An approach for using iterative learning for controlling the jet penetration depth in abrasive waterjet milling

Journal of Manufacturing Processes 22 (2016) 99–107

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Technical Paper

An approach for using iterative learning for controlling the jet penetration depth in abrasive waterjet milling A. Rabani a,∗ , J. Madariaga b , C. Bouvier c , D. Axinte a a Department of Mechanical, Materials and Manufacturing Engineering, Faculty of Engineering, University of Nottingham, University Park, Nottingham NG7 2RD, UK b IK4-Tekniker, Calle I˜ naki Goenaga, 5, 20600 Eibar, Gipuzkoa, Spain c Zeeko Ltd, 4 Vulcan Court, Vulcan Way, Coalville, Leicestershire LE67 3FW, UK

a r t i c l e

i n f o

Article history: Received 13 July 2015 Received in revised form 14 December 2015 Accepted 29 January 2016 Keywords: Abrasive waterjet Milling Iterative learning control Micro-waterjet

a b s t r a c t This paper presents a new methodology for controlling the jet penetration in abrasive waterjet milling. The generation of milled parts by means of abrasive waterjet is traditionally done by trial and error, relying mainly on the operator’s expertise. An Iterative Learning Control (ILC) based approach is proposed to improve the prediction and correction of depth of the generated jet footprints in a systematic way every time the same part is milled. This approach utilizes a P-type (proportional) ILC algorithm to control the milling process, and uses a geometrical model based on non-linear partial differential equations for predictions of footprints to provide a fast convergence to the process input and reducing errors. The experimental validations of this approach were carried out using a Titanium alloy (Ti6Al4V) workpiece where a slope-varying profile was generated using 220 mesh grit-size abrasives with 0.04 kg/min mass flow rate and 1380 bar pressure. The results show that the achieved depth accuracy is indeed improved by more than 50% after four iterations when using this approach; providing basis for generating precision freeform surfaces using abrasive waterjet machines in a controlled manner. © 2016 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

1. Introduction Abrasive waterjet (AWJ) is one of the fastest developing non-conventional machining processes. AWJ machining system is traditionally composed of water treatment unit, pump, high pressure intensifier, water jetting head (orifice, mixing chamber, focusing tube/nozzle), abrasive feed system, and supporting accessories such as hoses and control valves. In addition, a nozzle handling device, typically an automatic arm is required to manipulate the nozzle at pre-determined linear/angular speeds along the required pass. In the water jetting system an orifice (made of ruby/diamond) is used to form a water jet from the high pressure water. The high velocity water creates a vacuum in the mixing chamber which entrains the abrasives from a hopper into the mixing chamber. Thereafter the abrasives are accelerated and included in the water jet when passing through a focusing tube. Thus, the resultant focused, high-velocity three-phase (water, air and abrasives) plume is capable of virtually removing any materials through attrition by impacting abrasive particles due to its high energy,

∗ Corresponding author. Tel.: +44 01159514079. E-mail address: [email protected] (A. Rabani).

while water contributes to “wedging in” any cracks initiated by the abrasive particles. AWJ machining technology was commercially introduced in early 1980s [1]. Since then it is used for cutting [2], drilling [3], turning [4,5], peening [6], threading [7], polishing [8] and milling [9]. It benefits from various distinct advantages over other machining technologies such as virtually no thermal influence on the workpiece, high machining versatility to process virtually any materials, multi-axes operation and negligible cutting forces. Due to these advantages, AWJ technology is being widely used machining technique particularly for difficult-to-machine materials, for instance: ceramics [10,11], concrete [12], composites [13] and titanium based superalloys [14,15] and even diamond [16], where conventional machining is often not technically or economically feasible, as well as for artworks on various materials. However, AWJ machining has some drawbacks such as the generation of loud noise and a harsh working environment that require significant process knowledge to insure the part quality. AWJ machining process is influenced by several process parameters where a few out of many, such as water pressure, abrasive flow rate and jet feed speed can be precisely tuned. Therefore, development of robust models that can quantify the influence of these parameters on the jet footprint could yield high quality parts and productivity demanded by target industries.

http://dx.doi.org/10.1016/j.jmapro.2016.01.014 1526-6125/© 2016 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

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In AWJ milling the erosion process occurs as the result of the impact of the abrasive particles and the load impinged by the water. This process is regarded as dwell-time dependent because, for a given set of conditions, the jet feed speed is the key element of controlling the process, since the slower the jet is moved, the more particles will hit in a given area of the workpiece and therefore the final erosion will be higher. Extensive work has been carried out in order to exploit these characteristics to develop models for AWJ milling [17–19], achieving a good agreement between the model predictions and the actual generated footprints. These models aim to predict the effect of the jet on the workpiece under several conditions, such as different feed speeds, jet impact angles or overlapping distances between jet passes. This has allowed the generation of milling strategies and waterjet–specific model-based CAM systems that can assist in the machining of parts. There are other predictive modelling approaches for AWJ milling, finite element analysis [20–22], artificial intelligence algorithms [23,24], or statistical methods [25]. However, these are usually more time consuming and require a large set of experimental data as input. Furthermore, there is an inherent limit for the models as there are always phenomena difficult to capture (due to material inhomogeneities, jet reflections, etc.) occurring at the jet–workpiece interface, as shown recently with a new model to evaluate the stochasticity process [26]. Thus, these unforeseen contributing phenomena in the waterjet removal models may be relevant if a higher precision part is required. Alternatively, in order to improve the process precision, further investment on model development can rely on applying novel control techniques to the field. In this area, the use of conventional feedback control techniques presents some limitations due to the necessity of a reliable measurement of the process output, i.e. the footprint generated on the workpiece. Currently, to the authors’ knowledge there is no extended approach for gathering such information online at the time of processing due to the very harsh environment (e.g. high velocity abrasive particles/droplets reflecting from the target surface) where conventional monitoring techniques are difficult to implement. Some promising progress has been reported recently on this issue by using acoustic emission sensors [15,27] although the achievable accuracy still presents minimum errors in the order of 25%. Also the inaccuracies of the model used to characterize the abrasive process can limit the achievable part precision, even though the feedback measurement limitations have been solved. While progress has been made recently on characterizing the footprint dependence on process variables generated during AWJ milling [18], there are still other unconsidered phenomena that limit the achievable accuracy (e.g. stochasticity of the grit flow, dynamics of the waterjet machine, speed command treatment by commercial CNCs, etc.). One possible solution to this problem is to apply feed-forward control techniques such as Iterative Learning Controller (ILC). Being an offline executed control system there is no need for an online surface measurement (which practically is very difficult in the AWJ harsh environment); therefore, offline regular topographic measurements for characterizing part quality in production workshops can be used to feed the corrective algorithm and update the machining input. In this respect, it is beneficial to implement the part scanning system on the AWJ machine; a solution that is considered in the present work. In principle, abrasive waterjet milling involves layer-by-layer material removal so that the inspection can occur between each single pass. The acquired information from previous iterations can be utilized to overcome modelling inaccuracies by implicitly finding the inverse of the real process to be controlled. Note that this approach resembles the traditional craftwork style AWJ milling where the operator used to produce a part, check it and then tune the process parameters based on his knowledge and experience to obtain a better part on the next iteration. The ILC procedure is an upgraded and systematized way of operating,

where the use of available models and algorithm design methods are exploited to provide a fast convergence to a process input that reduces errors. As far as the authors are aware, this method has not been developed and implemented for controlling AWJ milling process to date. 2. Iterative learning control concept for AWJ milling Iterative Learning Control (ILC) is one of the most recent tools in control system design for achieving improved performance in dynamic systems. This technique provides advantages when the classical control theory is not able to achieve the required level of performance due to unconsidered dynamics, parametric uncertainties, disturbances or measurement noise [28]. In such cases ILC can help to improve the response of systems (processes, robotic manipulator trajectories, etc.) that execute the same operation over and over again [29]. The basic idea of ILC relies on the assumption that the generated errors will be repeated in a new iteration of the process if working conditions are kept unchanged [30]. Hence, they can be known beforehand and counteracted in consequence by adjusting the system input(s) adequately. The final objective would be to find an input sequence such that the output of the system is as close as possible to a desired output. On this basis, the ILC control theory tries to develop algorithms to progressively update the input signal so that the generated errors are reduced every time the operation is repeated. ILC has been successfully applied to the control of trajectories in robotic manipulators [31], machining processes [32], chemical processes [33], etc. ILC presents the potential to be applied to different systems as long as the abovementioned conditions are met. AWJ fits these conditions when the same part is milled several times (e.g. in short, medium or large series fabrication). Moreover, the conditions can be well met if the initial conditions are the same in each trial, i.e. the process always starts from an equal surface and the target footprint is being obtained in one run. The very last pass in a footprint generation process fits this description. Therefore, ILC could be applied to any AWJ process by focusing on the last pass. As a particularization of this condition, the potential benefits of ILC in AWJ could be more easily tested in a single-pass generated part (where the initial surface is always equal – untouched, flat surface – for all iterations). Fig. 1 shows a general model for ILC technique. During a trial (k) an input trajectory (uk ) is applied to the system, producing an output trajectory (yk ). Once the trial is over, the generated error (ek ) from the output trajectory and the reference trajectory (yref ), i.e. desired output, is analyzed and the ILC algorithm computes an updated input signal for the following trial (uk+1 ). It is expected that the new input signal generates smaller errors in the following iteration. In a well-designed ILC controller, applying this procedure several times reduces the output error progressively from iteration to iteration. It should be noted that in this technique the control algorithm is executed off-line between iterations and not on-line during the operation.

Fig. 1. Schematic of ILC concept.

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ILC technique for single input and single output systems can be generally described by: uk+1 (t) = L1 (t)uk (t) + L2 (t)ek (t)

(1)

yk (t) = G(t)uk (t) + y0

(2)

where, G(t) is the operator representing the system, and y0 is the response to initial conditions. L1 (t) and L2 (t) are operators applied to the previous cycle input and previous cycle tracking error respectively. The generated error in trial k is: ek (t) = yref (t) − yk (t)

(3)

The most common algorithms operate only on errors in the form of proportional [34], derivative [35] or proportional derivative ILC [36], leading the operator L1 (t) to the identity matrix. The proportional and derivative type controllers are the most widely utilized algorithms as they can perform quite robustly even with limited knowledge about the process. A P-type (proportional) ILC algorithm is proposed in this study to control the process. By employing P-type ILC controller the numerical differentiation of the error required by D-type ILC controllers is avoided so that noise effects can be kept minimal. The initial state error is assumed to be low so it is expected that the tracking error after learning will be small, thus avoiding the need for more difficult to tune PID-type ILC controller [37,38]. The proposed ILC algorithm can be expressed in general terms as: uk+1 (t) = uk (t) + ek (t)

(4)

where,  is the proportional ILC gain. In order to achieve a faster convergence, a model of the process will be used to dynamically change the proportional gain. The first aspect to take into account when applying ILC to AWJ is the definition of the control variable(s). In a general AWJ platform the tuneable parameters and variables affecting the jet footprint (here depth of the jet penetration) can be divided into two groups: (i) those related to the energy/intensity of the jet plume; (ii) those

Fig. 2. Footprint cross section profile (predicted and actual) using 220 mesh gritsize abrasives with 0.04 kg/min mass flow rate, 1380 bar pressure and 1000 mm/min feed speed.

[17–19]. In order to be able to solve the footprint profile prediction the model function can be linearized assuming the generated kerf slope is small. This small slope condition is true for high feed speeds of the jet and allows obtaining the etching rate from the model function by milling a calibration trench at high feed speed and measuring the generated trench profile [17]. The system is further characterized by making use of two open parameters that are calibrated against generated profiles: k and b. The k parameter stands for the jet velocity exponent and provides more or less weight to the material etching and it is indirectly related to the kinetic energy of the abrasive particles. The b parameter stands for the jet nozzle stand-off distance compensation that accounts for the loss of etching capacity when the distance to target surface increases (lower part of the footprint profile). It also compensates for unconsidered secondary effects in the model that occur at higher depths such as erosion due to jet reflections [18]. Therefore, footprint profiles for any speed could be predicted theoretically. The utilized geometrical model can be expressed as

⎧   ⎪ ˆ −εEˆ xˆ 2 + tˆ2 (1 + bZ) ⎪   ⎪ ⎪ ⎪ for − 1 − xˆ 2 ≤ tˆ ≤ 1 − xˆ 2 ⎛ ⎞ ⎪ k/2 2 ⎨  2 ∂Zˆ ˆ sin ˛ sin ˇ ∂Zˆ = ⎝1 + ∂ Z ⎠ +  ∂tˆ ⎪ ⎪ 2 ∂xˆ ⎪ cos2 ˇ + cos2 ˛ sin ˇ ∂tˆ ⎪ ⎪   ⎪ ⎩ 2 for 1 ≤ tˆ ≤ −

0

related to the motion of the nozzle/jet plume. The most dynamic (fastest) change of the footprint can be achieved with motion related variables (position, feed speed); therefore computed corrective actions by the ILC controller are applied on these variables. It should be noted that in general purpose AWJ machines, motion instructions are given through regular NC files where jet feed speed is commanded versus position rather than position versus time. In order to avoid major changes in the machine controller, the optimal input sought will not be a demanded nozzle position in time but a commanded feed speed at each position. Due to this operating procedure, time based events (e.g. periodic disturbances) could not be tackled unless such events are position related. In any case, the footprint errors obtained in a regular AWJ process are more likely to be generated due to an incorrect process set-up (e.g. feed speed definition), rather than other causes related to timely inconsistencies (e.g. random pressure build up on pump, irregular mass flow, etc.). A geometrical model, previously developed at the University of Nottingham that consists of a non-linear partial differential equation is used to find the footprint profile generated in the workpiece

1 − xˆ

and

(5)

1 − xˆ 2 ≤ tˆ ≤ 1,

ˆ represents the jet footprint profile, and b and k where, (ˆx, Z) ˆ are parameters of the model, and E(r) is an etching rate function that accounts for the erosion power of the jet. The details of this approach can be found in [18]. The output of the model is a ˆ x). A footprint profile generated footprint cross-section profile, Z(ˆ using Eq. (5) and the actual milled profile using 220 mesh grit-size abrasives with 0.04 kg/min mass flow rate, 1380 bar pressure and 1000 mm/min feed speed are illustrated in Fig. 2. Using the geometrical model in Eq. (5), a “removal model”, which consists of a table of jet feed speeds and corresponding predicted depths, can be generated. While there is no closed form equation that can be derived from the original model to directly relate these variables (jet feed speed and depth), depth estimation at non-prescribed/simulated feed speeds could be obtained through numerical interpolation. Hence, jet feed speed can be defined as a function of depth by fitting a curve to the numerically computed data and it was found that the removal model can be simplified and approximated in the depth/feed speed ranges of interest as:

v=

a h

(6)

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Fig. 3. Depth of the jet footprint versus jet feed speed: actual and fitted predicted data (Eqs. (5) and (6)) for garnet abrasives 220 mesh size with 0.04 kg/min mass flow rate and 1380 bar pressure.

where, h is the depth of the footprint, v is the jet feed speed and a is a constant to fit the equation to the simulated data for a set of process parameters (Fig. 3). This expression provides a relationship between the jet feed speed and the footprint depth. This formulation will allow performing fast computations when applying the ILC algorithm to control the milling process. ILC is a feedforward control technique that ultimately tries to find the inverse of the process to be controlled. If a model is available for the process, its inverse can be utilized as the most adequate controller for the system. However, this approach is only valid if the model accuracy and robustness is fully assured under any circumstances. Using Eq. (6), the jet feed speed corresponding to the target profile depth can be calculated. If the accuracy obtained is not enough for the application, enhancing the output requires either recalibration of the model, or further development of the model to consider relevant process phenomena that have been ignored. The ILC technique allows further improving the output of the abrasive waterjet milling process by making use of the information provided by the errors of previous iterations. Therefore, in this approach, the process output improves after each iteration. The model can be used not only to define the control signal explicitly but also to decide in which direction and how much a previously used command signal should be corrected. A straightforward approach for this is to use the gradient of the error in the direction of the control variable. Therefore: v = −

a h h2

milled with shallow depths, i.e.
a h h2

3. Methodology for implementing ILC Considering the described theoretical approach, a methodology for implementing ILC in AWJ milling can be split into 6 steps.

(7)

where, v and h are the changes in the jet feed speed and the depth of the footprint respectively. Eq. (7) shows the gradient that has been applied to the simplified model in Eq. (6). The corrections are applied to the jet feed speed by defining the weight of the proportional gain of the P-type ILC algorithm based on the depth error from the previous iteration. It can be noted that with the simplified model, large velocity changes will be produced at small depth errors. In fact, it is evident that for shallow depths, the tangent of the function can become very large. Occasionally, if the model is followed strictly, negative feed speeds can be expected. This could result from measurement errors, such as low level noise, which could be compounded with process errors and of similar amplitude at the low scale. In order to limit the effect of such large tangents a threshold can be set to determine when to apply a smaller weight for the corrections. The threshold can be defined as a depth limit H, such that when trenches are

(8)

Fig. 4. ILC execution flowchart.

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A step-by-step procedure is shown in the flowchart in Fig. 4; the main steps are detailed below: - Step 1 The geometrical model associated to Eq. (5) that describes the footprint profile is calibrated to the working scenario (grit type and flow, pressure, workpiece material, machine). The calibration procedure makes it possible to obtain the parameters b and k, as ˆ well as to estimate the etching rate function, E(r) for Eq. (5). - Step 2 The geometrical model is executed for a wide range of feed speeds to obtain theoretical (h, v) points from which the more simple jet velocity versus depth equation is approximated. Alternatively, this simplified model can also be obtained by fitting a series of experimental (h, v) points obtained by milling footprint at different jet velocities. - Step 3 An NC file is generated using the simplified model (Eq. (6)) and the goal trench profiles to be generated are considered as inputs. - Step 4: Once the NC file is executed and the workpiece milled, the trench is measured and the deviation from the goal profile is calculated.

103

- Step 5 Using the error deviation between the required and measured trench depths as input, the NC file is updated by applying feed speed corrections where necessary. Such feed speed corrections are computed based on the gradient of the simplified model calculated in Step 2 according to Eq. (8). A new NC file will be generated based on the new feed along with the feed corrections versus position relation. - Step 6 Repeat steps 4 and 5 until the deviation does not improve anymore. 4. Experiment design and considerations In order to test the applicability of ILC in AWJ milling the focus was set on features generated in a single pass, where the initial surface is always the same (untouched, flat surface) for all iterations; hence, the conditions for ILC are met. A continuously varying target profile was defined as a goal to be obtained through the ILC procedure described above. The goal is to create straight trenches with varying depths (three distinct footprint depths) along their longitudinal direction, e.g. piecewise linear profiles. The trench profile is formed by a flat section followed by a negative slope, continues

Fig. 5. Target trench profile for the AWJ milling experiments.

Fig. 6. Experimental abrasive waterjet machine set-up.

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very different erosion rates at the beginning and at the end of the slope. In very steep slopes the necessary velocity variations may not be achievable in the required space/time due to the dynamic limitations of the machine (unable to accelerate/decelerate quickly enough). Therefore, there will be a maximum achievable profile slope and acceleration in a single pass. Such limit slope and acceleration can be defined as:

   dh   dx  ≤ |s| dv ≤G dx

Fig. 7. In situ measurement system set-up; (a) STIL CL3 chromatic confocal imaging sensor head, (b) Ti6Al4V workpiece, (c) STIL CCS Prima4 chromatic confocal imaging controller, (d) workstation PC with dedicated software developed in LabVIEW for ILC implementation and measurements.

with a flat section and rises with a positive slope and finally finishes with a flat section as illustrated in Fig. 5. When defining a target profile, the limitations of the machine should be taken into account to make sure that the target profile can be achieved in a single pass. One example of difficult to achieve profiles is the case of very steep slopes in which the machine has to be able to alter the jet speed very quickly in order to generate

(9) (10)

where, s is the maximum profile slope and G is the maximum applicable velocity–distance gradient on the AWJ machine. Therefore, the achievable slope between two distinct feed speeds can be calculated by substituting Eqs. (6) and (10) in Eq. (9); hence:

 

|s| ≤ −

   v1 v2 aG

(11)

By dynamically testing the machine behaviour in accelerating/decelerating profiles the G can be evaluated. A gradient limit in the order of 200 mm/min was established for the testing of the waterjet machine in this study. Using this value as a reference, the removal model can be analyzed and the feasible part profile slopes can be computed. The feasible slopes are different depending on the velocity region in which the system is operating, most critical case being the high speed region in which a significant jet velocity

Fig. 8. (a) Measured (blue and red), mean (yellow), smoothed mean (purple) and target (black) profiles for the first iteration. (b) Deviation of the smoothed mean profile from the target profile. (For interpretation of reference to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 9. (a) Comparison between the jet feed speed used in the first iteration and the computed feed speed for the next iterations using the ILC algorithm. (b) Comparison of the footprints obtained during subsequent iterations.

change is necessary in order to generate a minimal variation in the footprint depth. The experimental trials were planned for AWJ milling of the target profile using garnet abrasives (220 mesh size) with 0.04 kg/min mass flow rate and 1380 bar jet pressure. The selected abrasive gritsize achieves a sufficient removal rate given the range of controlling feed speeds. In this case jet feed speeds ranging between 700 and 2750 mm/min were required (Fig. 3); so that the generated erosions range in depth from 0.07 to 0.27 mm (Fig. 5). According to the simplified removal model (Eq. (6)) a = 189.85 mm2 /min and the maximum achievable profile slope (Eq. (11)) for change in velocity from 2750 mm/min to 700 mm/min for transition from erosion depth of 0.7 mm to 0.27 mm is 0.0197 and for change in velocity from 700 mm/min to 1100 mm/min for transition from erosion depth of 0.27 mm to 0.17 mm is 0.0493. Therefore, care should be taken when defining the target profile so that a feasible part is designed to mill and not exceeding these maximum achievable slopes’ limits. The designed target profile is shown in Fig. 5 where the slopes of 0.0133 and 0.0143 were considered for transitions from 0.7 mm to 0.27 mm and from 0.27 mm to 0.17 mm respectively. By setting a plausible target profile the performance of the AWJ milling process can be tested. The experimental tests were conducted on Microwaterjet® F4 (Microwaterjet AG, Switzerland) equipped with a KMT streamline SL-V100D pump capable of providing a maximum operating pressure of 4137 bar (60,000 psi). In this experimental setup, filtrated (10 ␮m abs) water is pressurized by this ultra-high-pressure pump. It is then forced through a ruby orifice (ø0.3 mm) that makes the jet

travel to the nozzle body to create a partial vacuum drawing in garnet particles (220 mesh, ø45–125 ␮m) through the abrasive inlet. The abrasives are then accelerated by the jet energy in the mixing/focusing tube (ø0.8 mm) to form a high speed abrasive waterjet. The jet energy is set constant; hence, the following operating parameters were considered fixed: P = 1380 bar, ma = 0.04 kg/min, SOD = 3 mm (these are representative operating conditions that are employed in real abrasive waterjet processing). Also the jet was kept orthogonal to the initial flat surface while the feed speed was set as the control variable and was updated after each iteration. To ensure high repeatability of the testing conditions the following parameters have been closely monitored: the water pressure was gauged via an integrated sensor posed close to the mixing tube; the abrasive mass flow was delivered by a controlled mechanical abrasive metering system, which was calibrated before each test; the feed speed of the jet was monitored online via signal acquisition from the machine encoders (NUM controller). The experimental waterjet machine set-up is shown in Fig. 6. Prior to the tests, the predictions of the model in Eq. (5) were calibrated by establishing the simplified model as per Eq. (6) and following the procedure defined in the ILC flowchart in Fig. 4. Ti6Al4V was chosen as the specimen material since AWJ milling of components both for aerospace and medical industries could be targeted by the application of the proposed technology. After the calibration, an NC file was generated using the resulting simplified model as per Eq. (6) with a = 189.85 mm2 /min to compute jet feed speed versus position trajectories. Two parallel trenches were machined per iteration and averaged. For each test, the jet

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started outside the workpiece and was held stationary for 5 s to enable the abrasive flow to stabilize within the jet and the pump pressure to reach a steady state. The trenches were measured after each iteration using in situ chromatic confocal imaging system (STIL), obtaining high-density 3D measurements of each trench. The Zeeko metrology software suite (Zeeko Ltd., UK) was used to threshold, level (remove tilt using surrounding untouched area) and extract a single profile for each trench along its centre-line. The measurements results were introduced to the ILC algorithms developed in MATLAB software to generate a new NC file with compensated errors. Fig. 7 illustrates the in situ measurement system setup. As an example, the results from the two tests corresponding to the first iteration are shown alongside the target profile in Fig. 8a. In order to separate shape and roughness the measured profiles were averaged and filtered using a Gaussian filter with a cut-off of 4 mm. The smoothed mean profile was subtracted from the target profile to compute the errors (see Fig. 8b; negative values represent over erosion). Once the error for the current operation was obtained, then Eq. (8) was applied to generate feed speed corrections to reduce the error. 5. Results and validation The operation of making trenches using 220 mesh size garnet abrasives with 0.04 kg/min mass flow rate and 1380 bar jet pressure was repeated four times to progressively reduce the obtained footprint error. Fig. 9a compares the initial feed speed and the corrected feed speed using the ILC algorithm for the next iterations. It can be observed from the graph that a consequence in preparing for a high feed speed (at the beginning of the milled profile)

Table 1 Milled profiles errors statistics. Iteration

P–V (mm) R2 RMS (mm) Std (mm) Var (mm2 )

1

2

3

4

0.124 0.4250 0.072 0.030 0.000918

0.087 0.8860 0.0229 0.0211 0.000444

0.074 0.9180 0.0194 0.0186 0.000346

0.067 0.9260 0.0183 0.0166 0.000275

The average error reduction for iteration 4 is more than 50% compared to the 1st iteration. In subsequent repetitions of the same part by using the NC programme generated for iteration 4 profiles with the same accuracy can be achieved.

the machine, with its dynamic characteristic, requires accelerating beforehand; this is more evident with the increase of iterations from 1 to 4. It was observed that the resultant footprints get closer to the target profile from iteration to iteration. Fig. 9b shows the comparison of the footprints obtained in subsequent iterations during the ILC experiments. It can be observed that with the increase of iterations the deviation from the required slope decreases proving the fine continuous local adjustments of the feed speed by the ILC. It can also be seen that the errors associated to the milled profile compared to the target profile reduces very fast in the first iteration and converges to a minimum error after the 4th iteration. Fig. 10a shows the evolution of the maximum error in depth of the footprint along iterations. Statistical data such as R2 , Peak-to-Valley, RMS, standard deviation and variance of the error for milled profiles after each iteration are calculated from the smoothed mean profiles. These values are given in Table 1 and illustrated in Fig. 10b. It can be seen that milled profiles are progressively improved towards the target profile. 6. Conclusions Controlled depth abrasive waterjet milling is an immerging technique compared to its conventional counterpart milling systems, where their capabilities are limited, for generating 3D features in low-stiffness parts made of difficult-to-cut materials. In principle, the jet penetration is difficult to control through conventional feedback control techniques due to lack of precise measurements of process outputs and monitoring systems. Therefore, to generate complex geometries using AWJ milling feed-forward control techniques can offer reliable solutions. To address this possibility, this paper has described and implemented a method based on iterative learning to control the depth of jet penetration in AWJ milling. In this regard, this research has emphasized on the following aspects:

Fig. 10. (a) Evolution of the maximum footprint error along iterations. (b) Progressive improvement in errors of the milled profiles for 4 iterations.

• A new method for controlling the jet penetration depth in AWJ milling based on the design of an Iterative Learning Controller was proposed and implemented on 3-axis micro waterjet machine; this was achieved by a combination of in situ surface scanning at particular intervals of the AWJ milling process supported by mapping of machine dynamics so that the dwell-time dependency of the abraded features are kept under control. • The developed ILC technique allows improvements in the output (i.e. precision of the abraded surfaces) of the AWJ milling process by making use of the information provided by the errors of previous iterations (i.e. AWJ milling passes) and the prediction of the analytical model (as in Eq. (5) and simplified as in Eq. (8)) to enable corrections to the jet feed speed so that the final milled surface is produced with reduced dimensional error. • The effectiveness of the proposed technique in reducing the errors and increasing the precision in controlling the jet

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penetration depth was demonstrated on a micro-waterjet machine and evaluated by comparing the jet penetration output of four iterations for a designated trench; where it was proved the errors associated with the milled trenches can be reduced by more than 50%. • A step-by-step methodology was introduced for implementing ILC in AWJ milling that paves the way for the design and the generation of complex features with minimum errors while leading to a significant reduction of setup times by at least 200% when compared with the manual, trial and error, approach. The proposed approach of using ILC for controlling the jet penetration depth in AWJ milling seems feasible and its application to generate complex 3D shapes/freeforms in an automated manner for wide range of applications such as medical, aerospace and optics is promising. Acknowledgement The authors would like to acknowledge the funding support of EU FP7-NMP4 (Grant No. 229 155) for the works presented as a part of the ConforM-Jet project. References [1] Kovacevic R, Hashish M, Mohan R, Ramulu M, Kim TJ, Geskin ES. State of the art of research and development in abrasive waterjet machining. J Manuf Sci Eng 1997;119(4B):776–85. [2] Wang J, Wong WCK. A study of abrasive waterjet cutting of metallic coated sheet steels. Int J Mach Tools Manuf 1999;39(6):855–70. [3] Hashish M. Development of an AWJ deep hole drilling system for metals. In: Proc. 8th American waterjet conference. 1995. p. 26–9. [4] Axinte DA, Stepanian JP, Kong MC, McGourlay J. Abrasive waterjet turning—an efficient method to profile and dress grinding wheels. Int J Mach Tools Manuf 2009;49(3):351–6. [5] Carach J, Hloch S, Hlavacek P, Scucka J, Martinec P, Petru J, et al. Tangential turning of Incoloy alloy 925 using abrasive water jet technology. Int J Adv Manuf Technol 2015:1–6, http://dx.doi.org/10.1007/s00170-015-7489-0. [6] Arola DD, McCain ML. Abrasive waterjet peening: a new method of surface preparation for metal orthopedic implants. J Biomed Mater Res 2000;53(5):536–46. [7] Sheridan MD, Taggart DG, Kim TJ. Screw thread machining of composite materials using abrasive waterjet cutting. In: Symposium on nontraditional manufacturing processes for the 1990’s, vol. 68. ASME-Publications-PED; 1994. p. 421. [8] Zhu HT, Huang CZ, Wang J, Li QL, Che CL. Experimental study on abrasive waterjet polishing for hard–brittle materials. Int J Mach Tools Manuf 2009;49(7):569–78. [9] Hashish M. An investigation of milling with abrasive-waterjets. J Manuf Sci Eng 1989;111(2):158–66. [10] Momber AW, Mohan RS, Kovacevic R. An acoustic emission study of cutting bauxite refractory ceramics by abrasive water jets. J Mater Eng Perform 1999;8(4):450–4. [11] Srinivasu DS, Axinte D. An analytical model for top width of jet footprint in abrasive waterjet milling: a case study on SiC ceramics. Proc Inst Mech Eng Part B: J Eng Manuf 2011;225(3):319–35. [12] Momber AW, Mohan RS, Kovacevic R. On-line analysis of hydro-abrasive erosion of pre-cracked materials by acoustic emission. Theor Appl Fract Mech 1999;31(1):1–17.

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