Optimal control for chatter mitigation in milling—Part 2:Experimental validation

Jérémie Monnin a,n, Fredy Kuster b, Konrad Wegener b

a Mikron Agie Charmilles AG, Nidau, Switzerlandb IWF, ETH Zurich, Switzerland

a r t i c l e i n f o

Article history:Received 11 March 2013Accepted 11 November 2013Available online 9 December 2013

Keywords:MillingRegenerative chatterActive structural methodMechatronic systemOptimal control

a b s t r a c t

The productivity of milling operations in chatter-free conditions can be improved using active structuralmethods. This paper presents the use of a mechatronic system integrated into the spindle unit combinedwith two different optimal control strategies. The first one aims to minimize the influence of cuttingforces on tool tip deviations. The second one explicitly considers the process interaction and attempts tostabilize the overall closed-loop system for specific machining conditions. The first part of this two-partpaper describes the modeling and formulation used for both strategies. In this second part, the experimentalvalidation of the proposed concept is presented.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Different methods may be used to mitigate the occurrence ofchatter in milling operations. The use of active structural methodsis considered here where the objective is to adapt the dynamics ofthe machine tool structure between the tool and the workpiece sothat the chatter stability limit can be improved for the selectedmachining conditions. More specifically, in the present study, amechatronic system is integrated into the spindle unit of themachine. The performance of the active system depends on thecontrol strategy combined with this mechatronic system. Twodifferent strategies are investigated here. The first one, calleddisturbance rejection, does not explicitly consider the machiningconditions and attempts to minimize the influence of the exogen-ous disturbance forces coming from the process on the deviationsof tool tip from its nominal path. The second strategy, namedstabilization, tries to guarantee the stability of the overall closed-loop system formed by the machine tool interacting with themilling process and the active system for some specified machin-ing parameters.

Similar concepts of active spindle have been proposed by Ries,Pankoke, and Gebert (2006), Ries and Pankoke (2008), Dohneret al. (1997, 2004) and Kwan et al. (1997) where the appliedcontrol strategies correspond to disturbance rejection schemeattempting to damp the tool tip vibrations. Other stabilizing

strategies integrating the machining process into the controldesign have been developed by Mei, Cherng, and Wang (2006),Shiraishi, Yamanaka, and Fujita (1991), Chen and Knospe (2007)and van Dijk, van de Wouw, Doppenberg, Oosterling, and Nijmeijer(2010, 2012). However, their application to the milling process intypical machining conditions is still missing. The aim of thecurrent paper is to present the experimental validation of theproposed concept in representative milling conditions.

This paper corresponds to the second part of a two-part paper.In the first part (Monnin, Kuster & Wegener, 2013), the proposedactive structural control concept is presented. The modeling of thesystem formed by the mechatronic machine structure coupledwith the milling process is detailed. This model is used for theformulation of a generalized plant required for the design of thetwo different control strategies. Finally, both strategies are appliedto a simulation example pointing out their fundamentally differentworking principle and their inherent benefits and disadvantages.

In this second part, the conditions selected to realize themachining operations are first given. The criterion used to detectthe chatter occurrence during milling operation is defined. To per-form the control design, some parameters involved in the modelingof the system formed by the mechatronic machine structure inter-acting with the milling process need to be experimentally deter-mined. The procedures employed to identify these parameters arethen described. The control design based on the two aforementionedapproaches is also presented. Finally, the stability charts obtained inmachining conditions are compared with the predictions and theeffective influence of the control schemes over the milling processstability is evaluated.

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/conengprac

Control Engineering Practice

0967-0661/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.conengprac.2013.11.011

n Corresponding author. Tel.: þ41 32 366 14 80; fax: þ41 32 366 14 09.E-mail addresses: jeremie.monnin@georgfischer.com (J. Monnin),

kuster@iwf.mavt.ethz.ch (F. Kuster), wegener@iwf.mavt.ethz.ch (K. Wegener).

Control Engineering Practice 24 (2014) 167–175

2. Experimental conditions

2.1. Selected machining conditions

The validation of the concept is performed with the prototypespindle presented by Monnin et al. (2013). This spindle integrates

an actuating system composed of two pairs of piezoactuatorsworking in push–pull configuration and orthogonally located inthe radial plane over the front bearing (see Fig. 6). Every actuatoris driven by an analog hybrid voltage–charge amplifier com-manded by a PXI digital real-time system. Two uniaxial piezo-electric accelerometers nearly collocated with the actuatorsprovide the required information to the controller via a signalconditioning system. The active spindle is mounted on a MikronAgie Charmilles high performance milling center with an axesconfiguration AC–workpiece–tool–ZXY. The insert end mill shownin Fig. 1 is clamped on the spindle interface HSK-A63. This toolassembly has the specifications listed in Table 1.

The process stability is investigated over a range of spindlespeeds and axial depths of cut for a selected workpiece materialand some machining parameters. These parameters are given inTable 2.

2.2. Chatter detection

During the machining tests, the occurrence of chatter is moni-tored in order to evaluate the influence of the active system over abroad range of machining conditions and to make the comparisonwith the predicted stability lobes diagram (SLD). The signals ydelivered by both integrated sensors are used to this end. Inmilling process, forced and self-excited vibrations are coexistingduring a chatter occurrence. Their differentiation is made throughtheir frequency content. A chatter detection criterion based on thevariance of the Poincaré map, as proposed by Bayly et al. (2002), isused here. More specifically, by observing the variation of thesignals at every tooth passing period, it is possible to identify ifself-excited vibrations are superimposed to the forced vibrations.A certain threshold slim must be calibrated so that chatter is said tooccur if the standard deviation

sP4slim ð1Þ

with

sP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1NT

∑NT

k ¼ 1JyðkTzÞ�ykT J

22

sð2Þ

Fig. 1. Selected tool assembly for concept validation.

Table 1Specifications of the selected tool assembly.

Tool specifications Value

Diameter (mm) 40Free lengtha (mm) 175Number of teeth 5Helix angle (deg) þ10Entering angle (deg) þ90Insert corner radius (mm) 0.8

a The free length is the distance between thespindle nose and the tool center point.

Table 2Selected machining parameters. DOC: depth of cut;WP: workpiece.

Machining parameters Value

WP material Steel Ck45Machining type Up-millingFeed directiona YþFeed rate (mm/tooth) 0.08Radial DOC (mm) 25Spindle speed (rpm) 2000–3100

a Feed direction in machine coordinate system.

0 0.5 1 1.5

−100

−50

0

50

100Time signal

Time (s)

Sen

sor f

eed

(m/s

2 ) All samplesfSR samples

−100 0 100

−100

−50

0

50

100Radial vibrations

Sensor normal (m/s2)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

5

10

15

20

25

30Linear spectrum

Frequency (Hz)

Mag

. sen

sor (

m/s

2 ) FeedNormalfSR

k fSR

k fTP

Fig. 2. Chatter condition: spindle speed n¼2300 rpm, axial depth of cut ap ¼ 1:7 mm, open-loop. fSR: spindle revolution frequency; kf SR: spindle revolution frequencyharmonics; kf TP : tooth passing frequency harmonics.

J. Monnin et al. / Control Engineering Practice 24 (2014) 167–175168

and

ykT ¼1NT

∑NT

k ¼ 1yðkTzÞ ð3Þ

where the tooth passing period is Tz, the number of consideredtooth passing periods is NT and J � J2 is the Euclidean norm.

In Figs. 2 and 3, two examples of the signals obtained by theintegrated accelerometers during machining tests in chatter and,respectively, stable conditions are presented. Fig. 2 corresponds to anunstable machining conditionwith the prototype spindle in open-loopconfiguration, meaning that no action from the control system isgenerated. In this case, the resulting sP is equal to 60.6 m/s2. Fig. 3corresponds to the same machining condition but stabilized throughthe counteraction of the active system. In this case, the standarddeviation sP is equal to 5.4 m/s2.

The surface finish is also a good indicator of the chatter pre-sence. Fig. 4 shows the resulting peripheral surface correspondingto both previous figures. In the case of unstable condition, chattermarks are visible. In the lower plot, the waviness is represented.One notes that the undulations of the surface left during chatterare several times larger than in stable condition.

3. Model identification

In order to make stability predictions and design the model-based controllers, the models of the milling process and themechatronic structure of the active spindle coupled with the restof the machine, also called plant, must be determined.

The overall system can be represented by the flow chart ofFig. 5. The plant corresponds to the block G composed of four

0 0.5 1 1.5

−100

−50

0

50

100Time signal

Time (s)

Sen

sor f

eed

(m/s

2 ) All samplesfSR samples

−100 0 100

−100

−50

0

50

100Radial vibrations

Sensor normal (m/s2)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

5

10

15

20

25

30Linear spectrum

Frequency (Hz)

Mag

. sen

sor (

m/s

2 ) FeedNormalfSR

k fSR

k fTP

Fig. 3. Stable condition: n¼2300 rpm, ap ¼ 1:7 mm, closed-loop.

Fig. 4. Measured waviness of peripheral surface finish from machining tests where stable cut and chatter occur, corresponding to Figs. 3 and 2, respectively.

J. Monnin et al. / Control Engineering Practice 24 (2014) 167–175 169

transfer functions Gkj between its two inputs, j¼ u;w, and twooutputs, k¼ z; y. The milling process is represented by the block Pwhich is exciting the plant dynamics through the cutting forces w.These forces are the addition of forces fP generated by thekinematics of the process and a load wR induced by the tool tipdeviations z via the regenerative effect R. On the other side, thecontroller C determines the reference signals u delivered to theactuating system based on the information y provided by thesensors.

3.1. Plant model

The mathematical formulation of each model is detailed inMonnin et al. (2013). For a constant spindle speed, the plant model

is considered as a linear time-invariant (LTI) system of thefollowing state-space form:

_x ¼AxþBwu

� �ð4Þ

zy

" #¼ CxþD

wu

� �ð5Þ

where the state vector is x and the constant state-space matricesare A;B;C;D. These matrices are derived from frequency responsefunction (FRF) measurements realized on the non-rotating tool. Animpulse hammer is used to excite the tool center point (TCP) to getthe FRFs related to the process excitation w. An accelerometerlocated on the TCP allows the determination of the tool deviationsz after a double time integration. A sweep signal is applied at theinput of the actuating system to obtain the other FRFs. Fig. 6represents the plant with the different signals used for the modelidentification.

A FRFs matrix is obtained between the multiple inputs andmultiple outputs of the plant. The subspace method N4SID algo-rithm (Overschee & Moor, 1994) is used to derive a correspondingblack box LTI state-space model. Fig. 7 presents both measuredand fitted data of the FRFs matrix of the plant for a 24th ordermodel identified over the frequency range from 200 to 1800 Hz.This range corresponds to the zone where the most relevantnatural modes of the system are located. For convenience, onlythe direct and cross FRFs corresponding to the excitation inX-direction are plotted. The FRFs generated by the identifiedmodel show a satisfactory agreement over the selectedfrequency range.

The analysis of the plant model indicates that the critical modefor chatter occurrence corresponds to the first bending mode ofthe spindle shaft coupled with a deformation of the bearings andlocated around 700 Hz (see upper left-hand side chart of Fig. 7).The modal shape of this mode obtained by finite element modalanalysis is represented in Fig. 8. Theoretical and experimentalinvestigations demonstrate a relative insensitivity of the plantdynamics over the considered range of spindle speeds. This allowsus to consider the FRFs measurement made without rotation,and thus the identified model, as representative of the spindledynamics during machining.

Fig. 5. Flowchart of overall system. C: controller transfer function; G: plant transferfunction; Gkj: transfer function between the jth input and the kth output withj¼w;u and k¼ z; y; P: milling process; R: regenerative effect; fP: cutting forcescoming from process kinematics; u: reference signal delivered to actuating system;w: disturbance generated by milling process; wR: cutting forces coming fromregenerative effect; y: information provided by sensing system; z: tool center pointdeviations.

Fig. 6. Plant identification. TCP: tool center point.

J. Monnin et al. / Control Engineering Practice 24 (2014) 167–175170

3.2. Milling process model

As detailed in the part one of this contribution, the millingprocess model is based on the linear average cutting force modeldescribed by Altintas and Lee (1998). This model expresses thedifferential cutting forces df j generated on an infinitesimal cuttingedge of length dS on the jth tooth of the cutter as the sum of shearand edge cutting forces like

df j ¼ ðKchj dbþKe dSÞgj ð6Þ

where j¼ 1;…;Nt , with Nt being the total number of teeth of thecutter. The instantaneous chip thickness is hj, db is the infinitesi-mal uncut chip width and gj is a function determining whether thejth tooth is removing material or not. The shear and edge cuttingforce coefficients, Kc and Ke respectively, are determined by linearregression using the procedure presented in Altintas (2000) basedon cutting forces measured at different feed rates. Three feed ratesare used here and the feed, normal and axial cutting forcecomponents are measured in stable but representative conditionsusing a dynamometer table supporting the workpiece.

The shear and edge cutting force coefficients are identified byaveraging the cutting forces over several integer rotations of the

cutter. Fig. 9 shows the results of the cutting force measurementsand the linear regression used to derive the cutting forcecoefficients.

10−2

100

Mag

nitu

de (µ

m/N

)

Gzw,jX

ModelExp.

−200

0

200

Pha

se (d

eg)

10−4

10−2

Gyw,jX

Mag

nitu

de (V

/N)

500 1000 1500 2000−200

0

200

Frequency (Hz)

Pha

se (d

eg)

10−2

100

(µm

/V)

Gzu,jX

−200

0

200

10−2(V/V

)

Gyu,jX

500 1000 1500 2000−200

0

200

Frequency (Hz)

Fig. 7. Comparison of FRFs matrix with excitation in X-direction between experimental data and identified model (24th order model based on frequency range from 200 to1800 Hz in both directions). j¼ X;Y .

Fig. 8. Modal shape of critical bending mode of spindle shaft with tool assembly at 704 Hz obtained using finite element modal analysis.

0 0.02 0.04 0.06 0.08 0.1 0.12−150

−100

−50

0

50

100

Feed rate (mm/tooth)

Ave

rage

d cu

tting

forc

e (N

)

FeedNormalAxial

Fig. 9. Dependency of the averaged cutting forces with the feed rate using the toolwith only one insert, n¼2500 rpm and ap ¼ 3 mm. The dots represent the valuesobtained from the measurements and the lines represent the linear regressionleading to the identification of the average cutting force coefficients.

J. Monnin et al. / Control Engineering Practice 24 (2014) 167–175 171

4. Controllers design

Two different types of optimal model-based controllers, calleddisturbance rejection and stabilization schemes, are investigated.The first one considers the dynamics of the plant and attempts tominimize the deviations z of the TCP generated by the processdisturbance w independently of the selected machining condi-tions. This induces a reduction of the most critical resonancepeak of the transfer function Gzw which is reported on the SLD asan increase of the lowest critical depth of cut (DOC) over thewhole range of spindle speeds, as discussed in Monnin et al.(2013).

In addition to the plant dynamics, the stabilizing controllertakes the milling process model into account. For a selectedmachining condition, a LTI model of the coupled system formedby the plant and the regenerative effect is built using a Padéapproximation. In that case, the first control objective is toguarantee the stability of the overall closed-loop system. Theworking principle of this second type of controller is completelydifferent from the disturbance rejecting controller. Instead oftrying to decrease the resonance peak, the stabilizing controllertends to generate some additional resonances but at specificfrequencies corresponding to the tooth passing harmonics. Theseadditional resonance peaks shift the stability lobes along thespindle speed axis so that the selected machining condition islocated in a zone of high stability between two lobes.

The formulation and the details of the design procedure forboth control schemes are given in Monnin et al. (2013). Tosummarize, the disturbance rejecting controller is based on asimpler model and induce a broad positive influence over thestability lobes diagram. The stabilizing controller is more efficientbut requires a representative milling process model and needs anew synthesis for every different machining condition.

Both disturbance rejecting and stabilizing controllers are basedon the definition of a generalized plant GP integrating four weightingfunctions. These functions allow us to adjust the trade-off betweencontrol effort and performance. In a first step, frequency-independentweighting functions are selected. If the control objectives cannot bereached using these constant weighting functions, then higher orderfunctions are employed. The dimensioning of the controller consistsin choosing satisfactory values for the coefficients involved in theweighting functions.

From the state-space realization of GP , the optimal H2 or H1controllers are synthesized using the Robust Control toolbox ofMatlabs with the functions h2syn or hinfsyn, respectively.

The synthesized controller is implemented as a digital filter in areal-time PXI system using a sampling frequency of 20 kS/s. Inorder to be implementable, the closed-loop of the synthesizedcontroller with the mechatronic structure must be stable. This lattercondition is requested to guarantee the stability also when the cutteris not in interaction with the workpiece. In the case of stabilizationschemes, the process stability must also be guaranteed for depths ofcut smaller than the condition used for the control design. This canbe verified through robust stability analysis.

The order of the resulting synthesized controller is equal to theorder of the generalized plant GP . In the case of stabilizationscheme, its order is typically greater than 200. Due to numericalproblems coming from roundness errors, such high order transferfunctions cannot be implemented in a digital real-time controller.They first need to be reduced. An order reduction procedure basedon the balanced realization is used here.

In the following section, only the results obtained with the H2

optimal controller are presented. From a general observation madeduring the tests, the use of the H1 norm instead of the H2 doesnot lead to fundamentally different results. In some particularcases, depending on the type of encountered critical mode and the

considered machining parameters, one norm could achieveslightly better performance than the other but no general designcriterion can be established. So every case must be carefullyevaluated. More information on the resolution of optimal controlproblem using both norms can be found in Doyle, Glover,Khargonekar, and Francis (1989).

5. Results of experiments

5.1. Closed-loop frequency responses at tool tip

In order to verify the influence of the controller on the TCPdynamics, FRF measurements are made before performing themachining tests. The dynamic compliance functions at the tool tipare measured by exciting the non-rotating tool using an impulsehammer with and without the influence of the controller. Themeasured FRFs are compared with the predicted FRFs based on theidentified model and also based on the FRFs measured in open-loop and used for the plant model identification.

The closed-loop FRF at TCP Fzw is predicted by the open-loopFRFs matrix of the plant G and the FRF of the controller C using thefollowing equation:

FzwðωÞ ¼ GzwðωÞþGzuðωÞCðωÞSðωÞGywðωÞ ð7Þ

where S is the output sensitivity defined by

SðωÞ ¼ ðI�GyuðωÞCðωÞÞ�1 ð8Þ

and GðωÞ corresponds to the frequency response function GðiωÞwith i2 ¼ �1.

Fig. 10 compares the FRFs at the TCP obtained in open-loopwith the closed-loop FRFs using a disturbance rejecting controller.The maximum singular values (MSV) of the FRFs normalized to thepeak value in open-loop are represented. Both predictions in theupper chart announce a reduction of the critical resonance peakthrough the control action of approximately 30%. This is confirmedby the measured FRFs in the lower chart. More generally, theglobal behavior measured over the whole represented frequencyrange is very similar to the predictions. This allows us to claim thatthe predictions of the control influence at the TCP based on theidentified model are representative of the reality.

0

20

40

60

80

100

Predicted FRF at TCP

MS

V (%

)

Open−loop, based on modelClosed−loop, based on modelOpen−loop, experimentsClosed−loop, based on open−loop exp.

200 400 600 800 1000 1200 1400 1600 1800 20000

20

40

60

80

100Measured FRF at TCP

MS

V (%

)

Frequency (Hz)

Open−loopClosed−loop

Fig. 10. Predicted and measured FRFs at TCP obtained using disturbance rejectingH2 controller. MSV: maximum singular value.

J. Monnin et al. / Control Engineering Practice 24 (2014) 167–175172

5.2. Stability charts

First, the influence of the disturbance rejecting controller onthe milling process stability is investigated. To do so, the axial DOCis increased at several specific spindle speeds until reaching thelimit of stability where chatter is occurring. This is done in open-loop and closed-loop conditions. The obtained limits of stabilityare then compared with the predicted SLD based on the zerothorder approximation (ZOA) method developed by Altintas andBudak (1995). This method approximates with a constant thetime-varying coefficients of the resulting delay differential equa-tions describing the behavior of the machine dynamics coupledwith the milling process and uses the Nyquist stability criterion toderive the stability lobes.

Fig. 11 presents the experimental and predicted stability chartswith the influence of the disturbance rejecting controller. Thepredictions are based on the FRFs measured at the TCP andrepresented in the lower chart of Fig. 10. In the upper chart, thecritical axial DOC normalized to its minimal value predicted by theZOA method is represented in function of the spindle speed. Thecontinuous blue line corresponds to the predictions in open-loopcondition and the interrupted red line, in closed-loop. The dotsindicate the experimental results. The green circles correspond tostable machining tests for both open-loop and closed-loop condi-tions. The blue diamonds represent chatter conditions that arestabilized using the active system. The red squares correspond tomachining conditions where chatter occurs in both open-loop andclosed-loop configurations. Additionally, the black triangles indicatecases where the use of the active system is detrimental so that, froma stable condition in open-loop configuration, the process becomesunstable under the influence of the control action. Finally, the graycircles give information on the control effort indicating that duringthe corresponding machining test in closed-loop configuration, thelimits of the actuating system capabilities are reached. To facilitatethe comparison between experimental and predicted stability lobes,a continuous black line draws the experimental stability limit inopen-loop. Similarly for the experimental closed-loop results, ainterrupted black line is used. The stability predictions in bothopen-loop and closed-loop configurations are globally in accordancewith the experimental results, especially for the spindle speedscorresponding to the zones of lowest stability. This is also true forthe chatter frequencies represented in the lower chart where the

values identified during the machining tests, represented by theblue diamonds, are located close to the predictions. The activesystem is able to increase the lowest experimental critical DOC by55% and a maximal stability increase equal to 73% is locally achievedat 2300 rpm. However, the controller may also detrimentallyinfluences the process in the zones of higher stability, where chatteris generated by less dominant dynamics of the TCP. This is the casefor the spindle speeds of 2100 and 2800 rpm where a decrease ofthe stability is locally observed.

It is still to note that for most of the spindle speeds thecontroller reference output u reaches the limits of the actuatingsystem capabilities before attaining the limit of process stability.

To validate the stabilization scheme, the improvement of theprocess stability limit at several specific spindle speeds is inves-tigated. For each selected spindle speed, a stabilizing controller isdesigned in order to get the maximal stability improvement inaccordance with the capabilities of the actuating system.

Fig. 12 represents the stability lobes diagrams predicted by theZOA method based on the model used for the control design andthe resulting synthesized controllers. The magenta circle indicatesthe machining condition selected for the control design. One notesa general tendency to get, in practice, a slightly lower stabilizingperformance than expected. This is especially true in the regions ofhigher stability. For instance, at 2100 and 2900 rpm, the controllerdoes not succeed to stabilize the process up to the axial DOCselected for the control design. This is due to the fact that in theseregions, less dominant dynamics of the TCP becomes relevant forthe chatter stability. However, in these frequency ranges, theidentified model presents some gaps with the measured FRFs(see Fig. 7). These gaps are susceptible to induce such differencesbetween the predictions and the effective stability limit. In spite ofthat, for all investigated spindle speeds, an effective increase of thestability limit, representative of the tendency predicted by thecontrol design, is noticeable.

Fig. 13 shows the stability predictions based on the measuredFRFs at the tool tip for the case of a stabilizing controller designedat 2300 rpm. A good agreement is visible between the SLDs basedon the measured FRFs and those derived from the model used forthe control design (see middle left-hand side chart in Fig. 12).

Fig. 13 also compares the predictions with the experimentalresults obtained from machining tests. The comparison of theexperimental stability limit in open-loop (continuous black line)

50

100

150

200

250

300

350Stability chart

Spe

cific

axi

al D

OC

(%)

1800 2000 2200 2400 2600 2800 3000 3200 3400500

600

700

800

900

1000

Spindle rotational speed (rpm)

Cha

tter f

requ

ency

(Hz)

Fig. 11. Predicted and experimental stability charts obtained using disturbance rejecting controller. DOC: depth of cut, ZOA: zeroth order approximation method.(For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

J. Monnin et al. / Control Engineering Practice 24 (2014) 167–175 173

with the corresponding prediction, given by the blue line, indi-cates a quite good agreement. This is also verified by looking at thechatter frequency chart (lower plot of Fig. 13).

Due to the limitations imposed by the actuating system, thepredictions in closed-loop configuration cannot be fully verified.However, the machining condition at the spindle speed of 2300rpm selected for the control design is successfully stabilized by the

active system and shows accordance between predictions andexperiments. At this spindle speed, an increase equal to 91% of themaximal stable material removal rate, which is directly propor-tional to the axial DOC, is achieved between the open-loop andclosed-loop configurations.

At the other spindle speeds, where a stabilizing effect of theactive system is predicted, an effective increase of the stability

50

100

150

200

250

300

3502100rpm

Spe

cific

axi

al D

OC

(%)

ZOA open−loopZOA closed−loopStable with and without controlStabilized by controlChatter with and without controlSelected condition for control design

50

100

150

200

250

300

3502300rpm

Spe

cific

axi

al D

OC

(%)

1800

2000

2200

2400

2600

2800

3000

3200

3400

50

100

150

200

250

300

3502500rpm

Spindle rotational speed (rpm)

Spe

cific

axi

al D

OC

(%)

2700rpm

2900rpm

1800

2000

2200

2400

2600

2800

3000

3200

3400

3100rpm

Spindle rotational speed (rpm)

Fig. 12. Comparison between predicted stability charts based on plant model obtained using stabilizing controllers designed for different machining conditions andexperimental results. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

50

100

150

200

250

300

350Stability chart

Spe

cific

axi

al D

OC

(%)

1800 2000 2200 2400 2600 2800 3000 3200 3400500

1000

1500

Spindle rotational speed (rpm)

Cha

tter f

requ

ency

(Hz)

Fig. 13. Predicted and experimental stability charts obtained using stabilizing controller designed for 2300 rpm.

J. Monnin et al. / Control Engineering Practice 24 (2014) 167–175174

limit is observed. Also, for the spindle speeds predicting a detri-mental effect from the active system, effective destabilizing influ-ences are detected. This allows us to state that the correspondencebetween the SLD predictions and experiments is satisfactory.

The aforementioned experimental results obtained with bothcontrol strategies confirm the observations made from the simula-tion example in Monnin et al. (2013). More specifically, one notesthat the disturbance rejecting controller achieves an increase ofthe minimal critical axial depth of cut over the whole spindlespeed range but is susceptible to negatively influence the processstability in the regions of higher stability between two lobes. Thestabilizing controller is able to achieve greater performances thanthe disturbance rejecting controller but requires reliable millingprocess model.

This proves that the proposed concept can be applied for theimprovement of the productivity in chatter-free conditions and isworking in accordance with the predictions.

However, as every model-based concept, the quality of theunderlying models is determining for the achievable perfor-mances. From the obtained results, one notes that both types ofcontrollers are working according to the expectations in the lowerbounds of the SLD where the dominant dynamics of the TCP isdeterminant. In the zones of higher stability located between twolobes and influenced by the less dominant frequency range of thetool tip transfer function, the reliability of the stability predictionsand thus the performance of the controllers are reduced.

6. Conclusion

This paper investigates model-based optimal control strategies forthe mitigation of chatter in milling process using an active structuralcontrol system integrated into a high performance prototype spindle.Disturbance rejection and stabilization approaches are presented.The proposed concept is validated in representative machiningconditions. The disturbance rejecting controller improves the mini-mal critical axial depth of cut of 55%. Locally, the stabilization schemeis able to increase the productivity in stable condition up to 91%.

However, even if the validity of the proposed concept isdemonstrated, several aspects remain to be deeper studied. Themain one probably concerns its performance robustness regardingto modeling errors. Indeed, it is usually challenging to getsufficiently reliable and accurate stability predictions for the broadrange of machining conditions encountered in practice.

Acknowledgments

The authors address their thanks to Step-Tec AG for its supportin the realization of the experiments.

This work was supported by the Commission for Technologyand Innovation (CTI) of the Federal Department of EconomicAffairs of Switzerland.

References

Altintas, Y. (2000). Manufacturing automation: Metal cutting mechanics, machine toolvibrations, and CNC design. Cambridge University Press.

Altintas, Y., & Budak, E. (1995). Analytical prediction of stability lobes in milling.CIRP Annals – Manufacturing Technology, 44(1), 357–362.

Altintas, Y., & Lee, P. (1998). Mechanics and dynamics of ball end milling. Journal ofManufacturing Science and Engineering, 120(4), 684–692.

Bayly, P., Mann, B., Schmitz, T., Peters, D., Stépán, G., & Insperger, T. (2002). Effects ofradial immersion and cutting direction on chatter instability in end-milling. InProceedings of IMECE2002, ASME international mechanical engineering congressand exposition, New Orleans, Louisiana.

Chen, M., & Knospe, C. (2007). Control approaches to the suppression of machiningchatter using active magnetic bearings. IEEE Transactions on Control SystemsTechnology, 15(2), 220–232.

van Dijk, N., van de Wouw, N., Doppenberg, E., Oosterling, H., & Nijmeijer, H. (2010).Chatter control in the high-speed milling process using μ-synthesis. InProceedings of the American control conference, (pp. 6121–6126), Baltimore,MD, USA.

van Dijk, N., van de Wouw, N., Doppenberg, E., Oosterling, H., & Nijmeijer, H. (2012).Robust active chatter control in the high-speed milling process. IEEE Transac-tions on Control Systems Technology, 20(4), 901–917.

Dohner, J., Hinnerichs, T., Lauffer, J., Kwan, C.-M., Regelbrugge, M., & Shankar, N.(1997). Active chatter control in a milling machine. In SPIE proceedings(Vol. 3044, pp. 281–294), San Diego, CA, USA.

Dohner, J., Lauffer, J., Hinnerichs, T., Shankar, N., Regelbrugge, M., Kwan, C.-M., et al.(2004). Mitigation of chatter instabilities in milling by active structural control.Journal of Sound and Vibration, 269(1–2), 197–211.

Doyle, J., Glover, K., Khargonekar, P., & Francis, B. (1989). State-space solutions tostandard H2 and H1 control problems. IEEE Transactions on Automatic Control,34(8), 831–847.

Kwan, C.-M., Xu, H., Lin, C., Haynes, L., Dohner, J., Regelbrugge, M., et al. (1997).H1 control of chatter in octahedral hexapod machine. In Proceedings of theAmerican control conference (pp. 1015–1016), Albuquerque, NM, USA.

Mei, C., Cherng, J., & Wang, Y. (2006). Active control of regenerative chatter duringmetal cutting process. Journal of Manufacturing Science and Engineering, 128(1),346–349.

Monnin, J., Kuster, F., & Wegener, K. (2013). Optimal control for chatter mitigation inmilling—Part 1: Modeling and control design. Control Engineering Practice.http://dx.doi.org/10.1016/j.conengprac.2013.11.010.

Overschee, P. V., & Moor, B. D. (1994). N4SID: Subspace algorithms for theidentification of combined deterministic–stochastic systems. Automatica, 30(1),75–93.

Ries, M., & Pankoke, S. (2008). Increasing the stability of the milling process by anactive milling spindle. In Proceedings of the 1st international conference onprocess machine interactions (pp. 95–102), Hannover, Germany.

Ries, M., Pankoke, S., & Gebert, K. (2006). Increase of material removal rate with anactive HSC milling spindle. In Conference proceedings of the adaptronic congress,Göttingen, Germany.

Shiraishi, M., Yamanaka, K., & Fujita, H. (1991). Optimal control of chatter in turning.International Journal of Machine Tools and Manufacture, 31(1), 31–43.

J. Monnin et al. / Control Engineering Practice 24 (2014) 167–175 175