V L Krupenin

On the Description of Strongly Nonlinear Vibroconducting and Vibrogenerating Media

A quite general class of problems is considered, the analysis of which makes it possible to describe the mechanism of for the generation and propagation of broadband vibration in machine structures. Mathematical models based on the theory of strongly nonlinear continuous media with a complex structure are presented. Different types of models of a "linear carrier medium plus attached equipment containing impact pairs" class are presented. It is assumed that the impact is Newtonian. The necessary defining relations are given. An example of vibroconductor analysis with internal breaks is given.

1.

It is known [START_REF] Pal'mov | Kolebaniya uprugo-plasticheskikh tel (Elastic-Plastic Bodies Oscillation)[END_REF][START_REF] Artobolevskii | Vvedenie v akusticheskuyu dinamiku mashin (Introduction into Machines Acoustic Dynamics)[END_REF] that vibration is transmitted from point to point through vibroconductors (waveguides), which are the medium in which elastic waves, due to the presence of limiting surfaces, propagate in directions depending on the design features of the vibroconducting object and the character of interaction between structural elements. Recently developed methods of acoustic dynamics of machines, including the methods for analyzing the vibration propagation features and combating vibration are based on the assumption of linearity of vibroconductors or their weak nonlinearities. However, appearance of at least one break in a vibroconductor [START_REF] Krupenin | Vibroconductors equipped with impact elements and distributed vibroimpact systems[END_REF][START_REF] Astashev | Waves in distributed and discrete vibroimpact systems and strongly nonlinear mediums[END_REF], resulting in impacts of its design elements, causes a significant change in the dynamic qualities of the system as a whole, and the vibrocontucting medium is at the same time a vibrogenerating one. The same thing happens when elements directly connected to the vibroconductor collide. If similar parasitic impact pairs are rare and the vibroconductor is linear from the impact points to the observation points, then the generated broadband vibration after passage through actually present mechanical filters can be considered as certain high-frequency, low-intensity noise. If there is a larger number of impact pairs in any way attached to the vibroconductor and directly related to the vibration transfer process, the elastic transmitting medium transforms from linear to strongly nonlinear. In this case, the use of traditional calculation methods is problematical.

Linear (or close to linear) specific models of continuous media with a complex structure are considered in [START_REF] Pal'mov | Kolebaniya uprugo-plasticheskikh tel (Elastic-Plastic Bodies Oscillation)[END_REF]. One feature of the models is the presence of two original parts of the medium, namely: carrier and attached parts. Accordingly, the dynamic equations of such media consist of two groups. The first group of equations describes the carrier part, and the second group describes the attached part. Just like any multipolar mechanics model, here the understanding of point is substantially revised, the state of which can be defined by an arbitrary number of kinematic parameters. This approach, along with others, was fundamentally development in [START_REF] Belyaev | High-frequency dynamics of complicated engineering structures[END_REF], etc.

In [START_REF] Krupenin | Vibroconductors equipped with impact elements and distributed vibroimpact systems[END_REF][START_REF] Krupenin | To the theory of strongly nonlinear vibroguides[END_REF][START_REF] Krupenin | Model of strongly nonlinear vibroconducting medium with distributed impact element[END_REF][START_REF] Krupenin | To the problem on describing processes of nonlinear waves propagation through machines structures simulated by means of strongly nonlinear continuous mediums of complicated structure. Parts 1, 2[END_REF], strongly nonlinear models of a similar medium were considered, in which distributed impact elements were used to take into account multiple collisions in attached equipment components. Such models make it possible to describe the process of formation and propagation of vibratory fields in complex composite structures. In addition, the models make it possible to obtain a number of design formulas and significant defining relations. The need to access them is dictated by the fact that in machine structures, just multiple systematic collisions of subsystem elements are often responsible for the form of the generated global vibrofields and the vibroactivity of structures as a whole.

Returning to the two groups of equations of motion, we postulate the existence of an elastic (elastoviscous) carrier medium, i.e. the model contains the equations of motion of the carrier parts (the examples are as follows: the classical Lamé equation, the equation of longitudinal or transverse oscillations of rods, etc.), to which boundary conditions are added.

Assuming that it is as if the collisions are spread within a certain spatial domain, the equations of motion of attached equipment are added to the equations of motion of carrier parts. The coherence mechanism of both parts determines the global structure of the generated vibration field.

2. The representative case postulates the existence of some elastically dissipative carrier medium described by the displacement vector u(x, t), (u, x ∈ R 3 , x ∈ Ξ ⊆ R 3 , t ∈ R), being subject to the classical Lamé equation with a dissipative term (see [START_REF] Pal'mov | Kolebaniya uprugo-plasticheskikh tel (Elastic-Plastic Bodies Oscillation)[END_REF][START_REF] Krupenin | Model of strongly nonlinear vibroconducting medium with distributed impact element[END_REF]): [START_REF] Pal'mov | Kolebaniya uprugo-plasticheskikh tel (Elastic-Plastic Bodies Oscillation)[END_REF] where ρ is the density of the medium; λ, μ are the Lamé parameters describing the elastic properties of the medium; Ξ is the domain of the change in coordinates; Δ is the Laplacian; D[…] is the operator responsible to dissipative loss. Let the external (volume and surface) forces F = F 1 + F 0 , where F 1 is the specified vector; F 0 is the effect of attached systems containing the impact pairs.

Note that the carrier medium model can itself have a complex structure, so Eq. ( 1) can be replaced by a more common one. Equation (1) should satisfy to the boundary conditions.

Let the attached equipment be elastically amortized:

(

) 2
where it is assumed that the interactive (impact) pair consisting of two contacting linear stationary scleronomous subsystems A (1) (x) and A (II) (x is suspended in the each point of medium, each of these subsystems is described with the finite set of functions (x, t), j = 1, 2, …, N. For convenience, an identical number of degrees of freedom for both subsystems were assumed: the extra degrees of freedom can be fictitious. The points of suspension were identified in (2) as (x, t), and the point of contact is denoted as (x, t), n ≤ N. Let us assume that subsystems A (I, II) (x) are defined with families of distributed dynamic compliance operators (x; p) = O(p -2 ), p → ∞; indices q and k change from 1 to N; p = ∂/∂t, x, z ∈ Ξ ⊆ R 3 .

Here and below, we assume that interaction in every element is direct, central, and one-dimensional: let y 0 = -be the relative convergence of the points of contact and let the coordinate y 0 change along some axis. The interaction force Φ 1 (y 0 , ) = λΦ(y 0 ) + Φ 2 (y 0 , ), λ ≫ 1 is a large parameter [START_REF] Krupenin | Analysis of singularized motion equations of latticed vibroimpact 2D systems in renouncing Newton's hypothesis[END_REF]. Here, the first term in the right-hand side determines the elastic component of the interaction force, and the second term determines the dissipative component. The function Φ 1 determines the hypothesis of noninstantaneous (non-Newtonian) impact; it can directly depend on х, since the character of interaction can change from point to point. Let us assume that Φ is a threshold function [START_REF] Krupenin | To the theory of strongly nonlinear vibroguides[END_REF][START_REF] Babitsky | Vibration of Strongly Nonlinear Discontinuous Systems[END_REF]: where ψ(u) is continuously differentiable on the entire numerical axis, monotonically increases, and is a convex function for u ≥ 0 (Fig. 1); η(u) is a unit function. In the case when Newtonian interaction is rejected, the dissipative component Φ 2 is also described by some threshold function [START_REF] Krupenin | To the theory of strongly nonlinear vibroguides[END_REF][START_REF] Babitsky | Vibration of Strongly Nonlinear Discontinuous Systems[END_REF]. The methods for analyzing such systems are discussed in [START_REF] Krupenin | To the theory of strongly nonlinear vibroguides[END_REF][START_REF] Krupenin | Model of strongly nonlinear vibroconducting medium with distributed impact element[END_REF][START_REF] Krupenin | To the problem on describing processes of nonlinear waves propagation through machines structures simulated by means of strongly nonlinear continuous mediums of complicated structure. Parts 1, 2[END_REF][START_REF] Krupenin | Analysis of singularized motion equations of latticed vibroimpact 2D systems in renouncing Newton's hypothesis[END_REF][START_REF] Babitsky | Vibration of Strongly Nonlinear Discontinuous Systems[END_REF].

ρ = λ + μ + μ + + ( )graddiv [ ] , tt u u u D u F Δ = - - - - (I) (II) 0 1 1 2 1 ( , ) ( )[ ( , ) ( , )] ( )
For the points of suspension and interaction we have the operator equations

where for system A (I) in ( 3) and (4) we select the plus sign, and for system A (II) , we select the minus sign. Some functions describing an additional external effect can be introduced into these equations.

The communication mechanisms of the carrier and attached parts determine the structure of the global vibrofield. This approach does not take into account information on the features of specific parts of the system, nor effects whose appearance is possible only with allowance for the discreteness of the model.

3.

The proposed model is complicated for analysis and needs some simplifications. Let us give examples of more realistic models of this kind.

2D Structure of Carrier Part: Discrete Model

Let us consider a string lattice [START_REF] Krupenin | Analysis of singularized motion equations of latticed vibroimpact 2D systems in renouncing Newton's hypothesis[END_REF][START_REF] Krupenin | The way to generate models of multidimensional vibroimpact 2D systems of complicated structure[END_REF] composed of two mutually perpendicular families of linear elastic strings fastened at the ends and having lengths of L 1 and L 2 (Fig. 2). Each string is numbered using the indexes k = 0, 1, N 1 and q = 0, 1,…, N 2 . At the nodes are point perfectly solid bodies with masses m kq .

It is assumed that the lattice cells are identical and rectangular, but the lengths and widths of their sides are not equal among themselves and the lattice itself is anisotropic. The strings are inertia-free. Their fastenings at the nodes are considered perfectly hard, and the tensions are so large that possible changes during linear oscillations can be neglected.

Let the length of each horizontal side of cells be ΔL 1 , that of the vertical side be ΔL 2 ; let the tension of horizontal strings be T 1 , and that of vertical strings be T 2 . Thus, the dynamics of the lattice structure can describe by the displacement function of the lattice nodes u kq (t), k = 0, 1, …, N 1 , q = 0, 1, …, N 2 . Each of the functions u kq (t) changes along some axis perpendicular to the plane of static equilibrium of the lattice.

= ± Φ + (I,II) (I,II) (I,II) 0 0 (I,II) 1 1 1 1 , 2 1 1 1 ( ) ( ) ( ) ( , ) , n t y L pc xu L p y y f = ± Φ + (I,II) (I,II) (I,II) 0 0 (I,II) 1 1 , 2 1 ( ) ( ) ( ) ( , ) , nn nn t n y L pc xu L p y y f (I,II) 1,n f Fig. 2. 1 2 k N 1 N 2 1 2 λ Φ 2kq q
Let us consider that all actuating nonconservative forces are small. Let us denote them as εg kq (t, u kq , ), where ε is the small parameter. Since each particle lies on two strings, we obtain N the equations of motion N = (N 1 -1)(N 2 -1) [START_REF] Belyaev | High-frequency dynamics of complicated engineering structures[END_REF] where c 1, 2 = T 1, 2 /ΔL 1, 2 are the elasticity coefficients. The fastening boundary conditions can be written as [START_REF] Krupenin | Analysis of singularized motion equations of latticed vibroimpact 2D systems in renouncing Newton's hypothesis[END_REF] u kq = 0, at k = 0, N 1 ; q = 0, N 2 . Equations ( 5) represent the first group of equations; in the discrete model, they describe the carrier part of medium with a complex structure. Let the so-called impact oscillator be suspended in each nodal body of the lattice [START_REF] Babitsky | Vibration of Strongly Nonlinear Discontinuous Systems[END_REF] (Fig. 2). Then let w kq be the coordinate of the body forming the oscillator; M kq is its mass; Ω kq is a natural frequency; Δ kq are the values of the adjusting gaps (of interferences); k = 0, 1, …, N 1 , q = 0, 1, …, N 2 . Thus, the equations of motion of the attached equipment will be [START_REF] Krupenin | To the theory of strongly nonlinear vibroguides[END_REF] where f kq (t, w kq , ) are the additional forces acting on the oscillator.

It is seen that the forces in Eq. ( 5) are F kq = M kq (w kq -u kq ).

The interaction forces are determined by the threshold functions (see point 2 and [START_REF] Krupenin | Analysis of singularized motion equations of latticed vibroimpact 2D systems in renouncing Newton's hypothesis[END_REF]) Φ 1kq (w kq , ) = λΦ kq (w kq ) + Φ 2kq (w kq , ), λ ≫ 1. We believe that for λ → ∞, the considered hypothesis of interaction transforms into Newton's hypothesis: the impact is instantaneous; the impact force is set with a singular generalized function. This needs be postulated, since it is possible to construct its correct proof under the assumption of the elasticity of interaction [START_REF] Babitsky | Vibration of Strongly Nonlinear Discontinuous Systems[END_REF]. It is necessary to determine the displacements {u kq , w kq } for the specified forces {q kq , f kq }.

Continual Model

In system of equations ( 5), [START_REF] Krupenin | To the theory of strongly nonlinear vibroguides[END_REF], we turn to long waves, believing that at all k and q: m kq = m. Let the number of strings in both families be sufficiently large and quantities ΔL 1, 2 be small. At the same time, let the surface density γ = m/(ΔL 1 ΔL 2 ) and linear tensions θ x = T 1 /ΔL 1 , θ y = T 2 /ΔL 2 have orders O [START_REF] Pal'mov | Kolebaniya uprugo-plasticheskikh tel (Elastic-Plastic Bodies Oscillation)[END_REF]. System of equations (1) can be continualized, considering it as a finite-difference analog of continuous systems with distributed parameters. Let us consider a sufficiently smooth function u(х, у, t), coinciding (in this case) in the lattice nodes with the original functions u kq (t).

We replace the expressions (2u kq -u (k -1, q) -u (k + 1, q) ) and (2u kq -u (k, q -1) -u (k, q + 1) ) with the analogs Then, Eq. ( 5) can be represented as Here, instead of differences determining (1), the series in powers of ΔL 1 and ΔL 2 corresponding to them were written out. In this case, within the accuracy of small quantities of order O[(ΔL 1, 2 ) 2 in the original approximation, we turn to the anisotropic membrane equation

γu tt -θ x u xx -θ y u yy + … = εg(t, u, u t , …) + O[(ΔL 1, 2 ) 2 ].
If we consider the most popular isotropic membrane model when the stresses are θ x = θ y ≡ θ for ε = 0, then we arrive at the well-known and thoroughly studied equation of a classical membrane tightened along the contour of deflection, which is u = u(t, x, y):

u tt = c 2 (u xx + u yy ), c 2 = θγ -1 .
The boundary conditions for the lattice lead to the membrane being fastened on all its edges. If the lengths of its sides are l 1 and l 2 , than the limiting conditions are written as u(0, y, t

) = u(l 1 , y, t) = u(x, 0, t) = u(x, l 2 , t) = 0. kq u - + - + + - - + - - = ε + 1 ( 1, ) ( 1, ) 2 ( , 1) ( , 1) (2 )
(2 ) ( , , ) ,

kq kq kq k q k q kq k q k q kq kq kq kq m u c u u u c u u u g t u u F + Ω - +Φ = ε 2 1 ( ) (
- - = - -Δ - +Δ ( 1, ) ( 1, ) 1 1 2 2 ( , , ) ( , , ) ( , , )
, , ).

kq k q k q u u u u x y t u x L y t u x L y t - + - - = - -Δ - +Δ ( , 1) ( , 1) 2 2 2 2 ( , , ) ( , , ) ( ,
kq k q k q u u u u x y t u x y L t u x y L t γΔ Δ = Δ Δ θ + θ + Δ Δ θ + Δ Δ θ + … 3 1 2 1 2 1 2 3 1 2 ( ) ( )( ) ( / 1 2 ) ( )( ) ( /12) . tt x xx y yy x xxxx y yyyy L L u L L u u L L u L L u
Thus, taking into account the representations for force F appearing in (1), we obtain the following equation for the carrier part of the medium: [START_REF] Krupenin | Model of strongly nonlinear vibroconducting medium with distributed impact element[END_REF] Here the forces and other physical factors are replaced by their area distributions. Equations ( 6) are transformed as follows:

(8) Equations ( 7), [START_REF] Krupenin | To the problem on describing processes of nonlinear waves propagation through machines structures simulated by means of strongly nonlinear continuous mediums of complicated structure. Parts 1, 2[END_REF] give the simplest model of a two-dimensional strongly nonlinear medium with a complex structure. Generalization can be performed with allowance for more complex models of carrier media (plates, beam and rod lattices, etc.) and owing to more general models of attached equipment (point 2). In [START_REF] Krupenin | Random vibration fields in equipped multidimensional vibroimpulsive 2D systems[END_REF] at considering the discrete model exposed to the random broadband excitation, the lattice with the attached elastic chains of point bodies, provided with single-sided and double-sided catchers was studied. In the case of Newtonian interaction, the analysis of this model was partially carried out in [START_REF] Krupenin | The way to generate models of multidimensional vibroimpact 2D systems of complicated structure[END_REF].

Equipped Strongly Nonlinear Rod

A one-dimensional analog of model ( 1)-( 4) was studied in detail in [START_REF] Krupenin | To the theory of strongly nonlinear vibroguides[END_REF][START_REF] Krupenin | To the problem on describing processes of nonlinear waves propagation through machines structures simulated by means of strongly nonlinear continuous mediums of complicated structure. Parts 1, 2[END_REF]. Instead of Eq. ( 1) for a rod of length l, we have [START_REF] Krupenin | Analysis of singularized motion equations of latticed vibroimpact 2D systems in renouncing Newton's hypothesis[END_REF] where the boundary conditions are as follows: u(0, t) = 0; Eu x (l, t) = Pcosωt[u(l, t) = μcosωt]. The meaning of the notation is obvious. Relations (2)-( 4) retain their form.

Figure 3 shows the continual model of a strongly nonlinear vibroconductor with elastically amortized equipment, as well as the carrier part of the system and symbolically attached equipment (interacting in the subsystem buffers A (I) (x) and A (II) (x)).

Rod with Internal Breaks

Figure 4 shows a discrete model of a rod with internal breaks. For a large concentration of elastically attached systems, we can use the long-wave approximation. We postulate the existence of a one-dimensional medium with the determining equations [START_REF] Babitsky | Vibration of Strongly Nonlinear Discontinuous Systems[END_REF] (11)

= + + Ω -+ε 2 2 ( ) ( , ) ( , )( ) (, , )
.

tt xx yy u c u u M x y x y w u g t u u + Ω - +Φ =ε 2 1 ( , ) ( , , ) ( , ) ( , )[ ( , , ) ( , , )] ( , ) ( , , ).

tt t t M x y w t x y M x y x y w t x y u t x y w w f t w w

ρ - + - + - - + = ∈ (I) (II) 1 1 ( ) ( ) [ ] 0 [0 , ],
A (II) (x) u(x, t) u(x, t) u(x + Δx, t) A (I) (x)
where ρ and Е are the linear mass and Young's modulus of the rod; it is as though the interacting equipment is spread along the length of the rod; m is the linear mass of distributed impact elements; y 0 (х, t) is the distribution of their relative displacement. The boundary conditions are given after Eq. ( 9).

Note that use of this model is restricted, since it is possible to be outside the scope of continuous solutions; in certain frequency ranges, the model should be refined.

Remarks

The vibrofields in the considered objects are determined by two main mechanisms. The attached equipment acts on the carrier part like impact vibration dampers [START_REF] Pal'mov | Kolebaniya uprugo-plasticheskikh tel (Elastic-Plastic Bodies Oscillation)[END_REF]. An exception is narrow frequency ranges near the eigenfrequencies of linear oscillations, where this action is similar to linear dampers. The carrier part filters the passing vibration. Just these mechanisms, manifesting themselves in different ways at different points of carrier structures, can generate specific dynamic effects.

Ending the brief review of linear carrier medium models plus interactive attached equipment, we note that when writing the necessary equations of motion, one can use different assumptions on the structure of interacting elements. Two subsystems can contact, or attached equipment can interact with restraints, or there can be double-sided elements, etc.

4.

Let us assume that the interaction is elastic Φ 1 (y 0 , ) = λΦ(y 0 ) (point 2). If we accept the hypothesis that the interaction is instantaneous (λ → ∞), then the impact force is expressed by singular general functions. To analyze processes and systems in which it is important to take into account the finite duration of interaction, the method of singularization was proposed in [START_REF] Krupenin | Analysis of singularized motion equations of latticed vibroimpact 2D systems in renouncing Newton's hypothesis[END_REF][START_REF] Babitsky | Vibration of Strongly Nonlinear Discontinuous Systems[END_REF]; in accordance with which, taking into account the transition to the long-wave approximation, we write for a single interaction [START_REF] Krupenin | Random vibration fields in equipped multidimensional vibroimpulsive 2D systems[END_REF] where J λ (x) is the interaction momentum density; ϕ λ (х) is the distribution of instants of onset of interaction; τ λ (x) = 1/2t λ (x); t λ (x) is the distribution of times of interaction; δ(х) is the Dirac δ-function; and х ∈ Ξ 1 is the interaction region.

Let us denote by Δ(x) the distribution of coordinates of the onset (and, accordingly, end) of interaction:

(13) The momentum density and total interaction momentum are, respectively, (

The functions J λ (х) and ϕ λ (x) are determined during concretization of the problem and, finally, determine the form of the desired solutions. When considering Т-periodic processes, we express force ( 12) through a T-periodic δ-function (the Dirac comb) or dual Dirac comb in the case of a symmetric double-side catcher 5. The set of desired displacement fields of structural elements of medium has the form U = {u(x, t); y 0 (x, t); y 1 (x, t); …; y N (x, t)}, at that the first element describes the carrier part of medium and the rest depict the attached parts. In the considered models, the nonlinear forces are determined only by the presence of subsystems interacting among themselves or with fastened catchers. For the considered type of

0 t y λ λ λ λΦ ≈ δ -ϕ -τ 0 [ ( , )] ( ) [ ( ) ( )], y x t J x t x x λ λ λ ϕ = ϕ + = Δ 0 0 [ , ( )] [ , ( ) ( )] ( ). y x x y x x t x x λ λ λ λ Ξ = λΦ = ∫ ∫ 1 ( ) 0 0 ( ) [ ( , )] , ( ) . t x J x y x t dt I J x dx ∞ ∞ - - σ=-∞ =-∞ δ = σω = δ - ω = π ∑ ∑ T 1 1 ( ) exp( ) ( ), 2 / , K t T i t t KT T ∞ ∞ - - σ=-∞ =-∞ δ = σ+ ω = δ - -δ - - ω= π ∑ ∑ T/2 1 1 ( ) 1/2 exp[ (2 1) ) [ ( ) ( /2 )], 2 / . K t T i t t KT t T KT T Fig. 4.
models, one can indicate the structure of the set of elements U. The displacement field u(x, t) is summed as the result of the action of external (mass and surface) forces-let us denote it u 1 (x, t)-and the result of the action of interaction forces in the attached equipment, i.e.,

where V(x, z, t -s) is the corresponding Green's function determined by the structures of linear systems: the carrier and attached parts of the medium. For interacting elements of the medium (attached parts), we find representations of the type (16

)
where y q (x, t) is the displacement of the structural element of one of the interacting subsystems; H q (t -s) is the Green's function at the interaction point; the second term in the right-hand side of Eq. ( 16) is displacement occurring as a direct result of interaction at this point; the third term is displacement resulting from interaction, but appearing as a result of its motion of caused by other structural elements of the medium; Y q (x, z, t -s) is the Green's function. Now it is easy to write the relations for the relative coordinate and other elements of the medium. When analyzing periodic vibroimpact processes in Eqs. ( 15), ( 16), it is advisable to switch to periodic Green's functions (PFG). Examples of analyzing such models in the case of acceptance of analogs of Newton's hypothesis are given in [START_REF] Krupenin | Vibroconductors equipped with impact elements and distributed vibroimpact systems[END_REF][START_REF] Belyaev | High-frequency dynamics of complicated engineering structures[END_REF][START_REF] Krupenin | To the theory of strongly nonlinear vibroguides[END_REF][START_REF] Krupenin | Model of strongly nonlinear vibroconducting medium with distributed impact element[END_REF][START_REF] Krupenin | To the problem on describing processes of nonlinear waves propagation through machines structures simulated by means of strongly nonlinear continuous mediums of complicated structure. Parts 1, 2[END_REF]. For complication of models of the carrier parts of a medium, e.g., taking into account the microstructure, accordingly, the representations of displacements of the structural elements will change (they will be complex).

In [START_REF] Pal'mov | Kolebaniya uprugo-plasticheskikh tel (Elastic-Plastic Bodies Oscillation)[END_REF], a model of was considered medium in which the attached equipment was simulated with a continual set of isotropic oscillators. Similar constructions were also used in part for the models presented in this work, but owing to the strong nonlinearity of the attached part, to describe the frequency properties of the system, it is sufficient to give this part a finite number of degrees of freedom.

6.

As an example, we consider the problem of a vibroconductor with internal breaks (Fig. 4). Let the linear mass of impact elements be m(х) ≡ 1. We write Eqs. [START_REF] Babitsky | Vibration of Strongly Nonlinear Discontinuous Systems[END_REF], [START_REF] Krupenin | The way to generate models of multidimensional vibroimpact 2D systems of complicated structure[END_REF] as

(17) (18)
where the elastic interaction force belongs to a class of symmetric threshold functions, which is constructed using class {Φ} λ , described in point 2 as Φ(у 0 ) = Φ 1 (y 0 ) -Φ 1 (-y 0 ); Φ 1 (y 0 ) ∈ {Φ} Δ . We assume that the loading at the contact zones is linear: Let Δ(x) = const. We determine ε = O(λ -1/2 ). The boundary conditions are as follows:

(19) which corresponds to the case of fastening of the left end of the rod and the application to its right end of T-periodic force action; T = 2πω -1 .

In accordance with (15), ( 16) and [START_REF] Babitsky | Vibration of Strongly Nonlinear Discontinuous Systems[END_REF], we write the integral equations of symmetric T-periodic oscillations:

(20)

Ξ -∞ = + -Φ ∫ ∫ 1 0 0 1 1 ( , ) ( , ) ( , ,
) [ ( , ), ( , )] , and, in accordance with [START_REF] Krupenin | Model of strongly nonlinear vibroconducting medium with distributed impact element[END_REF], in this case ω 0 = ωa -1 , where a = ; the PFG χ 1 (t) is given by the series [START_REF] Babitsky | Vibration of Strongly Nonlinear Discontinuous Systems[END_REF] For 0 < t ≤ T/2, this series can be written in finite form: χ 1 (t) = [1/2t -1/4πω -1 ] [START_REF] Babitsky | Vibration of Strongly Nonlinear Discontinuous Systems[END_REF]. Outside the interval ]0, Т/2], this representation should continue on the periodicity based on symmetry conditions. The result is a periodic saw-tooth function ("triangular sine").

t t u x t u x t V x z t s y x s y x s dsdz -∞ Ξ -∞ = - -Φ + -Φ ∫ ∫ ∫ 1 0 0 1 1 0 0 1 ( , ) (
The PFG χ(х, z, t) corresponds to a linear wave operator and the type of boundary conditions being considered. Using [START_REF] Babitsky | Vibration of Strongly Nonlinear Discontinuous Systems[END_REF], we find (23) It turns out [START_REF] Krupenin | To the theory of strongly nonlinear vibroguides[END_REF][START_REF] Krupenin | To the problem on describing processes of nonlinear waves propagation through machines structures simulated by means of strongly nonlinear continuous mediums of complicated structure. Parts 1, 2[END_REF] that for PFG (23), another representation takes place:

(24)

Here, the functions (25) are determined by the Green's functions of the Sturm-Liouville problem, in this case, problems with the left fastened end and right nonfastened end [START_REF] Kamke | Differentialgleichungen reeller Funktionen[END_REF].

According to [START_REF] Krupenin | Random vibration fields in equipped multidimensional vibroimpulsive 2D systems[END_REF], for symmetric double-impact modes of motion [START_REF] Krupenin | Model of strongly nonlinear vibroconducting medium with distributed impact element[END_REF][START_REF] Krupenin | To the problem on describing processes of nonlinear waves propagation through machines structures simulated by means of strongly nonlinear continuous mediums of complicated structure. Parts 1, 2[END_REF], we write the force density of impact interaction as (26)

Taking into account the given relations, introducing (20) into (21), we get a double-functional representation [START_REF] Astashev | Waves in distributed and discrete vibroimpact systems and strongly nonlinear mediums[END_REF][START_REF] Krupenin | Model of strongly nonlinear vibroconducting medium with distributed impact element[END_REF][START_REF] Krupenin | To the problem on describing processes of nonlinear waves propagation through machines structures simulated by means of strongly nonlinear continuous mediums of complicated structure. Parts 1, 2[END_REF] of the form (27)

The double-functional representations of form (27) determined, under our assumptions, the sought displacements. For the accepted interaction hypotheses, we can assume for large momenta [START_REF] Krupenin | Analysis of singularized motion equations of latticed vibroimpact 2D systems in renouncing Newton's hypothesis[END_REF] that τ λ (x) = τ λ = π/2 = const(x). We use ( 13) and ( 22). We have

(28) = - - χ -λΦ ∫ /2 0 0 1 0 ( , ) ( , ) ( ) [ ( , )] , T y x t u x t t s y x s ds - = ω ω ω ω 1 1 1 0 0 0 ( , ) [ sin cos ][ cos ] , u x t P x t E l - ρ 1 E ∞ - - - =-∞ χ = - + ω + ω ∑ 1 2 2 1 ( ) (2 1) exp[ (2 1) ]. k t T k i k t ∞ - - = - - χ = ρ Ω Ω Ω - = + π Ω = + π = ρ < ≤ ∑ 1 1 0 1 1 ( , , ) ( ) ( cos1/4 ) sin( )sin( )cos[ ( 1/4 )], 1 [(2 ) (2 1) ], 1/2 (2 1) , / , 0 . 2 n n n n n n n n x z t lT T h x h z t T h l n a n l a E t T ∞ - + = χ = ρ + ω ∑ 2 1 2 1 1 ( , , ) 4( )
( , )cos[(2 1) ]. k k x z t Ta G x z k t - - = -ω ω ω - ω ≤ ≤ = -ω ω ω - ω ≤ ≤ 1 0 0 0 0 1 0 0 0 0 ( , ) [ cos ] cos ( )sin , 0 , ( , ) [ cos ] cos ( )sin , k k G x z k k k z l k x x l G x z k k k x l k z z x l λ λ λ Φ = δ -ϕ -τ 0 T / 2 ( ) ( ) ( )[ ( ) ( )]. y J x t t x x λ λ λ λ λ λ = - - χ -ϕ -τ - χ -ϕ -τ ∫ 0 1 1 0 ( , ) ( , ) ( ) [ ( ) ( )] ( ) [( , , ( ) ( )] . t y x t u x t J x t x x J z x z t x x dz λ - λ λ λ - λ = -χ χ ϕ -ϕ -ε ω χ ω ω ω ϕ ∫ 1 0 1 0 1 1 0 1 0 0 0 ( ) ( ) [ , , ( ) ( )] [ cos ] sin cos[ ( )]. t J x J J z x z x z dz P E l x x
Here, ω 0 = ωa -1 , χ 1 = χ 1 (τ λ ), J λ0 = -Δ/χ 1 (τ λ ) is the momentum density for the perfectly solid carrier part of medium; χ 1 (τ λ ) = 1/2τ λ -1/4πω -1 . The second condition determining the parameters of motion gives (14) or other equivalent relations. We have from formula (20) (29)

7.

Let us consider passage of a vibration near the first resonance frequency. It is shown in [START_REF] Krupenin | To the theory of strongly nonlinear vibroguides[END_REF][START_REF] Krupenin | Model of strongly nonlinear vibroconducting medium with distributed impact element[END_REF][START_REF] Krupenin | To the problem on describing processes of nonlinear waves propagation through machines structures simulated by means of strongly nonlinear continuous mediums of complicated structure. Parts 1, 2[END_REF] that in this case, in such systems, in-phase forms of nonlinear oscillations are achieved. Let the linear density of equipment be significantly less than the linear mass of the carrier rod ρ ≫ 1.

In the considered Hamiltonian case and in this frequency range, the phase values are as follows: ω 0 ϕ λ = 0, π. Under the action of real dissipation, only a larger value of the impact pulse can correspond to stable operation in the originating system (ρ → ∞), so for ω 0 <π/2l we select ω 0 ϕ λ = 0; meanwhile, for π/2l < ω 0 < π/l, we select ω 0 ϕ λ = π.

It should be especially emphasized that when analyzing the situation in above-resonance ranges ω 0 > π/l, it is necessary to introduce anti-phase forms of nonlinear oscillations of different kinds [START_REF] Krupenin | To the problem on describing processes of nonlinear waves propagation through machines structures simulated by means of strongly nonlinear continuous mediums of complicated structure. Parts 1, 2[END_REF]. For vibroconductors with internal breaks, the description of such forms requires special considerations going beyond the scope of this article, because it is necessary to refine the model.

Limiting ourselves to the most important case of synchronization of hammers near the first resonance range, we note that if ϕ = const, then Eq. ( 29) is the Fredholm equation. Believing ρ -1 to be small, outside the small neighborhood of ω 0 > 0.5π/l using the successive approximations method, we find the first approximation for the impulse density J λ (x), assuming that in the zero-order (generic) approximation J λ (x) ≈ J λ0 = -Δ/γ 1 (τ λ ).After calculations taking into account ( 23)-(28), we have From formula (15), with an accuracy to small quantities of highest order, we find From here, we can find that for ω 0 < π/2l, ϕ λ = 0, the vibroconductor is transparent to the fundamental tone and the structure is actively oscillating. If ω 0 > π/2l, ϕ λ = π, then the vibroconductor is locked and the fundamental tone of vibrations practically does not pass [START_REF] Krupenin | To the theory of strongly nonlinear vibroguides[END_REF][START_REF] Krupenin | To the problem on describing processes of nonlinear waves propagation through machines structures simulated by means of strongly nonlinear continuous mediums of complicated structure. Parts 1, 2[END_REF]. In both cases, the representative set of highest harmonic components is generated in the structure.

Fig. 1

 1 Fig. 1.

  Fig. 3.


Zayed Alsayed

Guillaume Bresson

Anne Verroust-Blondet

Fawzi Nashashibi

Failure Detection for Laser-based SLAM in Urban and Peri-Urban Environments

Simultaneous Localization And Mapping (SLAM) is considered as one of the key solutions for making mobile robots truly autonomous. Based mainly on perceptive information, the SLAM concept is assumed to solve localization and provide a map of the surrounding environment simultaneously. In this paper, we study SLAM limitations and we propose an approach to detect a priori potential failure scenarios for 2D laser-based SLAM methods. Our approach makes use of raw sensor data, which makes it independent of the underlying SLAM implementation, to extract a relevant descriptors vector. This descriptors vector is then used together with a decision-making algorithm to detect failure scenarios. Our approach is evaluated using different decision algorithms through three realistic experiments.

I. INTRODUCTION

The Simultaneous Localization And Mapping (SLAM) concept is widely considered to be the solution for making mobile robots truly autonomous. The idea behind SLAM is for a mobile robot to be able to incrementally build a map of its surroundings while estimating its pose (position and orientation) within this map. Since the formulation of its theoretical concept and its probabilistic solution, SLAM has been quickly adopted and developed by the scientific community. Various implementations of SLAM have been proposed, based on a variety of perception-sensors and applied to different fields for different types of platforms.

SLAM algorithms are designed to handle robot localization based on features extracted from the environment, also called landmarks, which are assumed to satisfy certain criteria, such as being unique, static and of sufficient number to estimate the robot's relative displacement.

If there appears to be an insufficient number of landmarks, it is put down to the fact that landmarks had been badly defined -which is not necessarily the case as the environment need not be limited to only one configuration. Autonomous cars could navigate in a wide range of scenarios where we must not be limited to exploiting a single or one set of specific types of information. In other words, the type of information content in the environment can change according to the local scenario.

Furthermore, SLAM approaches based on raw data, such as maximum-likelihood SLAM, which consider raw data, without extracting specific features, cannot avoid failure in non-salient (ambiguous) configurations, for example the tunnel passage scenario, where the environment configuration does not propose a unique/distinctive configuration to estimate the displacement. These ambiguous configurations also apply to landmark-based approaches.

Work done on SLAM generally focuses on combining other positioning information with SLAM in different ways, although to our knowledge, specifically detecting these odd scenarios a priori in order to ensure the operability of SLAM independently of any other information sources has not yet been explored. Work on failure detection has been carried out, although it focuses on comparing different sources of information rather than studying the limitations of the techniques themselves.

In this paper we propose an approach to detect potential SLAM failure scenarios based only on information perceived from the environment (SLAM input) only. Our approach is designed for 2D laser-based SLAM methods.

The rest of this paper is organized as follows: Section II presents the state of the art regarding SLAM algorithms and Fault Detection and Isolation (FDI) systems applied to localization. Then, Section III introduces limitations of the SLAM concept, illustrated by examples of 2D laser-based SLAM. Section IV presents our method to detect possible failure of laser-based approaches by processing laser data. Finally, Section V presents the results of our experiments and we conclude by giving some perspectives in Section VI.

II. STATE OF THE ART

Initial work on SLAM led to the foundation of a theoretical solution [START_REF] Bailey | Simultaneous localization and mapping (SLAM): Part II[END_REF][2] [START_REF] Cadena | Past, present, and future of simultaneous localization and mapping: Toward the robust-perception age[END_REF], although many issues emerged from practical implementations, and various methods have been proposed. These methods can essentially be classified according to the map representation, the estimation processes or the type of sensor used.

The first SLAM approaches used landmark-map representations and were based on Extended Kalman Filters [START_REF] Durrant-Whyte | Simultaneous localization and mapping: part I[END_REF] and Particle Filters [START_REF] Montemerlo | FastSLAM: A factored solution to the simultaneous localization and mapping problem[END_REF]. Other solutions used grid map representations [START_REF] Alsayed | PML-SLAM: a solution for localization in large-scale urban environments[END_REF][6] [START_REF] Vu | Vehicle perception: Localization, mapping with detection, classification and tracking of moving objects[END_REF] and were based on Likelihood Maximization. In these different approaches, perception data (the input of SLAM) are implicitly supposed to be static and unique in order to allow SLAM to operate properly.

The static world assumption is handled using strategies called SLAM-DATMO [START_REF] Wang | Ladar-based detection and tracking of moving objects from a ground vehicle at high speeds[END_REF][9] [START_REF] Vu | Vehicle perception: Localization, mapping with detection, classification and tracking of moving objects[END_REF], which integrate mobile object detection and tracking together with the SLAM process. Another trend is based on the transferable belief model [START_REF] Trehard | Credibilist Simultaneous Localization and Mapping with a LIDAR[END_REF] where the map representation makes it possible to deal directly with ambiguous information (dynamic/static). This method does not attempt to detect or track mobile objects, rather it treats them as ambiguous information to be very low weighted while maximizing the likelihood.

Error accumulation can be dealt with using a backward correction algorithm such as graph-SLAM [START_REF] Grisetti | A Tutorial on Graph-Based SLAM[END_REF], which rectifies the previous estimations when detecting loop-closing. Such approaches can correct drift caused by longterm operation and therefore reduce ambiguity, but require more information from the environment before rectifying or recovering from errors.

Generally, Fault Detection and Isolation (FDI) systems exploit information redundancy and measure the coherence of different sources together with a motion model for prediction to detect faults or failure of the sources. In [START_REF] Roumeliotis | Sensor Fault Detection and Identification in a Mobile Robot[END_REF] the authors analyze the residuals at the output of a bank of Kalman filters by thresholding, while in [START_REF] Goel | Fault detection and identification in a mobile robot using multiple model estimation and neural network[END_REF] they use a Neural Network to detect failure. In [START_REF] Sundvall | Fault detection for mobile robots using redundant positioning systems[END_REF] the authors provide some improvements to the method by measuring the coherence of different estimations. The authors of [START_REF] Morales | Vehicle localization in outdoor woodland environments with sensor fault detection[END_REF] combine fault detection with outlier detection based on the Normalized Innovation Squared (NIS) test. [START_REF] Jabbari | Application of Computational Intelligence for Sensor Fault Detection and Isolation[END_REF] use neural networks as approximators to predict the future state, and to detect failure. Their approach is designed for food transportation applications.

Wei et al. [START_REF] Wei | Camera/laser/gps fusion method for vehicle positioning under extended nis-based sensor validation[END_REF] present a fusion method for redundant localization information in order to build a consistent localization system. They start by a selection and validation step based on measuring the information coherence from every sensor, then the filtering is done using KF-like techniques (EKF, UKF, IF, UIF). Although promising, the evaluation method used does not give a clear idea about the accuracy of the method.

Localization could be achieved based on several techniques depending on each algorithm's limitations and operating range. Bresson et al. [START_REF] Bresson | A Cooperative fusion architecture for robust localization: Application to autonomous driving[END_REF] present a cooperative system based on a selection between several localization techniques, where the operating range of each algorithm is defined a priori on the test circuit. Experimental results prove the validity of such a methodology to achieve localization for autonomous driving.

In this paper, we study SLAM failure and ambiguity scenarios when considering 2D laser-based approaches. Then we propose to detect a priori these possible failure situations based on extracting relevant descriptors from raw perception data. Our approach does not put any assumption on how the underlying SLAM works.

III. SLAM FAILURE DESCRIPTION

By defining failure and non failure of SLAM we intend to analyze the limitations of SLAM algorithms and consequently define their operating range (where they are supposed to operate properly).

We define the term current scene as the configuration of the local environment perceived by SLAM. In practice, it is represented by the current sensor observation Z t jointly with its projection on the local map Z t ∪ M t-1 where M t-1 is the local SLAM map at previous time step. We define saliency as the general term which qualifies a landmark or a set of landmarks as being sufficient information for the SLAM process to calculate a unique solution that is coherent with the real displacement. Mobile objects are excluded from this definition as they invalidate the estimation.

Along the same lines, a salient scene is characterized either by containing at least one salient landmark that is both present in the current observation and on the map of the previous time step, or by a salient configuration of a set of landmarks that are shared between the current observation and the previous time step map. In contrast, a non-salient scene contains one or more landmarks that, taken as a whole, are not salient.

For the sake of clarity, Figure 1 illustrates some important failure and non-failure scenarios of 2D laser-based SLAM operating on a ground vehicle in urban and peri-urban environments. The roadside scenario in Figure 1(a) presents a potential non-salient landmark, although the presence of the boulder which is salient on the right side saves the situation and the scenario becomes salient. In Figure 1(c) adding a parallel roadside does not give any more information and the whole configuration remains non-salient. However, Figure 1(b) shows non equidistant roadsides where each side proposes a set of possible solutions, but the two together intersect at only one point. If salient landmarks are identical and regularly distributed, this may result in a non-salient scene, as we can see in Figure 1(e). Actually, this last scenario is tricky because it depends on the distances between the landmarks and the vehicle's maximum possible velocity. The presence of mobile objects in the scene, as shown in Figure 1(d), may mislead SLAM, as may vegetation or dust, as shown in Figure 1(f), due to the imprecision/vagueness of their position/measurement.

In the following section we present our approach to detecting SLAM failure.

IV. FAILURE DETECTION

Detecting SLAM failure and non-failure comes down to analyzing the saliency of the current scene. Our approach is suited to 2D laser-based SLAM, so we will draw on the different scenarios we presented in Section III. The general block diagram of SLAM with SLAM's Failure detector is illustrated in Figure 2. The diagram shows that the failure detection and SLAM module are independent and operate simultaneously. Therefore, the failure detection decision does not affect the operation of the SLAM algorithm but should guide the later processing.

A. Approach outline

To ensure the operability of 2D laser-based SLAM algorithms, it is necessary to first study the characteristics of different elements present in the scene. Our strategy consists of the following steps:

1. Clustering of the laser observation: the 2D laser points Z t will be distributed onto elements

S t = {E 1 t , E 2 t , ..., E J t }
where each E j t corresponds to a subset of the observation reflected from one element in the scene. The elements with non sufficient information will be considered as isolated points Z t = S t I t where I t groups all the isolated points from the current laser scan.

2. Detecting, tracking and filtering mobile objects S M t . In this way, we keep a list of remaining elements S R t with S t = S M t S R t . The issue of mobile object detection is beyond the scope of this paper, and we do not deal with it here.

3. Characterizing remaining elements: where the aim is to characterize and find a 2D geometric approximation for each remaining element E r t ∈ S R t . We approximate the element E r t either by line (first degree polynomial), circle (quadratic polynomial) or a cubic-spline (set of third degree polynomials). Following the order they are presented in, only one approximation will be validated based on two consistency measures.

4. Extracting current scene descriptors: based on element characterization, their point cloud and their approximation function, we define a set of descriptors D t to describe the whole scene and the relations between different non-salient elements.

5. Decision-making: the information extracted from the previous steps serves as input for failure detection algorithm. Several strategies are proposed.

B. laser observation clustering

An appropriate clustering method should consider the distance between points as a similarity criterion. In addition, the Euclidean distance between a couple of points separated by the same angle varies depending on how far the couple of points is from the sensor origin. In order to group points reflected from the same element, we use a modified version based on a connected component labeling algorithm [START_REF] Samet | Efficient component labeling of images of arbitrary dimension represented by linear bintrees[END_REF]. This derived version uses a circular neighboring window of variable size. Based on ρ which is the distance of the point considered to the sensor center, the window size is calculated as follows:

ω(ρ) = c × √ ρ ( 1
)
where c is an amplification coefficient determined experimentally.

Finally, elements with non sufficient information i.e. three or less points, will be considered as isolated points as they do not provide enough information. Thus Z t = ( J j=1 E j t ) I t where I t groups all the isolated points from the current laser scan, and S t = {E 1 t , E 2 t , ..., E J t } is the set of clusters.

C. Geometric characterization of the remaining elements

In order to extract an appropriate description of the scene for failure detection, we need to characterize the different elements composing the scene. A relevant characterization should focus on the geometric representation of salient and non-salient elements.

The saliency of each scene element E r t is directly related to its 2D approximation. Non-salient point configurations are commonly observed as long line-segments or an arc of a circle of large radius, hereafter referred to as a large-circle-arc. In contrast, salient elements have more complex forms, which we refer to as curves, such as open polygons e.g. a building with one or several corners, or some other non-regular but basically consistent shape.

Consequently, every element E r t in the scene will then be approximated by either:

1. a line-segment: using an algebraic linear regression [START_REF] Freedman | Statistical Models: Theory and Practice[END_REF]. Line-segments are non-salient elements, although they provide useful information but only along one axis, i.e. the distance between the vehicle and the roadside cf. Figure 1(a).

2. a circle-arc: using an algebraic circular regression following Taubin [START_REF] Taubin | Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation[END_REF] [START_REF] Al-Sharadqah | Error analysis for circle fitting algorithms[END_REF]. laser points distributed along an arc could suggest a non-salient configuration cf. Figure 1(c). 3. a cubic spline curve: using a smoothing cubic spline algorithm [START_REF] Reinsch | Smoothing by Spline Functions. II[END_REF], the aim is to handle all other shapes in order to qualify the element as {smooth-curve, noisy-curve or nonqualified-curve} according to the consistency of its point-cloud silhouette.

Following the order they were presented in, only one geometric approximation is chosen to represent an element. For this purpose, the approximation is validated through two consistency measures (δ 1 , δ 2 ) together with a couple of thresholds (τ 1 , τ 2 ) set experimentally. These consistency measures calculate the quantity and quality of dispersion of the pointcloud from the suggested approximation.

Assuming the element E r t , which is formed by a point-cloud of N laser impacts E r t = {z 1,r t , z 2,r t , . . . z N,r t }. Let us also assume that its 2D approximation is represented by A(i, E r t ) where i ∈ {1 : line -segment, 2 : circle -arc, 3 : curve}.

The function dist(z n,r t , A(i, E r t )) calculates the shortest distance between a laser endpoint z n,r t and a 2D approximation A(i, E r t ). Hence the consistencies are calculated as follows: The dispersion quality δ 1 is the mean of the cumulative distance of each point in the point-cloud from the approximated shape:

δ 1 (E r t , A(i, E r t )) = 1 N N n=1 dist(z n,r t , A(i, E r t )) (2)
The dispersion quantity δ 2 is the percentage of good fittings i.e. the normalized number of points which are under a certain distance σ (fixed according to the laser measurement accuracy) from the approximated shape:

δ 2 (E r t , A(i, E r t )) = 1 N × card {z n,r t ∈ E r t | dist(z n,r t , A(i, E r t )) < σ} (3)
To summarize, at the end of this step, each element E r t will have, in addition to its point cloud, an attributed geometric label in {line-segment, circle-arc, smooth-curve, noisycurve, non-qualified-curve}, its approximation function and its dispersion values (δ 1 , δ 2 ) to the selected approximation. This information will be used in the next step to extract a relevant description vector of the scene.

D. Descriptor extraction

Once the different elements that make up the scene have been characterized (a geometric representation and an appropriate function have been determined), we still need to determine how their organization in the environment may affect the SLAM process. As we have seen in Figure 1(c), when the scene contains non-salient elements of the same class (i.e. only line-segments or only circle-arcs) it is important to calculate how parallel the line-segments are, and how concentric the circle-arcs are.

According to this and the previous description, we propose a set of 20 descriptors D t = {d 1 t , d 2 t , . . . d 20 t } to decide whether the current scenario corresponds to SLAM failure or not. These descriptors are listed below:

1) Maximal difference between line-segments slopes: designed to measure the parallelism between line-segments. Let us consider having a number of Q line-segments sorted according to their slope values. Each element q ∈ {1, 2, . . . Q} is approximated by A(1, E q t ) y = a q x + b q .

d 1 t = f (a Q ) -f (a 1 ) (4)
with f : → is the function that takes the slope value as an argument and returns the corresponding incline angle.

2) Diameter of the centers of the large-circle-arcs: this descriptor measures the concentricity between circle-arcs. Let us consider having a number of K circle-arcs where element k ∈ {1, 2, . . . K} is approximated by

A(2, E k t ) (x -x k 0 ) 2 + (y -y k 0 ) 2 = r 2 . d 2 t = argmax p,k∈C g(p, k) ( 5
)
where C is the set of center points of circle-arcs and g : → is the function that returns the Euclidean distance between two points.

3) Raw observation number of points:

d 3 t = card(Z t ).

E. Decision making

All the previous processing results in transforming the raw observation Z t into a set of descriptors D t appropriate for decision making algorithms. For this purpose we propose to compare two strategies to detect SLAM failure:

1) Inference rules for SLAM failure detection: Based on a set of rules defined thanks to the previously described scenarios (Section III). The aim is to match the current scene description to know whether it could provoke a failure for SLAM or not. This strategy supposes that mobile objects were properly removed and noisy elements were correctly detected. In addition it favors the decision failure when the situation in question is not clear enough.

2) Supervised machine learning for SLAM failure detection: Machine learning algorithms require a training phase where the learning algorithm modifies the model's weights in order to learn how to take a decision when faced with different possible scenarios. A labeled database is mandatory in order to teach these models. The training examples for these algorithms are formed by the set of descriptors D i together with an adequate label:

T = {(D i , l i ) | l i ∈ {+1, -1}}
where l i = {+1} indicates that the current scene is labeled favorable for SLAM, and l i = {-1} indicates a failure for SLAM.

Once the training phase has been completed, the learnt model can be used for failure detection. In our study we have considered standard machine learning methods used for classification [START_REF] Alpaydin | Introduction to Machine Learning (Adaptive Computation and Machine Learning Series)[END_REF] [START_REF] Yoo | A comparison of logistic regression, logic regression, classification tree, and random forests to identify effective gene-gene and gene-environmental interactions[END_REF]. The algorithms we tested are the following:

1) Logit regression. 2) Multilayer perceptron (MLP). 3) Ensemble MLP. 4) Decision forests. 5) Adaptive Boosting.

V. DATASETS, EXPERIMENTS AND RESULTS

A. Labeling and Datasets

The approach we are presenting in this paper has not been explored before, consequently, benchmarks with appropriate data (i.e. labeled failure and favorable for SLAM) to evaluate our algorithm do not exist. Furthermore, automatic labeling cannot be an option due to the undefined behavior of SLAM when in a failure scenario. For these reasons we labeled our experimental data manually, frame by frame. This was done based on current laser scan views together with a local-map view. In practice, objective labeling is not an easy task due to the noise accompanying laser observations, the diversity of elements present in the environment, their geometric shape and identifying some specific configuration. For example, a human expert cannot estimate the precise incline difference between roadsides, and consequently the real impact it can have on the SLAM estimation. To mitigate this kind of bias the labeling was done by an expert in SLAM algorithms.

Several data sequences were acquired using a moving vehicle in a variety of urban and peri-urban environments. This car is equipped with five IBEO-Lux sensors deployed to cover a 360°field of view around the vehicle. Table I gives a quick statistical overview of the acquired datasets and test sequences. Failure Detection dataset built upon a variety of subsequences without mobile objects in the view. This database is intended mainly for training and contains only clear passages which required less difficulty to be labeled.

Plaisir & Amsterdam Circuits are two long and difficult sequences covering a diversity of scenarios containing mobile objects that have not been removed. The first circuit, acquired at Plaisir (France), is 14Km long, starting from a car park, passing round-abouts, dusty roads and rural areas with only one roadside visible in the laser view. The second circuit is more tricky for SLAM as the car was driving mainly on highways around Amsterdam. In order to avoid subjective labeling, confusing passages were omitted (not labeled) from these sequences.

In addition to these datasets, we use the sequence (01) from the KITTI odometry dataset [START_REF] Geiger | Vision meets robotics: The KITTI dataset[END_REF] that contains potential failure passages for 2D horizontal laser-based SLAM approaches (i.e. a highway with mobile cars).

B. Experiments

The validity of our approach was experimented through three test scenarios:

1. The first experiment is based on our Failure Detection dataset which is divided as follows: four fifths for training and one fifth for evaluation. It is important to mention that subsequences of the test-set and training-set were acquired from different environments and do not contain mobile elements.

2. The second experiment aims to evaluate the proposed approach in realistic difficult scenarios (a complete sequence) and its sensitivity to the presence of mobile objects (not removed). In this experiment we take the whole Failure Detection dataset as a learning-set and we evaluate our approach on the Plaisir and Amsterdam sequences.

3. The third experiment aims to evaluate the impact of Failure Detector on SLAM's estimation. We applied our Failure Detector using the best rated method from the previous experiments together with a PML-SLAM [START_REF] Alsayed | PML-SLAM: a solution for localization in large-scale urban environments[END_REF] implementation. This SLAM implementation is a maximum likelihood 2D laser-based SLAM approach that uses an occupancy grid representation.

C. Results and Discussion

The results of the first and second experiments are shown in Table II The FD-Test dataset evaluation gives considerably more satisfactory classification rates than on the Plaisir circuit. This is due to the presence of mobile objects, which can result in a higher False Positive rate, and to noisy passages that induce a considerably higher False Negative rate. These arguments are also valid for the Amsterdam circuit evaluation, except that there are many fewer noisy passages and less vegetation areas.

The results of the third experiment are illustrated in Figure 3 where the histogram shows the errors in estimating the displacement at each SLAM time step, with and without our Failure Detector. This KITTI test sequence contained initially 1101 frames which were annotated using AdaBoost as follows: 411 (37.33%) as Favorable and 690 (62.67%) as Failure, which is coherent as the car mainly navigated on a highway with several exits/junctions along the way. Note that for the failure detection evaluation, we only consider the estimation of SLAM for frames annotated Favorable which were immediately preceded by a Favorable frame. This reduces the number of frames considered to 266 (24.18%) estimations out of 1100.

SLAM failure can be seen in the histogram in Figure 3 represented mainly by the aberrant displacement estimations (i.e. errors of large value). If we consider that SLAM failure occurs when an error in displacement is greater than some tolerance value, Table IV shows the number of aberrant SLAM estimations with and without our Failure Detector. We observe that our Failure Detector identifies most of these aberrant estimations at a reasonable tolerance value, i.e. 4 aberrant estimations if the tolerated range of error in displacement is bounded by 15cm. In contrast, considering the same criterion, i.e. a 15cm tolerance value on estimation errors, we see that the Failure Detector also eliminates many frames which seem to give acceptable estimations i.e. 690 -119 = 571 frames. The reasons mentioned earlier about the subjective nature of labeling are still valid here. In addition, having a reasonable error does not necessarily imply that the scenario is favorable for SLAM, particularly with the presence of misleading elements such as mobile objects which were not removed.

VI. CONCLUSION AND PERSPECTIVES

In this paper we have presented an approach to detect and isolate odd configurations of the surrounding environment which could potentially mislead SLAM. As an example, the well-known tunnel scenario where localization using SLAM becomes a difficult, if not impossible task. State-of-the-art methods do not attempt to detect such limitations of SLAM algorithms.

Our approach is designed for 2D laser-based SLAM algorithms based solely on the analysis of the sensor data. First we cluster the laser observation into elements. Then each element is characterized by a 2D geometric approximation which is used together with laser points to build a relevant features vector. Finally, various standard decisionmaking algorithms are used and compared. The validity of our approach was proved through three experiments. AdaBoost learner succeeded in properly classifying (failure or favorable) 85.57% of scenarios. The last experiment showed that using Failure Detector together with SLAM helped to isolate almost all the aberrant estimations given by SLAM. In contrast, many other estimations were omitted even though they had a reasonable error range.

In future work, we plan to extend our technique and study the internal elements of the SLAM process in order to detect possible failure. In addition, it is worth studying the impact of using a training set which contains noisy passages.

Fig. 1 .

 1 Fig. 1. 2D laser-based SLAM failure and non failure scenarios from a ground vehicle in urban and peri-urban environments. Red ellipses represent the uncertainty area on position estimation induced from the present scenario.

Fig. 2 .

 2 Fig. 2. Block diagram of Failure Detector algorithm with SLAM

4 ) 6 ) 8 ) 13 ) 14 ) 17 ) 18 )

 46813141718 Number of isolated laser points: d 4 t = card(I t ). 5) Number of current scene elements and their cardinal. 7) Number of line-segment elements and their cardinal. 9) 10) Number of circle-arc elements and their cardinal. 11) 12) The number of smooth-curve elements and their cardinal. Number of noisy-curve elements and their cardinal. 15) 16) The number of non-qualified elements and their cardinal. Number of parallel line-segments and their cardinal. 19) 20) Number of concentric arc elements and their cardinal.

Fig. 3 .

 3 Fig. 3. Histogram of errors in displacement estimation by frame, with and without Failure Detection

  which summarizes the best correct classification rates per method over the different sequences. TableIIIgives more detailed information, as it shows the confusion matrix which is organized as follows: Negative refers to SLAM failure and Positive refers to SLAM non-failure. True Negative represents the percentage of frames which were annotated failure and classified failure by the decision making method. False Positive refers to failure frames that were wrongly classified favorable, while False Negative refers to the percentage of Favorable frames which were wrongly classified failure. Finally True Positive refers to favorable frames which were properly classified.The results show that AdaBoost gives the best results over the other methods, with considerably better correct classification rates on the FD-Test dataset and Amsterdam circuit. Random Forests, MLP and Ensemble MLP give comparable results. The logistic regression and Inference rules methods give more modest scores.

	C =	T rue N egative F alse P ositive F alse N egative T rue P ositive
	where

TABLE IV NUMBER

 IV OF ABERRANT ESTIMATIONS OF SLAM VS. SLAM +FD CONSIDERING DIFFERENT TOLERANCE ON DISPLACEMENT ERRORS

	Tolerance(cm)	5	10	15	20	30	40 50 65 80
	SLAM 439 176	119 105 93	74 52 17	2
	SLAM +FD	78	13	4	3	2	0	0	0	0