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Continuous identification for mechanistic force models in milling
9th IFAC Conference on Manufacturing Modelling, Management and 9th IFAC 9th IFAC Conference Conference on on Manufacturing Manufacturing Modelling, Modelling, Management Management and and Control 9th IFAC Conference on Manufacturing Modelling, Management and Control Available online at www.sciencedirect.com Control Berlin, Germany, August 28-30, 2019 9th IFAC Conference on Manufacturing Modelling, Management and Control Berlin, Germany, August 28-30, 2019 Berlin, Germany, August 28-30, 2019 Control Berlin, Germany, August 28-30, 2019 Berlin, Germany, August 28-30, 2019
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IFAC PapersOnLine 52-13 (2019) 1791–1796
Continuous identification for mechanistic Continuous identification for mechanistic Continuous identification for mechanistic Continuous identification for mechanistic force models in milling Continuous identification for mechanistic force models in milling force models in milling force models in milling models in milling ∗ ∗∗ ∗∗ ∗ ∗∗ M. Schwenzerforce ∗ S. Stemmler ∗∗ M. Ay ∗∗ T. Bergs ∗ D. Abel ∗∗
M. T. Bergs ∗ D. Abel ∗∗ M. Schwenzer Schwenzer ∗∗ S. S. Stemmler Stemmler ∗∗ M. M. Ay Ay ∗∗ ∗∗ T. Bergs ∗ D. Abel ∗∗ M. Schwenzer ∗ S. Stemmler ∗∗ ∗∗ M. Ay ∗∗ T. Bergs ∗ D. Abel ∗∗ ∗ M. Schwenzer S. Stemmler Ay T.Engineering Bergs D. (WZL), Abel for Machine Tools andM. Production ∗ Laboratory ∗ Laboratory for Machine Tools and Production Engineering (WZL), Laboratory for Machine Tools and Production Engineering (WZL), Aachen University, Campus-Boulevard 30, 52074 Aachen, ∗ RWTH Laboratory for Machine Tools and Production Engineering (WZL), Aachen University, Campus-Boulevard 30, ∗ RWTH RWTH Aachen University, Campus-Boulevard 30, 52074 52074 Aachen, Aachen, Laboratory for Machine and Production Engineering (WZL), Germany (e-mail: Tools [email protected]). RWTH Aachen University, Campus-Boulevard 30, 52074 Aachen, Germany (e-mail: [email protected]). ∗∗ Germany (e-mail: [email protected]). RWTH Aachen Campus-Boulevard 30, 52074 Aachen, of University, Automatic Control, RWTH Aachen University, ∗∗ Institute Germany (e-mail: [email protected]). ∗∗ Institute of Automatic Control, RWTH Aachen University, of (e-mail: Automatic Control, RWTH Aachen University, Germany [email protected]). Campus-Boulevard 30, 52074 Aachen, Germany ∗∗ Institute of Automatic Control, RWTH Aachen University, Campus-Boulevard 30, Aachen, Germany ∗∗ Institute Campus-Boulevard 30, 52074 52074 Aachen, Germany Institute of Automatic Control, RWTH Aachen University, Campus-Boulevard 30, 52074 Aachen, Germany Campus-Boulevard 30, 52074 Aachen, Germany Abstract: Force determines the product quality, the productivity and the safety of a milling Abstract: Force the quality, the and safety a milling Abstract: Force determines determines the product product the productivity productivity and the the safety of of milling process. Mechanistic force models are the quality, key to understand, optimize or control theaa cutting Abstract: Force determines the product quality, the productivity and the safety of milling process. Mechanistic force models are the key to understand, optimize or control the cutting process. Mechanistic force models are chip the quality, key to understand, optimize orcoefficients control the cutting Abstract: Force determines the product the productivity and the safety of a milling They combine the undeformed parameter with empiric tuning in a grayprocess. They Mechanistic force models are chip the key to understand, optimize orcoefficients control the cutting combine the undeformed parameter with empiric tuning in aa grayprocess. They combineforce the undeformed chip parameter with empiric tuningorcoefficients in cutting grayMechanistic models are the key to understand, optimize control the box model. Identifying those coefficients is costly in both, time and number of experiments. This process. They combine the undeformed parameter with empiric tuning coefficients in a graybox model. model. Identifying those coefficientschip is costly costly in both, both, time and number number of experiments. experiments. This box Identifying those coefficients is in time and of This process. They combine the undeformed chip parameter with empiric tuning coefficients in a graypaper introduces two recursive identification methods for force model identification: recursive box model. Identifying those coefficients is costly in both, time and number of experiments. This paper introduces two recursive identification methods for force model identification: recursive paper introduces two recursive identification force model recursive box model. Identifying thoseKalman coefficients is costly in both, time and number of experiments. This least squares and ensemble filters. Themethods model isfor nonlinear. The identification: ensemble Kalman filter paper introduces two recursive identification methods for force model identification: recursive least squares and ensemble Kalman filters. The model is nonlinear. The ensemble Kalman filter least squares and ensemble Kalman filters. The model isfor nonlinear. The ensemble Kalman filter paper introduces two recursive identification methods force model identification: recursive shows an extraordinary robustness against measurement noise and a fast convergence time – least squares and ensemble Kalman filters. The model is nonlinear. The ensemble Kalman filter shows an robustness against measurement noise aa fast convergence time –– shows an extraordinary extraordinary robustness againstThe measurement noise and andThe fast convergence time least squares and ensemble Kalman filters. model is nonlinear. ensemble Kalman filter depending on the selection of the ensemble size and the measurement noise. The recursive least shows an extraordinary robustness against size measurement noise and anoise. fast convergence time depending on the the selection selection of the the ensemble ensemble and the the measurement measurement The recursive recursive least– depending on of size and The least shows an robustness against measurement noise and anoise. fast convergence time – squares fitextraordinary serves as a benchmark but is highly sensitive to measurement noise. It isrecursive the first time depending on theas selection of the but ensemble size and theto measurement noise. The least squares fit serves a benchmark is highly sensitive measurement noise. It is the first time squares fit serves as a benchmark but is highly sensitive to measurement noise. It isrecursive theCopyright first least time depending on the selection of the ensemble size and the measurement noise. The that a continuous identification is examined for mechanistic force models in milling. squares fit serves asidentification a benchmarkisbut is highlyfor sensitive to measurement noise. It is theCopyright first time that a continuous examined mechanistic force in that examined mechanistic force models models in milling. milling. squares fit serves asidentification a benchmarkis is highlyfor sensitive to measurement noise. It is theCopyright first time c aa continuous 2019 IFAC that continuous identification isbut examined for mechanistic force models in milling. Copyright c 2019 IFAC c 2019 IFAC that a continuous identification is examined for mechanistic force models in milling. Copyright © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. c 2019, IFAC 2019 c 2019 IFAC Keywords: Ensemble Kalman filter, Recursive least squares, Manufacturing, Milling, Keywords: Ensemble Kalman filter, least squares, Manufacturing, Keywords: Ensemble Kalman filter, Recursive Recursive leastmodel squares, Manufacturing, Milling, Milling, Parameter identification, Identification, nonlinear identification Keywords: Ensemble Kalman filter, Recursive leastmodel squares, Manufacturing, Milling, Parameter identification, Identification, nonlinear identification Parameter identification, Identification, nonlinear model identification Keywords: Ensemble Kalman filter, Recursive least squares, Manufacturing, Milling, Parameter identification, Identification, nonlinear model identification Parameter identification, Identification, nonlinearan model identification 1. INTRODUCTION advantage regarding robustness. This work reduces the 1. an robustness. This work reduces the 1. INTRODUCTION INTRODUCTION an advantage advantage regarding regarding This work reduces and the identification effort androbustness. enables for more accurate 1. INTRODUCTION an advantage regarding robustness. This work reduces and the identification effort and enables for more accurate identification effort androbustness. enables for more accurate and 1. INTRODUCTION an advantage regarding This work reduces the Milling is a flexible and highly dynamic manufacturing more adaptive force models. identification effort and enables for more accurate and Milling is a flexible and highly dynamic manufacturing more adaptive force models. Milling to is produce a flexiblealmost and highly dynamic manufacturing more adaptiveeffort force models. identification and enables for more accurate and process arbitrary free-form surfaces. A Milling to is produce a flexiblealmost and highly dynamic manufacturing adaptive force models. process arbitrary free-form surfaces. A process to produce arbitrary free-form surfaces. A more Milling is intermittent a flexiblealmost and highly dynamic manufacturing 2. STATE models.OF THE ART cyclically cutting process takes place, which process to produce almost arbitrary free-form surfaces. A more adaptive force 2. cyclically intermittent cutting process takes place, which 2. STATE STATE OF OF THE THE ART ART cyclically intermittent cutting process takes place, which process to produce almost arbitrary free-form surfaces. A is overlaid with a translatory feed movement. This is 2. STATE OF THE ART cyclically intermittent cutting process takes place,This which is overlaid with aa translatory feed movement. is is overlaid with translatory feed movement. This is 2.can STATE OF THE ART different model In general, one distinguish between cyclically intermittent cutting process takes place, which characterized by a continuously varying thickness of the is overlaid with translatory feed movement. This is In thickness of one distinguish between different model characterized by aaa continuously varying In general, general, one can can distinguish between different characterized by continuously varying thickness of the the types in cutting technology (Ehmann et al., 1997): model is overlaid with a translatory feed movement. This is removed chip and with a varying force. Force is In general, one can distinguish between different model characterized by a continuously varying thickness of the types in cutting technology (Ehmann et al., 1997): removed chip and with a varying force. Force is in cutting technology (Ehmann et al., 1997): model removed chip by and with since a varying force. Force is the types In general, one can distinguish between different characterized a continuously varying thickness of crucial process parameter it affects the process safety types in cutting technology (Ehmann et al., 1997): removed chip and with a varying force. Force is the since it the safety • empirical (black-box), crucial process crucial process parameter it affects affects the process process safety in cutting(black-box), technology (Ehmann et al., 1997): removed chip parameter and with since a quality varying force. Force is the types (tool workpiece (geometric tolerances • empirical empirical (black-box), crucialbreakage), process parameter since it affects the process safety (tool breakage), workpiece quality (geometric tolerances •• analytical (white-box – only physical phenomena), (tool breakage), workpiece quality (geometric tolerances empirical (black-box), crucial process parameter since it affects the process safety through the deflection of the tool), and productivity (feed analytical (white-box • and analytical (white-box –– only only physical physical phenomena), phenomena), (tool breakage), workpiece quality (geometric tolerances empirical tool), and productivity (feed through the deflection of the through the models deflection of the tool), and productivity (feed • and analytical (black-box), (white-box – only physical phenomena), (tool breakage), workpiece quality (geometric tolerances rate). Force are the key to understand, optimize or and through the models deflection of the tool), and productivity (feed analytical (white-box – only physical phenomena), • mechanistic (gray-box relate to the chip cross secrate). Force are the key to understand, optimize or rate). Force areof the key to understand, optimize or and through the models deflection the tool), and productivity (feed even control the milling process. •• mechanistic (gray-box mechanistic (gray-box –– relate relate to to the the chip chip cross cross secsecrate).control Force models are the key to understand, optimize or and tion). even the milling process. even control the milling process. • tion). mechanistic (gray-box – relate to the chip cross secrate). Force models are the key to understand, optimize or tion). even control the milling process. • mechanistic (gray-box – relate to the chip cross secOnly recentlytheadvanced control methods are being aption). even control milling process. Usually mechanistic models are used due to their simOnly recently advanced control methods are apOnly recently advanced control methods are being being ap- Usually tion). plied to control the force in milling: adaptive control mechanistic models are used due to their simUsuallylow mechanistic models areand used dueinterpretability. to their simOnly recently advanced control methods are being ap- plicity, computational cost, their plied to the in milling: adaptive control plied to control control the force force in adaptive control Usuallylow mechanistic models areand used dueinterpretability. to their simOnly advanced control methods are being ap- plicity, (Altintas and Aslan, 2017) andmilling: model-based predictive computational cost, their plicity, low computational cost, and their interpretability. plied recently to control the force in milling: adaptive control Usually mechanistic models are used due to fundamentheir simMechanistic force models relate the force to the (Altintas and Aslan, 2017) and model-based predictive (Altintas and Aslan, 2017)etin and model-based predictive plicity, low computational cost, the andforce theirtointerpretability. plied to control the force milling: adaptive control control (MPC) (Stemmler al., 2017). Both use a semiMechanistic force models relate the Mechanistic force models relate the force tointerpretability. the fundamenfundamen(Altintas(MPC) and Aslan, 2017)et and model-based predictive plicity, low computational cost, and their tal undeformed chip parameters. The coefficients tune the control (Stemmler al., 2017). Both use a semicontrol (MPC) (Stemmler al., 2017). Both a semi- tal Mechanistic force models relate the force to the fundamenundeformed chip parameters. The coefficients tune the (Altintas Aslan, 2017)et model-based predictive empirical –and so-called “mechanistic”– model of use the process tal undeformed chip parameters. The coefficients tunetool, the control (MPC) (Stemmler et and al., 2017). Both use a semi- models Mechanistic force models relate the force to the fundamento a specific material of workpiece, of the empirical – so-called “mechanistic”– model of the process empirical – so-called “mechanistic”– model oforuse theeliminate process tal undeformed chip material parameters. The coefficients tunetool, the control (MPC) (Stemmler et al., 2017). Both a semiforce to reduce (Altintas and Aslan, 2017) models to a specific of the workpiece, of the models to a specific material of the workpiece, of the tool, empirical – so-called “mechanistic”– model of the process tal undeformed chip parameters. The coefficients tune the and to the tool geometry (Gradiek etworkpiece, al., 2004; Wang ettool, al., force to (Altintas and Aslan, 2017) or force to reduce reduce andovershoot Aslan,model 2017) ortheeliminate eliminate models to atool specific material of theet of theet empirical – et so-called “mechanistic”– process (Stemmler al.,(Altintas 2017) the of aofdesired force and to geometry (Gradiek al., Wang al., and to the the geometry etworkpiece, al., 2004; 2004; Wang ettool, al., force to reduce (Altintas andovershoot Aslan, 2017) or eliminate models to atool specific material of the of the 2014). Nevertheless, there(Gradiek exist exceptions where analytic (Stemmler et al., 2017) the of a desired force (Stemmler et al., 2017) the overshoot of a desired force and to the tool geometry (Gradiek et al., 2004; Wang et al., force to reduce (Altintas and Aslan,a2017) or tool eliminate reference. The coefficients represent certain state Nevertheless, there exist exceptions where analytic 2014). Nevertheless, there exist exceptions where analytic (Stemmler et al., 2017) the overshoot of a desired force 2014). and to the tool geometry (Gradiek et al., 2004; Wang et al., models (Budak et al., 1996; Fu et al., 2015; Campocasso reference. The coefficients represent a certain tool state reference. The coefficients represent a wears. certain tool state 2014). Nevertheless, there exist exceptions where analytic (Stemmler et al., 2017) the overshoot of a desired force and must be adapted when the tool Identifying models (Budak et al., 1996; Fu et al., 2015; Campocasso models (Budak et al., 1996; Fu exceptions et al., 2015; Campocasso reference. The coefficients represent a wears. certain Identifying tool state et 2014). Nevertheless, there exist where analytic al., 2017) or pure black-box models (Auerbach et al., and must be adapted when the tool and must The be adapted when the tool wears. Identifying models (Budak et al.,black-box 1996; Fu models et al., 2015; Campocasso reference. coefficients represent certain tool state et the coefficients of the mechanistic models is often al., 2017) or pure (Auerbach et al., et al., are 2017) or et pure black-box models (Auerbach etforce al., and specific must be adapted when the toola wears. Identifying models (Budak al., 1996; Fu et al., 2015; Campocasso the specific coefficients of the mechanistic models is often 2015) used. This paper focuses on mechanistic the specific coefficients of theapproaches mechanistic is often 2015) et al., are 2017) or pure black-box models (Auerbach etforce al., and must be adapted when the tool wears. cumbersome. Most existing aremodels notIdentifying real-time used. This paper on 2015) are used. This black-box paper focuses focuses on mechanistic mechanistic force the specific coefficients of theapproaches mechanistic models is often models. et al., 2017) or pure models (Auerbach et al., cumbersome. Most existing are not real-time cumbersome. Most existing approaches aremodels not real-time 2015) are used. This paper focuses on mechanistic force the specific coefficients of the mechanistic is often or on-line capable. New ideas are required to obtain a models. models. cumbersome. Most existing approaches are not real-time 2015) are used. This paper focuses on mechanistic force or on-line capable. New ideas are required to obtain a or on-line capable. New for ideas are required tomodels obtain ina The models. cumbersome. Most existing approaches are not continuous identification mechanistic force most popular mechanistic force models are the linear or on-line capable. New for ideas are required to real-time obtain ina The models. mechanistic force continuous identification most popular continuous identification for mechanistic forcetomodels models ina force The most popular mechanistic mechanistic force force models models are are the the linear linear or on-line capable. New ideas are required obtain cutting technology. model continuous identification for mechanistic force models in force The most popular mechanistic force models are the linear cutting technology. model cutting technology. force model continuous identification for mechanistic force models in The most popular mechanistic force models are the This paper introduces two recursive identification methFi,Altintas = b · (Kie + Kic · h) i ∈ t, r, p, linear (1) cutting technology. model This paper introduces two identification methF bb ·· (K K ii ∈ r, This paper introduces two recursive recursive identification meth- force i,Altintas = ie + ic ·· h) cutting technology. force model ods, namely recursive least squares (RLS) and ensemble F = (K + K h) ∈ t, t, r, p, p, (1) (1) i,Altintas ie ic This namely paper introduces two recursive identification meth- according to Altintas and Lee (1996) and the exponential ods, recursive least squares (RLS) and ensemble F = b · (K + K · h) i ∈ t, r, p, (1) i,Altintas ie Lee (1996) ic ods, namely recursive squares (RLS) and ensemble This paper introduces two identification meth- according to Altintas and and the exponential Kalman filter (EnKF).least RLSrecursive is the recursive formulation F = b · (K + K · h) i ∈ t, r, p, (1) according to Altintas and Lee (1996) and the exponential i,Altintas ie ic ods, namely recursive least squares (RLS) and ensemble formulation Kalman filter (EnKF). RLS is the recursive formulation according to Altintas and Lee (1996) and the exponential Kalman filterrecursive (EnKF).approach RLSsquares is the recursive formulation ods, namely least (RLS) and ensemble formulation of a popular and standard in identififormulation 1−m Kalman filter existing (EnKF).approach RLS is the recursive formulation according to Altintas and Lee i (1996) and the exponential of a popular existing and standard in identifiFi,Kienzle = ki · b · h1−mi i ∈ t, r, p (2) of a popular approach andrecursive standard in identifiKalman filter existing (EnKF). RLS is the formulation cation theory. EnKF is a fully nonlinear estimator with formulation F = ki · b · h1−mi ii ∈ of a popular existing approach and standard in identification theory. EnKF aa fully estimator with Fi,Kienzle ∈ t, t, r, r, p p (2) (2) i,Kienzle = ki · b · h1−mi cation theory.existing EnKF is is fully nonlinear nonlinear estimator with formulation of a popular approach and standard in identifiF = k · b · h i ∈ t, r, p (2) i,Kienzle i 1−m i cation theory. EnKF is a fully nonlinear estimator with F = k · b · h i ∈ t, r, p (2) i,Kienzle i cation theory. is a fully nonlinear estimator 2405-8963 © 2019 2019,EnKF IFAC (International Federation of Automatic with Control) Copyright © IFAC 1819Hosting by Elsevier Ltd. All rights reserved. Copyright © under 2019 IFAC 1819Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 1819 Copyright © 2019 IFAC 1819 10.1016/j.ifacol.2019.11.461 Copyright © 2019 IFAC 1819
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according to Kienzle (1952). It is typical for mechanistic force models to assume the same functional relationship of the uncut chip thickness h and the uncut chip width b for the tangential (index t), radial (index r), and the passive (index p) force component. The parameters Kie and Kic or ki and mi respectively, represent the material-specific coefficients that need to be identified. For the linear force model (1) often the coefficients themselves are assumed to vary with the uncut chip thickness h (Wan et al., 2007; Grossi, 2017; Wei et al., 2018; Wang et al., 2018) turning it into a nonlinear model. Therefore, this paper focuses on the exponential model (2) as in (Schwenzer et al., 2018), or similar formulations as in (Jayaram et al., 2001; Dotcheva et al., 2008; Wang et al., 2013; Adem et al., 2015; Zhang et al., 2017). In general, this work distinguishes between the method of • average forces, • instantaneous uncut chip thickness, and • averaged instantaneous uncut chip thickness.
The first method averages the force signal over a multiple of a revolution. The method is inspired by turning where the chip geometry does not change resulting in a static cutting force. A functional relationship is determined through linear regression for different cross sections of the chip, namely different uncut chip thicknesses h through different feeds (K¨ onig et al., 1982). To reproduce those conditions in the cyclically intermittent cut of the milling process, the force is averaged per revolution of the tool. Note that in theory, the arithmetic mean corresponds to the static component of a signal. To describe the force as a function of the uncut chip thickness F (h) as all mechanistic force models do, multiple cutting experiments have to be conducted with different feed rates. This is the most criticized drawback of this identification method, which is described by (Jayaram et al., 2001; Grossi, 2017; Zhang et al., 2017; Schwenzer et al., 2018). The method of instantaneous uncut chip thickness essentially describes a curve fit on the varying uncut chip thickness of a cut. They have been dominated by global optimization algorithms, such as evolutionary algorithm (Grossi, 2017), or particle swarm algorithm (Zhang et al., 2017); independently whether it is the linear Altintas force model (Grossi, 2017), or the exponential Kienzle force model (Zhang et al., 2017). Nevertheless, local optimization algorithms have a significant advantage in computation while still maintaining high accuracy (Schwenzer et al., 2018). Under certain restrictions, they are even online capable and – if constraint algorithms are used – can ensure technological plausibility. An interesting combination of both methods perform Wan et al. (2007). They average the force measures at the same instantaneous chip thickness h over every tooth, i.e. they average the measured force F h (ϕ) =
Nz −1 1 F (h(ϕ)) , Nz i=0
(3)
where Nz is the number of teeth and ϕ is the cutter rotation angle. This method combines the insensitivity to the radial deviation, which is obtained through averaging
the forces per revolution, and the high utilization rate of the force signal (resulting in a low number of required experiments) through using the instantaneous uncut chip thickness. Gonzalo et al. (2010) compare the method of average forces with the method of instantaneous forces. As a reference for the coefficients, they identify them in turning and define those values as the “single point of truth”. They conclude that the method of instantaneous forces shows a better conjunction to the “true” tuning coefficients. Nevertheless, the improvement is small but they argue that the method of instantaneous forces has a better physical credibility due to the correspondence to the turning coefficients. Adem et al. (2015) compare the linear Altintas (1) force model with an derivative of the exponential Kienzle force model (2), and the average forces method with a least squares curve fit. They conclude that the nonlinear force model is generally more accurate than the linear model and that the optimization-based curve fit results in more accurate models than the average forces approach. Wei et al. (2018) supports the first statement. 3. APPROACHES 3.1 Recursive least squares The RLS algorithm is a piecewise least squares fit, which uses the estimate of the previous time instance. In contrast to Schwenzer et al. (2018), this is not an discrete on-line curve fit but a recursive implementation of the method of instantaneous uncut chip thickness and works continuously. The output yˆk is estimated by the nonlinear system model f (·) and the corresponding estimated state vector x ˆk for time instance k (Strejc, 1979). By linearizing the model the measurement matrix MkT is derived and consequently the output is defined by xk ) ≈ MkT · x ˆk . (4) yˆk = f (ˆ Using the exponential force model (2) this translates to T ∂Fi k ∂ki i ∈ t, r, p. (5) · i yˆk = Fˆi,k ≈ ∂F i m i ∂m i
T
with the state vector x ˆk = [ki mi ] . The prediction of the state vector ˆk−1 · Gk · (zk − f (ˆ xk )) (6) x ˆk = x is based on the error between the prediction yˆk of the model and the measurement zk . The Kalman gain Pk−1 · Mk , (7) Gk = ρ + MkT · Pk−1 · Mk weights the influence of the error and depends on the covariance matrix 1 (8) Pk = · I − Gk · MkT . ρ The recursive approximation of the covariance matrix Pk is the particular advantage of this method over a classic curve fit since it limits the computational complexity. The initial value of the covariance matrix is set to (9) P0 = 105 · I
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with the identity matrix I. The forgetting factor 0 < ρ ≤ 1 weights the samples (Strejc, 1979) and is set to ρ = 0.98.
Table 1. Force coefficients for the Kienzle force model for X5CrNi18-10
3.2 Ensemble Kalman filter In essence, the EnKF is a particle filter with an Markov chain Monte Carlo simulation as its backbone to propagate the probability density of the states in time (Evensen, 2003). The EnKF approximates probability density through a finite number n of state vectors (Evensen, 1994). The ensemble members differ by their initial values representing the uncertainty in the initial state vector. As the number of points that underlie a certain probability distribution approaches infinity, the points exactly represent the probability density. The mean of the ensemble serves at the estimate at every sampling point – though first descriptions also offer to use the median for particularly large ensembles (Evensen, 1994). Assuming an ensemble matrix ˆ k|k−1 = x ˆ1k|k−1 , . . . , X
x ˆnk|k−1
(10)
with n individual state vectors x ˆ – different state vectors are distinguished by superscripts j ∈ [1, n]. The index k|k − 1 denotes a value at time instance k based on the information of the previous time instance k − 1.
In the case of identifying the Kienzle model (2) the state vector includes the measured force Fi and the coefficients which have to be estimated: T i ∈ t, r, p. (11) x = [Fi ki mi ] The EnKF follows the same idea as the Kalman filter (KF) which predicts and corrects the estimated states (Grewal and Adrews, 2008). In the prediction phase, the EnKF propagates every member of the ensemble forward in time using the nonlinear model function f (·). All ensemble members receive the same input uk : ˆ k−1|k−1 , uk . ˆ k|k−1 = f X (12) X
The spreading of the ensemble to the mean X k|k−1 represents the error variance: T ˆ k|k−1 − X k|k−1 . ˆ k|k−1 − X k|k−1 · X Pk|k−1 = X (13) This removes the need to propagate the covariance in time, which is expensive in memory and prone to rounding errors. This has been a particular problem in the beginnings of the KF. From an engineering perspective, this is of particular charm since the manual, experience-based tuning of the covariance matrix is omitted. In the update phase, the EnKF corrects its predictions evaluating the measurements x ˆjk|k = x ˆjk|k−1 + Gk · zk + εjk − Hk · x ˆjk|k−1 , (14)
with the measurement matrix Hk = [1 0 0]. The measurement vector zk is distorted by artificial noise εjk in order to find a robust solution and to provide an individual measurement to every ensemble member. The Kalman gain Gk weights the influence of the measurements −1 . (15) Gk = Pk|k−1 · HkT · Hk · Pk|k−1 · HkT + Rk
1793
upper bound lower bound X5CrNi18-10 (K¨ onig et al., 1982, s. 95)
kt
mt
kr
mr
1800 800 1700
0.6 0.05 0.18
1200 600 350
0.3 0.01 0.55
The covariance matrix of the measurement noise Rk simply consists of the artificial noise that is added to the measurements: Rk = εk · εT , (16) 1 k n εk = εk . . . , εk . (17)
This paper adds white Gaussian noise to the measurements. The power is 10.5 dB for the radial force, which corresponds to ca. 30 % of the power of the ideal input signal that is used for simulation. Making the added noise a function of the uncut chip thickness will be left for future examination. 3.3 Remark to the implementation Evensen (2003) recommends to create an ensemble by applying perturbations to a best-guess value. Therefore, we initialize the ensemble through a uniform random distribution within half of this range around the center of a technologically reasonable range of the coefficients, Tab. 1. Note, that a uniform random distribution does not reflect potential characteristics or correlations between the states. The same initial values are used for the RLS as well as the EnKF. To obtain a statistically validated result, Ninit = 1000 distinct initial ensembles were created with a maximum number of members of nmax = 100. Different ensemble sizes use the first n members of an initial ensemble. In this way, the same initial values were used for every ensemble size and combination of measurement noise to ensure consistency. Since the RLS only requires a single initial state vector, we take the nth initial member of the ensemble as the initial state vector for the RLS fit. This ensures that different initial values were used for every RLS fit and that in total the same number of fits were performed. At very low uncut chip thicknesses h < hcrit ploughing occurs. This is when a defined cutting breaks down and the cutting edge produces a plastic deformation instead. Since mechanistic force models are not defined in this region (K¨onig et al., 1982; Wan et al., 2008; Gonzalo et al., 2010), the identification is not performed for h < 0.05 mm. It is assumed that this improves the robustness, stability, and convergence time of the filter, because the artificial noise that the EnKF adds to its measurement input lets the filter measure a force value even if the uncut chip thickness is zero. The RLS fit breaks down at h ≈ 0 since the partial derivatives of the model (5) do not exist if h = 0. 3.4 Simulation for validation For validation, this paper simulates 42 revolutions of an orthogonal milling operation with a flat end-mill repre-
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Table 2. Tool and process parameters Tool geometry Diameter Number of teeth Helix angle Rake angle
RLS S/N: 50
Process parameter
D Nz
10 mm 2
β γ
45 ◦ 20 ◦
Feed Cutting velocity Depth of cut Width of cut
f vc
0.1 mm 2.44 m s−1
ap ae
2 mm 3 mm
×60
S/N: 15
×30
S/N: 5
S/N: 50,
S/N: 15,
S/N: 5
Fig. 1. One exemplary cutter revolution of the simulated tangential force including measurement noise of different signal-to-noise ratios senting a process of machining X5CrNi18-10 with a solid carbide tool, Tab. 2. The helix angle of the tool is approximated with N = 23 disk elements with a straight cutting edge as it is common practice, e.g. (Altintas and Lee, 1996). No radial deviation was assumed. The force measurements were simulated assuming the ideal coefficients given in Tab. 1, which were taken from one of the most extensive works on the determination of coefficients for the Kienzle force model (K¨onig et al., 1982). To account for measurement noise, artificial Gaussian noise with different Signal-to-noise ratios (S/N): 50, 15, and 5 was added to the simulated force signals, Fig. 1. This paper analyzes different ensemble sizes at different signal-to-noise ratios regarding accuracy, robustness, and convergence. For the sake of conciseness, we limit the results to the tangential component of the force and the coefficients (·)i = (·)t . Nevertheless, the approach equally applies to the radial and passive force component as well. All calculations were performed with the software MATLAB R2017a from The MathWorks on an AMD Ryzen7-2700 computer running Windows 10. For accuracy, we examine the mean absolute error per revolution for a certain ensemble l ∈ [1, Ninit ] (the superscript l now denotes an individual ensemble) 360 ◦ 1 l ∆Fi,abs,rev = [Fi,Kienzle (h(ϕ), b(ϕ), ki,1.1 , mi ) 360 ◦ 0 ◦ −Fi,ϕ dϕ] . (18) Using a sample frequency of fs = 10 kHz, a revolution translates to approximately 129 samples. Robustness is analyzed through the mean of the mean absolute error per revolution ∆Fi,abs,rev over all Ninit = 1000 initial ensembles – we will call it “mean average error per revolution”. Convergence is examined by defining the convergence time kcnv of an initial distribution l of the identification as the time step from which on the relative error (20) stays below 2 %:
Initial value group:
# revolutions / 10, 30,
×10
60,
100
Fig. 2. Evolution of the mean error per revolution of the RLS for different signal-to-noise ratios (top: S/N 50, middle: S/N 15, bottom: S/N 5). Size of the circles represent the scaled standard deviation lim
k>kcnv →∞
l < 2 %, ∆Fi,rel
∀k ≥ kcnv .
(19)
The relative error should be defined as l , mli ) − Fi F (ki,1.1 l ∆Fi,rel = . (20) Fi We will consider a fit as converged if the convergence time is reached within 90 % of the simulation steps kcnv < 4860. 4. RESULTS AND DISCUSSION To examine the two identification approaches, the results are merged into Fig. 3 and Fig. 2 showing the evolution of the mean of all Ninit = 1000 initial ensembles/values of the mean error per revolution Eq. 18. The circles indicate the scaled standard deviation within the Ninit groups; the scaling is 60, 30 and 10 for the signal-to-noise ratio of 50, 15 and 5 respectively. For the sake of clarity, the illustration is limited to four representative ensemble sizes. Fig. 2 shows that the RLS fit on average suffers from slow convergence with a high standard deviation but is insensitive to its initial value. This can be seen from the fact that the individual lines hardly differ from each other and are almost congruent. Furthermore, the signal-to-noise ratio seems to have a strong influence on the evolution of the mean average error per revolution. In particular the bottom chart of Fig. 2 stresses this observation showing that the error hardly converges – independent of the initial values (congruent lines and very small standard deviation). Fig. 3 depicts the results for the EnKF within the same range and with the same scaling. One can see that the filter converges in spite of the presence of noise and independently of the ensemble size. The ensemble size affects the deviation within the Ninit initial ensembles suggesting two implications:
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RLS
EnKF
S/N: 15
S/N: 5
kcnv / -
×60
1,500 1,000 500 0
×30
2,000 kcnv / -
S/N: 50
EnKF
2,000
×10
1,500 1,000 500 0
Ensemble sizes:
# revolutions / 10, 30,
60,
kcnv / -
2,000
100
Fig. 3. Evolution of the mean error per revolution of the EnKF for different signal-to-noise ratios (top: S/N 50, middle: S/N 15, bottom: S/N 5). Size of the circles represent the scaled standard deviation • the larger the ensemble size, the slower the convergence of the filter. To quantify the above findings, the convergence time is provided in Fig. 4 as boxplots. The whiskers represent 90 % of the convergence time; the box 50 %. The maximum and minimum is marked by the points and the median by the horizontal line. The RLS converges very slowly and showing a high variation within the Ninit = 1000 starting values independently of the initial value group. The higher the noise level the slower the (median) convergence and the larger the deviation. The RLS is not able to converge at a signal-to-noise ratio of 5. In contrast to this, the EnKF is able to converge at all noise levels and within a much shorter time. The charts support the above drawn conclusions but show a more differentiated result at higher noise levels. While at an almost ideal force measurement signal (S/N 50), the convergence time increases linearly with the ensemble size, higher noise levels profit from the inherent robustness of a larger ensemble size. In particular very small ensemble sizes (n = 10) tend to cause numerical instabilities in 5 %11 % of the cases – with only a small variation at different noise levels. As a compromise between fast convergence and high robustness to noise an ensemble size 40 ≤ n ≤ 60 can be suggested. 5. CONCLUSION This contribution describes a continuous identification of mechanistic force models in milling. Two methods are demonstrated using a nonlinear force model, Eq. 2. The recursive least squares estimator (RLS) extends the method of instantaneous chip thickness, which is in essence a curve fit, to a recursive formulation. The second method is based on an ensemble Kalman Filter (EnKF), which identifies the coefficients of the mechanistic force model.
1,500 1,000 500 0
10 30 50 70 90
10 30 50 70 90
Fig. 4. Convergence time of all 1000 initial ensembles represented as boxplots (whiskers: 5 %, 95 %; box: 25 %, 75 %, · maximum/minimum) for different signalto-noise ratios (top: S/N 50, middle: S/N 15, bottom: S/N 5) The results show that the RLS fit is prone noise in the measurements and suffers from convergence. The EnKF shows a remarkably insensibility to measurement noise and a fast convergence. The robustness increases with the ensemble size but so does also the convergence time at ideal measurements. A good compromise between convergence speed and robustness achieves an ensemble size of n = 50 members. So far, radial deviation was neglected and experiments have been simulated in order to provide a controlled environment. Future work will extend the continuous identification to jointly estimate a model of the force and of the radial deviation. This enables the presented approach to apply on a real measurements. An integration to a realtime target is the final step in order to use this approach for an on-line monitoring or control of the force in milling. ACKNOWLEDGEMENTS The authors would like to thank the German Research Foundation DFG for the kind support within the Cluster of Excellence “Internet of Production” (Project ID: 390621612). REFERENCES Adem, K.A.M., Fales, R., and El-Gizawy, A.S. (2015). Identification of cutting force coefficients for the linear and nonlinear force models in end milling process using average forces and optimization technique methods. The International Journal of Advanced Manufacturing Technology, 79(9), 1671–1687. doi:10.1007/s00170-0156935-3.
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Altintas, Y. and Aslan, D. (2017). Integration of virtual and on-line machining process control and monitoring. CIRP Annals - Manufacturing Technology, 66(1), 349– 352. doi:10.1016/j.cirp.2017.04.047. Altintas, Y. and Lee, P. (1996). A general mechanics and dynamics model for helical end mills. CIRP Annals - Manufacturing Technology, 45(1), 59–64. doi: 10.1016/S0007-8506(07)63017-0. Auerbach, T., Gierlings, S., Veselovac, D., Seidner, R., Kamps, S., and Klocke, F. (2015). Concept for a technology assistance system to analyze and evaluate materials and tools for milling. In Proceedings of the ASME Turbo Expo: Turbine Technical Conference and Exposition 2015, volume 6, V006T21A006. ASME. doi: 10.1115/GT2015-43209. Budak, E., Altinta, Y., and Armarego, E.J.A. (1996). Prediction of milling force coefficients from orthogonal cutting data. Journal of Manufacturing Science and Engineering, 118(2), 216–224. doi:10.1115/1.2831014. Campocasso, S., Poulachon, G., Bissey-Breton, S., Costes, J., and Outeiro, J. (2017). Towards cutting force evaluation without cutting tests. CIRP Annals - Manufacturing Technology, 66(1), 77–80. doi: 10.1016/j.cirp.2017.04.023. Dotcheva, M., Millward, H., and Lewis, A. (2008). The evaluation of cutting-force coefficients using surface error measurements. Journal of Materials Processing Technology, 196(1), 42–51. doi: 10.1016/j.jmatprotec.2007.04.136. Ehmann, K.F., Kapoor, S.G., DeVor, R.E., and Lazoglu, I. (1997). Machining process modeling: A review. Journal of Manufacturing Science and Engineering, 119(4), 655. doi:10.1115/1.2836805. Evensen, G. (1994). Sequential data assimilation with a nonlinear quasi-geostrophic model using monte carlo methods to forecast error statistics. Journal of Geophysical Research, 99, 10143. doi:10.1029/94JC00572. Evensen, G. (2003). The ensemble kalman filter: theoretical formulation and practical implementation. Ocean Dynamics, 53(4), 343–367. doi:10.1007/s10236-0030036-9. Fu, Z., Yang, W., Wang, X., and Leopold, J. (2015). Analytical modelling of milling forces for helical end milling based on a predictive machining theory. Procedia CIRP, 31, 258–263. doi:10.1016/j.procir.2015.03.013. Gonzalo, O., Beristain, J., Jauregi, H., and Sanz, C. (2010). A method for the identification of the specific force coefficients for mechanistic milling simulation. International Journal of Machine Tools and Manufacture, 50(9), 765–774. doi:10.1016/j.ijmachtools.2010.05.009. Gradiek, J., Kalveram, M., and Weinert, K. (2004). Mechanistic identification of specific force coefficients for a general end mill. International Journal of Machine Tools and Manufacture, 44(4), 401–414. doi: 10.1016/j.ijmachtools.2003.10.001. Grewal, M.S. and Adrews, A.P. (2008). Kalman Filtering: Theory and Practice using Matlab. John Wiley & Sons. Grossi, N. (2017). Accurate and fast measurement of specific cutting force coefficients changing with spindle speed. International Journal of Precision Engineering and Manufacturing, 18(8), 1173–1180. doi: 10.1007/s12541-017-0137-x.
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Continuous identification for mechanistic Continuous identification for mechanistic Continuous identification for mechanistic Continuous identification for mechanistic force models in milling Continuous identification for mechanistic force models in milling force models in milling force models in milling models in milling ∗ ∗∗ ∗∗ ∗ ∗∗ M. Schwenzerforce ∗ S. Stemmler ∗∗ M. Ay ∗∗ T. Bergs ∗ D. Abel ∗∗
M. T. Bergs ∗ D. Abel ∗∗ M. Schwenzer Schwenzer ∗∗ S. S. Stemmler Stemmler ∗∗ M. M. Ay Ay ∗∗ ∗∗ T. Bergs ∗ D. Abel ∗∗ M. Schwenzer ∗ S. Stemmler ∗∗ ∗∗ M. Ay ∗∗ T. Bergs ∗ D. Abel ∗∗ ∗ M. Schwenzer S. Stemmler Ay T.Engineering Bergs D. (WZL), Abel for Machine Tools andM. Production ∗ Laboratory ∗ Laboratory for Machine Tools and Production Engineering (WZL), Laboratory for Machine Tools and Production Engineering (WZL), Aachen University, Campus-Boulevard 30, 52074 Aachen, ∗ RWTH Laboratory for Machine Tools and Production Engineering (WZL), Aachen University, Campus-Boulevard 30, ∗ RWTH RWTH Aachen University, Campus-Boulevard 30, 52074 52074 Aachen, Aachen, Laboratory for Machine and Production Engineering (WZL), Germany (e-mail: Tools [email protected]). RWTH Aachen University, Campus-Boulevard 30, 52074 Aachen, Germany (e-mail: [email protected]). ∗∗ Germany (e-mail: [email protected]). RWTH Aachen Campus-Boulevard 30, 52074 Aachen, of University, Automatic Control, RWTH Aachen University, ∗∗ Institute Germany (e-mail: [email protected]). ∗∗ Institute of Automatic Control, RWTH Aachen University, of (e-mail: Automatic Control, RWTH Aachen University, Germany [email protected]). Campus-Boulevard 30, 52074 Aachen, Germany ∗∗ Institute of Automatic Control, RWTH Aachen University, Campus-Boulevard 30, Aachen, Germany ∗∗ Institute Campus-Boulevard 30, 52074 52074 Aachen, Germany Institute of Automatic Control, RWTH Aachen University, Campus-Boulevard 30, 52074 Aachen, Germany Campus-Boulevard 30, 52074 Aachen, Germany Abstract: Force determines the product quality, the productivity and the safety of a milling Abstract: Force the quality, the and safety a milling Abstract: Force determines determines the product product the productivity productivity and the the safety of of milling process. Mechanistic force models are the quality, key to understand, optimize or control theaa cutting Abstract: Force determines the product quality, the productivity and the safety of milling process. Mechanistic force models are the key to understand, optimize or control the cutting process. Mechanistic force models are chip the quality, key to understand, optimize orcoefficients control the cutting Abstract: Force determines the product the productivity and the safety of a milling They combine the undeformed parameter with empiric tuning in a grayprocess. They Mechanistic force models are chip the key to understand, optimize orcoefficients control the cutting combine the undeformed parameter with empiric tuning in aa grayprocess. They combineforce the undeformed chip parameter with empiric tuningorcoefficients in cutting grayMechanistic models are the key to understand, optimize control the box model. Identifying those coefficients is costly in both, time and number of experiments. This process. They combine the undeformed parameter with empiric tuning coefficients in a graybox model. model. Identifying those coefficientschip is costly costly in both, both, time and number number of experiments. experiments. This box Identifying those coefficients is in time and of This process. They combine the undeformed chip parameter with empiric tuning coefficients in a graypaper introduces two recursive identification methods for force model identification: recursive box model. Identifying those coefficients is costly in both, time and number of experiments. This paper introduces two recursive identification methods for force model identification: recursive paper introduces two recursive identification force model recursive box model. Identifying thoseKalman coefficients is costly in both, time and number of experiments. This least squares and ensemble filters. Themethods model isfor nonlinear. The identification: ensemble Kalman filter paper introduces two recursive identification methods for force model identification: recursive least squares and ensemble Kalman filters. The model is nonlinear. The ensemble Kalman filter least squares and ensemble Kalman filters. The model isfor nonlinear. The ensemble Kalman filter paper introduces two recursive identification methods force model identification: recursive shows an extraordinary robustness against measurement noise and a fast convergence time – least squares and ensemble Kalman filters. The model is nonlinear. The ensemble Kalman filter shows an robustness against measurement noise aa fast convergence time –– shows an extraordinary extraordinary robustness againstThe measurement noise and andThe fast convergence time least squares and ensemble Kalman filters. model is nonlinear. ensemble Kalman filter depending on the selection of the ensemble size and the measurement noise. The recursive least shows an extraordinary robustness against size measurement noise and anoise. fast convergence time depending on the the selection selection of the the ensemble ensemble and the the measurement measurement The recursive recursive least– depending on of size and The least shows an robustness against measurement noise and anoise. fast convergence time – squares fitextraordinary serves as a benchmark but is highly sensitive to measurement noise. It isrecursive the first time depending on theas selection of the but ensemble size and theto measurement noise. The least squares fit serves a benchmark is highly sensitive measurement noise. It is the first time squares fit serves as a benchmark but is highly sensitive to measurement noise. It isrecursive theCopyright first least time depending on the selection of the ensemble size and the measurement noise. The that a continuous identification is examined for mechanistic force models in milling. squares fit serves asidentification a benchmarkisbut is highlyfor sensitive to measurement noise. It is theCopyright first time that a continuous examined mechanistic force in that examined mechanistic force models models in milling. milling. squares fit serves asidentification a benchmarkis is highlyfor sensitive to measurement noise. It is theCopyright first time c aa continuous 2019 IFAC that continuous identification isbut examined for mechanistic force models in milling. Copyright c 2019 IFAC c 2019 IFAC that a continuous identification is examined for mechanistic force models in milling. Copyright © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. c 2019, IFAC 2019 c 2019 IFAC Keywords: Ensemble Kalman filter, Recursive least squares, Manufacturing, Milling, Keywords: Ensemble Kalman filter, least squares, Manufacturing, Keywords: Ensemble Kalman filter, Recursive Recursive leastmodel squares, Manufacturing, Milling, Milling, Parameter identification, Identification, nonlinear identification Keywords: Ensemble Kalman filter, Recursive leastmodel squares, Manufacturing, Milling, Parameter identification, Identification, nonlinear identification Parameter identification, Identification, nonlinear model identification Keywords: Ensemble Kalman filter, Recursive least squares, Manufacturing, Milling, Parameter identification, Identification, nonlinear model identification Parameter identification, Identification, nonlinearan model identification 1. INTRODUCTION advantage regarding robustness. This work reduces the 1. an robustness. This work reduces the 1. INTRODUCTION INTRODUCTION an advantage advantage regarding regarding This work reduces and the identification effort androbustness. enables for more accurate 1. INTRODUCTION an advantage regarding robustness. This work reduces and the identification effort and enables for more accurate identification effort androbustness. enables for more accurate and 1. INTRODUCTION an advantage regarding This work reduces the Milling is a flexible and highly dynamic manufacturing more adaptive force models. identification effort and enables for more accurate and Milling is a flexible and highly dynamic manufacturing more adaptive force models. Milling to is produce a flexiblealmost and highly dynamic manufacturing more adaptiveeffort force models. identification and enables for more accurate and process arbitrary free-form surfaces. A Milling to is produce a flexiblealmost and highly dynamic manufacturing adaptive force models. process arbitrary free-form surfaces. A process to produce arbitrary free-form surfaces. A more Milling is intermittent a flexiblealmost and highly dynamic manufacturing 2. STATE models.OF THE ART cyclically cutting process takes place, which process to produce almost arbitrary free-form surfaces. A more adaptive force 2. cyclically intermittent cutting process takes place, which 2. STATE STATE OF OF THE THE ART ART cyclically intermittent cutting process takes place, which process to produce almost arbitrary free-form surfaces. A is overlaid with a translatory feed movement. This is 2. STATE OF THE ART cyclically intermittent cutting process takes place,This which is overlaid with aa translatory feed movement. is is overlaid with translatory feed movement. This is 2.can STATE OF THE ART different model In general, one distinguish between cyclically intermittent cutting process takes place, which characterized by a continuously varying thickness of the is overlaid with translatory feed movement. This is In thickness of one distinguish between different model characterized by aaa continuously varying In general, general, one can can distinguish between different characterized by continuously varying thickness of the the types in cutting technology (Ehmann et al., 1997): model is overlaid with a translatory feed movement. This is removed chip and with a varying force. Force is In general, one can distinguish between different model characterized by a continuously varying thickness of the types in cutting technology (Ehmann et al., 1997): removed chip and with a varying force. Force is in cutting technology (Ehmann et al., 1997): model removed chip by and with since a varying force. Force is the types In general, one can distinguish between different characterized a continuously varying thickness of crucial process parameter it affects the process safety types in cutting technology (Ehmann et al., 1997): removed chip and with a varying force. Force is the since it the safety • empirical (black-box), crucial process crucial process parameter it affects affects the process process safety in cutting(black-box), technology (Ehmann et al., 1997): removed chip parameter and with since a quality varying force. Force is the types (tool workpiece (geometric tolerances • empirical empirical (black-box), crucialbreakage), process parameter since it affects the process safety (tool breakage), workpiece quality (geometric tolerances •• analytical (white-box – only physical phenomena), (tool breakage), workpiece quality (geometric tolerances empirical (black-box), crucial process parameter since it affects the process safety through the deflection of the tool), and productivity (feed analytical (white-box • and analytical (white-box –– only only physical physical phenomena), phenomena), (tool breakage), workpiece quality (geometric tolerances empirical tool), and productivity (feed through the deflection of the through the models deflection of the tool), and productivity (feed • and analytical (black-box), (white-box – only physical phenomena), (tool breakage), workpiece quality (geometric tolerances rate). Force are the key to understand, optimize or and through the models deflection of the tool), and productivity (feed analytical (white-box – only physical phenomena), • mechanistic (gray-box relate to the chip cross secrate). Force are the key to understand, optimize or rate). Force areof the key to understand, optimize or and through the models deflection the tool), and productivity (feed even control the milling process. •• mechanistic (gray-box mechanistic (gray-box –– relate relate to to the the chip chip cross cross secsecrate).control Force models are the key to understand, optimize or and tion). even the milling process. even control the milling process. • tion). mechanistic (gray-box – relate to the chip cross secrate). Force models are the key to understand, optimize or tion). even control the milling process. • mechanistic (gray-box – relate to the chip cross secOnly recentlytheadvanced control methods are being aption). even control milling process. Usually mechanistic models are used due to their simOnly recently advanced control methods are apOnly recently advanced control methods are being being ap- Usually tion). plied to control the force in milling: adaptive control mechanistic models are used due to their simUsuallylow mechanistic models areand used dueinterpretability. to their simOnly recently advanced control methods are being ap- plicity, computational cost, their plied to the in milling: adaptive control plied to control control the force force in adaptive control Usuallylow mechanistic models areand used dueinterpretability. to their simOnly advanced control methods are being ap- plicity, (Altintas and Aslan, 2017) andmilling: model-based predictive computational cost, their plicity, low computational cost, and their interpretability. plied recently to control the force in milling: adaptive control Usually mechanistic models are used due to fundamentheir simMechanistic force models relate the force to the (Altintas and Aslan, 2017) and model-based predictive (Altintas and Aslan, 2017)etin and model-based predictive plicity, low computational cost, the andforce theirtointerpretability. plied to control the force milling: adaptive control control (MPC) (Stemmler al., 2017). Both use a semiMechanistic force models relate the Mechanistic force models relate the force tointerpretability. the fundamenfundamen(Altintas(MPC) and Aslan, 2017)et and model-based predictive plicity, low computational cost, and their tal undeformed chip parameters. The coefficients tune the control (Stemmler al., 2017). Both use a semicontrol (MPC) (Stemmler al., 2017). Both a semi- tal Mechanistic force models relate the force to the fundamenundeformed chip parameters. The coefficients tune the (Altintas Aslan, 2017)et model-based predictive empirical –and so-called “mechanistic”– model of use the process tal undeformed chip parameters. The coefficients tunetool, the control (MPC) (Stemmler et and al., 2017). Both use a semi- models Mechanistic force models relate the force to the fundamento a specific material of workpiece, of the empirical – so-called “mechanistic”– model of the process empirical – so-called “mechanistic”– model oforuse theeliminate process tal undeformed chip material parameters. The coefficients tunetool, the control (MPC) (Stemmler et al., 2017). Both a semiforce to reduce (Altintas and Aslan, 2017) models to a specific of the workpiece, of the models to a specific material of the workpiece, of the tool, empirical – so-called “mechanistic”– model of the process tal undeformed chip parameters. The coefficients tune the and to the tool geometry (Gradiek etworkpiece, al., 2004; Wang ettool, al., force to (Altintas and Aslan, 2017) or force to reduce reduce andovershoot Aslan,model 2017) ortheeliminate eliminate models to atool specific material of theet of theet empirical – et so-called “mechanistic”– process (Stemmler al.,(Altintas 2017) the of aofdesired force and to geometry (Gradiek al., Wang al., and to the the geometry etworkpiece, al., 2004; 2004; Wang ettool, al., force to reduce (Altintas andovershoot Aslan, 2017) or eliminate models to atool specific material of the of the 2014). Nevertheless, there(Gradiek exist exceptions where analytic (Stemmler et al., 2017) the of a desired force (Stemmler et al., 2017) the overshoot of a desired force and to the tool geometry (Gradiek et al., 2004; Wang et al., force to reduce (Altintas and Aslan,a2017) or tool eliminate reference. The coefficients represent certain state Nevertheless, there exist exceptions where analytic 2014). Nevertheless, there exist exceptions where analytic (Stemmler et al., 2017) the overshoot of a desired force 2014). and to the tool geometry (Gradiek et al., 2004; Wang et al., models (Budak et al., 1996; Fu et al., 2015; Campocasso reference. The coefficients represent a certain tool state reference. The coefficients represent a wears. certain tool state 2014). Nevertheless, there exist exceptions where analytic (Stemmler et al., 2017) the overshoot of a desired force and must be adapted when the tool Identifying models (Budak et al., 1996; Fu et al., 2015; Campocasso models (Budak et al., 1996; Fu exceptions et al., 2015; Campocasso reference. The coefficients represent a wears. certain Identifying tool state et 2014). Nevertheless, there exist where analytic al., 2017) or pure black-box models (Auerbach et al., and must be adapted when the tool and must The be adapted when the tool wears. Identifying models (Budak et al.,black-box 1996; Fu models et al., 2015; Campocasso reference. coefficients represent certain tool state et the coefficients of the mechanistic models is often al., 2017) or pure (Auerbach et al., et al., are 2017) or et pure black-box models (Auerbach etforce al., and specific must be adapted when the toola wears. Identifying models (Budak al., 1996; Fu et al., 2015; Campocasso the specific coefficients of the mechanistic models is often 2015) used. This paper focuses on mechanistic the specific coefficients of theapproaches mechanistic is often 2015) et al., are 2017) or pure black-box models (Auerbach etforce al., and must be adapted when the tool wears. cumbersome. Most existing aremodels notIdentifying real-time used. This paper on 2015) are used. This black-box paper focuses focuses on mechanistic mechanistic force the specific coefficients of theapproaches mechanistic models is often models. et al., 2017) or pure models (Auerbach et al., cumbersome. Most existing are not real-time cumbersome. Most existing approaches aremodels not real-time 2015) are used. This paper focuses on mechanistic force the specific coefficients of the mechanistic is often or on-line capable. New ideas are required to obtain a models. models. cumbersome. Most existing approaches are not real-time 2015) are used. This paper focuses on mechanistic force or on-line capable. New ideas are required to obtain a or on-line capable. New for ideas are required tomodels obtain ina The models. cumbersome. Most existing approaches are not continuous identification mechanistic force most popular mechanistic force models are the linear or on-line capable. New for ideas are required to real-time obtain ina The models. mechanistic force continuous identification most popular continuous identification for mechanistic forcetomodels models ina force The most popular mechanistic mechanistic force force models models are are the the linear linear or on-line capable. New ideas are required obtain cutting technology. model continuous identification for mechanistic force models in force The most popular mechanistic force models are the linear cutting technology. model cutting technology. force model continuous identification for mechanistic force models in The most popular mechanistic force models are the This paper introduces two recursive identification methFi,Altintas = b · (Kie + Kic · h) i ∈ t, r, p, linear (1) cutting technology. model This paper introduces two identification methF bb ·· (K K ii ∈ r, This paper introduces two recursive recursive identification meth- force i,Altintas = ie + ic ·· h) cutting technology. force model ods, namely recursive least squares (RLS) and ensemble F = (K + K h) ∈ t, t, r, p, p, (1) (1) i,Altintas ie ic This namely paper introduces two recursive identification meth- according to Altintas and Lee (1996) and the exponential ods, recursive least squares (RLS) and ensemble F = b · (K + K · h) i ∈ t, r, p, (1) i,Altintas ie Lee (1996) ic ods, namely recursive squares (RLS) and ensemble This paper introduces two identification meth- according to Altintas and and the exponential Kalman filter (EnKF).least RLSrecursive is the recursive formulation F = b · (K + K · h) i ∈ t, r, p, (1) according to Altintas and Lee (1996) and the exponential i,Altintas ie ic ods, namely recursive least squares (RLS) and ensemble formulation Kalman filter (EnKF). RLS is the recursive formulation according to Altintas and Lee (1996) and the exponential Kalman filterrecursive (EnKF).approach RLSsquares is the recursive formulation ods, namely least (RLS) and ensemble formulation of a popular and standard in identififormulation 1−m Kalman filter existing (EnKF).approach RLS is the recursive formulation according to Altintas and Lee i (1996) and the exponential of a popular existing and standard in identifiFi,Kienzle = ki · b · h1−mi i ∈ t, r, p (2) of a popular approach andrecursive standard in identifiKalman filter existing (EnKF). RLS is the formulation cation theory. EnKF is a fully nonlinear estimator with formulation F = ki · b · h1−mi ii ∈ of a popular existing approach and standard in identification theory. EnKF aa fully estimator with Fi,Kienzle ∈ t, t, r, r, p p (2) (2) i,Kienzle = ki · b · h1−mi cation theory.existing EnKF is is fully nonlinear nonlinear estimator with formulation of a popular approach and standard in identifiF = k · b · h i ∈ t, r, p (2) i,Kienzle i 1−m i cation theory. EnKF is a fully nonlinear estimator with F = k · b · h i ∈ t, r, p (2) i,Kienzle i cation theory. is a fully nonlinear estimator 2405-8963 © 2019 2019,EnKF IFAC (International Federation of Automatic with Control) Copyright © IFAC 1819Hosting by Elsevier Ltd. All rights reserved. Copyright © under 2019 IFAC 1819Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 1819 Copyright © 2019 IFAC 1819 10.1016/j.ifacol.2019.11.461 Copyright © 2019 IFAC 1819
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according to Kienzle (1952). It is typical for mechanistic force models to assume the same functional relationship of the uncut chip thickness h and the uncut chip width b for the tangential (index t), radial (index r), and the passive (index p) force component. The parameters Kie and Kic or ki and mi respectively, represent the material-specific coefficients that need to be identified. For the linear force model (1) often the coefficients themselves are assumed to vary with the uncut chip thickness h (Wan et al., 2007; Grossi, 2017; Wei et al., 2018; Wang et al., 2018) turning it into a nonlinear model. Therefore, this paper focuses on the exponential model (2) as in (Schwenzer et al., 2018), or similar formulations as in (Jayaram et al., 2001; Dotcheva et al., 2008; Wang et al., 2013; Adem et al., 2015; Zhang et al., 2017). In general, this work distinguishes between the method of • average forces, • instantaneous uncut chip thickness, and • averaged instantaneous uncut chip thickness.
The first method averages the force signal over a multiple of a revolution. The method is inspired by turning where the chip geometry does not change resulting in a static cutting force. A functional relationship is determined through linear regression for different cross sections of the chip, namely different uncut chip thicknesses h through different feeds (K¨ onig et al., 1982). To reproduce those conditions in the cyclically intermittent cut of the milling process, the force is averaged per revolution of the tool. Note that in theory, the arithmetic mean corresponds to the static component of a signal. To describe the force as a function of the uncut chip thickness F (h) as all mechanistic force models do, multiple cutting experiments have to be conducted with different feed rates. This is the most criticized drawback of this identification method, which is described by (Jayaram et al., 2001; Grossi, 2017; Zhang et al., 2017; Schwenzer et al., 2018). The method of instantaneous uncut chip thickness essentially describes a curve fit on the varying uncut chip thickness of a cut. They have been dominated by global optimization algorithms, such as evolutionary algorithm (Grossi, 2017), or particle swarm algorithm (Zhang et al., 2017); independently whether it is the linear Altintas force model (Grossi, 2017), or the exponential Kienzle force model (Zhang et al., 2017). Nevertheless, local optimization algorithms have a significant advantage in computation while still maintaining high accuracy (Schwenzer et al., 2018). Under certain restrictions, they are even online capable and – if constraint algorithms are used – can ensure technological plausibility. An interesting combination of both methods perform Wan et al. (2007). They average the force measures at the same instantaneous chip thickness h over every tooth, i.e. they average the measured force F h (ϕ) =
Nz −1 1 F (h(ϕ)) , Nz i=0
(3)
where Nz is the number of teeth and ϕ is the cutter rotation angle. This method combines the insensitivity to the radial deviation, which is obtained through averaging
the forces per revolution, and the high utilization rate of the force signal (resulting in a low number of required experiments) through using the instantaneous uncut chip thickness. Gonzalo et al. (2010) compare the method of average forces with the method of instantaneous forces. As a reference for the coefficients, they identify them in turning and define those values as the “single point of truth”. They conclude that the method of instantaneous forces shows a better conjunction to the “true” tuning coefficients. Nevertheless, the improvement is small but they argue that the method of instantaneous forces has a better physical credibility due to the correspondence to the turning coefficients. Adem et al. (2015) compare the linear Altintas (1) force model with an derivative of the exponential Kienzle force model (2), and the average forces method with a least squares curve fit. They conclude that the nonlinear force model is generally more accurate than the linear model and that the optimization-based curve fit results in more accurate models than the average forces approach. Wei et al. (2018) supports the first statement. 3. APPROACHES 3.1 Recursive least squares The RLS algorithm is a piecewise least squares fit, which uses the estimate of the previous time instance. In contrast to Schwenzer et al. (2018), this is not an discrete on-line curve fit but a recursive implementation of the method of instantaneous uncut chip thickness and works continuously. The output yˆk is estimated by the nonlinear system model f (·) and the corresponding estimated state vector x ˆk for time instance k (Strejc, 1979). By linearizing the model the measurement matrix MkT is derived and consequently the output is defined by xk ) ≈ MkT · x ˆk . (4) yˆk = f (ˆ Using the exponential force model (2) this translates to T ∂Fi k ∂ki i ∈ t, r, p. (5) · i yˆk = Fˆi,k ≈ ∂F i m i ∂m i
T
with the state vector x ˆk = [ki mi ] . The prediction of the state vector ˆk−1 · Gk · (zk − f (ˆ xk )) (6) x ˆk = x is based on the error between the prediction yˆk of the model and the measurement zk . The Kalman gain Pk−1 · Mk , (7) Gk = ρ + MkT · Pk−1 · Mk weights the influence of the error and depends on the covariance matrix 1 (8) Pk = · I − Gk · MkT . ρ The recursive approximation of the covariance matrix Pk is the particular advantage of this method over a classic curve fit since it limits the computational complexity. The initial value of the covariance matrix is set to (9) P0 = 105 · I
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with the identity matrix I. The forgetting factor 0 < ρ ≤ 1 weights the samples (Strejc, 1979) and is set to ρ = 0.98.
Table 1. Force coefficients for the Kienzle force model for X5CrNi18-10
3.2 Ensemble Kalman filter In essence, the EnKF is a particle filter with an Markov chain Monte Carlo simulation as its backbone to propagate the probability density of the states in time (Evensen, 2003). The EnKF approximates probability density through a finite number n of state vectors (Evensen, 1994). The ensemble members differ by their initial values representing the uncertainty in the initial state vector. As the number of points that underlie a certain probability distribution approaches infinity, the points exactly represent the probability density. The mean of the ensemble serves at the estimate at every sampling point – though first descriptions also offer to use the median for particularly large ensembles (Evensen, 1994). Assuming an ensemble matrix ˆ k|k−1 = x ˆ1k|k−1 , . . . , X
x ˆnk|k−1
(10)
with n individual state vectors x ˆ – different state vectors are distinguished by superscripts j ∈ [1, n]. The index k|k − 1 denotes a value at time instance k based on the information of the previous time instance k − 1.
In the case of identifying the Kienzle model (2) the state vector includes the measured force Fi and the coefficients which have to be estimated: T i ∈ t, r, p. (11) x = [Fi ki mi ] The EnKF follows the same idea as the Kalman filter (KF) which predicts and corrects the estimated states (Grewal and Adrews, 2008). In the prediction phase, the EnKF propagates every member of the ensemble forward in time using the nonlinear model function f (·). All ensemble members receive the same input uk : ˆ k−1|k−1 , uk . ˆ k|k−1 = f X (12) X
The spreading of the ensemble to the mean X k|k−1 represents the error variance: T ˆ k|k−1 − X k|k−1 . ˆ k|k−1 − X k|k−1 · X Pk|k−1 = X (13) This removes the need to propagate the covariance in time, which is expensive in memory and prone to rounding errors. This has been a particular problem in the beginnings of the KF. From an engineering perspective, this is of particular charm since the manual, experience-based tuning of the covariance matrix is omitted. In the update phase, the EnKF corrects its predictions evaluating the measurements x ˆjk|k = x ˆjk|k−1 + Gk · zk + εjk − Hk · x ˆjk|k−1 , (14)
with the measurement matrix Hk = [1 0 0]. The measurement vector zk is distorted by artificial noise εjk in order to find a robust solution and to provide an individual measurement to every ensemble member. The Kalman gain Gk weights the influence of the measurements −1 . (15) Gk = Pk|k−1 · HkT · Hk · Pk|k−1 · HkT + Rk
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upper bound lower bound X5CrNi18-10 (K¨ onig et al., 1982, s. 95)
kt
mt
kr
mr
1800 800 1700
0.6 0.05 0.18
1200 600 350
0.3 0.01 0.55
The covariance matrix of the measurement noise Rk simply consists of the artificial noise that is added to the measurements: Rk = εk · εT , (16) 1 k n εk = εk . . . , εk . (17)
This paper adds white Gaussian noise to the measurements. The power is 10.5 dB for the radial force, which corresponds to ca. 30 % of the power of the ideal input signal that is used for simulation. Making the added noise a function of the uncut chip thickness will be left for future examination. 3.3 Remark to the implementation Evensen (2003) recommends to create an ensemble by applying perturbations to a best-guess value. Therefore, we initialize the ensemble through a uniform random distribution within half of this range around the center of a technologically reasonable range of the coefficients, Tab. 1. Note, that a uniform random distribution does not reflect potential characteristics or correlations between the states. The same initial values are used for the RLS as well as the EnKF. To obtain a statistically validated result, Ninit = 1000 distinct initial ensembles were created with a maximum number of members of nmax = 100. Different ensemble sizes use the first n members of an initial ensemble. In this way, the same initial values were used for every ensemble size and combination of measurement noise to ensure consistency. Since the RLS only requires a single initial state vector, we take the nth initial member of the ensemble as the initial state vector for the RLS fit. This ensures that different initial values were used for every RLS fit and that in total the same number of fits were performed. At very low uncut chip thicknesses h < hcrit ploughing occurs. This is when a defined cutting breaks down and the cutting edge produces a plastic deformation instead. Since mechanistic force models are not defined in this region (K¨onig et al., 1982; Wan et al., 2008; Gonzalo et al., 2010), the identification is not performed for h < 0.05 mm. It is assumed that this improves the robustness, stability, and convergence time of the filter, because the artificial noise that the EnKF adds to its measurement input lets the filter measure a force value even if the uncut chip thickness is zero. The RLS fit breaks down at h ≈ 0 since the partial derivatives of the model (5) do not exist if h = 0. 3.4 Simulation for validation For validation, this paper simulates 42 revolutions of an orthogonal milling operation with a flat end-mill repre-
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Table 2. Tool and process parameters Tool geometry Diameter Number of teeth Helix angle Rake angle
RLS S/N: 50
Process parameter
D Nz
10 mm 2
β γ
45 ◦ 20 ◦
Feed Cutting velocity Depth of cut Width of cut
f vc
0.1 mm 2.44 m s−1
ap ae
2 mm 3 mm
×60
S/N: 15
×30
S/N: 5
S/N: 50,
S/N: 15,
S/N: 5
Fig. 1. One exemplary cutter revolution of the simulated tangential force including measurement noise of different signal-to-noise ratios senting a process of machining X5CrNi18-10 with a solid carbide tool, Tab. 2. The helix angle of the tool is approximated with N = 23 disk elements with a straight cutting edge as it is common practice, e.g. (Altintas and Lee, 1996). No radial deviation was assumed. The force measurements were simulated assuming the ideal coefficients given in Tab. 1, which were taken from one of the most extensive works on the determination of coefficients for the Kienzle force model (K¨onig et al., 1982). To account for measurement noise, artificial Gaussian noise with different Signal-to-noise ratios (S/N): 50, 15, and 5 was added to the simulated force signals, Fig. 1. This paper analyzes different ensemble sizes at different signal-to-noise ratios regarding accuracy, robustness, and convergence. For the sake of conciseness, we limit the results to the tangential component of the force and the coefficients (·)i = (·)t . Nevertheless, the approach equally applies to the radial and passive force component as well. All calculations were performed with the software MATLAB R2017a from The MathWorks on an AMD Ryzen7-2700 computer running Windows 10. For accuracy, we examine the mean absolute error per revolution for a certain ensemble l ∈ [1, Ninit ] (the superscript l now denotes an individual ensemble) 360 ◦ 1 l ∆Fi,abs,rev = [Fi,Kienzle (h(ϕ), b(ϕ), ki,1.1 , mi ) 360 ◦ 0 ◦ −Fi,ϕ dϕ] . (18) Using a sample frequency of fs = 10 kHz, a revolution translates to approximately 129 samples. Robustness is analyzed through the mean of the mean absolute error per revolution ∆Fi,abs,rev over all Ninit = 1000 initial ensembles – we will call it “mean average error per revolution”. Convergence is examined by defining the convergence time kcnv of an initial distribution l of the identification as the time step from which on the relative error (20) stays below 2 %:
Initial value group:
# revolutions / 10, 30,
×10
60,
100
Fig. 2. Evolution of the mean error per revolution of the RLS for different signal-to-noise ratios (top: S/N 50, middle: S/N 15, bottom: S/N 5). Size of the circles represent the scaled standard deviation lim
k>kcnv →∞
l < 2 %, ∆Fi,rel
∀k ≥ kcnv .
(19)
The relative error should be defined as l , mli ) − Fi F (ki,1.1 l ∆Fi,rel = . (20) Fi We will consider a fit as converged if the convergence time is reached within 90 % of the simulation steps kcnv < 4860. 4. RESULTS AND DISCUSSION To examine the two identification approaches, the results are merged into Fig. 3 and Fig. 2 showing the evolution of the mean of all Ninit = 1000 initial ensembles/values of the mean error per revolution Eq. 18. The circles indicate the scaled standard deviation within the Ninit groups; the scaling is 60, 30 and 10 for the signal-to-noise ratio of 50, 15 and 5 respectively. For the sake of clarity, the illustration is limited to four representative ensemble sizes. Fig. 2 shows that the RLS fit on average suffers from slow convergence with a high standard deviation but is insensitive to its initial value. This can be seen from the fact that the individual lines hardly differ from each other and are almost congruent. Furthermore, the signal-to-noise ratio seems to have a strong influence on the evolution of the mean average error per revolution. In particular the bottom chart of Fig. 2 stresses this observation showing that the error hardly converges – independent of the initial values (congruent lines and very small standard deviation). Fig. 3 depicts the results for the EnKF within the same range and with the same scaling. One can see that the filter converges in spite of the presence of noise and independently of the ensemble size. The ensemble size affects the deviation within the Ninit initial ensembles suggesting two implications:
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• the larger the ensemble size, the more robust becomes the filter regarding the initial ensemble,
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RLS
EnKF
S/N: 15
S/N: 5
kcnv / -
×60
1,500 1,000 500 0
×30
2,000 kcnv / -
S/N: 50
EnKF
2,000
×10
1,500 1,000 500 0
Ensemble sizes:
# revolutions / 10, 30,
60,
kcnv / -
2,000
100
Fig. 3. Evolution of the mean error per revolution of the EnKF for different signal-to-noise ratios (top: S/N 50, middle: S/N 15, bottom: S/N 5). Size of the circles represent the scaled standard deviation • the larger the ensemble size, the slower the convergence of the filter. To quantify the above findings, the convergence time is provided in Fig. 4 as boxplots. The whiskers represent 90 % of the convergence time; the box 50 %. The maximum and minimum is marked by the points and the median by the horizontal line. The RLS converges very slowly and showing a high variation within the Ninit = 1000 starting values independently of the initial value group. The higher the noise level the slower the (median) convergence and the larger the deviation. The RLS is not able to converge at a signal-to-noise ratio of 5. In contrast to this, the EnKF is able to converge at all noise levels and within a much shorter time. The charts support the above drawn conclusions but show a more differentiated result at higher noise levels. While at an almost ideal force measurement signal (S/N 50), the convergence time increases linearly with the ensemble size, higher noise levels profit from the inherent robustness of a larger ensemble size. In particular very small ensemble sizes (n = 10) tend to cause numerical instabilities in 5 %11 % of the cases – with only a small variation at different noise levels. As a compromise between fast convergence and high robustness to noise an ensemble size 40 ≤ n ≤ 60 can be suggested. 5. CONCLUSION This contribution describes a continuous identification of mechanistic force models in milling. Two methods are demonstrated using a nonlinear force model, Eq. 2. The recursive least squares estimator (RLS) extends the method of instantaneous chip thickness, which is in essence a curve fit, to a recursive formulation. The second method is based on an ensemble Kalman Filter (EnKF), which identifies the coefficients of the mechanistic force model.
1,500 1,000 500 0
10 30 50 70 90
10 30 50 70 90
Fig. 4. Convergence time of all 1000 initial ensembles represented as boxplots (whiskers: 5 %, 95 %; box: 25 %, 75 %, · maximum/minimum) for different signalto-noise ratios (top: S/N 50, middle: S/N 15, bottom: S/N 5) The results show that the RLS fit is prone noise in the measurements and suffers from convergence. The EnKF shows a remarkably insensibility to measurement noise and a fast convergence. The robustness increases with the ensemble size but so does also the convergence time at ideal measurements. A good compromise between convergence speed and robustness achieves an ensemble size of n = 50 members. So far, radial deviation was neglected and experiments have been simulated in order to provide a controlled environment. Future work will extend the continuous identification to jointly estimate a model of the force and of the radial deviation. This enables the presented approach to apply on a real measurements. An integration to a realtime target is the final step in order to use this approach for an on-line monitoring or control of the force in milling. ACKNOWLEDGEMENTS The authors would like to thank the German Research Foundation DFG for the kind support within the Cluster of Excellence “Internet of Production” (Project ID: 390621612). REFERENCES Adem, K.A.M., Fales, R., and El-Gizawy, A.S. (2015). Identification of cutting force coefficients for the linear and nonlinear force models in end milling process using average forces and optimization technique methods. The International Journal of Advanced Manufacturing Technology, 79(9), 1671–1687. doi:10.1007/s00170-0156935-3.
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