Meitner Cadena
email: meitner.cadena@etu.upmc.fr

Revisiting extensions of regularly varying functions

Keywords: regularly varying function, slowly varying function, O-RV, large deviations, extreme value theory AMS classification: 26A12, 60F10

Revisiting extensions of regularly varying functions

Introduction

A positive and measurable function U defined on R + is a regularly varying (RV) function if

lim x→∞ U (t x) U (x) < ∞ (t > 0). (1)
If this limit equals 1, U is a slowly varying (SW) function. Classes RV and SV of regularly and slowly varying functions were introduced by Karamata [START_REF] Karamata | Sur un mode de croissance régulière des fonctions[END_REF] in 1930. Since then theory of these functions has been developed in many directions. Systematic treatment of this theory can be found in e.g. [START_REF] Bingham | Regular Variation[END_REF] and [START_REF] Seneta | Regularly Varying Functions[END_REF].

Extensions of RV functions have been obtained by letting (1) to vary. An early extension of this type was given by Avakumović in 1936 [START_REF] Avakumovi Ć | On a O-inverse theorem (in Serbian)[END_REF]. He introduced the class O-RV of O-regularly varying (O-RV) functions U which satisfy the following condition instead of (1):

0 < U * (t ) := lim x→∞ U (t x) U (x) ≤ lim x→∞ U (t x) U (x) =: U * (t ) < ∞ (t ≥ 1). (2)
Recently Cadena and Kratz [START_REF] Cadena | An extension of the class of regularly varying functions[END_REF] gave an extension of RV functions by also letting [START_REF] Cadena | O-regularly varying functions[END_REF] to vary, but they designed it in a different way to the previous one. They introduced the class M which consists in functions U satisfying the following condition instead of (1):

∃ρ ∈ R, ∀ǫ > 0, lim x→∞ U (x)
x ρ+ǫ = 0 and lim

x→∞ U (x) x ρ-ǫ = ∞. (3)
We have clearly RV O-RV and, for instance using Theorem 1 (see Corollary 1), RV M . There arises the natural question of how O-RV and M are related between them. We undertake this study helping us of characterizations of these classes: recalling well-known characterizations of O-RV and giving proofs of three characterizations of M , two of them provided in [START_REF] Cadena | An extension of the class of regularly varying functions[END_REF] and a new one given in this note.

Cadena and Kratz also introduced the following natural extensions of M .

M ∞ := U : R + → R + : U is measurable and satisfies ∀ρ ∈ R, lim x→∞ U (x)

x ρ = 0 (4)

M -∞ := U : R + → R + : U is measurable and satisfies ∀ρ ∈ R,

lim x→∞ U (x) x ρ = ∞ . ( 5
)
The new characterization given for M is extended to M ∞ and M -∞ . Relationships among M ∞ and M -∞ and O-RV are also investigated in this note.

This note is organized as follows. The main results are presented in the next section, introducing previously notations and definitions. First, the new characterizations of M , M ∞ , and M -∞ based on limits are given. Next, analyses of uniform convergence in these characterizations are presented and, finally, relationships among O-RV and M , M ∞ and M -∞ are shown.

All proofs are collected in Section 3. Conclusion is presented in the last section.

Main Results

For a positive function U with support R + its lower and upper orders are defined by (see e.g. [START_REF] Bingham | Regular Variation[END_REF])

µ(U ) := lim x→∞ log (U (x)) log(x) , ν(U ) := lim x→∞ log (U (x)) log(x) .
Throughout this note log(x) represents the natural logarithm of x.

We notice that the classes M , M ∞ , and M -∞ defined in (3), (4), and ( 5) are a bit weaker than the corresponding classes given in [START_REF] Cadena | An extension of the class of regularly varying functions[END_REF], and that each of them is disjoint from each other. Moreover, using straightforward computations, one can prove that ρ defined in (3) is unique, hence it will be denoted by ρ U , and one can show that ǫ > 0 in (3) can be taken sufficiently small. Additionally, one can prove that M is strictly larger than RV, for instance using Theorem 1 (see Corollary 1), and that M ∞ is related to the domain of attraction of Gumbel (see [START_REF] Cadena | An extension of the class of regularly varying functions[END_REF]).

The new characterizations of M , M ∞ , and M -∞ follow.

Theorem 1. Let U : R + → R + be a measurable function. Then

(i) U ∈ M with ρ U = -τ iff          ∀r < τ, ∃x a > 1, ∀x ≥ x a , lim t →∞ t r U (t x) U (x) = 0 ∀r > τ, ∃x b > 1, ∀x ≥ x b , lim t →∞ t r U (t x) U (x) = ∞. ( 6
) (ii) U ∈ M ∞ iff ∀r ∈ R, ∃x 0 > 1, ∀x ≥ x 0 , lim t →∞ t r U (t x) U (x) = 0. ( 7
) (iii) U ∈ M -∞ iff ∀r ∈ R, ∃x 0 > 1, ∀x ≥ x 0 , lim t →∞ t r U (t x) U (x) = ∞. ( 8
)
Example 1.

1. Consider a measurable and positive function U with support R + such that, for x ≥ x 0 with some x 0 > 1, U (x) = x log(x).

Noting that, for t , x > 1, lim

t →∞ t r U (t x) U (x) = lim t →∞ t r -1 log(x) log(t x) = 0 if r > 1 ∞ if r < 1,
provides, taking τ = -1 and applying Theorem 1, (i), U ∈ M with ρ U = 1.

Let U be a function defined by U

(x) := x sin(x) , x > 0. Writing t r U (t x) U (x) = t r +sin(t x) x sin(t x)-sin(x)
gives, for r ∈ R,

lim t →∞ t r U (t x) U (x) = ∞ and lim t →∞ t r U (t x) U (x) = 0.
Hence the necessary condition of Theorem 1, (i), is not satisfied and consequently gives U ∈ M .

It follows a consequence of Theorem 1. This result was proved by Cadena and Kratz in [START_REF] Cadena | An extension of the class of regularly varying functions[END_REF] combining a result provided in [START_REF] Haan | On regular variation and its applications to the weak convergence of sample extremes[END_REF] and another characterization of M (see Theorem CK later).

Corollary 1. RV M .

Note that, from Corollary 1, RV ⊆ M O-RV .

There are not common elements between O-RV and M under their definitions given in ( 2) and (3) respectively, but observing the characterization of M given in Theorem 1 one identifies the quotient U (t x) U (x), which appears in (2). The next example exploits this link to show a first relationship between O-RV and M .

Example 2. M ⊆ O-RV.

Let U be a function defined by U (x) := exp (log x) α cos (log x) β , x > 0, where 0 < α, β < 1 such that α + β > 1.

Prof. Philippe Soulier gave recommendations to correct an error in an early version of this example.

On the one hand, noting that, for x, t > e, using the changes of variable y = log(x) and s = log(t ) and observing that s → ∞ as t → ∞,

lim t →∞ t r U (t x) U (x) = lim s→∞ exp r s + (s + y) α cos (s + y) β -y α cos y β = lim s→∞ exp s r + 1 s 1-α 1 + y s α cos (s + y) 1/3 - y α s cos y β = 0 if r > 0 ∞ if r < 0,
provides, taking τ = 0 and applying Theorem 1, U ∈ M with ρ U = 0.

On the other hand, writing, for x > e and t > 0, using the previous changes of variables, with x such that log t x β = π 2 + 2kπ, for a given t ,

U (t x) U (x) = exp -log x α cos (log x) β = exp -y α cos ((π 2 + 2kπ) 1/β -s) β = exp (π 2 + 2kπ) 1/β -s α sin ((π 2 + 2kπ) 1/β -s) β -(π 2 + 2kπ) . Since (π 2 + 2kπ) 1/β -s β -(π 2 + 2kπ) → 0 as k → ∞, we have lim k→∞ (π 2 + 2kπ) 1/β -s α sin ((π 2 + 2kπ) 1/β -s) β -(π 2 + 2kπ) = lim k→∞ ((π 2 + 2kπ) 1/β -s) β -(π 2 + 2kπ) (π 2 + 2kπ) 1/β -s -α
, which is an indetermination of type 0 0. Then, applying L'Hopital's rule we have

lim k→∞ ((π 2 + 2kπ) 1/β -s) β -(π 2 + 2kπ) (π 2 + 2kπ) 1/β -s -α = lim k→∞ (2π) α/β ((π 2 + 2kπ) 1/β -s) β -(π 2 + 2kπ) k -α/β = lim k→∞ - β α (2π) α/β+1 ((π 2 + 2kπ) 1/β -s) β-1 (π 2 + 2kπ) 1/β-1 -1 k -α/β-1 ,
which is an indetermination of type 0 0. Then, applying again L'Hopital's rule we have

lim k→∞ ((π 2 + 2kπ) 1/β -s) β -(π 2 + 2kπ) (π 2 + 2kπ) 1/β -s -α = lim k→∞ s β(1 -β) α(α + β) (2π) α/β+2 ((π 2 + 2kπ) 1/β -s) β-2 (π 2 + 2kπ) 1/β-2 k -α/β-2 = lim k→∞ s β(1 -β) α(α + β) (2π) (α+β-1)/β k (α+β-1)/β = ∞ if s>0 -∞ if s<0.
Then, we get, for t > 1,

U * (t ) = lim x→∞ U (t x) U (x) = ∞,
and, for t < 1,

U * (t ) = lim x→∞ U (t x) U (x) = 0, which contradict (2), so U ∈ O-RV. In particular, U ∈ SV.
Next, the uniform convergences in x of limits given in ( 6), [START_REF] Cadena | An extension of the class of regularly varying functions[END_REF], and ( 8) are analyzed. To this aim, we will use the next two results.

Proposition 1. Let U : R + → R + be a measurable function. Then (i) If U ∈ M with ρ U = -τ, then there exists x 0 > 1 such that, for x 0 ≤ c < d < ∞, there exist 0 < M c < M d satisfying, for x ∈ [c; d], M c ≤ U (x) ≤ M d . (ii) If U ∈ M ∞ , then there exists x 0 > 1 such that, for c ≥ x 0 , there exist M c > 0 satisfying, for x ∈ [c; ∞), U (x) ≤ M c . (iii) If U ∈ M -∞ , then there exists x 0 > 1 such that, for d ≥ x 0 , there exist M d > 0 satisfying, for x ∈ [d; ∞), U (x) ≥ M d .
Proposition 2 (Given in [START_REF] Aranelovi Ć | An inequality for the Lebesgue measure[END_REF]). Let µ be the Lebesgue measure on R, A a measurable set of positive measure, and x n n∈N a bounded sequence of real numbers. Then, µ(A) ≤ µ lim n→∞ (x n + A) .

Now the results on uniform convergences are presented. Their proofs are inspired by [START_REF] Petkovi Ć | An Inequality for the Lebesgue Measure and its Applications[END_REF].

Theorem 2 (Uniform Convergence Theorem (UCT)). Let U : R + → R + be a measurable function. Then

(i) If U ∈ M with ρ U = -τ and r < τ, then, for any x a ≤ c < d < ∞ for some x a > 1, lim t →∞ t r sup x∈[c;d ] U (t x) U (x) = 0. (ii) If U ∈ M with ρ U = -τ and r > τ, then, for any x b ≤ c < d < ∞ for some x b > 1, lim t →∞ t r inf x∈[c;d ] U (t x) U (x) = ∞. (iii) If U ∈ M ∞ satisfying, for s > 1, U (x) ≥ M s for x ∈ [1; s] and some M s > 0, then, for r ∈ R and any constants x 0 ≤ c < d < ∞ for some x 0 > 1, lim t →∞ t r sup x∈[c;d ] U (t x) U (x) = 0. (iv) If U ∈ M -∞ satisfying, for s > 1, U (x) ≤ M s for x ∈ [1; s] and some M s > 0, then, for r ∈ R and any constants x 0 ≤ c < d < ∞ for some x 0 , lim t →∞ t r inf x∈[c;d ] U (t x) U (x) = ∞.
Note that UCT cannot be extended to infinite intervals. For instance, from the function U given in Example 2 we have that computing the supremum of the quotient U (t x) U (x) in x on [x 0 ; ∞), for any x 0 > 1, gives always ∞, and hence one cannot deduce that ρ U = 0.

The next results on O-RV, M , M ∞ , and M -∞ will be used to give more relationships between these classes. On O-RV we need:

Proposition 3 (see e.g. [START_REF] Karamata | Bemerkung über die vorstehende Arbeit des Herrn Avakumović mit, näherer Betrachtung einer Klasse von Funktionen, welche bei den Inversionssätzen vorkommen[END_REF], [START_REF] Seneta | Regularly Varying Functions[END_REF], [START_REF] Cadena | O-regularly varying functions[END_REF], [START_REF] Haan | Regular variation, extensions and Tauberian theorems[END_REF], and [START_REF] Bingham | Regular Variation[END_REF]). Let U : R + → R + be a measurable function.

Then the following statements are equivalent:

(i) U ∈ O-RV. (ii) There exist α, β ∈ R and x 0 > 1, c > 0 such that, for all t ≥ 1 and x ≥ x 0 , c -1 t β ≤ U (t x) U (x) ≤ ct α .
(iii) There exist functions η(x) and φ(x) bounded on [x 0 ; ∞), for some x 0 ≥ 1, such that

U (x) = exp η(x) + x 1 φ(y) d y y , x ≥ 1.
On M we need the next two characterizations of M given by Cadena and Kratz in [START_REF] Cadena | An extension of the class of regularly varying functions[END_REF]. For the sake of completeness of this note, we give them as Theorem CK and indicate their proofs. Part of these proofs are copied from [START_REF] Cadena | An extension of the class of regularly varying functions[END_REF].

Theorem CK. Let U : R + → R + be a measurable function. Then the following statements are equivalent:

(i) U ∈ M with ρ U = τ. (ii) lim x→∞ log (U (x)) log(x) = τ.
(iii) There exist b > 1 and measurable functions α, β, and δ satisfying, as x → ∞,

α(x) log(x) → 0, β(x) → τ, δ(x) → 1, such that U (x) = exp α(x) + δ(x) x b β(s) d s s , x ≥ x 1 for some x 1 ≥ b. Remark 1.
If F is the tail of a distribution F associated to a random variable (rv) X , some authors (see e.g. [START_REF] Nakagawa | Application of Tauberian Theorem to the Exponential Decay of the Tail Probability of a Random Variable[END_REF] and [START_REF] Nakagawa | On the Singularity of Laplace-Stieljes Transform of a Heavy-Tailed Random Variable[END_REF]) say that X is heavy-tailed if the limit

η := lim x→∞ log F (x) log(x)
exists and takes a negative value.

We notice that this characterization does not cover rvs with heavy tails satisfying η = 0 or with heavy tails for which such limit does not exist. Indeed, on the one side, from Theorem CK one has that η = 0 implies that F ∈ M with ρ F = 0, being a particular case of these functions the SV functions, which are considered heavy-tailed. On the other side, Cadena and Kratz presented in [START_REF] Cadena | An extension of the class of regularly varying functions[END_REF] families of tails F for which the limit lim

x→∞ log F (x) log(x)
does not exist, for instance the next tail defined by (see [START_REF] Cadena | An extension of the class of regularly varying functions[END_REF])

Let α > 0, β < -1, x a > 1, and define the series x n = x (1+α) n a , n ≥ 1, which satisfies x n → ∞ as n → ∞. It is not hard to prove that the tail F associated to a rv X and defined by

F (x) := 1 x ∈ [0; x 1 ) x α(1+β) n x ∈ [x n ; x n+1 ), ∀n ≥ 1 satisfies lim x→∞ log F (x) log(x) = - α(1 + β) 1 + α < -α(1 + β) = lim x→∞ log F (x) log(x) . Note that if -α(1 + β) (1 + α) < 1,
then the expected value of X is ∞, which means that X can be considered as a heavy-tailed rv.

We notice from the representations of U via O-RV and M given in Proposition 3, (iii), and Theorem CK, (iii), respectively, that a key difference between those representations is the presence of a bounded function under the integral symbol. Motivated by this observation, we built the next function belonging to O-RV but not to M . This aim is reached by building a bounded function φ such that the limit lim

             n-1 k=0 (-1) k e -k if n is odd n-1 k=1 (-1) k+1 e -k if n is even,
and one then gets

lim n→∞ log(U (x n )) log(x n ) =          1 1 + e -1 if n is odd e -1
1 + e -1 if n is even, which implies ν(U )-µ(U ) ≥ (1-e -1 ) (1+e -1 ) > 0, hence the limit lim The relationships of M ∞ and M -∞ with O-RV are simpler.

Proposition 5. For λ ∈ ∞, -∞ , M λ O-RV = .

Proofs

Proof of Theorem 1.

• Proof of the necessary condition of (i)

Assume U ∈ M with ρ U = -τ. Let r ∈ R such that r = 0.
-If r < τ Let 0 < ǫ < τ-r and δ > 0. By hypothesis, there exists a constant x 0 > 1 such that, for x ≥ x 0 , U (x) ≤ δx -τ+ǫ , and there exists x 1 > 1 such that, for x ≥ x 1 , U (x) ≥ x -τ-ǫ δ.

Hence, setting x a := max(x 0 , x 1 ), for x ≥ x a and t > 1,

t r U (t x) U (x) ≤ δ 2 t r (t x) -τ+ǫ x τ+ǫ = δ 2 t -τ+r +ǫ x 2ǫ ,
and the assertion then follows as t → ∞ since -τ + r + ǫ < 0.

-If r > τ Let 0 < ǫ < r -τ and δ > 0. By hypothesis, there exists a constant x 0 > 1 such that, for x ≥ x 0 , U (x) ≤ δx -τ+ǫ , and there exists

x 1 > 1 such that, for x ≥ x 1 , U (x) ≥ x -τ-ǫ δ.
Hence, setting x a := max(x 0 , x 1 ), for x ≥ x a and t > 1,

t r U (t x) U (x) ≥ 1 δ 2 t r (t x) -τ-ǫ x τ-ǫ = 1 δ 2 t r -τ-ǫ x -2ǫ ,
and the assertion then follows as t → ∞ since rτǫ > 0.

• Proof of the sufficient condition of (i)

Let δ > 0 and η > 0.

One the one hand, since τδ 2 < τ, by hypothesis, there exists a constant x a > 1 such that, for x ≥ x a , lim

t →∞ t τ-δ/2 U (xt ) U (x) = 0. Hence, given x ≥ x a , there exists t a = t a (x) > 1 such that, for t ≥ t a , t τ-δ/2 U (t x) ≤ ηU (x), or U (t x) (t x) -τ+δ ≤ η x τ-δ U (x) t δ/2 . ( 9
)
One the other hand, since τ + δ 2 > τ, by hypothesis, there exists a constant x b > 1 such that, for x ≥ x b , lim

t →∞ t τ+δ/2 U (xt ) U (x) = ∞.
Hence, given x ≥ max(x a , x b ), there exists t b = t b (x) > 1 such that, for t ≥ t b , t τ+δ/2 U (t x) ≥ ηU (x), or

U (t x) (t x) -τ-δ ≥ ηx τ+δ U (x)t δ/2 . ( 10
)
Combining ( 9) and ( 10), given x ≥ max(x a , x b ) and for t ≥ max(t a , t b ), and using the change of variable y = t x with y → ∞ as t → ∞, provide, for δ > 0, lim y →∞

U (y)

y -τ+δ = 0 and lim

y →∞ U (y) y -τ-δ = ∞, which implies that U ∈ M with ρ U = -τ.
• Proof of the necessary condition of (ii)

Let r ∈ R and η > 0. Set r ′ < -r . Since U ∈ M ∞ there exists a constant x 0 > 1 such that,

for x ≥ x 0 , U (x) ≤ ηx r ′ . Hence, for t > 1, t r U (t x) U (x) ≤ η t r +r ′ x r ′ U (x) ,
and the assertion then follows as t → ∞ since r + r ′ < 0.

• Proof of the sufficient condition of (ii)

Let r ∈ R. Taking r ′ < -r , by hypothesis, there exists a constant x 0 > 1 such that, for

x ≥ x 0 , lim t →∞ t r ′ U (xt ) U (x) = 0.
Hence, for η > 0, there exists a constant t 0 > 1 such that, for

t ≥ t 0 , t r ′ U (t x) ≤ ηU (x), or U (t x) (t x) r ≤ η U (x)
x r t r +r ′ .

Using the change of variable y = t x and noting that y → ∞ as t → ∞ give, for r ∈ R, being

r + r ′ > 0, lim y →∞ U (y) y r = 0, which means that U ∈ M ∞ .
• Proof of the necessary condition of (iii)

Let r ∈ R and η > 0. Set r ′ > -r . Since U ∈ M -∞ there exists a constant x 0 > 1 such that,

for x ≥ x 0 , U (x) ≥ ηx r ′ . Hence, for t > 1, t r U (t x) U (x) ≥ η x r ′ U (x) t r +r ′ ,
and the assertion then follows as t → ∞ since r + r ′ > 0.

• Proof of the sufficient condition of (iii)

Let r ∈ R. Taking r ′ < -r , by hypothesis, there exists a constant x 0 > 1 such that, for

x ≥ x 0 , lim t →∞ t r ′ U (xt ) U (x) = ∞.
Hence, for η > 0, there exists a constant t 0 > 1 such that, for

t ≥ t 0 , t r ′ U (t x) ≥ ηU (x), or U (t x) (t x) r ≥ η U (x) x r t -r -r ′ .
Using the change of variable y = t x and noting that y → ∞ as t → ∞ give, for r ∈ R, being

-r -r ′ > 0, lim y →∞ U (y) y r = 0, which means that U ∈ M ∞ .

Proof of Corollary 1.

Let U ∈ RV with tail index ρ. Then, for t > 1,

lim x→∞ t r U (t x) U (x) = t r +ρ ,
which implies that, for ǫ > 0, there exists a constant x 0 > 1 such that, for x ≥ x 0 ,

t r +ρ -ǫ ≤ t r U (t x) U (x) ≤ t r +ρ + ǫ.
Hence, setting τ = -ρ, gives, on the one hand, for r < τ,

-ǫ ≤ lim t →∞ t r U (t x) U (x) ≤ ǫ, which implies lim t →∞ t r U (t x) U (x)
= 0 taking ǫ arbitrary, and, on the other hand, for r > τ,

lim t →∞ t r U (t x) U (x) = ∞.
Therefore one has, applying Theorem 1, that U ∈ M with ρ U = ρ.

Finally, a function belonging to M but not to RV is for instance the function given in Example 2.

Proof of Proposition 1.

• Proof of (i)

Let ǫ > 0. By definition of U ∈ M with ρ U = -τ, there exist constants x a , x b > 1 such that, for x ≥ x a , U (x) ≤ x -τ+ǫ , and, for x ≥ x b , U (x) ≥ x -τ-ǫ .

So, for x ≥ x 0 := max(x a , x b ), x -τ-ǫ ≤ U (x) ≤ x -τ+ǫ . Hence, for any

x 0 ≤ c < d < ∞, one has, setting M c := min(c -τ-ǫ , d -τ+ǫ ) and M d := max(c -τ-ǫ , d -τ+ǫ ), that U satisfies M c ≤ U (x) ≤ M d for any x ∈ [c; d].
• Proof of (ii)

Let ǫ > 0. By definition of U ∈ M ∞ , there exists a constant x 0 > 1 such that, for x ≥ x 0 , U (x) ≤ x ǫ . Hence, for any c ≥ x 0 , one has, setting

M c := c ǫ , that U satisfies U (x) ≤ M c for any x ∈ [c; ∞).
• Proof of (iii)

Let ǫ > 0. By definition of U ∈ M -∞ , there exists a constant x 0 > 1 such that, for x ≥ x 0 , U (x) ≥ x ǫ . Hence, for any d ≥ x 0 , one has, setting

M d := d ǫ , that U satisfies U (x) ≥ M d for any x ∈ [d; ∞).

Proof of Theorem 2.

Let µ be the Lebesgue measure on R.

• Proof of (i)

Let U ∈ M with ρ U = -τ and let r < τ. Applying Theorem 1, (i), there exists x a > 1 such that, for x ≥ x a , lim

t →∞ t r U (t x) U (x) = 0. Let x a ≤ c < d < ∞.
Then using Egoroff's theorem (see e.g. [START_REF] Billingsley | Probability and Measure[END_REF]), there exists a measurable

A ⊆ [c; d] of a positive measure such that lim t →∞ sup x∈A t r U (t x) U (x) = 0.
Let us prove by contradiction that the previous limit holds on [c; d]. Then suppose that there exist ǫ > 0, x n n∈N ⊆ [c; d], and t n n∈N ⊆ R + such that t n → ∞ and

lim n→∞ t r n U (t n x n ) U (x n ) > ǫ. ( 11
)
By Proposition 2 one has, denoting log(A) = log(x) : x ∈ A and noting that log(A) has a positive measure,

µ lim n→∞ (log(A) -log(x n )) ≥ µ log A > 0,
which implies that there exist a constant log(u) ∈ R and a subsequence x n i i∈N ⊆ x n n∈N such that log(x n i ) + log(u) ∈ log(A), i.e. u x n i ∈ A. Note that u > 0.

By Proposition 1, (i), there exist 0

< M c ≤ M d < ∞ such that M c ≤ U (x) ≤ M d , x ∈ (c; d).
Hence, one then has

t r n i U (t n i x n i ) U (x n i ) = t n i u r U t n i u ux n i U (ux n i ) u r U (ux n i ) U (x n i ) ≤ t n i u r U t n i u ux n i U (ux n i ) u r M d M c .
Noting that [START_REF] Karamata | Bemerkung über die vorstehende Arbeit des Herrn Avakumović mit, näherer Betrachtung einer Klasse von Funktionen, welche bei den Inversionssätzen vorkommen[END_REF].

t n i u r U ((t n i u) ux n i ) U (ux n i ) → 0 since u x n i ∈ A and t n i u → ∞ as n i → ∞ provide t r n i U (t n i x n i ) U (x n i ) → 0 as n i → ∞, which contradicts
• Proof of (ii)

Let U ∈ M with ρ U = -τ and let r < τ. Applying Theorem 1, (i), there exists

x b > 1 such that, for x ≥ x b , lim t →∞ t r U (t x) U (x) = ∞. Let x b ≤ c < d < ∞
and let ǫ m m∈N be a strictly increasing sequence of positive numbers such that ǫ m → ∞ as m → ∞. Then using Egoroff's theorem, there exists a measurable

A m ⊆ [c; d], m ∈ N, of a positive measure such that lim t →∞ inf x∈A m t r U (t x) U (x) ≥ ǫ m .
Let us prove lim

t →∞ inf x∈[c;d ] t r U (t x) U (x) = ∞
by contradiction. Then suppose that there exist δ > 0, x n n∈N ⊆ [c; d], and t n n∈N ⊆ R + such that t n → ∞ and

lim n→∞ t r n U (t n x n ) U (x n ) < δ. ( 12
)
By Proposition 2 one has, denoting log(A m ) = log(x) : x ∈ A m , m ∈ N, and noting that log(A m ) has a positive measure,

µ lim n→∞ (log(A m ) -log(x n )) ≥ µ log A m > 0,
which implies, for m ∈ N, that there exist a constant log(u m ) ∈ R and a subsequence

x n m,i i∈N ⊆ x n n∈N such that log(x n m,i ) + log(u m ) ∈ log(A m ), i.e. u m x n m,i ∈ A m . Note that u m > 0 and c d ≤ u m ≤ d c, m ∈ N.
By Proposition 1, (i), there exist 0

< M c ≤ M d < ∞ such that M c ≤ U (x) ≤ M d for x ∈ (c; d).
Hence, one then has [START_REF] Nakagawa | Application of Tauberian Theorem to the Exponential Decay of the Tail Probability of a Random Variable[END_REF].

t r n m,i U (t n m,i x n m,i ) U (x n m,i ) = t n m,i u m r U t n m,i u m u m x n m,i U (u m x n m,i ) u r m U (u m x n m,i ) U (x n m,i ) ≥ ǫ m c d r M c M d , implying t r n m,i U (t n m,i x n m,i ) U (x n m,i ) → ∞ as m → ∞, which contradicts
• Proof of (iii)

Let U ∈ M ∞ and let r ∈ R. Applying Theorem 1, (ii), there exists x 0 > 1 such that, for

x ≥ x 0 , lim t →∞ t r U (t x) U (x) = 0. Let x 0 ≤ c < d < ∞.
On the one hand, by hypothesis, there exists a constant M d > 0 such that, for x ∈

U (t x) U (x) = 0.
• Proof of (iv)

Let U ∈ M -∞ and let r ∈ R. Applying Theorem 1, (iii), there exists x 0 > 1 such that, for

x ≥ x 0 , lim t →∞ t r U (t x) U (x) = ∞. Let x 0 ≤ c < d < ∞.
On the one hand, by hypothesis, there exists a constant M d > 0 such that, for x ∈

U (t x) U (x) = ∞.

Proof of Theorem CK.

Let U : R + → R + be a measurable function.

• Proof of (i) ⇒ (ii)

Let ǫ > 0 and U ∈ M with ρ U = τ. One has, by definition, that lim x→∞ U (x)

x ρ+ǫ = 0 and lim

x→∞ U (x) x ρ-ǫ = ∞.
Hence, there exists x 0 ≥ 1 such that, for x ≥ x 0 ,

U (x) ≤ ǫx τ+ǫ and U (x) ≥ 1 ǫ x τ-ǫ .
Applying the logarithm function to these inequalities and dividing them by log(x) (with

x
  γ(x) + x b log(U (s)) log(s) d s s log(x)   = lim x→∞ log(U (x)) log(x) = τ First, suppose τ = 0, then lim x→∞ log(U (x)) γ(x) log(x) + x b log(U (s)) log(s) d s s = 1,
and there exists x 0 > 1 such that, for x ≥ x 0 ,

δ U (x) := log(U (x)) γ(x) log(x) + x b log(U (s)) d s s ≥ 1 -ǫ > 0.
Setting x 1 := max(b, x 0 ) and defining the functions, for x ≥ x 1 , α U (x) := γ(x)δ U (x) log(x) and β U (x) := log(U (x)) log(x), the assertion follows. Now, suppose τ = 0. Define the function V (x) := xU (x), x > 0, which clearly satisfies lim x→∞ log(V (x)) log(x) = 1 = 0. Hence, applying to V the previous proof for U when τ = 0 gives that there exist x 1,V ≥ b V > 1 and measurable functions α V , β V , and δ V satisfying, as

x → ∞, α V (x) log(x) → 0, β V (x) → 1, δ V (x) → 1, such that, for x ≥ x 1,V , V (x) = exp α V (x) + δ V (x) x b V β V (s) d s s .
Defining, when τ = 0, the constant x 1,U := x 1,V and the functions α

U (x) := α V (x) + log(x) δ V (x) -1 , β U (x) := β V (x) - 1
, and δ U (x) := δ V (x), the assertion follows.

• Proof of (iii) ⇒ (i)

Suppose there exist b > 1 and measurable functions α, β, and δ satisfying, as x → ∞,

α(x) log(x) → 0, β(x) → τ, δ(x) → 1, such that U (x) = exp α(x) + δ(x) x b β(s) d s s , x ≥ x 1 for some x 1 ≥ b.
Let ǫ > 0 sufficiently small such that 2ǫ τ+ǫ 4 ≤ 1 and 2ǫ τ-ǫ 4 ≥ -1. Then there exist x a > 1 such that, for x ≥ x a , α(x) log(x) ≤ ǫ 4, x b > 1 such that, for x ≥ x b , β(x) -τ ≤ ǫ 4, and x c > 1 such that, for x ≥ x c , δ(x) -1 ≤ ǫ 2 /4.

On the one hand, writing, for x ≥ x 0 := max(b, x a , x b , x c ),

U (x) x τ+ǫ = exp -(τ + ǫ) log(x) + α(x) + δ(x) x b β(s) d s s = exp log(x) α(x) log(x) - ǫ 2 + δ(x) x b β(s) d s s -τ + ǫ 2 log(x) ≤ exp - ǫ 4 log(x) + δ(x) x 0 b β(s) d s s + δ(x) τ + ǫ 4 log(x) -log(x 0 ) -τ + ǫ 2 log(x)
and noting that

δ(x) τ + ǫ 4 -τ + ǫ 2 = δ(x) -1 τ + ǫ 4 - ǫ 4 ≤ - ǫ 8 give lim x→∞ U (x) x τ+ǫ = 0. ( 13
)
On the one hand, writing, for x ≥ x 0 := max(b, x a , x b , x c ),

U (x) x τ-ǫ = exp -(τ -ǫ) log(x) + α(x) + δ(x)
Combining ( 13) and ( 14) provides U ∈ M with ρ U = τ.

Proof of Proposition 4.

Let U : R + → R + be a measurable function.

Assume U ∈ O-RV and the limit lim Proof of Proposition 5. We will prove the proposition for λ = ∞. The proof for λ = -∞ is similar.

Let us prove it by contradiction. Assume there exists U ∈ M ∞ O-RV .

By assumption U ∈ M , we have, for ρ ∈ R and δ > 0, there exists x 0 > 1 such that, for x ≥ x 0 , U (x) ≤ cx ρ . Applying the logarithm function to this inequality, dividing it by log(x), x > 1, and taking the limit c > 0 such that, for all t ≥ 1 and x ≥ x 1 ,

c -1 t β ≤ U (t x) U (x) ≤ ct α .
Hence applying to these inequalities the logarithm function, dividing them by log(t ), t > 0, and taking the limit t → ∞ give lim t →∞ log(U (t )) log(t ) ≤ max |α|, |β| < ∞, which contradicts (15). The proposition is proved.

Conclusion

A new characterization of the class M introduced in [START_REF] Cadena | An extension of the class of regularly varying functions[END_REF], a strict larger class than the class of regularly varying functions (RV), was proved, and it was extended to the classes M ∞ and M -∞ . This characterization together with other two given by Cadena and Kratz in [START_REF] Cadena | An extension of the class of regularly varying functions[END_REF] allowed the study of relationships between M and the well-known class O-RV, another extension of RV. It was found that these classes satisfy M ⊆ O-RV and O-RV ⊆ M , and necessary conditions to have inclusions were provided. Relationships among O-RV and M ∞ and M -∞ were provided.

Note that any result obtained here can be applied to positive and measurable functions with finite support by using the change of variable y = 1 (x * U -x) for x < x * U where x * U is the endpoint of U defined by x * U := sup x : U (x) > 0 .

1 φ

 1 does not exist. Note that if this limit exists, then, applying Theorem CK, (iii), U ∈ M . Example 3. O-RV ⊆ M . Let U : R + → R + be a measurable function satisfying, for x ≥ 1, U (x) = exp x function φ has support [1; ∞) and is defined by φ(x) = 0 if x ∈ [1; e) or x ∈ I n with n odd 1 if x ∈ I n with n even, where I n = [e e n ; e e n+1 ), n ∈ N. On the one hand, applying Proposition 3, one has U ∈ O-RV. On the other hand, writing, for x > 1, using the change of variable y = log(s) log(x), log(x) d y gives, taking x n = e e n , n = 2, 3, . . ., log(U (x n )) log(x n ) = n-1 k=1 e k+1 /e n e k /e n φ e y e n d y =

Proposition 4 .

 4 and thus, applying Theorem CK, U ∈ M . Now we give another relationship between O-RV and M . Let U : R + → R + be a measurable function. If U ∈ O-RV and the limit lim x→∞ log(U (x)) log(x) exists, then U ∈ M .

[ 1 ;

 1 d], U (x) ≥ M d . On the other hand, by Proposition 1, (ii), there exists a constant M c > 0 such that, for x ∈ [c; ∞), U (x) ≤ M c . Combining these inequalities gives, for x ∈ [c; d], M d ≤ U (x) ≤ M c . Hence a proof similar to the one given to prove (i) can be done to conclude that lim x→∞ t r sup x∈[c;d ]

[ 1 ;

 1 d], U (x) ≤ M d . On the other hand, by Proposition 1, (iii), there exists a constant M c > 0 such that, for x ∈ [c; (x) ≥ M c . Combining these inequalities gives, for x ∈ [c; d], M c ≤ U (x) ≤ M d . Hence a proof similar to the one given to prove (ii) can be done to conclude that lim x→∞ t r inf x∈[c;d ]

  x τ-ǫ = ∞.

  Theorem CK gives U ∈ M with ρ U = lim x→∞ log(U (x)) log(x) .

  assumption U ∈ O-RV , applying Proposition 3, (i) ⇒ (ii), there exist α, β ∈ R and x 1 > 1,

Acknowledgments

The author gratefully acknowledges the support of SWISS LIFE through its ESSEC research program on 'Consequences of the population ageing on the insurances loss'.


Bernard Kamsu-Foguem
email: bernard.kamsu-foguem@enit.fr

Fonbeyin Henry Abanda

Experience modeling with graphs encoded knowledge for construction industry

Introduction

In the past, there has been no structured approach to learn from construction projects once they have been completed. Nowadays, the construction industry is adopting a sustainable strategy by using knowledge management concepts and techniques to improve the organization of knowledge from completed projects [START_REF] Forcada | Knowledge management perceptions in construction and design companies[END_REF]. Knowledge can be seen as the means to extract and capture the available information which can be used to provide better and efficient ways of improving processes, producing and/or delivering construction products (e.g., buildings). For instance, experiences that relate lessons learned of how well a product could be produced, maintained, used and so forth [START_REF] Goulding | Construction industry offsite production: a virtual reality interactive training environment prototype[END_REF]. Knowledge-based quality improvement involves experience feedback methodology that can be applied with three key methodological elements: (a) analysis with a set of tools; (b) a stepwise problem-solving method and (c) using common metrics [START_REF] Choo | Method and context perspectives on learning and knowledge creation in quality management[END_REF]. With regards to the first, recommended tools include statistical techniques (e.g., design of experiments and regression analysis), and non-statistical tools (e.g., flow charts and fishbone diagrams, or both). A stepwise problem-solving offers a systematic analysis and impacts learning and knowledge creation and consists of the following steps: problem identification, diagnosis, solution generation and implementation. Furthermore, stepwise problem-solving technique provides models for problems visualization and systemic understanding of their main characteristics and the reasoning options available. The interaction between continuous improvements, experience feedback processes and stepwise problem-solving procedures leads to knowledge creation from structured learning processes [START_REF] Zhang | Developing a knowledge management system for improved value engineering practices in the construction industry[END_REF]. Common metrics aid coordinate knowledge creation efforts by incorporating and aligning The Architecture, Engineering and Construction (AEC) industry is becoming increasingly knowledge intensive. Knowledge management has been hailed as an enabler for tapping this knowledge to improve efficiency in the AEC industry. In this study, the main concepts and benefits of knowledge management, relationships between knowledge management and continuous improvement have been examined. Furthermore, emphasis has been laid on knowledge management through experience feedback processes which constitute valuable assets for the AEC industry. This is done through the employment of ontologies and graph-based reasoning operations in eliciting and visualizing knowledge concepts in the AEC domain. The proposed approach which consists of two main aspects provides the opportunity to examine the conceptual and ontological knowledge models with associated reasoning. In the first, the concepts of knowledge management, their significance and application in the AEC field are discussed. The second deals with a methodological framework for the modeling and reasoning in the domain ontology. To facilitate automation, an ontology graph-based editor, Conceptual Graphs User Interface (CoGui) was used to model knowledge and encoded reasoning in the knowledge base. CoGui encodes knowledge as conceptual graphs and reasoning as graph operations that can be visualized in a logically precise way, based on domain ontologies. It emerged that CoGui could be very useful in acquiring information that can be used in collaboration with others to continuously improve information sharing and re-use. A French AEC company located in the Southwest region was employed as a case study in illustrating knowledge modeling through the experience feedback process. For each phase of experience feedback, actions that were implemented in the company are discussed. the problem-solving process aim at improving interaction and offer pragmatic solutions in work related situations. Despite the merits of the afore-mentioned experience feedback methodologies, common major challenges related to computing in civil and building engineering still exist. Tizani and Mawdesley [START_REF] Tizani | Advances and challenges in computing in civil and building engineering[END_REF] reported that (a) enhancing digital information modeling; (b) extending process modeling technologies; (c) improving decision-implication analysis, (d) integrating broader aspects of experience feedback processes are still quite common. Furthermore, in the field of civil engineering, studies about experience feedback tend to focus on structural aspects with little interest in preliminary stages of facilitating works. In France, experience feedback is more common in the field of industrial engineering than in AEC. The main reason is the automation of production systems in industrial engineering, compounded by the fact that industrial products often have similar characteristics.

It has been hailed that solutions for most knowledge management challenges lie in information and communication technologies (ICT). The increasing availability and quality of ICT; for example, the internet, underlying communication and emerging technologies are transforming the way people share knowledge and ideas. The ease of use and characteristics of these technologies have made them very popular in a very short period of time. However; compared to other industries, the AEC industry lags behind with regards to the successful implementation of ICT and in tapping the potential of the new and/or innovative technologies to improving productivity, achieving greater efficiency and higher quality [START_REF] Moum | What did you learn from practice today? Exploring experiences from a Danish R&D effort in digital construction[END_REF]. Thus, knowledge of products, services and/or processes from past AEC projects are hardly fully exploited for use in other projects leading to information duplications, inconsistencies and inefficiencies in delivering such projects. This study aims to employ an ontology-driven approach to develop a knowledge management model for construction projects. The model will serve as a knowledge-based decision support system. To facilitate understanding, the paper is divided into four sections. Section 2 provides a background of the state-of-the-art about knowledge management applications in the construction industry. Section 3 presents an ontology-driven approach for knowledge modeling in the construction AEC industry. In Section 4, an illustrative case study is employed to depict the methodology and concepts of experience feedback. The discussion and analysis of major issues covered in this manuscript are discussed in Section 5. The conclusion is presented in Section 6 by a way of summary of the key parts of the paper.

State-of-the-art

Generations of knowledge management applications in the AEC industry

Knowledge management is recognized as a key resource and a crucial enabler for continuous process improvement in construction projects. It has been widely acknowledged that one of the key sustainable advantages that a firm possesses comes from what it collectively knows, how efficiently it uses what it knows and how readily it acquires and uses new knowledge. Organizational knowledge could reduce the time spent on problem-solving and increases the quality of work. Recent developments in ICT have prompted organizations to utilize platforms such as corporate intranets and collaborative extranets for collaborative knowledge sharing. Companies are also implementing extranets to share information of experience feedback scenarios expressed for explicit knowledge capitalization and exploitation. In these situations, experience feedback is useful in terms of how well the organization works in many aspects, from learning knowledge from experience feedback information and using it in improving management strategies. The preceding statement underscores the difference between information and knowledge management. The right knowledge management adoption strategy should be put in place to develop and nurture core organizational competencies, and create intellectual capital by tapping or capturing experience feedback information. It has been argued that a true value creation culture can be found through the right combination of human networks, social, intangible and technology assets where issues such as change management, learning and trust must be blended successfully towards the vision of knowledge-enabled value creation. There has been an evolution of knowledge management over time [START_REF] Rezgui | Generations of knowledge management in the architecture, engineering and construction industry: an evolutionary perspective[END_REF]. The differences of the evolution are examined in Table 1.

In certain circumstances (e.g., the scarcity of specific data), it is not possible to move towards reliable analytical or statistical approaches [START_REF] Juha ´sova ´ | Stone masonry in historical buildings-Ways to increase their resistance and durability[END_REF][START_REF] Kovler | Properties of fresh and hardened concrete[END_REF] and experience feedback approach is then adopted as a more suitable alternative. However, Aamodt and Plaza [START_REF] Aamodt | Case-based reasoning: foundational issues, methodological variations, and system approaches[END_REF] recommend that there must be a reasonable number of experienced cases that are sufficiently similar to be grouped together for potential application of the reasoning process over its underlying knowledge for use in new cases or projects. This underpins the concept of ''experienced feedback''.

Knowledge management through experience feedback

The usefulness of information structuring and knowledge capitalization from experience feedback in solving engineering problems have been examined in Gardoni et al. [START_REF] Gardoni | Harnessing non-structured information and knowledge and know-how capitalisation in integrated engineering: case study at Aerospatiale Matra[END_REF][START_REF] Gardoni | Knowledge capitalisation based on textual and graphical semi-structured and non-structured information: case study in an industrial research centre at EADS[END_REF]. The experience feedback process consists of three essential steps (see Table 2): (i) the capitalization phase, (ii) the treatment phase, (iii) and the exploitation phase. In Table 2, the first two stages consist of capitalization and a synthesis of different stages of treatment in experience feedback. The third is about the exploitation phase of feedback for fault diagnosis systems, safety assessment (levels of protection), and forecasting changes in equipment over time.

In order to generate knowledge from experience feedback, a conceptual model [START_REF] Kamsu-Foguem | Knowledge formalization in experience feedback processes: an ontology-based approach[END_REF] incorporating problem-solving to generate explicit knowledge from experiences is presented in Fig. 1. In the AEC industry, the appropriate deployment of information structuring strategies with semantic knowledge can lead to efficient management of the key variables or factors that influence the built environment. For example, the use of semantic web and linked data to structure information for decision-making about energy, greenhouse gases, water or poor health have been reported in [START_REF] Abanda | Trends in built environment semantic Web applications: where are we today?[END_REF]. There are lessons to be learned from experiences in diverse contextual situations. An experienced actor could explore the knowledge in abundance covering different aspects of building construction to provide important work examples that is effective and instructive in construction activities. This approach involves the use of specialist information or knowledge from available records or sources where experts will exploit for use in their various applications such as diagnosis [START_REF] Chang | Knowledge-based navigation system for building health diagnosis[END_REF], risk analysis [START_REF] Liu | A fuzzy risk assessment approach for occupational hazards in the construction industry[END_REF], safety assessment and security [START_REF] Theodossopoulos | A review of analytical methods in the current design processes and assessment of performance of masonry structures[END_REF]. In fact, the conceptualization of the shared representations significantly influences the manner in which the collaborative work takes place, as evidenced by their contribution in the exchange of information and generated knowledge. Ontology provides a better way to explicitly specify concepts of any domain [START_REF] Gruber | A translation approach to portable ontology specifications?[END_REF]. Therefore, in order to gain a better understanding of domain and to be able to make comprehensive analysis of the experience feedback processes in the AEC domain, an ontologydriven approach is adopted. Furthermore, ontologies provide means to formalize domain knowledge in a way that makes it accessible, shareable, reusable and machine processable.

Ontology-driven approach for knowledge modeling in AEC

In this section, an ontology-driven approach for knowledge modeling in AEC is proposed. This approach provides opportunities for sustained continuous improvement and re-use of knowledge in AEC industry. In software engineering the task of rendering information to an end-user through the implementation of any computer system often involves three separate layers: the presentation, the application or business logic and data (data management processes) layers (see Fig. 2). This is often called the hybrid system, client-server architecture or 3-tier architecture. Access to the presentation layer is often through the use of a graphical user interface (GUI) or a web browser. As argued by [START_REF] Alonso | Web Services Concepts, Architectures and Applications[END_REF], the implementation of the 3-tier model in its full generality and the invocation of access via the internet or web browser is called an Ntier model. The N-tier model has the following advantages: adaptability, encapsulation, re-use and quality. The implementation will be performed according to each technology involved in a given layer. In order to demonstrate how each layer feeds the other, the data layer implementation will be conducted before the application layer. It is important to note that within the scope of this study, the focus will only be on the data and business logic layers. The development of a user-friendly interface (i.e. the presentation layer) is reserved as part of future study.

With regards to the data layer, the main technology in it is the CoGui graphical knowledge representation model which can store information about any domain of discourse [START_REF] Sowa | Conceptual Structures: Information Processing in Mind and Machine[END_REF]. The data captured in CoGui is from external documents and storage systems which could be non-structured, semi-structured or structured. Once edited into CoGui, the CoGui knowledge model can be converted into different file/data model formats (e.g., Resource Description Framework (RDF)) for other uses such as in application development. The model can also be enriched with conceptual graph rules and constraints to capture real life situations [START_REF] Kamsu-Foguem | Conceptual graph-based knowledge representation for supporting reasoning in African traditional medicine[END_REF][START_REF] Kamsu-Foguem | Ontological view in telemedicine[END_REF][START_REF] Kamsu-Foguem | Verifying a medical protocol with temporal graphs: The case of a nosocomial disease[END_REF][START_REF] Kamsu-Foguem | Conceptual graph operations for formal visual reasoning in the medical domain[END_REF][START_REF] Kamsu-Foguem | Graph-based reasoning in collaborative knowledge management for industrial maintenance[END_REF][START_REF] Kamsu-Foguem | Using conceptual graphs for clinical guidelines representation and knowledge visualization[END_REF][START_REF] Kamsu-Foguem | Knowledge description for the suitability requirements of different geographical regions for growing wine[END_REF][START_REF] Kamsu-Foguem | Requirements modelling and formal analysis using graph operations[END_REF][START_REF] Ruiz | Knowledge reuse integrating the collaboration from experts in industrial maintenance management[END_REF][START_REF] Ruiz | Generating knowledge in maintenance from Experience Feedback[END_REF]. The rules and constraints are based on Concepts Types, Relationship Types and Individuals. With regards to the business logic layer, its main purpose is to extract relevant parts from the CoGui experience feedback knowledge base, stored in the form of a CogXML file format. This data is obtained from the data layer. It contains the conceptual graphs base ''CG base'' that communicates with the ''Conceptual Graph Manager''. The ''Projection Engine'' is a reasoning engine based on the semantic mapping between conceptual graphs involving both adjacent concepts and relations. The constraint checking is a procedure for verification of the essential requirements and expectations in the CG base. The ''Inference Engine'' is a reasoning and deductive process applying rules (if their conditions are true) to the available facts in order to generate new conclusions. In fact, the rules are applied only when required conditions are in compliance with considered facts and/or available hypotheses. The crucial aspect of the architecture presented in Fig. 2 is the ontology development using CoGui. All what is required to be undertaken in the data and business logic layers can be undertaken within the CoGui environment. In the ensuing sections, the ontology development procedures using CoGui will be examined.

Graph-based ontology development technique

To develop the ontology, a graph-based technique has been used to represent knowledge about the domain of interest. This technique enables real-time changes to be made to find faster and clearer solutions for different applications [START_REF] Dieng | Conceptual graphs for semantic web applications[END_REF]. For any knowledge representation, it is important to establish the requirements to be fulfilled by the application for which the knowledge model is being developed for. To this end, the following requirements have been set:

A wider application field to allow the satisfaction of certain needs of expanding the ground in which knowledge can be shared. This constitutes the first big step for having a properly positioned experience feedback process in a continuous improvement environment. A formal conceptualization of a collaborative work system should be considered for purposes of shared representations in supporting the process of collective knowledge creation, evaluation and learning from experiences.

A simple and clear procedure to manage and exploit the appropriate information. It is highly advisable to go for the less crowded and the most explicit way of data represention and visualization A feasible implementation of the solution to reality: This implies that the solution should bring more benefits than problems. It is known that the full exploitation requires a period of fine tuning in terms of use and operational application.

The AEC industry is noted for generating huge amount of information from construction projects. For this information to be effectively shared, a well-organized method is required to minimize ambiguity. Hence, the rationale for adopting an ontology approach to disambiguate inconsistencies and facilitate understanding and reasoning. As explained earlier, an ontology formally represents knowledge as a set of concepts within a domain, and the relationships between pairs of concepts. It can be used to model a domain and support reasoning about entities. From this perspective, ontology provides a viable and efficient technique of sharing experience feedback knowledge. As argued in [START_REF] Sowa | Knowledge Representation: Logical, Philosophical, and Computational Foundations[END_REF], one of the best ways to employ ontology to capture the above requirements is to adopt conceptual graph representation. To this end, CoGui, a graphical and development ontology editor is used in modeling a real world application of experience feedback in the construction domain.

Knowledge modelling with conceptual graphs

Conceptual Graphs user interface (CoGui1 ) is a free graph-based visual tool, developed in Java, for building conceptual graph knowledge bases represented in CogXML format, compatible with Cogitant [START_REF] Baget | Logical, graph based knowledge representation with CoGui[END_REF]. CoGui makes it possible to create, edit and control the structure and content, of a knowledge base. The knowledge base can be serialized in the XML format called CogXML. Such a knowledge base is composed of an ontological knowledge, a factual knowledge represented by a set of conceptual graphs and procedural knowledge described by conceptual graph rules (see Fig. 3). The conceptual graph can be simple or nested for embedded descriptions of knowledge representation.

Ontology representation

According to [START_REF] Chein | Graph-based Knowledge Representation: Computational Foundations of Conceptual Graphs[END_REF], the basic components of an ontology is the vocabulary, which consists of: A hierarchy of concept types (also called concepts), which represents the kinds of entities in the application domain. These concepts could be defined to be disjoint or to have an intersection; A hierarchy of relation types (also called relations), which represents the kinds of relationships between entities, and may have any number of arguments; A set of individuals (or individual names) and graphs associated with individuals, which represent knowledge about these individuals; A set of conceptual graph rules takes the form of an implication between an antecedent (body) and a consequent (head). The intended meaning is read as: whenever the conditions specified in the antecedent holds, then the conditions in the consequent must also hold.

Conceptual graphs formalism

Conceptual graphs were introduced by Sowa as a diagrammatic system of logic with the purpose ''to express meaning in a form that is logically precise, human readable and computationally tractable'' [START_REF] Sowa | Conceptual graphs for a database interface[END_REF]. Conceptual graphs encode knowledge as graphs and thus can be visualized in a natural way [START_REF] Sowa | Knowledge Representation: Logical, Philosophical, and Computational Foundations[END_REF]:

The vocabulary, which can be seen as a basic ontology, is made of hierarchies of concepts and relations. All other kinds of knowledge are based on the representation of entities and their relationships. This representation is encoded by a labeled graph, with two kinds of nodes, respectively corresponding to entities and relations. Edges link an entity node to a relation node. A rule expresses implicit knowledge of the form: ''if hypothesis, then conclusion'', where the hypothesis and conclusion are both basic graphs. Using such a rule consists of adding the conclusion graph (to some fact) when the hypothesis graph is present. There is a one-to-one correspondence between some concept nodes of the hypothesis with concept nodes of the conclusion. Two nodes in correspondence refer to the same entity. These nodes are said to be connection nodes. The knowledge encoded in rules can be made explicit by applying the rules to specified facts.

Having discussed the graph-based ontology development process in Section 3, in Section 4, the steps will now be applied to a case study.

An illustrative modeling case in the French context

A French AEC company located in the southwest region has been used as a case study to illustrate knowledge modeling of the experience feedback process. This company is active in two sectors: public works and building. The main activities often undertaken by the company are public works (e.g., roads, railroads, bridges, water supply, sewage and electrical grid) and buildings (e.g., universities and hospitals). This study focused on facilitating works as defined by the UK New Rules of Measurement (NRM) [START_REF]NRM1. Order of cost estimating and cost planning for capital building works[END_REF]. The facilitating works were related to six tramway projects being constructed by company X (for data protection purposes, X is assumed to be the name of the company) in the Southeast region of France. Facilitating works include specialist works which normally need to be completed before any building works can commence (e.g., major demolition works, soil stabilization works and/or temporary diversion of mains drainage). The key challenges associated with construction of tramway platforms were the quality and safety needs, the regulatory constraints and minimum traffic disruptions. The rationale for choosing the UK NRM is because of its wellestablished structure and popularity of its use in other countries. It provides definitions for different work elements in a construction project than most standard measurement methods. By adopting the NRM standards, it is easier to map concepts elicited from feedback processes. In the adopted modeling approach, an individual (e.g., experience feedback manager) in the company responsible for experience feedback treatment undertook the following tasks discussed in the ensuing paragraphs.

Firstly, information from company's project are gathered or elicited. This can be achieved in two ways [START_REF] Wenger | Cultivating Communities of Practice[END_REF]. The experience feedback manager completes a record of information gathered from the treatment of experienced cases, and then conducts interviews with key actors of projects having potential interesting lessons that can be learned. Based on these, the feedback manager established records of collected information in order to generate useful knowledge. Secondly, the structured or semi-structured forms of experience feedback activities prepared by the feedback manager which can also be obtained and prepared by any actors through interviews are saved in a database. This is stored for purposes of knowledge reuse in AEC [START_REF] Lin | Enabling the creation of domain-specific reference collections to support textbased information retrieval experiments in the architecture, engineering and construction industries[END_REF]. After completing this collection of information, it is analyzed to determine the type, richness and its usefulness. The identification of different types of information depends on the associated knowledge and also on their importance. Examples of categories include information gathered associated with onsite practical problems (e.g., lack of safety on site) and simple information that can be useful afterwards (e.g., use of a new material). With regards to the tramway projects, some actions were undertaken to elicit knowledge about experience feedback processes. The actions implemented are summarized in Table 3.

On the one hand, in the engineering phase of a construction of the tramway projects the capitalization concerns data collection from specific projects related to design phase (new materials, used efficiency coefficients, particular production mechanisms.). The treatment is about the different ways used to address the facilitating work challenges and the search for reasonable solutions. The exploitation defines the means within for information retrieval and the dissemination of the interesting results of lessons learnt. On the other hand, in the engineering phase of a construction project the capitalization is for the execution and coordination of the studies by the architectural and engineering team and for the actual works. The treatment concerns the actions undertaken in finding solutions to recurrent problems and more feasible alternatives. The exploitation of feedback is achieved through meetings with the aim of preventing problems, or to deliver solutions.

Upon collecting the information as described in Table 3, the following key ontological concepts elicited were developed and will be discussed in the ensuing sections.

Concept types

Based on the definition of the facilitating works [START_REF]NRM1. Order of cost estimating and cost planning for capital building works[END_REF], concepts or classes with their respective sub-concepts were abstracted and modeled in CoGui (see Fig. 4).

Based on Fig. 4, the first levels in the hierarchy are Toxic/ Hazardous/contaminated material treatment, Major demolition works, Temporary support to adjacent structures, Specialist groundworks, Temporary diversion works and Extra-ordinary site investigation works. These top level concepts also contain subconcepts. Some relations may be established between the concepts and used for the modeling of factual and procedural knowledge in conceptual graphs. This can facilitate automated reasoning in experience feedback processes.

Relation types

Fig. 5 depicts the relationships (Logical operators, Comparison operators, Usual relations and Temporal relations) between concepts, including their respective branches created. Logical operators (such as AND, NOT, OR and XOR) are used in combining two or more relational expressions into a single expression that evaluates to either true or false. Comparison operators (Equal, Less than and Greater than) can be used to compare two concepts with the logical true and false results. Usual relations (such as Agent, Attribute and Object) refer to the construction of sentences in terms of subject, verb and complement in the common language with active and passive components. Temporal relations (such as After, before and during) are used for the temporal descriptions of events.

Facts and rule applications

After creating the concept and relation types, the relevant constraints and rules are created to facilitate understanding, only rules related to four sub-elements of the third level will be examined. The elements are ''Toxic or hazardous material removal'', ''Demolition works'', ''Soil stabilization measures'' and ''Archaeological investigation''. Toxic or hazardous material removal is a concept describing the removal, employing special safety measures, of toxic or hazardous material prior to demolition or refurbishment works. Fig. 6 represents the modeling of an associated rule in the conceptual graph formalism. The rule in Fig. 6 means, if a certain project has components (object) with a toxicity value greater than a given threshold level then the object should be detoxified.

Logical expression: 9x 9y 9z (Project (x) ^ Toxicity (y) ^ Upper Threshold (z) ^ Attribute (x, y) ^ Superior (y, z) ! 9u (Project

(x) ^ Detoxification (u) ^ Object (u, x))
Soil stabilization measures is a concept describing the stabilization or improvement of soil bearing capacity or slip resistance of existing ground to facilitate construction by injecting or otherwise

Table 3

Experience feedback activities conducted in the case study.

Execution phases of a construction project.

Type of experience feedback

Activities conducted

Engineering

Capitalization Data collection from specific projects related to design phase (new materials, used efficiency coefficients, particular production mechanisms.) Interview with site workers, foremen, site supervisors and construction managers.

Treatment

The person is seeking alone some solutions to the problems during its working time. Talks with others affected by the problem for collaborative problem solving. Exploitation Ascertained and recorded solution. Solution transmitted (orally, by mail, by phone . . .) to individuals involved in the study phase.

Construction

Capitalization Collected data: Passed from the foreman to site supervisors (security issues, problems on a machine, a material used, bad equipment . . .).

Amendments made during quality/safety meetings. Execution and coordination of the studies by the architectural and engineering team and for the actual works.

Treatment

The actors are trying to find solutions to recurrent problems Exploitation Memorandum, diagrams, or any other practical documents. Safety Data Sheet. Meetings to prevent existing problems, or to deliver solutions. Quality/security manager of the considered area informed of the progress (by e-mail, phone . . .).

introducing stabilising materials, by power vibration, by soil nailing or by ground anchors. Fig. 7 represents the modeling of an associated rule in the conceptual graph formalism. The rule in Fig. 7 means that if the value of the soil physical stability where the project is being undertaken is less than a given threshold value, a soil stabilization measure should be performed which consists of improving the soil bearing capacity and/or slip resistance of project. Physical archaeological investigation works are site based excavation works in search of artefacts and the like carried out by the main contractor under the guidance and instruction of an archaeologist. Fig. 8 represents the modeling of an associated rule in the conceptual graph formalism. The rule in Fig. 8 means that if a project is located on an archaeological grounds, then excavations works and the search of archaeological artefacts should be undertaken.

Logical expression

:: 9x 9y (Project (x) ^ Archaeological investi- gation (y) ^ Element (x, y) ! 9u 9v (Project (x) ^ Excavation works (u) ^ Search of artefacts (v) ^ Element (x, u) ^ Object (u, v))
Demolition works concern the taking down to ground level and removing complete buildings/structures or parts of buildings/ structures, including services, fittings and finishes. Fig. 9 represents the modeling of an associated rule in the conceptual graph formalism. The rule in Fig. 9 means that if major demolition works are to be undertaken on a project's site, all services' networks should be disconnected.

Logical expression: 9x 9y (Project (x) ^ Major demolition works (y) ^ Element (x, y) ! 9u (Project (x) ^ Network disconnection (u) ^ Element (u, v) ^ Mains Services ({water, gas, electrici-ty}) ^ Object (u, {water, gas, electricity}))

Based on facts presented in Figs. 6 and7, an illustration of the general principle regarding the application of any rule will be examined. This is presented in Fig. 10, where the selection of a fact activates an applicable rule. The application of the two previous facts and rules concerns the facilitating works for the construction of a tramway management platform in the outskirts of Toulouse.

The platform provides services supporting the coordination and the monitoring of tramway vehicles serving the community including the remote parts.

The rules 6-9 are implemented in CoGui. After the fact and rules are selected, the simulation is launched. The simulation results will serve as a basis to highlight issues that need to be addressed.

As an example, the right of Fig. 10 is the output of the execution of the facts on the left of the same rule. It is important to note that the visibility of the simulated results in CoGui was very poor, and could not be read when the screen was snapped. Consequently, the output on the right of Fig. 10 is coloured green just to indicate that it is an actual output from CoGui.

Discussions

In the case study examined, a key observation that can be made about the procedure is the fact that challenges associated with the facilitating works can be modeled using conceptual graphs and enriched with rules to facilitate reasoning. By developing an ontology knowledge base that includes feedback processes, it is possible to re-use in future in solving a large majority of problematic situations. An example of a practical problem solved is the decision to stabilize the soil if the soil bearing capacity is lower than a given threshold value (Fig. 7). Reasoning in the knowledge based is facilitated by both the Projection and Inference engines in CoGui. In addition to strengths of these engines, CoGui conceptual graph-based structure makes it easy to organize and sort information that can be used in corporations for continuous improvement in construction projects. In France, the application of experience feedback in improving organizations' efficiency is more common in the field of industrial engineering than in AEC sector. The main reason is because of the automation of production systems in industrial engineering compounded by the fact that industrial products often have similar characteristics. Thus, it is not uncommon to find experience feedback applications in industrial repairs and maintenance management ( [START_REF] Kamsu-Foguem | Knowledge-based support in non-destructive testing for health monitoring of aircraft structures[END_REF], [START_REF] Kamsu-Foguem | Graph-based reasoning in collaborative knowledge management for industrial maintenance[END_REF]; Potes [START_REF] Ruiz | Knowledge reuse integrating the collaboration from experts in industrial maintenance management[END_REF]). While experience feedback process can easily be applied in industrial production/engineering, because most of its tasks/ products are often repetitive and standardized, its application in AEC is very challenging. This is because each AEC project is often unique in nature. With regards to renovation/refurbishment projects, the challenge is even enormous as each existing building structure has different characteristics and different levels of defects requiring different appraisal strategy or surveying techniques before recommendation for repairs and maintenance. That notwithstanding, and given the knowledge intensive nature of AEC projects, developing an ontology knowledge base about experience feedback processes provides an opportunity to structure knowledge that can be re-used in solving problems on different projects.

Conclusion and related works

The rationale for this study stemmed from the fact that knowledge is an important asset of an organization including AEC companies; many AEC companies are yet to tap on the full potential of knowledge from already executed projects. Companies are aware of situations in their organizations in which costly errors have been made because knowledge was not available when and where it was needed and because employees did not know how to interpret or use the information available to them. These situations are mainly due to the project-based nature of the construction industry and the fact that knowledge is embedded in social relations. Nevertheless, for those big companies who might be carrying out international projects, strategies to mobilize knowledge is critically important. In construction organizations, time is often associated with the need to deliver projects according to schedule. Many construction organizations believe that their organizational structure requires heavy procedures and long processing times to exploit domain knowledge. Employees may be willing to share knowledge, but the pressure to deliver projects under tight schedule and budget makes them have very low appetite to take on additional responsibility for knowledge management activities. In this manuscript, it was revealed that useful knowledge on AEC projects and businesses or organizations can be helpful in resolving persistent, complex and workload problems. Eliciting experience feedback knowledge is core to re-using existing knowledge that can improve organizational efficiency in product (e.g., buildings) delivery. This is enshrined in the methodology for knowledge capitalization from semi-structured information and unstructured information in industrial engineering domain [START_REF] Gardoni | Harnessing non-structured information and knowledge and know-how capitalisation in integrated engineering: case study at Aerospatiale Matra[END_REF][START_REF] Gardoni | Knowledge capitalisation based on textual and graphical semi-structured and non-structured information: case study in an industrial research centre at EADS[END_REF]. Graphical conceptual modeling is very powerful as it allows lessons learned to be captured and represented textually and graphically, facilitating visualization. This improves communication to users who may be interested in consuming knowledge about feedback from previous projects [START_REF] Silvestre | Ceramic tiling in building fac ¸ades: inspection and pathological characterization using an expert system[END_REF]. In this study, we adopted the N-tier model to develop the ontology knowledge base containing experience feedback knowledge from tramway projects constructed in the Southeast region of France. The focused of the paper was on the data and business logic layers. As part of future study, the development of a user-friendly interface will be investigated. It is also important to note that, although the ontology developed is based on tramway projects in Southeast France, the concepts and rules are generic and can be re-used on any other tramway projects in any country. Such a move should only focus on facilitating works.

Fig. 1 .

 1 Fig. 1. Experience model incorporating problem-solving technique (Kamsu Foguem et al. [15]).

Fig. 2 .

 2 Fig. 2. N-tier architecture supporting the used framework.

Fig. 3 .

 3 Fig. 3. A pictorial representation of the modeling components.

Fig. 4 .

 4 Fig. 4. Concept types of facilitating works.

Fig. 5 .

 5 Fig. 5. Relation types.

Fig. 6 .

 6 Fig. 6. A rule modeling a situation for the detoxification.

Fig. 7 .

 7 Fig. 7. A rule modeling a situation for the soil stabilization measures.

  Logical expression: 9x 9y 9z (Project (x) ^ Soil Physical stability (y) ^ Lower Threshold (z) ^ Attribute (x, y) ^ Inferior (y, z)) ! 9u 9v 9w (Project (x) ^ Soil stabilization measures (u) ^ Bearing capacity (v) ^ Slip resistance (w) ^ Object (x, u) ^ Element (u, v) ^ OR (v, w)).

Fig. 8 .

 8 Fig. 8. A rule modeling a situation for the archaeological investigation.

Fig. 9 .

 9 Fig. 9. A rule modeling a situation for the major demolition works.

Fig. 10 .

 10 Fig. 10. Selection of a fact for the application of the relevant rules.

Table 1

 1 Three generations of knowledge management in the AEC industry.

	Generations of KM criteria	Generation 1: knowledge sharing	Generation 2: knowledge	Generation 3: knowledge value creation
			conceptualization and nurturing
	Underpinning ICT	Human interpretable knowledge	Semantic based systems(articulated	Systems managing knowledge as an
		systems (knowledge embedded	around the use of Building Information	asset(leveraging the intellectual and
		in documents requiring human	Modeling or ontology)	social capital of an organization)
		interpretation)
	Socio-technical	Information and communication	Human and organizational factors	Human networks, social capital,
	dimension	technology		intellectual capital, technology assets,
				and change management
	Lifecycle focus	Software application focus	Discipline focus	Total lifecycle and AEC domain focus
	Knowledge perspective	Condition perspective emphasizing
		access to information

Knowledge process perspective focusing on knowing and acting

Capability perspective: knowledge is a capability with a view of creating value

Table 2

 2 Experience feedback process in line with continuous improvement[START_REF] Rakoto | Elaboration and exploitation of lessons learned, 93[END_REF].

	Phases of	Stages of	Definition of the stage
	experience	problem
	feedback	solving
	Capitalization	Problem	Identification of problem
	Treatment	Reflection	Analysis of problem
		Origins	Investigation of causes
		Guide	Search for possible solutions
		Reaction	Selection of the most
			adequate solution
	Exploitation	Execution	Implementation of the solution
		Monitoring	Monitoring of actions and
			verification of the effectiveness
		Standardization	Generalization to prevent the
			recurrence of similar problems

http://www.lirmm.fr/cogui/

Acknowledgement

This work has been undertaken as part of the collaborative research funded through two organisations whose goals are to encourage interdisciplinary research work. Firstly, the authors were awarded a grant by the Ecole Nationale d'Inge ´nieurs de Tarbes through the ''Projet Soutien a ´la Mobilite ´Internationale (SMI) 2014'' programme. Secondly, the authors were awarded another grant by the Oxford Brookes University through Central Research Funding Scheme. The authors gratefully acknowledge the financial support received.


El Houssein

Chouaib Harik

Franc ¸ois Guérin
email: francois.guerin@univ-lehavre.fr

Frédéric Guinand
email: frederic.guinand@univ-lehavre.fr

Jean-Franc ¸ois Brethé
email: jean-francois.brethe@univ-lehavre.fr

Hervé Pelvillain
email: herve.pelvillain@univ-lehavre.fr

A Decentralized Interactive Architecture for Aerial and Ground Mobile Robots Cooperation

Keywords: Unmanned Aerial Vehicles, Unmanned Ground Vehicles, Air-Ground-Cooperation, non-linear kinematic controller, DES algorithm

This paper presents a novel decentralized interactive architecture for aerial and ground mobile robots cooperation. The aerial mobile robot is used to provide a global coverage during an area inspection, while the ground mobile robot is used to provide a local coverage of ground features. We include a human-in-the-loop to provide waypoints for the ground mobile robot to progress safely in the inspected area. The aerial mobile robot follows continuously the ground mobile robot in order to always keep it in its coverage view.

I. INTRODUCTION

Unmanned Aerial and Ground Vehicles (UAVs, UGVs) cooperation attracts increasingly the attention of researchers, essentially for the complementary skills provided by each type to overcome the specific limitations of the others. UAVs provide a global coverage and faster velocities, and UGVs provide a higher payload and stronger calculation capabilities, adding a local coverage for the unseen areas from an aerial perspective. In addition, deploying both types together ensure faster and more reliable results within a shorter frame of time compared to the deployment of a single type of mobile robots.

Air-Ground-Cooperation (AGC) in mobile robots systems can envisage a large panel of applications. Our focus in this paper is on Surveillance or Inspection (SI) missions. The authors in [START_REF] Ridley | Tracking in decentralised air-ground sensing networks[END_REF], [START_REF] Grocholsky | Information driven coordinated air-ground proactive sensing[END_REF], [START_REF] Grocholsky | Cooperative air and ground surveillance[END_REF] use a Decentralized Data Fusion (DDF) technique [START_REF] Manyika | Data Fusion and Sensor Management: a decentralized information-theoretic approach[END_REF] for exploring a specific area. The task of both air and ground based nodes is to make observations of terrain features and identify moving or stationary targets.

The authors in [START_REF] Chaimowicz | A framework for the scalable control of swarms of unmanned ground vehicles with unmanned aerial vehicles[END_REF] use AGC for target searching missions. Their testbed is composed of three layers: a high altitude UAV (a planner) to determine the motion of the mobile robots based on the desired goals, medium altitude UAVs (Blimps) to track the group of Unmanned Ground Vehicles (UGVs) and try to maintain them in their field of view, and the UGVs that navigates for exploration or other tasks. Another interesting work can be found in [START_REF] Chaimowicz | Aerial shepherds: Coordination among uavs and swarms of robots[END_REF], where the authors present a system that allow to deploy a significant number of ground mobile robots monitored by aerial mobile robots (aerial shepherds) for target searching. An urban environment surveillance application can be found in [START_REF] Hsieh | Adaptive teams of autonomous aerial and ground robots for situational awareness[END_REF]. The authors of this work cover a large set of technological topics from mapping to communication and collaborative navigation using multiple aerial and ground mobile robots. The authors in [START_REF] Herbert | Decentralized cooperative control of heterogeneous vehicle groups[END_REF] also use a group of UAVs to provide an aerial coverage to a group of UGVs used to clear a path through that area. We can find in [START_REF] Tanner | Cooperation between aerial and ground vehicle groups for reconnaissance missions[END_REF] a surveillance scheme including a group of six UGVs and four UAVs. An interagent cohesion and separation and a velocity synchronization scheme were combined into a control and communication strategy for mobile robots navigation. Another work of the same authors [START_REF] Herbert | Switched uav-ugv cooperation scheme for target detection[END_REF] present an AGC scheme for target detection mission. The size of the navigation area is related to the number of the deployed mobile robots. A set of grid points is defined for UGVs. An interagent potential force based approach is used for the UGVs navigation towards the grid points. Once the UGVs reached their grid points the aerial mobile robots start scanning the field. The authors in [START_REF] Stentz | Integrated air/ground vehicle system for semi-autonomous off-road navigation[END_REF], [START_REF] Stentz | Real-time, multi-perspective perception for unmanned ground vehicles[END_REF] use a UAV as a remote sensor that flies ahead the UGV to provide geo-referenced 3D geometry, in order to help the UGV for navigating in the area avoiding obstacles.

In the previous related works, a first flight of the drone over the inspected area is always performed before the beginning of the mission. A traversability map is then processed to provide a trajectory to the UGV. The main contribution of our work consists in providing a real-time navigation scheme: the UAV flies over the area to provide a global coverage, and to assist a UGV in real-time to navigate safely avoiding obstacles. To provide the real-time navigation scheme without the need to have a per-processed map, a human operator is used to provide waypoints by monitoring the global and local coverages, making the architecture interactive, and applicable to real scenarios. Indeed, for numerous reasons, monitoring, surveillance and inspection of industrial sites belonging to Seveso category (highly critical sites) cannot be performed without a constant and careful attention of human experts.

The paper is organized as follows: In section II we introduce the problem statement and define the objectives of our work. Section III presents the UGV and UAV controllers design. We discuss in section IV and V the simulation and experimental results, and we conclude in section VI the present work and discuss future perspectives.

II. PROBLEM STATEMENT

A UAV is used to monitor a given area where a UGV is present to perform the inspection task. The images taken from the camera mounted on the UAV are sent continuously to the ground station. The images are processed in real time to provide localization information to the UGV to navigate in the area through the on-the-go waypoints selected by a human operator (Figures 1 and2). To successfully inspect a given area within a reasonable frame of time, global and local coverages must be acquired simultaneously, which can be possible if we deploy both types of mobile robots (aerial and ground) simultaneously. In this direction, our architecture is composed of a UAV equipped with a down-facing camera, and a UGV equipped with a horizontal-facing camera. Both video flows are sent continuously to a ground station running our developed Human-Machine-Interface (HMI). A human operator supervises the video flows, and selects progressively navigation waypoints for the UGV. As the UGV navigates through the given waypoints, the UAV follows continuously the UGV in order to keep it in its coverage view (the center of the image plan) using visual servoing.

A. Hardware configuration

We consider in this work a non-holonomic, unicycle-like, Wheeled Mobile Robot (WMR) that has been designed in the GREAH laboratory of the University of Le Havre (France) [START_REF] Franc ¸ois Guerin | double exponential smoothing for predictive vision based target tracking of a wheeled mobile robot[END_REF]. The UAV is a quadcopter (Phantom 2 Vision) developed by DJI [START_REF] Dji | Quadcopter phantom 2 vision[END_REF] with a sufficient payload to carry an open platform device (android phone) that is used to take the video flow and send it to the ground station continuously via Wifi (IEEE 802.11). The received flow is processed to extract the position and orientation (pose estimation) of the UGV relative to the selected waypoint.

B. Pose estimation of the UGV

To successfully navigate in a given area, the relative pose of the UGV should be known (orientation and location). The authors of AGC applications ( [START_REF] Choi | Collaborative Tracking Control of UAV-UGV[END_REF], [START_REF] Chaimowicz | Experiments in multirobot air-ground coordination[END_REF], [START_REF] Vidal | Probabilistic pursuit-evasion games: theory, implementation, and experimental evaluation[END_REF], [START_REF] Kim | A hierarchical approach to probabilistic pursuit-evasion games with unmanned ground and aerial vehicles[END_REF]) used a single colored marker to define the relative location of the UGV to the UAV, and a digital compass to communicate the orientation which generally gives inaccurate measurements (digital compass is sensible to electromagnetic field variations). To overcome this issue, we added a second marker to also estimate its orientation.

We consider X c and Y c the axis of the image (camera) taken from the UAV (Figure 3). In order to locate the UGV, we used a color tracking algorithm to extract the position of colored markers: R c (red marker) fixed on the center of the driving wheels, and R h (blue marker) fixed on the head of the UGV. The position (in pixels) of R c and R h along X c and Y c axis are respectively: Y Rc , X Rc and Y Rh , X Rh . We assume that the center of the camera (center of the image) is the center of the UAV, thus we assume that the camera frame (X c ,Y c ) and the UAV inertial frame (X D , Y D ) are superimposed. The orientation α of the UGV to a given waypoint w, can be computed as α = θ -β with:

θ = atan( Y w -Y Rc X w -X Rc ) β = atan( Y Rh -Y Rc X Rh -X Rc ) (1)
Note that by using the colored markers on the UGV, we can extract its orientation without knowing the orientations of the UAV and the UGV in the world frame. The distance d that separates the UGV and the waypoint w has to be expressed in the world frame. It can be written as follows:

d = (x w -x Rc ) 2 + (y w -y Rc ) 2 (2)
Where x w , y w , x Rc , y Rc represent (in meter) the coordinates of the R c marker and the waypoint in the UAV or camera frame previously supposed to be the same. They can be computed using the classical Pinhole model of the camera [START_REF] Franc ¸ois Guerin | commande conjuguée d'un robot mobile, modélisation dynamique et vision artificielle[END_REF]. The example is given for the waypoint:

X w .z Y w .z z = G x 0 0 z.X 0 0 G y 0 z.Y 0 0 0 1 0    x w y w z 1    (3)
G x = f ∆x G y = f ∆y
Where: f : Focal distance -∆ x , ∆ y : Dimensions of a pixel. X 0 , Y 0 : Projection (in pixels) of the camera optical center in the image plan. They are assumed to be null. X w , Y w : Projection (in pixels) of the waypoint in the image plan.

We note as well that using the pinhole model we can estimate the altitude of the UAV using the distance between the two colored markers (red and blue).

C. Design of the UGVs controller

The UGV receives d and α via a radio module from the ground station. The goal is to use these information to generate u and r that respectively represent the longitudinal and angular velocities of the UGV (Figure 4). The control objective is to regulate the distance d and the steering angle α as follows: Let us consider the following distance errors on X R and Y R axes respectively:

lim t→∞ d(t) = L lim t→∞ α(t) = 0 (4)
e X R = d.cos(α) -L e Y R = d.sin(α) (5)
The time derivative of the distance errors [START_REF] Chaimowicz | A framework for the scalable control of swarms of unmanned ground vehicles with unmanned aerial vehicles[END_REF], which depends on both the UGV's velocities (u, r) and the target's velocities along the X R and Y R axes, denoted as T V X R and T V Y R , are given by:

ėX R = ḋ.cos(α) -d. α.sin(α) = T V X R -u ėY R = ḋ.sin(α) + d. α.cos(α) = T V Y R -rL (6)
Where rL corresponds to the linear velocity of R h (blue marker).

T V X R and T V Y R are null since the target of the UGV is a waypoint. We propose then the following error dynamic equations:

ėX R = -K.e X R = -u ėY R = -K.e Y R = -r.L (7)
Where K > 0 is the proportional gain of the UGV's controller. It is chosen to get a first order dynamic behavior for the control objective:

ėX R + K.e X R = 0 ėY R + K.e Y R = 0 (8)
The numerical value given to K allows to define the decreasing speed of the distance errors e X R and e Y R . Note that for large values of the distance d, a saturation is implemented to consider the limitations of the electrical drives that control the UGV's left and right wheels.

Starting from [START_REF] Hsieh | Adaptive teams of autonomous aerial and ground robots for situational awareness[END_REF] we propose the following non-linear kinematic controller:

u = K.(d.cos(α) -L) r = K.(d.sin(α))/L (9)

D. Design of the UAV controller

To hover the UAV over the UGV during its waypoints navigation, we need to control the UAV movements along X D and Y D axis (Figure 3) since the UGV moves on a 2D plan (assumed to be flat). We need to develop a navigation controller that takes as input the location of the UGV (R c ), and generates the necessary pitch and roll angles to keep the UGV at the center of the image plan (visual servoing).

We consider four inputs φ d , θ d , ψ d and z d . They represent respectively the desired roll, pitch, yaw angles, and the desired altitude. Note that the UAV attitude (roll, pitch, yaw, altitude) is controlled by the internal autopilot. We suppose that the UAV flies at a fixed altitude, and a fixed yaw: z d = z and ψ d = 0. z is chosen to have a sufficient coverage (3) of the inspected area (z min < z), and to keep the track of the UGV (z < z max ). z min and z max can be defined experimentally.

To design the UAV controller, we used the dynamic model of a quadcopter [START_REF] Bouabdallah | Full control of a quadrotor[END_REF]. Starting from this dynamic model, the movements along X D and Y D axis can be described as follows:

ẍD = (cosφ d sinθ d cosψ d + sinφ d sinψ d ) 1 m U 1 ÿD = (cosφ d sinθ d sinψ d -sinφ d cosψ d ) 1 m U 1 (10)
ẍD and ÿD correspond to the acceleration along X D and Y D axis respectively. ( 10) can be as:

ẍ = u X D 1 m U 1 ÿ = u Y D 1 m U 1 (11)
With:

u X D = (cosφ d sinθ d cosψ d + sinφ d sinψ d ) u Y D = (cosφ d sinθ d sinψ d -sinφ d cosψ d ) (12)
where u X D , u Y D represent the orientations of the total thrust (U 1 ) responsible for linear motions of the quadcopter along X D and Y D axis. m is the UAVs weight.

To allow the UAV to keep the UGV in its coverage view (the center of the image plan), let us consider the following distance errors (Figure 3):

e X D = X Rc -X 0 e Y D = Y Rc -Y 0 (13)
Note that X 0 , Y 0 are assumed to be null. The control objective is to regulate the distance errors as follows:

lim t→∞ e X D (t) = 0 lim t→∞ e Y D (t) = 0 (14)
To fulfill this control objective, we propose the following error dynamic equations:

ëX D = -K 1 XD e X D -K 2 XD ėX D = T A XD -ẍD ëY D = -K 1 Y D e Y D -K 2 Y D ėY D = T A Y D -ÿD (15)
Where K (1,2),(XD,Y D) are positive gains. They are chosen to get a second order dynamic behavior without damping and oscillations for the control objective:

ëX D + K 2 XD ėX D + K 1 XD e X D = 0 ëY D + K 2 Y D ėY D + K 1 Y D e Y D = 0 (16)
T A XD and T A Y D are the accelerations of the UGV along X D , Y D axis. They are neglected compared to ẍD and ÿD since the UAV moves much faster than the UGV. Starting from [START_REF] Choi | Collaborative Tracking Control of UAV-UGV[END_REF], we propose the following controller:

u X D = m U 1 (K 1 e X D + K 2 ėxD ) u Y D = m U 1 ((K 1 e Y D + K 2 ėyD ) (17)
To find the desired roll and pitch angles needed to track the UGV, we combine [START_REF] Vidal | Probabilistic pursuit-evasion games: theory, implementation, and experimental evaluation[END_REF] in [START_REF] Stentz | Real-time, multi-perspective perception for unmanned ground vehicles[END_REF] taking a fixed yaw angle ψ = 0, the desired angles are then:

φ d = asin(-u Y D ) θ d = asin( u X D cosφ d ) (18)
We note that the steady state error will be null (when the UGV stops) since it is a position control.

E. DES algorithm

Furthermore, the Double Exponential Smoothing (DES) algorithm [START_REF] Jr | an experiment comparing double exponential smoothing and kalman filter-based predictive tracking algorithms[END_REF] has been implemented to smoothen the uncertain measurements given by the vision system. The DES algorithm runs approximately 135 times faster with equivalent prediction performances and simpler implementations compared to a Kalman filter. The DES algorithm at time instant n.T e , where T e is the sampling period and n is the discrete-time index, implemented for the distance d, the steering angle α, and the projection of the red marker X Rc , Y Rc is given as follows:

S mn = γ m .m n + (1 -γ m ).(S mn-1 + b mn-1 ) (19)
b mn = λ m .(S mn -S mn-1 ) + (1 -λ m ).b mn-1 (20)
Where: m n is the value of (d,α,X Rc , Y Rc ) at nth sample instant. S mn is the smoothed values of (d,α,X Rc , Y Rc ). b mn is the trend value of (d,α,X Rc , Y Rc ).

Equation [START_REF] Franc ¸ois Guerin | commande conjuguée d'un robot mobile, modélisation dynamique et vision artificielle[END_REF] smooths the value of the sequence of measurements by taking into account the trend, whilst (20) smooths and updates the trend.

The initial values given to S mn , b mn are:

S m1 = m 1 b m1 = m 2 -m 1 (21)
Usually, γ (0 ≤ γ ≤ 1) is called the data smoothing factor and λ (0 ≤ λ ≤ 1) is called the trend smoothing factor. A compromise has to be found for the values of γ and λ. High values make the DES algorithm follow the trend more accurately whilst small values make it generate smoother results. The smoothed values are used instead of the direct noisy measurements in the proposed controllers. DES algorithm has been used successfully in a previous work [START_REF] Franc ¸ois Guerin | double exponential smoothing for predictive vision based target tracking of a wheeled mobile robot[END_REF] for vision based target tracking.

IV. UGV RESULTS

We will present in this section simulation and experimental results concerning only the UGV, the UAV is piloted by a professional.

A. Simulation results

We simulated a real-like scheme to navigate in a given area avoiding obstacles. For the UGV we have used the dynamic model fully described in [START_REF] Franc ¸ois Guerin | double exponential smoothing for predictive vision based target tracking of a wheeled mobile robot[END_REF] and [START_REF] Franc ¸ois Guerin | commande conjuguée d'un robot mobile, modélisation dynamique et vision artificielle[END_REF]. Simulations have been carried out with Matlab-Simulink software (sampling period: 10 ms). For the UGV non-linear kinematic controller we used the following parameters: L = 15cm, K = 0.1.

Figure 9 shows that the UGV is able to follow the given waypoints, which validates the proposed controller [START_REF] Tanner | Cooperation between aerial and ground vehicle groups for reconnaissance missions[END_REF]. The results concerning the control objective (4) have been checked experimentally.

B. Experimental results

We created a user interface running on the ground station using Processing [START_REF][END_REF]. The human operator can supervise in real time both video flows received from UAV and UGV (Figure 5). The experiments were carried out with a smartphone mounted under the quadcopter. For safety reasons, the UAV was piloted by a professional, but we carried out as well other experiments without a pilot using an AR Drone 2.0. Videos can be found in [START_REF]Experimentation 1: Pilot aided flight[END_REF] for the experiment with a pilot, and [START_REF]Experimentation 2: Autonomous flight[END_REF] for the newer version without a pilot . We present in this section only simulation results of the UAV following the UGV in its waypoints tracking because we do not yet have access to the UAV's position measurements in the inertial frame (for example VICON MX system). That is why quantitative results are presented for the simulation trials only. Please refer to the links mentioned in section IV for experiments videos.

A. Simulation result

We used for simulation the parameters (table I) of an opensource quadcopter developed by [START_REF][END_REF]. The parameters have been obtained thanks to experimentations fully described in [START_REF] Houssein | stabilisation par backstepping d'un quadrirotor[END_REF]: Figure 9 shows that the UAV takes-off from an initial position (-6,-9) and flies to hover at 3m altitude, we suppose that at this height the UGV with an initial position (-8, -8) is in the coverage view of the UAV. We can see that the UGV moves followed by the UAV. Figures [START_REF] Herbert | Switched uav-ugv cooperation scheme for target detection[END_REF] show that the UAV changes its pitch and role angles according to the motion of the UGV.

VI. CONCLUSION AND FUTURE WORK

We presented in this paper a novel decentralized interactive architecture, where an air-ground-cooperation is deployed for area inspection. A UAV is used to provide a global coverage where a UGV is located to provide a local coverage. A human operator is introduced to provide waypoints for the UGV in order to safely navigate in the area. Our future work goes towards implementing an autonomous waypoints selection through free-space navigation extraction, which allow the testbed to be extended in more than one UGV, and let the human operator focus on other coordination tasks.

Figure 1 .

 1 Figure 1. Problem statement

Figure 2 .

 2 Figure 2. Overall architecture

Figure 3 .

 3 Figure 3. The UGV in the image plan

Figure 4 .

 4 Figure 4. Design of the UGV controller

Figure 5 .

 5 Figure 5. Aerial and ground perception on the HMI. Left: The global view from the UAV. Right: the local view from the UGV.

Figure 6 .

 6 Figure 6. Waypoint tracking experimentation.

Figure 6

 6 Figure 6 illustrate the UGV experimentation: The pilot (1) manipulates the UAV (2) that takes and send to the ground station a video flow (3). The operator (4) selects new waypoints to guide the UGV (5).

Figures 7 and 8 ,

 8 Figures 7 and 8, represent the distance d and the steering angle α variations over time for a rectangular path selected by a human operator through the mission (send a new waypoint each time the UGV reaches the old one). Each peak on both graphs represent a new click by the user, which gives a new distance d and a new steering angle α. We can see that according to (7) the distance and the steering angle decrease exponentially and respectively to d = L = 0.15 m and α = 0 • which confirms the efficiency of our developed controller (9).

Figure 7 .Figure 8 .

 78 Figure 7. Distance variation

Figure 9 .Figure 10 .

 910 Figure 9. Inspection scheme simulation

ACKNOWLEDGMENTS

The authors would like to thank Le Havre Town Council CODAH for their support under research grants, and "DroneXTR" for piloting the drone. They would like as well to thank Upper Normandie Region through their support to PCMAI project.