Gianluca Antonelli
email: antonelli@unicas.it

Elisabetta Cataldi
email: e.cataldi@unicas.it

Filippo Arrichiello
email: f.arrichiello@unicas.it

Paolo Robuffo

Stefano Chiaverini
email: chiaverini@unicas.it

Antonio Franchi
email: afranchi@laas.fr

† G Antonelli

‡ P Robuffo

≀ A Franchi

Adaptive Trajectory Tracking for Quadrotor MAVs in Presence of Parameter Uncertainties and External Disturbances

The paper presents an adaptive trajectory tracking control strategy for quadrotor Micro Aerial Vehicles (MAVs). The proposed approach, while maintaining the common assumption of an orientation dynamics faster than the translational one, removes the assumption of absence of external disturbances and of Geometric Center coincident with the Center of Mass. In particular, the trajectory tracking control law is made adaptive with respect to the presence of external forces and moments (e.g., due to wind) and to the uncertainty of parameters of the dynamic model, such as the position of the center of mass. A stability analysis is presented to analytically support the proposed controller, while numerical simulations are provided in order to validate its performance.

I. INTRODUCTION

Over the last years the robotics community experienced a substantial increase of interest in the Micro Aerial Vehicles (MAVs) field. In particular, quadrotor MAVs have become more and more widespread in the community as experimental platform for testing novel 3D planning, control and estimation schemes in real-world indoor and outdoor conditions. Indeed, in addition to being able to take-off and land vertically, quadrotors can reach high angular accelerations thanks to the relatively long lever arm between opposing motors.

The development of effective flight controllers and motion planning strategies has been one of the primary objectives in MAV research over the last decade, see e.g. [START_REF] Michael | The GRASP multiple micro-UAV testbed[END_REF]- [START_REF] Hua | Introduction to feedback control of underactuated vtol vehicles: A review of basic control design ideas and principles[END_REF]. Together with the constant improvements in the miniaturization of Micro Electro-Mechanical Systems and sensors (MEMS) and in the computational power of microcontrollers, this has led to impressive achievements by employing quadrotor MAVs as robotics platforms: planning and control for aggressive flight maneuvers [START_REF] Mellinger | Trajectory generation and control for precise aggressive maneuvers with quadrotors[END_REF] and collective control of multiple small-and micro-quadrotors [START_REF] Kushleyev | Towards A Swarm of Agile Micro Quadrotors[END_REF], [START_REF] Franchi | Shared control: Balancing autonomy and human assistance with a group of quadrotor uavs[END_REF] are just a few examples.

Being the quadrotor an underactuated mechanical system (indeed, only four control inputs are available despite the six dimensions of its configuration), a common strategy is to control its 3D position and yaw angle, i.e., the so-called flat outputs for the system [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF]. The addition of tilting propellers to the classical quadrotor model has nevertheless been recently explored in [START_REF] Ryll | A novel overactuated quadrotor uav: Modeling, control and experimental validation[END_REF] in order to increase actuation capabilities and to gain full controllability of the 6-Degrees Of Freedom (DOF) of the quadrotor pose in space.

Robustness of the flight controller performance is a fundamental feature for any MAV application. Integral-based actions can be used to counteract external disturbances, such as wind and presence of small loads. Nevertheless, an adaptive/integral action may result in an additional disturbance when the nonlinearities of the model are not properly taken into account, see, e.g., [START_REF] Antonelli | On the use of adaptive/integral actions for 6-degrees-offreedom control of autonomous underwater vehicles[END_REF] for analogous problems in the marine context. In this sense, at the best of our knowledge the only adaptive control for MAVs has been proposed in [START_REF] Palunko | Adaptive control of a quadrotor with dynamic changes in the center of gravity[END_REF], [START_REF] Palunko | Agile load transportation. Safe and efficient load manipulation with aerial robots[END_REF].

The goal of this paper is to present a novel adaptive control scheme for quadrotor MAVs able to take into account the effects of constant exogenous forces and moments, as well as to cope with the presence of unknown dynamic parameters (e.g., the position of the CoM). Preliminary results of this work were presented in the papers [START_REF] Antonelli | Adaptive trajectory tracking for quadrotor MAVs in presence of parameter uncertainties and external disturbances[END_REF] and [START_REF] Antonelli | Experimental validation of a new adaptive control scheme for quadrotors MAVs[END_REF]; in particular, the work [START_REF] Antonelli | Adaptive trajectory tracking for quadrotor MAVs in presence of parameter uncertainties and external disturbances[END_REF] contains the stability analysis and numerical validation of the proposed adaptive control law, while [START_REF] Antonelli | Experimental validation of a new adaptive control scheme for quadrotors MAVs[END_REF] reports the experimental results. In this paper we extend these works by elaborating and discussing on the effects obtained when placing the origin of the body-fixed frame at the center of mass rather then at other more convenient locations, such as the quadrotor geometrical center. Moreover, here we present a more complete simulative case study considering both the cases of changing payload mass (such as during a transportation mission) and the presence of wind.

II. MODELING A. Kinematics

Let us define a frame Σ b , O bx b y b z b fixed to the MAV rigid body, from now on the body-fixed frame. The rigid body pose is described by its position and orientation with respect to a reference frame Σ I , O Ixyz that here is assumed as an inertial, North-East-Down, earth-fixed reference frame.

Let η 1 = x y z T ∈ R 3 be the position of O b expressed in Σ I , and η 2 ∈ R 3 be the vector collecting the set of body Euler-angle coordinates in Σ I η 2 = φ θ ψ T . Among the possible combinations, the roll, pitch and yaw angles are selected, i.e., the set of successive elementary rotations around x, y and z in Σ I [START_REF] Siciliano | Robotics: modelling, planning and control[END_REF]. The two vectors can be collected as

η = η T 1 η T 2
T representing the overall six-dimensional position vector of the MAV. The vectors η1 and η2 are time derivatives of η 1 and η 2 respectively (i.e., η1 is the velocity of O b expressed in Σ I and η2 is the derivative of the Euler-angles expressed in Σ I ).

Let's define ν 1 = u v w T as the linear velocity of O b with respect to O I , expressed in Σ b , then it holds:

ν 1 = R b I η1 , (1)
where R b I (η 2 ) is the rotation matrix expressing the transformation from the Σ I to Σ b .

Let's define ν 2 = p q r T as the angular velocity of Σ b with respect to Σ I expressed in Σ b . Following [START_REF] Siciliano | Robotics: modelling, planning and control[END_REF], the relation between the vectors η2 and ν 2 can be expressed by the transformation matrix T ∈ R 3×3 , i.e. :

ν 2 = T (η 2 ) η2 . (2)
The two velocity vectors can be collected in the sixdimensional velocity vector ν defined as

ν = ν T 1 ν T 2 T . x y z O I O GC O CoM f 1 f 2 f 3 f 4 r C ω t,1 ω t,2 ω t,3 ω t,4
l Fig. 1. Variables related to the actuation system with, in particular, the two possible placements for the origin of the body-fixed reference frame (Geometric Center (GC) or Center of Mass (CoM))

B. Dynamics

As done, among the others, in [START_REF] Michael | The GRASP multiple micro-UAV testbed[END_REF]- [START_REF] Kushleyev | Towards A Swarm of Agile Micro Quadrotors[END_REF], we model the quadrotor dynamics using the equations of motion for a single 6-DOF rigid body, i.e. we do not consider here the effects of the spinning propellers on the main body dynamics. When placing the origin O b of the body-fixed reference frame, two choices are naturally possible: the Geometric Center (GC) of the quadrotor, i.e., the intersection between the two thrusters axes, or the Center of Mass (CoM) of the quadrotor structure (in Fig. 1 we mark as O GC as O CoM the respective two possible placements of O b ).

The rigid body dynamics of a quadrotor w.r.t. the body-fixed frame placed in the GC is given, in matrix form, by:

M ν + C(ν)ν + g(η 2 ) = τ + τ W , (3)
where τ ∈ R 6 represents the generalized input forces acting on the vehicle. In detail, let us partition τ as

τ = τ T 1 τ T 2 T ,
where the vector τ 1 = X Y Z T collects the linear forces acting on the rigid body expressed in the body-fixed frame, and the vector τ 2 = K M N T collects the moments acting on the rigid body expressed in the body-fixed frame.

The system inertia matrix M ∈ R 6×6 is constant, symmetric and positive definite, i.e., Ṁ = O, M = M T > O. Its unique parametrization takes the form:

M = mI 3 -mS(r b C ) mS(r b C ) I O b , (4)
where

r b C = r C,x r C,y r C,z T ∈ R 3
is the center-ofmass position expressed in body-fixed frame, I 3 is the (3 × 3) identity matrix, I O b is the inertia tensor expressed in the bodyfixed frame, S(•) is the skew-symmetric matrix representing the cross product operator [START_REF] Siciliano | Robotics: modelling, planning and control[END_REF], and m ∈ R + is the total mass of the vehicle. Explicit expressions for C can be found in, e.g., [START_REF] Fossen | Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles[END_REF]. The gravity generalized force g ∈ R 6 , acting at the center of mass r b C , is represented in body-fixed frame by:

g(η 2 ) = - mR b I g I mS(r b C )R b I g I , (5)
where g I = 0 0 g T m/s 2 and g = 9.81. When placing the body-fixed frame at the CoM, this expression results simplified since the last three terms of the vector (i.e., the moment components) are null.

The term τ W ∈ R 6 represents external forces/moments acting on the vehicle due to disturbances such as wind. To this purpose, let define as γ W ∈ R 6 a vector of constant parameters; within a time interval in which the external disturbances can be considered as constant, their effect can be modeled in the vehicle-fixed frame as

τ W = Φ W (η 2 )γ W = R b I O 3×3 O 3×3 R b I γ W ( 6
)
where the (6×6) regressor matrix Φ W expresses the force and moment coordinate transformation between the two frames.

By exploiting the linearity in the parameters, it is possible to rewrite (3) as:

Φ( ν, ν, η 2 )γ = τ (7)
where γ ∈ R 16 collects the following dynamic parameters: the mass (1 param.), the first moment of inertia (3 param.), the inertia tensor (6 param.) and the disturbance γ W (6 param.).

The first three rows of equation ( 7) can be rewritten with respect to the variables expressed in the inertial frame η, η by exploiting ( 1)-( 2) and the corresponding time derivatives, according to the guidelines of robotics textbooks, e.g., [START_REF] Siciliano | Robotics: modelling, planning and control[END_REF]- [START_REF] Fossen | Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles[END_REF]. In particular, on the base of the diagonal structure of the M 1,1 block of eq. ( 4), they can be decomposed in the terms Φ xy ∈ R 2×16 and φ z ∈ R 1×16 denoting respectively the first two rows and third row of Φ in [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF] expressed in the inertial frame, i.e.,:

  Φ xy ( ẍ ÿ , η2 , η, η 2 ) φ z (z, η2 , η, η 2 )   γ = R I b τ 1 . (8)
For the sake of space, in this work we will omit further details on the system dynamics. However, the controller presented in the following will intentionally only consider the parameters affecting the steady-state error; these will be listed case-by-case during the next developments.

C. Thrusters

Quadrotors are equipped with four thrusters aligned along the z b axis placed at position p b t,i ∈ R 3 , for i = 1, . . . , 4. Each thruster provides a force and a moment according to

f i = bω 2 t,i , τ t,i = dω 2 t,i for i = 1, . . . , 4
where ω t,i is the angular velocity of the i-th rotor, b and d are the thrust and drag coefficients. Figure 1 reports the common motor position, positive rotation directions and relevant variables. In this figure both the GC and CoM body-fixed frames are highlighted.

When calculating the input forces acting on the vehicle, for both GC-and CoM-centered models it holds

τ 1 = 0 0 Z c T = 0 0 4 i=1 -f i T . (9)
The moment components τ 2 , on the other side, differ between the GC-and CoM-centered models since they are computed with respect to different axes. With a proper numbering of the thrusters, for the GC-centered model it holds:

τ 2 = l(f 2 -f 4 ) l(f 1 -f 3 ) τ t,1 -τ t,2 +τ t,3 -τ t,4 T (10)
where l is the arm length.

For the CoM-centered model, assuming r b C = r C,x r C,y r C,z T = 0 T , the computation of the moments is not anymore as symmetric as in [START_REF] Palunko | Adaptive control of a quadrotor with dynamic changes in the center of gravity[END_REF]. Indeed, a value r C,x = 0 and r C,y = 0 introduces distortion effects along the pitch and roll directions, respectively. After few computations one has [START_REF] Antonelli | Adaptive trajectory tracking for quadrotor MAVs in presence of parameter uncertainties and external disturbances[END_REF] while the yaw torque (N) is not affected by the CoM position. Due to the distributive property over the addition of the cross product, it is then easy to verify that

K = l(f 2 -f 4 ) + r C,y (f 1 + f 2 + f 3 + f 4 ), (11) M = l(f 1 -f 3 ) -r C,x (f 1 + f 2 + f 3 + f 4 ),
  p b t,i -   0 0 r C,z     ×   0 0 -f i   = p b t,i ×   0 0 -f i   ∀ i (13)
and thus the displacement along z b of the center of gravity does not modify the moment contribution.

Since the dynamics of the low-level motor controller can be typically neglected with respect to the vehicle motion time scale, it is possible to express the mapping from the angular velocities to the force-torque acting on the vehicle as:

    Z K M N     = B     ω 2 t,1 ω 2 t,2 ω 2 t,3 ω 2 t,4     = Bu (14)
where B can assume the forms

BGC =    -b -b -b -b 0 bl 0 -bl bl 0 -bl 0 d -d d -d    , or (15)
BCoM=    -b -b -b -b brC,y b(l + rC,y) brC,y -b(l -rC,y) b(l -rC,x) -brC,x -b(l + rC,x) -rC,x d -d d -d   (16)
depending on the choice of a GC or CoM-centered model, respectively. It is worth noting that, for the GC-centered model, matrix B does not depend on the CoM coordinates also in presence of a non-zero offset of the CoM with respect to the GC, i.e., when r b C = 0.

D. Discussion

The reported analysis shows that the selection of where to place the origin of the body-fixed frame should be properly assessed when considering quadrotors with the CoM and the GC placed at different locations (as, e.g., when carrying external loads or because of imprecise calibration). Indeed, when opting for a CoM-centered model, the inertia matrix and gravity terms result simplified but, on the other hand, the computation of the thrust-generated moments shows a dependence on the (typically unknown) r C .

Conversely, the opposite choice of resorting to a GC-centered model solves the sensing part and simplifies the computation of the thrust-related moments but, on the other hand, introduces both an undesired coupling in the inertia matrix, and a moment contribution in the gravitational term. However, again, these effects cannot be compensated in the controller as they would require the knowledge of r C that, on the contrary, is in most cases unknown.

Neglecting the displacement among the GC and the CoM in the feedback control law introduces a perturbation effect that can be evaluated as follows. Denoting with the subscript c the force components elaborated by the feedback control assuming a GC-centered model, the actual forces components applied on the quadrotor in a CoM-centered model are

    Z K M N     =B CoM B -1 GC     Z c K c M c N c     =     1 0 0 0 -r C,y 1 0 0 r C,x 0 1 0 0 0 0 1         Z c K c M c N c     ; (17)
thus, a perturbation effect proportional to the offset is introduced in the K and M components; such perturbation might become significant for MAVs carrying a payload. Therefore, the ability of being robust w.r.t. the CoM position by means of an online adaptive scheme is clearly beneficial for an effective flight control. The next section discusses a possible solution.

III. QUADROTOR ADAPTIVE CONTROL

We present here an adaptive control law for quadrotor position and yaw tracking based on the GC-centered model. The proposed solution consists of a set of interconnected control algorithms that allow to separately deal with horizontal/vertical position and yaw tracking, while managing the underactuated nature of the quadrotor structure exploiting the common assumption of an orientation dynamics significantly faster than the translational one. Desired force and torque are then used to compute the motors' square velocities u as

u = B -1 Z c K c M c N c T (18)
where B = B GC as defined in [START_REF] Fossen | Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles[END_REF].

A. Altitude control

By denoting z d as the desired altitude, let us define the error variables z = z d -z ∈ R, s z = ż + λ z z ∈ R with λ z > 0, and γz = γ -γz ∈ R 16 , where γ z represents the combined contribution of the gravity and the vertical wind effects. The Lyapunov candidate function

V z > 0 is V z (s z , γz ) = m 2 s 2 z + 1 2 γT z K γ,z γz (19)
with K γ,z > O being the adaptive gain. The time derivative of V (s z , γz ) is given by V (s z , γz ) = s z mz d -mz + mλ z ż -γT z K γ,z γz (20) that, making mz explicit from eq. ( 8), can be rewritten as

V (s z , γ) = s z φ z d γ -0 0 1 R I b τ 1 -γT z K γ,z γz (21) in which φ z d = φ z (z d + λ z ż, η2 , η, η 2 )
and where φ z is as defined in eq. ( 8).

Assuming φ, θ = ±π/2, the scalar function V is made negative semidefinite by selecting

Z c = 1 cos φ cos θ φ z d γz + k v,z s z γz = K -1 γ,z φ T z d s z (22)
yielding, after some algebra,

Vz (s z , γz ) = -k v,z s 2 z . (23)
We can now prove the system stability in a Lyapunov-like sense using Barbȃlat's Lemma [START_REF] Khalil | Nonlinear Systems[END_REF]. Since • V z (s z , γz ) is lower bounded;

• Vz (s z , γz ) ≤ 0;

• Vz (s z , γz ) is uniformly continuous; then Vz (s z , γz ) → 0 as t → ∞ and thus s z → 0. In particular, as usual when resorting to adaptive control techniques, one cannot prove asymptotic stability of the whole state since only boundedness of γz can be guaranteed.

It is interesting to implement a simpler version of the controller aimed at compensating the sole persistent dynamic terms, i.e., those terms preventing a null steady-state error [START_REF] Antonelli | On the use of adaptive/integral actions for 6-degrees-offreedom control of autonomous underwater vehicles[END_REF]:

Z c = 1 cos φ cos θ (γ z + k vz s z ) γz = k -1 γ,z s z (24)
in which γ z ∈ R embeds the combined contribution of the gravity and the vertical wind effects.

B. Horizontal position control

Vxy = s T xy Φ xy d γ -R T z c φ s θ -s φ Z c -γT xy K γ,xy γxy (26) with Φ xy d = Φ xy (η d,xy + λ xy ηxy , η2 , η, η 2 ) ∈ R 2×16 and
and where Φ xy is as defined in eq. ( 8). Similarly to the analysis performed in the altitude control case, according to (9) the sole Z c component is considered as possible body-fixed force provided by the actuators. The scalar function Vxy can be made negative semidefinite by selecting the virtual inputs φ d and θ d , for the orientation control so as to solve

c φ d s θ d -s φ d = 1 Z c R z Φ xy d γxy + k v,xy s xy , γxy = K -1 γ,xy Φ T xy d s xy . (27)
In this case too, by considering only the persistent dynamic terms as in [START_REF] Antonelli | On the use of adaptive/integral actions for 6-degrees-offreedom control of autonomous underwater vehicles[END_REF], the controller reduces to:

c φ d s θ d -s φ d = 1 Z c R z γxy + k v,xy s xy γxy = k -1 γ,xy s xy ( 28
)
where γxy ∈ R 2 represents the sole wind effect (supposed constant in the inertial frame), and φ d and θ d are sent to the orientation controller as the desired angles.

C. Orientation control

The orientation control receives as input the desired roll and pitch φ d , θ d from the horizontal position control (see Section III-B) and the desired yaw ψ d , and it outputs the torque components

K c M c N c T .
When resorting to a GC-centered model, see ( 14)-( 15), the moments generated by the controller can be transformed into the thruster angular velocities without the need of a known CoM position. On the other hand, at hovering, the vehicle is experiencing the moments from [START_REF] Kushleyev | Towards A Swarm of Agile Micro Quadrotors[END_REF] computed for null roll and pitch, i.e., g| φ=0,θ=0 = mg r C,y -r C,x 0

T . This effect needs to be properly compensated and, thus, an integral or adaptive action is required to estimate the CoM position rC in order to achieve null error at steady-state. The yaw dynamics, instead, appears to be decoupled from the remaining DOFs and it is not affected by the CoM displacement. The orientation control can be structured, as an example, as a set of independent Proportional-Derivative (PD) controllers of the different orientation components with gravity compensation:

K c = P D(φ) + rC,y Z c , M c = P D(θ) -rC,x Z c , N c = P D(ψ) with ṙC,x ṙC,y = k rC θ d -θ φ -φ d .
It is worth remarking that neither the altitude nor the yaw control loops are affected by the offset between CoM and GC; the convergence to a steady-state value for Z c can thus be assumed. In any case, roll and pitch control can be designed by considering the estimation error as an external, constant, disturbance and several adaptive control laws for attitude control may thus be used to the purpose.

IV. SIMULATIONS Numerical simulations in a wide range of operative conditions have been run in order to verify the effectiveness of the proposed adaptive controller. Here we report the results of numerical simulation in Matlab [START_REF]MATLAB, version 7.10.0[END_REF] environment gained with the sole reduced controller (i.e., the one ensuring a null steady-state error detailed in (24) and (28)) in order to prove that the persistent terms have been effectively compensated by resorting to a minimal set of five dynamic parameters; namely, {γ z , γ xy , r C,x , r C,y }, where γ z ∈ R represents the combined contribution of the gravity and the vertical wind effects, γ xy ∈ R 2 the wind horizontal effect, and r C the center-of-mass position expressed in body-fixed frame.

In order to properly test the proposed adaptive control strategy, we consider the case of a quadrotor picking up and transporting an unknown object while being affected by a non-constant disturbance term due to wind. Table I reports the dynamic parameters of the simulated quadrotor (without considering the possibly transported object) and, when applicable, the initial values used by the adaptive controller. reports the control gains. The sampling time of the sensors and the controller have been set to T = 1 ms.

We assume that the grasped object is a point mass of 0.5 kg and that the grasping point in the O GC frame is 0.13 0.11 0.00 T m; thus, when the object is grasped, the position of the CoM of the system in the O GC frame becomes r C = 0.07 0.05 0.00 T m.

The mission has been decomposed in the following time stages: [0 -10] s: keep a hovering position in order to let the adaptive controller extinguish the initial transient; Figure 2.a shows the components of the quadrotor position (x-magenta, y-black, z-red) and its desired values (in gray). For sake of comparison, we performed the same mission using in one case the adaptive controller and in another case a non-adaptive controller that, instead, it is composed of a PID controller for both the vertical and horizontal control. Figure 2.b shows the norm of the position error for both the cases (the adaptive case in blue and the non-adaptive case in green) from which one can note the better performance of the proposed adaptive controller. It is worth noticing that the large errors in the z-direction around the time 40 s and 90 s are due respectively to the grasp and the release of the transported object that is simulated via an instantaneous change of the weight and of the position of the CoM.

Figures 2.c-d show in blue the roll and pitch angles and in gray the desired values obtained as outputs of the horizontal position control, see Sec. III-B; it can be noticed how the roll and pitch angles computed by the horizontal position control, in presence of constant input, converge to constant non-zero values in order to counteract for the external wind.

Figure 2.e shows the simulated and the estimated force along z-axis; as expected, when the quadrotor grasps the

a) η 1 [m] (b) η 1,d -η 1 [m] (c) φ d , φ [rad] (d) θ d , θ (
(e) γzr object, the force along the z-axis changes due to the change of the total weight. One can also appreciate how the controller correctly estimates the total force.

The effective (gray) and estimated (blue) forces along the x-axis (solid-line) and y-axis (dashed-line) are shown in Figure 2.f; in this case, the controller ensures a null position error at steady-state. One can also appreciate that during the perturbation the z-axis remains almost constant.

Figure 2.g shows the effective (gray) and the estimated (blue) position of the CoM, where the x-axis components are the solid-lines and the y-axis components are the dashed-lines. It is possible to appreciate that the estimation of the CoM position is quite accurate even if the CoM position changes during the mission due to the grasp/release of the object. In order to evaluate the effectiveness of the proposed control law in different conditions, we repeated the simulations by considering different weights of the transported object. Table III reports, for the considered cases, the weights of the objects and the position of the CoM expressed in body-fixed frame. Figure 3 reports the norms of the position errors and it shows that, during the transient, the errors increase with the weight of the object, while in the state steady the errors converge to zero in all the cases.

In Figure 4 to better appreciate the performances of the proposed control the error of the proposed control (blue) and the PID control (green) are reported.

Results of an experimental validation of the proposed method can be found in [START_REF] Antonelli | Experimental validation of a new adaptive control scheme for quadrotors MAVs[END_REF] (details are here omitted for lack of space).

V. CONCLUSION

In this paper, an adaptive control scheme for quadrotor MAVs has been presented. By first discussing some modeling insights, we have shown how most of the literature resorts to a dynamic model, here labelled as CoM-centered model, that assumes the CoM coincident with the GC; when this assumption is not true, e.g. for MAVs carrying payloads or manipulators, neglecting the offset between CoM and GC implies some errors in the assumed placement of the onboard sensing and actuation system with respect to the CoM. This may cause unpredicted disturbance terms that are not easy to compensate since the exact CoM position is typically unknown. The adaptive control approach proposed in this paper is able to cope with an unknown CoM location, with uncertainties in MAV mass and with the possible presence of external disturbances. The proposed control approach is analytically supported by a stability analysis and its performance are validated via extensive numerical simulation results.

  By denoting x d , y d the desired horizontal position, let us define ηxy = x d -x y d -y T ∈ R 2 , s xy = ηxy +λ xy ηxy ∈ R 2 with λ xy > 0, and γxy = γ -γxy ∈ R 16 , where γ xy represents the wind horizontal effect. We consider the Lyapunov candidate function V xy > 0 V xy (s xy , γxy ) = 1 2 ms T xy s xy + 1 2 γT xy K γ,xy γxy (25) where K γ,xy > O is the adaptive gain. The time derivative of V xy (s xy , γxy ) is given by

  [10 - 20] s: move toward the object position; [20 -40] s: remain in hovering without carrying the object; 40 s: grasp the object; [40-80] s: remain in hovering while carrying the object; [80-90] s: go back to the initial position; 90 s: release the object; [90 -100] s: remain in hovering. Please notice that the system is wind affected in the time intervals [30-40] s and [50-70] s.

Fig. 2 .

 2 Fig. 2. Simulation results -(a) Quadrotor position along the different axis x-magenta, y-black, z-red; (b) Norm of the position error adaptive control (in blue) and the no-adaptive control (in green); (c) Desired (gray) and the effective values of the quadrotor roll; (d) desired (gray) and the effective value of the quadrotor pitch; (e) effective (gray) and estimated (blue) force along the z-axis; (f) effective (gray) and estimated (blue) forces along the x, y-axis, the solid line the x-axis and the dashed line the y-axis; (g) effective (gray) and estimated (blue) position of the CoM, the solid line the x-axis and the dashed line the y-axis.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Simulation results -Norm of the position error of the proposed control for four simulations with different mass attached, in blue 0.5 kg, green 0.8 kg, red 1.0 kg and cyan 1.2 kg.

Table II

 II

				simulated		initial estimate
	mass		m	1.50 kg		1.49 kg
	inertia		I Ob	.025I 3 kg m 2	not used in the reduced controller
	length		l	30 cm		known
	drag coeff.	b	1 Ns 2 /rad 2		known
	thrust coeff.	d	1 Nms 2 /rad 2	known
	center of mass r C,b	5 3 0	T cm	0 0 0	T cm
				TABLE I
			DATA USED IN THE SIMULATION
	λz kv,z kγ,z	1.1 3.0 1.3 k v,φθψ k p,φθψ	λxy kv,xy kγ,xy 10.0 40.0	1.5 2.0 1.0	mass kg 0.5 0.8 1.0 1.2	r C,x m r C,y m 0.070 0.050 0.077 0.057 0.082 0.062 0.085 0.065
		k rC	5.0			TABLE III
	TABLE II GAINS USED IN THE SIMULATION	MASSES OF THE ADDED OBJECT AND THE COM DISPLACEMENT.

ACKNOWLEDGMENTS

The research leading to these results has received funding from the European Community's 7th FP under grant agreement No.287617 (IP project ARCAS) and from the European Community's Horizon 2020 Program under grant agreement No. 644271 (IP project AEROARMS).


G Casalino
email: g.casalino@poliba.it

P Guglielmi

V D Lorusso

M Mortello

P Peyre

D Sorgente

Laser offset welding of AZ31B magnesium alloy to 316 stainless steel

Keywords:

In this paper, the feasibility of using a fiber laser to perform a dissimilar metal joining was explored. AZ31B magnesium and 316 stainless steel were autogenously joined in butt configuration. The weldability between different materials is often compromised by a large difference in thermal properties and poor metallurgical compatibility. Thus, the beam was focused onto the top surface of the magnesium plate, at a certain distance from the interfaces (offset), and without using any interlayer or groove preparation. Such a method was called laser offset welding (LOW). Results proved a very good capability. The ultimate tensile strength exceeded the value of 100 MPa, since a resistant and thin layer of hard intermetallic compounds is formed within the fusion zone. The rupture was observed within the magnesium side, far from the centerline. The metallurgy of fusion zone indicated the effectiveness of phases coalescence, without mixing at liquid states. LOW was demonstrated to be a promising technique to join dissimilar metal welds, being capable to produce an effective bonding with good tensile strength.

Introduction

The capability to join magnesium alloys to steel will facilitate increased use of Mg alloys in the automotive and aerospace industries. The ultra-light magnesium has demonstrated outstanding potential for ultra-light metal construction design. Achieving versatile and tailored properties in one composite part is highly desirable. Hence, reliable welding process would enable broader deployment of magnesium alloys in structures that require integration with steel components for vehicle weight reduction.

The inherent characteristics of Magnesium alloys include strong tendency to oxidize, low absorptivity of laser beams, high thermal conductivities, high coefficients of thermal expansion, low melting and boiling temperatures. Moreover, wide solidification temperature ranges, high solidification shrinkage, a tendency to form low melting-point constituents, low viscosity, low surface tensions, high solubility of hydrogen in the liquid state, and absence of a color change at the melting point are landmarks of Mg and its alloys. Most of these characteristics make Mg alloys hard to be welded by laser.

In particular, Magnesium alloys can have a relatively large freezing range of about 420 • -620 • C. Thus, there is considerable risk of hot-cracking during any fusion welding process. Therefore, no accepted method for metallurgical bonding is available.

Solid state joining of Mg alloys and steel was made with success using both ultrasonic (US) and friction stir welding (FSW) processes. The performance of adhesive bonds was also improved by using solid-state techniques.

Chen and Nakata studied the effect of tool geometry on microstructure and mechanical properties of friction stir lapwelded AZ31 Mg alloy and steel. [START_REF] Wei | Microstructures and mechanical properties of magnesium alloy and stainless steel weld-joint made by friction stir lap welding[END_REF] also investigated the friction stir lap welding of magnesium alloy AZ31 and stainless steel (SUS302) and they found a shear strength of the lap joint, with maximum value closed to the shear strength of butt joint of friction stir welded magnesium alloy. [START_REF] Jana | Friction stir welding of magnesium alloy to steel: a preliminary investigation[END_REF] studied FSW of AZ31 Mg alloy to steel in lap configuration, by developing the concept of using a third metal, such as Zn, at the interface to get successful results. [START_REF] Elthalabawy | Eutectic bonding of austenitic stainless steel 316L to magnesium alloy AZ31 using copper interlayer[END_REF] also studied the eutectic bonding of austenitic stainless steel 316L to magnesium alloy AZ31 using a copper interlayer. In this application, the bonding took place thanks to the occurrence of solid state diffusion of Cu into the magnesium alloy, eutectic phase formation, interlayer dissolution, and isothermal solidification.

Diffusion brazing has been used to join advanced alloys by Saha andKhan (2006, 2009). This method can be a viable alternative for joining Mg alloys-steel dissimilar metals together. Yuan 2013) investigated the diffusion-brazing of Mg alloy (AZ31) and stainless steel (AISI 304L) using a pure copper interlayer. They achieved joints free from defects and reported microhardness peaks in the transition zone due to Mg-Cu intermetallic formation. [START_REF] Liu | The mechanisms of resistance spot welding of magnesium to steel[END_REF] studied the resistance spot welding (RSW) of AZ31B Mg alloy to DP600 steel using Zn interlayer in the form of Zn coated steel sheet. They found that a pre-existing transition layer of Fe 2 Al 5 between the Zn coating and the steel improved wetting and bonding between the steel and the magnesium alloy. [START_REF] Wahba | Laser welding of AZ31 B magnesium alloy to Zn-coated steel[END_REF] studied the laser lap welding of 3 mm thick AZ31B magnesium alloy to 1.2 mm thick Zn-coated steel. An unstable welding process was reported due to different physical properties between the two materials (particularly in keyhole mode when the laser beam penetrates into the steel specimen). [START_REF] Liu | Microstructural characteristics of lap joint between magnesium alloy and mild steel with and without the addition of Sn element[END_REF] investigated joining of AZ31B Mg alloy to Q235 steel with the addition of a 0.3 mm thick Sn interlayer, using the same hybrid process. The addition of Sn eliminated the gaps distributed along the fusion zone-steel interfaces as the presence of Sn improves the wettability of the Mg to the steel. Also [START_REF] Tan | Microstructural characteristics and mechanical properties of fiber laser welded brazed Mg alloy stainless steel joint[END_REF] proposed the dual-beam laser welding-brazing process for Magnesium alloy and stainless steel welds using a Mg-Al-Zn filler wire.

The laser brazing technology was studied both by [START_REF] Nasiri | Effect of Zn interlayer on brazeability of AZ31B-Mg alloy to steel sheet[END_REF] and Naisiri and Zhou (2015). They predicted the early stage phase formation in the steel-interlayer-Mg alloy system during laser brazing, bonding mechanisms in the steel-interlayer-Mg alloy joints, and investigated the mechanism of wetting in steelinterlayer-Mg alloy system. The benefits of using Al-12Si, Ni and Sn interlayers were also explored. [START_REF] Chen | Fibre laser welding of dissimilar alloys of Ti-6Al-4 V and Inconel 718 for aerospace applications[END_REF] joined Magnesium alloy and stainless steel by dual-beam laser welding-brazing process. The authors found that joints fractured at the interface or fusion zone at low/high heat input due to weak bonding or porosity formation at fusion zone. Some other authors have recently proposed the use of the fiber laser for welding dissimilar metal alloys, like [START_REF] Casalino | Laser-arc hybrid welding of wrought to selective laser molten stainless steel[END_REF] and [START_REF] Chen | Fibre laser welding of dissimilar alloys of Ti-6Al-4 V and Inconel 718 for aerospace applications[END_REF].

In this work, the dissimilar laser welding between AZ31B magnesium and 316 stainless steel, by adopting an alternative technique, was presented. Such a novel method, also referred as laser offset welding (LOW), consisted in focusing an Yb-YAG laser source onto the top surface of one of the two materials to be welded, at a specific distance from the interfaces, which is called offset. Neither interlayer nor groove preparation is necessary. This study aims to contribute to extend the adaptability of the above mentioned technique for producing various dissimilar metal welds, since previous studies, conducted by [START_REF] Tomashchuk | Direct keyholelaser welding of aluminum alloy AA5754 to titanium alloy Ti6Al4 V[END_REF] and [START_REF] Casalino | Yb-YAG laser offset welding of AA5754 and T40 butt joint[END_REF], already demonstrated outstanding results in Al-Ti assemblies.

In the present case, the stainless steel plate, which presented higher melting point and larger laser weldability, was irradiated with high energy density in order to promote a stable keyhole mode welding. Then, the heat is transmitted to the Mg sheet without any mixing of liquid phases, leading to the formation of a very thin and regular interface along the weld centerline. The metallurgical bonding that develops during the process and the influence of some main welding conditions on the joint's strength, integrity, and microstructures were explored. The results obtained exhibited a promising feasibility of the LOW technique to produce dissimilar welds between AZ31B magnesium and 316 stainless steel, as demonstrated by good tensile properties and microstructural analysis. In spite of the challenging weldability between the two materials, the metallurgical bonding has resulted effective. Under optimal conditions, the ultimate tensile strength (UTS) even exceeded 100 MPa. This result was consistent with microstructural analysis, which exhibited the formation of a resistant and thin layer of intermetallic compounds within the fusion zone.

Materials and methods

Material properties and weld configuration

In this study, dissimilar assemblies between austenitic stainless steel AISI 316 and magnesium AZ31 sheets, were realized in butt configuration. The size of the sheets was 100 mm length by 50 mm width by 3 mm thickness. Despite the large difference in the tensile properties between the two materials, the weld was performed by adopting the same thickness for both plates. Such a choice was consistent with the main purpose of the investigation, which aimed to explore the process dynamics and the metallurgical bonding. Thus, although the structure would benefit from the use of a thinner steel plate, the priority of this study was to explore the effect of the keyhole along the overall plate thickness. The chemical composition and physical properties of the base metals are listed in Tables 1 and2, respectively.

The welding procedure: LOW technique

The welds were produced by an innovative and promising technique, also referred as laser offset welding (LOW), in which the

- 3.5 - 0.3 - balance - 0.5 - - 0.1 0.005

Table 2

Properties of the as-received materials: ultimate tensile strength (UTS), yield stress (YS), Young module (E), elongation to fracture%(A%), Vickers microhardness (HV), thermal conductivity (K), melting point (Tm), density (), specific heat capacity (c). laser source is focused at a certain distance (offset) from the sheets' contact line. This was possible thanks to the selective and highly focused action of a disk laser, which provides for the keyhole regime fusion in the base structure of the irradiated material.

UTS (MPa) YS (MPa) E (GPa) A% HV K (W/(m.K)) Tm ( • C) (g/cm 3 ) c (J/g
A schematic drawing of the technique is shown in Fig. 1. The need to explore a new procedure to build dissimilar joints derives from some drawbacks that are intrinsically related to fusion welding methods. Despite the successful outcome of traditional fusion based methods for welding a wide range of metal materials, they exhibit high weakness during dissimilar joining. On one hand, the larger the difference in thermal properties, the harder the challenge to accomplish a satisfactory weld. Further, the poor mutual solubility between different metals could hinder the formation of stable coalescence of phases. On the other hand, the formation of brittle intermetallic compounds and mixing of liquid phases weakens the mechanical properties of the joints. Moreover, vaporization phenomena occur when magnesium is irradiated by high-density energy sources. The LOW technique overcomes the limits of the previously mentioned methods, by keeping the fusionaffected interface within thin portion of material and without any direct irradiation of magnesium base sheet. The process was carried out by focusing the laser beam on the stainless steel side, then making the keyhole to travel along a line parallel to the sheets interface. The high melting point metal is melted and then it wets the low melting point one. A thin layer of intermetallic compound forms thanks to heat propagation by conductive and convective heat transfers (see Fig. 2).

Set-up of welding system

Integrated instruments were combined to accomplish the process. The whole apparatus set-up mainly comprises a laser system with multi-axes machine, a shielding gas system and a workbench, equipped with clamps and supporting table (see Fig. 3). A disk Yb-YAG laser, with a wavelength of 1030 nm and a maximum available power of 10 kW, was used in continuous wave regime. The active gain medium consisted in a 200 m diameter Yb optical fiber. Collimating lens and focusing lens with a focal length of 120 mm and 250 mm respectively were adopted for beam delivering. Preliminary tests were carried out to determine the focal plane and focal depth, in order to ensure minimum spot diameter at high energy density. Then a beam profiler was adopted to detect the spatial intensity profile at the focal plane. A focus spot of about 400 m diameter (1/e 2 width) near-Gaussian distribution was positioned on the top surface of the stainless steel sheet according to the LOW principle.

The sheets were fixed to the support by using four clamps. The workbench is schematically presented in Fig. 4.

Plan of the experiment

Preliminary bead-on-plate tests were conducted to find suitable values of process parameters and to identify key issues that could affect the repeatability of the process. In order to reduce the thermal contact resistance, the sheets were prepared by machining at low milling speed, grinding and cleaning with acetone.

A camera inspected the gap between the sheets. It was reported an average gap of about 35 m, which provides for a good thermal coupling. The misalignment and gaps between the sheets are critical problems during autogenous laser welding processes. This aspect is even more detrimental for dissimilar joining, in which the metallurgy could be highly sensitive to small displacements. An optimal clamping and an accurate sheet preparation could potentially minimize these concerns. High-precision cutting methods for interfaces and in situ monitoring of plates displacement could be adopted. However, in the industrial environment, the coupling between the sheets could be exasperated by other factors, including the length of the welds or the capability of the clamping system available. Furthermore, a greater accuracy in the coupling could increase the time and costs of the process. As a consequence, the present study was conducted by adopting a common clamping system, and sheets were prepared according to standard mechanical procedures, in order to approximate the feasibility of the present technique for industrial purposes.

Trial sets of experiments were performed for ensuring satisfactory penetration and avoiding lack of fusion zones. The laser power and the travel speed were kept constant at 2.5 kW and 1.8 m/min respectively. This results in a thermal input (defined as the ratio of the power to the welding speed) of about 83 J/mm. The laser offset was varied on three levels, i.e. 0.3, 0.4 and 0.5 mm, in order to explore its influence on the microstructural properties along the intermetallic layer. In fact, the specific objective of the present work is to study the weldability between magnesium alloy and stainless steel by direct laser irradiation as a function of the offset distance. Visual macro inspections and optical and electron microscopy. Both microhardness and tensile test were performed to analyse the static properties.

Metallographic analysis and mechanical testing

Metallographic specimens were prepared according to standard procedures. Samples were sectioned, mounted in a conductive resin and polished mechanically. Since the weld was made up of two different materials, a tailored procedure for chemical etching was designed, which included two steps:

• Firstly, the magnesium side was etched by immersing the whole specimen surface in a 10% aqueous solution of oxalic acid. This action did not have effect on the steel side, but provided for microstructures in the magnesium fusion zone. • Secondly, an electrolytic polishing/etching machine (STRUERS Lectropol-5) was adopted for electrolytic etching with the same acidic solution, leading to over-etching the magnesium side (that lost contrast) and chromatic contrast in the steel side.

The microstructure was studied by optical microscopy (OM) and scansion electron microscopy (SEM). The analysis of chemical composition was carried out by Energy-dispersive X-ray spectroscopy (EDS). Vickers micro-hardness measurements were carried out on a HighWood HWMMT-X7 hardness tester with a 200 gf load.

Tensile tests on the base material and on the joints were carried out on a universal electromechanical tensile test machine (INSTRON 4485) equipped with pneumatic grips. In order to investigate the strain behavior of the joint, all tests were assisted by an optical strain measurement system (ARAMIS by GOM), based on the digital image correlation. The image is discretized in a cells grid and the software automatically recognizes the cells during the test, calculating displacements and strains. The acquisition of local strain allows acquiring the strain value in a very small area of the specimen. Two 1.3 MPixel sensors were used with the ARAMIS strain measurement software. An acquisition rate of 1 Hz was used during the tests. Before testing, a random black speckle pattern was applied over a previously white painted mat surface, in order to enable data acquisition by cameras.

Microstructure analysis and microhardness

In Fig. 5, cross sections of the joints welded with 0.3 and 0.4 mm laser offset are depicted: the fusion zone in the steel can be observed on the right portion of the image whilst the fusion zone of the magnesium alloy is on the left side. A catastrophic lack of fusion was registered for sample 3, which exhibited poor mechanical properties.

No further description will be reported on sample 3, since its outcome was not successful. Because of the excessive offset distance, the thermal energy at the interface was not sufficient to produce the weld. As a consequence, the steel weld pool collapsed and wetted the solid state magnesium. The interaction between materials in two different states just promoted a cohesive bonding without generating any phases' coalescence. This phenomena is not amongst the topics of the current investigation. Threshold limits for laser offset is a fundamental aspect that will be object of ongoing studies.

As regards samples 1 and 2, the heat input produced a keyhole in the steel and the fusion wall reached the melted magnesium alloy. Due to the asymmetry in the heat conduction, the fusion zone of the steel moved towards the magnesium side where the interface hindered the heat conduction and made this zone overheated. In both conditions reported in Fig. 5, a pronounced excess of metal in the upper part (weld reinforcement) can be seen on the steel side. On the magnesium side (left in Fig. 5) there was a strong undercut both on the top and on the bottom of the weld bead. This weld morphology can be related to the shrinkage during the re-solidification at the liquid-liquid interface. During the heat transfer from the steel to the magnesium side, a steel downstream started from point A at the interface (Fig. 5a). A narrow zone due to the melt flow formed in the molten-to basematerial transition zone. Fig. 6 gives a close-up of the re-solidified structure. The reader can see ␣ and ␤ magnesium phases and interposed steel tongues. The same downstream structure can be observed in Fig. 5b, but the starting point is not in the plane of the picture. That phenomenon can be explained with overheating of some contact points between steel and magnesium, which favored the steel melt flow into the magnesium side.

In Fig. 7, results of the hardness measurement are reported. The two different conditions lead to similar hardness profiles along the weld bead.

The hardness in the magnesium side is almost constant and very close to the value of the base metal (approximately 56 HV). In the interface, there is a significant increase of hardness until an intermediate value between the magnesium alloy and the steel hardness. In the steel side the hardness in the fusion zone is slightly higher than that in the base metal (187 HV and 178 HV, respectively) and a peak value in the heat affected zone is found (approximately 195 HV). No significant difference between the two different welding conditions was found. Fig. 8 shows the microstructure evolution with the distance from the weld centerline.

During welding, a transition zone formed adjacent to the complete fusion zone. It was a partially melting zone, which was heated to the temperature range between the liquidus and solidus temperature. The grains were very small and the volume fraction of equiaxed grains accordingly increased with respect to the base material.

Many precipitates concentrated in the fusion zone, in a distribution that tended to grow from a few scattered particles to densely packed coarser ones, which is favored from the total content of Al in the Mg alloy. Al has the main influence on the formation of particles as [START_REF] Quan | CO2 laser beam welding of dissimilar magnesium-based alloys[END_REF] explained. The coarse dendrites of primary ␣-Mg with the grain size of a few micrometers were observed throughout the fusion zone, which indicated a normal dendrite growth mode, by [START_REF] Guo | Microstructure formation and mechanical properties of AZ31 magnesium alloy solidified with a novel mechanical vibration technique[END_REF].

In the base material, the grains are equiaxed and twins are not observed. The grain size decreased from the base material towards the fusion zone.

Fig. 9 shows the intermetallic edge after welding and the eutectic, which was ␤-Mg x Al y /Mg (x and y depending on Al content), in its typical tortuous and interconnected distribution as it was explained by [START_REF] Wang | Three dimensional (3D) microstructural characterization and quantitative analysis of solidified microstructures in magnesium-Based alloys[END_REF].

From a crystallographic point of view, it is not possible for pure magnesium to nucleate on steel due to the very large lattice mismatching of Fe and Mg, which has called for the use of Ni-interlayer, see [START_REF] Nasiri | Interfacial microstructure of diode laser brazed AZ31 B magnesium to steel sheet using a nickel interlayer[END_REF]. A nano-interlayer probably formed (<0.5 m). The formation of a bond between the two alloys was probably facilitated by the deferred thermal cycle in the Magnesium side with respect to the steel one. Solid state inter-diffusion between Fe and Mg during the welding process did not occur.

The porosity in the weld, which was one of the major concerns during laser beam welding of magnesium alloys, was absent. The loss of alloying elements at high laser power density was avoided by the irradiance incident upon the steel side of the joint.

Tensile test

The tensile test results demonstrated that after welding, both the strength and the elongation decreased. The reduction of the tensile strength can be related to the irregular shape of the weld bead that affected the load-bearing capability during tensile testing.

The ultimate tensile strength (UTS) of the joints obtained in the 0.3 mm offset and 0.4 mm offset conditions are shown in Fig. 10. The sample obtained with 0.5 mm offset showed no resistance. Three samples for each condition were tested and the average UTS was also displayed.

The maximum strain values in the tensile direction, registered along the area acquired by the ARAMIS system, are reported. A standard deviation of 5.08 and 2.58 mm was observed for 0.3 mm offset and 0.4 mm offset conditions, respectively, which indicates a low dispersion of data values. In all the specimens, the maximum values were found in the welded joint and the base material underwent a low deformation.

All welds fractured in the magnesium side, quite far from the inter-layer. Figs. 11 and12 show the macro effect of fracture on the joints obtained for 0.3 and 0.4 mm offset, respectively.

In Fig. 13, the strain profile along the middle section is reported for both the conditions analysed.

fracture and SEM analysis

Fig. 14 shows a SEM image of the Mg side of the fracture surface for 0.3 mm offset.

SEM examinations revealed that cracks initiated from the surface or near surface welding defects at low stress (points 1 and 3 in Fig. 14). Those points were characterized by lack of fusion due to under-heating. The crack propagation through the grains along several crystallographic planes caused the river marking. Then, the cracks formed plateaus and connecting ledges and stopped until the stress rose to an ultimate one. Meanwhile, the Mg deformed in a ductile manner. Eventually, the fracture occurred with dimplelike characteristics at a relative low stress, which was due to the sub-surface defects (points 4 and 5).

Cleavage-like brittle fracture with flat facets in conjunction with river marking was also observed (Figs. 15 and16). The river marking was caused by the crack propagation, see [START_REF] Chowdury | Microstructure and mechanical properties of fiber-laser-welded and diode-laser-welded AZ31 magnesium alloy[END_REF].

The EDM analysis of some points of the fracture surface of sample 1 are shown in Figs. 17-20. The composition (wt%) of elements was also reported.

The points 1 and 3, where Magnesium was absent, presented a cleavage-like brittle fracture. In those points, the rupture was initiated by surface defects and rapidly propagated in the magnesium side of the weld where dimple-like rupture surface was observed (points 2 and 4).

Conclusions

The study of the offset fiber laser welding of magnesium alloy AZ31 to the 316 stainless steel gave the following evidences.

• A thin layer, which dimension was in the order of micron, formed in the weld zone, without the need for any interposed interlayer. as revealed by the SEM analysis. • Steel melt flows into the magnesium side were observed. They may depend on the contact condition between the two plates, which can induce local overheating. • Although zones of lack of fusion and surface defects were observed on the fractured surface, the tensile test indicated that the UTS was close to 100 MPa. • In the zone where the interlayer well formed, the rupture was ductile. A flat facets-to-dimple transition zone was observed. • The loss of alloying elements at high laser power density was avoided, since the laser beam stroke the steel side of the joint.

As a number of drawbacks in Magnesium to stainless steel fusion laser welding were avoided without interposing any interlayer, it has been demonstrated that the LOW technique has the potential to produce the Mg-steel weld. Future work to improve the geometry of welds should validate this statement.

Fig. 1 .

 1 Fig. 1. Schematic illustration of LOW technique procedure.

Fig. 2 .

 2 Fig. 2. Schematic drawing for describing the joining mechanism by LOW technique.

Fig. 3 .

 3 Fig. 3. LOW experimental set-up.

Fig. 4 .

 4 Fig. 4. Schematic drawing to overview the workbench set-up.

Fig. 5 .

 5 Fig. 5. Micrographs of the cross section of the AZ31-SS LOW joint. 0.3 mm offset 1 (a) and 0.4 mm offset (b).

Fig. 6 .

 6 Fig. 6. Microstructure of the steel downstream in the magnesium side at two different magnification.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Hardness profile and values along the weld in two different process conditions.

Fig. 9 .

 9 Fig. 9. Mg-steel interface (a) and evidence of the eutectic (b) at two different SEM magnification.

Fig. 10 .

 10 Fig. 10. Ultimate tensile strength (UTS) (a) and maximum nominal strain% (b) of the joints.

Fig. 11 .

 11 Fig. 11. Macros of the ruptured sample 1.

Fig. 12 .

 12 Fig. 12. Macros of the ruptured sample 2.

Fig. 13 .

 13 Fig. 13. Major strain profiles in different instants during the tensile test on the specimens welded at 0.3 mm offset (a) and 0.4 mm offset (b).

Fig. 14 .

 14 Fig. 14. Fracture surface and EDS point analysis.

Fig. 15 .

 15 Fig. 15. Flat facets-to-dimple transition zone (125×).

Fig. 16 .

 16 Fig. 16. Flat facets-to-dimple transition zone (500×).

Fig. 17 .

 17 Fig. 17. Chemical composition at point 1 (Fig. 14).

Fig. 18 .

 18 Fig.18. Chemical composition at point 2 (Fig.14).

Fig. 19 .

 19 Fig.19. Chemical composition at point 3 (Fig.14).

Fig. 20 .

 20 Fig. 20. Chemical composition at point 4 (Fig. 14).

Table 1

 1 Chemical composition of the as-received materials (weight%).

		C	Al	Cr	Mn	Mo	Mg	Ni	Zn	P	S	Si	Fe
	AISI 316	0.08	-	18	2	3	-	14	-	0.045	0.03	1	balance
	AZ31