Pisey Keo

Clémence Lepourry

Hugues Somja
email: hugues.somja@insa-rennes.fr

Frank Palas

Behavior of a new type of shear connector for U-shaped steel-concrete hybrid beams

Keywords: Composite structure, steel-encased concrete hybrid beam, angle-shaped shear connector, square bar shear connector, push-out test

This paper presents an investigation of the behavior of a new type of shear connectors used in Ushaped steel-concrete hybrid beams. Besides the role in transferring the force between concrete and steel material, this new type of shear connectors, welded on the upper flange of the U-shaped steel beam, serves to maintain the shape of the steel cross-section during concrete encasement. Several forms of shear connectors can be used such as L-shaped or square cross-section. The experimental investigation of the behavior of these shear connectors through asymmetrical push-out tests is presented in this paper. A finite element model has been developed in order to identify the stress behavior of the connectors and the surrounding concrete. The FE model is validated by comparing its results against experimental data and then used to perform a parametric study. Based on the parametric study results, an analytical formula for calculating the force transfer capacity of the shear connector is proposed.

Introduction

In composite construction, profiled steel sheets have been successfully used as permanent and integral formwork for the underneath of reinforced concrete slabs [START_REF] Wright | The use of profiled steel sheeting in floor construction[END_REF][START_REF] Porter | Design recommendations for steel deck floor slabs[END_REF][START_REF] Rackham | Composite slabs and beams using steel decking: best practice for design and construction, Metal Cladding & Roofing Manufacturers Association in partnership with the[END_REF]. Profiled steel sheets serve not only as a concrete formwork during construction but also as principal tensile reinforcements for bottom fibers of the composite slab, offering an economic design solution over a plywood formwork. Acting compositely with reinforced concrete slab, profiled steel sheets produce a considerably stiffer and stronger floor system than many others. Consequently, the weight and size of primary structures as well as foundations can be reduced. Over the last few years, the profiled steel sheet has been introduced as permanent formwork and integral shuttering for the sides of reinforced concrete beams [START_REF] Oehlers | Composite profiled beams[END_REF][START_REF] Oehlers | Flexural strength of profiled beams[END_REF][START_REF] Uy | Ductility of Profile Beams. Part I: Experimental Study[END_REF][START_REF] Uy | Ductility of Profile Beams. Part II: Analytical Study[END_REF]. More recently, a U-shaped steel profile with infill concrete has been developed [START_REF] Liu | Experimental investigation on the flexural behavior of steel-concrete composite beams with U-shaped steel girders and angle connectors[END_REF]. This new composite configuration can be considered as one of the most recent developments in steel-concrete composite beam construction which provides several advantages: more ductility compared to reinforced concrete beams with the same flexural strength, increase in shear strength and shear ductility, increase in span to depth ratio, reduction of concrete creep and shrinkage, and decrease in site labor costs. Although the infill concrete in the U-shaped steel profile increases the weight of the beam, it helps mitigate local buckling of the steel webs and improve strength and stiffness of the beam. In addition, fire resistance can be assessed by ensuring that the reinforced concrete beam, surrounded by the profiled steel section, can resist the fire loads.

The overall behavior of composite steel-concrete members strongly depends on shear connection between steel and concrete encasement which may be accomplished by three main shear transferring mechanisms: chemical bonding (bond between cement paste and surface of the steel), friction (assumed to be proportional to the normal force at the interface) and mechanical interaction (due to embossments, ribs or shear connectors). The role of shear connection is essential; without it there is no collaboration between steel and concrete material. It limits the slip that may occur along the steel-concrete interface. Thus, ensuring a resumption of longitudinal shear, it allows to obtain a composite section with two components working together. However, superposition of force transfer mechanisms is not generally permitted. The experimental data indicate that direct bearing or shear connection often does not initiate before the direct bond interaction has been breached. Moreover, limited experimental data is available regarding the interaction of direct bearing and shear connection via steel anchors. Therefore, the shear connection between the two materials is supposed to be ensured only by the mechanical devices, commonly headed studs. The behavior of headed stud shear connectors has been investigated by numerous researchers worldwide by conducting push-out experimental tests [START_REF] Viest | Investigation of stud shear connectors for composite concrete-steel T-beams[END_REF][START_REF] Chinn | Pushout Tests on Lightweight Composite Slabs[END_REF][START_REF] Valente | Experimental analysis of shear connection between steel and lightweight concrete[END_REF] and/or by numerical simulation [START_REF] Lam | Behavior of headed stud shear connectors in composite beam[END_REF][START_REF] Ellobody | Performance of shear connection in composite beams with profiled steel sheeting[END_REF][START_REF] Nguyen | Finite element modeling of push-out tests for large stud shear connectors[END_REF]. Although the common type of shear connector is headed stud, some older generations of shear connectors such as channel, angle or square bar shear connectors have been increasingly interested by many researchers [START_REF] Abe | Flexible shear connectors for railway composite girder bridges[END_REF][START_REF] Maleki | Behavior of channel shear connectors, Part I: Experimental study[END_REF][START_REF] Maleki | Behavior of channel shear connectors, Part II: Analytical study[END_REF][START_REF] Pashan | New design equations for channel shear connectors in composite beams[END_REF][START_REF] Baran | An experimental study on channel type shear connectors[END_REF][START_REF] Shariati | Behaviour of C-shaped angle shear connectors under monotonic and fully reversed cyclic loading: an experimental study[END_REF][START_REF] Shariati | Comparison of behaviour between channel and angle shear connectors under monotonic and fully reversed cyclic loading[END_REF][START_REF] Soty | Formulation for shear force-relative displacement relationship of L-shape shear connector in steel-concrete composite structures[END_REF][START_REF] Shariati | Experimental assessment of angle shear connectors under monotonic and fully reversed cyclic loading in high strength concrete[END_REF] over the last decade. The installation of these older generations of shear connectors is not expensive since the procedure is similar to that used for beam stiffeners or connection components, where a specific welding equipment with high voltage is not required [START_REF] Pashan | New design equations for channel shear connectors in composite beams[END_REF]. Recently, angle connectors have been used in steel-concrete composite beams with U-shaped steel girders [START_REF] Liu | Experimental investigation on the flexural behavior of steel-concrete composite beams with U-shaped steel girders and angle connectors[END_REF]. However, no analytical models were readily available for calculating the shear force transfer capacity of this shear connector type, used in this new composite beam configuration, as yet.

In this paper, the behavior of two types of shear connector for U-shaped steel-concrete hybrid beams (USCB), see Fig. 1 is investigated. Besides the role in transferring the force between the two materials, these shear connectors, welded on the upper flange of the U-section, serve to maintain the shape of the steel cross-section during concrete encasement. Several forms of the connector cross-section can be used such as L-shaped or square cross-section. The L-shaped angle connectors, in the absence of top flange, could be cheaper and more economical compared to the channel ones. In general, a hoop reinforcement should be provided for the L-shaped angle connector while being used in classical composite beams, in order to limit the uplift of the concrete slab [START_REF]Eurocode 4: design of composite steel and concrete structures-Part 1.1: General rules and rules for buildings[END_REF]. However, it is not the case for L-shaped angle connectors used in U-shaped steel-concrete hybrid beams in consideration of the connector part fully embedded in the concrete. This new type of shear connection is not covered in present norms of composite structures and it requires an investigation on its behavior and on force transfer mechanisms. This paper presents the experimental investigation of the behavior of these shear connectors through asymmetrical push-out tests. Two different types of the shear connector cross-section are considered: square and L-shaped section. To get further insight into the force transfer mechanism while using L-shaped shear connectors as connector devices in USCB, a finite element model is developed. The later is validated by comparing its results against four experimental data tests. Based on a parametric study using FE model, an analytical design formula for shear connectors is proposed.

Experimental program

An experimental program is developed to quantify the strength and deformation capacities of shear connectors as well as to gain an insight into the force transfer mechanism in the Ushaped steel-concrete hybrid beams. A modification of the typical push-out test setup proposed by Eurocode 4 [START_REF]Eurocode 4: design of composite steel and concrete structures-Part 1.1: General rules and rules for buildings[END_REF], which is usually adopted for classic shear stud connectors, is made to represent the real situation of the U-shaped steel-concrete hybrid beams. Primarily, two experimental tests with two different types of connector cross-section are conducted in order to choose an appropriate connector type to be used in USCB; as stated earlier, the square and L-shaped connector crosssection are considered. Next, three additional tests are conducted for the selected appropriate connector type, L-shaped shear connector in this case. In the following, the detailed description of the experimental program is presented.

Geometric description of the test configuration

The test setup consists of a universal testing machine with a capacity of 1500 kN, a U-shaped steel-concrete hybrid beam specimen, reaction blocks and bracing systems, see Fig. 2 and Fig. 3. The actuator is fixed to one end of the steel beam. It is attached to a 40 mm-thick steel plate that serves as a platform for push-out test specimens. The 40 mm thick steel plate is welded to two 6 mm-thick gussets, which are connected to the webs of steel beam at one end via twelve 20 mm-diameter high-strength bolts. At the other end of the specimen, the back side surface of the concrete floor is put in contact with a rigid steel beam. The specimen is positioned horizontally on 2 vertical supporting steel columns where at the top surface of their end-plate (with slotted holes), two greased PTFE plates are placed, allowing the horizontal displacement of the steel beam to occur. This test setup is similar to the one used by Lowe et al. [START_REF] Lowe | Characterization of the Splitting Behavior of Steel-concrete Composite Beams with Shear Stud Connection[END_REF] and Topkaya et al. [START_REF] Topkaya | Composite shear stud strength at early concrete ages[END_REF]. The advantage of this setup is that there is only one slippage interface, compared to two interfaces for a typical push-out test.

Test specimens

The specimen consists of a U-shaped steel beam, two precast-slabs, a reinforced concrete beam encased in a U-shaped steel beam and several shear connectors. The internal surfaces of U-shaped steel beam are greased before concrete encasement to ensure that the force transfer between the two materials is done only through shear connectors. The geometry of the specimen is the following. U-shaped steel beam is 500 mm high, 300 mm wide and 6 mm thick. The webs are stiffened by horizontal and vertical plates in order to avoid a premature collapse of the specimens due to web buckling. The precast slabs are 70 mm thick and 450 mm wide. They are posed on each steel flange. The grid rebar ST35 is used for precast and slab reinforcement in which the rebar diameter is 7 mm and the spacing in transverse and longitudinal direction is 300 mm and 100 mm, respectively. The detailed description of specimen cross-section and reinforcement is presented in Fig. 4c and Fig. 4d, respectively. The connectors are welded on the flanges of the steel beam. Five specimens with different types of connector are evaluated. The first specimen denoted "PO-S20", Fig. 4a, has 4 shear connectors with square-shaped section of 20×20 mm. The second specimen denoted "PO-L40", Fig. 4b, has 3 shear connectors with L-shaped section of 40×40×4 mm. The third to fifth specimen denoted respectively "PO-L50a", "PO-L50b" and "PO-L50c" have 3 shear connectors with L-shaped section of 50×50×5 mm. The spacing between each connector for all specimens is 300 mm.

Materials

The concrete for RC beam-floor has a strength class of C25/30. The concrete characteristics at the day of test are determined on cylinder samples with dimensions of 11 × 22 cm. Due to the different usage of concrete in precast slab and RC beam-floor, the concrete characteristics of the precast slab are measured separately. Coupon samples are taken from the steel sheet, forming the U-section, and from the shear connector (L-angle and square section) used in the push-out specimens. Results of material characteristic tests are summarized in Table 1. It is worth mentioning that PO-L50a,b,c specimens are tested at an early age of the concrete in order to have a concrete strength at the day of test that is about 60% to 80% of concrete characteristic strength, as required by EC4 [START_REF]Eurocode 4: design of composite steel and concrete structures-Part 1.1: General rules and rules for buildings[END_REF]. (d) Specimen rebar detailing. Figure 4: Test specimen.

f cm (MPa) f cm (MPa) f y (MPa) f u (MPa) f y (MPa) f u (MPa) PO-S20 27

Loading procedure and instrumentation

The specimens are tested under monotonic loading. The compressive load is applied horizontally and monotonically on the specimen via a hydraulic jack. Monotonic loading involves a slow increment of the load using the displacement-control method with a rate of 0.015 mm/s. For PO-S20 and PO-L40 specimens, the loading is split into 3 stages. At first stage, the load is firstly increased monotonically up to 215 kN (value estimated for service limit state design) before being decreased to zero. Thereafter, the loading is resumed at the second stage and increased up to 300 kN (value estimated for ultimate limit state design) before being decreased once again to zero. At the final stage, the reloading is applied until the failure of the specimens. On the other hand, PO-L50(a,b,c) specimens are subjected to 25 loading-unloading cycles with a maximum load of 328 kN (value estimated for service limit state design) and a minimum load of 50 kN before being submitted to incremental loading up to failure. The force reading is recorded by the universal machine. To measure the relative horizontal displacement and vertical separation of the components, displacement sensors are installed. The measurement of the relative horizontal displacement (slip) between the concrete and the steel beam are made through six LVDT displacement sensors that are installed at both sides of the beam (3 at each side of the beam), at the location where the connectors are placed, see Fig. 5. Four LVDT displacement sensors (2 at each side) are installed to measure the vertical separation between the reinforced concrete and the steel beam (uplift). The displacement readings are recorded through a data acquisition system connected to the displacement transducers.

Test results

This section presents the experimental test results by highlighting the failure modes and the load-slip/uplift response of the specimens.

Failure mode

For PO-S20 and PO-L40 specimens, no concrete cracking is observed during the test. The failure mode of both specimens is associated with the failure of the connectors and/or the local concrete crushing. For PO-S20 specimen, the failure is due to the local concrete crushing. The failure mode observed in PO-L40 specimen is the connector fracture, see Fig. 6a, that is associated with the plastic deformation of connectors due to shear near the fillets between the angles of the cross-section and at the leg angle welded on the flange of the steel beam. This mode of failure leads to a large ductility in the force-slip response. The end of the plastic yielding is due to the fracture of the connector and of the welds. A failure mode similar to PO-L40 specimen is observed for PO-L50(a,b,c) specimens, see Fig. 6b. However, it is observed that at a load level near the ultimate load, concrete cracking appeared on top surface of concrete slab, on back surface in contact with rigid steel beam and on front surface at the junction between precast slab and concrete infill of PO-L50(a,b,c) specimens. Nevertheless, concrete cracking did not lead to total failure of the specimens. For PO-L50c specimen, the fracture of all connectors happened simultaneously. After removing the U-shaped steel beam from PO-L50c specimen, it appears that the concrete experienced inelastic deformations or local crushing/splitting, see Fig. 7. The gap, produced by local concrete crushing, allowed the connector to deform. Because of the deformation occurring in shear connector, the overall behavior of the L-shaped shear connector used in USCB is ductile.

Load-displacement response

The relation between the push-out load and the mean slip as well as the mean uplift between the concrete beam-floor and the steel beam is illustrated in Fig. 8. The load-mean slip curve of PO-S20 specimen shows that after the ultimate load, the load capacity drops down rapidly. For specimens with L-shaped shear connectors, load reduction occurred in a slow rate after reaching the ultimate load. It proves that the specimens with L-shaped shear connectors present a ductile behavior while the one with square bar shear connectors, PO-S20, exhibits less ductility. Due to the lack of ductility, which is required in design codes for partial shear connection design, PO-S20 solution for USCB is withdrawn and no further discussed in this paper. The ultimate load and corresponding slip of each specimen are reported in Table 2 in which the ultimate slip determined by using EC4 criterion are also provided. Besides, it can be seen that at 80% of ultimate load after peak, the vertical separation between concrete and steel beam for each specimen is less than half of the longitudinal slip. It shows that according to EC4 [START_REF]Eurocode 4: design of composite steel and concrete structures-Part 1.1: General rules and rules for buildings[END_REF], the shear connectors have a sufficient capacity in resisting the vertical separation between the concrete component and the steel beam.

Finite element simulations

In order to get further information, a numerical simulation is performed on ABAQUS-Explicit. Since the use of square bar shear connectors is not a solution for partial shear connection design in U-shaped steel-concrete hybrid beam, FEA is performed only for specimens with L-shaped shear connectors. All main components that may affect the behavior of shear connection are considered in the model. By taking advantage of the symmetry, only a half of the specimen is simulated.

Finite element type and mesh

The components are modeled in separate parts as presented in Fig. 9. The concrete component is meshed with combined solid elements C3D8R and C3D6R available in ABAQUS library. The former element type is an 8-node brick element and the latter is a 6-node linear triangular prism element. Each node has three translational degrees of freedom. C3D8R element is also used for meshing shear connectors and welds. The steel beam is meshed with 4-node shell (S4R) element while the reinforcement bars are meshed with beam elements (B31). The overall mesh size of the concrete component, steel beam, shear connectors, welds and of the rebars are 20 mm, 20 mm, 1 mm, 2 mm and 20 mm, respectively. For concrete and steel beam components, a refined mesh is used at zones in contact with shear connectors.

Constraints and contact interactions

Once all parts of the model are positioned together into an assembly, appropriate constraints are used to describe the interaction between components. The surfaces in contact between shear connectors and welds are tied together. The bottom surfaces of the latter are tied to the surface of the beam flange. This is equivalent to the actual push-out specimen where shear connectors remain tied to the steel beam flange via welding. For rebars which are placed inside the concrete beam-floor, the embedded constraint is applied. In this constraint, the translational DOF of the nodes on the rebar elements are constrained to the interpolated values of the corresponding DOF of the concrete element. The slip and debond of the rebar is ignored. Contact interactions are applied for the remaining contact surfaces. The interaction properties of the contact interactions used in the model are defined by the tangential and normal behavior to the surfaces. Since the interior surfaces of the U-shaped steel beam are greased, the frictionless for tangential behavior is used at the contact surfaces between steel beam and other components. The penalty friction formulation is used for tangential behavior of the remaining contact surfaces with a coefficient of friction equal to 0.3. The default normal behavior (hard contact) is assumed for all contact surfaces. This normal behavior allows a minimum penetration of the slave surface into the master surface. For all contact pairs, the penalty contact formulation is used.

Loading and boundary conditions

Since ABAQUS/Explicit is a dynamic analysis program, a static solution is required for pushout test simulation. To obtain a quasi-static solution from the explicit dynamic analysis, the kinetic energy is suggested to be controlled within 5%-10% of the internal energy [27]. This criterion can be reached by a slow loading rate. Different loading rates with displacement controlled are attempted in the simulation and the optimum rate is found to be 0.33 mm/s. Moreover, the displacement loading is slowly applied by means of a smooth amplitude function to reduce the dynamic effect of the inertial forces at the beginning and at the end of loading. Besides, the semi-automatic mass scaling method in ABAQUS/Explicit is adopted to achieve a balance between solution time and accuracy.

The boundary condition applied to the median plane of the model (symmetric in Z-axis) is based on the symmetry, as shown in Fig. 10a. The translational displacement in Y-direction is restrained at nodes on the bottom flange of steel beam (the position of vertical supports). The loading is applied by introducing an axial displacement in X-direction at steel cross-section that is tied using rigid body constraint. The reference point of the latter is fixed in all remaining degrees of freedom. At the other end of the specimen, the rigid plate representing the contact surface between concrete floor and rigid steel beam defined in Fig. 10b (surface observed during the experimental tests) is fixed in all directions at its reference point.

Concrete material model

The concrete constitutive model used in [START_REF] Alfarah | New methodology for calculating damage variables evolution in Plastic Damage Model for RC structures[END_REF] is adopted. The nonlinear behavior of the concrete material in compression and tension is presented by a uniaxial compressive stress-strain curve and a tensile stress-crack width relationship, see Fig. 11a and Fig. 11b, respectively. The compressive stress-stain curve is split into three parts. The first part is assumed to be elastic when the stress is lesser than 0.4f c , where f c is the compressive cylinder strength of concrete. The elastic modulus is determined by: The second part is a nonlinear parabolic portion ranging from the proportion limit stress 0.4f c to the peak stress f c , described by the following expression [START_REF]EN 1992-1-1, Eurocode 2: Design of Concrete Structures: Part 1-1: General Rules and Rules for Buildings[END_REF]:

E c = 22000 (f c ) 1/3 . (1
σ c = kη -η 2 1 + (k -2) η f c (2)
where σ c is the concrete compressive stress (MPa); k is a constant expressed by k = 1.05E c ε c1 /f c in which ε c1 = 0.7f 0.31 c /1000; and η is the ratio of stain to peak strain, η = ε c /ε c1 . The third part of the curve which corresponds to the descending segment is given by [START_REF] Krätzig | An elasto-plastic damage model for reinforced concrete with minimum number of material parameters[END_REF]:

σ c = 2 + γ c f c ε c1 2f c -γ c ε c + ε c γ c 2ε c1 -1 (3)
where

γ c = π 2 f c ε c1 2 G ch l eq -0.5f c ε c1 (1 -b) + b f c E c
2 in which b is the ratio of compressive plastic strain to crushing strain, b = ε pl c /ε ch c ; G ch is the crushing energy per unit area; and l eq is the characteristic length. The latter depends on the mesh size, the type of finite element and the crack direction [30; 31]. For the validation of the FE model, the characteristic length is taken equal to 220 mm which is the height of concrete cylinder specimen.

For un-cracked concrete subjected to tension, a linear stress-strain relationship up to the tensile strength, f t , is adopted. Moreover, the modulus of elasticity in tension is assumed to equal to the one in compression. The following nonlinear stress-crack width relationship is adopted for cracked concrete [START_REF] Hordijk | Tensile and tensile fatigue behaviour of concrete; experiments, modelling and analyses[END_REF]:

σ t f t = 1 + c 1 w w c 3 exp -c 2 w w c - w w c 1 + c 3 1 exp(-c 2 ) ( 4
)
where f t is the tensile strength determined by f t = 0.3(f c -8) 2/3 (MPa); w is the crack width (mm); w c is the critical crack opening (mm) given by w c = 5.14G F /f t in which the fracture energy G F (N/mm) per unit area can be calculated as G F = 0.073f 0.18 c ; and the constants c 1 and c 2 are equal to 3 and 6.93, respectively. The tensile stress in concrete can be expressed in terms of tensile strain using the strain-crack opening relationship as follow [START_REF] Alfarah | New methodology for calculating damage variables evolution in Plastic Damage Model for RC structures[END_REF]:

ε t = f t /E c + w/l eq (5)
In push-out test simulations, the Concrete Damaged Plasticity model available in ABAQUS material library is adopted. In this model, yielding parts of the concrete in tension and compression are treated separately. The parameters used to define the concrete damaged plasticity are the following. The material dilation angle (ψ), eccentricity ( ) are taken as 15 • and 0.1, respectively. The ratio of biaxial compressive strength to uniaxial compressive strength (σ b0 /σ c0 ) is taken as 1.16.

Steel material models

The stress-strain relationship of shear connectors is represented by a tri-linear curve, see Fig. 12. The initial regime is assumed to be elastic followed by a stage of yielding and finally a branch of strain hardening. The stress-strain behaviors in tension and compression are assumed to be the same. The nominal or engineering ultimate strain, ε nom u , was set to be 20%. It is worth mentioning that when defining plasticity data in ABAQUS, true stress and true strain must be used for a simulation with large strains. The relationship between true strain and nominal strain is established by

ε true = ln(1 + ε nom ) (6)
The relationship between true stress and nominal stress and strain is

σ true = σ nom (1 + ε nom ) (7)
Table 3 shows the true stress-strain data for the L-shaped shear connectors as input in ABAQUS. Besides, an elastic-perfectly plastic model is adopted for U-shaped steel beam and reinforcing bars. The modulus of elasticity and the Poisson's ratio of all steel materials used in the model are 210 GPa and 0.3, respectively.

Validation of finite element model

In order to verify the developed finite element model, the push-out tests presented previously are modeled using the assumptions described above. The shear connection capacity obtained from the tests (P T est ) and finite element analysis (P F E ) as well as the load-slip and load-uplift response of the test specimens, including its failure modes, are investigated. Table 4 shows a comparison of the capacities of shear connection obtained experimentally and numerically. It can be seen that a good agreement between both results has been achieved. The experimental load-slip and load-uplift curve is compared with the numerical curve obtained from the finite element analysis, see Fig. 13 and Fig. 14. Generally, a good agreement between the pairs has been attained. It is shown that the finite element models successfully predicted the shear connection capacity and the stiffness of the L-shaped shear connectors used for USCB. Fig. 15 and Fig. 16 show the stress contour and the deformed shape of the shear connector at different load levels up to failure for PO-L40 and PO-L50a,b,c specimen, respectively. From the stress contours, it is shown that the local failure of concrete in compression occurs in front of both bottom leg angle and the fillet between the angles of the L-shaped shear connector. Moreover, the double shearing of the L-shaped shear connector is observed.

Parametric study

The following parametric study is performed by evaluating key parameters, for instance the concrete strength (f c ), the cross-section of L-angle, the length of the shear connector welded to the steel beam flange (L w ), as well as the different thicknesses of concrete slab (H c ). The global geometry of PO-L40 specimen is adopted. Three cross-sections of shear connectors are evaluated: L30 × 30 × 3, L40 × 40 × 4 and L50 × 50 × 5. The parametric study is summarized in Table 5. At least three different values are considered for each evaluated variable.

The effect of concrete compressive strength is shown in Fig. 17. It is clear that the shear bearing capacity of the connector increases with increasing concrete strength. Fig. 18 shows the effects of the other evaluated parameters on shear capacity of the L-shaped shear connectors. The vertical axis in Fig. 16 represents the ratio of the ultimate load for each case-study to the mean value of those corresponding to the three cases for an evaluated variable. As expected, the cross-section of the L-shaped shear connector influence significantly the shear capacity of the connector. In fact, a larger L-angle cross-section has a larger flange thickness and a larger size of fillet that provide a higher shear resistance. It can be also seen that the shear capacity of the shear connector is directly related to the length of weld, L w ; a longer length of the weld provides a larger shear capacity of the connector. Furthermore, a larger thickness of concrete slab provides a larger bearing strength. All the forces presented on vertical axis in Fig. 17 The experimental tests show that the failure of tested specimens is associated with the local concrete crushing and with the plastic deformation of connectors due to shear near the fillets between the angles of the cross-section and at the leg angle welded on the flange of the steel beam. As a consequence, the shear connector resistance may be provided by two parts: shear resistance of the connector and concrete contribution. The former is characterized by two shear plans while the latter is assumed to be idealized by a rectangular stress block. The latter assumption for concrete pressure around shear connectors is made in order to calibrate the formulation. The equation predicting the shear bearing capacity of one shear connector is then given by:

P u = 2(A s1 + A s2 ) × f u √ 3 + 2K c × A c × f c (8)
where the factor 2 represents the two sides of the connector welded on the top flanges of the Ushaped steel beam; A s1 , A s2 and A c are defined in Fig. 19; f u is the ultimate strength of the shear connector; f c is the compressive strength of concrete; and K c is a calibration factor on concrete strength due to local pressure effect. The calibration factor K c is defined by the following expression obtained from a regression analysis of the parametric study: in which h a is the height of the L-shaped cross-section. Table 6 shows the comparison between the ultimate loads given by proposed design equation against FEA and experimental results. The first four values in the table represent experimental results. P s and P c represent the contribution of L-shaped shear connector and concrete to the shear capacity of the connector, respectively. It is shown that there is a good agreement between the results obtained from the proposed formulation and FEA results. It is worth mentioning that the case studies with the concrete strengths 10 MPa and 60 MPa have not been considered in the regression analysis. Indeed, such concrete classes are out of scope for usual buildings; moreover the results of the FE model should be confirmed by experimental testing with similar concrete resistance before being used.

K c = 19 -38.1 h a H c (9)

Conclusion

Push-out experimental tests have been conducted on L-shaped and square bar shear connectors used in U-shaped steel-concrete hybrid beams. The experimental tests showed that:

-L-shaped shear connectors provided a ductile behavior while the square bar shear connector presented less ductility. The ductility of the former, similar to the one of classical studs, was associated with the plastification of the connectors before failure.

-The slip at load level descending to 90% of ultimate load is larger than 6 mm, the minimal requirement of ductility to be considered as ductile following EC4 [START_REF]Eurocode 4: design of composite steel and concrete structures-Part 1.1: General rules and rules for buildings[END_REF]. As square connectors did not meet this requirement, they were not considered as adequate and they have not been further investigated.

To identify the stress behavior of the L-shaped shear connector and of the surrounding concrete, a FE model was developed. It gave the key features in developing a new design equation for Lshaped shear connectors used in U-shaped steel-concrete hybrid beams. It showed that the shear connector resistance may be provided by two parts: concrete contribution and shear resistance of the connector. The latter is associated with the plastic deformation of connectors due to shear near the fillets between the angles of the cross-section and at the leg angle welded on the flange of steel beam. The former is due to the local pressure effect in the concrete block. To take into account this effect, a calibration factor obtained from a regression analysis has been introduced. It has been shown that the estimated ultimate loads are in good agreement with FEA and experimental results.

Figure 1 :

 1 Figure 1: U-shaped steel-concrete hybrid beam (USCB).

Figure 2 :

 2 Figure 2: Schematic of push-out test setup.

Figure 3 :

 3 Figure 3: Side view of push-out test setup.

  (a) Specimen with square bar connectors. (b) Specimen with L-shaped connectors. Specimen cross-section detailing (dimension in mm).

Figure 5 :

 5 Figure 5: LVDT sensor installation.

Figure 6 :

 6 Fracture of L-shaped shear connector.Local concrete failureBottom flange of Lshaped shear connector

Figure 7 :

 7 Figure 7: PO-L50c: concrete local crushing/splitting.

Figure 8 :

 8 Figure 8: Load-mean slip/uplift response.

Figure 9 :

 9 Finite element model.

  Symmetric plane, vertical supports and load application point. Rigid plate (b) Support representing contact surface with the rigid beam.

Figure 10 :

 10 Boundary conditions.

Figure 11 :

 11 Concrete constitutive model.

Figure 12 :

 12 Figure 12: Engineering stress-strain relationship model for shear connectors.

Figure 13 :

 13 Figure 13: Comparison of FEA against experimental results of PO-L40 specimen.

Figure 14 :

 14 Figure 14: Comparison of FEA against experimental results of PO-L50a,b,c specimen.

  and Fig. 18 correspond to the total applied load (a) Load level 254 kN (b) Load level 560 kN (c) Load level 634 kN

Figure 15 :

 15 Figure 15: Stress contours and deformed shapes of FE model for PO-L40 specimen at various load levels.

4 .

 4 Proposal of a new equation predicting the shear capacity of shear connectorTo the best of our knowledge, there is currently no available formulation for determining the shear bearing capacity of the L-shaped shear connector used in USCB. To fill in this gap, a new Printed using Abaqus/CAE on: Tue Aug 08 14:45:24 Paris, Madrid (heure d'été) 2017 Printed using Abaqus/CAE on: Tue Aug 08 14:49:45 Paris, Ma (a) Load level 347 kN Printed using Abaqus/CAE on: Tue Aug 08 14:44:11 Paris, Madrid (heure d'été) 2017 Printed using Abaqus/CAE on: Tue Aug 08 14:50:30 Paris, Ma (b) Load level 855 kN Printed using Abaqus/CAE on: Tue Aug 08 14:42:44 Paris, Madrid (heure d'été) 2017 Printed using Abaqus/CAE on: Tue Aug 08 14:51:24 Paris, Ma (c) Load level 999 kN

Figure 16 :

 16 Figure 16: Stress contours and deformed shapes of FE model for PO-L50 specimen at various load levels.

Figure 17 :

 17 Concrete strength effect on shear bearing capacity of 3 L50-shear connectors.

  Connector ultimate strength.

  Concrete deck thickness.

Figure 18 :

 18 Effects of evaluated parameters on shear capacity of the L-shaped shear connectors.

Figure 19 :

 19 Figure 19: Shear failure plans and concrete local pressure zones.

Table 1 :

 1 Mean material properties.

	Specimen	Concrete	Precast	Connectors	U-shaped

Table 2 :

 2 Ultimate load and corresponding slip.

	Specimen	PO-S20 PO-L40 PO-L50a PO-L50b PO-L50c
	P max (kN)	748.6	637.1	932.1	916.5	975.9
	δ max (mm)	2.743	5.054	5.081	3.032	3.341
	δ u,EC4 (mm)	4.09	8.90	15.49	9.59	11.04

Table 3 :

 3 Steel connector properties.

		L40×40×4		L50×50×5
	Yield stress (MPa) Plastic strain Yield stress (MPa) Plastic strain
	430.88	0.0000	325.50	0.0000
	438.80	0.0182	330.03	0.0138
	660.00	0.1792	534.00	0.1798

Table 4 :

 4 Comparison of ultimate loads obtained from FEA against experimental results.

	Specimen PO-L40 PO-L50a PO-L50b PO-L50c
	P T est (kN)	637.1	932.1	916.5	975.9
	P F E (kN)	634.67	999.14	999.14	999.14
	P F E /P T est	1.00	1.07	1.09	1.02

Table 5 :

 5 Summary of case-studies.

		L-angle		Concrete	Length of welds
	Cross-section f y (MPa) f u (MPa) f c (MPa) h c (mm)	L w (mm)
		235	360	35	200	40
		335	455	35	200	40
	L30 × 30 × 3	355	510	35	200	40
		430	550	35	200	40
		235	360	42	200	30
		430	550	27	200	20
		430	550	27	200	30
	L40 × 40 × 4	430 235	550 360	27 42	200 200	40 30
		430	550	35	200	30
		430	550	42	200	30
		235	360	42	200	30
		235	360	35	200	30
		235	360	27	200	30
		355	510	27	200	20
		355	510	35	200	20
		355	510	42	200	20
		325	445	10	200	40
		325	445	27	200	40
	L50 × 50 × 5	325	445	35	200	40
		325	445	42	200	40
		325	445	60	200	40
		325	445	27	150	40
		325	445	35	150	40
		325	445	42	150	40
		325	445	27	120	40
		325	445	35	120	40
		325	445	42	120	40
	means of a set of data for a regression analysis.

Table 6 :

 6 Comparison of the ultimate loads given by proposed design equation against experimental and FEA results.

	L angle	f u	f c	H c	L w	K c	P s	P c	P u,th P F E; exp	P u,th P u,F E
		(MPa) (MPa) mm mm		(kN) (kN) (kN)	(kN)
	L50a	445	20.27 200	40	9.48	764	230	994.3	932.1	1.07
	L50b	445	21.29 200	40	9.48	764	242 1005.9	916.5	1.10
	L50c	445	22.71 200	40	9.48	764	258 1022.1	975.9	1.05
	L40	550	26.97 200	20 11.38 518	147	665.0	637.1	1.04
	L30	360	42	200	30 13.29 250	301	551.2	582.3	0.95
	L30	360	35	200	40 13.29 287	335	622.1	640.6	0.97
	L30	445	35	200	40 13.29 355	335	689.9	709.4	0.97
	L30	510	35	200	40 13.29 407	335	741.8	748.7	0.99
	L30	550	35	200	40 13.29 439	335	773.7	755.7	1.02
	L40	550	27	200	30 11.38 594	221	815.1	863.9	0.94
	L40	550	27	200	40 11.38 670	295	965.1	900.5	1.07
	L40	360	42	200	30 11.38 389	344	732.9	750.8	0.98
	L40	550	35	200	30 11.38 594	287	880.7	921.4	0.96
	L40	550	42	200	30 11.38 594	344	938.0	971.7	0.97
	L50	360	42	200	30	9.48	556	358	913.7	953.5	0.96
	L50	360	35	200	30	9.48	556	298	854.1	928.8	0.92
	L50	360	27	200	30	9.48	556	230	785.8	869.6	0.90
	L50	510	27	200	20	9.48	699	153	852.2	933.7	0.91
	L50	510	35	200	20	9.48	699	199	897.7	979.2	0.92
	L50	510	42	200	20	9.48	699	239	937.5	970.7	0.97
	L50	445	27	200	40	9.48	764	307 1070.8 1040.0	1.03
	L50	445	35	200	40	9.48	764	398 1161.8 1115.2	1.04
	L50	445	42	200	40	9.48	764	478 1241.4 1190.4	1.04
	L50	445	27	150	40	6.30	764	204	968.0	953.6	1.02
	L50	445	35	150	40	6.30	764	265 1028.4 1059.0	0.97
	L50	445	42	150	40	6.30	764	318 1081.4 1089.5	0.99
	L50	445	27	120	40	3.13	764	101	865.1	850.6	1.02
	L50	445	35	120	40	3.13	764	131	895.1	933.6	0.96
	L50	445	42	120	40	3.13	764	158	921.3	977.5	0.94
										moy	0.99
										std	0.05


Zayed Alsayed

Guillaume Bresson

Anne Verroust-Blondet

Fawzi Nashashibi

2D SLAM Correction Prediction in Large Scale Urban Environments

Simultaneous Localization And Mapping (SLAM) is one of the major bricks needed to build truly autonomous mobile robots. The probabilistic formulation of SLAM is based on two models: the motion model and the observation model. In practice, these models, together with the SLAM map representation, do not model perfectly the robot's real dynamics, the sensor measurement errors and the environment. Consequently, systematic errors affect SLAM estimations.

In this paper, we propose two approaches to predict corrections to be applied to SLAM estimations. Both are based on the Ensemble Multilayer Perceptron model. The first approach uses successive estimated poses to predict the errors, with no assumptions on the underlying SLAM process or sensor used. The second method is specific to 2D likelihood SLAM approaches, thus, the likelihood distributions are used to predict the corrections, making this second approach independent of the sensor used. We also build a hybrid correction module based on successive estimated poses and the likelihood distributions.

The validity of both approaches is evaluated through two experiments using different evaluation metrics and sensor configurations.

I. INTRODUCTION

SLAM simultaneously estimates a mobile robot's pose (position and orientation) and incrementally builds a map of its surroundings.

The theoretical formulation of the SLAM concept led to establishing a solution that is subdivided into two recursive steps. The first is the time update step, where the state is propagated over an elapsed time using a motion model in order to build a prediction. The second step, the measurement update, consists in integrating the sensor observations into the prediction through an observation model in order to compute an estimation.

Due to the non-linearity of the SLAM solution, SLAM methods suffer from systematic errors related to the linearization of its models. Work on overcoming these errors has been carried out, but has usually focused on assisting KFlike based SLAM approaches. In contrast, we aim to improve the accuracy by estimating a correction to be applied to the SLAM output based on relevant information available from the SLAM algorithm.

In this paper, we propose two approaches. The first approach is designed for 2D SLAM methods, i.e. independently of the underlying SLAM process and sensor used, where we aim to reduce the errors due to the approximations in dynamic modeling. The second approach is designed to handle errors related to the SLAM map model and the sensor measurement errors based on likelihood distributions, which makes it suitable for 2D Likelihood SLAM methods (i.e. no assumptions on the sensor used). Both approaches are based on the Ensemble Multilayer Perceptron (EMLP) which is a Supervised Artificial Neural Network model. In such a model, an off-line learning phase is required to build an operational model. Once the model has been built, it can be used for correction prediction with negligible execution time.

The rest of this paper is organized as follows: Section II presents methods from the state-of-the-art to overcome systematic SLAM errors. Section III highlights the impact of errors on incremental localization methods and motivates our evaluation metric choice. Section IV presents our methods to predict corrections for 2D SLAM approaches and 2D Maximum Likelihood SLAM approaches. Finally, Section V presents the results of our experiments and we conclude by giving some perspectives in Section VI.

II. STATE OF THE ART

In its probabilistic form [START_REF] Bailey | Simultaneous localization and mapping (SLAM): Part II[END_REF] [START_REF] Durrant-Whyte | Simultaneous localization and mapping: part I[END_REF], the SLAM problem is solved via two recursive steps. The first solutions were based on Kalman Filtering (KF) and KF-like processes. The Kalman Filter [START_REF] Sasiadek | Sensor data fusion using Kalman filter[END_REF] assumes a linear motion and observation models together with Gaussian error distributions. In spite of the validity of the process to solve SLAM, the performances obtained deteriorate rapidly because of the linearity of the models, which does not reflect the reality.

To overcome these issues, the filter was extended to nonlinear models by approximating them using first order Taylor expansion, which results in the well known Extended Kalman Filter (EKF) [START_REF] Davison | MonoSLAM: Real-time single camera SLAM[END_REF]. Then, the Unscented Kalman Filter (UKF) [START_REF] Martinez-Cantin | Unscented SLAM for large-scale outdoor environments[END_REF] uses a set of well defined candidates (named sigmas) around the predicted position which allows us to avoid calculating the Jacobian matrices.

The Particle Filter (PF)-based SLAM [6][7][8] uses a large set of candidates (particles) in order to cover all the possibilities in the navigation space.

To improve the accuracy of SLAM algorithms, the authors of [START_REF] Mouragnon | Real time localization and 3D reconstruction[END_REF][10] combine more robust algorithms and apply a bundle adjustment to refine the estimations of an on-line SLAM approach which gives less accurate results. Another possibility is to use other sensor information, such as in [START_REF] Zhang | LOAM: Lidar odometry and mapping in realtime[END_REF] [START_REF]Visual-lidar Odometry and Mapping : Low-drift , Robust , and Fast[END_REF] where visual data are mainly used to propagate the estimations more frequently; the estimations are then refined using laser data.

Another line of research aims to improve SLAM accuracy by considering the errors due to the model's linearization. Indeed, [START_REF] Choi | Measurement Noise Estimator Assisted Extended Kalman Filter for SLAM Problem[END_REF] includes a measurement noise estimator to assist the SLAM algorithm which is based on an Extended Kalman Filter. M.Choi et al. [START_REF] Choi | Neural network-aided extended Kalman filter for SLAM problem[END_REF] use a neural network aided Extended Kalman filter in order to perform SLAM. The Kalman Filter is used for both the estimation of the mobile robot state and the learning of the neural network weights. The same approach was extended by using another type of neural network in [START_REF] Yu | Recurrent neural network and extended Kalman filter in SLAM problem[END_REF]. Similar work is presented in [START_REF] Choi | Enhanced SLAM for a mobile robot using extended Kalman Filter and neural networks[END_REF] and [START_REF] Panah | Enhanced SLAM for Autonomous Mobile Robots using Unscented Kalman Filter and Neural Network[END_REF] where they both propose to enhance filter-based SLAM with neural networks. K.Choi et al. [START_REF] Choi | Enhanced SLAM for a mobile robot using extended Kalman Filter and neural networks[END_REF] use a Multilayer Perceptron (MLP) neural network with an EKF-SLAM. In contrast, Panah et al. [START_REF] Panah | Enhanced SLAM for Autonomous Mobile Robots using Unscented Kalman Filter and Neural Network[END_REF] use a Radial Basis Function (RBF) neural network together with a UKF-SLAM approach.

However, the experiments in these papers are validated only on simulation data and are dedicated exclusively to KF-like SLAM approaches.

Similar approaches, which couple neural networks to KF, were proposed to Fuse INS with GPS data [START_REF] Sharaf | Online INS/GPS integration with a radial basis function neural network[END_REF][19] [20][21]. In [START_REF] Zhang | A low-cost GPS/INS integration based on UKF and BP neural network[END_REF] the neural network was mainly used to predict INS errors when GPS is not available. The KF performs the fusion of INS with GPS or INS with the predicted correction when there is no GPS information available. Although such an approach is interesting, it has a major drawback in that the predicted errors are re-injected as inputs for the next error prediction, which results in a system which amplifies the errors caused by prediction and leads the whole approach to diverge, as can be seen in the experiments.

In this paper, we propose two approaches to predict systematic errors affecting SLAM estimations. The first approach is suitable for 2D SLAM approaches independently of the underlying estimation process and sensor used. The second approach is suitable for 2D Maximum Likelihood SLAM techniques independently of the sensor used. Both approaches are based on Ensemble MLP neural network models and make use of relevant information available from SLAM without additional processing.

III. ERROR PROPAGATION AND SLAM EVALUATION

A review of localization approaches, specifically those based on SLAM, highlights the difficulty of evaluating the reliability of the method in large-scale environments.

Measuring the drift by comparing the estimated localization with the ground truth could be fair when using a global positioning approach. This, however, is not the case for raw SLAM approaches (i.e. based only on exteroceptive sensors data and without a GNSS receiver) which operate in an incremental way. Indeed, the models used in SLAM algorithms (e.g. the non-linearity of motion and observation models in KF-based SLAM approaches, or discretization and map representation for grid-based SLAM approaches), which are largely induced by angular errors, have a strong impact on the divergence of the global localization. The impact of linear and angular drifts is illustrated in Figure 1.

X 0 X t X t ˆXt Fig.

1.

Impact of linear and angular drift on incremental localization approaches. In green, the ground truth of trajectory. Below, examples of linear and angular drift (red arrow) occurring at different levels on the estimated path.

Burgard et al. [START_REF] Burgard | A comparison of SLAM algorithms based on a graph of relations[END_REF] and Kummerle et al. [START_REF] Kümmerle | On measuring the accuracy of SLAM algorithms[END_REF] compare SLAM approaches and measure their accuracy based on differences in relative poses. For this purpose the data has to be synchronized and must have a high level of accuracy. Making use of the same metrics, Geiger et al. [START_REF] Geiger | Are we ready for Autonomous Driving? The KITTI Vision Benchmark Suite[END_REF] perform the comparison at fixed travel lengths, which reduces the reliance on the synchronization and accuracy of the ground truth. However, measuring errors at the frame level allows us to isolate errors at the frame where they occur. Consequently, this metric is more convenient for building the correction prediction that we present in Section IV.

Navigation in a 2D space implies a state vector x t comprising the position (x t , y t ) and the heading θ t at the current time t. Thus the drift affecting the localization estimation could be due to errors in position and/or in heading.

In practice, considering errors on each axis independently (i.e. the error xt which affects the position following the x axis independently of the error yt on the y axis) implicitly involves a high correlation to the heading and its errors in both position components. In contrast, by considering lineardisplacement errors dt independently of the angular errors θt , reduces the correlation. We choose to use similar metrics to those in [START_REF] Kümmerle | On measuring the accuracy of SLAM algorithms[END_REF], however, we separate translation errors from angular errors (as proposed in [START_REF] Geiger | Are we ready for Autonomous Driving? The KITTI Vision Benchmark Suite[END_REF], but we apply it at the frame level), which results in two elementary values, ( dt , θt ) respectively, instead of combining them into a single measure. Let δ t,t-1 = x t x t-1 be the relative transformation which transforms the ground truth pose x t into x t-1 by means of the inverse operator of the standard motion composition , and accordingly for the estimated SLAM path δt,t-1 = xt xt-1 . where x t denotes the ground truth pose and xt the estimated pose.

dt = lin(δ t,t-1 δt,t-1 ) (1) θt = rot(δ t,t-1 δt,t-1 ) (2)
where lin(.) and rot(.) are used to separate the relative translational and rotational components respectively.

IV. SLAM CORRECTION PREDICTION

Predicting systematic errors affecting SLAM estimations requires isolating these errors at the same frame where they occur. Our approach predicts a correction that we systematically apply to the output of SLAM in order to reduce the raw SLAM estimation errors. The general block diagram of SLAM together with the correction estimation is illustrated in Figure 2. The corrected estimation x t is calculated using Equation 3 by combining the raw SLAM estimation xt and the predicted correction ˆ t .

x t = xt ⊕ ˆ t (3)
Our approach relies on an Ensemble MLP model [START_REF] Perrone | Improving Regression Estimation: Averaging Methods for Variance Reduction with Extensions to General Convex Measure Optimization[END_REF][26] to predict the corrections ˆ t . In such a model, an off-line training phase is required to build an operational model. Once the model has been built, it can be used for correction prediction with negligible execution time. The important feature of Artificial Neural Networks (ANN) comes from their ability to learn, memorize and map relations for a set of inputs to expected outputs [START_REF] Hashem | Approximating a function and its derivative using MSE optimal linear combination of trained feed forwrd neural networks[END_REF].

We use two kinds of information available from the SLAM process: 1. Successive estimated poses (c.f. Section IV-B). 2. The likelihood distribution (c.f. Section IV-C).

Accordingly, the Correction Prediction Module is denoted:

• +dS when using successive SLAM poses as input.

• +ML when using the likelihood distribution information.

• +dSML when the correction prediction involves both successive SLAM poses and the likelihood distribution. Consequently, depending on the information exploited, our approach could be suitable for 2D SLAM techniques or for Maximum Likelihood SLAM approaches.

A. The Artificial Neural Network Model

An Ensemble MLP [START_REF] Hashem | Approximating a function and its derivative using MSE optimal linear combination of trained feed forwrd neural networks[END_REF] [START_REF] Perrone | Improving Regression Estimation: Averaging Methods for Variance Reduction with Extensions to General Convex Measure Optimization[END_REF] network is formed of multiple MLP models organized in parallel with their respective outputs combined in order to produce an improved model [START_REF] Windeatt | Ensemble MLP classifier design[END_REF]. Assuming that each MLP model may make different errors, the idea is to pool together the results from all the MLP models to find a composite system that outperforms any individual (base) MLP. In contrast to MLP, Ensembles solve local minima and parameter selection for optimal performance.

Moreover, the architecture of our model is of the form:

ξ × [I × λ × 2]
where the number 2 refers to the number of outputs (i.e. predicted translation and rotation error components), λ is the number of neurons in the hidden layer, ξ the size of the Ensemble (i.e. the number of individual MLPs in the network), and I denotes the number of inputs which depends on the information used to feed the model. The parameters λ and ξ are both determined experimentally.

B. Correction Prediction for 2D SLAM

Consecutive pose estimations provide information about the velocity profile of the mobile robot, and more specifically about the small oscillations that are mainly due to the linearization and/or discretization of the observation model. Taking the example of a 2D SLAM approach which uses a grid map, such as PML-SLAM [START_REF] Alsayed | PML-SLAM : a solution for localization in large-scale urban environments[END_REF], the estimated path alternately deviates (oscillating) from the real path. These oscillations are directly related to the measurement errors affecting the sensor observation and the discretization of the map. Consequently, this affects the probability of making an observation given a specific position on the environment map, which represents the measurement model.

Furthermore, the sign and the amplitude of errors on heading and linear displacement may change according to the velocity profile. For example, we observed that the heading error is amplified during a turning maneuver. In addition, the sign of heading errors changes according to the direction in which the turn is made. These errors are induced from the motion model, which describes a probability of a displacement when applying a command input given the previous pose. Therefore, the amplitude and sign of errors are related to the dynamic modeling considered in the motion model.

Our approach to predict linear and angular drifts (ˆ dt , ˆ θt ) makes use of a limited number n of the previous raw SLAM displacements { δt,t-1 , δt-1,t-2 , δt-2,t-3 , . . . } where each displacement δt,t-1 is represented by two components {lin( δt,t-1 ), rot( δt,t-1 )}. Therefore, I = 2 × n with n determined experimentally.

C. Correction Prediction for 2D Likelihood SLAM

Likelihood SLAM approaches incorporate the sensor observations in pose estimation by finding the candidate which maximizes the likelihood probability of an observation on the map. This is done by exploring the candidate space in which the robot evolves (i.e. the map seen from a given pose).

The likelihood distribution over candidate space reflects how distinctive (Figure 3(b)) and how ambiguous (Figure 3(c)) the situation could be for the SLAM algorithm. Some examples of the likelihood distributions from PML-SLAM [START_REF] Alsayed | PML-SLAM : a solution for localization in large-scale urban environments[END_REF] implementation are illustrated in Figure 3. (c) corridor scenario Fig. 3. Likelihood distribution over position candidates for different scenarios. In practice, for a 2D environment, the candidate space is of three as it includes the orientation as well as the position. For this reason, we represent the best likelihood probability over all orientation values at a given position.

In Figure 3(b) the distribution clearly suggests a unique maximum with a significant difference over the entire candidate area. However, depending on the environment map, the velocity profile and the likelihood measure, the likelihood distribution shape changes (cf. Figure 3(a)). Thus, the simple choice of the pose which maximizes the likelihood over the distribution is subject to errors. These errors are induced by the approximations in the observation model, the map representation, and the observation measurement errors. In contrast, in Figure 3(c) the distribution indicates the presence of the mobile robot in a corridor scenario where multiple solutions could be possible. Hence, the simple choice of the candidate that has the maximum score may not indicate the real position.

The likelihood distributions could contain important information about the estimated pose uncertainty involved in the Maximum Likelihood process. In our work, we aim to exploit this information to predict a correction for Likelihood 2D SLAM approaches. For this purpose, and in order to use the maximum amount of information, we use the full likelihood distribution which is expressed in a 4D space (i.e. composed of the three pose components and the likelihood probability). However, this will include a lot of useless information which, on the one hand, involves extra training time because of the important length of the input vector and, on the other hand, misleads the ANN model from processing the useful information. For these reasons, we transform the full likelihood distribution to another space by applying Principal Component Analysis (PCA) [START_REF] Daffertshofer | PCA in studying coordination and variability: A tutorial[END_REF] [START_REF] Shlens | A Tutorial on Principal Component Analysis[END_REF] on its probability values. This transformation leads to re-projecting the likelihood probabilities in another space of the same dimension as the initial one, except that the relevant information in the new space (i.e. PCA space) could be reconstructed from a smaller number of axes. Therefore, the information size is reduced to the number of relevant axes in the PCA space.

In practice, taking the example of a 2D likelihood SLAM approach, such as PML-SLAM [START_REF] Alsayed | PML-SLAM : a solution for localization in large-scale urban environments[END_REF], the candidate space is naturally discretized because of the map representation and the likelihood process. The approach explores an area of 13 candidate positions, each of which is associated to 65 possible heading angles, which results in a total of 845 candidates. Applying PCA leads to transforming the full likelihood distribution from the space of the 845 dimensions to another space of the same size but where the relevant information could be reconstructed through only 24 components. Therefore, the input size of the corresponding correction model I = 24.

D. Correction Prediction for 2D Likelihood SLAM: a Second Method

The information presented in subsection IV-B is generic and so it can be used together with the likelihood distribution to build another correction prediction model suitable for 2D Likelihood SLAM approaches. The input vector will be the sum of the previous displacements together with the reduced full likelihood distribution transformed into PCA space.

V. DATASETS, EXPERIMENTS AND RESULTS

A. Datasets

Our approach is based on supervised machine learning, which means an off-line learning phase is required to build an operational model. Several datasets suited to SLAM benchmarking are available on-line. Bresson et al. [START_REF] Bresson | Simultaneous Localization And Mapping: A Survey of Current Trends in Autonomous Driving[END_REF] give a summary of existing benchmarks. We chose to work mainly with the KITTI dataset [START_REF] Geiger | Vision meets robotics: The KITTI dataset[END_REF] to train our models. In addition, we test the models on data acquired using our platforms (see Figure 4). The KITTI dataset covers a variety of scenarios mainly collected in urban environments. In addition, the ground truth is synchronized and of high accuracy. It initially contains a total of 54 sequences sorted according to the categories: city, residential and road environments. We eliminate 4 tracks after analyzing the ground truth quality, we choose 4 tracks randomly for testing, and we keep 46 sequences for training. The names of the eliminated tracks are : city: 2011 09 Then, in order to validate our approach under realistic and more complex conditions where the underlying sensor characteristics are different (i.e. the resolution and the frequency), we acquire data from a circuit around Bordeaux Lake (France) which covers different types of scenarios in urban and peri-urban environments with a travel length of about 9km. We use a car equipped with five IBEO-Lux sensors deployed to cover a 360°field of view around the vehicle.

A clear statistical overview of the datasets used in both training and in the experiments is given in Table I. For the KITTI dataset we perform a horizontal cut in the 3D laser data in order to simulate a 2D Laser-sensor to be adapted as input to the SLAM implementation used in our experiments.

We use the PML-SLAM [START_REF] Alsayed | PML-SLAM : a solution for localization in large-scale urban environments[END_REF] implementation which is a 2D Maximum Likelihood SLAM approach based on laser data that uses a grid map representation. Using this approach, we are able to compare the different models as the underlying SLAM implementation is the same.

B. Experiments

The models are trained on the 46 tracks extracted from the KITTI dataset. The validation of our approach is tested through two test scenarios:

1. The first experiment is based on KITTI dataset test tracks. We compare the performances obtained using the different Correction Prediction modules presented in this article. Moreover, we use this experiment to find the best parameters (i.e. λ and ξ) of the correction prediction model.

2.

The second experiment is based on the Bordeaux Circuit using the validated parameters from the first experiment. The challenges arising from the test are multiple: the data are collected using our platforms and thus the underlying sensor characteristics are different. In addition, the environment is more complex and the driving conditions are different.

C. Results and Discussion

The results obtained from the first experiment on the KITTI test tracks are illustrated independently for each track. The tables illustrate different error measures: the Mean Absolute Error (MAE), which expresses the mean value of the disagreement between the estimations and the ground truth. The Mean Signed Error (MSE) which expresses the deviation of the estimator errors from zero (i.e. around which point the estimator errors are centered), the Root Mean Squared Error (RMSE) which expresses the mean standard deviation between the estimated values and the ground truth. The Normalized Mean Absolute Error (NMAE) and the Normalized Root Mean Squared Error (NRMSE) are normalized to the corresponding errors of the raw SLAM on the same track. This gives a percentage of the improvement made by the Correction Module on each error measure. The lower the percentage is, the better the Correction Prediction Module is. Moreover, the difference of the absolute mean signed deviation (dMSD) shows the amplification / reduction of the general estimator bias after applying the corrections. It is calculated as the MSD of the rectified SLAM minus the MSD of raw SLAM. Thus, a negative value sign means that correction prediction reduces the bias, while a positive one means that a more important bias affects the estimation after applying the corrections.

Based on the NRMSE and MAE measures, we clearly see that the correction module enhances the accuracy of SLAM estimations. Furthermore, the SLAM+dSML module generally performs better than the other Correction Predictors.

The NMAE measure shows that error amplitudes generally become smaller, from 7% to a 30% attenuation on the amplitude of linear displacement error and from 4% to 9% attenuation on the amplitude of heading errors using Correction Prediction rather than raw SLAM.

Based on the MSE metric, the center of the linear errors when using the corrections of the proposed approach is slightly closer to zero. This desirable effect means that by using the correction module, linear drift is better centered on zero. Conversely, the correction brings the angular errors center slightly farther from zero. The main reason for this behavior is the unbalanced training dataset in terms of drift scenarios.

Moreover, we can observe from Table IV that corrections based on the likelihood distribution induce a slightly amplified heading error. This could be due to the absence of a similar scenario in the training dataset.

The global path of PML-SLAM, the SLAM+dSML (i.e. SLAM with correction prediction based on successive SLAM poses and likelihood distribution) path and the ground truth path, are illustrated in Figures 5 and6. A comparison between the PML-SLAM estimated path (illustrated in gray), the ground truth path (yellow) and the SLAM+dSML rectified using the proposed approach (in blue).

The path of rectified SLAM using the correction prediction is almost superimposed on the ground truth path in Figure 5 and Figure 6(c). Furthermore, in Figure 6(b), the rectified path and the endpoint of the rectified path are closer to the ground truth path and endpoint than the path estimated by raw SLAM. In contrast, in Figure 6(a) the raw SLAM final endpoint is closer to the ground truth than the rectified SLAM+dSML. In fact, raw SLAM performs estimation errors that make the path deviate from the ground truth, but later on, other errors reduce the global localization error.

Reducing errors occurring at each time-step along the trajectory should theoretically bring the estimated path closer to the ground truth. However, this is not necessarily always the case, especially in complex and long circuits because of the incremental nature of the underlying approach (c.f. Figure 1). In other words, the accuracy measure cannot be reduced to how far the endpoint of the estimated trajectory is compared to the ground truth. The validated parameters for each module which were used to obtain the illustrated results are summarized in Table VI. Although these parameters gave the lowest global error results on the KITTI test tracks, the global error does not Fig. 6. A comparison between PML-SLAM estimated path (illustrated in gray), the ground truth path (yellow) and the rectified SLAM using the proposed approach (in blue) based on previous displacements and the likelihood distribution change greatly when using other configurations (i.e. when varying (λ, ξ)). Thus, the results obtained using the approach we propose can be considered quite stable.

The second experiment on the Bordeaux Circuit was performed using the validated models and parameters of the previous experiment. Table VII summarizes the error values based on the same metrics as those used in the first experiment.

We clearly see that the corrections improve the heading accuracy considerably (nearly 20% vs. 10% in the first experiments based on the NRMSE measure). In fact, the Velodyne used to acquire the KITTI dataset has a higher angular resolution 0.09°separating laser beams vs. 0.25°for the IBEO-Lux laser sensors used in the Bordeaux Circuit acquisition. Consequently, the use of the IBEO-Lux laser sensors causes a greater angular error on the bare SLAM than the use of Velodyne. Therefore, the correction module had adapted to handle the errors induced from the underlying sensor to overcome the subsequent additional amount of errors in the heading. The error histograms of PML-SLAM and SLAM+dSML with corrections are illustrated in Figure 7. The histogram emphasizes the number of frames in which a given amount of errors occurs. We see that the rectified estimation histograms peaks are much higher and the bins at the margins are less tall than on the error histograms of the bare PML-SLAM. In other words, applying the corrections in SLAM estimations attenuates the errors.

VI. CONCLUSION AND PERSPECTIVES

In this paper, we have presented an approach to predict corrections for 2D SLAM methods. Based on two types of relevant information (successive estimated poses and the likelihood distribution), our strategy could be suitable both for 2D SLAM methods (independently of the sensor used and the underlying SLAM approach) and for 2D Maximum Likelihood SLAM approaches (also independently of the underlying sensor). The proposed strategy performs separately from the SLAM module and operates in parallel where the proposed corrections are applied a posteriori to the SLAM estimations. The correction modules are based on MLP Ensembles which can be trained once off-line.

The validity of our approach was proved through two experiments. We observed an improvement (attenuation of the error) of approximately 20% to 30% in linear displacement errors, and of approximately 10% in heading errors, both based on the NRMSE measure. In addition, we can see that the model's global errors do not change greatly when changing the Ensemble and MLP parameters. Thus, the solution gives a stable behavior to the model parameters. Furthermore, in the second experiment, our approach showed a robustness to additional errors induced from using a lower resolution sensor. Indeed, we managed to reduce the heading errors by 20% when using a laser with 0.25°resolution vs. 10% when using a laser sensor with 0.09°. In contrast, the modules built showed that angular error corrections are less centered to zero than when using SLAM alone. This could be due to a training bias because right turn maneuvers are more frequent in the dataset than left turns.

In future work, we plan to extend our approach and make use of the probabilities of the particles as relevant information in the case of PF-based SLAM approaches, and the sigmas probabilities from the UKF-based SLAM in order to build more suitable correction prediction models for these approaches. Moreover, we aim to evaluate the gain in error attenuation when fusing multiple localization sources. Finally, we believe it is worth studying a homogeneous learning dataset in terms of scenarios and its impact on the quality of the resulting correction modules.

Fig. 2 .

 2 Fig. 2. Block diagram of Correction Prediction algorithm together with SLAM

  100

Fig. 4 .

 4 Fig. 4. VEDECOM Institute experimental platform. The car is equipped with 5 IBEO-Lux laser scanners with a high resolution IMU fused with an RTK-GPS receiver.

Fig. 5 .

 5 Fig.5. KITTI test track 2 (residential). A comparison between the PML-SLAM estimated path (illustrated in gray), the ground truth path (yellow) and the SLAM+dSML rectified using the proposed approach (in blue).

Fig. 7 .

 7 Fig. 7. Error histograms of the SLAM (gray bins) vs. the SLAM+dSML (blue bins) with the correction prediction on the Bordeaux Circuit. The histograms superposed, thus the dark blue area means a shared area between the SLAM with and without corrections.

  The selected Test sequences from the KITTI dataset are named as follows: { city: 2011 09 26 drive 0117 sync, residential: 2011 09 26 drive 0064 sync, residential: 2011 09 30 drive 0020 sync, road: 2011 09 29 drive 0004 sync, } that we tag from Track 1 to 4 respectively.

	26 drive
	0013 sync; city: 2011 09 26 drive 0093 sync; road: 2011 10
	03 drive 0042 sync; road: 2011 10 03 drive 0047 sync.

  Table II illustrates errors on KITTI test track 1, Table III shows errors on test track 2, Table IV shows errors on track 3 and Table V for the KITTI test track 4. Each table is divided into two parts, the upper part details linear displacement errors d expressed in cm, while the lower part gives the angular component of the error θ , expressed in degrees.

TABLE II ERRORS

 II TABLE SHOWING PERFORMANCES OBTAINED USING DIFFERENT MODULES ON KITTI TEST TRACK 1 (2011 09 26 DRIVE 0117).

	lin(cm)	MAE	RMSE	MSD	NMAE	NRMSE	dMSD
	SLAM	1.78	2.38	-0.13
	+dS	1.41	1.94	0.021	79.28%	81.66%	-0.109
	+ML	1.63	2.18	0.0004	91.39%	91.74%	-0.1296
	+dSML	1.39	1.94	-0.085	77.84%	81.80%	-0.045
	rot(°)	MAE	RMSE	MSD	NMAE	NRMSE	dMSD
	SLAM	0.043	0.055	-0.004
	+dS	0.041	0.053	0.041	96.46%	96.67%	+0.037
	+ML	0.039	0.051	0.039	92.14%	91.63%	+0.035
	+dSML	0.038	0.049	0.038	91.27%	90.36%	+0.034
				TABLE III
	ERRORS TABLE SHOWING PERFORMANCES OBTAINED USING DIFFERENT
	MODULES ON KITTI TEST TRACK 2 (2011 09 26 DRIVE 0064).
	lin(cm)	MAE	RMSE	MSD	NMAE	NRMSE	dMSD
	SLAM	1.43	1.87	-0.298
	+dS	1.29	1.85	-0.072	90.47%	98.86%	-0.226
	+ML	1.38	1.96	-0.130	93.02%	92.49%	-0.168
	+dSML	1.07	1.51	-0.145	74.87%	80.53%	-0.153
	rot(°)	MAE	RMSE	MSD	NMAE	NRMSE	dMSD
	SLAM	0.045	0.06	-0.006
	+dS	0.044	0.059	0.044	98.31%	98.64%	+0.038
	+ML	0.044	0.07	0.044	92.17%	92.25%	+0.038
	+dSML	0.041	0.055	0.041	91.74%	91.97%	+0.035

TABLE IV ERRORS

 IV TABLE SHOWING PERFORMANCES OBTAINED USING DIFFERENT MODULES ON KITTI TEST TRACK 3 (2011 09 30 DRIVE 0020).

	lin(cm)	MAE	RMSE	MSD	NMAE	NRMSE	dMSD
	SLAM	2.01	2.61	-1.166
	+dS	1.87	2.40	-0.383	93.08%	92.11%	-0.783
	+ML	1.75	2.27	-0.382	87.12%	86.85%	-0.784
	+dSML	1.64	2.15	-0.166	82.02%	82.52%	-1
	rot(°)	MAE	RMSE	MSD	NMAE	NRMSE	dMSD
	SLAM	0.056	0.083	-0.008
	+dS	0.055	0.082	0.055	99.65%	99.52%	+0.047
	+ML	0.057	0.083	0.057	102.01% 101.97% +0.049
	+dSML	0.056	0.084	0.056	101.07% 101.59% +0.048
				TABLE V
	ERRORS TABLE SHOWING PERFORMANCES OBTAINED USING DIFFERENT
	MODULES ON KITTI TEST TRACK 4 (2011 09 29 DRIVE 0004).
	lin(cm)	MAE	RMSE	MSD	NMAE	NRMSE	dMSD
	SLAM	1.30	1.62	0.105
	+dS	0.99	1.25	0.156	76.29%	77.07%	+0.051
	+ML	1.26	1.57	0.405	96.95%	96.69%	+0.3
	+dSML	0.89	1.13	0.125	69.00%	69.69%	+0.02
	rot(°)	MAE	RMSE	MSD	NMAE	NRMSE	dMSD
	SLAM	0.036	0.044	-0.004
	+dS	0.034	0.042	0.034	95.24%	95.83%	+0.03
	+ML	0.031	0.038	0.031	86.87%	87.37%	+0.027
	+dSML	0.031	0.038	0.031	87.23%	87.52%	+0.027

TABLE VII ERRORS

 VII TABLE SHOWING PERFORMANCES OF SLAM WITH AND WITHOUT USING THE CORRECTION PREDICTION MODULE ON THE BORDEAUX CIRCUIT.

	lin(cm)	MAE	RMSE	MSD	NMAE	NRMSE	dMSD
	SLAM	2.76	3.87	-0.36
	+dSML	2.44	3.61	-0.72	88.26%	93.20%	+0.36
	rot(°)	MAE	RMSE	MSD	NMAE	NRMSE	dMSD
	SLAM	0.063	0.089	-0.006
	+dSML	0.048	0.071	0.04	76.47%	79.15%	+0.034