Circular tests for HSM machine tools: Bore machining

Keywords: circular interpolation, HSM, kinematic machine tool behaviour, jerk, feed rate limitation

Today's High-Speed Machining (HSM) machine tool combines productivity and part quality. The difficulty inherent in HSM operations lies in understanding the impact of machine tool behaviour on machining time and part quality. Analysis of some of the relevant ISO standards and a complementary protocol for better understanding HSM technology are presented in the first part of this paper. These ISO standards are devoted to the procedures implemented in order to study the behavior of machine tool. As these procedures do not integrate HSM technology, the need for HSM machine tool tests becomes critical to improving the trade-off between machining time and part quality. A new protocol for analysing the HSM technology impact during circular interpolation is presented in the second part of the paper. This protocol which allows evaluating kinematic machine tool behaviour during circular interpolation was designed from tests without machining. These tests are discussed and their results analysed in the paper. During the circular interpolation, axis capacities (such as acceleration or Jerk) related to certain setting parameters of the numerical control unit have a significant impact on the value of the feed rate. Consequently, a kinematic model for a circular-interpolated trajectory was developed on the basis of these parameters. Moreover, the link between part accuracy and kinematic machine tool behaviour was established. The kinematic model was ultimately validated on a bore machining simulation.

Introduction

Over the past decades, the benefits and disadvantages of high-speed machining (HSM) have been presented in detail. As a result of advances in numerically-controlled machine tools, the productivity and quality of several processes have been considerably improved, e.g. end milling at high rotation speeds has become a costeffective manufacturing process for producing parts with high precision and a high surface quality [START_REF] Fallböhmer | Highspeed machining of cast iron and alloy steels for die and mold manufacturing[END_REF], [START_REF] Mawussi | Experimental characterization of a high speed machine tool[END_REF]. Several companies have thus decided to combine the benefits of high-speed machining with flexible machine tools.

In the automotive industry for example, the parts machined are castings, cylinder heads and gearboxes. Based on these parts, typical machining features can be extracted [START_REF] Tapie | Elaboration d'un protocole de simulation d'usinage en fraisage de pièces prismatiques: évaluation des performances d'une machine U.G.V[END_REF] [4] such as bores, holes, drillings, tapered holes and planes. The machining of these features involves two types of displacements: displacements along the spindle axis and plane displacements normal to the spindle axis. The work presented in this paper is focused on bore machining which involves a circular interpolation of two axes during plane displacements. High accuracy and a high surface roughness are required for bores. Nevertheless, it is difficult to determine HSM impact on the machining of the corresponding features.

In this paper, an original approach to analyse HSM machine tool behaviour during circular interpolation is presented. This approach is divided into two types of circular interpolated test types: circular displacements, and spiral displacements composed of arcs of circles. Tests results have been compared to highlight the HSM impact in terms of machine tool behaviour and part quality. Differences between programmed, simulated and measured feed rates are presented. These differences are induced by the acceleration and jerk (acceleration derivative) limitations. Moreover, some numericallycontrolled unit settings can induce feed rate limitations. Arc length and the transition between two circular blocks also stand out as significant parameters in the feed rate ultimately reached.

Standard procedures

Machine tool test procedures can be classified into 7 categories [START_REF] Wilhelm | A comparison of international standards for performance testing of machining centres[END_REF]: environmental effects, machine performance, geometric accuracy, volumetric performance, cutting performance, machining test parts and multi-function. Some of these procedures have been developed in national or international standards. The international standards ISO 230-1998 and ISO 10791-1998 propose test procedures during circular interpolation. Two types of procedures are presented in these international standards: measurements without machining, and test parts for machining procedures.

Procedures without machining

Circular tests are suggested in both the ISO 230-4 [START_REF]Acceptance Code for Machine Tools[END_REF] and ISO 10791-6 [7] standards. The procedures described in these standards are based on the circular interpolation solicitation of machine tool axes. Feed rates and diameters are also given. To analyse machine tool behaviour, performance indexes are defined and some reference data are given.

The index for circularity deviation G (Fig. 1a) is defined as the minimum distance between two concentric circles around the entire trajectory. It can be evaluated as the maximum radial deviation according to the square-fit method for circles.

The index for radial deviation (Fig. 1b) is defined as the distance between the real trajectory and the nominal trajectory. The centre of the nominal trajectory is obtained by centring measurement devices on the machine tool. The indexes and data given in ISO standards provide an efficient database for conventional machine tool behaviour. However, analysis of these indexes shows their incompatibility with HSM technology. For example, the indicated feed rates and performance indexes defined in these ISO standards correspond only to conventional machining. The proposed approach integrates new procedures compatible with HSM displacement, and the ISO 230-4 and ISO 10791-6 standard indexes.

Test part procedures

Test parts are advised in the ISO 10791-7 standard [8]. The first standard part is defined in order to qualify machine tool accuracy for the finishing operation of a plane. The second standard part is defined to qualify machine tool accuracy for the finishing operations of different features (Fig. 2) which are intended to control the machine tool in various displacements such as linear and circular interpolated tool paths. These test parts are specified for machine tool acceptance and may be used for comparing different machine tool structures [START_REF] Geldart | A direct comparison of the machining performance of a variax 5 axis parallel kinetic machining centre with conventional 3 and 5 axis machine tools[END_REF].

Nomenclature

ISO standard indexes r Jerk vector t r

Trajectory tangent vector at a given point n r

Trajectory normal vector at a given point In this standard, recommendations are given for both roughing operation and cutting tool parameters which nonetheless are made available for conventional milling operations. Accuracy indexes and accuracy reference data are also provided for conventional milling.

In the standard part dedicated to circular interpolation, tests are aimed at evaluating machine tool accuracy by means of part quality. Although part quality and machining time [START_REF] Vidal | Démarche de conception d'un centre d'usinage grande vitesse pour la production de pièces en grande série[END_REF] constitute the main objectives in the automotive industry, machining time has not been set in the standard.

The work presented in this paper relies upon circular interpolation associated with a test part designed for the car manufacturer RENAULT [START_REF] Mawussi | Experimental characterization of a high speed machine tool[END_REF]. Test procedures designed in the scope of this work allow evaluating three characteristics: machine tool accuracy, machining feature quality and machining time. In addition, the link between indexes associated to these characteristics is established within an HSM context. In the following section, the specific constraints associated with HSM technology are analysed.

HSM technology

HSM technology has impacts in various machining topics, such as cutting tools [START_REF]Modern Metal Cutting[END_REF], tool path strategies [START_REF] Dragomatz | A classified bibliography of literature on NC milling path generation[END_REF] [13], machine tool structures [START_REF] Geldart | A direct comparison of the machining performance of a variax 5 axis parallel kinetic machining centre with conventional 3 and 5 axis machine tools[END_REF] [START_REF] Bohez | Five-axis milling machine tool kinematic chain design and analysis[END_REF] and numerical control units (NCU). Dugas in [START_REF] Dugas | Simulation d'usinage de formes complexes[END_REF] gave a classification of the HSM technology that separates machining capacity into three fields: machine tool limits, NCU limits and cutting tool limits.

The tests presented in this paper are designed to highlight HSM performance related to the first two fields. Machine tool limits are closely associated with axes kinematics. NCU limits are introduced, in particular via the numerical axis control loop.

Machine tool limits

Each machine tool axis features has its own capacity. For a given axis i, this capacity is represented not only by the stroke, but also by kinematic parameters such as maximum feed rate (V mi ), acceleration (A mi ) and jerk (J mi ). The less dynamic axis thus imposes its kinematic skills during an interpolated displacement and then defines the limit of the feed rate. Three types of feed rate limitations are induced by axis kinematics.

According to the first limitation, the feed rate limit value (V t ) is determined by (V mi ), the maximum feed rate of axis i that has imposed its feed rate capacity. The (V t ) calculation for the X-and Y-axis interpolation is detailed in Section 5.

The second limitation may be imposed by the maximum acceleration (A mi ) of the i axis. At a given point on a circle trajectory with a radius R r , acceleration is composed of a tangential acceleration in the tangent direction ( t r ) and a normal acceleration in the normal direction ( n r ). The unit vector of these two directions constitutes a Frenet frame ( t r , n r ) at the given point. Using this frame, (eq. 1) yields both the acceleration expression and the corresponding value of the feed rate (V f ), as defined on the tool axis.

(eq. 1)

n R V t dt V d A r f f r r r ⋅ + ⋅ = 2 ) ( and t V V f r r ⋅ =
When feed rate is in the steady state, the corresponding acceleration is only normal (A n ) and constant (eq. 1). (A n ) is thereby limited by the lowest maximum acceleration (A mi ) of the i axis. The (A n ) calculation is shown in Section 5. The corresponding feed rate limitation value (V An ) is thus given by (eq. 2).

(eq. 2)

n r An A R V × =
The third limitation case can be imposed by the maximum jerk (J mi ) of the i axis. The jerk is obtained by taking the derivative of acceleration. The derivative of (eq. 1) therefore yields the expression for jerk (eq. 3). When feed rate is constant, according to (eq. 1), it can be deduced that jerk is only tangential (J t ), with its value given by (eq. 3). (J t ) is limited by the maximum jerk (J mi ) of the i axis. The (J t ) calculation is also presented in Section 5. The corresponding feed rate value (V Jt ) is thus deduced from (eq. 3) and given by (eq. 4).

(eq. 3)

2 3 r f t R V J dt n d A n dt dA J = ⇒ ⋅ + ⋅ = r r r (eq. 4) 3 / 1 2 r t Jt R J V × =
The feed rate (V s ), which equals the minimum resulting from the three limitation situations (eq. 5), is called the "static look-ahead".

(eq. 5)

) , , min(

Jt An t s V V V V =

NCU optional functions

In the HSM context, the numerical controller unit (NCU) is intended to anticipate the trajectory in order to command axes in real time. This NCU anticipation skill depends upon its processing skill and dynamic lookahead, which represents the number of blocks read in advance by the NCU. These two characteristics are established either by NCU construction technology or by basic machine tool settings.

The NCU tool path processing capacity is an important parameter. Trajectory anticipation and realtime command are in fact both integrated into this parameter. Moreover, this processing capacity depends on the machine tool interpolator cycle time. The trajectory is typically divided into small segments processed in accordance with this cycle time. If a segment length is too small, the NCU reduces the feed rate in order to allow enough time for axes to cover the segment with a time equal to or greater than the interpolation cycle time.

For a given arc length ) (

s e r R α α - ×
, a minimum feed rate (V tcy ) is thus computed using the interpolator cycle time (t cy ) (eq. 6).

(eq. 6)

cy s e r tcy t R V ) ( α α - × =

NCU specific capabilities

The realization of the tests on a high-speed machine tool imposes the use of some HSM specific modes offered by the SIEMENS 840D NCU [16] [17] used in the scope of the work. Since these HSM specific modes have an impact on the machine tool behaviour, they are defined and analysed in this section.

3.3.1

Continuous path mode The tool path is locally modified at each block transition to remove tangential discontinuity. Axis displacements are smoother and machining time and cutting conditions are improved. Yet the tool path is locally modified by inserting additional curves at the block transition. For purposes of curvature discontinuity, only the acceleration jump has been authorised. This mode is very often added with a dynamic lookahead in order to improve efficiency. The tool path is then analysed by anticipation and an upper feed rate is generated.

3.3.2

Acceleration mode The axis may be controlled using various acceleration modes: "brisk", "soft" and "drive" (Fig. 3).

In the brisk mode, axes move at maximum acceleration until the feed rate set point is reached. Machining time is optimised; however shocks in the machine tool structure are present, so part quality might be affected.

In the soft mode, axes move at constant acceleration until the feed rate set point is reached. This mode minimises shocks in the machine tool structure. Machining accuracy and part quality are better than in the brisk mode, although with increased machining time and lower productivity.

The drive mode combines both the brisk and soft modes. Axes move at maximum acceleration until reaching a feed rate limit, which has been defined within a numerical controller registry. Acceleration is then reduced according to machine tool limits until the feed rate set point has been reached.

3.3.3

Feed forward mode When the feed forward mode is activated, the pursuit closed-loop error is reduced to zero, in accordance with the tangential feed rate of displacement. Both displacement accuracy and part quality increase.

3.3.4

Experimental HSM configuration The final HSM configuration adopted for the test integrates the continuous path mode with the tangential displacement mode (G64), the "soft" acceleration mode and the feed forward mode. This configuration has been validated from previous test results derived in [START_REF] Tapie | Elaboration d'un protocole de simulation d'usinage en fraisage de pièces prismatiques: évaluation des performances d'une machine U.G.V[END_REF] and those proposed by CETIM [START_REF] Cetim | Technical report addressed to RENAULT SA, Caractérisation dynamique pour centres d'usinage[END_REF]. This choice is intended to achieve the fixed objectives of using machine tool structures sparingly and simulating industrial machining conditions.

Experimental framework

The protocol applied is based on tests without machining and then validated in bore machining simulations.

Materials

Experiments were conducted on a MIKRON UCP 710 machine tool equipped with a vertical spindle and a SIEMENS 840 D NCU. The machine tool and NCU characteristics are listed below in Table 1.

The machining simulation was performed on the test part designed for RENAULT (Fig. 4). The experiments presented in this paper have been limited to 25-and 80mm diameter bore machining in circular interpolation.

Performance indicators

In order to analyse tests results, three performance indicators have been defined: accuracy, kinematics and time.

For the tests carried out without machining, accuracy is evaluated from the index for circularity deviation as well as from the indexes for radial deviation defined in ISO standards (Section 2.1). This evaluation step is carried out using data processing programmes previously developed. Tool positions are measured by reading output encoder data from the machine tool-integrated oscilloscope. This type of accuracy measurement has already been validated in previous tests [START_REF] Tapie | Elaboration d'un protocole de simulation d'usinage en fraisage de pièces prismatiques: évaluation des performances d'une machine U.G.V[END_REF]. For part machining tests, accuracy is characterised by circular bore deviation which is evaluated using a coordinate measurement machine.

Kinematic indicators, such as accuracy counterparts, are evaluated with data stemming from the integrated oscilloscope. The associated indexes are: feed rate, acceleration and jerk.

Machining times are evaluated by means of internal clock measurements, which had previously been validated using a chronometer.

Tests without machining

5.1

Test protocol In order to highlight the behaviour of machine tools without machining, three circular-interpolated trajectories following several tests were designed (Fig. 5). The first trajectory is a simple circle whereas the two others are spirals made up of half circles and quadrant. The spiral trajectories have been compared with different positions for a curvature radius variation in the X-Y plane. Various spiral inclinations have thus been tested.

Different feed rates and circle radii are tested herein to highlight the impact of radius values on machine tool behaviour (Table 2). Feed rate values have been chosen based on current industrial recommendations. For simple circular trajectories, values are chosen depending on both tool diameter (Ø 20 mm) and test part bore dimensions (Ø 25 and Ø 80 mm). The circle radii generated by tool centring come to: R r =2.5 mm and R r =30 mm. For spiral trajectories, a sampling interval close to the simple trajectory has been selected. This interval represents the various bore radii defined in automotive parts. Furthermore, the definition of two different curvature radius jump values allows to analyse its impact on machine tool behaviour.

Simple circular trajectories

Kinematic index analyses highlight the limitation phenomena (Table 3). The programmed feed rate is not always reached and a constant feed rate is imposed. These two types of behaviour are highlighted by the measured feed rate. Circularity index analyses show the impact of machine tool kinematics, i.e. accuracy decreases as feed rate increases. In addition, the radial deviation indexes indicate the presence of a centrifugation effect, which extends over the tool axis and out of the trajectory. In order to understand these phenomena, it is proposing to simulate the static lookahead of the machine tool and then compare the measured feed rate with the 6 m/min programmed feed rate for 2.5-mm and 30-mm circle trajectories.

Static look-ahead analysis

A numerical simulation of the machine tool behaviour has been carried out for both 2.5-mm and 30mm circular interpolation in the XY plane. This simulation represents the feed rate limitation imposed by maximum axis kinematics (V mi , A mi , J mi ). These limitations are calculated for each angular position of the circle (α c ).

The maximum feed rate (V mx =V my ) is 30 m/min for the X and Y axes (Table 1). The feed rate limitation (V t ) is represented (Fig. 6) for the entire circular trajectory and then calculated for a given angular position (α c ), using (eq. 7).

(eq. 7)

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ) sin( ; ) cos( min c my c mx t V V V α α
For this complete circular interpolation (0°< α c < 360°), the minimum feed rate value imposed by the axes is 30 m/min in the axes direction (90° and 270° for the X-axis, 0° and 180° for the Y-axis). In these angular positions, only one axis is interpolated; the feed rate limitation is thus limited by the maximum axis feed rate (V mx =V my =30 m/min); for other angles, V t increases. The maximum feed rate is represented in Fig. 6 and amounts to 42.4 m/min for angular positions of 45°, 135°, 225° and 315°. For these positions, the two axes are interpolated and the limitation feed rate is given by (eq. 7):

V t =min[V mx /cos(45°), V my /sin(45°)] = 42.4 m/min. The measured feed rates (Table 3) lie beyond the scope of the simulated feed rate limits (Fig. 6). Therefore it can be concluded that axis feed rate capacities (V mi ) are not the cause of feed rate limitations measured on the machine tool evaluated. Besides, the maximum feed rate tested is less than each maximum axis feed rate, hence the limit feed rate cannot be imposed by axis feed rate capacity. Feed rate (m/min) Fig. 6. Axis feed rate capacity limitation for 2.5-mm and 30-mm radii The maximum accelerations for the X and Y axes of the machine tool (Table 1) are: A mx =2.5 m/s 2 , A my =3 m/s 2 . For the entire circular trajectory at a given angular position (α c ), the normal acceleration (A n ) is calculated with (eq. 8):

(eq. 8) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ) cos( ; ) sin( min c my c mx n A A A α α
The normal acceleration (A n ) during a circular trajectory is represented in Fig. 7 (dashed line). In this figure two minimum values for the normal acceleration can be observed. The first minimum value of 2.5 m/s 2 is obtained for the angular positions of 90° and 270°. For these positions, only the X-axis is interpolated, thus the normal acceleration is limited by the maximum acceleration of the X-axis (A mx ). The second minimum value of 3 m/s 2 is reached for the angular positions of 0° and 180°. For these positions, only the Y-axis is interpolated, hence the normal acceleration is limited by the maximum acceleration of the Y-axis (A my ).

The maximum normal acceleration is obtained for the angular positions of 39.8°, 140.2°, 219.8° and 320.2°. For these angular positions, both the X-and Y-axes are interpolated with the maximum axis acceleration. For instance, the first maximum normal acceleration is obtained as follows:

A n = A mx /sin(39.8°) = A my /cos(39.8°).
This first angular position value is thus given by: tan(α c ) = A mx /A my The calculation can be adapted to determine each angular position for the maximum angular position.

The corresponding feed rate limited by maximum axis acceleration (V An ) is then calculated according to (eq. 2) and represented (Fig. 7, solid line) for a radius of 2.5 mm. For this simulation, it is assumed that the steady-state feed rate is reached for each angular position. The measured feed rates (Table 3) lie beyond the scope of the simulated feed rate limits (Fig. 7). It can be concluded that axis feed rate capacities (A mi ) do not cause the feed rate limitation measured on the machine tool. The same conclusion can be drawn for a 30-mm radius circle. As a result of the axis jerk skills, J mx =5 m/s 3 and J my =5 m/s 3 (Table 1). The tangential jerk (J t ) is calculated for a given angular position (α c ) using (eq. 9): (eq. 9)

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ) sin( ; ) cos( min c my c mx t J J J α α
The tangential jerk (J t ) during a circular trajectory has been represented in Fig. 8 and Fig. 9 (dashed line). The minimum value of 5 m/s 3 is obtained for angular positions of 0°, 90°, 180° and 270°. For the 0° and 180° positions, only the Y-axis is interpolated, hence tangential acceleration is limited by the maximum jerk of the Y-axis (J my ). For the 90° and 270° angular positions, only the X-axis is interpolated, so the tangential jerk is limited by the maximum jerk of the X-axis (J mx ).

The maximum tangential jerk is reached for angular positions of 45°, 135°, 215° and 315°. For these angular positions, both the X-and Y-axes are interpolated with maximum axis jerk. In the first angular position for example, the tangential jerk value is obtained as follows:

J t = J mx /cos(45°) = J my /sin(45°) = 7.07 m/s 3 .

The corresponding feed rate limited by the maximum axis jerk (V Jt ) is then calculated according to (eq. 4) and represented (Fig. 8, solid line) for a 2.5-mm radius. For this simulation, it is assumed that the steady-state feed rate is achieved at each angular position. The measured feed rate limit (Table 3) of 2.01 m/min lies within the scope of the simulated feed rate limits (Fig. 8); thus axis feed rate capacities (J mi ) can cause the feed rate limitation measured on the machine tool for an interpolated circle of 2.5-mm radius. The feed rate limited by maximum axis jerk (V Jt ) is calculated from (eq. 4) for a 30-mm radius and represented in Fig. 9. The measured feed rate (Table 3) of 10.53 m/min remains within the scope of simulated feed rate limits (Fig. 9), which allows to conclude that axis feed rate capacities (J mi ) can be the cause of feed rate limitation measured on the machine tool for a 30-mm interpolated circle.

In the case of a 30-mm circle trajectory, it can be observed that programmed feed rates of 6 m/min and 9 m/min are indeed reached (Table 3). These feed rates are below the scope of the feed rate limited by maximum axis jerk: min(V Jt ) = 9.9 m/min (Fig. 9). It is thus possible to conclude that during circular interpolation, NCU compares the programmed feed rate with the static look-ahead given in (eq. 5). When the programmed feed rate lies within the scope of a static look-ahead, NCU generates new feed rate set points. For the machine tool tested, the static look-ahead analysis shows that tangential jerk may be the cause of feed rate limitation. Each axis capacity therefore influences the generation of the feed rate law. The static look-ahead analysis however remains to be insufficient for explaining the constant imposed feed rate. To understand jerk influence and imposed constant feed rate more precisely, the feed rate law is analysed in the following section.

5.2.2

Kinematic law analysis To evaluate the feed rate law influence, the records obtained from the machine tool are analysed. The feed rate and acceleration curves analysed have been recorded during one circular interpolation at 6 m/min of a 30-mm radius.

The corresponding measurements are given in Fig. 10. For this circle, the programmed feed rate has been reached. The kinematic law generated by the numerical controller is divided into three phases.

Phase A: The NCU generates an increase in tangential acceleration. During this phase, acceleration increases according to a linear law. The derivative of this law yields a constant jerk of 5 m/s 3 . The tangential acceleration increase is thus generated using the minimum tangential jerk available in a circle trajectory laid out by numerical analysis (Fig. 9): min(J t ) = 5 m/s 3 . A tangential acceleration decrease is then generated in order to reach the programmed feed rate. This decrease is calculated by the NCU to avoid excess of speed. The tangential acceleration law generated is also linear with a constant derivative of -5m/s 3 equal to the minimum tangential jerk in a circular trajectory (Fig. 9): min(J t ) = 5 m/s 3 . During phase A, the feed rate increases according to two parabolic laws (Fig. 10).

Phase B: The programmed feed rate is reached. The tangential acceleration and jerk become zero. Acceleration therefore is only normal (eq. 1) and constant. Using (eq. 2), this normal acceleration can be calculated A n = (6 m/min) 2 /30 mm = 0.333 m/s 2 . The survey (Fig. 10) supports the constant normal acceleration hypothesis; its value is approximately 0.335 m/s².

Phase C: Phase C is symmetric to phase A. A second case is tested with a circle of 2.5-mm radius at 6 m/min. The corresponding measurements are shown in Fig. 11. For this circle, the programmed feed rate is not reached and remains limited to 2.01 m/min. The kinematic law generated by the numerical controller is also divided in this case into three phases.

Phase A: Same behaviour as in the first case.

Phase B:

The tangential acceleration becomes zero and a constant feed rate is reached. The survey shows a feed rate of 2.01 m/min and a normal acceleration of 0.444 m/s²; these values meet (eq. 2). Therefore it can be presumed that the NCU calculates a new feed rate set point based on the static look-ahead. This new feed rate set point is lower than the simulated one represented in Fig. 7 (min(V An ) = 2.55 m/min). Any normal acceleration limitation phenomenon occurs; this is validated by the axis acceleration presented in Fig. 11. The limit could thus be due to tangential jerk limitation, as previously predicted. During this phase, the feed rate reached is constant at 2.01 m/min (Fig. 11). This latest feed rate set point is higher than the simulated one represented in Fig. 8 (min(V Jt ) = 1.9 m/min) and calculated to avoid the feed rate variations visible in Fig. 8. The corresponding limit tangential jerk value (J t ) is 6 m/s 3 , as given in (eq. 4). It may be now concluded that during a circle displacement, the feed rate set point is not merely being processed according to the static look-ahead (V s ).

Phase C: Same behaviour as in the first case.

Conclusion 1: Controlled jerk behaviour

In both cases analysed, during phases A and C, the feed rate law is generated on the basis of the minimum tangential jerk reachable in the circle (Fig. 8 and Fig. 9), according to the maximum axis jerk (J mi ). During phase B, the central jerk is null, which allows to conclude that during circular interpolation, the tangential jerk value is controlled by NCU in order to generate a feed rate limited by the maximum axis jerk (J mi ).

5.2.4

Conclusion 2: Controlled feed rate set point Within specific machine tool registries, a curvilinear jerk (J curv = 10 m/s 3 ), a rate assigned to the tangential jerk (r jct = 60%) and a rate assigned to the central jerk (r jcc = 40%) are set. These values strengthen the tangential jerk, given in (eq. 4): J curv × r jct = 0.6 × 10 = 6 m/s 3 , as determined for the 2.5-mm radius case. The feed rate is limited by the tangential jerk settings J tcurv (eq. 10) for the machine tool evaluated. The corresponding feed rate (VJtcurv) is then given by (eq. 11).

(eq. 10) J tcurv =J curv × r jct (eq. 11)

3 / 1 2 r curv t Jtcurv R J V × =
The minimum feed rate allowed by numerical controller settings (V st ) is given by (eq. 12). (V tcy ) is the feed rate limited by interpolation cycle time, as previously presented in Section 3.2.

Program data

Current block R r , V prog , α e , α s

Axis Capacities

V mi , A mi , J mi

Limitation on current arc of circle

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = ) sin( ; ) cos( min c my J c mx J t J α α ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ) cos( ; ) sin( min c my c mx n A A A α α ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ) sin( ; ) cos( min c my c mx t V V V α α for α c ∈[α s , α e ]

Jerk and feed rate limitations in steady state

NCU settings

J curv , r jct , t cy

3 / 1 2 r tcurv jtcurv jct curv tcurv R J V r J J × = ⇒ × = n r An A R V × = 3 / 1 2 r t Jt R J V × = ) , , , , , min( tcy t An Jt Jtcurv prog st V V V V V V V = cy s e r tcy t R V ) ( α α - × = Program data Current block R r , V prog , α e , α s

Axis Capacities

V mi , A mi , J mi

Limitation on current arc of circle

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = ) sin( ; ) cos( min c my J c mx J t J α α ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ) cos( ; ) sin( min c my c mx n A A A α α ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ) sin( ; ) cos( min c my c mx t V V V α α for α c ∈[α s , α e ]

Jerk and feed rate limitations in steady state

NCU settings

J curv , r jct , t cy

3 / 1 2 r tcurv jtcurv jct curv tcurv R J V r J J × = ⇒ × = n r An A R V × = 3 / 1 2 r t Jt R J V × = ) , , , , , min( tcy t An Jt Jtcurv prog st V V V V V V V = cy s e r tcy t R V ) ( α α - × =

Non-inclined spiral trajectories

In this section, the circular interpolation of spiral trajectories composed of semicircles and quarter-circles are analysed. Such trajectories are performed in order to assess the influence of arc length and transition between two arcs. The semicircular spiral trajectory is defined in Fig. 13 by means of arcs (1-2), (2-3), (3-4), (4-5) and [START_REF] Wilhelm | A comparison of international standards for performance testing of machining centres[END_REF][START_REF]Acceptance Code for Machine Tools[END_REF]. The quarter-circle trajectory is depicted in Fig. 13 by arcs (1-1'), (1'-2), (2-2'), (2'-3), (3-3'), (3'-4), (4-4'), (4'-5), (5-5') and (5'-6).

5.3.1

Maximum arc path feed rate When a programmed feed rate of 6 m/min is set for the semicircular spiral, for the first arc radius (10 mm), the programmed feed rate is not reached (Fig. 14) (V f = 5.06 m/min), and a steady state with a constant feed rate is imposed. However the programmed feed rate does get reached for the three subsequent radii. These observations are confirmed by the feed rate processed using the flowchart presented in (Fig. 12).

For the quarter-circle spiral, the programmed feed rate of 6 m/min is not reached for the first arc radius (10 mm) (Fig. 15), and the constant feed rate phenomenon is not observed. There is not enough time to reach the feed rate set point. The same phenomenon is observed for 12-mm and 14-mm radii (Fig. 15). Moreover, the feed rate set point calculation (Fig. 12) yields: V f = 5.06 m/min for Rr = 10 mm, V f = 5.71 m/min for Rr = 12 mm, and V f = 6 m/min for Rr = 14 mm. Such values are greater than the maximum feed rate reached for these radii (Fig. 15). For a 16-mm radius, the programmed feed rate is reached and a steady state (phase B) with a constant feed rate is observed. For the next radius values, the programmed feed rate is also reached and neither an acceleration phase (A) nor a deceleration phase (C) exist.

5.3.2

Transition block feed rate Transitions between arcs of circle involve feed rate decreases (2,3,4,5 on Fig. 14). If the first two transitions feed rates are compared (2 and 3, Fig. 14), the feed rate is higher for the second transition. For these two transitions, the curvature radius jump remains the same (5 mm). The transition block feed rate value is thus correlated with curvature before (R r1 ) and after (R r2 ) the transition block.

These surveys meet the curvature discontinuitycrossing model (eq. 13) proposed in [START_REF] Pateloup | Corner optimization for pocket machining[END_REF].

(eq. 13)

2 1 2 1 r r t r r fr R R t J R R V - × × × = δ , where ) ) sin( ; ) cos( min( α α my mx t J J J =
The machine tool parameter δt represents the time allocated for the machine tool to cross a transition block. According to Siemens' documentation [START_REF] Siemens | Doc on web sinumerik 840D, Description of Functions Basic Machine, Continuous Path Mode, Exact Stop and Look Ahead[END_REF], the time required to cross a transition between two blocks is limited to the interpolator cycle time which is 12 ms for the machine tool tested.

Comparisons between the model (eq. 13) and the surveys result in a 0.21% mean relative error with a standard deviation of 0.22%.

In Fig. 15, after a 16-mm radius, transition blocks do not generate any feed rate loss (V fr = V prog = 6 m/min). It can be deduced that if the block feed rate (V f ) before the transition is lower than the feed rate allowed for crossing the transition (V fr ), the transition does not generate any feed rate loss.

Inclined spiral trajectories

In the trajectories shown in Fig. 13, transitions between arcs correspond to angular positions where tangential jerk given by the maximum axis jerk (J mi ) is minimal (Fig. 9, dashed line). In order to highlight the impact of transition block placement on the feed rate generation law, each spiral trajectory analysed in the previous section is inclined. Only the quarter spiral trajectory is presented in this section as it is shown in Fig. 16 with α = 30° or 45°.

Fig. 17. Comparison of inclined spirals: quarter-circle at 12 m/min

Transition block analysis

To illustrate the transition block impact, quartercircle spirals at 12 m/min with two different angular positions (30°, 45°) are tested and compared with noninclined spirals (Fig. 17). For instance, the comparison between transition feed rates, for the same transition block R r = 14 mm to R r = 16 mm, yields: V fr (0°) < V fr (30°) < V fr (45°). This phenomenon can be predicted using the curvature discontinuity model defined in [START_REF] Pateloup | Corner optimization for pocket machining[END_REF]. To cross a transition block, the feed rate is assumed to be constant, thus acceleration is normal (eq. 1). The jerk calculation (eq. [START_REF] Bohez | Five-axis milling machine tool kinematic chain design and analysis[END_REF] shows that during a transition block crossing, the jerk is tangential.

(eq. 14) n R V A r f r r 2 - = and dt A d J ) ( r r = , dt n d r is collinear to t r , hence t J J t r r ⋅ =
; then it can be deduced (eq. 15) in the case of X and Y interpolation at a given transition block with the angular position α.

(eq. 15)

If °≠ 0 α , then ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ) sin( ; ) cos( min α α my mx t J J J .
According to X-and Y-axis maximum jerk (Table 1), thus it can be written the following: J t (0°) = J my < J t (30°) = 2 × J my / 3 < J t (45°) = J my 2

× = J mx 2 ×
To make the transition from a 14-mm to a 16-mm curvature radius discontinuity, a feed rate (V fr ) of 4.92 m/min is calculated (eq. 13) for 0°, a feed rate of 5.29 m/min is calculated for 30°, and a feed rate of 5.85 m/min for 45°. In the surveys conducted (Fig. 17), it is observed for 0°, 30° and 45° feed rates of 4.92 m/min, 5.29 m/min and 5.86 m/min, respectively.

5.4.2

Arc path feed rate analysis Fig. 18 shows a close-up on the first arc radius, Rr = 10 mm in Fig. 17. For a radius of 10 mm (Fig. 17 and Fig. 18), phase B of the feed rate law has been deleted. A comparison between the maximum feed rate values reached has resulted in: V f (0°) < V f (30°) < V f (45°). This observation may be simply explained by the feed rate values at arc extremities. In the case analysed herein, the starting transition feed rates are the same and null, whereas the ending feed rates differ, as previously observed and explained in Section 5.4.1 by the model of (eq. 13). For a higher ending feed rate (V fr (0°) < V fr (30°) < V fr (45°)), the arc length being covered to slow movement (phase C) is lower, therefore making the arc length covered to accelerate movement (phase A) greater; consequently, the feed rate reached is higher at the end of the acceleration phase (A).

Conclusion 3: Block length and block transition impact

From spiral trajectories tests impact of block length on the feed rate law have been highlighted. The presence of steady state with a constant feed rate (phase B) is detected if axis skills and arc length are sufficient to reach the feed rate set point obtained using the algorithm presented in (Fig. 12).

The curvature jump value and the angular position of the transition block both have an influence on the feed rate crossing value between two blocks. This influence is evaluated using the model described in (eq. 13). Moreover, feed rates at block start and block end are significant parameters in feed rate law generation. The presence of a steady-state feed rate (phase B) is closely correlated with these parameters and the block length. This phenomenon, which is illustrated in Fig. 17, can be observed for an arc radius of 18 mm at the block start and block end: V fr (0°) < V fr (30°) < V fr (45°). For spirals inclined at 30° and 45°, the constant steady-state feed rate (phase B) is reached, whereas for the noninclined spiral the steady state does not exist. In the first two cases, the arc length is sufficient to attain the feed rate set point given by the algorithm presented in (Fig. 12). However the steady state is longer for a 45° inclination than for a 30° inclination (Fig. 17). To generate the acceleration (phase A) and deceleration (phase B), the length covered is correlated with the difference between the feed rate set point (V f ) and the transition block feed rates (V fr (α e ),V fr (α s )). This difference is higher for a 30° inclination than for 45°.

In the non-inclined spiral case, this difference is more obvious than in the other two cases. Axis skills prove insufficient to reach the feed rate set point with respect to arc length. The steady state (phase B) is therefore deleted and only the acceleration (A) and deceleration (C) phases actually exist.

Global kinematic behaviour

For the circular trajectory, the jerk limitation is highlighted through feed rate surveys and a static lookahead analysis: the curvilinear jerk (J curv ) and curvilinear jerk rate, allowed up to the tangential jerk (r jct ), are used to generate the feed rate set point, as specified in the flowchart shown in Fig. 12. In addition, analysis of the feed rate law shows that the machine tool axes are controlled during circular interpolation according to the lowest of their maximum jerk (J mi ).

Complementary tests using spiral trajectories were carried out to better understand machine tool behaviour, in particular the impact of the angular position of the transition block. The influence of arc length is also highlighted in the feed rate generation law. Finally, based on these tests, the kinematic model of the machine tool tested during circular interpolation and curvature discontinuity crossing was developed. This model implementation can be decomposed as follows:

Stage 1: Feed rate set point calculation The feed rate set point is calculated from specific settings of the machine tool using the flowchart given in Fig. 12.

Stage 2: Extremity feed rate calculation

Circular block start and end feed rates are calculated from their angular positions in the interpolation plane. This calculation is based on the discontinuity crossing model (eq. 13) and the controlled jerk kinematic law generation given by Fig. 19.

Stage 3: Controlled-jerk, kinematic law generation Both the acceleration phase (A in Fig. 10 and Fig. 11) and deceleration phase (C in Fig. 10 and Fig. 11) are calculated according to arc length and extremity feed rate values (stage 1). During this stage, the existence of a constant feed rate phase (B in Fig. 10 and Fig. 11) can be deduced from the arc length. If phase B exists, the constant feed rate reached during this phase is given by stage 1, which is represented in Fig. 19. If phase B does not exist, the maximum feed rate reached (V f ) at the end of the acceleration phase (A) is calculated using (eq. 16). (V frs ) is the transition feed rate value between the previous and current blocks, while (V fre ) is the transition feed rate between the current and subsequent blocks.

(eq. 16)

( ) ( ) J L V V V V V V V V rs f f rs f f re f f re f f × = + × - + + × -
The acceleration phase (A) and deceleration phase (C) implemented are based on the feed-rate controlled-jerk model detailed in [START_REF] Erkormaz | High speed CNC system design. Part I: jerk limited trajectory generation and quintic spline interpolation[END_REF].

Arc length covered during phase A:

Current Block i (Radius R ri , End angle α ei , Sart angle α si , V progi ) Previous Block i-1 (Radius R ri-1 , End angle α ei-1 , Sart angle α si-1 , V progi-1 ) Following Blocki+1 (Radius R ri+1 , End angle α ei+1 , Sart angle α si+1 , V progi+1 ) Current steady state feed rate calculation V sti given by flowchart Fig. 12 Previous steady state feed rate calculation V sti-1 given by flowchart Fig. 12 Following steady state feed rate calculation V sti+1 given by flowchart Fig.

12

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - × × × = - - - 1 1 1 ; min ri ri ts ri ri sti frs R R t J R R V V δ with ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ) sin( ; ) cos( min s my s mx ts J J J α α ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - × × × = + + + 1 1 1 ; min ri ri te ri ri sti fre R R t J R R V V δ with ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ) sin( ; ) cos( min e my e mx te J J J α α ( ) J V V V V L fre sti sti fre A - + = ( ) J V V V V L frs sti sti frs C - + = R r x(α s -α e ) = L A + L C
Steady state (phase B) exists, its feed rate value is V sti

Yes

Maximal feed rate (V f ) is given by the solution of (eq.

16)

Feed rate (Vf) is limited by arc length No Steady state (phase B) does not exist, maximal feed rate value is V f Arc length covered during phase C: Arc length covered during phase A:

Current Block i (Radius R ri , End angle α ei , Sart angle α si , V progi ) Previous Block i-1 (Radius R ri-1 , End angle α ei-1 , Sart angle α si-1 , V progi-1 )

F o l l o w i n g Blocki+1 (Radius R ri+1 , End angle α ei+1 , Sart angle α si+1 , V progi+1 ) Current steady state feed rate calculation V sti given by flowchart Fig. 12 Previous steady state feed rate calculation V sti-1 given by flowchart Fig. 12 Following steady state feed rate calculation V sti+1 given by flowchart Fig.

12

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - × × × = - - - 1 1 1 ; min ri ri ts ri ri sti frs R R t J R R V V δ with ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ) sin( ; ) cos( min s my s mx ts J J J α α ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - × × × = + + + 1 1 1 ; min ri ri te ri ri sti fre R R t J R R V V δ with ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ) sin( ; ) cos( min e my e mx te J J J α α ( ) J V V V V L fre sti sti fre A - + = ( ) J V V V V L frs sti sti frs C - + = R r x(α s -α e ) = L A + L C
Steady state (phase B) exists, its feed rate value is V sti

Yes

Maximal feed rate (V f ) is given by the solution of (eq.

16)

Feed Current steady state feed rate calculation V sti given by flowchart Fig. 12 Previous steady state feed rate calculation V sti-1 given by flowchart Fig. 12 Following steady state feed rate calculation V sti+1 given by flowchart Fig.

12

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - × × × = - - - 1 1 1 ; min ri ri ts ri ri sti frs R R t J R R V V δ with ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ) sin( ; ) cos( min s my s mx ts J J J α α ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - × × × = + + + 1 1 1 ; min ri ri te ri ri sti fre R R t J R R V V δ with ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ) sin( ; ) cos( min e my e mx te J J J α α ( ) J V V V V L fre sti sti fre A - + = ( ) J V V V V L frs sti sti frs C - + = R r x(α s -α e ) = L A + L C

Bore-bearing machining simulation

This simulation objective is to validate the test protocol developed in the previous section and to highlight the centrifugation effect observed in Section 5.1.

The simulation has been divided into two phases: circular interpolations are first experimented without machining, then both small (Ø 25 mm) and large (Ø 80 mm) bores are machined. Simulation parameters are chosen on the basis of industrial experience (Table 4).

During these experiments, circular approaches are added to machining tool path (Fig. 20) (arcs 1-2 and 3-4). The radius value is 1.5 mm for small bores and 20 mm for large ones. Furthermore, the same circular clearances are programmed after machining. The radial depth of cut which is 0.5 mm, is also chosen according to industrial applications. The axial depth of cut is 9 mm. The corresponding experimental results are presented in Table 5.

Small-bore machining

The feed rate path analysis shows a limitation phenomenon. For each programmed feed rate, a 2 m/min limit appears (Fig. 21). As it was previously identified in simple circular interpolation, the machine tool settings (J curv , r jct ) impose a 2-m/min feed rate. This limited feed rate is then supported by the circularity indexes, with bore circularity being closed (Table 5). Moreover, for each test, machining time is the same: the cutting force exerts no impact on the machining feed rate.

The analysis of quarter-circle approach and clearance shows a feed rate decrease (1,2,3,4 in Fig. 21). This decrease is generated by curvature discontinuity between the approach-derived circle and the bore circle (2 and 3 in Fig. 21). The discontinuity between a 1.5 mm and 2.5 mm circle radius (approach tool path for the small bore) introduces a transition block-crossing feed rate of 0.9 m/min. This value is validated by the surveys in (1 in Fig. 21). This phenomenon is due to the tangential discontinuity between a segment tool path and the quarter-circle. A segment is in fact added between the circular approach paths and the bore centres. The phenomenon is the same for clearance tool path (4 in Fig. 21). This can be explained by activating the continuous path mode. A smoother tool path is thus generated by means of an added curve in the transition block for tangential discontinuity.

Large-bore machining

In this case, the feed rate analysis shows that the first two programmed feed rates have been reached (Fig. 22), whereas a feed rate limit of 10.55 m/min is derived for the third feed rate (Table 5). As previously determined with the simple circular interpolation, the machine tool settings (J curv , r jct ) imposed a 10.55 m/min feed rate for a 30-mm radius circle.

The quarter-circle approach analysis highlights a feed rate decrease; this decrease is generated by a transition block between the approach-derived circle and the bore circle. The discontinuity between a 20-mm and 30-mm radius occurs at a feed rate crossing of 3.6 m/min, which has been confirmed by surveys. In addition, feed rate decreases are generated by the approach and the clearance tool path, as previously explained for smallbore machining.

Accuracy index analysis has confirmed the centrifugation effect. Circularity measurements show this phenomenon in bore shapes. As shown in the simple circular trajectory, a higher feed rate induces accuracy loss. Yet time indexes are better at a higher feed rate. Productivity is thus improved and the main difficulty encountered in HSM bore machining is to be able to anticipate accuracy losses as a function of cutting condition. Fig. 22. 80-mm bore machining: Kinematic indexes for 3 feed rates

Kinematic model validation

Bore machining simulation has confirmed the phenomena highlighted in tests without machining: feed rate limitation, and centrifugation effects. Moreover, machine tool kinematic behaviour is closely correlated with part accuracy and productivity. The predicted machine tool behaviour during tool path generation thus proves to be highly significant. Machining time, feed rate loss and potential impact on part accuracy should all be analysed by means of kinematic machine tool behaviour simulation.

The model developed using tests without machining, has been validated on the basis of these tests. By means of very simple tests without machining, significant machine tool parameters can be highlighted and easily integrated into the kinematic model.

Conclusion

HSM technology influences both part quality and machining time. Experts need to strike a balance between part quality and machining time with respect to this technology. Standards are not sufficient to assist experts for solving this problem. In this context, test procedures that enhance machine tool accuracy, the machining quality and machining time evaluation must be further developed. An original approach to analyse HSM machine tool behaviour during circular interpolation is presented in this paper.

The test procedure elaborated in this approach is based on machine tool and NCU limits. In HSM context kinematic parameters (axis feed rate, acceleration and jerk) can imply feed rate limitation. Besides, NCU processing capacity and specific capabilities influence feed rate law generation and imply feed rate loss.

To emphasize the influence of these HSM parameters, test types without machining have been designed herein. Both circular and spiral trajectories have been performed and assessed. Test analyses rely upon three types of indicator: accuracy (F min ,F max ,G), kinematics (feed rate, acceleration, jerk), and time indexes.

The test procedure has been applied to the evaluation of a given machine tool. The results obtained underscore the link between part accuracy, machining time, NCU behaviour and kinematics parameters. Influence of the jerk (acceleration derivative) has been emphasised. This parameter is controlled during circular displacements by means of NC code, the numerically-controlled unit and kinematic capacities of axes. Thus, the feed rate set point processed by NCU can be lower than the programmed feed rate. Besides, a centrifugation effect has been emphasized. Accuracy indicators underline the loss of part quality when the maximal reached feed rate is higher. Thus, to maintain a given part quality, sometimes the programmed feed rate should be reduced. Besides, for a given interpolated arc radius, a circularity index limit can be induced by feed rate limitation.

Based on these tests, a kinematic machine tool model was developed. This model was then validated through bore machining and phenomena underlined during simple tests without machining were confirmed. The developed kinematic model has allowed to simulate each bore machining time and to anticipate loss of bore quality. Besides, this kinematic model has permitted to analyse respect of cutting condition.

2 Fig. 1 .

 21 Fig. 1. a: Index for circularity deviation (G); b: Indexes for radial deviation (F min ,F max )

Fig. 4 .

 4 Fig.4. Renault test part[2]

Fig. 5 .

 5 Fig. 5. Trajectories without machining

Fig. 7 .

 7 Fig. 7. Axis acceleration skills and feed rate limitation for the 2.5-mm radius

Fig. 8 .Fig. 9 .

 89 Fig. 8. Static look-ahead: Axis jerk skills for 2.5-mm radius

Fig. 10 .

 10 Fig. 10. Surveys conducted during 30-mm circular interpolation at 6 m/min

Fig. 11 .

 11 Fig. 11. Surveys conducted during 2.5-mm circular interpolation at 6 m/min

Fig. 12 .

 12 Fig. 12. Feed rate set point calculation flowchart

Fig. 14 .Fig. 15 .

 1415 Fig. 13. Spiral trajectories

  Fig. 16. Inclined spiral trajectories

Fig. 18 .

 18 Fig. 18. 10-mm arc radius interpolation

Fig. 19 .

 19 Fig. 19. Controlled-jerk kinematic law generation

Fig. 20 .

 20 Fig. 20. Bore machining trajectory

Fig

  Fig. 21. 25-mm bore machining: Kinematic indexes

Table 1 .

 1

	5-axis HSM centre MIKRON UCP710
	parameters
	Machine Tool	MIKRON UCP 710	Spindle	Step Tec
	Workspace	710 x 550 x 500 mm	Tool holder	HSK 63A
	Maximum feed rate (V mi )	30 m/min (X,Y,Z)	Spindle Speed	18,000 rev/min
	Maximum	2.5 m/s 2 (X)	Power	16 kW

HSM functions

Brisk, Soft, Drive, Dynamic Lookahead, Feed forward

Table 2 .

 2 Test conditions

Trajectory

Simple circle Semicircular spiral

Quarter-circle spiral Interpolation radius R r

(mm)

Table 3 .

 3 Feed rate and accuracy indexes

	Programmed feed rate (m/min)	6	9	12	24
	Measured	(R r = 2.5)	2.01	2.01	2.01	2.01
	feed rate (m/min)	(R r = 30)	6.01	9.01	10.53	10.53
		Programmed
	R r (mm)	feed rate	G (μm)	F max (μm)	F min (μm)
		(m/min)
	2.5	6 9	43 43	80 80		-11 -11

Table 4 .

 4 Simulation parameters

		Tool
	Type		Carbide flat end mill
	Diameter		20 mm
	Number of Tooth		4
			Cutting conditions
	Features	Cutting speed	Feed per tooth	Programmed feed rate
		530 m/min	0.2 mm	6,748 mm/min
	Bore diameter -80 mm	750 m/min		9,549 mm/min
		940 m/min		11,968 mm/min
	Bore diameter -25 mm	470 m/min		5,984 mm/min
		580 m/min		7,385 mm/min

Table 5 .

 5 Accuracy and time indexes

	Bore	Circularity	T wm (1)	T m (2)
	Ø 80, 530 m/min	26 µm	1.76 sec	1.77 sec
	Ø 80, 750 m/min	56 µm	1.36 sec	1.35 sec
	Ø 80, 940 m/min	71 µm	1.28 sec	1.27 sec
	Ø 25, 470 m/min	25 µm	0.53 sec	0.53 sec
	Ø 25, 580 m/min	20 µm	0.54 sec	0.54 sec
	Ø 25, 670 m/min	20 µm	0.53 sec	0.53 sec
	(1): time without machining	(2): machining time

(eq. 12)

( )

The feed rate set point is controlled by three main parameters: NCU settings, NC code, and axis capacities. The complete algorithm for feed rate set point processing is detailed in Fig. 12; this algorithm is based on the presence of a steady-state feed rate (phase B); i.e. axis skills are sufficient to reach the feed rate set point that has been processed according to arc length. The feed rate set point calculated in this manner must respect the axis skills represented by the static look-ahead (V s ) for the angular position (α c ) obtained when feed rate becomes constant (Limitation on current arc of circle in Fig. 12). As governed by NCU settings for jerk limitation, the feed rate set point reached in the steady state (V st ) is then processed (Jerk and feed rate limitations in steady state in Fig. 12).

For a circular trajectory of R r = 2.5-mm radius at V prog = 6 m/min: J tcurv = 6 m/s 3 and V Jtcurv = 2.01 m/min, it can be deduced: α e = 0°< α c = 33.5°< α s = 360, hence V Jt (α c ) = V Jtcurv , V An (α c ) = 5.7 m/min, V t (α c ) = 36 m/min, V tcy = 78.5 m/min, which yields: V st = V Jtcurv = 2.01 m/min. For a circular trajectory of R r = 30 mm at V prog = 6 m/min, according to the feed rate set point calculation flowchart: V st = V prog = 6 m/min.


Yasmina Ziad
email: yasmina.ziad@etu.univ-amu.fr

Nathalie Fabbe
email: nathalie.fabbe-costes@univ-amu.fr

Upstream supply chain vulnerability assessment: a collaborative research project with a car manufacturer

Keywords: Upstream supply chain, Vulnerability assessment, Automotive industry

This paper focuses on the concept of supply chain (SC) vulnerability and how to assess it. In the context of an action research project, we conducted an SC vulnerability analysis for an automotive manufacturer. Our goal is to help SC managers prevent risks and SC disruptions, which in turn can improve the robustness of the SC. We propose a number of categories of vulnerability, which are drawn from both the literature and from practice. Using a specific SC case, we present the steps of our assessment approach and discuss the different categories used, how to assess them and how to assess overall SC vulnerability.

Introduction

Recent events have left their mark on global supply chains (SCs), revealing once again their vulnerability to risk and constant uncertainty. The electronic components crisis, for example, caused major losses for companies in the automotive sector, which were severely impacted by the shortage of parts. The automotive industry lost more than $200 billion and about eleven million fewer vehicles were produced [START_REF] Walsh | How auto companies are adapting to the global chip shortage[END_REF]. Beyond the various crises and uncertainties in the environment, SC risk is also due to how entities interact in the SC. According to [START_REF] Chapman | Identifying and managing supply chain vulnerability[END_REF], lack of visibility, misapplication of just-in-time practices, and inaccurate forecasting are factors of risk. The 3rd Supply Chain Risk Barometer (2022) recently ranked the lack of supplier capacity and sourcing problems in the first and fourth places of their top-10 list of SC risks. This same barometer also shows that in 2021 (p.10), 69% of the crises experienced by the companies participating in the study were caused by problems related to suppliers of tier-1, tier-2, tier-3 and beyond, while only 1% of these crises were caused by customers. The barometer also states (p.5) that "it is therefore urgent for companies to gain some insight into how they can accurately assess their vulnerability and strengthen their strategies and control systems". In terms of financial impacts, [START_REF] Hendricks | The effect of supply chain glitches on shareholder wealth[END_REF] found that announcements of SC failures due to parts shortages were associated with an abnormal decrease in shareholder value of 8.16%.

A large body of academic literature on SCM has focused on SC risks from different perspectives [START_REF] Zsidisin | Do Perceptions Become Reality? The Moderating Role of Supply Chain Resiliency on Disruption Occurrence[END_REF]. However, SC vulnerabilities have received less attention, and even less work has been done on vulnerability assessment to proactively reduce a firm's exposure to risk [START_REF] Ekanayake | Identifying supply chain vulnerabilities in industrialized construction: an overview[END_REF][START_REF] Sharma | Supply chain vulnerability assessment for manufacturing industry[END_REF]. Since identifying and reducing SC vulnerability can reduce the number of disruptions and their impact [START_REF] Wagner | A comparison of supply chain vulnerability indices for different categories of firms[END_REF] and increase the robustness of the SC system [START_REF] Asbjørnslett | Assessing the Vulnerability of Supply Chains[END_REF] as well as its ability to bounce back after a disruption, it is important to develop a methodology for vulnerability assessment, especially for the upstream SC of a global company. Recent crises have revealed that SC managers underestimate the vulnerability of their upstream SC and are not sufficiently prepared for disruptions down the chain [START_REF] Blackhurst | Supply chain vulnerability assessment: A network based visualization and clustering analysis approach[END_REF]. As a result, recent papers have proposed vulnerability assessment tools and/or methods [START_REF] Blackhurst | Supply chain vulnerability assessment: A network based visualization and clustering analysis approach[END_REF][START_REF] Sharma | Supply chain vulnerability assessment for manufacturing industry[END_REF][START_REF] Wagner | A comparison of supply chain vulnerability indices for different categories of firms[END_REF], though very few have studied vulnerabilities all the way back to raw materials. Our goal is to develop a methodology to assess the vulnerability of upstream SCs and answer the following research question: how can a global industrial company assess the vulnerabilities of its entire upstream SC?

Based on action research, this paper focuses on the identification and analysis of vulnerability, which is defined as the exposure of the SC to internal or external hazards. The scope of the analysis in this study is part of a carmaker's upstream SC. For the carmaker, the upstream SC is a strategic area for its future development, especially as it is the part most affected by crises.

In this paper we describe what we did to assess the vulnerability of the upstream SC for a car manufacturer, the difficulties we encountered and how we dealt with them. Based on the literature and a real case study, we show how we analysed the vulnerable points in the upstream SC of an automotive component by adding an analysis of raw materials, which play a critical role in the many shortages occurring at the moment. The objective of this vulnerability analysis is to help SC managers prevent risks and avoid SC disruptions. It could therefore be used to improve the resilience and robustness of the SC.

Literature review Vulnerability: definition, drivers and assessment

Vulnerability is a multidisciplinary concept that applies to individuals, populations and ecological systems. According to the Merriam Webster dictionary, vulnerability is the capacity to be physically or emotionally injured and is synonymous with being exposed or susceptible. It is derived from the Latin "vulnus", meaning injury. In order to explore the concept of vulnerability, we draw on the climate change literature where it has been studied extensively [START_REF] O'brien | What's in a word? Conflicting interpretations of vulnerability in climate change research[END_REF]. In the sixth report of the Intergovernmental Panel on Climate Change (IPCC) [START_REF] Begum | Point of departure and key concepts[END_REF], the panel defined vulnerability as the propensity or predisposition to experience adverse effects. The authors of the report consider that vulnerability is composed of three dimensions: exposure, sensitivity or susceptibility to hazards, and the lack of adaptive capacity to cope with them. Exposure is defined as a set of characteristics that can be negatively affected. Sensitivity reflects the extent to which the system is negatively affected by climate variability or change. Finally, adaptive capacity or adaptability pertains to an adjustment process that helps to mitigate negative consequences or derive benefits. Thus, the system may be less vulnerable if it is less exposed or less sensitive to certain hazards and/or if it has adaptive capacity.

Ontological questions aside, vulnerability is recognized in the SCM literature as important due to the numerous events that regularly alter or stop SC flows. Thus, SC vulnerability is often defined as the exposure of the SC to disruptions [START_REF] Chapman | Identifying and managing supply chain vulnerability[END_REF][START_REF] Garnett | Vulnerability of the United Kingdom's food supply chains exposed by COVID-19[END_REF], due to internal or external risks. In other papers, vulnerability is defined as an inherent characteristic of SCs [START_REF] Deshpande | An exploratory study into manufacturing supply chain vulnerability and its drivers[END_REF][START_REF] Wagner | An empirical investigation into supply chain vulnerability[END_REF]. Recently, [START_REF] Deshpande | An exploratory study into manufacturing supply chain vulnerability and its drivers[END_REF], drawing on the work of [START_REF] O'brien | Why different interpretations of vulnerability matter in climate change discourses[END_REF] in relation to the climate change literature, proposed the following definition of the concept of SC vulnerability: "It is an inherent characteristic of the SC (independent of hazards) and is defined by its present inability to cope with and adapt to the impacts of hazards. SC vulnerability is the result of contextual conditions of an SC, which act as drivers of SC vulnerability. SC vulnerability drivers determine the exposure, sensitivity, adaptive capacity and potential adaptive responses of an SC to hazards." [START_REF] Deshpande | An exploratory study into manufacturing supply chain vulnerability and its drivers[END_REF] definition recognizes that vulnerability, which is a characteristic of the SC, would be present even in the absence of risk.

According to [START_REF] Wagner | An empirical investigation into supply chain vulnerability[END_REF], vulnerability is a multi-faceted concept and there are many factors that contribute to vulnerability. Indeed, [START_REF] Peck | Reconciling supply chain vulnerability, risk and supply chain management[END_REF] considers that the vulnerability of SCs originates at several levels in a systemic way: value stream / product / process, asset and infrastructure dependencies, organizations and interorganizational networks and the environment in which SC organizations operate. Vulnerability factors are preconditions for the manifestation of risk [START_REF] Deshpande | An exploratory study into manufacturing supply chain vulnerability and its drivers[END_REF][START_REF] Sharma | Supply chain vulnerability assessment for manufacturing industry[END_REF]. Among these factors, [START_REF] Wagner | An empirical investigation into supply chain vulnerability[END_REF] empirically proved that certain characteristics, such as dependence on certain customers and suppliers, the degree of single sourcing, and dependence on global sourcing, increase the vulnerability of the SC to certain risks. In a recent paper, [START_REF] Sharma | Supply chain vulnerability assessment for manufacturing industry[END_REF] also found that certain practices adopted by firms increased the SC's exposure to risk. The authors identified 26 predominant vulnerability factors in the manufacturing industry. Using an analytic hierarchy process (AHP), they showed that the most critical vulnerability factors are critical parts suppliers, supplier location, long SC lead times, process improvement, and poor strategic alignment in the SC. Similarly, through interviews with different companies, [START_REF] Pettit | Ensuring Supply Chain Resilience: Development of a Conceptual Framework[END_REF] identified seven categories of vulnerability factors: turbulence, deliberate threats, external pressures, resource limitations, sensitivity, connectivity, and supplier/customer disruption.

Few papers analyze SC vulnerabilities compared to those on risk analysis [START_REF] Blackhurst | Supply chain vulnerability assessment: A network based visualization and clustering analysis approach[END_REF][START_REF] Sharma | Supply chain vulnerability assessment for manufacturing industry[END_REF] and even fewer propose an approach to assess upstream SC vulnerabilities. A significant body of literature proposes various tools and methodologies for risk assessment at different points in the SC and with different scopes [START_REF] Deshpande | An exploratory study into manufacturing supply chain vulnerability and its drivers[END_REF][START_REF] Zsidisin | Do Perceptions Become Reality? The Moderating Role of Supply Chain Resiliency on Disruption Occurrence[END_REF]. However, according to [START_REF] Asbjørnslett | Assessing the Vulnerability of Supply Chains[END_REF], the difference between a risk analysis and a vulnerability analysis lies in its purpose. While risk analysis focuses on the identification and evaluation of a negative event and its impact on the organization, vulnerability analysis focuses on the mission of the system and its survival. [START_REF] Asbjørnslett | Assessing the Vulnerability of Supply Chains[END_REF] developed a seven-step approach to evaluate different vulnerability scenarios for a production system. With a different approach, [START_REF] Blackhurst | Supply chain vulnerability assessment: A network based visualization and clustering analysis approach[END_REF] propose a combination of two methods of mathematical and graphical analysis to assess the vulnerability of SCs to disturbances on the basis of a structural analysis. Their proposed method is to reconstruct the propagation path of disruptive events, which can unveil the vulnerabilities arising from SC design, including SC structure, connectivity and dependence. [START_REF] Liu | An Integrated Method of Supply Chains Vulnerability Assessment[END_REF] also relied on an analytical method in combination with relational theory to address the shortcomings of the failure mode and effect analysis (FMEA) method. An enhanced version of the FMEA process allowed the authors to analyze the vulnerability of five SC processes: planning; sourcing, making, delivering and returning.

Based on existing vulnerability definitions and related drivers, and considering current approaches for assessing SC vulnerability, the objective of our research is to develop a dedicated method to analyze the vulnerability of a car manufacturer's upstream SC. Given the high vulnerability of today's SCs to disruptions [START_REF] Wagner | A comparison of supply chain vulnerability indices for different categories of firms[END_REF], and the lack of a grounded analysis of the vulnerability of a given SC, conducted in collaboration with practitioners, our approach aims to contribute to a better understanding of the difficulties and solutions associated with vulnerability assessment in the upstream SC in a real-world context.

Design/methodology/approach

The automotive industry plays a major role in the global economy and mobility has become a strategic issue in many countries. The large number of upstream and downstream SC actors, combined with constant cost-cutting has contributed to SC complexity and globalization. Over the past three years, companies in the automotive sector have been very much at the mercy of the Covid-19 pandemic and, even more so, the crisis in electronic components. According to the 3rd SC Risk Barometer, only 9.7 million vehicles were sold in Europe in 2021, one of the lowest volumes since 1990. Moreover, in the coming years, the automotive industry will face several challenges: accelerating the transition to electric vehicles to meet the European Commission's plan to end sales of internal combustion engine vehicles by 2035, addressing declining global volumes, and the development of shared mobility, etc.1 In addition to these challenges, the industry has been caught in a massive financial trap by the ongoing semiconductor crisis, with the confirmed risk of shortages of certain raw materials.

Our study is based on a collaborative industry-academia research project with a French automotive company. The "risk of SC failure" and "global warming risk" were identified by the company's Corporate Risk Management department as two of the most significant risks to the company in a global risk mapping exercise. In order to reduce the "risk of SC failure", the car company suggested that the researchers work on a methodology to improve knowledge of their SC and to better understand its vulnerabilities. One of the authors of this article is involved in an action research project with the company, while the other participated in a number of key meetings during the study.

Frequent non-directive one-on-one interviews were held with various managers from the automotive company, as well as regular meetings with a group of experts (from the same company) dedicated to developing an approach to SC vulnerability assessment. We interacted with managers from different backgrounds: SC, purchasing, raw material specialists, the engineer in charge of each car component, and the corporate risk manager.

The data was collected in real time using a diary book. We chose strategic car components as a starting point. Together with various expert groups in the company, we tried to trace the flow of suppliers and components back to the raw materials and we identified sources of vulnerability. Based on this experience, we propose a six-step vulnerability analysis and a set of categories derived from both literature and practice. Ultimately, the case demonstrates how challenging it is to assess the vulnerability of an upstream SC.

Upstream supply chain vulnerability assessment

To illustrate our approach, this paper looks at the supply chain of a given component. Together with a group of experts, we chose a component called an exhaust gas differential pressure sensor. It is used to measure the difference in pressure between two points in a system. The differential pressure sensor is part of the emission control system fitted to diesel engine vehicles to meet European emissions regulations. More than half of the cars expected to be sold by the end of 2023 will be equipped with this technology. As a result, any potential failure in this sub-SC will have an impact on more than half of the company's annual vehicle sales. The component is an assembly of several subcomponents, including an electronic component, a plastic box and a steel box. The approach to analyzing and attempting to assess the vulnerability of the sensor's SC can be broken down into six steps, which are outlined below.

Step (1): Defining the scope of the vulnerability assessment

There are different starting points for an SC analysis. On the upstream side, it is possible to start by analysing a supplier and its SC, or to start from a given component and/or raw material. With our group of experts, we decided to identify a strategic component as the starting point for our analysis. In other words, we chose a component with a strong impact in case of subcomponent shortage, difficult to replace quickly. In terms of scope, we chose to focus on suppliers' production facilities. Indeed, the data on the product supply network is limited beyond tier-1 to the location of sub-suppliers' production plants.

Step (2): Development of the categories, sub-categories and criteria for component analysis

As shown in other articles on SC vulnerability assessment (for example [START_REF] Deshpande | An exploratory study into manufacturing supply chain vulnerability and its drivers[END_REF][START_REF] Sharma | Supply chain vulnerability assessment for manufacturing industry[END_REF], the assessment approaches are based on a set of categories and their criteria. The categories we used in our assessment came from two sources. Some of the categories were identified in the literature on vulnerability factors presented above. The others were selected by the experts on the basis of categories used in tier-1 supplier assessment documents. Table 1 presents the categories and sub-categories together with their respective definitions. Due to lack of space, it was impossible to include the different assessment criteria for each sub-category.

Almost all of the selected categories are already used by the manufacturer to evaluate tier-1 suppliers. These categories have been identified in the literature. Only one subcategory was added by the expert group, relating to product substitution lead time in case of parts shortages, which we did not find in the academic literature. The lead time for product substitution refers to the time it takes to obtain an alternative or to obtain an alternative from a second source [START_REF] Norrman | Ericsson's proactive supply chain risk management approach after a serious sub-supplier accident[END_REF].

H Production lead time**

The manufacturing lead time implies a long process lead time and the high number of decision points in the process increases the complexity of the business process [START_REF] Sharma | Supply chain vulnerability assessment for manufacturing industry[END_REF]).

Supplier's capacity ** I Dedicated lines**

The number of dedicated lines, i.e. those which have a given percentage of their capacity reserved for the production of the component, and the total number of production lines that may be called on. These are important parameters in assessing the level of SC vulnerability as they represent production constraints and reflect the availability of production factors [START_REF] Pettit | Ensuring Supply Chain Resilience: Development of a Conceptual Framework[END_REF]. Step (3): Upstream SC mapping Any company whose SC is known for its complexity must have a mapping system. SC mapping has become an essential tool both for risk prevention and for crisis management. The manufacturer gave us access to its SC tier-n mapping documentation, in particular its Component Supply Chain Charts (CSCC). These documents are completed by tier-1 suppliers following the supplier selection process. When a CSCC is received, it is first loaded into a file storage system and then into an SC mapping system. From the information in the CSCC, it is possible to know part of the SC structure. In our case, the tier-1 supplier mentions 13 sub-components, of which 12 components come from its own suppliers. During our review of the CSCC, we found that the supplier did not provide any information on the raw materials and their origin. To overcome these difficulties, we suggested analysing the MDS (material data sheet) of assemblies and materials provided by the tier-1 supplier. We analysed the MDS for the studied component and extracted information about the raw materials considered critical by the manufacturer, and we included the criticality rating in the vulnerability categories. This means that we did not have to evaluate the raw materials, we just had to identify them in order to assign a criticality ranking, which is mentioned in the raw material criticality assessment.

In fact, most critical raw materials are analysed by a dedicated department within the manufacturer. For confidentiality reasons, it is not possible to disclose this list of critical raw materials. However, the most critical raw materials in the world are identified in the academic and professional literature. For example, cobalt is generally considered to be a critical material due to its scarcity and its concentration in a critical geographical area: the Democratic Republic of the Congo. Furthermore, in terms of analysis methodology, the approach taken by the manufacturer's experts is similar to that of Yale University analyses [START_REF] Helbig | How to evaluate raw material vulnerability-An overview[END_REF]. The evaluation criteria are price, delivery and ESG data (environmental, social and governance). Analysing critical raw materials allows the company to prioritise the measures to be taken to reduce the risk of raw material shortages in the medium and long term.

Step 4: Data collection for missing information. Not all of the information requested was available internally, so we had to send a questionnaire to the tier-1 supplier to collect information relating to the following categories: life cycle, process, production technology, lead time and production lines.

Step 5: Category assessment by experts It was decided that the categories would be assessed by in-house experts. The expert group decided to leave the assessment of vulnerability to the commodity experts, i.e. the engineer, the key account manager and finally the supply planner. One of the researchers organized the meetings and prepared the information needed, i.e. collected the supplier documentation for the selected component and checked the CSCC to see if there was any missing information. We also had a meeting with the key account manager and the engineer to verify the information concerning the tier-n suppliers and their location, as this had not been done previously.

Step 6: Supply chain vulnerability assessment: analysis of categories The starting point of the analysis is the examination of the available alternatives (A). It is well known that, from a risk perspective, double sourcing is less risky than monosourcing. The differential pressure sensor is a component for which there is no known alternative at any stage of the SC. Regarding the life cycle and the technology (B) used in the production process, the supplier did not mention the possibility of a downward trend in production of the component. The technology will continue to be used next year, although according to the manufacturer's experts, there will be some changes due to new regulations regarding the emissions control system. With regards to external pressure (C), the experts pointed to new regulations as part of the political/regulatory changes mentioned by [START_REF] Pettit | Ensuring Supply Chain Resilience: Development of a Conceptual Framework[END_REF], which can be a factor of vulnerability.

For the SC structure (limited to suppliers' factories), the facilities were mainly located (D) in Southeast Asia. Of the 13 suppliers, five were in Malaysia, with the remainder spread between Singapore, the Philippines and China. The manufacturer has its own risk classification of countries based on two criteria: geopolitical and environmental risk. Overall, 8 out of 13 suppliers were located in countries vulnerable to environmental or geopolitical risk or both. Supplier concentration (E) was considered a highly critical parameter, given the significant presence of several suppliers in the same geographical area. Similarly, the complexity (F) of the SC (which we assess by looking at the number of actors involved) was also considered to be highly critical. In addition, the tier-1 supplier has not documented its entire SC, which extends well beyond its 13 suppliers.

The production substitution lead time (G) varies from one component to another. For the evaluation of this category, we were not able to collect all the information on the different subcomponents. The experts estimated that if there were a shortage of the electronic component, the substitution lead time would be quite high, between 6 and 12 months. Similarly, we were not able to obtain precise information on production lead times (H), except for the tier-1 supplier.

In terms of supplier capacity, production lines (I) were only dedicated to the parts closest to the tier-1 supplier; the further away from tier-1, the fewer lines were dedicated to the component studied. Regarding the number of production lines (J), the tier-1 supplier had only one line dedicated to the production of the part.

Variation in demand (K) is known by the supply planner, who has full information about the SC's response to such variation. This was very common at the time due to the crisis in electronic components. Therefore, this parameter was considered very sensitive.

Based on our case study of an upstream SC, we have developed an approach to vulnerability analysis through action research. The steps we have highlighted represent the main steps followed. Due to space constraints, it is not possible to elaborate on the criteria for the analysis in Table 1. However, we can provide further information on request. By developing the criteria, we were able to create a multi-criteria matrix for the company which facilitates the analysis and presentation of SC weaknesses. The case of the differential pressure sensor shows the complexity of the approach. The SC for the sensor appeared to be highly vulnerable to disruption, as discussed with the experts. To take the analysis further, other parameters such as inventory levels, the presence or absence of an additional production line in another geographical area need to be considered. Our approach thus provides a preliminary methodology to study the level of SC vulnerability. Further studies with other components and more intensive data collection should enable us to better explore the vulnerability of the entire upstream SC.

Discussion

By conducting this vulnerability assessment for the SC of a differential pressure sensor, we were able to identify vulnerabilities in the SC of a component that is used in over a million vehicles. There are several key steps in our approach. Mapping the SChighlighted by [START_REF] Harland | Risk in supply networks[END_REF] -is a crucial element of our approach. For the case we studied, the mapping provided by the manufacturer gives us a partial vision of the SC structure. However, it is still very complicated for companies to obtain a complete vision, be it upstream or downstream [START_REF] Fabbe-Costes | The map is not the territory": a boundary objects perspective on supply chain mapping[END_REF]. Similarly, from the perspective of the focal firm, the 'visible horizon' is limited and subject to constraints the further one moves away from the focal firm, as argued by [START_REF] Carter | Toward the Theory of the Supply Chain[END_REF] and echoed in [START_REF] Fabbe-Costes | The map is not the territory": a boundary objects perspective on supply chain mapping[END_REF]. Thus, for the case we studied, since the suppliers themselves do not, in most cases, hold all the information about their tier-n, the mapping is limited to the information that the tier-1 supplier has itself or wants to communicate to the focal firm. Similarly, in analysing the different mapping sheets of the subcomponents, we noticed that the quantity and quality of information varies from one component / supplier to another. However, this does not prevent a vulnerability analysis from being carried out to identify strategic or more vulnerable suppliers, beyond first-tier suppliers, even though the mapping is partial. Regarding the categories we evaluated, most had already been identified in the literature. Another category, related to substitution lead times, was proposed by the expert group. In our opinion, this category is very useful for assessing the vulnerability of the upstream chain: it serves to assess the time needed to validate a second technical alternative in case of a shortage. Furthermore, the studied component is sourced from a single supplier at all levels of the chain. This represents a significant dependency. While mono-sourcing has its advantages, it leaves little room for finding an alternative solution if the alternative must undergo very long quality-control processes [START_REF] Wagner | A comparison of supply chain vulnerability indices for different categories of firms[END_REF]. In terms of the quality of the information found, there is still much to be done in terms of transparency of information. Similarly, there is very little data available at an individual company level, so a collaborative approach to data collection and analysis with suppliers would be a more interesting way of anticipating weaknesses and triggering corrective action. Likewise, the various mapping and risk analysis tools available on the market are becoming increasingly powerful, and can overcome the complexity of non-digital data collection and analysis, but vulnerabilities are context-dependent [START_REF] Deshpande | An exploratory study into manufacturing supply chain vulnerability and its drivers[END_REF] and the mapping provided by external tools is aggregated from different companies, which may be irrelevant to a given company.

Conclusion

Our findings have both academic and managerial implications. Firstly, our literature review showed that very few studies deal with upstream SC vulnerability analysis. By conducting this action research, we found new insights into the difficulties and solutions of upstream SC analysis, rooted in the reality on the ground. From a managerial point of view, our approach allowed us to demonstrate to the various experts we consulted at the manufacturer and some of its suppliers the complexity of analyzing vulnerabilities beyond the tier-1 supplier and the imperfections of SC mapping as well as of available data. It also illustrated the problem of missing data and the lack of transparency between SC actors. Our research has certain limitations. As few components were investigated in this stage of our research, it is impossible to generalize the results. Moreover, the choice of categories was not based on the Delphi method or a quantitative prioritizing method such as AHP, which limits our study and constitutes an avenue for future research. Indeed, an approach whereby vulnerability would be analyzed through categories and scoring criteria would allow us to create an analytical tool with relevant scoring to assess the overall vulnerability of the upstream SC, which is one of our objectives in the next stage of our research.

  by the manufacturer can make the SC vulnerable due to other constraints[START_REF] Pettit | Ensuring Supply Chain Resilience: Development of a Conceptual Framework[END_REF]. (*): Derived from the literature. (**): Derived from the meetings and mentioned in the literature. (***): Derived from the meetings and not mentioned in the literature.

Table 1 -

 1 vulnerability categories and sub-categories

	Category		Sub-category	Category definition
				The lack of alternatives makes SCs vulnerable.
				Dependence can originate from customer or
		A Alternatives**	supplier due to single sourcing or single
				customer, for example, which in turn increases
				risk exposure (Zsidisin et al., 2000).
	Product characteristic**	B	Life cycle and technology used in the production system**	Product lifecycles are much shorter. The same applies to the technology used to produce the product, which creates vulnerabilities (Pettit et al., 2010).
		C	External pressures**	Influences not specifically targeted at the business that constrain or prevent the business from operating (Pettit et al., 2010).
	(SC) structure (Restricted to supplier's facilities)*	D E Density* Location of supplier's plant ** F Complexity*	The design characteristics of the SC increase the severity of an SC disruption (Craighead et al., 2007) as well as the location of the facilities, which in turn makes the facilities of the SC vulnerable to disruptions.
			Product
		G	substitution lead
			time***
	Lead time **

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