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Bayesian optimization for autonomous process set-up in turning
G Model CIRPJ 507 No. of Pages 7
CIRP Journal of Manufacturing Science and Technology xxx (2019) xxx–xxx
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CIRP Journal of Manufacturing Science and Technology journal homepage: www.elsevier.com/locate/cirpj
Bayesian optimization for autonomous process set-up in turning Markus Maiera,* , Ruben Zwickerb , Mansur Akbarib , Alisa Rupenyana,c , Konrad Wegenerb a
inspire AG, Technoparkstrasse 1, 8005 Zurich, Switzerland Institute of Machine Tools and Manufacturing (IWF), ETH Zurich, Leonhardstrasse 21, 8092 Zurich, Switzerland c Automatic Control Laboratory, ETH Zurich, Physikstrasse 3, 8092 Zurich, Switzerland b
A R T I C L E I N F O
A B S T R A C T
Article history: Available online xxx
Constrained Bayesian optimization with Gaussian process models is applied to optimize the turning of 1.4125 steel bars by considering tool life, machining time, and surface roughness, in addition to costs of insert, operator, and machine. Feed rate and cutting speed are the optimized process parameters with constant depth of cut, both for a simulated process and for on-machine experiments on a micromachining lathe. The results demonstrate that even without prior knowledge of the process, this method successfully recommends physically reasonable process parameters within a limited number of experiments. The flexibility of the method is demonstrated by providing optimized process parameters for various cost parameters and various requirements for the constraints, based on experimental data. © 2019 CIRP.
Keywords: Turning Process optimization Bayesian optimization Process set-up Gaussian process models Manufacturing Machine tools
Introduction The selection of optimal process parameters in turning is crucial because it directly influences the cost efficiency and the workpiece quality. Process parameters, such as feed rate, cutting speed, and depth of cut are normally selected carefully based on previous knowledge about the process, coolant, machine, cutting tool and response of the workpiece in machining or in the traditional way by experienced operators. The initial process parameters are adapted depending on the required quality of the workpiece, tool wear, machine tool capability, manufacturing time and process requirements. In particular, for the turning process in micromachining applications, the final product quality is of high importance and small variations in the process parameters can lead to unacceptable products. Finding the right combination of parameters requires specific expertise, or sometimes brute-force search and long series of experimental trials. Automatic approaches can optimize the performance of a given sequence of operations to manufacture a desired workpiece by selecting the optimal or most feasible ranges of process parameters. Table 1 shows an overview of existing optimization methods and objectives in turning. A detailed description of turning process optimization can be found in Ref. [1]. In general, the optimization objective can be reformulated as a cost minimization problem subjected to constraints. Formally the optimized values x can be
* Corresponding author. E-mail address: [email protected] (M. Maier).
determined as the arguments of the minima (argmin) of total production costs C T ðx Þ subjected to (s.t.) constraints cðx Þ: ð1Þ x ¼ arg min C T x ; s:t: c x cmax x2x
where x are the optimization parameters, cmax are the maximum allowed constraints and x is the design space of machining parameters. Several methods exist to optimize turning processes. Empirical model-based approaches such as those used in Refs. [2,3] for optimization are suitable for specific optimization tasks where the experimental set-up is fixed. However, these methods have difficulties to optimize set-ups outside of the trained scope such as new workpiece, tool and machine combinations. Experimental design methods, such as the Taguchi method used in Refs. [4–6], are applicable for the optimization of new setups. The Taguchi method uses fractional factorial design test plans created with orthogonal arrays to investigate the influence of different variables at specified levels (see Ref. [7] for a detailed introduction to the Taguchi method). The choice of the number of levels specifies the shape of the unknown function e.g. specifying two levels implies a linear relation. The main disadvantage of the Taguchi method is that the experimental parameters must be defined before the optimization — knowledge gained during the tests is ignored. A powerful global optimization method is Bayesian optimization [8]. It is a sequential design strategy for optimizing an a priori unknown objective function that is expensive to evaluate, such as a time-consuming physical measurement, or a complex simulation [9]. Bayesian optimization is used in various applications such as hyperparameter tuning [10], photovoltaic power plants [11], and
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Table 1 State of the art optimization methods and objectives in turning. Source
Method
Objective
Wang et al. [2] Vijayakumar et al. [3] Yang and Tarng [4] Nian et al. [5] Nalbant et al. [6]
Genetic algorithm Ant colony algorithm Taguchi method Taguchi method Taguchi method
Objective function including surface roughness, cutting force, tool-life, material removal rate and chip breakability Cost function including cutting cost by actual time, machine idling cost, tool replacement cost and tool cost Optimization of tool life and surface roughness Optimization of tool life, cutting force and surface finish Optimization of surface roughness
robotics [12]. It is common to combine Bayesian optimization with Gaussian process (GP) models which are used as surrogate models of the unknown objective and constraint functions. The GP model uses available samples to provide an estimation of the cost function at new parameter configurations [13]. The mean and the uncertainty estimation of the GP model are used in Bayesian optimization to select sample points were experiments are highly informative and beneficial for the optimization [9]. In turning processes, Bayesian inference has been successfully used in the prediction of tool life [14,15]. In Ref. [16], Bayesian optimization has been successfully applied to process parameter optimization of a simulated turning process, and it is demonstrated that for a simulated turning process, the constrained Bayesian optimization [17], where GP model estimations of the cost and constraint function are used, performs better than Bayesian optimization without constraints. In this work suitability and convergence of Bayesian optimization is investigated on a simulated turning process. Afterwards an on-machine process parameter optimization is demonstrated. Gaussian process models and bayesian optimization In this section, Gaussian processes based on Ref. [13], and Bayesian optimization as proposed in Refs. [9,18] are introduced. Rasmussen and Williams [13] define Gaussian processes as follows: “A Gaussian process is a collection of random variables, any finite number of which have a joint Gaussian distribution”. The GP model is fully specified with its mean mðx Þ and covariance 0 function k x ; x .The covariance function, often called kernel, specifies the smoothness of the modeled function. Several different kernel functions exist. A detailed overview can be found in Ref. [13]. Common kernels are exponential and Matérn kernels. Exponential kernels model a very smooth function. For Matérn kernels the parameter n determines the smoothness of the kernel – a higher n leads to a smoother function. In this study Matérn kernels with n = 5/2 and automatic relevance determination (ARD) were used. ARD uses different kernel length scale parameters for each dimension of the input data. The used kernel can be calculated as: pffiffiffi pffiffiffi 5 0 kM ðx; x Þ ¼ s 2f 1 þ 5r þ r2 expð 5rÞ ð2Þ 3
T
s 2t x ¼ k x; x k x
ðK þ s 2N I Þ1 k x
ð4Þ
where I is the identity matrix, the vector k ðx Þ contains covariance terms between x and x 1 : t and the covariance matrix K is defined as: 0 1 kðx 1; x 1Þ kðx 1; x tÞ B C .. .. K ¼@ ð5Þ A . } . kðx t; x 1Þ
kðx t; x tÞ
In this work it was assumed that the Gaussian process mean function is zero. In this way no prior knowledge is given to the GP model. Fig. 1 shows the working principle of Bayesian optimization. The estimated mean and variance function of the GP models are used to optimize an acquisition function, which is used as a cheaper way to evaluate the unknown objective. The acquisition function needs to balance the discovery of regions with high model uncertainty (exploration) and regions with high model prediction values (exploitation). The next experiment is conducted for parameters where the acquisition function reaches a maximum. Afterwards the data from the experiments is used to calculate an updated GP model. This procedure is repeated until a stopping criterion is reached. Various acquisition functions can be used for the optimization of the cost and to determine where to conduct the next
where s 2f is the signal variance, the distance r between x and x is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ffi x x0 P 1 x x0 ,P ¼ diagðs 21 ; s 22 ; . . . ; s 2D ) is a diagr¼ 0
onal matrix with length scale parameters s 21 ; s 22 ; . . . ; s 2D , and D is the dimension of the input space. Based on the GP model and available pairs of measurements ~ 0; s 2 ; the mean ðx 1 : t; y 1 : tÞ corrupted with Gaussian noise eN N
mt and variance s 2t of the joint distribution of the observed target
values and the function values at the test locations can be calculated for unknown test inputs x , using the following predictive equations for GP regression:
T
mt x ¼ k x
ðK þ s 2N I Þ1 y 1 : t
ð3Þ
Fig. 1. Working principle of Bayesian optimization.
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experimental evaluation (see Ref. [9] for a detailed summary). A common acquisition function is the expected improvement (aEI Þ: F ðZ Þ þ s c x fðZ Þ ð6Þ aEI ðxÞ ¼ y xþ mc x
Z¼
mc ðx Þ yðxþ Þ s c ðx Þ
ð7Þ
where y xþ is the best observed value, mc ðx Þ is the predicted
mean cost calculated using (3), s 2c ðx Þ is the predicted cost variance calculated using (4), FðZÞ is the cumulative distribution function of pffiffiffiffiffiffiffi R Z 2 a standard normal distribution F ðZ Þ ¼ 1= 2p 1 et =2 dt and fðZÞ is the probability density function of a standard normal distribupffiffiffiffiffiffiffi 2 tion fðZ Þ ¼ 1= 2p eZ =2 . Gartner et al. [17] extended the expected improvement acquisition function to constraint optimization which was used in this study. ð8Þ aEIC ðxÞ ¼ pf x aEI ðxÞ aEI ðx Þ is the expected improvement without constraints (6) and pf ðx Þ is the probability that x is feasible. The probability that x is feasible can be calculated as follows: pf x ¼
1 pffiffiffiffiffiffiffi s 2p
Z l e
ðtmcon Þ2 2s 2 con
ð9Þ
dt
1
where l is the maximum constraint value. The predicted mean mcon and variance s 2con of the constraint are calculated using (3) and (4). Methodology Cost function The optimization task in this study is to produce parts with minimal total production costs C T (U) and produce workpieces which fulfill the roughness constraint Ra;max : ð10Þ x ¼ arg min C T x ; s:t: Ra x < Ra;max x2x
where x is a vector of the optimized parameters cutting speed vc and feed rate f . The cost unit is assumed to be Swiss francs but for the sake of universality is denoted as U. In this study the manufacturing cost function was assumed to be: C I þ ti ðC MH þ C o Þ ð11Þ C T ðxÞ ¼ tc ðxÞ C MH þ Tðx Þ where tc is the cutting time and T is the tool life (see Ref. [19] for a detailed introduction to cost calculations in manufacturing). The total cost combines cost of machining time and tool replacement cost. The used cost and constraint parameters are listed in Table 2.
Table 2 Cost and constraint parameters for optimization. Parameter
Description
Value
C MH CI ti Co Ra;max
Machine hour-rate Cost per cutting edge Time to change worn out insert Machine operator cost Maximum roughness
60 U/h 10 U 10 min 100 U/h 0.8 mm
3
Experimental set-up The used feedstock were steel bars made of 1.4125 (AISI440C, X105CrMo17) with an initial diameter of 7 mm. The longitudinal turning was performed with constant depth of cut of 0.3 mm. The bars were cut 6 times with a cutting length of 10 mm for each cut. This results in a final workpiece of length 10 mm and a final diameter of 3.4 mm. A Swiss-type CNC-Lathe (Deco Sigma 8 made by Tornos SA) was used. The machine is equipped with an automatic bar feeder Robobar SBF-210 from Tornos SA. As cooling fluid Blasomill 15 mineral oil from Blaser Swisslube AG with a constant flow rate of 5.8 l/min was used. The cutting insert type of VCGX-FN 120302 from Diametal AG, made of coated cemented carbide (Diametal specification: carbide M10/30 with coating D60, type number 388.948) with a corner radius of 0.2 mm was used. The tool was mounted in a 12 12 mm, right-hand tool holder of type SVAC 90 from Diametal AG. The roughness Ra of the machined parts was measured parallel to the axis of the cylinder using a MarSurf PS1 (from Mahr). The roughness values of all available parts were measured. In a conservative approach the average of the 5% highest roughness values was used for the on-machine optimization. For the simulated process the roughness measurement was simplified - only the roughness of the first part was measured which neglects the influence of wear. The tool life of the inserts was determined by maximum flank wear land width (VB) measurements using a Leica DCM3D microscope. The tool was considered worn out if VBmax>0.07 mm. In this study tool life and roughness measurements were conducted by an operator. The maximum cutting speed was limited at 80 m/min to avoid bar feeder vibrations. Based on recommendations from the tool manufacturer Diametal AG [20] the maximum feed rate was set to 0.1 mm/rev which is half the tip radius of the indexable insert per revolution. The minimal parameters (f min = 0.01 mm/rev and vc;min = 10 m/min) were selected to provide a wide range while still avoiding very slow experiments. Optimization implementation The optimization was implemented in MATLAB using the gpml library [21] for Gaussian process regression and [17] for constrained Bayesian optimization. Matérn 5/2 kernels with ARD (2) were used to model the covariance cost and roughness function because in Ref. [16] their suitability for turning applications was demonstrated. ARD uses different length scale parameters for cutting speed and feed rate because it is known that feed rate and cutting speed influence cost and roughness differently. As explained by Ref. [13], the marginal log-likelihood was maximized to select the hyperparameters of the GP model. Maximizing the marginal log-likelihood balances data fit and model complexity. The constrained expected improvement acquisition function (8) was maximized to determine the next experimental parameters as proposed by [17]. Fig. 2 summarizes the work-flow used for on-machine optimization. Data collected from the turning process was used to calculate the cost of the current experiment. These results were used to update the GP cost model. Roughness measurements of the workpiece were used to update the GP roughness model. The estimations of both GP models were used by the constrained Bayesian optimization algorithm to determine the next measurement point. This procedure was repeated until convergence or the maximum number of experiments was reached. Some experiments with slow cutting speed and slow feed rate took very long while already violating the required roughness constraints. To shorten the experiments they were
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Fig. 2. Work-flow of Bayesian optimization in turning.
stopped when the roughness value was already 20% higher than Ra;max and the share of cost to replace the insert was below 10% of the total production cost.
reached when the convergence limit e is fulfilled, the predicted optimal cutting speed varies within less than 0.5 mm/min over three iterations and the predicted feed rate can vary within less than 0.003 mm/rev over three iterations.
Convergence criterion Results The best experimental point is defined where the posterior mean cost function ðmC Þ is minimal and the posterior mean roughness function ðmRa ) fulfills the maximum allowed roughness (Ra;max ). ð12Þ x min ¼ argminmC x s:t mRa x < Ra;max The cost uncertainty of the estimated optimal parameters xmin is used as a measure of convergence, 2s c ðx minÞ < e
ð13Þ
where s c ðx minÞ is the standard deviation at the current best mean and e is the convergence limit. The model estimates that 95% of all samples conducted at the optimal values will be within mC ðx minÞ e. To improve the robustness of the convergence criterion, convergence can be only reached if three consecutive values are below the convergence limit. The value for the convergence limit should be set as close as possible to the combined process and measurement uncertainty because the uncertainty of the estimated cost and roughness cannot be minimized below this value as the convergence criterion cannot be met if the combined uncertainty is higher than the convergence limit. The convergence limit was set to 0.15 U which corresponds to 10% of an assumed final cost of 1.5 U. Full convergence can be
Simulated process optimization The suitability and convergence of the Bayesian optimization method was tested on a model, based on curve fitting of experimental data. Note that for these experiments only the first workpiece was used for roughness measurements. Fig. 3 shows the model fits for the tool life and the surface roughness. The model of the tool life T is based on curve fitting using the Taylor equation (14) [22]. The surface roughness Ra has a quadratic dependence on the feed rate [19] and is modelled according to Eq. (15). Cutting speed vc is in m/min, Tool life T in min, Feed rate f in mm/rev, and surface roughness Ra in mm. T ¼ 455735 v2:6685 c
ð14Þ
Ra ¼ 186:1f
ð15Þ
2
Fig. 4 shows the result of the Bayesian optimization method tested on the simulated process. Note that for the simulated process the models (14) and (15) are used to evaluate the tool life and the surface roughness – instead of experiments on the machine. To initialize the algorithm, the first two experiments are conducted at random points. The red line on Fig. 4 shows the
Fig. 3. Model of simulated turning process. Left: tool life as a function of cutting speed. Right: surface roughness as a function of feed rate.
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Fig. 4. Optimization of simulated process.
estimated optimal costs, and the shaded green surface shows the 95% confidence interval of the estimated optimal cost. The uncertainty is high in the beginning and decreases after additional experiments were available. After 18 iterations full convergence is reached. A comparison to a brute force grid search (dotted magenta line), where all parameter combinations for a grid size of 0.1 m/min for cutting speed and 0.001 mm/rev for feed rate were tested, confirmed the validity of the estimated optimal parameters. The blue line shows the current cost of the experiments. The peaks of the current value, such as the peak at the 11th iteration, indicate exploration of regions with high uncertainty (cost uncertainty before experiment was 4 U for these parameters), resulting in high experimental costs. Until the 9th iteration the uncertainty of the estimated optimal cost is high, and the optimal costs do not converge to the optimal costs from brute force search because the GP model is not very accurate. The GP model is updated after each iteration. At the 11th iteration the model improves significantly, resulting in an accurate prediction of the unknown objective function afterwards and in a decrease of the uncertainty. The inaccurate modelling of the unknown objective function in the beginning may happen for a low number of experiments because the model hyperparameters such as kernel length scale parameters, signal variance and noise variance were determined automatically based on maximizing the marginal log-likelihood. The maximum of the expected improvement function decreases overall with a sudden jump at iteration 10 due to the model improvement. On-machine optimization In a next step the optimization is tested on the lathe. In this case both the tool life and the surface roughness are determined experimentally. Figs. 5 and 6 show the results of the optimization. The optimization is again started with two random configurations of parameters. The roughness constraint is violated for both trials. The algorithm used the third and fourth experiments to explore edges of the design space. The third measurement point fulfills the surface roughness constraint but at high costs of 5.23 U. Most of the experiments between 5th and 12th iteration are already located near the estimated optimal parameters. Full convergence is reached after 11 iterations, corresponding to optimal cutting speed of 16.1 m/min and optimal feed rate of 0.026 mm/rev. The estimated optimal cutting speed is rather low because due to manual change of the inserts the costs to change an insert are assumed to be high (compare with Table 2). To validate the convergence criterion additional experiments were performed. The 12th experiment fulfilled the
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Fig. 5. On-machine optimization.
Fig. 6. Conducted experiments as a function of feed rate and cutting speed. Experiment with lowest cost was the 12th experiment.
roughness constraint and showed the lowest measured cost of 2.22 U. This measurement point is very close to the estimated optimal parameters. The algorithm used the experiments 13–16 to explore regions with high uncertainty (high feed rates and high cutting speeds). From the 9th to the 16th iteration the estimated optimal speed was within an interval of 0.5 m/min width and the estimated optimal feed was within an interval of 0.003 mm/rev width. It might be that the current measured cost is below the estimated minimal costs because the measured cost might be low, but the constraint is not fulfilled for this experiment and can therefore not be considered optimal (see for example iteration 13 in Fig. 5). Fig. 6 shows the experiments for which the constraint was fulfilled, and those, where it was exceeding the critical value Ra;max . The maximum of the expected improvement Eq. (8) also shows the convergence behavior. Fig. 7 shows the predicted mean and 95% confidence interval of roughness and cost from the on-machine cutting trials. At low cutting speeds the manufacturing costs are high because of the slow process, while at high cutting speeds the costs increase due to high
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Fig. 7. Predicted mean cost (top left), predicted mean roughness (top right), 95% confidence interval of cost (bottom left), and 95% confidence interval of roughness (bottom right) after 16 experiments.
wear. At high feed rates the costs decrease since more parts can be produced during the lifetime of the tool. The roughness depends on the feed rate and cutting speed. At cutting speeds between 20 m/min and 50 m/min the roughness increases due to built-up edge on the tool. The influence of the built-up edge is known in literature, see Ref. [19] for comparison. The highest cost and roughness uncertainties are reached in an area where the estimated mean roughness is much higher than the allowed roughness. The presented results demonstrate the applicability of Bayesian optimization for process set-up in turning. To further automate the process set-up a promising direction is to use a handling system to exchange workpiece and tool between the machine and the measurement equipment and connect the tool wear and roughness
measurement with the machine to exchange data. It might even be possible to measure roughness and tool wear on-line, such as demonstrated in Ref. [23] for roughness and in Ref. [24] for tool wear. The experimental data can also be used to estimate optimal process parameters for different system variables and constraint requirements (see Table 3). Only the time to change an insert, the machine operator cost, and the allowed surface roughness were changed – the other parameters were the same as those listed in Table 2.The first parameter set corresponds to the baseline parameters as specified in Table 2. First the cost parameters were changed assuming an automatic change of the insert instead of a manual change. In this case it was assumed that the machine operator cost is reduced from 100 U/h to 0 U/h and the time to change
Table 3 Estimated optimal process parameters for different constraint requirements and system variables. Cost & constraint parameters
Estimated optimal parameters
Estimated cost (U)
C O = 100 U/h ti = 10 min Ra;max = 0.8 mm C O = 0 U/h ti = 0.5 min Ra;max = 0.8 mm C O = 100 U/h ti = 10 min Ra;max = 1.2 mm
f = 0.026 mm/rev vc = 16.1 m/min
2.46 0.15
f = 0.071 mm/rev vc = 79.8 m/min
0.88 0.09
f = 0.037 mm/rev vc = 18.4 m/min
1.14 0.24
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an insert is reduced from 10 min to 30 s. The algorithm recommends a higher cutting speed which is expected, since according to the Taylor equation (14) tool wear increases with an increase in cutting speed and changing a worn-out tool is less costly for an automatic machine. The estimated optimal cutting speed is very close to the upper cutting speed limit of the optimization domain. Therefore, in this case the cutting speed is limited by bar feeder vibrations. Another scenario is the loosening of the maximum roughness constraint from 0.8 mm to 1.2 mm. In that case the algorithm recommends an increase in feed rate, in line with the increase of the kinematic surface roughness, as higher feed rates correspond to higher surface roughness according to the model in Eq. (15). Conclusion Optimization of process parameters in turning of new workpiece materials, new cutting inserts, new cutting oil and new machine tool with performing least experimental trials is a challenge in the manufacturing community. Constrained Bayesian optimization was successfully applied to determine the optimal process parameters in turning with only a few experimental trials. First, Bayesian optimization was demonstrated on a simulated turning process. After 18 iterations the estimated optimal cutting speed and feed rate converged to the brute force results with very low uncertainty, fulfilling the convergence criterion. These results clearly demonstrate the capability of Bayesian optimization to find optimal parameters in a limited number of experiments. Subsequently, optimizationwas performed on a micromachining lathe, with an optimization objective to minimize production costs considering tool life, machining time and required surface roughness, in addition to costs of insert, operator, and machine. After 11 iterations full convergence was reached and the Bayesian optimization algorithm found optimal cutting speed of 16.1 m/min and optimal feed rate of 0.026 mm/rev. The GP models of the cost and the roughness which are purely derived from measured data display physically plausible dependence on the optimization variables. The GP model of roughness has even captured the influence of build-up edge which is mostly neglected in simple models of roughness. This accurate capturing of physical phenomena, combined with the limited number of experiments, clearly shows the potential of Bayesian optimization for autonomous process set-up in turning. Based on the available experiments the algorithm has also proposed plausible process parameters for different constraint requirements and system variables which demonstrates the flexibility of Bayesian optimization. In the future it is planned to further improve the efficiency of Bayesian optimization. Promising directions of efficiency improvement are to incorporating prior knowledge from data measured at similar conditions and/or using prior knowledge from physical, stochastic and empirical models. Declaration of Competing Interest None.
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Bayesian optimization for autonomous process set-up in turning Markus Maiera,* , Ruben Zwickerb , Mansur Akbarib , Alisa Rupenyana,c , Konrad Wegenerb a
inspire AG, Technoparkstrasse 1, 8005 Zurich, Switzerland Institute of Machine Tools and Manufacturing (IWF), ETH Zurich, Leonhardstrasse 21, 8092 Zurich, Switzerland c Automatic Control Laboratory, ETH Zurich, Physikstrasse 3, 8092 Zurich, Switzerland b
A R T I C L E I N F O
A B S T R A C T
Article history: Available online xxx
Constrained Bayesian optimization with Gaussian process models is applied to optimize the turning of 1.4125 steel bars by considering tool life, machining time, and surface roughness, in addition to costs of insert, operator, and machine. Feed rate and cutting speed are the optimized process parameters with constant depth of cut, both for a simulated process and for on-machine experiments on a micromachining lathe. The results demonstrate that even without prior knowledge of the process, this method successfully recommends physically reasonable process parameters within a limited number of experiments. The flexibility of the method is demonstrated by providing optimized process parameters for various cost parameters and various requirements for the constraints, based on experimental data. © 2019 CIRP.
Keywords: Turning Process optimization Bayesian optimization Process set-up Gaussian process models Manufacturing Machine tools
Introduction The selection of optimal process parameters in turning is crucial because it directly influences the cost efficiency and the workpiece quality. Process parameters, such as feed rate, cutting speed, and depth of cut are normally selected carefully based on previous knowledge about the process, coolant, machine, cutting tool and response of the workpiece in machining or in the traditional way by experienced operators. The initial process parameters are adapted depending on the required quality of the workpiece, tool wear, machine tool capability, manufacturing time and process requirements. In particular, for the turning process in micromachining applications, the final product quality is of high importance and small variations in the process parameters can lead to unacceptable products. Finding the right combination of parameters requires specific expertise, or sometimes brute-force search and long series of experimental trials. Automatic approaches can optimize the performance of a given sequence of operations to manufacture a desired workpiece by selecting the optimal or most feasible ranges of process parameters. Table 1 shows an overview of existing optimization methods and objectives in turning. A detailed description of turning process optimization can be found in Ref. [1]. In general, the optimization objective can be reformulated as a cost minimization problem subjected to constraints. Formally the optimized values x can be
* Corresponding author. E-mail address: [email protected] (M. Maier).
determined as the arguments of the minima (argmin) of total production costs C T ðx Þ subjected to (s.t.) constraints cðx Þ: ð1Þ x ¼ arg min C T x ; s:t: c x cmax x2x
where x are the optimization parameters, cmax are the maximum allowed constraints and x is the design space of machining parameters. Several methods exist to optimize turning processes. Empirical model-based approaches such as those used in Refs. [2,3] for optimization are suitable for specific optimization tasks where the experimental set-up is fixed. However, these methods have difficulties to optimize set-ups outside of the trained scope such as new workpiece, tool and machine combinations. Experimental design methods, such as the Taguchi method used in Refs. [4–6], are applicable for the optimization of new setups. The Taguchi method uses fractional factorial design test plans created with orthogonal arrays to investigate the influence of different variables at specified levels (see Ref. [7] for a detailed introduction to the Taguchi method). The choice of the number of levels specifies the shape of the unknown function e.g. specifying two levels implies a linear relation. The main disadvantage of the Taguchi method is that the experimental parameters must be defined before the optimization — knowledge gained during the tests is ignored. A powerful global optimization method is Bayesian optimization [8]. It is a sequential design strategy for optimizing an a priori unknown objective function that is expensive to evaluate, such as a time-consuming physical measurement, or a complex simulation [9]. Bayesian optimization is used in various applications such as hyperparameter tuning [10], photovoltaic power plants [11], and
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Table 1 State of the art optimization methods and objectives in turning. Source
Method
Objective
Wang et al. [2] Vijayakumar et al. [3] Yang and Tarng [4] Nian et al. [5] Nalbant et al. [6]
Genetic algorithm Ant colony algorithm Taguchi method Taguchi method Taguchi method
Objective function including surface roughness, cutting force, tool-life, material removal rate and chip breakability Cost function including cutting cost by actual time, machine idling cost, tool replacement cost and tool cost Optimization of tool life and surface roughness Optimization of tool life, cutting force and surface finish Optimization of surface roughness
robotics [12]. It is common to combine Bayesian optimization with Gaussian process (GP) models which are used as surrogate models of the unknown objective and constraint functions. The GP model uses available samples to provide an estimation of the cost function at new parameter configurations [13]. The mean and the uncertainty estimation of the GP model are used in Bayesian optimization to select sample points were experiments are highly informative and beneficial for the optimization [9]. In turning processes, Bayesian inference has been successfully used in the prediction of tool life [14,15]. In Ref. [16], Bayesian optimization has been successfully applied to process parameter optimization of a simulated turning process, and it is demonstrated that for a simulated turning process, the constrained Bayesian optimization [17], where GP model estimations of the cost and constraint function are used, performs better than Bayesian optimization without constraints. In this work suitability and convergence of Bayesian optimization is investigated on a simulated turning process. Afterwards an on-machine process parameter optimization is demonstrated. Gaussian process models and bayesian optimization In this section, Gaussian processes based on Ref. [13], and Bayesian optimization as proposed in Refs. [9,18] are introduced. Rasmussen and Williams [13] define Gaussian processes as follows: “A Gaussian process is a collection of random variables, any finite number of which have a joint Gaussian distribution”. The GP model is fully specified with its mean mðx Þ and covariance 0 function k x ; x .The covariance function, often called kernel, specifies the smoothness of the modeled function. Several different kernel functions exist. A detailed overview can be found in Ref. [13]. Common kernels are exponential and Matérn kernels. Exponential kernels model a very smooth function. For Matérn kernels the parameter n determines the smoothness of the kernel – a higher n leads to a smoother function. In this study Matérn kernels with n = 5/2 and automatic relevance determination (ARD) were used. ARD uses different kernel length scale parameters for each dimension of the input data. The used kernel can be calculated as: pffiffiffi pffiffiffi 5 0 kM ðx; x Þ ¼ s 2f 1 þ 5r þ r2 expð 5rÞ ð2Þ 3
T
s 2t x ¼ k x; x k x
ðK þ s 2N I Þ1 k x
ð4Þ
where I is the identity matrix, the vector k ðx Þ contains covariance terms between x and x 1 : t and the covariance matrix K is defined as: 0 1 kðx 1; x 1Þ kðx 1; x tÞ B C .. .. K ¼@ ð5Þ A . } . kðx t; x 1Þ
kðx t; x tÞ
In this work it was assumed that the Gaussian process mean function is zero. In this way no prior knowledge is given to the GP model. Fig. 1 shows the working principle of Bayesian optimization. The estimated mean and variance function of the GP models are used to optimize an acquisition function, which is used as a cheaper way to evaluate the unknown objective. The acquisition function needs to balance the discovery of regions with high model uncertainty (exploration) and regions with high model prediction values (exploitation). The next experiment is conducted for parameters where the acquisition function reaches a maximum. Afterwards the data from the experiments is used to calculate an updated GP model. This procedure is repeated until a stopping criterion is reached. Various acquisition functions can be used for the optimization of the cost and to determine where to conduct the next
where s 2f is the signal variance, the distance r between x and x is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ffi x x0 P 1 x x0 ,P ¼ diagðs 21 ; s 22 ; . . . ; s 2D ) is a diagr¼ 0
onal matrix with length scale parameters s 21 ; s 22 ; . . . ; s 2D , and D is the dimension of the input space. Based on the GP model and available pairs of measurements ~ 0; s 2 ; the mean ðx 1 : t; y 1 : tÞ corrupted with Gaussian noise eN N
mt and variance s 2t of the joint distribution of the observed target
values and the function values at the test locations can be calculated for unknown test inputs x , using the following predictive equations for GP regression:
T
mt x ¼ k x
ðK þ s 2N I Þ1 y 1 : t
ð3Þ
Fig. 1. Working principle of Bayesian optimization.
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experimental evaluation (see Ref. [9] for a detailed summary). A common acquisition function is the expected improvement (aEI Þ: F ðZ Þ þ s c x fðZ Þ ð6Þ aEI ðxÞ ¼ y xþ mc x
Z¼
mc ðx Þ yðxþ Þ s c ðx Þ
ð7Þ
where y xþ is the best observed value, mc ðx Þ is the predicted
mean cost calculated using (3), s 2c ðx Þ is the predicted cost variance calculated using (4), FðZÞ is the cumulative distribution function of pffiffiffiffiffiffiffi R Z 2 a standard normal distribution F ðZ Þ ¼ 1= 2p 1 et =2 dt and fðZÞ is the probability density function of a standard normal distribupffiffiffiffiffiffiffi 2 tion fðZ Þ ¼ 1= 2p eZ =2 . Gartner et al. [17] extended the expected improvement acquisition function to constraint optimization which was used in this study. ð8Þ aEIC ðxÞ ¼ pf x aEI ðxÞ aEI ðx Þ is the expected improvement without constraints (6) and pf ðx Þ is the probability that x is feasible. The probability that x is feasible can be calculated as follows: pf x ¼
1 pffiffiffiffiffiffiffi s 2p
Z l e
ðtmcon Þ2 2s 2 con
ð9Þ
dt
1
where l is the maximum constraint value. The predicted mean mcon and variance s 2con of the constraint are calculated using (3) and (4). Methodology Cost function The optimization task in this study is to produce parts with minimal total production costs C T (U) and produce workpieces which fulfill the roughness constraint Ra;max : ð10Þ x ¼ arg min C T x ; s:t: Ra x < Ra;max x2x
where x is a vector of the optimized parameters cutting speed vc and feed rate f . The cost unit is assumed to be Swiss francs but for the sake of universality is denoted as U. In this study the manufacturing cost function was assumed to be: C I þ ti ðC MH þ C o Þ ð11Þ C T ðxÞ ¼ tc ðxÞ C MH þ Tðx Þ where tc is the cutting time and T is the tool life (see Ref. [19] for a detailed introduction to cost calculations in manufacturing). The total cost combines cost of machining time and tool replacement cost. The used cost and constraint parameters are listed in Table 2.
Table 2 Cost and constraint parameters for optimization. Parameter
Description
Value
C MH CI ti Co Ra;max
Machine hour-rate Cost per cutting edge Time to change worn out insert Machine operator cost Maximum roughness
60 U/h 10 U 10 min 100 U/h 0.8 mm
3
Experimental set-up The used feedstock were steel bars made of 1.4125 (AISI440C, X105CrMo17) with an initial diameter of 7 mm. The longitudinal turning was performed with constant depth of cut of 0.3 mm. The bars were cut 6 times with a cutting length of 10 mm for each cut. This results in a final workpiece of length 10 mm and a final diameter of 3.4 mm. A Swiss-type CNC-Lathe (Deco Sigma 8 made by Tornos SA) was used. The machine is equipped with an automatic bar feeder Robobar SBF-210 from Tornos SA. As cooling fluid Blasomill 15 mineral oil from Blaser Swisslube AG with a constant flow rate of 5.8 l/min was used. The cutting insert type of VCGX-FN 120302 from Diametal AG, made of coated cemented carbide (Diametal specification: carbide M10/30 with coating D60, type number 388.948) with a corner radius of 0.2 mm was used. The tool was mounted in a 12 12 mm, right-hand tool holder of type SVAC 90 from Diametal AG. The roughness Ra of the machined parts was measured parallel to the axis of the cylinder using a MarSurf PS1 (from Mahr). The roughness values of all available parts were measured. In a conservative approach the average of the 5% highest roughness values was used for the on-machine optimization. For the simulated process the roughness measurement was simplified - only the roughness of the first part was measured which neglects the influence of wear. The tool life of the inserts was determined by maximum flank wear land width (VB) measurements using a Leica DCM3D microscope. The tool was considered worn out if VBmax>0.07 mm. In this study tool life and roughness measurements were conducted by an operator. The maximum cutting speed was limited at 80 m/min to avoid bar feeder vibrations. Based on recommendations from the tool manufacturer Diametal AG [20] the maximum feed rate was set to 0.1 mm/rev which is half the tip radius of the indexable insert per revolution. The minimal parameters (f min = 0.01 mm/rev and vc;min = 10 m/min) were selected to provide a wide range while still avoiding very slow experiments. Optimization implementation The optimization was implemented in MATLAB using the gpml library [21] for Gaussian process regression and [17] for constrained Bayesian optimization. Matérn 5/2 kernels with ARD (2) were used to model the covariance cost and roughness function because in Ref. [16] their suitability for turning applications was demonstrated. ARD uses different length scale parameters for cutting speed and feed rate because it is known that feed rate and cutting speed influence cost and roughness differently. As explained by Ref. [13], the marginal log-likelihood was maximized to select the hyperparameters of the GP model. Maximizing the marginal log-likelihood balances data fit and model complexity. The constrained expected improvement acquisition function (8) was maximized to determine the next experimental parameters as proposed by [17]. Fig. 2 summarizes the work-flow used for on-machine optimization. Data collected from the turning process was used to calculate the cost of the current experiment. These results were used to update the GP cost model. Roughness measurements of the workpiece were used to update the GP roughness model. The estimations of both GP models were used by the constrained Bayesian optimization algorithm to determine the next measurement point. This procedure was repeated until convergence or the maximum number of experiments was reached. Some experiments with slow cutting speed and slow feed rate took very long while already violating the required roughness constraints. To shorten the experiments they were
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Fig. 2. Work-flow of Bayesian optimization in turning.
stopped when the roughness value was already 20% higher than Ra;max and the share of cost to replace the insert was below 10% of the total production cost.
reached when the convergence limit e is fulfilled, the predicted optimal cutting speed varies within less than 0.5 mm/min over three iterations and the predicted feed rate can vary within less than 0.003 mm/rev over three iterations.
Convergence criterion Results The best experimental point is defined where the posterior mean cost function ðmC Þ is minimal and the posterior mean roughness function ðmRa ) fulfills the maximum allowed roughness (Ra;max ). ð12Þ x min ¼ argminmC x s:t mRa x < Ra;max The cost uncertainty of the estimated optimal parameters xmin is used as a measure of convergence, 2s c ðx minÞ < e
ð13Þ
where s c ðx minÞ is the standard deviation at the current best mean and e is the convergence limit. The model estimates that 95% of all samples conducted at the optimal values will be within mC ðx minÞ e. To improve the robustness of the convergence criterion, convergence can be only reached if three consecutive values are below the convergence limit. The value for the convergence limit should be set as close as possible to the combined process and measurement uncertainty because the uncertainty of the estimated cost and roughness cannot be minimized below this value as the convergence criterion cannot be met if the combined uncertainty is higher than the convergence limit. The convergence limit was set to 0.15 U which corresponds to 10% of an assumed final cost of 1.5 U. Full convergence can be
Simulated process optimization The suitability and convergence of the Bayesian optimization method was tested on a model, based on curve fitting of experimental data. Note that for these experiments only the first workpiece was used for roughness measurements. Fig. 3 shows the model fits for the tool life and the surface roughness. The model of the tool life T is based on curve fitting using the Taylor equation (14) [22]. The surface roughness Ra has a quadratic dependence on the feed rate [19] and is modelled according to Eq. (15). Cutting speed vc is in m/min, Tool life T in min, Feed rate f in mm/rev, and surface roughness Ra in mm. T ¼ 455735 v2:6685 c
ð14Þ
Ra ¼ 186:1f
ð15Þ
2
Fig. 4 shows the result of the Bayesian optimization method tested on the simulated process. Note that for the simulated process the models (14) and (15) are used to evaluate the tool life and the surface roughness – instead of experiments on the machine. To initialize the algorithm, the first two experiments are conducted at random points. The red line on Fig. 4 shows the
Fig. 3. Model of simulated turning process. Left: tool life as a function of cutting speed. Right: surface roughness as a function of feed rate.
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Fig. 4. Optimization of simulated process.
estimated optimal costs, and the shaded green surface shows the 95% confidence interval of the estimated optimal cost. The uncertainty is high in the beginning and decreases after additional experiments were available. After 18 iterations full convergence is reached. A comparison to a brute force grid search (dotted magenta line), where all parameter combinations for a grid size of 0.1 m/min for cutting speed and 0.001 mm/rev for feed rate were tested, confirmed the validity of the estimated optimal parameters. The blue line shows the current cost of the experiments. The peaks of the current value, such as the peak at the 11th iteration, indicate exploration of regions with high uncertainty (cost uncertainty before experiment was 4 U for these parameters), resulting in high experimental costs. Until the 9th iteration the uncertainty of the estimated optimal cost is high, and the optimal costs do not converge to the optimal costs from brute force search because the GP model is not very accurate. The GP model is updated after each iteration. At the 11th iteration the model improves significantly, resulting in an accurate prediction of the unknown objective function afterwards and in a decrease of the uncertainty. The inaccurate modelling of the unknown objective function in the beginning may happen for a low number of experiments because the model hyperparameters such as kernel length scale parameters, signal variance and noise variance were determined automatically based on maximizing the marginal log-likelihood. The maximum of the expected improvement function decreases overall with a sudden jump at iteration 10 due to the model improvement. On-machine optimization In a next step the optimization is tested on the lathe. In this case both the tool life and the surface roughness are determined experimentally. Figs. 5 and 6 show the results of the optimization. The optimization is again started with two random configurations of parameters. The roughness constraint is violated for both trials. The algorithm used the third and fourth experiments to explore edges of the design space. The third measurement point fulfills the surface roughness constraint but at high costs of 5.23 U. Most of the experiments between 5th and 12th iteration are already located near the estimated optimal parameters. Full convergence is reached after 11 iterations, corresponding to optimal cutting speed of 16.1 m/min and optimal feed rate of 0.026 mm/rev. The estimated optimal cutting speed is rather low because due to manual change of the inserts the costs to change an insert are assumed to be high (compare with Table 2). To validate the convergence criterion additional experiments were performed. The 12th experiment fulfilled the
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Fig. 5. On-machine optimization.
Fig. 6. Conducted experiments as a function of feed rate and cutting speed. Experiment with lowest cost was the 12th experiment.
roughness constraint and showed the lowest measured cost of 2.22 U. This measurement point is very close to the estimated optimal parameters. The algorithm used the experiments 13–16 to explore regions with high uncertainty (high feed rates and high cutting speeds). From the 9th to the 16th iteration the estimated optimal speed was within an interval of 0.5 m/min width and the estimated optimal feed was within an interval of 0.003 mm/rev width. It might be that the current measured cost is below the estimated minimal costs because the measured cost might be low, but the constraint is not fulfilled for this experiment and can therefore not be considered optimal (see for example iteration 13 in Fig. 5). Fig. 6 shows the experiments for which the constraint was fulfilled, and those, where it was exceeding the critical value Ra;max . The maximum of the expected improvement Eq. (8) also shows the convergence behavior. Fig. 7 shows the predicted mean and 95% confidence interval of roughness and cost from the on-machine cutting trials. At low cutting speeds the manufacturing costs are high because of the slow process, while at high cutting speeds the costs increase due to high
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Fig. 7. Predicted mean cost (top left), predicted mean roughness (top right), 95% confidence interval of cost (bottom left), and 95% confidence interval of roughness (bottom right) after 16 experiments.
wear. At high feed rates the costs decrease since more parts can be produced during the lifetime of the tool. The roughness depends on the feed rate and cutting speed. At cutting speeds between 20 m/min and 50 m/min the roughness increases due to built-up edge on the tool. The influence of the built-up edge is known in literature, see Ref. [19] for comparison. The highest cost and roughness uncertainties are reached in an area where the estimated mean roughness is much higher than the allowed roughness. The presented results demonstrate the applicability of Bayesian optimization for process set-up in turning. To further automate the process set-up a promising direction is to use a handling system to exchange workpiece and tool between the machine and the measurement equipment and connect the tool wear and roughness
measurement with the machine to exchange data. It might even be possible to measure roughness and tool wear on-line, such as demonstrated in Ref. [23] for roughness and in Ref. [24] for tool wear. The experimental data can also be used to estimate optimal process parameters for different system variables and constraint requirements (see Table 3). Only the time to change an insert, the machine operator cost, and the allowed surface roughness were changed – the other parameters were the same as those listed in Table 2.The first parameter set corresponds to the baseline parameters as specified in Table 2. First the cost parameters were changed assuming an automatic change of the insert instead of a manual change. In this case it was assumed that the machine operator cost is reduced from 100 U/h to 0 U/h and the time to change
Table 3 Estimated optimal process parameters for different constraint requirements and system variables. Cost & constraint parameters
Estimated optimal parameters
Estimated cost (U)
C O = 100 U/h ti = 10 min Ra;max = 0.8 mm C O = 0 U/h ti = 0.5 min Ra;max = 0.8 mm C O = 100 U/h ti = 10 min Ra;max = 1.2 mm
f = 0.026 mm/rev vc = 16.1 m/min
2.46 0.15
f = 0.071 mm/rev vc = 79.8 m/min
0.88 0.09
f = 0.037 mm/rev vc = 18.4 m/min
1.14 0.24
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an insert is reduced from 10 min to 30 s. The algorithm recommends a higher cutting speed which is expected, since according to the Taylor equation (14) tool wear increases with an increase in cutting speed and changing a worn-out tool is less costly for an automatic machine. The estimated optimal cutting speed is very close to the upper cutting speed limit of the optimization domain. Therefore, in this case the cutting speed is limited by bar feeder vibrations. Another scenario is the loosening of the maximum roughness constraint from 0.8 mm to 1.2 mm. In that case the algorithm recommends an increase in feed rate, in line with the increase of the kinematic surface roughness, as higher feed rates correspond to higher surface roughness according to the model in Eq. (15). Conclusion Optimization of process parameters in turning of new workpiece materials, new cutting inserts, new cutting oil and new machine tool with performing least experimental trials is a challenge in the manufacturing community. Constrained Bayesian optimization was successfully applied to determine the optimal process parameters in turning with only a few experimental trials. First, Bayesian optimization was demonstrated on a simulated turning process. After 18 iterations the estimated optimal cutting speed and feed rate converged to the brute force results with very low uncertainty, fulfilling the convergence criterion. These results clearly demonstrate the capability of Bayesian optimization to find optimal parameters in a limited number of experiments. Subsequently, optimizationwas performed on a micromachining lathe, with an optimization objective to minimize production costs considering tool life, machining time and required surface roughness, in addition to costs of insert, operator, and machine. After 11 iterations full convergence was reached and the Bayesian optimization algorithm found optimal cutting speed of 16.1 m/min and optimal feed rate of 0.026 mm/rev. The GP models of the cost and the roughness which are purely derived from measured data display physically plausible dependence on the optimization variables. The GP model of roughness has even captured the influence of build-up edge which is mostly neglected in simple models of roughness. This accurate capturing of physical phenomena, combined with the limited number of experiments, clearly shows the potential of Bayesian optimization for autonomous process set-up in turning. Based on the available experiments the algorithm has also proposed plausible process parameters for different constraint requirements and system variables which demonstrates the flexibility of Bayesian optimization. In the future it is planned to further improve the efficiency of Bayesian optimization. Promising directions of efficiency improvement are to incorporating prior knowledge from data measured at similar conditions and/or using prior knowledge from physical, stochastic and empirical models. Declaration of Competing Interest None.
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