDMA (Time Division Multiple Access) is a collision-free multiple access technique whereby users share a transmission medium by being assigned and using (one at a time) time slots assigned previously.

TDMA is more suitable for real time communications, since it guarantees a upper bound delay before transmitting a data frame.

C. Network Analysis

To evaluate the performance of CSMA/CA and TDMA protocols on the quadrotor consensus problem, extensive simulations have been run using the network simulator NS2. As it is shown in Figures 2 and3, it is clear that the packet drops for TDMA protocol are almost null. We attribute the packet drops shown in Figure 2 to the synchronization phase during the simulation initialization. Unlike TDMA, CSMA/CA shows a higher rate of packet drops due to its random broadcasting nature, see Figure 3. almost constant rate. This can be confirmed by observing Figure 4. On the other hand, taking into account the packet drop rate at each quadrotor, it is expected that CSMA/CA would show a variable time delay and the maximum time delay is much higher than TDMA. The evolution of the average end-2-end time delay over time for the CSMA/CA protocol is shown in Figure 5.

IV. QUADROTOR CONSENSUS OVER WIRELESS NETWORKS

In this section, we can introduce a Network-aware average consensus control for a multi-quadrotor system over a wireless network. By taking into account the phenomena discussed above, the packet dropout process will be considered as a source of time delays in the wireless network. As shown in [START_REF] Lian | Performance Evaluation of Control Networks: Ethernet, ControlNet and DeviceNet[END_REF], the end-2-end time delay τ is given by the difference τ dst -τ src which in turns depends on the preprocessing time τ pre , wait time τ wait , transmission time τ tx and the post processing time τ post . Since the nearest neighbor approach assumes that each vehicle communicates only with immediate neighbors that are in its radio range, and assuming that a packet loss implies the re-transmission of the data that has been lost, we consider the packet dropout process, as part of the transmission time delay which has been defined as the frame time τ f rame and the propagation time τ prop , more details see [START_REF] Lian | Performance Evaluation of Control Networks: Ethernet, ControlNet and DeviceNet[END_REF].

Then, we propose to use the following multi-quadrotor consensus control as in [START_REF] Chopra | Output Synchronization of Mechanical Systems[END_REF]

ūi (t) = - ∑ j∈Ni (x i -x j (t -τ ji )) (17)
Using the control law (17) instead of ( 11), the quadrotor consensus control is given as

F i = -k 1,i żi -k 2,i ( ∑ j∈Ni (z j (t -τ ji ) -z i ) -z d ) + 1 cos(ϕ i ) cos(θ i ) (18)
τ ψ,i = -k 3,i ψi -k 4,i ( ∑ j∈Ni (ψ j (t -τ ji ) -ψ i ) -ψ d ) (19) τ θ,i = -σ 4 ( θi + σ 3 ( θi + θ i + σ 2 ( θi + 2θ i -ẋi (20)
+ σ 1 ( θi + 3θ i -3 ẋi - ∑ j∈Ni (x j (t -τ ji ) -x i ))))) τ ϕ,i = -σ 4 ( φi + σ 3 ( φi + ϕ i + σ 2 ( φi + 2ϕ i + ẏi (21)
+ σ 1 ( φi + 3ϕ i + 3 ẏi - ∑ j∈Ni (y j (t -τ ji ) -y i )))))
Let us consider the Lyapunov function as in [START_REF] Chopra | Output Synchronization of Mechanical Systems[END_REF]

V = N ∑ i=1 ∑ j∈N ∫ t t-τji x T j (χ)x j (χ)dχ + 2(V 1 + ... + V N ) (22)
where V i is the storage function for each quadrotor vehicle in the platoon. Differentiating ( 22),

V = -2 N ∑ i=1 S i (x i )- N ∑ i=1 ∑ j∈Ni (x j (t-τ ji )-x i ) T (x j (t-τ ji )-x i ) (23)
From [START_REF] Hu | Stability Robustness of Networked Control Systems with Respect to Loss[END_REF] we have that

lim t→∞ |(x j (t -τ ji ) -x i )| = 0 (24)
From [START_REF] Ling | Soft Real-Time Scheduling of Networked Control Systems with Dropouts Governed by a Markov Chain[END_REF], it is clear that the solution of the average consensus to the average of the initial conditions depends on the network medium access algorithm, which in turns determines the values of the τ ji for each quadrotor. Figure 6 shows the performance of the average consensus considering τ ji = τ kj . It can be observed that the average consensus is the same than the one for the perfect communication case. The difference is that the convergence time for the equal time delay case is larger than for the perfect communication case.

Figure 7 shows the performance of the average consensus Now, lets take a look at the average consensus performance using a wireless network. The network scenario consisted of 10-quadrotors sharing information over a cyclic topology. A quadrotor application was developed, a UDP transport protocol was modified to exchange information with the quadrotor application and the MAC layer as shown in Figure 8. The following MAC algorithms: CSMA/CA and TDMA From Figures 9 and10, it is possible to observe that the solution for the average consensus as well as its convergence time are affected by the network media access control algorithm. It can be concluded that the average consensus using a CSMA/CA protocol presents a larger convergence time due to both packet drops and time delay in the end-2end transmission among quadrotors. Also, due to the fact that the access to the transmission medium is assigned randomly, it is evident that some quadrotors will transmit their positions to their neighbors before their counterparts. This implies that the difference between any two τ ji can be large enough such that some quadrotors will evolve faster than the others. Then, the quadrotor with the smallest τ ji will update its position at a highest rate than the others. Figure 11 shows the performance of the average consensus for a small platoon of 4 quadrotors using different medium access control algorithms such as: CSMA/CA, TDMA, constant time delay (τ ji > 0), and perfect communication.

The simulation results are summarized in Table I Figure 12 shows the performance of the average consensus for the heading subsystem of a 2-quadrotor system using CSMA/CA.

VI. CONCLUSIONS AND FUTURE WORK

A nonlinear control based on nested saturations and a single integrator consensus control considering time delay for flight formation of mini rotorcraft was developed. We remark that TDMA is more suitable for real time communications than CSMA/CA, since it guarantees a upper bound delay before transmitting a data frame. CSMA/CA is less suitable suitable for real time communications, since it does not guarantee an upper bound for the delay before sending a data frame. Extensive simulations were run in order to show the performance of the developed control scheme. Future work in this area includes experimental tests on mini rotorcraft using real-time embedded control systems and to develop an optimized medium access control algorithm for mobile robot systems.

Fig. 1 .

 1 Fig. 1. Multiple quadrotor consensus using perfect communication

Fig. 2 .

 2 Fig. 2. Packet drop using TDMA

Fig. 3 .

 3 Fig. 3. Packet drop using CSMA/CA

Fig. 4 .

 4 Fig. 4. Packet time delay using TDMA

Fig. 5 .

 5 Fig. 5. Packet time delay using CSMA/CA

Fig. 6 .

 6 Fig. 6. Multiple quadrotor consensus with τ ji = τ kj

Fig. 7 .

 7 Fig. 7. Multiple quadrotor with τ ji ̸ = τ kj

Fig. 8 .

 8 Fig. 8. Multiple quadrotor wireless network stack were analyzed using the network simulator NS2. Figures 9 and 10 show the performance of the MAC protocols CSMA/CA and TDMA respectively.

Fig. 9 .

 9 Fig. 9. Multi-agent consensus using IEEE 802.11

Fig. 10 .

 10 Fig. 10. Multi-agent consensus using TDMA

  [1] V.Gazi and Kevin M. Passino, "Stability Analysis of Swarms", IEEE Trans. on Autom. Control, Vol. 48, No. 4, pp.692-697, 2003.

Fig. 11 .

 11 Fig. 11. Comparison of convergence time for different MAC algorithms