domain: spi phys

text:
Silya Achat
email: silya.achat@onera.fr

Julien Marzat
email: julien.marzat@onera.fr

Julien Moras
email: julien.moras@onera.fr

Curved Surface Inspection by a Climbing Robot: Path Planning Approach for Aircraft Applications

Keywords: Coverage Path Planning, Cell Decomposition, Curved Surface Inspection, Sensor Constrained Inspection

This paper presents a path planning method for a climbing robot used for the exterior inspection of an aircraft. The objective is to plan a covering path with the least possible overlap, while respecting constraints related to an embedded sensor, a power cable, and the robot mechanical efforts. To achieve this, a semantic 2D grid model of the aircraft is first created by unfolding and labeling a 3D mesh reference model. An obstacle-based area decomposition method is then applied to divide this 2D discrete space into inspection areas. Inspection segments that satisfy the sensor constraints are then sampled and rearranged based on the order of the areas. The connections between the inspection segments are finally determined by a weighted A * search approach, so as to limit the gravity-induced mechanical efforts on the robot.

INTRODUCTION

Aircraft maintenance is crucial to ensure passenger safety, which includes inspecting their external surfaces to trigger condition-based maintenance operations. Early detection of defects, especially on the fuselage, is necessary to prevent them from causing further damage to the structure. These operations are generally carried out manually by operators, but in recent years the use of remotely operated robotic means has been studied, in particular unmanned aerial vehicles [START_REF] Bugaj | Unmanned aerial vehicles and their use for aircraft inspection[END_REF]. They allow the fast observation of areas that are difficult to access, while reducing the risks taken by technicians. However, these aerial platforms present some drawbacks. First, they require skilled pilots which limits their development and the use of fleets. Second, they cannot deploy all types of measurement payloads, in particular those requiring proximity such as ultrasound measurements. Finally, they present a risk of damage to the inspected aircraft in case of control loss.

The use of autonomous climbing robots represents an alternative to avoid these issues [START_REF] Fernandez | Design of a system for aircraft fuselage inspection[END_REF]. Indeed, by moving directly on the surface of the aircraft, these robots will be able to conduct different types of observation (visual, sonar, etc.) while not requiring a qualified pilot. Different locomotion and adherence technologies can be considered such as wheeled robots with ducted fans [START_REF] Qin | Design and research of a wall climbing robot with ducted fan for aircraft appearance inspection[END_REF][START_REF] Andrikopoulos | On model-based adhesion control of a vortex climbing robot[END_REF] or pneumatic legged con-figurations [START_REF] Wang | Switching motion control of the climbing robot for aircraft skin inspection[END_REF], however magnetic wheels [START_REF] Starbuck | Consistent state estimation on manifolds for autonomous metal structure inspection[END_REF] are not applicable for aircraft surfaces. These climbing robots are generally quite slow, therefore a critical point is to optimize the inspection path allowing to cover the area so that it is the shortest, while taking into account the inspection constraints of the embedded sensors (in range and direction of observation), the motion of the robot (mechanical constraint of adherence or winding of a power cable) and prohibited areas such as the fins.

This paper presents the definition of a method for planning the inspection path for such climbing robots to inspect the fuselage of an aircraft, considering numerous operational and technical constraints. The numerical performances of the procedure are evaluated on a 3D mesh model, as illustrated in Figure 1. This new method relies on a decomposition in areas delimited by obstacles of arbitrary shapes from an unfolded 2D grid model. Inspection segments are defined in these areas with sensor constraints taken into account, and are finally connected using a weighted A * search procedure to provide the global inspection path. Numerous studies have addressed the problem of computing a covering path for applications such as surface inspection [START_REF] Wu | Path planning for surface inspection on a robot-based scanning system[END_REF][START_REF] Liu | Coverage path planning for robotic quality inspection with control on measurement uncertainty[END_REF], as well as in the context of autonomous vacuums [START_REF] Hasan | Path planning algorithm development for autonomous vacuum cleaner robots[END_REF], industrial painting robots [START_REF] Wu | Robotic spray painting path planning for complex surface: boundary fitting approach[END_REF][START_REF] Chen | Automated industrial robot path planning for spray painting process: A review[END_REF], or problems related to the traveling salesman [START_REF] Rai | Research paper on travelling salesman problem and its solution using genetic algorithm[END_REF][START_REF] Dahiya | Literature review on travelling salesman problem[END_REF]. These paths can be computed on flat or curved surfaces. In the case of curved surfaces, there are two possible approaches: either a path is directly computed in the 3D space based on a 3D model, or a representative 2D parametrization is first extracted from the 3D model to exploit a planar model for path planning.

Various strategies can be applied to obtain a path that fully covers the surface to be inspected. Sampling-based methods generate random paths, leading after many iterations to the complete scanning of a given area [START_REF] Sörme | A comparison of path planning algorithms for robotic vacuum cleaners[END_REF]. On the other hand, deterministic methods can be more time-efficient by dividing the area of interest into subareas delimited by obstacles, called cells. This section presents a state of the art of the methods related to the above-mentioned issues, namely path planning methods of 3D structures, different methods for constructing a representative 2D model from a 3D one, and previous works dealing with the decomposition of a zone into sub-areas from obstacles.

Some techniques bypass the 2D representation of space and allow for the construction of covering paths directly on the curved surfaces to be treated. This is the case in the work presented in [START_REF] Mineo | Introducing a novel mesh following technique for approximation-free robotic tool path trajectories[END_REF] where parallel paths are calculated on the surface of a 3D mesh model of a curved piece to be followed by a painting robot. The objective is to create paths spaced by a constant distance d from an edge of the piece. Planes Π i are defined to be perpendicular to the reference edge, containing regularly spaced points P i . To find the points located at a distance d from the reference edge, we have to simply move from P i towards the successive intersection points with a mesh of the surface: the curvilinear distance between P i and an intersection point is the sum of the distances between each intersection point. The point on plane Π i at a distance d from P i is determined by linear interpolation between the two successive intersection points having curvilinear distances from P i respectively greater and lower than d.

Studies on the inspection of large metallic structures such as ship hulls by small differential-drive robots with magnetic wheels [START_REF] Starbuck | Consistent state estimation on manifolds for autonomous metal structure inspection[END_REF] have focused on establishing a planar model repre-sentative of a cylinder by stereographic projection. This model has the advantage of being continuous and therefore exploitable for state estimation by a Manifold Invariant Extended Kalman Filter. Indeed, an unrolled model in cylindrical coordinates would present a discontinuity at the angular position ±π. However, in the context of planning an inspection path, this kind of cylindrical parametrization would be more appropriate as it involves less deformation of distances between points on the surface. Moreover, this discontinuity of the unrolled model could be exploited to manage directly the non-winding constraint of a robot's power cable during path planning.

Coverage Path Planning (CPP) is a well-known discipline, and a relevant state-of-the-art is presented in [START_REF] Galceran | A survey on coverage path planning for robotics[END_REF]. A common first step among the methods described is the decomposition of space into cells defined by the obstacles present in the space. The Boustrophedon Decomposition method [START_REF] Choset | Coverage path planning: The boustrophedon cellular decomposition[END_REF] divides a continuous space into cells based on the vertices of the obstacles, which requires the obstacles to be modeled as polygons. This constraint is lifted by the Morse-Based Boustrophedon Cellular Decomposition method [START_REF] Acar | Morse decompositions for coverage tasks[END_REF], which defines cells based on critical points on the obstacles depending on the inspection direction. Other methods exploit a discrete environment model represented as a grid, such as the Grid Based Coverage using wavefront algorithm [START_REF] Zelinsky | Planning paths of complete coverage of an unstructured environment by a mobile robot[END_REF] and Spiral Spanning-Tree Coverage [START_REF] Gabriely | Spanning-tree based coverage of continuous areas by a mobile robot[END_REF].

The method described in [START_REF] Zelinsky | Planning paths of complete coverage of an unstructured environment by a mobile robot[END_REF] decomposes the space into a grid of uniform cells, where those in contact with an obstacle are not considered by the path planning algorithm. The starting and ending cells are defined, and the algorithm begins the path search from the ending cell. The idea is to propagate a wave from the ending cell to the starting cell, assigning each encountered cell its distance from the ending point. Thus, the cell at the ending point is assigned the index 0. Then, at each iteration, this index is incremented and assigned to all neighboring cells that have not yet been visited. The process is iterated until the starting cell is reached. A covering path is then derived by starting at the starting cell and moving at each iteration to the neighboring cell with the highest assigned number (breaking ties randomly if necessary). Possible applications of this method are adapted to autonomous vacuum cleaners, lawn mowers or security robots.

Another method [START_REF] Gabriely | Spanning-tree based coverage of continuous areas by a mobile robot[END_REF]) is applicable online for a household robot equipped with onboard position, orientation and a range sensor. A decomposition of a large proportion of the free space into so-called mega cells of size 2D is performed. Each mega-cell is divided into four sub-cells of the size of the robot D, which are visited only once by the planned path. It is assumed that a square-shaped planar tool is attached to the robot, and is required to cover the entire surface of a workspace. A constraint is that the robot can only move in directions orthogonal to the four sides of this tool which cannot rotate. Starting from an initial cell, the next direction of movement is determined by selecting the unvisited mega cell in its neighborhood (a mega cell is considered visited if at least one of its sub-cells has been visited) in counterclockwise order, resulting in spiraling paths. During this first phase, a spanning tree is built by connecting the mega cells in their order of visit until each mega cell has been visited. Thus, at the end of this recursion, the path will have traversed the outer part of the spanning tree built online. In a second phase, the inner part of the spanning tree is updated by passing through the remaining unvisited sub-cells of the visited mega cells.

However, the use of a sensor during the robotic inspection of an aircraft's surface requires specific observation directions. This excludes the use of the aforementioned Grid Based Coverage using wavefront algorithm [START_REF] Zelinsky | Planning paths of complete coverage of an unstructured environment by a mobile robot[END_REF] and Spiral Spanning-Tree Coverage [START_REF] Gabriely | Spanning-tree based coverage of continuous areas by a mobile robot[END_REF] path planning methods. The Mesh Following Technique [START_REF] Mineo | Introducing a novel mesh following technique for approximation-free robotic tool path trajectories[END_REF] method could address this constraint but does not handle obstacle avoidance. An additional constraint comes from the robot's power cable, which should not wrap around the structure. This suggests the use of a 2D unfolded model rather than a 3D one, where the angular discontinuity directly enforces this constraint, unlike a 2D continuous model similar to [START_REF] Starbuck | Consistent state estimation on manifolds for autonomous metal structure inspection[END_REF]. Finally, cell decomposition methods such as [START_REF] Choset | Coverage path planning: The boustrophedon cellular decomposition[END_REF] and [START_REF] Acar | Morse decompositions for coverage tasks[END_REF] are relevant strategies for optimizing the order of coverage inspection paths. However, these methods would need to be adapted to a discretized environment to be compatible with path planning methods that exploit grids.

PROPOSED APPROACH

Problem formulation

The problem considered is to generate a path P as the ordered set of waypoints {W n } that a climbing robot must traverse so that its onboard sensor can scan entirely a given aircraft surface (the focus is put here on the fuselage). The path generation solution must respect constraints related to the sensor embed-ded on the robot (here, a laser sensor which must scan a locally flat segment is considered), prevent the power cable from tangling, and minimize the gravityinduced efforts on the robot.

By a space transformation, the inspection zone can be represented in 2D and discretised into a semantic grid G of dimensions i max × j max cells.

The objective then becomes to find a minimallength, ordered set of adjacent traversable cells Q = {q 1 , q 2 , ..., q n } ⊂ G such that if the centre of the sensor line projected in G sequentially traverses each cell q ∈ Q , the sensor will have scanned all the traversable cells. The sensor line is thus considered to be perpendicular to the direction of travel in the grid.

Definition 3.1 ( Cell q ). Square of a 2D grid with coordinates (i, j) to which a semantic label l is associated. The cell is traversable depending on a userdefined rule on the label defined in Table 1.

Definition 3.2 ( Segment ). Ordered set of adjacent cells q. Definition 3.3 ( Waypoint W ). Each cell q ∈ G of the 2D grid corresponds to a waypoint W defined in the 3D space.
First, a method is presented for creating the twodimensional semantic grid G corresponding to a simplified and unfolded model of an airplane from a 3D mesh model format, by parametrizing the external surface of the airplane in cylindrical coordinates. Then, a method for planning parallel inspection segments on this unfolded 2D representation is described, such that each segment corresponds to a subsampling of cells q belonging to the grid representing the aircraft surface, and such that the set of these segments S respects the inspection sensor constraint and ensures complete coverage of the surface. The final set Q is obtained by ordering and connecting the segments using a weighted A * procedure whose cost takes into account the efforts. The 3D path P is finally given by associating to each cell q n ∈ Q its corresponding waypoint W n .

2D Aircraft Model from Unfolded 3D Representation

The outer surface of an aircraft, excluding wings and control surfaces, can be parametrized in cylindrical coordinates with the radius variable representing the surface of interest. Based on this principle, this section proposes a method for creating a 2D semantic model of an aircraft corresponding to the unfolded model in cylindrical coordinates, which can then be used for path planning of inspection. This discontinuity in the unfolded model implicitly prevents the robot from following a path that would wrap around the aircraft, which would be problematic with respect to the power cable. In this work, planning on (or under) the wings is not addressed, since these flat surfaces do not require any particular preprocessing and path generation is straightforward using a Boustrophedon method [START_REF] Bähnemann | Revisiting boustrophedon coverage path planning as a generalized traveling salesman problem[END_REF].

Starting from a 3D mesh model of an aircraft, the vertices are extracted. To obtain a point cloud describing only the surface of the aircraft in a more homogeneous manner, the first step is to interpolate points between the vertices. To take into account the semantics of the different parts of the aircraft, each point in the cloud has been manually labeled as indicated in Table 1. A central axis ⃗ z has also been defined, so as to convert the points from Cartesian coordinates to cylindrical coordinates (cf. Figure 2).

The 3D point-cloud is defined in Cartesian frame

P cart = {P i cart } for i ∈ 0 N
where N is the number of points of the point-cloud. A semantic label l i is associated to each point of the cloud. We define the cloud P cyl from P cart through the cylindrical coordinate transformation: ∀P cart = x y z , P cyl = ρ θ z where:

ρ = x 2 + y 2 θ =          arctan( y x ) if x > 0 arctan( y x ) + sign(y) • π if x < 0 π 2 • sign(y) if x = 0 and y ̸ = 0 indeterminate else
An intermediate grid in cylindrical coordinates is defined with steps δθ and δz. It is used to define the two following mappings for each cell:

• R (θ, z), which contains a radius ρ as a function of θ and z (see Figure 3a)

• L(θ,z), which describes the corresponding label l as a function of θ and z (see Figure 3b )

For a given cell of coordinates θ f z f and size δθ δz T we consider the set of points

P f = {P i ∈ P cyl } such that θ ∈ θ f ± δθ 2 and z ∈ z f ± δz 2 . An index i f is then computed as follows i f = arg max P i ∈P f (ρ i ) R (θ, z) = ρ i f L(θ,z) = l i f
If the set P f contains several points with identical ρ, the tie is broken using a majority vote on the labels of the set.

In the aircraft point-cloud considered, points labeled "Winglets", "Engines", and "Rear Stabilizer", Figure 2: Semantic point cloud obtained after vertex interpolation and manual labeling from an aircraft 3D mesh that are considered as non-traversable by the robot and which can be easily truncated from the model, were excluded. The "Wing" label points with a radius larger than that of the surrounding fuselage are also excluded, not only because path planning in this area is not addressed as justified previously, but also because their shape is not expected to be compatible with the surface representation based on θ and z. Indeed, several "Wing" labeled points could have highly variable radius, and only the one with the largest radius would be retained in the model, which is not representative of the actual wing surface. Moreover, this would result in the projection of the "Wing" label onto fuselage areas located in the same cell, which would exclude these areas from the inspection path calculation. From the perspective of the fuselage inspection, the resulting "Wing" points are considered as non-traversable obstacles around which the inspection should be conducted.

The semantized coordinates of a point (ρ θ z) T and the associated label l in the semantic point cloud can be directly obtained from the grid indices (i j) T in the unfolded discrete model through the inverse transformation:

         θ = i • δθ z = j • δz ρ = R (θ, z) l = L(θ,z) (1)
In the model studied, the z-axis origin has been repositioned such that the points labeled as "Rear Stabilizer" are excluded from the 2D grid. This inverse transformation makes it possible to reconstruct a 3D point cloud from the two surface models, as illustrated in Figure 3c. This will allow us then to obtain the waypoints in the 3D space from the set of q cells in the 2D grid. Three 2D grid layers of dimension i max × j max can now be defined, as follows.

Definition 3.4 ( Label Grid G ). The label at coordi-nates (i, j) of the semantic grid G is valued as:

G(i, j) = L(i • δθ, j • δz)
(2) Definition 3.5 ( Traversability Grid B ). B is a binary grid where

B(i, j) = 1 if G(i, j) is traversable 0 otherwise (3)
Then, to avoid collisions between the inspection robot and obstacles, a dilation is applied to B so that the cells with a non-traversable label are dilated by a circular structuring element with radius equal to ∆ j, which is the length of the sensor field projected onto the grid.

Definition 3.6 ( Obstacle Grid O ).
O is an obstacle grid of dimensions i max × j max obtained by using the Connected-Component Labeling algorithm [START_REF] Chang | A linear-time component-labeling algorithm using contour tracing technique[END_REF] on the grid B. As a result, neighbor cells q ∈ O corresponding to obstacles are assigned the same index, and cells corresponding to free space are assigned 0.

Inspection segments

From the two-dimensional grid model of the aircraft, a full-coverage path should be determined to provide successive inspection points fulfilling the required objectives and constraints. The climbing robot must perform a full coverage of the surface while avoiding obstacles such as hatches and wings. The robot's sensor field of view should also be aligned with the surface to carry out its mission. This results in the need for the inspection paths to be wrapped around the aircraft's longitudinal axis, each of them remaining in a plane defined by a constant z coordinate. In a 2D grid, this corresponds to scanning the cells in a direction collinear with ⃗ i. The definition of the inspection segments is performed thanks to the binary grid B. They are sampled along the i axis between labels considered as obstacles, and they are spaced from each other along the j axis by a distance of ∆ j = floor l sensor 2 • δz , where l sensor is the width scanned by the sensor on the surface. This ∆ j parameter could be chosen smaller than this value in order to pre-compensate for path tracking errors by the robot. Each cell q of a segment (corresponding to coordinates (i, j)) is stored in a vector p k , which corresponds to a path identified by index k (see Figure 4), and the collection of these segments is stored in a set of inspection segments S.

Definition 3.7 ( Inspection Segment p k ). A segment p k of index k ∈ N * , and of first cell q b and last cell q e with respective coordinates (i b , j b ) and (i e , j e ), verifies: • ∀q 1 , q 2 ∈ p k , j 1 = j 2 := j k

• if i b > 1, then B(i b -1, j k ) = 0 • if i e < i max , then B(i e + 1, j k ) = 0
The collection of these segments is initially stored in a set of inspection segments S.

Obstacle-Based Area Decomposition in 2D Discrete Space

The grid G is partitioned into ordered areas a m with respect to obstacles from the grid O. This will be used to connect the inspection segments contained in these areas.

Definition 3.8 ( Area a m ). The m-th area a m is a set of traversable cells located between groups of nontraversable cells with the same obstacle index, respectively denoted by n ol on the left border and n or on the right border. The areas are primarily ordered along ascending i indices, and then along the j indices. If there exists an area a i , i < m that has the same n ol and n or indices as a m , then a third index cpt is used to distinguish them. This defines the unique identifier vector α m = [n ol n or cpt] of the area a m . The ordered A stores all the identifiers α m .

The obstacle indices n ol and n or take their values from the grid O, and if there is no obstacle (i.e. a traversable cell is located at the minimum or maximum i index on a given j-th line) then the default assigned index is -1.

Remark. The variable cpt ∈ N is a counter to distinguish areas that are located between the same obstacles of index n ol and n or but are not connected along the j-axis.

All the segments p k that belong to an area a m are traversed in ascending order of j to determine the ordered set of cells Q m . It should be noted that this assumes that the very starting cell of the global path will be in position (i = 1, j = 1). Let q b and q e be the starting and ending cells of p k . To determine the area a m to which a path p k belongs, we query the labeled obstacle grid O to find the obstacle identifiers n ol and n or respectively neighbors of q b and q e . Indeed, we keep track of the last j-coordinate of the last added path to each area a m , and this data is stored in a vector last j area at the index m. Let q last be the last cell q of the last segment p k added in Q m . If q last is closer in the grid to q e than q b , the next segment p k = {q 1 , q 2 , ..., q N } is reversed. This operation consists of reordering p k in descending order of indices n, resulting in a segment p k = {q N , q N-1 , ..., q 1 }. This procedure iterates until all the segments p k ∈ S have been inserted into Q . As a result, on average, one inspection segment over two is reversed to sweep the surface in a Boustrophedon pattern.

Algorithm 1 describes the corresponding pseudocode, and an illustration of the clustering is given in Figure 5. The remaining step of the method is to connect the inspection segments together to form the final global path Q . When there are no obstacles to the left of an area, n ol defaults to -1 (and vice versa for n or ). In this example, we can observe that a 1 and a 2 are differentiated by their value of cpt since the connection criterion is not satisfied, same for a 3 and a 6 .

Algorithm 1: Ordering of the inspection segments q last = Q {m}.end().end()

Input: set S, grid O Output: set Q 1 sets A ← {}, Q ← {[]} 2 Vector last j area ← [] 3 for p k in S do 4 Integer cpt ← 0 5 cells q b ← p k .begin(), q e ← p k .end() 6 if q b .i > 1 then 7 n ol ← O(q b .i -1, q b . j) 8 else 9 n ol ← -1 10 if q e .i < i max then 11 n or ← O(q e .i + 1, q e . j)
25
if distance(q last , q e ) < distance(q last , q b ) then 26 p k .reverse();

27 Q {m}.pushBack(p k ) 28 last j area[m] ← q b . j

Path Connection between Inspection Segments that minimize Gravity Constraints

The robot's configuration implies that it experiences the least amount of effort when its vertical axis is parallel to gravity, that is to say when it adheres to a horizontal surface. Indeed, the robot tends to slip the most when it adheres to a vertical surface, which requires it to deliver more power to adhere more strongly to the wall and thus increase the frictional force that keeps it in place. It will therefore expend less energy adhering to the surface at the top ( θ = π 2 ) and bottom of the air-craft ( θ = -π 2 ). Moreover, intermediate cells need to be inserted outside of the inspection segments so that the robot can move from an inspected area to an uninspected one, and it can also move from one segment to another within the same area. It is preferable for the robot to pass through the locations where the gravitational force is energetically less constraining. For this purpose, a weighted version of the A * algorithm is used to plan the connecting segment (defined below) linking the last cell of the previous inspected segment q last and the first cell q b of a not-yet-inspected segment. Definition 3.9

( Connecting Segment d k 1 ,k 2 ). A Con- necting Segment d k 1 ,k 2 of indices k 1 , k 2 ∈ N * is
a segment that connects two inspection segments p k 1 and p k 2 . Let q b d and q e d be the first and last cell of d k 1 ,k 2 , and q b k and q e k be the first and last cell of p k . Then d k 1 ,k 2 verifies q b d = q e k 1 and q e d = q b k 2 .

The computation process evaluates the cost f associated to the starting cell and other neighboring cells successively until reaching the q b cell (the neighborhood is defined as 8-connectivity). These costs between a currently evaluated node q cur and one of its neighbors q nei are defined as:

g(q nei , L) = g(q cur , L) + c(q nei , L) • ∥ξ nei -ξ cur ∥ (4) f (q nei , q b , L) = g(q nei , L) + ∥ξ b -ξ cur ∥ (5)
where ξ cur , ξ nei , ξ b are the discrete indices (i, j) associated with cells q cur , q nei , q b , respectively. The weighting function is defined as : 6) where (θ, z) are the non-discretized coordinates associated to q nei . θ is defined such that θ ∈ [-π; π] and such that θ = -π 2 at the bottom of the aircraft, and θ = π 2 at the top. This cost factor gives more weight to the distance between a node q cur and its neighbor q nei if it has an angular position θ far from ± π 2 . It would also be possible to introduce a cost function that prioritizes a path crossing through certain labels over others, for example, in cases where avoiding passage over windows is preferable, without being an absolute requirement. This is a way to generate a semanticaware path from the available graph information, as previously proposed in [START_REF] Achat | Path planning incorporating semantic information for autonomous robot navigation[END_REF].

w(q nei , L) = 1 + |cos(θ)| if L(θ,z) is traversable ∞ otherwise (
All the inspection segments previously ordered by areas are traversed in this order to be connected together in Q . Thus, the Global Path Q contains the subsets Q m ranked in ascending order of m such as In the end, a comprehensive global path which includes all the inspection segments is obtained, with additional connecting paths (see Figure 6). The spacing between these segments is at most equal to half of the sensor's field of view (by construction of the 2D grid), ensuring that the entire space to be inspected is covered. The inverse transformation (1) enables to project back the global inspection path in the initial 3D space as illustrated in Figure 7.

d k 1 ,k 2 (brown). In this example, Q = {Q 1 , Q 2 , . . . , Q 7 }, with Q 1 = {p 1 , d 1,2 , p 2 , d 2,3 }, Q 2 = {p 3 , d 3,6 , p 6 , d 6,4 }, Q 3 = {}, Q 4 = {p 4 , d 4,5 }, Q 5 = {p 5 , d 5,7 }, Q 6 = {p 7 , d 7,8 }, Q 7 = {p 8 , d 8,9 , p 9 }. ∀Q m ∈ Q and ∀p k ∈ Q m , if p k belongs to a m . The p k are ranked in Q m in ascending order of k. Every subsets Q m ∈ Q contains also connecting segments d k 1 ,k 2 between every p k such as ∀d k 1 ,k 2 ∈ Q m : ∃!p k ∈ Q m |q b d = q e k ,

NUMERICAL EXPERIMENTS

Path Planning Unitary Tests

This section aims to evaluate the previously presented path planning method, specifically focusing on the effectiveness of sorting inspection paths into areas prior to the connection process. The developed algorithms have been implemented in C++ and run within a Ubuntu 20.04 WSL Environment on a computer equipped with Intel Core i5-10400H CPU and 16GB RAM. Calculation times are detailed in Table 2. The following Table 3 presents the numerical parameters associated to the 2D grid generated from the aircraft 3D mesh. Table 4 presents the characteristics of the inspection segments that have been defined in the 2D grid. It should be noted that in our case, the length of the sensor's field of view is such that its half-length in the 2D grid ∆ j is equal to 1. This implies that the robot must pass through all the grid cells q associated with traversable labels. A comparison was conducted between the computed inspection paths in the two following cases:

• "With sorting by area": the global path is computed with the proposed path planning method.

• "Without sorting by area": the global path is computed by connecting the inspection segments together, still avoiding obstacles, but without prior area sorting (Section 3.4).

In both cases, the length of the computed path, as well as the number of connected waypoints were recorded (see Table 5). It turns out that sorting the inspection segments by the areas defined between obstacles allows us to halve the total distance traveled, and decrease by 93% the number of interpolated waypoints to connect these segments.

ROS Simulation

For illustration purposes and to eventually close the gap with real-world deployment, a ROS simulation has been set up to test the path-following behaviour by a climbing robot on the surface of an aircraft. A wheel-based robot equipped with an adhesion system similar to the Vortex Climbing Robot (Andrikopoulos et al., 2019) is simulated using the ROS/Gazebo environment (see Figure 10) 1 . The adhesive force of the robot is emulated by applying forces on the robot in the opposite direction of the local surface normal.

For this purpose, a model of the surface normal vectors of the airplane N (i, j) has been established from a meshed model, in a similar manner to the models R (i, j) and L(i, j). Note that N is a grid of dimen-1 Video: https://tinyurl.com/OneraClimbingRobotPlanning sions i max × j max × 3. A representation of the surface normal vectors of the airplane is depicted in Figure 9.

In order to perform path tracking control, the 3D position and orientation of the robot (x r y r z r quat) T are associated with a 2D position and orientation in the plane defined by (θ, z). The orientation of the robot in this plane corresponds to the projected yaw angle ψ, and its position is defined by θ and z. The odometry data provides the position of the robot in the 3D space, namely x r , y r and z r , as well as its orientation quat parametrized as a quaternion. Thus, the robot's state in the 2D plane is described by the coordinates (θ z ψ) T . This robot local planar state is determined as follows. Firstly, the unit vector ⃗ v corresponding to the robot's orientation in the 3D space is computed using the quat value. Let P 1,cart be the origin of the vector ⃗ v (thus P 1,cart is located at the center of the robot in the 3D space) and let P 2,cart be the point located at its endpoint. These two points can be rewritten in cylindrical coordinates similarly to Section 3.2, which gives P 1,cyl and P 2,cyl . The coordinates of these points in the local 2D plane are P 1 = (ρ 1 , θ 1 , z 1 ) and P 2 = (ρ 2 , θ 2 , z 2 ), which allows us to determine the state of the robot as:

  θ z ψ   =   θ 1 z 1 atan2(z 2 -z 1 , ρ 2 • θ 2 -ρ 1 • θ 1 )   (7)
Due to the discontinuity at θ = ±π, the calculated value of the angle ψ can be incorrect when the points P 1,cyl and P 2,cyl are located on different sides of this boundary. To address this issue, the value of θ 2 is adjusted within the range [-2π; +2π] based on θ 1 ∈ [-π; +π]. The cells q = (i, j) of the precomputed offline global inspection path Q are converted into waypoints with coordinates θ w z w ρ w = i • δθ j • δz R (i • δθ, j • δz)

The path tracking performed by the robot is a waypoint-based navigation method based on a unicycle model. Thus, the robot aligns itself towards the next waypoint by adjusting its yaw angle in the 2D plane with the angular set-point ψ s = atan2(z wz, ρ w • θ wρ • θ) using a proportional angular controller. A velocity controller is then applied once the angular error is below a small threshold. This control logic leads to the expected correction in the 3D space and the simulated robot model was successfully able to track the inspection path on the surface of the aircraft (Figure 10).

CONCLUSIONS AND PERSPECTIVES

A method for planning a covering path on a curved surface in the context of an automated inspection mission has been proposed in this paper. A 2D unfolded model of this surface was established using a parametrization of an input 3D model in cylindrical coordinates. It was discretized to obtain two surface functions, one describing the shape of the surface and the other its semantics. Inspection segments were sampled in this 2D space, taking into account various constraints related to the robot. A method for decomposing this discrete space into areas has been defined to order these segments efficiently. Connection segments have been inserted between the inspection segments and calculated to pass through locations that minimize the robot's efforts using a weighted A * search procedure. This method has been tested numerically on a representative aircraft 3D mesh where the relevance of sorting the inspection path segments by areas was demonstrated, and in a Gazebo simulation with a simplified climbing robot model. In the context of aircraft inspection, approximating its surface with a cylindrical parametrization presents some limitations, e.g. to exclude the wings in the model. It would be interesting to investigate in future work the extension of the path planning method to surfaces of different shapes, or the possibility of dividing a 3D model into multiple sub-models that can be individually approximated by a collection of geometrical primitives.

Figure 1 :

 1 Figure 1: Illustration of a path planned for a climbing robot performing aircraft fuselage inspection

  Figure 3: 3a : Airplane Surfacic Model R (θ, z), 3b : Airplane Semantic Unfolded Model L(θ,z) , 3c : 3D Semantic Point Cloud reconstructed by inverse transformation (1)

Figure 4 :

 4 Figure 4: Definition of the inspection segments p k visualized on the G(i, j) grid (simplified version without dilatation of the obstacles). The robot is represented in grey, and the field of its inspection sensor is shown in white. The direction of the robot's movement is represented by a black arrow.

Figure 5 :

 5 Figure 5: Example of areas numbering based on their semantics and surrounding obstacles.When there are no obstacles to the left of an area, n ol defaults to -1 (and vice versa for n or ). In this example, we can observe that a 1 and a 2 are differentiated by their value of cpt since the connection criterion is not satisfied, same for a 3 and a 6 .

14

  Vector α ← n ol n or cpt 15 Integer m ← A. f ind(α) b . jlast j area[m]| ̸ = ∆ j and m not empty do 21 cpt ← cpt + 1 22 α ← n ol n or cpt 23 m ← A. f ind(α) 24

Figure 6 :

 6 Figure 6: Final global path Q (black) vizualized in the 2D unfolded discrete space (simplified version). It is composed of inspection segments p k (red) and connecting segments

  with q b d the first cell of d k 1 ,k 2 and q e k the last cell of p k . d k 1 ,k 2 is inserted at the index following p k in Q m .

  Zoomed front bottom view Figure7: Global planned path with full coverage. Inspection segments are represented in red, and connecting segments that limit gravity constraints in brown.

Figure 8 :

 8 Figure 8: Computed inspection path in the case "Without sorting by areas". In this case, the length of connecting paths (represented in brown) is significantly increased in comparison to the path illustrated in Figure 7.

Figure 9 :

 9 Figure 9: 3D field of surface normal vectors to the airplane obtained through the inverse transformation of the grid N

Table 1 :

 1 Semantic labels and related traversability

	Label l i	Label Name	Color	is traversable?
	l 1	Nose	dark green	0
	l 2	Fuselage	blue	1
	l 3	Window	magenta	1
	l 4	Engine	grey	0
	l 5	Rear Stabilizer	cyan	0
	l 6	Wing	yellow	0
	l 7	Winglet	orange	0
	l 8	Hatch	green	0

Table 2 :

 2 Calculation times

	Sampling of inspection segments p k (ms)	0.54
	Algorithm 1 (ms)	0.61
	Calculation and insertion of connecting segments (ms)	53

Table 3 :

 3 Numerical characteristics of the test grid

	Longitudinal resolution δz (metres)	0.24
	Angular resolution δθ (degrees)	3.6
	Grid dimensions i max × j max	101 × 226
	Number of areas	33

Table 4 :

 4 Numerical characteristics of the inspection segments generated from the test grid

	Number of inspection segments sampled	300
	Inspection segments cumulative	19282
	length in pixels (in metres)	(3639 m)
	Average spacing between two consecutive waypoints (metres)	0.19
	Half-length of the sensor's range ∆ j in cells l sensor 2 in metres	1 cell (0.4 m)

Table 5 :

 5 Comparison of interpolated paths with and without sorting inspection paths into areas

		Without	With
	Method	sorting by	sorting by
		areas	areas
	Number of interpolated waypoints	19234	1343
	Path length (in metres)	8206	4034

ACKNOWLEDGMENTS

This research was partially supported by DGAC France Relance project EXAM, in the frame of the NextGenerationEU program.


Côme Legrand

Guillaume Fromentin

Gerard Poulachon

Mickael Rancic

Gérard Poulachon

Development of an enhanced burr

Keywords: Burr modelling, Burr accumulation, Planing experiment, Burr measurements

come

Introduction

The modelling of burrs formation is a real issue on critical parts. In the industry, it is necessary to remove them for insuring fatigue resistance. These deburring operations induce extra costs, explaining the need of reliable models to minimize burrs size. This study focuses on two types of exit burrs. They are presented on Fig. 1 and named according to Régnier et al. [START_REF] Régnier | Fundamental study of exit burr formation mechanisms during orthogonal cutting of AlSi aluminium alloy[END_REF]. The first type corresponds to burrs without chamfer (cf. Fig. 1a). They are described by Gillespie and Blotter [START_REF] Gillespie | The formation and properties of machining burrs[END_REF] as the long ones found at the end of a cut. During a broaching operation, cut thickness is usually small and often smaller than the root thickness of the burrs without chamfer. This configuration may lead to burr accumulation according to Régnier et al. [3]. Thus, the large number of teeth on a broach could conduct to a situation where each tooth, one after the other, pushes the matter outside of the part, resulting in a burr growth by accumulation. The second type of burrs, which is discussed in the last part of this article, concerns burr with chamfer (cf. Fig. 1b) resulting of a crack propagation while the tool moves outside of the workpiece, preventing a burr without chamfer to be shaped.

Hashimura et al. [START_REF] Hashimura | Analysis of burr formation mechanism in orthogonal cutting[END_REF] describe the steps of burr formation and the critical moment when a crack may initiate or not to form a burr with or without chamfer for one cut. According to the authors, the type of burr depends on the work material behaviour: ductile (without chamfer burrs) or fragile (with chamfer burrs). Their observations are done at very low speed (25 μm/s) in a scanning electron microscope (SEM) during copper and Al-2024-O aluminium alloy machining. A finite element analysis based on their observations is presented in parallel to give results that could help the development of future mechanical models. Régnier et al. [START_REF] Régnier | Phenomenological study of multivariable effects on exit burr criteria during orthogonal cutting of AlSi alloys using principal components analysis[END_REF] observe during the machining of a cast aluminium alloy (AlSi7Mg0.3 + 0.5Cu) that both types of burr may appear on the same part exit edge due to the microstructure heterogeneity and that the proportion of the burr type can be controlled by changing the cut thickness in orthogonal cutting. Olvera and Barrow [6] explain also that the axial depth of cut in milling can change the type of burr and that it exists a transition value of the cut thickness parameter. Low cut thickness is associated with burr without chamfer, and high ones with the other type. According to observations, a small burr (circled in Fig. 1b) may appear at the end of the crack propagation area after the formation of a burr with chamfer.

Aurich et al. [START_REF] Aurich | Burrs-analysis, control and removal[END_REF] show in their bibliography review that the main parameters that influence burr formation are work material, tool geometry, wear, machining parameters (especially cut thickness and cutting speed), and shape of the workpiece at tool exit. The three first parameters are also very influent on cutting forces. Several studies [START_REF] Niknam | Analytical modelling of slot milling exit burr size[END_REF][START_REF] Bourlet | Analysis and modeling of burr formation during the plane milling of cast aluminum alloy using polycrystalline diamond tools[END_REF][START_REF] Régnier | Experimental analysis and burr formation modeling in the AlSi7Mg0.3+0.5Cu alloy -Application to orthogonal cutting and milling[END_REF] introduce burr geometrical models based on cutting force or their mechanistic formulation. Such modelling approach shows good results but does not consider the mechanic of burr formation and only gives results for one output.

Few mechanical models exist to describe the burr formation during orthogonal cutting in the known literature. They are based on the plastic hinge theory.

Ko and Dornfeld [START_REF] Ko | A study on burr formation mechanism[END_REF] introduce a model of burr formation based on observations of plasticine and copper machining, two ductile materials, at very low speed. These observations lead to distinguish three steps during burr formation:

(i) Burr formation initiation. When the tool reaches the part exit edge, there is a point (A) where the chip stops growing and the burr formation starts (cf. Fig. 2a). Pekelharing [START_REF] Pekelharing | Cutting tool damage in interrupted cutting[END_REF] also observe this phenomenon in AISI 1045 steel. The tool cutting edge is then at a distance w of the part exit surface. At this point, the energy previously used to generate the chip, now develops the burr. The previous positive shear plane at the chip root is transformed to a negative shear plane at an angle β 0 with the machined surface. It starts at the tool tip and finishes at a point, which becomes the centre of a plastic hinge (B), at the part exit surface.

(ii) Burr development. As the tool moves forward, the negative shear plane rotates, the plastic hinge centre staying at the same position. The material above this plane is considered to rotate. The maximal strain is supposed to be at the tool tip. The deformation is composed of plastic shear and bending. (iii) Burr formation. The burr is shaped by the rotation of the negative deformation plane that pushes the material outside of the part. As the plane rotates, the strain increases. If a fracture occurs, it will propagate along the negative deformation plane and create a burr with chamfer. The modelling of crack initiation for burr with chamfer is discussed in a following paragraph. Figure 2b introduces the final burr shape description that is used in this article.

Ko and Dornfeld [START_REF] Ko | A study on burr formation mechanism[END_REF] determine w and β 0 considering the minimum energy principle and the energy conservation. In addition, Ko and Dornfeld [START_REF] Ko | Burr formation and fracture in oblique cutting[END_REF] introduce the effect of the tool cutting edge inclination angle into their previous burr model.

Chern and Dornfeld [START_REF] Chern | Burr/breakout model development and experimental verification[END_REF] use the same approach for the burr formation description but introduce the hypothesis of the incompressibility of the triangles ABC and BCE. This methodology for the β 0 calculation is assumed to be more accurate than the one proposed in [START_REF] Ko | A study on burr formation mechanism[END_REF]. They also introduce the influence of the part exit angle Φ e . The results are in good agreement with experimental measurements for copper and two aluminium alloys. Toporov et al. [START_REF] Toropov | A new burr formation model for orthogonal cutting of ductile materials[END_REF] describe the 2D negative deformation curve not with a straight line but with a slip line that have the shape of an arc of a circle. The burr incompressibility hypothesis is also used. The results are in agreement with experimental observations for pure copper. The model also takes into account the tool exit angle Φ e .

Niknam and Songmene [START_REF] Niknam | Modeling of burr thickness in milling of ductile materials[END_REF] use a similar mechanical model, as the one described in [START_REF] Ko | A study on burr formation mechanism[END_REF], to predict burr thickness when milling two aluminium alloys. They show good agreements between experimental results and calculated values in an operation which is not planning.

To conclude, a high link between chip and burr formation exists. The mechanical modelling of burr formation seems to be relevant with experimental results close to modelled values in ductile materials. The key parameters appear to be cut thickness, cutting forces, chip shear angle and tool exit angle.

The present study focuses on the understanding and the modelling of burr formation and its accumulation after multicut. The configuration used is orthogonal cutting of nickel based alloy. A novel geometrical approach of burr formation is proposed considering the burr accumulation process. The developed methodology based on a plastic hinge mechanical model allows the quantification of burr height, root thickness and added volume by each cut, during accumulation. The effect of the cutting geometry, the cut thickness and the orientation of the part exit surface is discussed thanks to an experiment developed to follow after several cuts the burr growth with an in situ camera and a LASER profilometer. The model has been extended to evaluate the occurrence of crack initiation leading to burr with chamfer. This work proposes a new outlook on the burr formation mechanism and complete the models already developed in the known literature.

Experimental setup and measurements

Experimental setup and parameters

The experimental setup is presented on Fig. 3. It allows carrying out a planing operation in a 3-axes gantry milling machine DMG DMC-85 V along its X-axis. The moving sample is maintained in a holder which is attached to the

Burr height measurement

Z-axis of the milling machine. The tool stays fixed in the machine environment and is attached to a tool holder located on a Kistler 9257 force sensor. It measures the cutting forces during the planing operations in the three axes (X, Y, Z) of the milling machine. A Basler acA19210-155um camera is used to capture a picture of the burr side after a planing cut. It has a 2.3 MP CMOS sensor. A telecentric lens is added to the camera to limit optical aberrations. The camera axis is parallel to the Y machine axis. In addition, a LJ-V7060 Keyence LASER profilometer located on the machine table allows scanning the burr shape, all along the sample exit edge. The LASER beam located in the XZ machine plane has a measurement repeatability of 0.4 μm and 5 μm along the X and Z axes respectively. The third scan direction is obtained by moving the sample with the Y-axis, and the linear encoder of the machine is used to trig the LASER profilometer every 4 μm increment.

This experimental setup allows to characterise the burr geometry between each cut, and then to evaluate its accumulation. The capture of the burr side picture and the burr scan takes time, so it only has been recorded for some selected cut numbers.

The experimental procedure is the following one:

(i) Scan with the LASER profilometer of the initial sample exit surface before any cut, (ii) Single planing operation with forces measurement, (iii) Capture of the burr profile picture with the camera, in the case of the selected cut, (iv) Three dimensional scan of the burr at the sample exit edge, in the case of the selected cut, (v) Go back to ii for a new cut.

It appears this first global scan of the burr is contaminated by some non-consistent points, mainly due to the sharp edges of the sample and the reflections on its holder. The first step of the process presented on Fig. 4a consists of deleting these points to clean the scans.

On Fig. 4b, a best-fit plane is determined thanks to the point cloud of the initial surface scan of the sample before the 1st cut. This plane becomes the reference for the next analyses, with its own coordinate system.

One hundred consecutive profiles, in the XZ machine plane, are extracted from the centre part of each burr scan and then rotated in the reference exit plane coordinates system. Figure 4c shows a graph where the one hundred selected profiles are in the exit plane coordinates system. The point having the maximum altitude (along the Z exit plane axis) determines the burr height for each profile. The mean value and the standard deviation of the one hundred local burr heights are then calculated and plotted on a graph, as on Fig. 4d. This graph shows the burr height as a function of the accumulated cut thickness and presents the results of the burr accumulation. The accumulated cut thickness of a pass is defined as the sum of the cut thickness of all the previous cuts and of the current one. The error bars represent plus and minus 1.96 times the standard deviation. They do not take into account directly the measurement uncertainty of the LASER profilometer but represent the variability of the burr height calculation from the 3D scan. The device measurements uncertainties are considered to be of an inferior order.

On a large majority of the scans, the burr is homogeneous all along the sample exit edge. The few samples where it is not the case are not taken into account in the following discussions. The problem is linked to the difficulty to produce sharp edges during the samples manufacturing, and not to material heterogeneity like Régnier et al. observed in [START_REF] Régnier | Fundamental study of exit burr formation mechanisms during orthogonal cutting of AlSi aluminium alloy[END_REF].

The machining parameters used during this study are presented in Table 1.

Four sample geometries presented on Fig. 5 have been designed to provide six various exit plane orientations. Indeed, the geometries 1 and 4 on Fig. 5

Analysis of parameters effect on burr height accumulation

Thirty-one experiments of burr accumulation were conducted at the cutting speed of 2 m/min. Each one includes numerous cuts, e.g., one hundred for some with a cut thickness equals to 10 μm. First observations show that the burrs are homogeneous along the sample exit edge and their height is constant after each cut. The burr matter is not push in a particular area of the sample exit edge even if the cutting edge inclination angle (λ s ) is not equal to 0° or the secondary exit angle is different of 90°. Figure 6 shows experimental curves of burr height evolution as a function of the accumulated cut thickness. Three zones are observed (cf. Fig. 6a). Starting from low to high accumulated cut thickness, the 1st one corresponds to a significant increase of the height after each pass, which is followed by the 2nd zone corresponding to a threshold. During the 3rd one, the height increases again but with a lower rate compared to what could be observed during the 1st zone.

Figure 6a shows that for a same principle exit angle, the cut thickness (h) has the greatest influence on the burr height. Its increase from 10 to 50 μm leads to a burr height rise. The change of cutting geometry with rake angles (γ n ) between 2 and 14° and cutting edge inclination angles between 0 and 10° have only little influence. The variation of the cutting edge radius (r β ) from 10 to 30 μm has no significant influence.

The increase of the principle exit angle induces a rise of burr height, as shown on Fig. 6b. The configurations where the principle exit angles are 80° and 100° also have a secondary exit angle of 80°, not 90° as the other ones. Experiments with a Φ ep angle equals to 70° and a 30 μm or 50 μm cut thickness have been conducted, but are not represented. Indeed, it leads to burrs with chamfer and no measurable burr height.

Figure 6c introduces the influence of the secondary burr exit angle with samples having a principle exit angle of 90°. This parameter seems to have an influence here. However, it is also observed that the curves of the samples with a secondary exit angle of 90° have a delay before starting. After inspections, these samples have an unwilling small chamfer at the exit edge resulting locally in a decrease of the principle exit angle to a value lower than 90°. The cut thickness being smaller than these chamfers height that certainly has an influence on the burr height by changing the local exit angle. This phenomenon may also explain the differences of the curves at greater accumulated cut thickness and prevent from giving a real quantification on the effect of the secondary exit angle. Nevertheless, this effect appears to be lower than the impact of the primary exit angle. This unwilling chamfer is not as important on the other samples geometries.

The normalization is the same for all the graphics of the article.

Analysis of burr morphology during its accumulation

Figure 7 shows the bases of all the following analysis. This curve has already been presented on Fig. 6a, showing the burr height as a function of the accumulated cut thickness. The side burr pictures taken just after the various pass are matched with the curve points. First of all, two different phases of the burr development process can be identified. The first one is linked to the two first zones of the burr height evolution and its stabilisation (cf. 2.2.). This 1st phase corresponds to the burr formation before its tip touches the sample exit free surface, and the second one starts just after. The burr grow freely only during the 1st phase which is then of great interest for the developed model presented in Sect. 3.1. On Fig. 7, the burr seems rotating while accumulating. During the 2nd phase, the burr tip touches the sample exit free surface, which explains the burr height increase. Moreover, the burr seems to grow in this phase as a spiral with an initial condition corresponding to the shape of the burr at the end of the first phase.

Figure 8 shows some of the pictures of Fig. 7 during the 1st phase of burr development. The machined surface and the free exit plane of the sample are depicted with two (γn2°, λs0°)_h30_rβ10 (γn2°, λs0°)_h30_rβ10 (γn2°, λs10°)_h30_rβ10 (γn5°, λs6°)_h10_ rβ10 (γn5°, λs6°)_h30_ rβ10 (γn5°, λs6°)_h50_ rβ10 (γn8°, λs0°)_h10_ rβ10 (γn8°, λs0°)_h30_ rβ10 (γn8°, λs0°)_h30_rβ10 (γn8°, λs0°)_h30_rβ10 (γn8°, λs0°)_h30_ rβ30 (γn8°, λs0°)_h50_rβ10 (γn8°, λs0°)_h50_ rβ10 (γn8°, λs0°)_h50_ β30 (γn8°, λs10°)_h30_rβ10 (γn14°, λs0°)_h30_rβ10 (γn14°, λs10°)_h30_rβ10 It is observed that the burr rotates between the two circles as long as the burr tip does not touch the exit plane while accumulating during this multiple passes experiment.

A similar behaviour appears for all the different configurations and that explains the threshold part of the burr height curves shown on Figs. 6 and7. This phenomenon takes place just before the burrs tips touch the samples exit planes. Burr circular growth is the base of the model presented in the next paragraph.

Development of a burr accumulation model before contact with the workpiece

Burr accumulation model in orthogonal cutting

In the bibliography, very few indications are given on burr accumulation at the part exit edge. The models previously presented only deal with burr formation for one cut of the tool. Nevertheless, when those methodologies are applied to describe burrs evolution after several cuts, they start rotating. Figure 9 shows the different steps that may be expected during the burr accumulation process, when the cut thickness is smaller than the burr root thickness. Schäfer in his burr description [START_REF] Schäfer | Entgraten: Theorie, Verfahren[END_REF] notices that a burr root radius exist, corresponding to the internal circle in Fig. 9. Régnier et al. [3] observe during a burr root accumulation experiment in a cast aluminium alloy that the curve of burr height as a function of the accumulated cut thickness (i.e. number of cut), reaches a threshold. In parallel, the images taken during the experiment show burr without chamfer rotating as the one observed in Fig. 8.

The modelling path proposed by the previews mechanical models goes from the tool tip to the plastic hinge centre and then the calculation of the burr size. The proposed model in this article use a similar methodology, but on the other direction, starting from the burr heights measurements during accumulation, which gives interesting results.

The geometrical configuration of the burr accumulation model defined before the burr tip touches the exit plane is exposed on Fig. 10. The graphical representation takes place in a plane normal to the sample exit edge.

The model is composed of a large and a small circle, which are respectively tangent to the machined surface and the exit plane. Both are concentric with their centre located on a line starting from the sample exit edge and defined by an angle α with the exit plane. According to experiments, this angle is considered to be constant and equal to 20°. The experimental burr is supposed to be situated between both circles. ψ, the local exit angle, is the angle between the exit plane and the line normal to the machined surface. The positive direction is considered anti-clockwise. The ψ angle is linked with the principle and secondary exit angles by the formula (1). This equation results from the dot product of the normal vector of the exit plane and the normal vector of the machined surface.

The θ angle is the parameter that drives the burr tip location. Three different areas are identified as follows:

Inside the 2nd area, the burr is contained between the two circles and the burr height increases when θ decreases. The third area corresponds to the situation when θ may change but the burr height has reached a threshold and does not change anymore. The 1st area is the 1st stage of the burr growth where the volume of burr added by each cut is a function of θ. However, considering the values of the ψ and α angles during the conducted experiments, the size of this 1st area is considered negligible compared to the ones of the 2nd and 3rd area. According to the observations, this 1st area is filled after one or two cuts (cf. Fig. 8). The 1st zone of the curves evolution (cf. 2.2) corresponds to the 1st and 2nd area of the model. The 2nd zone of the curves (the threshold) is linked to the 3rd area.

For simplification purpose, the modelled burr is considered to be the red curve on Fig. 10, following a straight line in the 1st area of the model, and then the external circle in the 2nd and 3rd areas.

By construction, the internal circle radius depends on the external circle one, by the following formula:

The volume of matter added to the burr, after each cut, is considered to be constant (the 1st area of the model being neglected). This implies that the parameter θ can be described by a linear function with respect to the cut number (C n ): ε allows adapting the small variation of the samples geometry due to their manufacturing. The θ 0 angle is equal to the following:

(1)

Ψ = arccos[ cos Φ .sin es √ sin 2 Φ ep + cos 2 ep .sin2( es ) ] - 2
(2) The burr model is defined with five coefficients. α and ψ are inputs but Δθ /cut , R ext and ε need to be identified thanks to a best-fit algorithm on experimental curves (presented in paragraph 3.2).

⎧ ⎪ ⎨ ⎪ ⎩ 1 st area if 𝜃 > 𝜋 2 2 nd area if 𝜋 2 ≥ 𝜃 ≥ -Ψ 3 rd area if -Ψ > 𝜃 (3) R int = R ext . sin( ) cos( -Ψ) (4) = Δ ∕cut .C n + 0 + (5) 0 = 2 + -Ψ
Burr height (Bh) can then be calculated by the following:

The location of the plastic hinge centre of the phenomenological model is situated at a vertical distance of half the cut thickness above the point (T) which is at the tangency between the exit plane and the inner circle (cf. Fig. 10).

Burr root thickness (Br) can be calculated by the formula:

It is the distance taken parallel to the sample exit surface from point T to the part exit edge (cf. Fig. 10).

Identification of the burr model coefficients and their modelling

The coefficients Δθ /cut and R ext of the burr accumulation model presented in the paragraph 3.1 need to be determined. A best-fit algorithm is used to minimise the differences to the square between the measurements, corresponding to the burr heights as a function of the accumulated cut thickness, and the Bh modelled values.

The ε parameter gives one additional degree of freedom and allows adjusting the starting point of the best-fit curves corresponding to the local small differences of the samples initial exit edges due to their manufacturing.

(6) 11 is an example of the results given by the algorithm, the curves are the fitted ones and the points represent the measurements. Not all of the measured points are considered, only the ones corresponding to the first phase of the burr development (cf. Fig. 7). The model underestimates the threshold value and overestimates the burr height just before, due to the facts that the burr is modelled as a line without any thickness. Actually, the burr is thick, and the tip is round, resulting in a change of location of the point used to measure the burr height after each cut (first at the tip, then on the side of the burr, cf. Fig. 8). In contrast, the point used for the burr height calculation is always the same during the first zone of the curve evolution: at the tip of the red curve on Fig. 10.

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ B h = R ext .[ � cos � 𝜋 2 + Ψ � + sin(𝛼) cos(𝛼-Ψ) � -tan � 𝜃 -𝜋 2 � .cos(Ψ)] if 𝜃 > 𝜋 2 B h = R ext .(cos(𝜃 + Ψ) + sin(𝛼) cos(𝛼-Ψ) ) if 𝜋 2 ≥ 𝜃 ≥ -Ψ B h = R ext .(1 + sin(𝛼) cos(𝛼-Ψ) ) if -Ψ > 𝜃 (7) B r = R int tan( ) Figure
This methodology allows the determination of the different coefficients (α, ψ, Δθ /cut , ε and R ext ) needed for the burr accumulation model as a functions of the various experimental parameters (that could be γ n , λ s , r β , h, Φ es , Φ ep …).

To determine the burr geometries and develop the different analyses presented in this article, the expressions for the coefficients Δθ /cut and R ext need to be modelled. The ε coefficient being only defined to adjust the small variations of the samples exit edges due to their manufacturing; it is not interesting to model it.

According to the previous discussions, h and the angle ψ are very influent on burr height. Based on experimental results, two following equations are proposed: Formulations that are more complex have been tried, taking into account the cutting geometry (rake and cutting edge inclination angles), the cutting edge radius and also the 11 Three burr height evolutions with the modelled curves identified by best-fit methodology primary and secondary exit angles but the gain in accuracy is not significant. Moreover, this would imply more coefficients to identify. The proposed formulations are close to a cutting force model, as discussed in the introduction.

(8) R ext = k R0 + k Rh .h .(1 + k RΨ .Ψ) (9) Δ ∕cut = k Δ0 + k Δh .h .(1 + k ΔΨ .Ψ) Fig.
Twenty-eight experiments of burr accumulation (thirtyone conducted minus three leading to burr with chamfer) have been used to identify the coefficients of Δθ /cut and R ext models (Eqs. 8 and 9). Each experiment corresponds to numerous cuts. Figure 12 shows the relative error made by the two models of R ext and Δθ /cut compared to the values identified thanks to the best-fit algorithm on the twentyeight experiments. The models residual degrees of freedom are twenty-five.

The R ext model gives good results with more than 60% of experiments with modelling errors lower to 10%. The Δθ /cut model is a little bit less reliable with around 50% of experiments with modelling errors inferior to the same limit.

Comparison to experimental results and other models

Chern and Dornfeld [START_REF] Chern | Burr/breakout model development and experimental verification[END_REF] and Toropov et al. [START_REF] Toropov | A new burr formation model for orthogonal cutting of ductile materials[END_REF] presented models to determine burr height and root thickness after one cut. Figure 13 compares the results of the burr accumulation model and the ones developed by them, on three different configurations. Burr heights on Fig. 13 are measured after one cut and burr root thicknesses after few cuts to improve the measurement accuracy. Indeed, according to the model presented in this article, Br is not influenced by burr accumulation.

However, burr height measurements after the first cuts are very dependent on the initial state of the samples and on the planing preparation process, the cut thicknesses chosen being small. By considering the entire burr growth process during accumulation, the model developed in this article is less dependent to this issue.

Burr height and root thickness calculations from the models developed in [START_REF] Chern | Burr/breakout model development and experimental verification[END_REF] and [START_REF] Toropov | A new burr formation model for orthogonal cutting of ductile materials[END_REF] have been adapted to fit the definitions presented in Fig. 2.

The burr model developed in this article gives more accurate results on burr heights and root thicknesses even after only one cut. As a reminder, burr height measurements during burr accumulation are the input of the model. There is no material behaviour assumption for these two outputs. It has to be noticed that [START_REF] Chern | Burr/breakout model development and experimental verification[END_REF] and [START_REF] Toropov | A new burr formation model for orthogonal cutting of ductile materials[END_REF] models, as explained in the introduction, are linked to burr formation in ductile materials. Toropov and Ko [START_REF] Toropov | A new burr formation model for orthogonal cutting of ductile materials[END_REF] Rela ve error on Bh (absolute) Fig. [START_REF] Ko | Burr formation and fracture in oblique cutting[END_REF] Comparison after one cut of a the burr height and b the burr root thickness relative errors, for three different configurations between the presented model and the ones of [START_REF] Chern | Burr/breakout model development and experimental verification[END_REF] and [START_REF] Toropov | A new burr formation model for orthogonal cutting of ductile materials[END_REF] by each cut. To estimate the fracture occurrence with some of the methodologies based on the plastic hinge presented in the introduction, the tool distance at initiation and the initial negative deformation angle have to be quantified. The incompressibility assumption is used to determine the tool distance initiation. On Fig. 2, the area defined by the points A, C, D and E is supposed to be equal to the additional surface (S /cut ) added by each cut to the burr. Thanks to the burr accumulation model, it is determined that:

The area of ACDE is equal to: This allows the w length calculation:

The chip shear angle Φ can be determine by different methodologies; here, the next formulation has been chosen: where γ is the rake angle. F c is the cutting force along v c and F h the second force component along the cut thickness direction during an orthogonal experiment. The subscript "cut" means that only the forces components due to the chip contact on the rake face is taken into account (known as the "cut effect" in [18]).

Considering the negative deformation curve to be the segment AB on Fig. 2a [START_REF] Chern | Burr/breakout model development and experimental verification[END_REF], the initial negative deformation angle is as follows:

The calculation of w and β 0 from the burr accumulation model allows using different mechanical models presented in the bibliography as shown in the next paragraph.

Crack initiation criteria

The different steps of the burr formation have been presented in the introduction. The possibility of a crack initiation and propagation along the negative deformation line to create a burr with chamfer has been mentioned. It may occur if the strain at the tool tip becomes strong enough. Different criteria for crack initiation during burr formation exist in the bibliography. The deformation during burr [START_REF] Régnier | Experimental analysis and burr formation modeling in the AlSi7Mg0.3+0.5Cu alloy -Application to orthogonal cutting and milling[END_REF]

S ∕cut = R 2 ext -R 2 int . Δ ∕cut 2. = R 2 ext . 1 - sin 2 ( ) cos 2 ( -Ψ) . Δ ∕cut 2 (11) A ACDE = w.h - h 2 2.tan(Φ) + h 2 2
.tan(Ψ)

(12) w = S ∕cut h + h 2.tan(Φ) - h 2 .tan(Ψ) (13) = 4 - 1 2 .(arctan( F c,cut .sin( ) + F h,cut .cos( ) F c,cut .cos( ) -F h,cut .sin( ) ) -) (14
) 0 = B r .cos(Ψ) w -B r .sin(Ψ)
formation is considered to be composed of shear and bending deformation. Chern and Dornfeld [START_REF] Chern | Burr/breakout model development and experimental verification[END_REF] predict crack initiation if the equivalent strain A at point A (cf. Fig. 2) is bigger than a constant ε f . A is considered to be the average of the Von Mises and Tresca equivalent strain. The calculations are as follows:

where γ A is the shear strain on the negative shear line, at point A. A 0 and A f are the initial and final areas of a specimen during a tensile test.

Ko and Dornfeld [START_REF] Ko | Analysis of fracture in burr formation at the exit stage of metal cutting[END_REF] use the variation of the deformation angle from β 0 to β when the tool moves forward for the calculation of the maximum strain max at the moving tool tip:

If max becomes higher to a material dependant constant, the fracture is supposed to initiate at the angle β and then propagating through point B. Thus, between the moment when the burr initiates and the fracture occurs, the burr grows a little bit.

Modelled strain and burr with chamfer occurrence

No material characterisation has been performed during this study but the different strains A and max can be calculated for different cases in Fig. 14 (respectively a and b). It is then compared to the experimental appearance of a burr with or without chamfer, the last one corresponding to a crack propagation case. The green dots represent the configurations leading experimentally to burrs without chamfer and the orange ones to burrs with chamfer.

For comparison purpose, the integral boundary β in the calculation of max is chosen equal to 70°. Indeed, it is the maximum value reached when the tool is at the sample exit edge when Φ ep is at an angle of 70°.

Those criteria do not succeed to estimate the occurrence of a burr with chamfer but can give an estimation. Configuration (ii), in Fig. 14c, leads to a burr without chamfer whereas (iii), with a slightly lower calculated strain, conducts to a burr with chamfer. However, it is observed that the start of the curve slope (i) in Fig. 14c is at higher rate than

(15) a = 1 2 ( A √ 3 + 2 3 A ) (16) f = ln( A 0 A f ) (17) A ≈ cot 0 -cot( + 0 ) (18) max = ∫ 0 √ 2 3 . � 6 2 + 3 2.tan 4 ( ) d
the one of (ii). The second configuration may be close to a situation resulting to a burr with chamfer. Figure 15 is a burr micrograph of a configuration leading to a burr with chamfer. The two circles of the burr accumulation model are drawn with the calculated initial negative deformation plane. The first observation that can be made is that the modelled internal circle is tangent to the burr root, although the model coefficients have been identified on experiments that have generated burr without chamfer. The second observation is that the real crack propagation plane does not go through the burr plastic hinge centre B. This observation refutes the constructions of the mechanical models presented before. This can explain the difficulty for the models to evaluate if a burr with chamfer is going to initiate or not.

Hashimura et al. [START_REF] Hashimura | Analysis of burr formation mechanism in orthogonal cutting[END_REF] in their observations of the burr formation steps described this phenomenon on Fig. 16. It explains why a small burr often appears at the extremity of the burr with chamfer (cf. Fig. 1). It corresponds to the burr root that initiated during the cut, just before the crack starts.

A better understanding of the crack propagation plane behaviour is needed to improve the evaluation of the conditions leading to a burr with chamfer.

Conclusions

This article gives a new point of view on burr formation thanks to an experiment allowing in situ burr observations and burr height measurements. Indeed, a camera and LASER profilometer located on the machine table, enable burr characterisations as closely as possible of the area of interest without any part disassembly. It allows observing the influence on burr height during accumulation of numerous machining parameters (normal rake and cutting edge inclination angles, cutting edge radius and cut thickness) and workpiece configuration (principle and secondary exit angles). Cut thickness and part exit plane inclination are the most influent parameters that can be used to reduce burr height and root thickness.

A model based on the formation of a plastic hinge has been developed without any need of material characterisation unlike the previous methodologies developed in the known literature. This model gives good results on the quantification of burr height, burr thickness and volume added to the burr by each cut. It is only based on burr height measurements during accumulation, which are simple data to obtain.

The model has been extended to determine the tool distance at initiation and the initial negative deformation angle. This has been used to evaluate the occurrence of crack initiation leading to burr with chamfer. The methodology still needs to be improved, but it reveals the fact that the crack propagation plane does not go through the plastic hinge centre. The rotation of the burr during accumulation appears on the observations made by Régnier et al. [3] in an aluminium alloy AlSi7Mg0.3 + 0.5Cu. An outlook to this work could be the generalisation of the present study with other workpiece materials and tool materials.

A second outlook is to apply the methodology presented in this article to three-dimensional machining operations, as milling, with higher cutting speed. Liu et al. [START_REF] Liu | Experimental investigations on flank contour accuracy of milling groove for Zr-based bulk metallic glass using mesoscale milling[END_REF] find during milling operations of a zirconium-based alloy that the increase in feed (i.e., cut thickness) results in bigger burrs, in accordance with the results in the present study. Liu et al. [START_REF] Liu | Experimental investigations on flank contour accuracy of milling groove for Zr-based bulk metallic glass using mesoscale milling[END_REF] show also that the increase in cutting speed reduces the burrs size. That may be linked to a modification of the negative shear plane by the deformation rate.

Fig. 1 Fig. 2

 12 Fig. 1 Burr types observed in a cast aluminium alloy (AlSi7Mg0.3 + 0.5Cu) with a high-speed CCD camera PHOTRON SA-Z. a Burr without chamfer (h = 0.03 mm, γ = 7°). b Burr with chamfer (h = 0.1 mm, γ = 7°). Adapted from Régnier et al. [3]

Fig. 3

 3 Fig. 3 Experimental setup for burr and chip analysis

Fig. 4

 4 Fig. 4 Analysis methodology for burr height characterisation from scanning. a Scan cleaning process b burr profiles extraction c burr height calculation and d graphical representation of burr accumulation

  relating to the state of the art. Thus, the principle exit angle changes from 70 to 110° and the secondary from 70 to 90°.Each presented experiment started from new samples. The test designations used in this article are the following ones: (γ n (°), (λ s °))_h(μm)_r β (μm)_ Φ ep (°)_Φ es (°).

)

  ep = 80°) ) ep = 100°) ) ep = 90°) ) es = 70°) ) es = 80°) ) es = 80°0 °°°F ig. 5 Four sample designs leading to six different exit plane orientations: (Φ ep = 70°, Φ es = 90°), (Φ ep = 80°, Φ es = 80°), (Φ ep = 90°, Φ es = 90°), (Φ ep = 90°, Φ es = 70°), (Φ ep = 100°, Φ es = 80°), (Φ ep = 110°, Φ es = 90°)

Fig. 6

 6 Fig. 6 Influence on burr height of a the cutting geometry and cut thickness, b principal exit angle and c secondary exit angle. Some configurations are repeated

  A third line at 20° from the exit plane is also plotted with two concentric circles. Their common centre is on the previous line. The small circle is tangent to the exit plane, the larger one is tangent to the machined surface. Their radius values are the same on all the different pictures.

Fig. 7 Fig. 8

 78 Fig. 7 Burr height evolution as a function of the accumulated cut thickness (γ n = 8°, λ s = 0°, h = 30 μm, Φ ep = 110°, Φ es = 90°)

Fig. 9 Fig. 10

 910 Fig. 9 Burr accumulation process expected before contact with the workpiece

Fig. 12

 12 Fig.[START_REF] Pekelharing | Cutting tool damage in interrupted cutting[END_REF] Relative error (Rel error) made by the determination of a R ext and bΔθ /cut

  h = 30 µm_ψ = 20°h = 50 µm_ψ = 20°h = 30 µm_ψ = 0°(

Fig. 14 Fig. 15 Fig. 16

 141516 Fig.[START_REF] Chern | Burr/breakout model development and experimental verification[END_REF] Comparison of the calculated strains with the burr with chamfer occurrence, a ε A , b ε max . Green dots for burr without chamfer appearance and orange ones to burr with chamfer. c burr height evolution for configuration (i) and (ii)

Table 1

 1 Machining parameters used for the experiments

	Parameters	Level
	Normal rake angle (γ n )	2°; 8°; 14°C
	utting edge inclination angle (λ s )	0°; 5°; 10°C
	ut thickness (h)	10 μm; 30 μm; 50 μm
	Cutting edge radius (r β )	10 μm; 30 μm
	Cutting tool material	High-speed steel

4 Burr fracture determination 4.1 Quantification of the parameters needed for the estimation of burr fracture occurrence

			Rela ve error made by the Rext quan fica on
	Percentage of the 28	experiments	0% 20% 40% 60% 80%	Rel	10%<Rel	20%<Rel	30%<Rel
				error<=10%	error<=20%	err<or=30%	error<=40%
			Rela ve error by the Δθ/cut quan fica on
	Percentage of the 28	experiments	0% 10% 20% 30% 40% 50% 60%
				Rel	10%<Rel	20%<Rel	30%<Rel error
				error<=10%	error<=20%	error<=30%
							The burr accumulation model developed in this article can
							directly quantify burr height, root thickness and added volume

Availability of data and material

The data and material used are not publicly available.

Code availability

The code used is not publicly available.

Author contribution All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed mainly by Côme Legrand. The first draft of the manuscript was written by Côme Legrand, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Funding The presented study received research support from the company Safran Aircraft Engines.